and mesoscale modeling of thermal convection, internal waves and

Budapest University of Technology and

Economics

Micro- and mesoscale modeling ofthermal convection, internal waves

and cloud formation in theatmosphere

Author: Supervisor:Norbert Rácz Dr. Gergely Kristóf

A Thesis Submitted in partial fullfillment for the Degreeof Doctor of Philosophy

in theFaculty of Mechanical Engineering

Department of Fluid Mechanics

June 8, 2015

Nyilatkozat

Alulírott Rácz Norbert kijelentem, hogy ezt a doktori értekezést magamkészítettem és abban csak a megadott forrásokat használtam fel. Minden olyanrészt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva másforrásból átvettem, egyértelműen, a forrás megadásával megjelöltem.

Aláírás:

Dátum:

i

Abstract

The present thesis is about the simulation of micro- and mesoscale flows using gen-eral purpose pressure based computational fluid dynamical (CFD) solvers. CFDsolvers are already widely used in urban studies such as in the modeling of the ven-tilation of urban areas, pollution transport studies, wind farm design or calculationof wind loads on different human made constructions. These solvers are capable ofhandling complex topography, building structures and they have wide variety ofturbulence and physical models, effective numerical techniques and parallelizationhowever. With a recent transormation method it is possible to further extendthe limits of such solvers to the simulation of micro- and mesoscale atmosphericflows with arbitrary stratification. A novel approach was also proposed where theextended CFD solver was coupled with a bulk microphysical model in order todescribe wet adiabatic processes in the atmosphere. The technical viability andcapabilities of the technique are demonstarted in this thesis through the modelingof thermal convection, internal gravity waves and the formation of natural- andhuman made clouds in and above the atmospheric boundary layer.

ii

Acknowledgments

I would like to express my appreciation and thanks to my supervisor Dr. GergelyKristóf, you have always helped and motivated me during the coarse of this work.I would like to thank for the support of the Department of Fluid Mechanics (DFM),especially Tamás Lajos and János Vad who gave me the possibility to finish thisproject.Special thanks goes to the members of DFM who were very kind to me and en-coureged me during my PhD studies.At the end I would like express appreciation to my beloved wife who pushed meforward when I lost my motivation.

iii

ContentsAbstract ii

Acknowledgments iii

Contents v

List of Figures viii

List of Tables ix

1 Introduction 1

2 Modeling stratified atmospheric flows and thermal convection – prov-ing the technical viability and accuracy of a new modeling approach 42.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Meteorological outlook, equations and numerical solution . . . . . . 5

2.2.1 Filtering of acoustic modes . . . . . . . . . . . . . . . . . . . 62.2.2 Planetary boundary layer (PBL) and surface-layer schemes,

turbulence, closure problem . . . . . . . . . . . . . . . . . . 72.2.3 First order closure . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Higher order closures . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Adaption of general purpose CFD solvers . . . . . . . . . . . . . . . 102.3.1 Stratification and turbulence models . . . . . . . . . . . . . 122.3.2 System of transformations . . . . . . . . . . . . . . . . . . . 132.3.3 Reference profiles . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Volume sources . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.5 A simplified model version for water-tank experiments . . . 162.3.6 Nesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Validation of the dynamical core of the model . . . . . . . . . . . . 182.4.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Validation on laboratory scale urban heat island circulation . . . . . 212.5.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

v

2.6 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 24

3 Modeling internal gravity waves by using a pressure based CFD solver 263.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Introduction and literature survey . . . . . . . . . . . . . . . . . . 273.3 Validation with the analytic solution of gravity wave propagation . 28

3.3.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Validation with laboratory scale gravity wave experiments . . . . . 313.4.1 Statistical performance measures . . . . . . . . . . . . . . . 323.4.2 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 323.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Model comparison with a full scale event . . . . . . . . . . . . . . . 373.5.1 The Boulder windstorm case study . . . . . . . . . . . . . . 373.5.2 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 393.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


4 Modeling phase change and moisture transport with a bulk microphys-ical model using a general purpose pressure based CFD solver 464.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Overview of existing tools . . . . . . . . . . . . . . . . . . . . . . . 494.4 Description of the bulk microphysical model . . . . . . . . . . . . . 51

4.4.1 Transport equaitons . . . . . . . . . . . . . . . . . . . . . . 514.4.2 Droplet growth by activation and diffusion . . . . . . . . . . 534.4.3 Partial and total evaporation of droplets . . . . . . . . . . . 54

4.5 Validation with a rising thermal in a dry stable atmosphere . . . . . 554.5.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 564.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Validation with the rise of a moist thermal . . . . . . . . . . . . . . 574.6.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 574.6.2 results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Validation with the Bugey 1980 field campaign . . . . . . . . . . . . 604.7.1 Geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 614.7.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 614.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


5 Summary, conclusions and outlook 675.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 67

vi

5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Thesis points 711. Thesis: Modeling stratified atmospheric flows and thermal convection

– proving the technical viability and accuracy of a new modelingapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2. Thesis: Modeling internal gravity waves by using a pressure basedCFD solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3. Thesis: Modeling phase change and moisture transport with a bulkmicrophysical model using a general purpose pressure based CFDsolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Bibliography 75

vii

List of Figures

2.1 Nesting of domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Nonlinear density current . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Validation with laboratory scale urban heat island circulation, mean

flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Validation with laboratory scale urban heat island circulation, u

and w velocity components . . . . . . . . . . . . . . . . . . . . . . . 232.5 Validation with laboratory scale urban heat island circulation, T

profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Validation with gravity wave propagation, the analytic solution . . 313.2 Laboratory scale gravity waves, elongation of waves . . . . . . . . . 333.3 Validation with gravity wave propagation, normalized wave length

and amplitude data . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Validation with gravity wave propagation, mean flow field Nh/U =

0.69 and Nh/U = 3.33 . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 The Boulder windstorm field survey . . . . . . . . . . . . . . . . . . 383.6 Validation with the Boulder windstorm survey, mean flow field . . . 43

4.1 Rise of a dry thermal, mesh sensitivity of the solution . . . . . . . . 574.2 Rise of a dry thermal, sensitivity of the solution to the explicit

viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Rise of a moist thermal, time evolution . . . . . . . . . . . . . . . . 594.4 Rise of a wet thermal, mesh sensitivity of the solution . . . . . . . . 604.5 Rise of a wet thermal, sensitivity of the solution to the explicit

viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6 The Bugey 1980 field campaign, ambient conditions . . . . . . . . . 634.7 Validation with the Bugey 1980 field campaign, LWC field . . . . . 644.8 Validation with the Bugey 1980 field campaign, comparison with

aircraft data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

viii

List of Tables

2.1 Nonlinear density current, CFD and reference calculations . . . . . 20

3.1 Laboratory scale gravity waves, investigated cases . . . . . . . . . . 343.2 Validation with gravity waves, statistical metrics . . . . . . . . . . . 373.3 Validation with the Boulder windstorm survey, statistical measures 41

4.1 The Bugey 1980 field campaign, tower exit conditions . . . . . . . . 62

ix

1 Introduction

A clear trend can be seen in the development of mesoscale meteorological codestowards the usage of higher resolution numerical models incorporating multiplephysical effects in order to better describe the atmosphere, to give higher resolutionmodels for urban environments or to give higher fidelity forecasting. This processis well reflected in the “urbanization” of several mesoscale meteorological modelswhere more and more fine scale physical effects are introduced.The urban heat island circulation, which largely affects the ventilation and ther-mal comfort of large cities is a good example for the urbanization of such codes.Many researchers investigated the phenomena numerically by different purposedeveloped solvers and by mesoscale meteorological models, such as MM5, Meso-NH and WRF, which utilize urbanized canopy models and surface energy bal-ance calculations. An excellent agreement with temperature measurements canbe achieved by using fine tuned subgrid-scale surface parameterizations however,these models still do not allow the exact analyses of contaminant transport in anurban atmosphere as local immission levels are strongly dependent on fine flowstructures such as urban canyon effects, building-specific lofting, and eddies in thewakes of buildings.Boundary conditions for the CFD domain can be taken from mesoscale simulationsby utilizing one-way or two-way model nesting. The drawbacks of this approachare the numerical errors and model uncertainties introduced by the sudden changein physical description and interpolation of variables between grid interfaces withdifferent resolutions. When both close- and far-field flow features are important,it may be more reasonable to use a single framework for the physical descriptionfor the entire domain.There is another potential approach however for going towards fine scale namelywhen modern computational fluid dynamical (CFD) solvers are adapted to handlemesoscale effects. General purpose CFD solvers are already widely used in urbanstudies such as in the modeling of the ventilation of urban areas, pollution trans-port studies, wind farm design or calculation of wind loads on different humanmade constructions. These solvers are capable of handling complex topography,building structures and they have wide variety of turbulence and physical models,effective numerical techniques and parallelization.

1

In order to further extend these wide range of functionalities a model transfor-mation has been developed (Kristóf, Rácz and Balogh, 2009) for general purposeCFD solvers to make them capable of handling mesoscale effects. The atmosphericstratification, the Coriolis force and baroclinicity are taken into account by usingadditional source terms in the conservation equations of the solver.The model uses only a single unstructured grid, and a uniform physical descriptionfor close- and far-field flow avoiding interpolation errors and model uncertaintiesdue to model nesting. The authors intended purpose, furthermore, is to createa more general method, which is easy to implement in any CFD solver allowingprogrammable user defined volume sources in the governing equations. This newapproach can be applied in several areas of practice, but before the applicationof the method, it is an important step to validate the model and to understandthe capabilities of the technique. Therefore the model validation of the differentmodules representing the different source terms applied to the governing equationswill be presented in the following chapters with each focusing on one importantelement of the model.In the first set of cases the focus was on the proper implementation of the Corioliforce, the compressibility model and the implementation of the energy equationsource term. The model was validated with the analytical solution of the Ekmanspiral, a well-known two-dimensional benchmark solution of a spreading densitycurrent for testing compressible fluid solvers and with the results of a water-tankexperiment involving thermal convection in a stratified medium. The latter vali-dation cases have the advantage of that they have more control over the measuredparameters due to the nature of the measuring device. However, they have lowand limited Reynolds number range, the cases are mainly hydrostatic, the verticalextent is limited to a certain height, and turbulence modeling is limited to a fixedturbulence viscosity model.The purpose of the second chapter is to further extend the validation cases charac-terized by non-hydrostatic and strong nonlinear situations. Differences comparedto previous validations are the complex topography, the extended simulation do-main height, incorporating the tropopause, and that all of the source terms areactivated. Results of gravity wave simulations compared against the correspondinganalytic solution, small scale water-tank experiments and full scale observationswill be presented. The simulation of such cases has been found to be ideal fortesting and evaluating mesoscale numerical models due to the presence of complexflow patterns and wave breaking phenomena.The transformation method in its early state essentially handled dry adiabaticprocesses, however several applications require the inclusion of a proper moisturemodel. In the third chapter the extension of the current transformation method

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with a humidity transport and phase change model will be shown accompaniedwith the validation against the rise of idealized two-dimensional dry and wet ther-mals and a full scale three dimensional wet cooling tower plume formation char-acterized by different environmental stratifications.

3

2 Modeling stratified atmosphericflows and thermal convection –proving the technical viability andaccuracy of a new modelingapproach

2.1 Overview

General purpose Computational Fluid Dynamics (CFD) solvers are frequently usedin small-scale urban pollution dispersion simulations without a large extent of ver-tical flow. Vertical flow, however, plays an important role in the formation of localbreezes, such as urban heat island induced breezes that have great significance inthe ventilation of large cities. The effects of atmospheric stratification, anelasticityand Coriolis force must be taken into account in such simulations. We introducea general method for adapting pressure based CFD solvers to atmospheric flowsimulations in order to take advantage of their high flexibility in geometrical mod-eling and meshing. Compressibility and thermal stratification effects are taken intoaccount by utilizing a novel system of transformations of the field variables andby adding consequential source terms to the model equations of incompressibleflow. Phenomena involving mesoscale to microscale coupled effects can be ana-lyzed without model nesting, applying only local grid refinement of an arbitrarylevel. Elements of the method are validated against an analytical solution, resultsof a reference calculation, and a laboratory scale urban heat island circulationexperiment. The new approach can be applied with benefits to several areas ofapplication.

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2.2. METEOROLOGICAL OUTLOOK, EQUATIONS AND NUMERICALSOLUTION

2.2 Meteorological outlook, equations and numericalsolution

In this section a brief overview will be given on the governing equations, numericalmethods, and parameterizations applied in numerical weather prediction (NWP)and CFD models in order to show the similarities and differences between themeteorological and engineering assumptions.The Navier-Stokes (N-S) equation describes all types of fluid motion of our interest.It can be seen that today’s NWP and CFD models are based on these equationsof different forms.Modern numerical forecast models are based on a formulation of the dynamicalequations, which is essentially the formulation proposed by Richardson [116]:

dvdt

= −2Ω× v− 1ρ∇p+ g + Fr, (2.1)

where v,Ω, ρ, p,g and Fr are the velocity vector, angular velocity of the Earth,air density, pressure, gravity term, and the frictional force, respectively. Thecentrifugal force is combined with gravitation in the gravity term g. This formof the N-S equation is basic to most work in dynamic meteorology and solvedtogether with the continuity and thermodynamic equations and the equation ofstate in NWP-s. After expanding the components of Eq. (2.1), one arrives tothe eastward, northward, and vertical components in spherical coordinate system,respectively [72]:

du

dt− uvtanϕ

a+ uw

a= −1

ρ

∂p

∂x+ 2Ωvsinϕ− 2Ωwcosϕ+ Frx (2.2)

dv

dt− u2tanϕ

a+ vw

a= −1

ρ

∂p

∂y+ 2Ωusinϕ+ Fry (2.3)

dw

dt− u2 + v2

a= −1

ρ

∂p

∂z+ 2Ωucosϕ+ Frz (2.4)

where u, v, w are the velocity components, a is the curvature of the Earth, ϕ is thelatitude, and Fr,i are the components of the frictional force. The terms standingwith 1/a in Eqs. (2.2) – (2.4) are so-called curvature terms, and they arise dueto the curvature of the Earth and often neglected in midlatitude synoptic scalemotions.

5


The precise form of equations depends on the vertical coordinate system chosen aswell, such as pressure coordinates, log pressure coordinates, sigma coordinates, hy-brid coordinates, etc. [85, 89, 157]. Furthermore, the variables may be decomposedinto mean and perturbation components. Equation systems using perturbationvariables reduce the truncation errors in the horizontal pressure gradient calcu-lations, in addition to reducing machine rounding errors in the vertical pressuregradient and buoyancy calculations. For this purpose, new variables are defined asperturbations from a hydrostatically-balanced reference state, and reference statevariables are defined to satisfy the governing equations for an atmosphere at rest.[166] As an example, in the hydrostatic model version of the Fifth-GenerationPenn State/NCAR Mesoscale Model (MM5) [60], the state variables are explicitlyforecasted while in the non-hydrostatic model version [42] or in the more advancedWeather Research and Forecasting modeling system (WRF) [165], pressure, tem-perature, and density are defined in terms of a reference state and perturbationcomponents. When Reynolds averaging is applied, various covariance terms willappear in the system, representing turbulent fluxes. For many boundary layers,the magnitudes of the turbulent flux terms are of the same order as the other termsin Eqs. (2.2) – (2.4). In these cases one cannot neglect these fluxes even if it isnot of direct interest.There are tendencies towards higher spatial resolution models, but the resolutionof most NWP models is yet too coarse to resolve boundary layer eddies, andparameterizations of them are usually necessary as the complete energy cascadecannot be resolved. (The High Resolution Limited Area models (HIRLAM) [120],or ALADIN operate on horizontal grids in the range of 1–10 km. [77]) Thus anumber of turbulent mixing and filtering formulations were developed in the past.Some of these filters are for numerical reasons. For example, divergence dampingfilters acoustic modes from the solution. Other filters represent sub-grid processesthat cannot be resolved on the given spatial resolution.

