Turbulent bluff body flows modeling using OpenFOAMtechnology

Dmitry Lysenko †

e–mail: dmitry.lysenko@ntnu.no

Ivar S. Ertesvag† and Kjell Erik Rian ‡

†Department of Energy and Process EngineeringFaculty of Engineering Science

The Norwegian University of Science and Technology,Trondheim‡Computational Industry Technologies AS, N-7462 Trondheim, Norway

Summary Plane turbulent bluff-body flows were numerically analysed using compressible for-mulation of the conventional URANS approach. The low-Reynolds-numberk−ǫ turbulence modelof Launder and Sharma was applied for the closure problem. Numerical simulation was carried outusing the state-of-the-art OpenFOAM technology for the three popular test problems in fluid dy-namics : 1) plane laminar compressible flow (Re=140, M=0.2) over a circularcylinder; 2) turbulentbluff-body flow in the channel (Re = 17500, M = 0.03) replicating the Fujii lab-test conditions;3) turbulent bluff-body flow in the channel (Re = 45000, M = 0.05) replicating the Volvo testrig. Satisfactory agreement between numerical and measured data for themain integral and localflow characteristics was achieved which indicates on the adequacy and accuracy of the establishednumerical method, implemented in the OpenFOAM library, for prediction plane fully-developedturbulent bluff-body flows.

Introduction

The OpenFOAM toolbox was originally developed as a hi-end C++classes library (Field Op-eration and Manipulation) for a broad range of fluid dynamicsapplications and quickly becamevery popular in industrial engineering as well as in academic research. The basic numericalapproach for Navier-Stokes equations solution is based on the so-called projection procedure(i.e. the well-known physical variables splitting technique) in the frame of the factorized finite-volume method (FVM). One of the three main algorithms (SIMPLE, PISO and PIMPLE) canbe chosen for the pressure-velocity coupling. The wide listof the numerical schemes and math-ematical models implemented in OpenFOAM provides robustness and accuracy of this technol-ogy for a wide spectrum of fluid dynamics problems. Nevertheless, in spite of many attractivefeatures, the OpenFOAM toolbox has some disadvantages, as well. The most crucial are: the ab-solute lack of default settings and the absence of the quality certification and, as a consequence,the absence of high-quality documentation and references.The huge amount of different nu-merical schemes, algorithms and mathematical models creates the illusion that any problem canbe solved. Actually, the available catalogs of mathematical models are not perfect and manyof them are subject to further research. Moreover, the acceptability of mathematical models forsolving complex (multi-physics) problems has yet to be analyzed. The limits for application ofmost of the models are also not clearly understood. Thus, themost important problem of anynumerical investigation is to estimate the adequacy of numerical simulations.

This paper presents some results dedicated to the validation of the mathematical method forplane turbulent separated flows modeling using OpenFOAM technology. First, the numericalmethod based on the state-of-the-art transient compressible URANS solver with the definitenumerical schemes, boundary conditions and the two-equation turbulence model was estab-lished. Second, the method was tested for a laminar unsteadyflow over a circular cylinder. Atthe third step, the chosen generalized setup was applied forthe numerical simulations of the

two test problems, which represents fluid mechanics of the plane turbulent bluff-body flows.All these tests were numerically replicated using ANSYS FLUENT with a quite similar numer-ical method. Detailed face-to-face comparison of the numerically predicted and experimentallymeasured data showed satisfactory agreement between them.Based on such optimistic results,the established numerical method may be extended for the simulations of the turbulent reactingflows.