2.2.1 Filtering of acoustic modes

The complete equations of motion (Eqs. (2.2) – (2.4)) describe all types andscales of atmospheric motion. The elimination of terms on scaling considerationshas an important advantage of simplified mathematics and filtering of a range ofunwanted type of motions.Theseunwanted type of high-frequency acoustic modes would limit the time stepduring the calculation. To circumvent, different time discretization techniques aredeveloped, see, e.g., the Runge-Kutta time-split scheme in [190] or [53]. The ef-ficiency of the time-split scheme arises from that the large time step ∆t is much

6


larger than the acoustic time step ∆τ , so the most costly calculations are onlydone in the less-frequent large steps. There are less efficient methods than theleapfrog-based models (e.g. in MM5 or WRF) resulting typically a factor of twosmaller time step. In the non-hydrostatic model of MM5, a semi-implicit schemebased on Klemp and Wilhelmson [90] is used to filter the acoustic waves, whilein the hydrostatic model, a split-explicit scheme based on Madala [117] is used tofilter gravity waves from the solution. The time differencing in MM5 is extensivelydiscussed in Grell et al. [61]. In highly complicated systems, also involving pol-lution transport and chemical reactions, efficient operator splitting methods aredeveloped recently to reduce computational time. The method is well spread inrelated fields of applied mathematics [55] and in circulation and pollution models.[71, 48]

2.2.2 Planetary boundary layer (PBL) and surface-layerschemes, turbulence, closure problem

NWP and air pollution models must contain proper treatment for the PBL since itcouples energy, momentum, and mass transfer between the land and atmosphere.The importance of its modeling is increasing nowadays due to environmental re-quirements, including human health, urban air quality, local and global warmingtrends, or homeland security problems. The PBL parameterization is especiallyimportant for predicting pollutant transport and dispersion [115].In order to solve the equations of motion (Eqs. (2.2) – (2.4)), closure assumptionsmust be made to approximate the unknown fluxes as a function of known quantitiesand parameters [72]. Turbulent fluxes provide a lower boundary condition for thevertical transport done in the PBL schemes. The PBL schemes determine the fluxprofiles within the well-mixed boundary layer and the surface layer providing ten-dencies of temperature, moisture, and horizontal momentum in the entire column.Thus, when a PBL scheme is activated, explicit vertical diffusion is de-activatedassuming that the PBL scheme will handle the process. Most PBL schemes ap-ply dry mixing, but can also include moisture effects in the vertical stability thatdetermines the mixing. The schemes are one-dimensional, since the horizontallarge gradients cannot be resolved, and assume a clear scale separation betweensub-grid eddies and resolved eddies. This assumption is less clear below a certaingrid size, where boundary layer eddies may start to be resolved. In these situa-tions the scheme is replaced by a fully three-dimensional local sub-grid turbulencescheme such as the turbulent kinetic energy (TKE) diffusion scheme (see in theWRF model: [92, 129]). Numerical simulations of PBL have been performed bymany authors in the past resulting in different model complexity ranging from very

7


simple zero dimensional parameterization to 3-D high resolution models. In thefollowing sections a brief description will be given on the commonly used closuremodels.

2.2.3 First order closure

Operational NWP, emergency response, and air-quality models usually are of aRANS type (solving Reynolds averaged Navier-Stokes equations), and they employfirst- or 1.5-order turbulence closures. The simplest turbulent transport parame-terization is the first-order closure based on the K-theory [195, 73, 173, 33]. It isrobust and requires low computational resources but gives poor approximation inthe boundary layer, where the scale of typical turbulent eddies is strongly depen-dent on the distance to the surface and static stability. In many cases the mostintense eddies have scales comparable to the boundary layer depth, and there themomentum and heat fluxes are not proportional to the local gradient of the mean.In much of the mixed layer, heat fluxes are positive even with neutral conditions.To improve the model behavior, alternative approaches have been developed. Anexample is the modified first-order closure [178, 179, 76]. This scheme employs acounter-gradient flux for heat and moisture in unstable situations. It uses enhancedvertical flux coefficients in the PBL, and the PBL height is determined from acritical bulk Richardson number (Ri). It handles vertical diffusion with an implicitlocal scheme, and it is based on local Ri in the free atmosphere (Rifree).In spite of its drawbacks, the first-order closure based models remain the mostpopular parameterization for stable conditions, although alternative approacheshave been used for the daytime convective period, e.g., the Blackadar (BK) non-local mixing scheme [14, 198] and non-local K-theory [76]. The BK scheme is afirst-order hybrid local and nonlocal scheme in which eddy diffusivity, a functionof the local Ri number, is applied to the stable and forced convective regimes,while nonlocal mixing is used for free convective cases. The BK scheme’s closureis based on an expression for the mass that is exchanged between individual layersin the boundary layer.Recent improvements of first order schemes include the Yonsei University model(YSU, [75]) or the asymmetrical convective models ACM1 ([141]) and ACM2 [140].YSU is based on the K profile for the convective cases as a function of local windshear and local Rifree. It also considers non-local mixing by adding a non-localgradient adjustment term to the vertical diffusion equation. Moreover, it containsadditional terms to describe entrainment at the top of PBL proportional to thesurface flux. For stable cases, the original mixing coefficient [75] is replaced by anenhanced diffusion based on the bulk Ri number between the surface and top layers

8


[74]. The ACM1 and ACM2 models are modifications of the BK scheme. In ACM1the symmetrical downward transport of BK scheme is replaced by an asymmetricallayer-by-layer model. In order to produce more realistic vertical profiles, the ACM2model, a combination of local and non-local closures, was introduced [140], thatadds an eddy diffusion component to the non-local transport. With this additionthe ACM2 scheme can better represent the shape of the vertical profiles in thenear surface region. Both ACM1 and ACM2 models are available in the MM5 andWRF solvers.

2.2.4 Higher order closures

Given the known drawbacks of these simpler models, new approaches are developedwith increasing complexity of turbulence description. The Gayno–Seaman (GS)scheme is a 1.5-order local closure scheme that computes eddy diffusivity based onlocal vertical wind shear, stability, turbulent kinetic energy (TKE) predicted bya prognostic equation [162], and length scale. The Eta PBL scheme, also knownas the Mellor-Yamada-Janjic (MYJ) scheme [84, 122, 82, 83], is a level-1.5 localclosure scheme that computes vertical eddy diffusivity based on TKE predictedby a prognostic equation as a function of local vertical wind shear, stability, andturbulence length scale. The effects of the viscous sub-layer are taken into accountthrough variable roughness length for temperature and humidity [201]. Other moresophisticated schemes are based on ensemble-averaged turbulence models (Xue etal., 1996) with varying orders of closure (e.g., [121, 194, 5]). These schemes oftenperform remarkably well under horizontally homogeneous conditions in model-ing the horizontally (ensemble) averaged profiles of quasi-conservative quantities.However, the 3-D structure of the boundary layer is not predicted well. They aremore complicated, and require solving equations of higher order moments, limitingthe practical application [105]. Several researcher proposed modifications to theoriginal Mellor-Yamada model improving the master length scale equation [176]or the pressure-strain, pressure –temperature covariance closures. [130]Other researchers, however, showed that there is little gain in accuracy with in-creasing scheme complexity using different turbulence parameterizations. Increas-ing the complexity of the turbulence parameterization not just increased the com-putational resources but did not show obvious improvement, sometimes producingequally poor, if not worse, predictions [199] of the simulated mean and turbulentproperties in the boundary layer, even around complex regions [13].Traditional PBL schemes were adequate for flat, horizontally homogeneous sur-faces under steady-state conditions with different stratification. They cannot cope,however, with increased-resolution models that require more detailed and accurate

9

2.3. ADAPTION OF GENERAL PURPOSE CFD SOLVERS

representations of physical processes [7]. Therefore, different approaches are de-veloped where the calculation of eddies was done explicitly in the PBL using 3-Dhigh resolution models. Since only a small portion of the turbulence is handledby the subgrid scale (SGS) scheme, the results are less sensitive to turbulence clo-sure assumptions. The first LES models and work in this area was pioneered byDeardorff [38, 37]. For PBL applications, LES models typically require horizontalresolutions on the order of 100 m (see, e.g., ARPS, MESO-NH codes), and theyare typically used for research applications (e.g., [188]). Deardorff [39] designeda simplified 1.5-order closure scheme that requires the solution of only one addi-tional prognostic equation for the SGS turbulent kinetic energy, where the eddycoefficient was assumed to be proportional to the square root of TKE. This schemehas been widely used by researchers [90] to handle SGS turbulence in cloud-scalemodels. Such models have a horizontal resolution in the order of 1 km, and theyare expected to resolve cloud structures and turbulent eddies.

2.3 Adaption of general purpose CFD solvers

CFD tools have been in use for decades with success for solving engineering relatedproblems involving broad range of physics. Spatial scales are extended to urbanscales to handle flows involving pollution dispersion and can be even extendedto mesoscale problems by the presented transformation method. Therefore, it isinteresting to show the main aspects and some of the current problems of CFDsolvers in this field.In engineering CFD, continuity, momentum, and energy equations are usuallysolved based on the finite volume method in an unsteady conservative form. Be-cause of numerous advantages of the finite volume method, it is widely spreadamong commercial and open source fluid mechanical solvers. Although the in-stantaneous Navier–Stokes equations exactly define all fluid flow, it is essentiallyimpossible to solve these equations for turbulent flows over domains of significantspatial scale. Therefore, the exact equations are often Reynolds averaged to createa set of equations that can be solved for the spatial scales of engineering interest.The current adaption method was developed for the commercial fluid mechanicalsolver ANSYS-FLUENT, but it can be implemented in other solvers as well, havinguser defined function (UDF) capabilities such as the commercial codes ANSYS-CFX and StarCD or the open source solver Openfoam. Through UDF-s, theuser can modify the governing equations of the CFD code by adding appropriatesource/sink terms to the equations.The governing equations (Eqs. (2.5)–(2.7)) are solved by using the Boussinesqapproximation (Eq. (2.10)) for the density.

10


∇v = 0, (2.5)

∂

∂t(ρ0v) +∇(ρ0v⊗ v) = −∇p+∇τ + (ρ− ρ0)g + F, (2.6)

∂

∂t(ρ0cpT ) +∇(ρ0vcpT ) = ∇KT∇T + ST , (2.7)

∂

∂t(ρ0k) +∇(ρ0vk) = ∇

(µTσk∇k

)+Gk +Gb − ρ0ε+ Sk, (2.8)

∂

∂t(ρ0ε)+∇(ρ0vε) = ∇

(µTσε∇ε)

+ρ0C1Sε−ρ0C2ε2

k +√νε

+C1εε

kC3εGb+Sε, (2.9)

ρ = ρ0 − ρ0β(T − T0), (2.10)

In the equation system v, p, ρ, T are the transformed field variables of velocity,pressure, density, and temperature. cp and β are the specific heat capacity of dryair at constant pressure and the thermal expansion coefficient.From the velocity vector (v = ui + vj + wk ) only the vertical component wasaffected by the transformation. τ contains the viscous and turbulent stresses, g =−gk is the gravitational force per unit mass, and g = 9.81[Nkg–1]. In the presentedsystem the turbulent transport is modeled by the realizable k–ε turbulence modelwith full buoyancy effects (Eqs. (2.8) – (2.9)) developed by Shih et al. [163]. σk andσε are the turbulent Prandtl numbers for k and ε , respectively. The turbulentviscosity µt and the turbulent heat conduction coefficient Kt are evaluated onthe basis of turbulence kinetic energy (k) and dissipation rate (ε) fields [104].The constant values of C1ε, C2ε, the expressions of C1and C3ε, the turbulenceproduction and buoyancy terms Gk and Gb modulus of mean rate-of-strain tensorS can be referred either from CFD literature [163] or from software documentation[6] of the applied simulation system. ρ0 and T0 are reference (sea level) values ofdensity and temperature.Volume sources ST , Sk, and Sε in Eqs. (2.7)–(2.9), as well as vector in Eq. (2.6),are functions of local values of field (prognostic) variables, and used for adjustingthe model to handle mesoscale effects. The components of the Coriolis force areincluded in F through Su, Sv and Sw. Reynolds averaging introduces other un-knowns into the equation system, the so-called Reynolds stresses. Since there is

11


no existing general formula for the description of these stresses, large number ofengineering turbulence models were developed applicable to certain flow features,but there are no turbulence models that would be generally applicable to all kindof turbulent flows. The modeling of these virtual stresses is still a major part ofthe today’s turbulence modeling science.Different versions of two equation models are used in CFD for turbulence mod-eling. The most popular are versions of the k–ε model or other two equationmodels. These versions are denoted as linear eddy viscosity models, which adoptthe Boussinesq approximation, implying an isotropic eddy viscosity. There wereattempts using two equation models in NWP codes as well. An example is theimplementation of a version of the k-ε model by Hanjalic and Launder [63] in theMESO-NH meteorological code. It was not used extensively since, according tothe authors, the current version did not give satisfactory results. The disadvantageof these eddy viscosity models is that they give an unrealistic representation of thenormal turbulent stresses. This representation of the turbulent stresses producesfairly good results as long as only one of the turbulent stresses is dominant in themomentum equations. Further downside is that in more complicated flows, wheremore than one of the normal stresses is important, the ability of a turbulencemodel to predict normal stress anisotropy becomes significant. This motivates thedevelopment and application of nonlinear k–εmodels [54] and other more advancedmodels, such as LES or hybrid RANS-LES models.

2.3.1 Stratification and turbulence models

One can find several works on the application of turbulence models in neutralconditions [155, 15, 70]. Stable stratification, however, causes drainage flows overuneven topography, intermittent turbulence, low-level jets, gravity waves, flowblocking, intrusions, and meandering, thus posing challenges in modeling of stablestratification [105].Several problems can be identified regarding the maintenance of turbulence profileswith distance [80] dissipating turbulence early downstream of obstacles [68]. Treat-ment of these inconsistencies is the modification of the model constants of existingmodels [46], adding source terms to the turbulent dissipation rate (TDR) equationfor ε or a non-constant formulation for the C1ε parameter [51, 144]. Vendel et al.[182] proposed an inlet pressure profile and flux condition for the ground in orderto define an appropriate downwind boundary condition for the stable or unstablecases.Although CFD models are currently slow to be used for real-time emergency re-sponse, they can be used for planning purposes and to guide parameterizations

12


of real-time wind flow models. A good example of a dispersion model that is pa-rameterized based on the CFD results is the Quick Urban and Industrial Complexdispersion modeling system [192]. As computing power has become more afford-able, CFD has become an increasingly valuable tool for studying urban flow. Thesemodels explicitly account for building geometry and require minimal parameteri-zations [8, 10].

2.3.2 System of transformations

In this subsection the transformation method will be introduced briefly, the fulldescription can be found in [99].The proper representation of Coriolis force, compressibility, and stratification ef-fects is achieved by a system of transformations. These adjust the governingequations solved by the CFD solver through user defined volume source or sinkterms. The CFD solver operates with transformed field variables (Eqs. (2.5) –(2.10)). Relations between untransformed physical quantities (T, p, ρ, z, w) andtransformed ones (T , p, ρ, z, w) are defined by Eqs. (2.11) – (2.14).

T = T − T0 + T , (2.11)

p = ρ

ρ0p+ p = Ce−ςzp+ p, (2.12)

ρ = ρ− ρ0 + ρ, (2.13)

z = −1ςln(e−ςzref − ς

C(z − zref )

), (2.14)

w = ρ0

ρw = wC−1eςz, (2.15)

where ς is the density parameter described by Eq. (2.18) and zref is a referencealtitude (Eq. (2.20)). Zero subscript denotes values at ground level. The verticalextent of the atmosphere is “compressed” below a well-defined bound (C/ς) de-scribed by Eq. (2.14), thus z → C/ς, when z → ∞, where C, described by Eq.(2.19), acts as a switch between the description of stratosphere and troposphere.The model utilizes an (x, y, z) Cartesian coordinate system. The Jacobian of thetransformation of coordinate z can be calculated according to Eq. (2.16). J → 1

13


when z → 0, and therefore the geometrical transformation has a negligible effectclose to ground.

J = dz

dz= C−1exp(ςz), (2.16)

In this zone, the original form of the Cartesian equations existing in the CFDsolver gives a good enough description for the flow close to the surface.

2.3.3 Reference profiles

The relationship between the absolute physical quantities and the field variablesin the CFD solver are based on the reference profiles (distinguished by over-bars)Eqs. (2.21) – (2.29). It can be optimized to have the least possible deviation fromthe hydrostatic equilibrium and can simplify the specification of the initial con-ditions. These terms depend only on the vertical coordinate using approximationof the polytrophic atmosphere, as according to Eqs. (2.21) – (2.29). The originaltransformation expressions [99] that were valid below an altitude of 11 km havebeen extended to incorporate the tropopause and the lower stratosphere up toan altitude of 25 km. For simplicity, the following double valued constants areintroduced to describe these layers:

γ =γt, when z < ztp

0, when z ≥ ztp, (2.17)

ς =ςt, when z < ztp

ςs, when z ≥ ztp, (2.18)

C =1, when z < ztp

e(ςs−ςt)ztp , when z ≥ ztp, (2.19)

where subscripts t and s denote values for the troposphere and stratosphere, andztp is the tropopause altitude. In the stratospheric region we use ztp as a referencealtitude, therefore:

zref =0, when z < ztp

ztp, when z ≥ ztp, (2.20)

Using the above notations, the reference profiles for the troposphere will read as:

14


T t = T0 − γtz, (2.21)

pt = p0

(T0 − γtzT0

) gRγt

, (2.22)

ρt = ρ0e−ςtz, (2.23)

ςt = −(g −RγtzRγT

)ln(

1− γtz

T0

). (2.24)

The reference profiles in the stratospheric region are based on an isothermal tem-perature profile, therefore:

T s = Ttp, (2.25)

ps = ptpe−ςs(z−ztp), (2.26)

ρs = ρtpe−ςs(z−ztp), (2.27)

ςs = g

RTtp, (2.28)

By using the above notations, the universal density profile reads:

ρ = ρ0Ce−ςz, (2.29)

The constant values in Eqs. (2.21) – (2.29) can be optimized for a given case, butthe standard ICAO (Manual of the ICAO Standard Atmosphere, Doc 7488 1993)temperature and pressure profiles, based on the following constants, have beenfound to be suitable references for simple applications: γt = 0.0065 °C m–1, T0 =15 °C, p0= 101325 Pa, ρ0=1.225 kg m-3, R = 287.05 J kg-1 K-1, ςt = 10-4 m-1

2.3.4 Volume sources

The volume sources (Eqs. (2.30) – (2.35)) have been calculated according to Eqs.(2.21)–(2.29) by utilizing the UDF capability of the system.

15


Su = ρ0fv − ρ0 ˜lwJ, (2.30)

Sv = −ρ0fu, (2.31)

Sw = ρ0(J2 − 1

) (luJ−1 + β

(˜T − T0

)g)

+ ρ0luJ−1 + ςJ

( ˜p− ρ0 ˜ 2w), (2.32)

ST = −ρ0cpw (Γ− γ) J, (2.33)

Sk = −βg µtPrt

(Γ− γ) , (2.34)

Sk = −C1εC3εε

kβg

µtPrt

(Γ− γ) . (2.35)

In the above expressions, µt and Prt are the turbulent viscosity and turbulentPrandtl number, f = 2Ωsinϕ and l = 2Ωcosϕ are the Coriolis parameters, ϕ isthe average latitude, and Ω is the angular velocity of Earth.Moisture transport is not taken into account in the mathematical model, thereforethe dry adiabatic temperature gradient Γ appearing in Eqs. (2.33) – (2.35), is cal-culated according to the assumption of a dry adiabatic process: Γ = g/cp =0.00976°C m–1.

2.3.5 A simplified model version for water-tank experiments

Validation against small scale water tank experiments requires the adjustment ofthe transformation expressions. The working medium is liquid and the verticalextent of the device is small, therefore the atmospheric pressure variation withthe vertical coordinate is negligible. Compressibility is not taken into accountconsequently (Γ = 0, z = z, and ςt = 0). The vertical reference profiles forpressure and density are ρt = ρ0, pt = p0. Due to the very weak turbulencecharacterizing these experiments, a laminar approach or LES method should beused instead of the k-ε model. The Coriolis force usually has a negligible effect.The adiabatic expansion can also be neglected in such experiments, and therefore,Γ = 0 can be assumed. These assumptions will simplify the source term acting onthe energy equation:

ST,incomp = ρ0cpwγ. (2.36)

16


The values of γ = −dT/dz , cp, β, and T0 for water-tank simulations can be cal-culated according to the temperature or density gradient of the fluid, maintainingthe same Brunt-Väisälä frequency and material properties of the experiment.

2.3.6 Nesting

In NWP models the simplest method is the “one-way” downward nesting, wherethe outer model provides time dependent 3D boundary conditions to the innerhigher resolution domain. In some cases it can be provided, if the mesoscale modelhorizontal resolution is as fine as a few hundred meters. The one-way nesting tech-nique, in the Advanced Regional Prediction System (ARPS), allows adjustments invertical resolution between grids, which is important for LES mode, where the gridaspect ratio should be kept close to unity. In more advanced “two-way” nesting,the nested finer domain and the outer coarser domain interact at every time step ofthe outer coarse grid and the microscale model provides lower boundary-conditionto the outer domain. “With current computer capacities, this can be done withinsmall parts of the mesoscale domain, which creates inconsistencies with the re-maining parts of that domain.” [7] The current implementation of two-way nestingschemes in WRF or MM5 does not allow higher vertical resolution in the innernesting levels, because the number of vertical levels must be the same for all levels[124]. Another problem that also exists in the CFD field is the velocity and lengthscale changes across the boundaries of nesting levels. This is especially importantwhen the complex high resolution topography generates nonlinear motions closeto these boundaries. One may also expect the damping of high frequency mo-tions passing through the boundary and as a consequence providing an improperupstream turbulence field for the inner microscale model.

17

2.4. VALIDATION OF THE DYNAMICAL CORE OF THE MODEL

Figure 2.1. Schematic of domain nesting in NVP models (left) and arbitrary meshrefinement in the CFD tool (right).

The advantage of our CFD approach is that it allows arbitrary mesh refinements inthe simulation domain (see Fig. 2.1), resulting a continuous change of field variablesaround the finely resolved region. The possibility of nesting also exists in theCFD solver allowing different mesh resolution either horizontally or vertically onboth sides of the domain interface, but one can avoid it with the existing meshrefinement options [6].

2.4 Validation of the dynamical core of the model

The proper representation of Coriolis force, compressibility and stratification ef-fects in the model are tested and demonstrated in the following examples. Eachcase focuses on one individual element of the model.The implementation of the Coriolis force has been tested [99] on a simple testproblem in order to compare results of the present model with the analyticalsolution of Ekman [47].A comparison between simulations of a two-dimensional density current generatedby a descending cold air blob has been made by Straka et al. [172]. The highlyresolved grid-independent benchmark solution used in this study is still a standardtest case for compressible flow simulation codes [152].

18


2.4.1 Geometry and mesh

In this test case, a cold air bubble released at high altitude descends, and aftercolliding with the ground spreads along the surface in the x direction, generatingvortices via Kelvin-Helmholtz instability in the shear layer. In correspondencewith the benchmark solution, a constant viscosity is used and the Coriolis force isset to zero. Free slip boundary conditions (symmetry) are used on each side of thedomain. The domain extends 25.6 km in the horizontal and 6.4 km in the verticaldirections. Potential temperature is constant in the model atmosphere, exceptwithin the ellipsoid shaped cold bubble where the local temperature is decreasedby ∆T . As in the reference case [172], model constants are set toR = 287 J kg-1K-1, cp=1004 J kg-1K-1, µt/ρ0=75 m2s-1, g=9.81 m s-2, T0=Θinit=300K, p0=105Pa, and

∆t =0 if L > 1−15(cos(πL) + 1)/2[K] if L ≤ 1

, (2.37)

L =√(

x− xcxr

)2+(z − zczr

)2, (2.38)

with xc= 0, xr= 4 km, zc=3 km, zr=3 km.Equidistant 100-m resolution is used in the transformed space of the present simu-lation. In correspondence with the constant potential temperature reference profileused in the test problem, the values of γ = Γ = g/cp and ς=0.9 e-4 m-1 have beenchosen.One advantage of the present transformation system [99] is its ability to easily al-ternate between compressible and incompressible models. The simulation has beenrepeated with an incompressible model by setting z = z and ς=0, and setting allsource terms to Su = Sv = Sw = ST = Sk = Sε = 0, but, in general, stratificationand adiabatic expansion can be taken into account in incompressible simulationsby utilizing ST .

19


2.4.2 Results

Figure 2.2. Nonlinear density current calculated by compressible (a-c) and by in-compressible (d-f) versions of the simulation method. Contour plots of the poten-tial temperature Θ (a, d), velocity components u (b, e) and w (c, f) at t = 900 s.Contour intervals are 1 K and 2 m s-1.

REFQ25 LES100 CFDA100 CFDB100

Front location (m) 15509 15200 15753 16816umax(m s-1) 34.72 32.49 29.17 33.36umin(m s-1) -15.31 -13.64 -16.56 -12.51wmax(m s-1) 13.04 12.15 17.69 17.34wmin(m s-1) -16.89 -15.07 -19.63 -17.82

Θ′max(K) 0.00 0.01 0.00 0.00Θ′min (K) -10.00 -10.40 -9.00 -8.097p′max (hPa) 1.74 1.43 1.12 1.21p′min(hPa) -5.21 -4.29 -6.45 -6.04

Table 2.1. Maximum and minimum values of u, w, Θ′ = Θ−Θ, p′ = p−p and frontlocation of the density current at 900 s for the grid converged (25-m resolution)reference solution of Straka et al. [172] (REFQ25), large-eddy simulation of Reinertet al. [152] (LES100), the present model (CFDA100) and the present model withincompressible option (CFDB100).

20

2.5. VALIDATION ON LABORATORY SCALE URBAN HEAT ISLANDCIRCULATION

Simulation results from the compressible version of the present model (Fig. 2.2/a–c) agree well with the corresponding plots of Reinert et al. [152] and also fall withinthe range of the results of other compressible solvers published by Straka et al.[172]. Of the numerical data listed in (Tab. 2.1), the front location has specialimportance because it can be considered as a time integrated quantity, unlike theother results that are more sensitive to numerical fluctuations. The front locationobtained from the compressible version of the present model is in strong agreementwith the reference solution. It is interesting to note that there is little differenceobserved between the compressible and incompressible cases.

2.5 Validation on laboratory scale urban heat islandcirculation

The urban heat island circulation and its interaction with a sea breeze was in-vestigated by Cenedese and Monti [26] using a water tank with stable thermalstratification created by hot and cold heat exchanger plates at the top and thebottom of the tank. Thermal convection was generated by a thin layer (D = 100mm diameter) circular electric heater plate placed on the bottom of the tank. Thevelocity field was measured with a particle image velocimetry (PIV) device, allow-ing high spatial resolution in the vertical symmetry plane, while the temperatureprofiles were measured using probes in a vertical arrangement above the heaterplate. The experiment was started at t = 0 s by turning on the electric heater towhich a constant power density of 1250 W m-2 was applied. Instantaneous veloc-ity fields, measured in the mid-plane via PIV, were averaged in the time interval850–1150 s. The same procedure has been used in the corresponding numericalsimulation.


Due to the weak turbulence characterizing the-water tank experiments, the k-εmodel has been switched off and the large-eddy simulation (LES) method hasbeen used; therefore, source terms in Eqs. (2.8) and (2.9) were not tested. TheCoriolis force had a negligible effect in these experiments and therefore Ω = 0in the simulation. Variable density effects are also not taken into account in thelaboratory experiments, consequently Γ = 0, z = z, ς = 0 and p = p0 − ρ0gz havebeen applied in the simulation. The only operational part of the transformationsystem was the heat source according to the following simplified expression:

21


ST,incomp = ρ0cpwγ (2.39)

Note that γ = −dT/dz is negative in this case. Model constants were taken fromthe water tank experiment: γ = – 200 K m-1, cp = 4180 J kg-1 K-1,β = 0.00025 K-1, T0 = 300K.The number of cells was 920,000, the timestep was 0.05 s, the typical value of theCourant number was 0.4, and the non-dimensional cell wall distance was y+ =0.2 (y+ = yu∗/ν in which y is the wall distance, u∗ is the friction velocity, and νis the kinematical viscosity of water). A bounded central differencing scheme [6]was used for spatial discretization. Subgrid-scale stresses were calculated by usingthe Smagorinsky model [50]. Simulations were run a small cluster consisting of 6AMD64 Athlon 3000+ CPU cores. The total flow time of 1200 s was calculatedin approximately 130 hours wallclock time using this machine.Results of the LES simulation in comparison to experimental data are plottedon Fig. 2.3 – Fig. 2.5, where zi denotes the height of the mixing layer, U and Ware velocity scales with U = (gβDH0/ρ0cp)1/3, Fr = U/ND, W = UFr, N =(gβdTa/dz) 1

2 , and the dimensionless temperature is T ′ = (T − T )/(Tx=0,z=0 − T ),in which T is the unperturbed (initial) temperature profile.

2.5.2 Results

According to qualitative comparison of Fig. 2.3 with the work of Cenedese andMonti (Fig. 6b in [26]), the simulation results for the time-averaged velocity fieldare in agreement with the measured vector plot. The width and maximum heightof the thermal plume, as well as the direction and the magnitude of velocity vectors,are very similar. As shown in Fig. 2.4, the u component along a vertical line at theperiphery of the heater plate corresponds closely with the measured values. Themeasured w component along the axis of the heater is slightly overestimated by thesimulation, but the tendency and the top of the plume is well predicted. Measureddimensionless temperature profiles (Fig. 2.5) have been remarkably reproduced bythe simulation, demonstrating the correct representation of thermal stratificationby the volume heat source ST , the only active element of the transformation systemin this simulation.