Overview of the fluid mechanics of plane (2D) bluff-body flows

The structure of the bluff-body flow field consists of three regions: the boundary layer along thebluff-body, the separated free shear layer, and the wake. One can find the precise descriptionof the detailed fluid dynamics for such type of flows, for example in [25]. Here, we providesome limited description, closely following [25]. A boundary layer (hereafter BL) is formedon the bluff-body leading edge and finalized with flow separation at the bluff-body base. Shearlayers are started in the points of flow separation at the baseand organized recirculation bubblebehind the bluff-body when top and bottom layers are merged with each other and begin tointeract. This recirculation flow region starts to form the wake. For Reynolds numbers with alength scale equal to the obstacle base, Re< 200000, the boundary layer may be assumed aslaminar (analogue to a circular cylinder and referred to as the ‘sub-critical regime’), and thedynamics of the downstream flow field is largely driven by the shear layer and wake processesalone. Both asymmetric vortex shedding (the Benard/von Karman instability) and convectiveinstabilities (Helmholtz instability) of the separated shear layer may exist [25]. The main goalof this study is to provide methodical validation of the numerical method for accurate numericalpredictions of the plane turbulent bluff-body flows.

Brief description of numerics

Preface

The main emphasis of this work put on the problem of validation and verification of a numericalmethod implemented in the OpenFOAM toolbox. However, for more consistency and validityof results, ANSYS FLUENT technology was used as well. The numerical methods that werechosen for predictions were quite similar. In both codes, a so-called pressure-density solver,based on the projection method [2] for a solution of compressible URANS equations, was used.The modified low-Reynolds-numberk − ǫ turbulence model of Launder and Sharma (hereafterLSKE) was chosen as the ‘baseline’ model in OpenFOAM. This model has been implementedin ANSYS FLUENT as an undocumented feature, however, we stayed at the Realizablek − ǫmodel (hereafter RKE) for our simulations as ‘baseline’ for the FLUENT code. Special atten-tion was drawn to the treatment of a near-wall region. In all cases the low-Reynolds-numberformulation was used. With this purpose so-called ‘dampingfunctions’ were developed in theLSKE model, and a two-layer approach was implemented for theRKE model.

OpenFOAM

The OpenFOAM code [31] v.1.7.1 was used for numerical simulations. The standard solverrhoPisoFOAM was utilized for unsteady compressible Reynolds-averaged Navier-Stokes equa-tions (URANS) modeling based on the finite-volume (FVM) factorized method [7] and thepredictor-corrector PISO algorithm [14]. Typically threeand one (or two) iterations were setfor a PISO loop and for non-orthogonal corrections, respectively.

The generalized fully second-order setup (in space and in time) was used for all simulations. TheNVD type differencing scheme – GAMMA [16] withγ = 0.1 was applied for all convective

terms approximation. A second order implicit Euler method (BDF2 formula, [7]) was usedfor time integration together with dynamic adjustable timestepping technique to guarantee thelocal Courant number less then CFL<0.5. Preconditioned (bi-) conjugate gradient method [10]with incomplete-Cholesky preconditioner (ICCG) by Jacobs [15] was used for solving linearsystems with the local accuracy of10−7 for all depended variables at each time step. The under-relaxation factors (0.3 for pressure and 0.7 for all other terms) were set to prove stability ofcalculations.

The modified low-Reynolds-numberk−ǫ turbulence model of Launder and Sharma was chosen[19] for Navier-Stokes equations closure. Such approach does not require the specification ofthe ‘wall-functions’ as used in high-Reynolds-number formulations to describe the near wallregion treatment. So-called ‘damping functions’ were introduced and incorporated ink − ǫmodel by Jones and Launder (1972). It was later re-optimisedby Launder and Sharma [19],and demonstrated satisfactory results in many applications. The wide acceptance of such formu-lation gradually granted the status of the benchmark for low-Reynolds-numberk− ǫ turbulencemodel [3]. In this work the model was applied with the following differences in the sink termof ǫ equation compare to the ‘baseline’ LSKE model [19]:

1. Low-Reynolds-number functionf2 was slightly changed:

f2 = 1− 0.3 exp(

−min(

Re2τ , 50))

,

where Reτ – is a turbulent Reynolds number.