22


Figure 2.3. Computed flow field in the symmetry plane (the time interval of aver-aging is 300 s)

Figure 2.4. Dimensionless profiles of horizontal (left) and vertical (right) velocitycomponents, symbol: measured data, solid line: LES results

23

2.6. SUMMARY AND CONCLUSIONS

Figure 2.5. Dimensionless profiles of the temperature perturbation along the axisof the heater plate, LES results (left) and measured data (right)

2.6 Summary and conclusions

An extension of the physical model used in general purpose CFD solvers has beendeveloped in order to simulate mesoscale atmospheric phenomena together withfinely structured microscale flow around complex geometrical features. The ulti-mate purpose of this model is the simulation of the urban heat island circulationand flow around buildings with pollution dispersion, using only one single unstruc-tured grid and a uniform physical description for close and far-field flow.The new method has been formulated in a transformation system, based on ref-erence profiles fulfilling the hydrostatic equilibrium. The reference temperatureprofile must be linear, such as the standard ICAO atmosphere. Density is approx-imated with an exponential function of the vertical coordinate in order to reducethe complexity of the transformation expressions.The addition of volume sources to the standard set of incompressible transportequations is necessary for the correct description of hydrostatic pressure drivendensity variation, adiabatic cooling in a thermally stratified atmosphere, and theCoriolis force.In this study the ANSYS-FLUENT simulation system was used, but users of othergeneral purpose CFD packages can easily adopt the model if the simulation systemallows user-defined source terms to be functions of local velocity, pressure, andtemperature.The model was successfully validated with the analytical solution of the Ekmanspiral, a well-known two-dimensional benchmark solution for testing compressible

24


fluid solvers, and with the results of a water-tank experiment involving thermalconvection in a stratified medium. Validation of the proper treatment of turbulenttransport requires further comparisons to full-scale atmospheric measurements.The implementation of the moisture transport is also a necessary step toward thepractical application of the method. The latter two points will be addressed inmore details in the following chapters.

25

3 Modeling internal gravity wavesby using a pressure based CFDsolver

3.1 Overview

Mountain wave phenomena have been simulated by using a well-known generalpurpose computational fluid dynamic (CFD) simulation system adapted to at-mospheric flow modeling. Mesoscale effects have been taken into account witha novel approach based on a system of transformations and customized volumesources acting in the conservation and governing equations. Simulations of lin-ear hydrostatic wave fields generated by a two-dimensional obstacle were carriedout, and the resulting vertical velocity fields were compared against the corre-sponding analytic solution. Validation with laboratory experiments and full-scaleatmospheric flows is a very important step toward the practical application of themethod. Performance measures showed good correspondence with measured dataconcerning flow structures and wave pattern characteristics of non-hydrostatic andnonlinear mountain waves in low Reynolds number flows. For highly nonlinear at-mospheric scale conditions, we reproduced the well-documented downslope wind-storm at Boulder in January 1972, during which extreme weather conditions, witha wind speed of approximately 60 m s–1, were measured close to the ground. Theexistence of the hydraulic jump, the strong descent of the stratospheric air, wavebreaking regions, and the highly accelerated downslope wind were well reproducedby the model. Evaluation based on normalized mean square error (NMSE), frac-tional bias (FB), and predictions within a factor of two of observations (FAC2)show good model performance, however, due to the horizontal shift in the flowpattern, a less satisfactory hit rate and correlation value can be observed.

26

3.2. INTRODUCTION AND LITERATURE SURVEY

3.2 Introduction and literature survey

In the first validation steps (the implementation of the energy source term andthe Coriolis force was investigated), large-eddy simulations (LES) of small scalethermal convection problems were carried out in order to simulate urban heatisland circulation problem.These validation cases have the advantage of that they have more control over themeasured parameters due to the nature of the measuring device. However, theyhave low Reynolds number range, the cases are mainly hydrostatic, the verticalextent is limited to a certain height, and applied a fixed turbulence viscosity model.The purpose of this section is to further extend the validation cases characterizedby non-hydrostatic and strong nonlinear situations. Further differences comparedto previous validations are the complex topography, the extended simulation do-main height incorporating the tropopause, and that all of the source terms areactivated (see Eqs. 2.30 – 2.35). Results of gravity wave simulations comparedagainst the corresponding analytic solution, small scale water-tank experiments[62], and full scale observations [109] will be presented. The simulation of suchcases has been found to be ideal for testing and evaluating mesoscale numericalmodels due to the presence of complex flow patterns and wave breaking phenom-ena.Atmospheric waves form when stable air flow passes over an obstacle. Fluid parcelstend to return into their original height due to the restoring forces caused bystratification. Various types of oscillatory flow responses occur depending on thestate of stratification and geometric parameters [72]. One can see two divisions ofmountain waves, vertically propagating and trapped lee waves. Mountain wavesvertically propagating over a barrier may have horizontal wavelengths of manytens of kilometers, even reaching the lower stratosphere [110]. The vertical prop-agation of trapped lee waves is limited to a certain height due to the presence ofa highly stable layer when waves can be reflected in such situations. In general,gravity waves can modify the local weather situation near mountains: they cancreate rotor motions, hydraulic jumps, and they have the capability of concen-trating momentum on the lee slopes, or occasionally leading to violent downslopewindstorms. [88, 164] Lee waves can be a potential hazard for wave gliders, byproducing rotors or clear air turbulence. Flow beneath the wave crests can beextremely turbulent, thus causing a potential hazard for low-level aviation as well[131]. The simulations of atmospheric scale flows around mountains can also haveeconomical importance when the future location of a wind farm is to be estimated[126, 113, 135].One can observe mountain waves [169] with the help of various types of clouds,

27

3.3. VALIDATION WITH THE ANALYTIC SOLUTION OF GRAVITY WAVEPROPAGATION

altocumulus or wave clouds at wave crests. Rotor clouds may dye some parts ofthe wave field if the appropriate amount of moisture is present. In some cases theyare marked by regularly spaced clouds, and may be of great help to the flight ofgliders [107].Several researchers examined mountain waves and established theories to describethe basic phenomena [161, 112, 40, 170]. One can also find experimental worksdealing with the examination of flows around small scale, simplified, two-dimensionalisolated obstacles [62]. Gravity waves can be a good basis for the validation of at-mospheric simulation models, as their structure strongly depends on the stateof stratification. Indeed, one can see numerous works dealing with the valida-tion of mesoscale models against mountain waves [44, 177, 197], such as the onescaused by a severe downslope windstorm event of Boulder. Similar events canoccur in mountainous regions all over the world [32, 12]. Postevent analysis ofsevere windstorms are often performed to study the future predictability of suchevents and also to test model performance and validity. A recent study dealt withthe November 19, 2004 windstorm in the High Tatras in Slovakia [164]. The au-thors concluded that an increased 2.5 km resolution mesoscale model can forecastdownslope windstorms.In the following section simulation results will be shown in comparison with an-alytical solutions, water-tank experiments, and full scale downslope windstormobservations.

3.3 Validation with the analytic solution of gravitywave propagation

The application of the model transformation will be presented in this section.Results will be compared against the analytical solution of linear, hydrostaticmountain wave field. Nonlinear and non-hydrostatic simulation cases are thencompared to small scale water-tank experiments, and finally, a full scale nonlinearand non-hydrostatic downslope windstorm case will be shown against an inter-comparison study of NWP models and field measurements.


The simulations were compared with the analytic solution of a linear hydrostaticwave field. A Witch of Agnesi curve was used for the relief geometry,

28


z(x) = ha

x2 + a2 , (3.1)

where h and a are the obstacle height and half width, respectively. This geometryhas been used extensively in the literature [193], as an analytic solution can bederived for this shape. In order to obtain comparable results with water-tankexperiments, the following material properties and working conditions were used:ρ0 = 998 kg m–3, T0 = 27 °C, β = 0.000207 K–1, –γ = 832 K m–1, U = 0.25 m s–1.The Reynolds number, based on the inlet flow speed (U) and the obstacle height(h), was about 2500; therefore a laminar model was applied. Compressibilityand the Coriolis force were not taken into account. Free-slip wall and symmetryboundary conditions were used at the lower and upper boundaries. A constantvelocity and static pressure profile were prescribed as inlet and outlet conditions.The model was initialized with constant T = T0, p = p0 , and uniform horizontalvelocity U = U0.Based on the linear wave theory (Eq. 3.2), an analytic solution can be found forhydrostatic problems with a constant Scorer parameter (l). The Scorer parameteris usually a function of the vertical coordinate, and it is derived from the waveequation for atmospheric gravity waves with an assumption of a two-dimensional,non-viscous, adiabatic flow [168, 45]. This parameter is most often used to forecastthe existence of trapped lee waves, which can be expected when l decreases stronglywith height. This is especially true if l decreases suddenly in the mid-tropospheredue to the presence of a stable layer dividing the troposphere into a highly anda weekly stable layer. The square root of the parameter l (Eq. 3.4) has unitsof wave number. The wave number of the lee wave (l) lies between lupper of theupper layer and llower of the lower layer. Wide mountain ranges generate verticallypropagating waves of wave numbers less than lupper. Small obstacles, that forcewave numbers greater than llower, produce waves that will vanish with height.For hydrostatic problems, the development of the full analytic solution has beensummarized by several researchers [4, 168]. The combined governing equationswere resulted in the deep Boussinesq equation (Eq. 3.2) for w. The Scorer param-eter l is defined by Eq. 3.2 and Eq. 3.3:

∂2(√

ρ0(z)ρ0(z=0)w

)∂x2 +

∂2(√

ρ0(z)ρ0(z=0)w

)∂z2 − l2(z)

√√√√ ρ0(z)ρ0(z = 0)w

= 0, (3.2)

l2(z) = N2

U2 −1U

d2U

dz2 −1U

dlnρ

dz

dU

dz− 1

4

(dlnρ

dz

)2

− 12

(d2lnρ

dz2

), (3.3)

29


where w is the perturbation vertical velocity, N =√

gρ0

∂ρ(z)∂(z) is the Brunt-Väisälä

frequency, U is the inlet flow speed, ρ is the fluid density, and z is the verticalcoordinate, respectively. For incompressible flow only the first two terms remain.

l2(z) = N2

U2 −1U

d2U

dz2 (3.4)

The first term on the right-hand side dominates in our case, and additionally, if auniform velocity profile (U = U0) and Brunt-Väisälä frequency (N) are used, Eq.3.4 simplifies to:

l = N

U(3.5)

With the help of the Scorer formula, flows can be categorized on the basis of thefollowing two dimensionless quantities:

Fx = a l, Fz = h l (3.6)

When Fx >> 1, the flow is essentially hydrostatic, while non-hydrostatic effectsbecome important when Fx ~ 1 [160]. Regarding the linearity, when Fz << 1, theflow is linear and nonlinear effects dominate, when the value approaches to Fz ~1. The borders are, however, not well-defined, and that is why one can find testcases marked as moderately hydrostatic or moderately nonlinear [177]. Calculatingthese dimensionless quantities using a = 2 m and h = 0.01 m, one can find thatour setup is linear and hydrostatic with l = 5.32 m–1, Fx = 10.64, and Fz = 0.0532.In the case of hydrostatic flow, an analytic solution exists (Eq. 3.7) for the verticalvelocity field w(x, z), based on Eq. 3.2:

w(x, z) =

√√√√ρ0(z = 0)ρ0(z) Uha

(x2 − a2)sin(lz)− 2xacos(lz)(a2 + x2)2 (3.7)

3.3.2 Results

The vertical velocity contours have been plotted against the corresponding analyticsolution on Fig. 3.1 The agreement between the two solutions is very good, withthe magnitude of the vertical velocity in the numerical solution being only slightlysmaller. The locations of velocity maxima, in the case of the CFD results, areshifted only moderately downward in the vertical direction.

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3.4. VALIDATION WITH LABORATORY SCALE GRAVITY WAVEEXPERIMENTS

Figure 3.1. Contour plots of the vertical velocity (w [mm s–1]), obtained from theanalytic solution based on Eq. 3.7 (left) and from CFD calculations (right). Thesolution is obtained for U = 0.25 m s–1, l = 5.32 m–1, N = 1.3 s–1, a = 2 m, h =0.01 m. The vertical and horizontal scales are normalized by the mountain height(h) and half width (a) respectively. Dashed lines represent negative values.

In the next section, a set of test cases have been prepared and the results arecompared against experiments that cover a hydrostatic and nonlinear parameterrange. The statistical performance measures applied for the comparison are alsodescribed briefly.

3.4 Validation with laboratory scale gravity waveexperiments

Among the experiments focused on topography-induced gravity waves, one canfind investigations dealing with symmetric and asymmetric obstacles, looking atthe effect of asymmetry [62], determining surface drag by numerical simulations[88] and numerically examining the wave breaking characteristics [41, 2]. Thecommon feature of the former experiments is the experimental apparatus. Theyuse mainly the water as working medium, since in water it is relatively easy togenerate and maintain stable stratification for a longer period of time. Usually,this is done by using layered salt water. The achievable range of Reynolds numberis limited to approximately 103. In spite of this fact, these methods are widelyused, since there is more control on the parameters and conditions, than in realatmosphere.

31


3.4.1 Statistical performance measures

To quantitatively evaluate the output of the model with observations, Hanna et al.[69, 65] recommend the use of the following statistical performance measures [28]:the fractional bias (FB), the normalized mean-square error (NMSE), the fractionof predictions within a factor of 2 of observations (FAC2), and the hit rate (HR).Chang and Hanna [27] suggest that a good model would be expected to have about50% of the predictions within a factor of 2 of the observations (i.e., FAC2 > 0.5), arelative mean bias within 30% of the mean (i.e., –0.3 < FB < 0.3), a relative scatterof about a factor of 2 or 3 of the mean (i.e., NMSE < 4), and a hit rate above 66%(HR > 0.66) with an allowed deviation of D = 0.25. The absolute value of themodel’s fractional bias (FB) is reasonably good if it is less than 0.25. A tendencytoward overprediction of wind speeds is seen, with the fractional bias regularlybetween 0 and –0.25 [115]. The perfect model would have the following idealizedperformances: FAC2 = HR = 1 and FB = NMSE = 0. In air quality modeling,typical values of the above statistical measures have been defined as acceptable formodel evaluation and also correspond to a model acceptance criterion [27].


Two-dimensional simulations of internal gravity waves have been carried out by us-ing symmetric and asymmetric obstacles, in correspondence with the experimentsperformed by [62]. Obstacle shapes have been characterized by the following func-tion:

z(x) = aexp(−b|x|2k), (3.8)

where a, b, and k are shape parameters and x is the horizontal coordinate of theobstacle. Both the symmetric and asymmetric shapes can be described by Eq.3.8, by prescribing different parameters for the upstream and downstream part ofthe mountain barrier. Measurements were performed in a narrow plexi glass tankof 2.4 m × 0.087 m × 0.4 m filled with linearly stratified salt water, by towingthe obstacles along the bottom of the tank at a constant speed. The range of theexperimental parameters (obstacle height h = 0.02 – 0.04 m, towing velocity U =0.01 – 0.15 m s–1, and Brunt-Väisälä frequency N = 1.09 – 1.55 s–1) correspondsto an atmospheric flow up to an elevation of 5 – 10 km for an obstacle height of600 m and a wind speed of 10 – 70 m s–1 for a large range of hydrostatic state andlinearity.The flow was considered unsteady, incompressible, and two-dimensional in thesimulation. Due to the low Reynolds number range, a laminar model was used.

32


The domain was discretized with a structured quad grid, using 150 x 800 quadelements. Second order time discretization, pressure staggering option (PRESTO)for pressure interpolation [6], and second order upwinding methods were used whensolving the momentum and energy equations. Compressibility was turned off inthis case by using the transformation described in sec. 2.3.5.

3.4.3 Results

Typically for higher flow velocities, we observed flow separation during the simula-tion of symmetric steep lee-side obstacles. The separation induced bubble modifiedthe shape of the leeward side in this case, quasi elongating the obstacle (Fig. 3.2),consequently changing the hydrodynamic characteristics of the barrier. Therefore,these cases were excluded from the comparison.

Figure 3.2. Computed streamlines of the velocity field for U/Nh = 0.7 (left) andU/Nh = 1 (right) showing the elongation of the separation bubble and of the wavelengths for a symmetric obstacle.

In the numerical model, flow enters into the domain with a uniform inlet velocityprofile. A moving bottom wall, for simulating the stationary bottom wall of the ex-periment was used. No visible disturbances were observed during the experimentson the water surface, consequently a rigid lid (symmetry) with free slip boundarycondition was applied in the simulation model. Avoiding the interaction betweenupward propagating and reflected wave fronts, data extraction was made after ashort transient period, before interference could occur in the wave field. This pe-riod was estimated from the vertical group velocity [62], which was proportionalto the towing speed. In those cases where steepening and wave breaking occurred,samples were excluded from the averaging of amplitudes and wavelengths.In Tab. 3.1, the simulations were categorized into four groups based on Fx =Na/U and Fz = Nh/U non-dimensional numbers as described by Eq. 3.6. Thesegroups are nonlinear (NL), moderately nonlinear (M–NL), hydrostatic (H), and

33


moderately non-hydrostatic (M–NH) cases. For all cases, the mountain height h,was 0.04 m, the leeward half width a, was 0.13 m, the Brunt-Väisälä frequencyN , was 1.33 s–1 and U was the incoming flow velocity corresponding to the towingvelocities of the obstacle in the survey.

Case No. U(m s–1) Na/U Nh/U Vertical structure Linearity1 0.016 10.80 3.33 H NL2 0.032 5.31 1.643 0.037 4.63 1.434 0.041 4.15 1.285 0.048 3.60 1.116 0.053 3.24 1.007 0.077 2.23 0.69 M–NH M–NL8 0.106 1.62 0.509 0.160 1.08 0.33

Table 3.1. List of cases and parameters for the simulation of the gentle leeward sideobstacle. Vertical structure and linearity categories are: H – hydrostatic, M-NH –moderately non-hydrostatic, NL – nonlinear, M–NL – moderately nonlinear

Wavelengths and amplitudes were measured in multiple positions above and down-stream of the obstacle, by locating the wave fronts and measuring the distancebetween the fronts. The error bars in Fig. 3.3 show the standard deviation of boththe measured and simulated values. Circular wave fronts assumed in the case whenlee waves are generated by a moving point source [183]. As shown from the upperpart of Fig. 3.4, the shape of the wave fronts has been strongly affected by theobstacle shape, even in the M–NL cases. Here the centers of the wave fronts weregradually shifted toward negative coordinates, indicating strong wave dispersion.The large error bars of the measurements (Fig. 3.3) represent not only the limitedresolution caused by the visualization technique (dying of different layers), but arealso the consequence of the aforementioned strong dispersion of lee waves.It was observed that our model performs well in the H–NL range for steeper lee-side obstacles, like the one presented in Fig. 3.2. However, close to the M–NHrange with the same obstacle, results tend to differ from the experiments. Thisdifference may be explained by boundary layer separation effects. At an increasedvelocity, the separation bubble that developed behind the obstacle elongated andcaused the virtual mountain half width (mountain half width plus the horizontal

34


size of the separation) and consequently the normalized wave length to almostdouble. (See the wave field in Fig. 3.2)

Figure 3.3. Normalized averaged wave length (λ/h) (left) and normalized averagedwave amplitude (A/h) (right) as a function of non-dimensional horizontal flowvelocity, in the case of the gentle leeward slope. The dotted line represents thewave length obtained from a linear theory, based on λ = 2πU/N ([161]). Filledsymbols represent the measurements of Gyüre and Jánosi [62]. Error bars indicatethe error originating from the data extraction technique and wave dispersion.

35


Figure 3.4. Contours of computed streamlines and wave fields from the experiments(courtesy of Gyüre and Jánosi [62]) at Nh/U = 0.69 (the upper two) and Nh/U= 3.33 (the lower two) non-dimensional flow velocities.

According to experimentalists [147, 62], the side walls have a negligible overalleffect on the experimental results, however, the exact location and size of theseparation were not captured properly, which can possibly be explained by the3-D structure of the flow characterizing the laboratory experiments, due to theboundary layer on the side walls interacting with the separation bubble. In thecase of a 3-D narrow obstacle, the separation bubbles were smaller, as the strat-ified flow tended to flow around the obstacle, instead of flowing over it. The

36

3.5. MODEL COMPARISON WITH A FULL SCALE EVENT

prediction of the location of separation is currently a difficult topic, it requires themodeling of transitional turbulence and is beyond the scope of this investigation.Avoiding separation by using a gentle leeward side obstacle, however, gave bothqualitative and quantitative agreements concerning the normalized amplitudes andwavelengths (Fig. 3.3). Tab. 3.2 shows the performance measures, indicating goodmodel performance. At low inlet velocities (bottom of Fig. 3.4), where wave break-ing and rotors occurred, the flow structures, characterized by a high nonlinearity,were also captured properly.

Validation metric Abbreviation Limit λ/h A/h ClassificationCorrelation coefficient R >0.8 0.95 0.99 goodFractional bias FB ±0.3 0.317 0.21 goodNormalized mean squareerror

NMSE 0–4 0.12 0.05 good

Hit rate HR >0.66 0.75 1 goodFraction of predictionswithin a factor of two ofobservations

FAC2 >0.5 1 1 good

Table 3.2. Statistic metrics of normalized wavelength (λ/h) and amplitude (A/h)for water-tank studies using CFD and experimental data of Gyüre and Jánosi [62].Definition and the applied limits for the statistic metrics are described by Changet al. [28]

3.5 Model comparison with a full scale event

3.5.1 The Boulder windstorm case study

A relatively well documented and studied event occurred during the winter of1972 near Boulder, Colorado, where a severe wind storm, with a strong descentof air originating from the higher atmosphere, caused significant damage to theenvironment [109, 21]. The strong tropospheric descent is well reflected in thepotential temperature contours in Fig. 3.5a, where the contours become denserclose to the ground. The accompanying near ground downwind was also reportedas being especially severe. A hydraulic jump and waves also developed above anddownstream of the mountain (Fig. 3.5a).

37


Figure 3.5. Analysis of the potential temperature (a) and the horizontal velocitycomponent (b) from aircraft flight data and sondes taken on January 11, 1972. Air-craft tracks are shown by dashed lines with the locations of significant turbulenceindicated by plus signs [88]

The purpose of the present simulation is to show that the model is able to capturethe main features and nonlinearities of such flows, for example the existence of thementioned nonlinearities. This event became a standard test case of researchers(see ,e.g., [197]) for developing simulation models, or to reproduce and understandthe related mechanisms of such phenomena. A recent study dealt with the de-velopment of a severe thunderstorm in Budapest. With high resolution numericalmodeling using MM5, they were able to reproduce the main features of the severeconvective storm [78].

38


Different assumptions are made during similar case studies, in order to reducecomputational cost, taking the advantage of the quasi two-dimensionality of theflow, or simply investigating each flow phenomena separately. Although mountainflows are generally three-dimensional, two-dimensionality can be a good assump-tion for the downslope windstorm case, as the continental divide is long comparedto its cross section. This is acceptable only if the large scale flow is being inves-tigated and the microstructure has only negligible influence on the main flow. Insome cases these structures have an effect, e.g., by modifying the horizontal posi-tion of the hydraulic jump. This is due to the increased surface friction resultingfrom a non-slip boundary. In spite these differences in the treatment of the lowerboundary, the main features of the flow will still remain.


The elevation of the realistic terrain of the E–W oriented section was derived froma 3 arc second resolution SRTM (Shuttle Radar Topographic Mission) database.The section was then interpolated for the latitude of 40.015 N. The longitudinalcoordinates were between 107.100 W and 103.900 W, giving an approximately 270km wide section. The new coordinates were transformed onto the Universal Trans-verse Mercator (UTM) coordinates and were interpolated onto an approximately1.5 arc second grid using a bicubic spline method.Test cases using an idealized geometry have also been examined, where a simplifiedtwo-dimensional model has been used. In spite of the fact, that the real mountainhas a plateau-like shape, due to the upstream influence and partial blocking, theupstream mountain profile does not affect significantly the upstream flow. Due tothis fact, a symmetric mountain profile was used in several works [88], as well asin this study. The profile described by Eq. 3.1 was used for modeling both theupstream and downstream sections of the simplified geometry. Here a = 10 kmand h = 2 km are the mountain half width and height, respectively. The initialtemperature and velocity profiles were based on the intercomparison study of Doyleet al. [41]. They found that these initial conditions were more appropriate for wavebreaking tests, and more realistic than the conditions used in earlier studies [137].Conditions favorable for downslope windstorms are usually characterized by strongcross mountain winds, and by the presence of a stable layer at the appropriateheight [43]. Both of these conditions were present in the studied situation.The solution was found to be sensitive to the grid resolution [41], therefore, inthis study a relatively higher resolution grid was used. An equidistant grid wasapplied with a size of 1000 m and adapted to 250 m in two steps in the vicinityof the mountain. In the vertical direction the mesh size decreased down to 10

39


m resolution close to the ground. The domain extended from 2 to 25 km inthe vertical and – 115 to 120 km in the horizontal direction, with the mountaincrest positioned at x = 0 km. The top boundary was defined as symmetry withzero normal velocity, and the bottom was defined as a free-slip wall. A standardnon-reflective boundary condition (outflow) was applied as an outlet. This meansthat the internal pressure field has been extrapolated to the outlet surface, andtherefore, no further information on the outlet velocity and pressure profiles wererequired by the system.Second order time discretization, pressure staggering option (PRESTO) for pres-sure interpolation [6], and second order upwinding methods were used in the mo-mentum and energy equations. Compressibility was introduced with the help ofthe transformation used for model adaption. Moist dynamics and the Coriolisforce were not considered during the calculations.

3.5.3 Results

By using the ANSYS–FLUENT 13 simulation system, the model was integratedfor a non-dimensional period of time of t∗ = Ut/a = 43.2, based on an average inletflow speed of 30 m s–1, which corresponds to a 4-hour flow time. The results shownin this section are obtained at t∗= 32.4 time instant. Results were quantitativelycompared to several mesoscale meteorological codes using statistical performancemeasures described in Section 3.2. The summary of the comparison can be seen inTab. 3.3. Based on FB, NMSE, and FAC2, the idealized model can be consideredacceptable. Negative FB values indicate lower predicted overall horizontal velocity.The correlation coefficient and hit rate, however, show values under the limit. Thehit rate was calculated with 5 m s–1 absolute and 10% relative deviation for thevelocity and 1 K and 5% for the temperature, respectively, since large deviationswith a factor of 2 differences were realized among the different intercomparisioncases presented by Doyle et al. [41]. Chang and Hanna [27] suggest that the modelperformance should not be judged based only on the performance measures buttogether with the comparison of flow patterns or the time evolution of the flowfield.

40


Validation metric Abbreviation Limit U Θ ClassificationCorrelation coefficient R >0.8 0.154 0.96 not

sufficientFractional bias FB ±0.3 –0.07 –0.056 goodNormalized mean squareerror

NMSE 0–4 0.396 0.01 good

Hit rate HR >0.66 0.29 0.56 notsufficient

Fraction of predictionswithin a factor of two ofobservations

FAC2 >0.5 0.7 1 good

Table 3.3. Statistic metrics calculated for horizontal velocity (U) and potentialtemperature (θ) for the Boulder windstorm case study comparing CFD and NWP[41] model results

According to the experiments of Doyle et al. [41], the results were highly timedependent, and therefore, unsteady simulations were executed and time-averagingwas applied on the results. An averaging interval varying from 10 to 60 minuteswas applied during the simulation, and it was found that the value and locationof the maximum velocity were not affected. The horizontal position of the up-stream edge of the hydraulic jump however changed significantly. The lee-slopewind magnitude varied considerably among the model simulations presented byDoyle et al. [41], and these differences were also reflected in the horizontal posi-tion of the hydraulic jump. The location of the upstream edge in some models,e.g., the Durran and Klemp [44] model (DK83) or the Eulerian/semi-Lagrangianmodel (EULAG) was positioned immediately after the lee slope, similarly to ourresults, while in other cases it was positioned 10 – 25 km downstream of that(e.g., MESONH, RAMS, or RIMS models of the same intercomparison). This par-tially explains the lower HR and R values due to the horizontal shift in the flowpattern. Regarding the downslope wind, a higher value of maximum horizontalvelocity was realized by the simulation than that of the onsite measurements. Att∗= 32.4, the location of the maximum velocity peak was found at approximately8.5 km downstream of the mountain crest at a height of 7 km (Fig. 3.6). Withfurther integration, the locations of the maxima were shifted downstream to ap-proximately 16 km beyond the crest and to a lower height of 4 km (bottom ofFig. 3.6). The magnitudes were not changed considerably, stabilizing at around 66m s–1. Regarding the wind speed close to the ground, within the lowest 50 m, 51

41


– 59 m s–1 was realized, depending on the position along the mountain lee side.The near ground values of velocity magnitude were in fair agreement both withthe observations and the meteorological models. The high instantaneous velocitypeaks at higher altitudes obtained from the simulation could be partly caused bythe lack of moisture transport and phase change processes. According to DK83,the wave response could be even 50% lower if a proper treatment for moist airflow is applied. This could significantly affect the properties of wave breaking, andconsequently the development of a low level jet.Several mesoscale codes simulated multiple breaking regions with the strongestones located above the hydraulic jump at an altitude of 12 km and above (seethe model results of RAMS, MESONH models presented by Doyle et al. [41]).Wave steepening regions were obtained by CFD for the ideal and real mountainprofiles at a similar altitude (between 10 and 15 km). Using the idealized geometry(Fig. 3.6), this region was weaker. The altitude of the maximum steepness abovethe hydraulic jump in the breaking region was at about 12 km with the highestamplitude waves being of 2 – 2.5 km.The present results show smooth downstream isolines. In some of the cases ofDoyle et al. [41], the downstream part of the hydraulic jump was oscillatory, whilein other cases the isolines were smooth. Among the results of different mesoscalemeteorological solvers, depending on the amount of applied eddy diffusivity (Kd),different flow structures were obtained. Smaller Kd values usually resulted insmaller wave structures.

42


Figure 3.6. Contours of horizontal velocity component (left panel) and potentialtemperature (right panel) for an idealized (top) and a real (bottom) mountainshape. The pictures show the results at t∗= 32.4 non-dimensional flow time. Ver-tical coordinates were magnified by a factor of 12 for better visualization. Poten-tial temperature and velocity contour intervals are 8 °C and 5 m s–1, respectively.Light and dark grey areas indicate regions characterized by turbulent kinetic en-ergy higher than 5 m2 s–2 and between 5–25 m2 s–2.

The stratospheric air descent reached the altitude of 5 – 6 km using the real topog-raphy which correlates well with the non-hydrostatic mesoscale vorticity (TVM)model solution of Thunis and Clappier [177], or the model comparisons presented

43


by Doyle et al. [41] (CUMM, RAMS, RIMS). The descent was even stronger andpropagated to lower altitudes in the case of the simplified geometry accompaniedby high TKE values (see Fig. 3.6) . The highest peaks, even reaching 117 m2 s–2,can be found around the upstream edge of the hydraulic jump and at the wavesteepening regions at higher altitudes. The maximum value of TKE is reportedapproximately 15 – 20 m2 s–2 in atmospheric rotors (Doyle et al., 2002) and evenreaches higher values in the case of extremely severe turbulence.Differences in the magnitude of the horizontal velocity may also be related to thetwo-dimensional treatment of the topography if compared to the real situation.In three spatial dimensions, the flow can pass around the obstacle, while in 2-D it is forced over it. The test results of Doyle et al. [41] confirmed that asignificant reduction in wave breaking could be found for the 3-D case. Some ofthe studies dealing with gravity wave evolution used a frictionless lower boundary[154]. Concerning the bottom boundary condition, a more realistic treatment ofthe ground surface could result in a slower movement and different horizontallocation of the hydraulic jump.