2. The model constantCǫ3 was updated from the ”baseline” valueCǫ3 = 2 to

Cǫ3 =

(

−0.33− 2

3Cǫ1

)

.

ANSYS FLUENT

Using the factorized FVM [7] we solved a URANS system for a compressible viscous fluid witha second order accuracy in space and in time. The linear system of equations was solved withGauss-Seidel smoother or the Incomplete Lower Upper (ILU) decomposition smoothers, whichwas accelerated by an algebraic multi-grid (AMG) technique, based on the additive-correctionstrategy [11]. The convective terms were represented according to the Leonard quadratic up-wind scheme (QUICK) [13]. The velocity and pressure fields were matched with a centeredcomputational template within the spirit of Rhee and Chou [23]. A SIMPLEC [30] and PISO[14] pressure-correction procedures and an implicit iteration algorithm with fixed time-steppingfor physical-time integration were used.

To close the system of equations we used the Realizablek − ǫ model of Shih [26]. Originallythis model was developed in high-Reynolds-number formulation. However, two-layer approachhas been employed in FLUENT to specify both theǫ and the turbulent viscosity (µτ ) in thenear-wall cells. The main idea of this approach is to subdivide a fluid domain in a vicinity ofa wall into viscosity-affected and fully-turbulent regions. For this purpose, turbulent Reynoldsnumber, Rey, is introduced:

Rey =ρy

√k

µ,

wherey – is a wall-normal distance, calculated at the cell centers.The boundary between thesetwo regions is defined at Re∗

y ≡ 200. In the fully-turbulent region (Rey > Re∗y) Realizablek− ǫ

model is used. In the viscosity-affected near wall region (Rey < Re∗y) the one equation modelof Wolfshtein [33] is employed for calculation of theµt, which is computed from:

µt = ρCµlµ√k. (1)

Theǫ field in the viscosity-affected region is defined from

ǫ =k3/2

lǫ. (2)

The length scales (lµ andlǫ) from 1–2 are calculated according to Chen and Patel [1]:

lµ = yC∗l (1− exp (−Rey/Aµ))

lǫ = yC∗l (1− exp (−Rey/Aǫ)) . (3)

Constants in eq. 3 are taken from [1]:C∗l = kC

−3/4µ , Aµ = 70, Aǫ = 2C∗

l . The blendingfunctions for a smooth transition between the near-wall algebraically predictedǫ andµt andtheir values obtained from a solution of the Realizablek − ǫ model transport equations in theouter region are implemented according to Jongen [18].

Boundary and initial conditions

The following boundary conditions were applied. Inlet: fixed values for velocity, temperature,turbulence kinetic energy and dissipation rate; pressure -zero gradient. Outlet: non-reflecting[22] boundary conditions (NRBC) for pressure and zero gradients for velocity, temperatureand turbulent properties. Bluff-body and channel walls weretreated as isothermal no-slip con-ditions. The upper and lower buffer domain boundaries were used as symmetry planes. Theturbulence intensity at the inlet was set equal to4% that is common for the typical wind tunnels.The characteristic scale of the turbulence was set equal to the bluff-body diameter (or base). Themolecular viscosity and the thermal conductivity were taken to be constant. The Prandtl numberwas assumed to be 0.75, and the ratio of specific heats is 1.4 (the ‘ideal gas’). The initial condi-tions, at the moment of timet∗ = 0, corresponded to the conditions of sudden stopping of thebluff-body in a fluid flow, i.e., the input conditions were extended to the whole computationalregion.

Mesh independence

It is common knowledge that the disagreement between experimental data and numerical resultsis determined by two groups of errors (apart from experimental errors): 1) ‘model’ errors dueto the inadequate assumptions made in selecting one turbulence model or another and 2) ‘dis-cretization’ errors caused by the inadequate resolution ofthe employed finite-element grids andcomputational methods. Whereas the errors of the first group are assumed to be ‘systematic’under certain assumptions, e.g., for a fixed computational methodology, ‘discretization’ errorsare controlled by the method of adaptation (increase in the resolution) of a computational grid.The grid independence of results is confirmed by:

• the application of the two different CFD technologies (the commercial code ANSYSFLUENT and the open-source OpenFOAM toolbox);

• the use of the projection method [2] for a solution of compressible URANS equations (orso-called pressure-based solver) with the quite similar implementation in both codes;

• the use of different grid topologies (structured vs. unstructured) and cell types (quadrilat-eral vs. triangular);

• satisfactory agreement in the distribution of the localy-velocity component obtained at afixed reference point at three grids with gradually decreased cell size (by factor 2).

Analysis of results

Laminar unsteady flow over circular cylinder (Re = 140, M = 0.2)

Methodical investigation of an unsteady laminar flow arounda circular cylinder at a Reynoldsnumber, Re= 140 and Mach number, M= 0.2 was carried out with the goal of validation,verification and understanding of the numerical methods andtheir capabilities implemented inOpenFOAM. The unsteady laminar flow was simulated in two different formulations (based onURANS approach): incompressible and compressible.

Two types of the grids were used: unstructured, based on tetrahedral elements and structuredcurvilinear (polar) O-type.

Figure 1: Descriptions of computational grids: curvilinear O-type orthogonal (a,b) and unstructured tri-angular (c,d).

O-type grid (Fig. 1,(a-b): the configuration of the region surrounding the cylinder from thecenter had the form of a circle. A cylinder of diameter,D = 0.1 m is located in the center of thecomputational domain. The integration domain had a radial extension of40 × R (or 20 × D),which was chosen based on preliminary investigations [13] and it’s assumed sufficient forincompressible flow simulation. The grid points were clustered in the vicinity of the cylinder.The obstacle as well as the outer boundary profiles were divided into 325 equal intervals. Radialstates were divided into 325 intervals with an expansion factor in the radial direction of 1.020.Fig. 1,(b) displays a zoom of the grid in the vicinity of the cylinder. However, for compressiblesimulations such grid with radial extension of40×R is not sufficient [21] and far-field boundarytypically is chosen toR = 80 − 100 calibers (or sometimes, additional buffer domain is usedwith the domain extension up toR = 500− 750). So for compressible laminar flow simulation

updated O-type grid with radial extension100×R (or 50×D) was used with size of[600×600].The expansion factor for radial states was the same as in the first grid.

Triangular grid (Fig. 1,(c,-d): the computational region represented the rectangle of a sizeL ×H = 6.5 m× 2 m. For consistency with previous results, the computationaldomain replicatedsolutions discussed by Isaev et al. [13], where generally satisfactory results were obtained forthis problem. A cylinder of diameter,D = 0.1 m was located at a distance of 17.5 calibers(or cylinder diameters,D) from the inlet and symmetrically relative to the upper and bottomboundaries. The obstacle was surrounded by a structured ring grid (so called viscous BL) with aminimum nearwall step size ofF = 5× 10−6 m. The cylinder profile was divided intoN = 100equal intervals. The cell size at the outer boundaries was fixed and equal to0.015 m, leading tosmooth grid refinement in the vicinity of a cylinder.

Table 1: The parameters of the designed computational grids.

id Mesh type Size Domain1 Unstructured triangular 0.327× 106 L×H = 6.5 m× 2 m2 O-type quad 325× 325 40×R3 O-type quad 600× 600 100×R

The time history of the liftCl and dragCd coefficients are presented in Fig. 2. Results obtainedboth at structured and unstructured grids with incompressible/compressible formulations are inquite good agreement between each other. The discrepanciesfor the drag coefficient distribu-tions (Fig. 2,(b) may be associated with distinct mesh topologies resulted in the slightly differentpressure field prediction. Signals obtained at the O-type grids (1-2) are quite close to each otherin opposite to another curve (3) related with the unstructured triangular mesh. Isaev et al. [13]discussed the same numerical results using FLUENT and the in-house VP2/3 code. The meandrag coefficient was determined in the rangeCd = 1.34−1.36. These values agree with the DNSresults reported by Inoue and Hatakeyama [12] and with the data obtained by Muller [21]. Theamplitudes of the lift and drag coefficients areC