In this chapter, simulations were presented around more complex geometrical fea-tures, idealized barriers, and real terrain, demonstrating the capabilities of theCFD based approach.Simulations of linear hydrostatic waves were compared to an analytical solution,and it was stated that for this regime the code behaved well, and an excellentagreement was found.Secondly, the simulation results were compared to water-tank experiments of two-dimensional mountain waves with different degrees of nonlinearity and hydrostac-ity. A good agreement was found based on the statistical performance measuresand flow pattern comparison concerning the measured and simulated wavelengthsand amplitudes. The model evaluation for lambda/h gave a correlation coeffi-cient of 0.95, fractional bias 0.317, normalized mean square error of 0.12, a hitratevalue of 0.75 and 100% of predictions within a factor of two of observations. Theequivalent results for A/h were 0.99 (R), 0.21 (FB), 0.05 (NMSE), 1 (HR), and100% (FAC2). The model was also able to capture well the location and size ofthe appearing nonlinear structures, such as the rotor that was formed behind theobstacle.Simulation of the Boulder windstorm case is ideal for testing and evaluatingmesoscale numerical models, therefore it was chosen as the third object of analysis.

44


The simulation results were compared to the on-site observations and a series ofmodeling experiments that were presented by Doyle et al. [41]. Model evaluationhas demonstrated reasonable agreement with measurements for potential temper-ature (Θ). The model evaluation statistics gave a correlation coefficient of 0.96and fractional bias –0.056, normalized mean square error of 0.01, a hit rate valueof 0.56, and 100% of predictions within a factor of two of observations. The equiv-alent results for horizontal velocity (U) were 0.154 (R), –0.07 (FB), 0.396 (NMSE),0.29 (HR) and 0.7 (FAC2). Results obtained from our simulations are encouragingwith regard to the predictability of a low level, highly accelerated channel flow,and upper level wave breaking. Close to the ground a very strong lee-side windwas realized, accompanied by a well-defined hydraulic jump downstream.Using only one single unstructured grid and a uniform physical description forclose- and far-field flow, one can take the advantage of the model adaption inthe simulation of mesoscale atmospheric phenomena. In the same model, one caninvestigate the finely structured microscale flow around complex geometrical fea-tures, such as flow around buildings with pollution dispersion or to study the close-and far-field of micro- and mesoscale phenomena and its effects to the environment.

45

4 Modeling phase change andmoisture transport with a bulkmicrophysical model using ageneral purpose pressure basedCFD solver

4.1 Overview

Wet cooling tower plume formation has been simulated by using a well-knowngeneral purpose computational fluid dynamic (CFD) simulation system adaptedto atmospheric flow modeling. Mesoscale effects have been modeled with a novelapproach based on a system of transformations and customized volume sourcesacting in the conservation and governing equations. Moist dynamics are takeninto account with a bulk microphysical model that was recently implemented intothe solver.The microphysical scheme has been validated against numerical experiments ofidealized two-dimensional dry and wet thermals. Validation with full-scale at-mospheric flows is a very important step toward the practical application of themethod. Therefore model performance has also been compared to onsite mea-surements for the formation of a large wet cooling tower plume. Results obtainedfrom our simulations are encouraging with regard to the predictability of cumuluslike plume structures with complex thermal stratification, the overall liquid watercontent along the plume axis and also the turbulent fluctuations caused by thevertical movements in the plume.

4.2 Literature survey

The transformation in its current state essentially handles dry adiabatic processes,however several applications require the inclusion of a proper moisture model.

46

4.2. LITERATURE SURVEY

Therefore in the present paper the extension of the current transformation methodwith a humidity transport and phase change model will be shown accompaniedwith the validation against the rise of idealized two-dimensional dry and wet ther-mals and a full scale three dimensional wet cooling tower plume formation withdifferent environmental stratifications.Several researchers investigated the behavior of cooling tower plumes from differentaspects in the past decades. Wet cooling towers are widely used in the powergeneration industry since it is relative easy to build and cheap to operate especiallyin regions where limited water resources are available for cooling purposes. [3]Most of the earlier studies are from the early 1970 related to the design, construc-tion and operation of nuclear and coal fired power plants. The performance ofcooling towers is an important topic as the energy demand is growing. Few per-cent increase in overall efficiency in power generation could lead to high amountof total energy savings. Therefore number of researches investigated the effectof changes in environmental conditions to the tower performance. Al-Waked andBehnia [3] and Lohasz and Csaba [111] studied the effects of crosswind, otherresearchers [91, 134, 11] studied temperature and humidity inversion with CFDmethods under different operating conditions. These effects are important sincethey could lower the efficiency of cooling towers [187]The visibility was not a great concern at that time, instead the plume rise of dryand wet plumes were investigated [66, 189]. Wet cooling towers however do nothave much control over the visible plume [181] The exhaust of the tower is generallysaturated and during certain weather conditions cannot be absorbed completelyby the surrounding air, as a result, it will appear as fog and visible to human eye.Another important aspect regarding wet cooling tower operation is the preventionof the growth of legionella bacteria in the cooling water. The various legionellaspecies are the cause of Legionnaires’ disease in humans and the transmission isvia the exposure to aerosols. The bacteria could live and travel hundreds of metersor even kilometers from the source. [59]Nowadays environmental impacts are becoming more and more important not onlyconcentrating on toxic materials but also on the visibility of water vapor plumes.The reduction of visibility conditions, the local reduction of solar radiation, andthe interaction with low level clouds, in particular the fog, are in concern nowa-days. The latest literature review about cooling towers shows different options forreducing and manipulating the visible plume using hybrid cooling towers, wet–drycooling towers, dry cooling towers and so on, depending on the need and demand.The formation of a visible water vapor plume is directly related to water massfraction, temperature, exhaust velocity of the tower and also to ambient mete-orological conditions. In more recent studies Wang et al. [185, 184] and Xu et

47

4.2. LITERATURE SURVEY

al. [196] investigated the control and abatement of plumes emitted by large com-mercial buildings by reheating the exhaust with heat pumps or solar collectors.Sturman and Zawar-Reza [174] predicted the yearly visibility of a stack plume ofa planned industrial site with an atmospheric mesoscale pollution model (TAPM)by providing boundary conditions from the meteorological code. Presotto et al.[145] investigated the possibility of reducing the plume visibility by lowering theexit temperature of a petrochemical refinery.There is usually no contaminant involved, but there is a risk of the plume’s materialreturning to the ground level causing local fogging, ice formation or entrainmentof saturated air into other adjacent towers. Mokhtarzadeh-Dehghan et al. [125]and Spillane and Elsum [171] investigated the occurrence of rain and fog and thepossibility of plume blow-out by strong winds. Their investigation showed thatthe fog can extend to the ground in cases where the plume interacts with the wakeof the tower and the ambient temperature is very low.Number of researchers worked on the effect of drift deposition as it could be ob-jectionable due to human health hazards. Their purpose was to investigate theeffects of ambient conditions and absolute humidity, droplet output temperature,and the affected area [114]. Drift of small water droplets from mechanical andnatural draft cooling towers can contain salt particles, water treatment chemicalsand bacteria. Meroney [123] recommends a CFD protocol to correct drift anddeposition predictions provided by current analytic models to take into accountbuilding effects.Wet cooling tower plumes can also play a role in the formation of different kinds ofhydro-meteors. Campistron [23] and Huff [79] studied snowfalls caused by coolingtowers and they found that the rate of precipitation can even be enhanced by afactor of two.The increasing computational power makes the CFD based approach more andmore affordable. Several investigations showed that CFD models are valid in pre-dicting the flow field, performance or drift deposition predictions of microscale flowaround cooling towers. [8, 123, 10] We have successfully extended the capabilitiesof CFD solvers [99, 150] in order to simulate atmospheric scale flows. In the fol-lowing sections a further enhanced model version will be shown that is capable ofpredicting moist dynamics through the implementation of condensation and phasechange models.In the next chapter an overview will be given about the existing tools for modelingplume dispersion and about their advantages and limitations, a full descriptionof the mesoscale model extension that also takes into account moist dynamics.In further subsections the model validation will be shown against idealized two-dimensional calculations and atmospheric scale flows done by meteorological codes

48

4.3. OVERVIEW OF EXISTING TOOLS

and also with field measurements.

4.3 Overview of existing tools

Theories for describing the heat, mass and momentum transfer inside naturaldraft cooling towers have long been established by authors in the early 19th-s.These works [108, 156] include also heat transfer due to vaporization and thereforeapplicable to the prediction of wet tower performance. These analysis and simpli-fications are still in use today and several numerical models have been developedbased on this study in order to describe transfer processes inside cooling towers.[3]Wide range of model complexity can be found in the literature regarding themodeling of plume formation outside of the tower. These models include simpleralgebraic models towards more complex integral models, atmospheric dispersionmodels or CFD based approaches. Commonly used plume models are based onconservation equations describing the entrainment processes of ambient air alongthe plume axis.Several authors [67, 64] studied the dynamics of plume motion and developednumerical models. They [167, 191, 34] identified three phases, the initial, interme-diate and final phases of plume rise. They developed a rise model for jets that isalso applicable to ambient conditions with stable stratification.Widely used models e.g. the analytical models of Briggs [19, 20] and Weil [189]describe reasonably well the first phase of plume rise near the source, however it isvalid only for constant density gradient and wind speed [20]. With the detrainmentconcept of Netterville [132] the transitional region and the leveling can also bedescribed. Davidson [36, 35] developed a formulation that is able to predict bothplume rise and dilution.The entrainment is often modeled by using different empirical coefficients [158].One reason of difference between model results of different formulations is due tothe differences in empirical coefficients applied to obtain the analytic solution.Early models did not account for phase change of water during the plume develop-ment and neglect the effect of turbulence to the entrainment. Gangoiti et al. [52]and Janicke and Janicke [81] have developed Gaussian dispersion type models thatincluded prediction of condensation and is also applicable to complex wind fields.Condensation and evaporation of droplets is important since it changes buoyancyby introducing and removing latent heat during the rise and leveling. The lack ofplume condensation causes the underestimation of plume rise in simpler models.

49

4.3. OVERVIEW OF EXISTING TOOLS

Another usual limitation is the improper handling of the complex atmosphericstratification and wind fields. Simplifications often assume winds depending onlyon the height above the ground, or constant wind speed and direction along theplume axis. In the case of complex input meteorological conditions data are oftenextrapolated from nearby meteorological stations to represent ambient conditionsat the source. However this causes accuracy problems [145]. Simpler models couldgive good results in certain conditions when the lower few 100 meters of the at-mosphere is linearly stratified. However in the cases of high exhaust temperaturesthe plume can penetrate deeply into the atmosphere crossing different layers withdifferent lapse rates. This is typical for stack plumes of wet scrubbers with highexhaust temperatures.The proper estimation of plume rise height is especially important when the deposi-tion of plume particles or ground level concentration is to be calculated. Policastroet al. [142] compared drift deposition models to experimental data and found thatthe existing models did not perform well.The formulation of Briggs [20] extended with empirical coefficients for plume tra-jectory are in use in environmental protection regulatory models. [52] For shortrange transport, the modifications to the EPA Point, Area Line Source Algo-rithm (PAL2.1), the EPA Industrial Source 5 Complex Short Distance 3 (ISCST3)model, and the Argonne National Laboratory Seasonal/Annual Cooling Tower Im-pact (SACTI) model ([24, 143]) are used. Some of the limitations described abovewere addressed later in the development of ISC-PRIME and AERMOD-PRIMEmodels in order to replace ISC3 series models ([159, 138]). The current trend inmodeling for urban air pollution is focused on the improvement of advection inatmospheric dispersion models (atmospheric dispersion models TAPM, Ausplumeand CALPUFF [22]) and integrating them with local scale models (see e.g. theEUROTRAC-2 subproject SATURN; [128]).These models are currently acceptedby regulatory authorities.In the last decade new generation of Gaussian dispersion models were introduced inEast European countries, with a better description of real physical processes in theatmospheric boundary layer. Examples are the Danish OML model [133], or theBritish UK-ADMS model [25]. These often contain integrated systems for differentpurposes, street canyon model, Gaussian plume model, Eulerian grid model anda dispersion model. These model families could give reasonable results in the farfield however it could give under or overestimation by a factor of two when complextopography and buildings are considered and near filed concentration is a concern.[119, 133])CFD based tools have also been used recently to assess plume visibility [22]. Cur-rent CFD models however have limited capabilities. They were mostly used to

50

4.4. DESCRIPTION OF THE BULK MICROPHYSICAL MODEL

investigate the wind field under steady state conditions, not representing the spa-tial and temporal variability of the meteorological fields especially around complexterrain (as recognized by [22]). They often include assumptions for the verticalatmospheric profiles that do not reflect real conditions (as cautioned by Presottoet al. [145]), and frequently exclude effects of surface vegetation and soil. Theseissues have been addressed by introducing the transformation method describedin [99, 150] and [10].Several authors treated moisture by means of various warm cloud microphysicalmodels of different complexity. These approaches range from simple single-moment[86], two-moment bulk parameterizations ([200, 30, 127]) to more complex binmicrophysics schemes. [49, 93, 1] Typically in bulk microphysics, the liquid wateris separated into two categories: non-precipitable cloud water and precipitable rainwater [86].

4.4 Description of the bulk microphysical model

4.4.1 Transport equaitons

The extended commercial CFD solver we used is essentially closed source but it isallowed to setup and use arbitrary number of passive scalar equations during thesolution. Therefore in order to describe the phase change, three additional scalarequations (Eqs. 4.1 – 4.3) were considered for the number concentration of cloudcondensation nuclei (in order to track depletion and entrainment effects) and thenumber of condensed water droplets Nc together with total water content of airqt. Rain water is currently neglected since condensation will determine the growthof drops smaller than about 20 µm and no larger droplets are expected during theinitial rise of a thermal or during the cooling tower plume rise.

∂ρnCCN∂t

+ ∂

∂xi

(ρuinCCN − Γ∂nCCN

∂xi

)= (SnCCN )act + (SnCCN )evap + (SnCCN )sed,

(4.1)

∂ρqt∂t

+ ∂

∂xi

(ρuiqt − Γ∂qt

∂xi

)= (Sqt)act + (Sqt)evap + (Sqt)sed, (4.2)

∂ρNc

∂t+ ∂

∂xi

(ρuiNc − Γ∂Nc

∂xi

)= (SNc)act + (SNc)evap + (SNc)sed, (4.3)

51


qt = qv + ql, (4.4)

where qv and ql are the specific humidity of water vapor and liquid water per unitmass. Γ is the diffusion coefficient of the given scalar.In order to simulate mean sizes and concentrations of droplets in wet plumes,the scheme needs to represent processes that would affect the droplet spectraof the plume. (R.H.S. of Eqs. 4.1 – 4.3) These are the activation of droplets oncondensation nuclei, condensation/evaporation of droplets, entrainment of ambientair [186, 136, 18, 175, 103] and additional activation of droplets [186, 139].The relationship between droplet number (Nc) and liquid water content (ql ) isdetermined by using a log-normal droplet size distribution function (Eqs. 4.5 –4.6):

nC(r) = Nc

rσc√

2πexp

(− 1σ2c

(lnr

r0

)2), (4.5)

r3 = r0exp(3

2σ2c

)and ρql = 4

3πr33ρwNc, (4.6)

where r is the droplet radius, r0 is the distribution median value, σc the logarithmicstandard deviation, r3is the mean mass radius and ρw is the density of water.Atmospheric aerosols and cloud condensation nuclei play an important role inthe condensation of droplets and evolution of clouds. In this study, the methoddescribed by Cohard et al. [31] was chosen to describe the relationship between su-persaturation and the nucleated number of droplets (NCCN ) as it is computation-ally more efficient, gives more robust estimates [56] and requires less programmingefforts.

NCCN = CskF

(µ; k2 ; k2 + 1;−βs2

), (4.7)

where C = 3270 cm-3 is a parameter of the nucleation process, s is the supersatu-ration, F is the hyper-geometric function and µ = 0.7, β = 136 and k = 1.56 areparameters characterizing the aerosol distribution of a continental type of air mass.Eq. 4.7 is the extension of the simple and famous power law formula of Twomey[180] but improved to give better results from weak to strong supersaturations.

52


Given the number of nucleated drops the source term due to nucleation can becalculated assuming heterogeneous nucleation. [146]

(SNc)act = 1∆tmax (NCCN,max −Nc, 0), (4.8)

where ∆t is the numerical time step and here NCCN,max is the maximum concen-tration of condensation nuclei that can be activated and Ncis the number concen-tration of condensed droplets already available in the given cell. It is assumedthat condensation and evaporation proceeds through changes of ql without any ef-fects on Nc [153] except during activation of CCN and total evaporation of smallerdroplets.

4.4.2 Droplet growth by activation and diffusion

Once droplets are activated the two primary components of their growth are thevapor diffusion and collision-coalescence. Since rain water is neglected only growthby diffusion is considered. The critical radius of newly formed droplets can becalculated according to the Köhler theory. (see e.g. [146])

(rcrit)act = 43MwσwRρwT (s/100), (4.9)

whereMw and σw are the molecule mass and surface tension of the water substance,R and T are the universal gas constant and physical temperature of ambient air.The corresponding source term due to activation for the liquid water content is:

(Sql)act = 4πρw3ρ r3

crit

(∂Nc

∂t

)act

, (4.10)

The condensation rate can be calculated based on [146]:

(Sql)cond/evap = 4πr(s− A+B)fvG(T, p) (Pruppacheretal., 1998), (4.11)

53


where A,B and fv express curvature, solute and ventilation effects and G(T, p) isa term dependent on local thermodynamic variables.

A = 2σwRvTρwr

, B = ενφsMwma

Ms(m− ρwma/ρs), G(T, p) = Lv

kaT

(LvRvT

− 1)

+ RvT

psatDv

,

(4.12)

here ε is the water-soluble fraction of the aerosol, the product νΦs is the van’tHoff factor, ρs and ma are the aerosol density and mass, m is the droplet mass,Mv and Ms are the molecular mass of water and solute, Lv is the latent heat ofevaporation, Rv is the specific gas constant of water vapor, ka and Dv are themodified thermal conductivity and diffusivity of air and water vapor, psat is thesaturated vapor pressure over flat water surface at environmental temperature.

4.4.3 Partial and total evaporation of droplets

Droplets evaporate once convected to unsaturated regions (s < 0). This process isdescribed based on the work of Chaumerliac et al. [29]. Due to partial evaporation,drops with a size smaller then a certain threshold (rcrit) will be removed from thesystem, therefore resulting a decrease in Nc. The corresponding source term is:

(SNc)evap = 1∆t

rcritˆ

0

Nc

rσc√

2πexp

(− 1

2σ2c

(lnr

r0

)2)dr, (4.13)

where rcrit can be calculated as follows:

rcrit =√(−2A−1

3 s∆t), (4.14)

A3 = ρwRvT

psatDv

+ LvρwkaT

(LvRvT

− 1), (4.15)

Total evaporation of droplets will occur when the calculated mass of evaporateddrops is larger than the mass present in the given volume. In this case Nc = 0and the local CCN number is regenerated. The sedimentation terms on the RHSof Eqs. 4.1 – 4.3 can be used to describe the settling of droplets, however the

54

4.5. VALIDATION WITH A RISING THERMAL IN A DRY STABLEATMOSPHERE

settling terms are not considered here as it is expected to be small in certainplume dispersion cases. [17]The superstauration s is calculated based on the saturation pressure (Eq. 4.16).It is calculated for flat water surfaces based on the formula of Bolton [16]. Theexpression is accurate to 0.3 % for -35 °C < T < 35 °C temperature range.

psat = 6.112 exp( 17.67T

243.5 + T

), (4.16)

where T and psatare in °C and hPa.The following expressions give the connection between water vapor mass fractionand vapor pressure:

pv =(

qv0.622 + qv

)p, (4.17)

where pv and p are the vapor partial pressure and the pressure of moist air. Finally,saturation is defined as:

s = pvpsat− 1. (4.18)

In the following chapter the transformation and the microphysics scheme will bevalidated against the simulation of an idealized two dimensional rising thermaland three dimensional full scale plume dispersion cases.

4.5 Validation with a rising thermal in a dry stableatmosphere

It is common in the development of microphysical schemes that researchers testtheir models’ behavior with idealized two dimensional simulations. A frequentlyused generic test case is a rising thermal simulation in a stably stratified environ-ment. Two types are common in the literature, one where the development of thethermal is initiated by surface heating (see e.g. [87]) and another type when aninitial perturbation (potential temperature, vapor content etc.) is placed in the

55

4.5. VALIDATION WITH A RISING THERMAL IN A DRY STABLEATMOSPHERE

domain at a certain height. The latter one described by Grabowsky et al. [57] willbe used here as a test case for validation as it allows shorter computation due tonot simulating the initial development phase of the thermal.In order to separate the moist dynamics from discretization effects, the rise of adry thermal in a Bousinesq fluid was simulated first.


The model domain was 3.6 km wide in the horizontal and 2.4 km in the verticaldirection with equidistant mesh resolution. Different grid sizes of 20, 10, 5, 2.5and 1.25 m were tested in order to see mesh sensitivity of the solution. The lateralboundaries were defined as periodic, the top and bottom boundaries were definedas free slip adiabatic walls. The initial dish shaped perturbation was placed atx = 0 km and z = 0.8 km with a radius of 500 m and 0.5 K higher initial tem-perature than the constant 287 K ambient. Third order MUSCL schemes (Mono-tone Upstream-Centered Schemes for Conservation Laws, ([106]) were used whensolving the momentum and energy equations as they can effectively suppress thenon-physical oscillations with the introduction of adaptive numerical dissipationinto numerical solutions. All transformation source terms in Eqs. 2.6 – 2.9 wereturned off in this case. The total flow time was 8 min with a time step of 1 s. Re-sults were compared to standard nonoscillatory MPDATA simulations (presentedby Margolin et al., in section 4)

4.5.2 Results

At 20 m and lower resolutions results were starting to degrade rapidly, the inter-facial eddies were smeared out (not shown here). A converged solution could beobtained at higher resolutions, the 5, 2.5 and 1.25 m cases were virtually identicalFig. 4.1. The viscosity was explicitly defined and was varied among the differ-ent cases. Fig. 4.2 shows that the results are less sensitive to the changes of thepredefined viscosity in the investigated range.Overall a good correspondence can be found regarding the gross features, the riseheight and size of the final shape of the bubble agrees well. There are differencesin the fine details though, the size and position of interfacial eddies slightly differfrom the MPDATA solutions.

56

4.6. VALIDATION WITH THE RISE OF A MOIST THERMAL

Figure 4.1. Mesh sensitivity of the developed interfacial eddies. From left to right20 m, 5 m, 1.25 m equidistant mesh resolution, viscosity was 0.5 m2 s-1 for allcases.

Figure 4.2. Sensitivity of the developed interfacial eddies to the viscosity. Fromleft to right the viscosity was 1, 0.5, 0.1 m2 s-1 and the mesh resolution 2.5 m forall cases.

4.6 Validation with the rise of a moist thermal


Model behavior changed significantly when moist dynamics were activated. Thedomain in this case was the same as above with an equidistant resolution in bothspatial directions of 40, 20, 10, 5, 2.5 m. The initial perturbation of relativehumidity (RH = 100 %) was placed at 0.8 km height with a diameter of 200 mthat was smoothly relaxed to the environmental value within approximately 250m radius. The environmental base state stability and the relative humidity wereset to dlnΘ/dz = 1.3 e-5 m-1 and 20 % and the perturbation relaxation of RH wasdefined as:

57


RH = 20% + 80%cos2(π

2r − 200

100

), 200m < r < 300m. (4.19)

The lateral, the top and bottom boundaries were defined as periodic and freeslip adiabatic walls with a temperature of 289 K at the lower surface. The eddyviscosity of the air was explicitly defined and varied between 0.25 and 2 m2 s-1among the simulation cases. Third order MUSCL schemes, Pressure StaggeringOption (PRESTO) [6] were used when solving the momentum, energy equationsand for pressure interpolation. Results were compared to series of non-oscillatoryMPDATA solutions presented in section 3 in [57]

4.6.2 results

The general formation of the liquid water field of the thermal is illustrated inFig. 4.3 During the initial 4 min of rise a very good match can be found. Thisperiod was weakly affected by the mesh resolution or the predefined viscosity. (notshown here) At higher resolutions (5 m and 2.5 m), the model predictions werefairly close to the semi-Lagrangian and Eulerian MPDATA solutions, the overallshape and rise height are similar, interfacial eddies were well resolved. Differencescan be found in the fine details though, the exact position and shape of interfacialeddies are somewhat different but still close to the semi-Lagrangian model resultsthat is superior in capturing the interface instability (see section 4a of Grabowskyand Smolarkiewicz [58]). At low spatial resolutions the CFD model still capturedthe overall shape, but the quality degraded quickly, no interfacial eddies could befound at 20 m resolution. However the erosion of cloud water field was not asstrong as the Eulerian MPDATA model showed at the same resolution.Overall the model output on Fig. 4.4 and Fig. 4.5 suggest that the results areconverging to a common solution and it is in a good agreement with the MPDATAresults. Differences are possibly due to the different numerical schemes and thelack of precipitation scheme in the CFD model.

58


Figure 4.3. Development of the initial water vapor perturbation. Isolines of ql fieldat t = 2, 4, 6 and 8 min of the moist thermal rise. The viscosity and the meshresolution are 0.5 m2 s-1 and 2.5 m in each panel. The contour intervals for ql are0.05, 0.1, 0.2, and 0.3 g kg-1

59

4.7. VALIDATION WITH THE BUGEY 1980 FIELD CAMPAIGN

Figure 4.4. same as Fig. 4.3 but for different mesh resolutions after 8 min of thermalrise. From left to right the mesh sizes are 20, 10, 5 and 1.25 m. The viscosity wasfixed to 0.5 m2 s-1.

Figure 4.5. same as Fig. 4.4 but for different viscosities. Mesh resolution was 2.5m, viscosity was 0.1 and 0.25 m2 s-1 (from left to right).

4.7 Validation with the Bugey 1980 field campaign

In this chapter the CFD simulation results will be compared to full scale mea-surement data that was collected during a large measurement survey around wet

60


cooling towers of a nuclear power plant around Bugey, France. Radio-soundings,droplet spectra, airborne measurements, measurements describing the ambientthermodynamic states as well as photographic records are available for compari-son. [17]


During the campaign very different plume shapes were observed from which two ofthe characteristic cases were selected for comparison. The first one was character-ized by high wind shear, conditionally unstable stratification and an upper layerwith high relative humidity. The second one had lower ambient wind speed, morestable stratification and as a result, smaller horizontal plume extent characterizedby sharp plume bent-over.The domain in both cases covers a 10 km x 4 km area with a total height of 4 km.An equidistant grid was used in the simulations with a resolution of 40 m thatwas adaptively refined in two steps around the tower and the plume, resulting aminimum grid size of 10 m in critical areas.

4.7.2 Boundary conditions

The interpolated initial and boundary profiles for the domain as well as the plumeexit conditions, the vertical velocity, temperature, turbulence profiles and liquidwater content were based on the radio-soundings and previous calculations of [17](see Fig. 4.6 for the atmospheric profiles and Tab. 4.1 for tower exit conditions)Turbulent kinetic energy k and its dissipation ε at the tower exit were set to 1.7m2 s-2 and 0.07 m2 s-3 in both cases. Each tower had an aproximate heigth of 128m with a diamater of 60 m. Droplet number concentration at the tower exit wasdeduced from measurements, i.e. Nc = 104 cm-3. The nucleation parameterizationwas after Cohard et al. [31] using parameters for a continental type of air massi.e. C = 3270 cm−3, k = 1.56, µ = 0.70, β = 136 and σc = 0.28. The time stepwas 1 s and the total integration time was 3600 s in both cases.

61


Tower exitconditions

ql[g kg-1] ∆T [°C] w[m s-1] Tenv[°C] penv[Pa]

4E-W 5E-W 4E-W 5E-W 4E-W 5E-WMarc 11 0.889 0.719 18.34 17.77 3.8 3.73 4.44 97771Marc 12 0.8 0.8 18.2 17.7 3.8 3.7 3.31 97994

Table 4.1. Plume exit conditions for 11 and 12 March 1980. The exit conditionsfor the towers no. 4E-W and 5E-W are given separately. The subscript “env”refers to environmental values. The tower exit temperature ∆T is given comparedto the environmental values. w and ql are the vertical velocity component and theliquid water content respectively at the tower exit.

62


Figure 4.6. Ambient initial and boundary conditions for the 11 March 1980 (top)and 12 March 1980 (bottom) cases. Left panel: ambient temperature. Right panel:wind speed (solid line) and wind direction (dashed line)

4.7.3 Results

Moist air is injected at the tower exit into the atmosphere characterized by strongshear with a high relative humidity upper layer, resulting in a periodic cumuluslike plume formation farther downstream of the tower. This phenomena is wellcaptured by the CFD code compared to visual observations and existing calcula-tions (see chapter 4 and appendix A. of [17]). The oscillation in the LWC fieldobserved as rising thermals on Fig. 4.7 shows a wave length of about the same asthe model output of the MERCUR code presented in [17] and of the observations.

63


The first thermal like structure appears closer downstream of the tower in theCFD model, that could be explained by differences in the turbulence level, and bythe effect of higher resolution mesh in critical areas.

Figure 4.7. Volume rendering of the LWC field of the simulated plumes, 11 March1980 (top) and 12 March 1980 (bottom). The field is transparent below ql = 0.01g kg-1 and opaque when higher than 1.0 g kg-1 with 0.1 g kg-1 steps in the opacity.

A quantitative comparison is shown on Fig. 4.8 where CFD model output is plot-ted against available aircraft data. In order to obtain valuable results on plumeformation and dispersion it is important to have good initial data on the verti-cal thermal structure as well as wind shear and humidity. The temperature fieldwas well captured by the simulation, it follows the ambient temperature profileand the aircraft data lies well between the simulated extrema. The maximumsimulated values of LWC field were also compared to recorded data during thecampaign. Although aircraft data for LWC was not available for comparison forthe initial rise, in the layer between 900 m and 1800 m the simulation compareswell, tendencies in liquid water field are well captured for the high wind case witha slight overestimation along the plume axis. During aircraft data sampling, datafor instantaneous vertical velocity was also collected. This cannot be compareddirectly to simulation output as vertical velocity from URANS simulations doesnot contain the fluctuation component of the wind speed. However it is possibleto deduce approximate values from the turbulence kinetic energy field. The survey

64


shows significant fluctuations in the velocity field which is also well reflected in thesimulation results.The simulated plume height could be extracted from Fig. 4.8 where the liquid watercontent is starting to drop significantly. The simulation shows slight overestimationfor the high wind case, it gave values between 2000 m and 2500 m while duringthe survey a height of ~1950 m and ~2250 m was observed.In general, the model showed good overall performance with the CFD model im-proved with the bulk microphysical scheme. Further steps will include the imple-mentation of sedimentation and precipitation schemes.


A transformation method was developed in the past in order to extend the capa-bilities of commercial CFD solvers with mesoscale effects. In this chapter furtheradvances were shown. A bulk warm microphysical scheme was implemented in thesolver in order to simulate humidity transport and phase change in atmosphericflows.Model results were successfully validated with the rise of idealized two-dimensionaldry and wet thermals. We have then applied the model to the simulation of wetplume formation originated from a cooling tower of a large nuclear power plant.Results obtained from our simulations are encouraging with regard to the pre-dictability of cumulus like plume structures in the far field of the tower formedin complex wind field and thermal stratification, the overall liquid water contentalong the plume axis and also the turbulent fluctuations caused by the verticalmovements in the plume.

65


Figure 4.8. Simulation results for 11 March (top panels) and 12 March 1980 (bot-tom panels) compared to the aircraft measurements. On the left: Maximum (solidline) and minimum (dashed line) of the simulated temperature. Symbols: mea-sured temperature average through the plume. Center: predicted maximum of theLWC field (solid line) at each level. Symbols: ‘x’ and ‘+’ represent the measuredminimum and maximum of the LWC. Right panels: simulated vertical velocity w(solid lines: max. in black, min in grey). Doted and dashed lines show the fluc-tuation components. Symbols: ‘x’ and ‘+’ represent the minimum and maximuminstantaneous vertical velocity observed at each flight level.

66

5 Summary, conclusions and outlook


General purpose computational fluid dynamical (CFD) solvers are frequently usedin small-scale urban pollution dispersion simulations without taking into accounta large extent of vertical flow. Vertical flow, however, plays an important rolein the formation of local breezes, such as urban heat island induced breezes thathave great significance in the ventilation of large cities and consequently couldbe an important factor in civic design. The effects of atmospheric stratification,anelasticity and Coriolis force must be taken into account in such simulations. Atransformation method described by Kristóf et al. [91] was implemented in orderto prove the applicability of general purpose CFD solvers for modeling stratifiedatmospheric flows and thermal convection. In order to accomplish this, a set oftest cases was defined for investigating each effect separately. Investigations werebased on non- dimensional quantities characteristic for the geometry and of thestratification, each case focusing on one key element of the transformation method.Firstly the source term acting on the energy equation was investigated by usinglarge eddy simulation on laboratory scale water tank experiments simulating urbanheat island circulation phenomena. This is a core element since it is responsiblefor the treatment of atmospheric stratification. It was shown with temperatureprofiles and PIV data that with this new modeling approach thermal circulationphenomena can be modeled with an accuracy reasonable in the engineering prac-tice. The behavior of the dynamical model core was further investigated withthe simulation of a nonlinear density current in an atmospheric scale. Using thisexample the significance of taking into account compressibility effects was demon-strated with regard to the model results, moreover it was shown that the effect ofcompressibility can also be taken into account with this new method.It is not common in the literature that one uses CFD solvers for the modeling ofinternal gravity waves, except a solver specific implementation. [112] This areahowever has a great engineering significance from the point of view of aviationhazard, reduction of damages caused by severe downslope windstorms or the es-timation of the location of future wind-farms. Simulation of gravity wave propa-gation moreover allows the investigation of flows characterized by wide range of

67


hydrostatic and nonlinear states and gives a good opportunity for the evaluationof model performance. It was shown that the transformation method is applica-ble to the modeling of gravity waves and with this established the modeling ofatmospheric problems of this class using engineering tools. However by modelingatmospheric scale gravity waves using the original transformation formulation ofKristóf et al. (2009) a significant deviation from measurement results was foundin terms of horizontal velocity and potential temperature field, therefore a refine-ment of the source terms of the original transformation method was proposed byusing a variable temperature reference profile. It was shown based on the simu-lation of full scale downslope windstorm phenomenon that the application of theimproved transformation method leads to more reasonable model results relevantfor engineering practice than he original formulation. A series of validation caseswere defined for models that incorporate gravity wave phenomena. Model perfor-mance was demonstrated with the simulation of a hydrostatic and linear gravitywave propagation based on nondimensional quantities of hydrostatic and nonlinearstates compared first to the analytic solution of the flow field.The model performance was quantitatively evaluated with the simulation of thelaboratory scale phenomena, concerning the measured and simulated wavelengths,amplitudes and the appearance of nonlinear structures. The model evaluation forλ/h gave a correlation coefficient of 0.95, fractional bias 0.317, normalized meansquare error of 0.12, a hitrate value of 0.75 and 100% of predictions within a factorof two of observations. The equivalent results for A/h were 0.99 (R), 0.21 (FB),0.05 (NMSE), 1 (HR), and 100% (FAC2) that corresponds to excellent modelperformance for both variables.Furthermore it was shown that pressure based CFD solvers using the improvedtransformation method are able to predict full atmospheric scale nonlinear gravitywave propagation and the accompanying downslope windstorm event, where thecompressibility of the air also plays an important role. It was demonstrated thatwith this novel method a similar accuracy can be reached compared to widelyused mesoscale meteorological codes. The model evaluation statistics for potentialtemperature (Θ) gave a correlation coefficient of 0.96 and fractional bias –0.056,normalized mean square error of 0.01, a hit rate value of 0.56, and 100% of pre-dictions within a factor of two of observations. The equivalent results for horizon-tal velocity (U ) were 0.154 (R), –0.07 (FB), 0.396 (NMSE), 0.29 (HR) and 0.7(FAC2). The poor prediction and hitrate value for the horizontal velocity is duethe horizontal shift in the overall flow pattern. Recommendations for the statis-tical evaluation states however that results should be judged based not only onperformance measures but together with the overall flow pattern.The original formulation proposed in [91] essentially handles dry adiabatic pro-cesses. However practical applications require the modeling of moist dynamics.

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5.2. OUTLOOK

The built in models in general purpose CFD solvers for the treatment of moistureare not compatible with the Bousinesq density model used by the transforma-tion method or they require a density based solver. Therefore a new method wasdeveloped where the advantages of the original transformation method was com-bined with an efficient bulk microphysical model for describing phase change andmoisture transport. The presented technique is novel in the CFD field since itallows both the computation of micro- and mesoscale stratified flows in a singleframework with arbitrary stratification and moist dynamics with reasonable com-putational efforts due to the bulk microphysical approach. The model performanceof a multi-moment bulk microphysical scheme in a general purpose pressure basedCFD solver was demonstrated, where the number of activated condensation nuclei,the condensed water droplets and the total water content was described with ad-ditional scalar equations. The number of activated droplets was estimated basedon heterogeneous nucleation. Supersaturation was calculated based on the localthermodynamic field variables.The implemented microphysical model is suitable in such cases where the temper-ature of hydrometeors does not reach the freezing level, e.g. in numerical experi-ments of cumulus clouds with lower vertical extent. It was demonstrated with thesimulation of two-dimensional dry rising thermal calculations that the CFD modelis able to predict the evolution of interfacial eddies of thermals and less sensitive tothe changes in the predefined viscosity than widely used mesoscale meteorologicalcodes. Secondly it was shown that the model extended with mesoscale and moistdynamic capabilities is able to predict the shape and hydrometeor properties ofwet rising thermals. Model results are in line with previous calculations with slightover prediction in the liquid water field. This novel combination of the mesoscaletransformation method and bulk microphysical approach makes pressure basedCFD solvers capable of predicting moist dynamics with arbitrary ambient stratifi-cation. The model performance was demonstrated with the simulation and on-sitemeasurement of temperature, liquid water content and vertical velocity field of afull scale cooling tower plume dispersion study. Results indicate that this newapproach reproduces the measurement data, and the model is valid for differentenvironmental conditions, from stable to conditionally stable stratification fromlow to high ambient wind shear.

5.2 Outlook

In general the presented results using the transformation method indicate satis-factory predictions for velocity, with hit rate values above 60% and even higher inthe laboratory scale simulations. However performance of the model for turbulent

69

5.2. OUTLOOK

kinetic energy is difficult to judge due to difficulties in the atmospheric scale mea-surements. It was observed and expected that the usage of any variant of k − εmodels would lead to limited performance. Therefore, the implementation of non-linear k − ε turbulence model should be investigated, as it was already proposedin the literature. This requires the reformulation of the original transformationexpressions in order to make it compatible with the description of the stratifica-tion models. The implementation would be beneficial where recirculation zonesare expected in the flow as it was pointed out in the laboratory scale internal wavepropagation study.The advantages of using CFD with the discussed mesoscale transformation methodis clear. There are further points that need to be addressed though in future stud-ies. Urban physics are usually parametrized in mesoscale models [118], on theother hand one can also obtain good results with the distributed drag force ap-proach in CFD combined with the resolved flow field around building structures.[10, 9]. This way a less detailed representation of building objects can be usedin rural areas and higher resolution in the desired locations. Parametrization ofmoisture and temperature in soil is required in NVP models in order to describeland-surface processes. The moisture transport model described in Chapter 4 alsorequires boundary data at surface boundaries. It is currently possible to supplysuch input based on measurement data, however the implementation of a propersoil model is desirable in order to supply boundary data of humidity and tempera-ture field. The presented moisture transport model is able to capture the evolutionof shallow cumulus clouds and equivalent structures in the atmosphere. Furthersteps are required to describe processes that involve coagulation, deposition orevolution of rain droplets in clouds with more prominent vertical extension.One of the target applications is the modeling of local circulations that also de-pends on mesoscale effects. In urban heat island studies the surface energy balance,the calculation of short and long wave radiation is a crucial point. It is possibleto address these question by using the available built in radiation models of theCFD tool or by modifying the governing equations with custom radiation models.A clear trend can be seen in the evolution of NVP models towards the usage ofhigher resolution by using nested mesoscale models or even nesting a CFD model atthe level of the highest resolution. Data exchange interfaces are exist and routinelyused in NVP models. The implementation and application of interfaces betweenthe investigated CFD model and mesoscale or global scale models requires furthersteps. In operational NVP models fast computation is a must. One can expectlonger computations with the CFD tool due to the physics involved e.g. moretransport equations due to moisture transport, on the other hand more preciseresults. The presented CFD tool therefore does not serve operational purposes, itcan be considered as a research tool.

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6 Thesis points

The scientific results of the present work are summarized in the following thesispoints.

1. Thesis: Modeling stratified atmospheric flows andthermal convection – proving the technical viabilityand accuracy of a new modeling approach

General purpose computational fluid dynamics (CFD) solvers are frequently usedin small-scale urban pollution dispersion simulations without a large extent ofvertical flow. Vertical flow, however, plays an important role in the formation oflocal breezes, such as urban heat island induced breezes that have great significancein the ventilation of large cities and consequently could be an important factor incivic design. The effects of atmospheric stratification, anelasticity and Coriolisforce must be taken into account in such simulations.a) A transformation method described by Kristóf et al. [99] was implementedin order to prove the applicability of general purpose CFD solvers for modelingstratified atmospheric flows and thermal convection.b) A set of test cases was defined and evaluated for separate effect investigation,based on non-dimensional quantities characteristic for the geometry and of thestratification, each case focusing on one key element of the transformation method.c) The source term acting on the energy equation was investigated by using largeeddy simulation on laboratory scale water tank experiments simulating urban heatisland circulation phenomena. It was shown with temperature profiles and PIVdata that with this new modeling approach thermal circulation phenomena can bemodeled with reasonable accuracy relevant for engineering practice.d) The behavior of the dynamical model core was investigated with the simulationof a nonlinear density current in an atmospheric scale. Using this example thesignificance of taking into account compressibility effects was demonstrated with

71

regard to the model results, moreover it was shown that the effect of compressibil-ity can be taken into account with reasonable accuracy with this new method.

Publications related to this thesis statement:[99, 98, 97, 95, 101, 94, 102, 148]

2. Thesis: Modeling internal gravity waves by usinga pressure based CFD solver

It is not common in the literature that one uses CFD solvers for the modeling ofinternal gravity waves, except a solver specific implementation. [126] This areahowever has a great engineering significance from the point of view of aviationhazard, reduction of damages caused by severe downslope windstorms or the es-timation of the location of future wind-farms. Simulation of gravity wave propa-gation moreover allows the investigation of flows characterized by wide range ofhydrostatic and nonlinear states and gives a good opportunity for the evaluationof model performance. It was shown that the transformation method is applica-ble to the modeling of gravity waves and with this established the modeling ofatmospheric problems of this class using engineering tools.a) By modeling atmospheric scale gravity waves using the original transformationformulation of Kristóf et al. (2009) a significant deviation from measurementresults was found in terms of horizontal velocity and potential temperature field,therefore a refinement of the source terms of the original transformation methodwas proposed by using a variable temperature reference profile.b) It was shown based on the simulation of downslope windstorm phenomena thatthe application of the improved transformation method leads to more reasonablemodel results relevant for engineering practice.c) A series of validation cases were defined and evaluated for models that incor-porate gravity wave phenomena. Model performance was demonstrated with thesimulation of a hydrostatic and linear gravity wave propagation based on nondi-mensional quantities of hydrostatic and nonlinear states.The model performance was quantitatively evaluated with the simulation of thelaboratory scale phenomena concerning the measured and simulated wavelengths,amplitudes and the appearance of nonlinear structures. The statistical modelevaluation showed excellent model performance.Furthermore it was shown that pressure based CFD solvers using the improvedtransformation method are able to predict full atmospheric scale nonlinear grav-ity wave propagation and the accompanying downslope windstorm event. It was

72

demonstrated that with this novel method a similar accuracy can be reached com-pared to widely used mesoscale meteorological codes.

Publications related to this thesis statement:[150, 151, 96, 100]

3. Thesis: Modeling phase change and moisturetransport with a bulk microphysical model using ageneral purpose pressure based CFD solver

The formulation proposed in [99] essentially handles dry adiabatic processes. How-ever practical applications require the modeling of moist dynamics. The built inmodels in general purpose CFD solvers for the treatment of moisture are notcompatible with the Bousinesq density model used by the transformation methodor they require a density based solver. The presented technique is novel in theCFD field since it allows both the computation of micro- and mesoscale stratifiedflows in a single framework with arbitrary stratification and moist dynamics withreasonable computational efforts due to the bulk microphysical approach.a) The model performance of a multi-moment bulk microphysical scheme in ageneral purpose pressure based CFD solver was demonstrated, where the numberof activated condensation nuclei, the condensed water droplets and the total watercontent was described with additional scalar equations. The number of activateddroplets was estimated based on heterogeneous nucleation. Supersaturation wascalculated based on the local thermodynamic field variables.b) The implemented microphysical model is suitable in such cases where the tem-perature of hydrometeors does not reach the freezing level, e.g. in numerical exper-iments of cumulus clouds with lower vertical extent. It was demonstrated with thesimulation of two-dimensional dry rising thermal calculations that the CFD modelis able to predict the evolution of interfacial eddies of thermals and less sensitive tothe changes in the predefined viscosity than widely used mesoscale meteorologicalcodes. Secondly it was shown that the model extended with mesoscale and moistdynamic capabilities is able to predict the shape and hydrometeor properties ofwet rising thermals. Model results are in line with previous calculations with slightoverprediction in the liquid water field.c) This novel combination of the mesoscale transformation method and bulk mi-crophysical approach makes pressure based CFD solvers capable of predictingmoist dynamics with arbitrary ambient stratification. The model performancewas demonstrated with the simulation and on-site measurement of temperature,

73

liquid water content and vertical velocity field of a full scale cooling tower plumedispersion study. Results indicate that this new approach reproduces the mea-surement data, and the model is valid for different environmental conditions, fromstable to conditionally stable stratification from low to high ambient wind shear.

Publications related to this thesis statement:[99, 150, 149, 96]

74

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