′

l = 0.48−0.53 andC′

d = 0.023−0.026. Thesenondimensional force amplitudes are in good agreement with[12], who reportC

′

l = 0.52 andC

′

d = 0.026, respectively, for inlet freestream Mach number M= 0.2. Muller [21] using a high-order finite difference method (HOFDM), gotC

′

l = 0.5203 andC′

d = 0.02614, respectively.

The Strouhal number, i.e. the nondimensional frequency of the vortex shedding, was computedto St= 0.18− 0.184. Inoue and Hatakeyama [12] found the value of St= 0.183 in their DNS,who used a compact HOFDM to solve the 2D compressible Navier-Stokes equations. Muller[21] predicted the value of St≈ 0.1831 in the similar conditions. Williamson [32] discussed indetail the Strouhal-Reynolds number relationship for the laminar shedding regime (47 < Re<200). Over a period of 100 years [32], beginning with the vortex frequency measurements ofStrouhal (1878), there has existed of the order of20% disparity among the many measurementsof Strouhal number vs. Reynolds number, for this regime. The more recent results show thesingle St–Re function and have an agreement to the1% level of St–Re relationship for laminarparallel shedding using different techniques, such as a wind tunnel facility and a water facilityknown as a towing tank. The generalized St–Re curve proposed by Williamson [32] has the

Figure 2: Time evolution of the lift (a) and drag (b) coefficients calculated with different numericalsetups (1 – compressible flow, O-type mesh, 2,3 – incompressible flow, O-type and unstructured meshes,respectively) and comparison of time averaged pressure coefficient distribution over the cylinder’s base(c) obtained numerically (1 – compressible flow, O-type mesh, 2,3 – incompressible flow, O-type andunstructured meshes, respectively, 4 – DNS [12]) and experimentally (5-6 – [8]). The contours (d) ofinstantaneous radiated density gradient field (256 values from 0.0 to 0.025with an equal step).

following equation:

St=A

Re+B + C ·Re

A = −3.3265, B = 0.1816, C = 1.6× 10−4,

which leads to an experimental Strouhal number, St= 0.18. Williamson [32] provides the plotof the base suction coefficient (−Cpb) over wide range of Reynolds numbers, as well also. ForRe=140, the experimental value of−Cpb = 0.84 can be determined and correlates well with thenumerically predicted values of−Cpb = 0.842 − 1.060. The distribution of the time-averagedpressure coefficient over a circular cylinder is presented in Fig 2,c. The numerical results ob-tained for compressible flow (1) matched well with DNS [12]. Data from incompressible flowcalculations obtained at different grids (2-3) are very close to each other one the one hand,and on the other hand – close to experimental results by Groveet al. [8]. The gap betweenthe distributions of mean pressure coefficient (1-3) can be explained by the difference in theimplementation of NS equations solution algorithms in compressible and incompressible flowformulations. The DNS as well as the current compressible simulation were carried out for theMach number M= 0.2. Experimentally measured values by Grove et al. [8] are close to theresults obtained with incompressible flow assumption. Thus, we can expect that the last one wasobtained in ambient conditions with weak compressibility (M < 0.1). It should be noticed thatthere is some underprediction of integral parameters and fluctuations values in the force coef-ficients, predicted by FLUENT (see Table 2). The main reason for such lower values may betreated as a result of excessive numerical scheme dissipation when the dynamic mesh adaptationalgorithm is applied.

Finally, Fig. 2,(d) represents an instantaneous field of the radiated density gradient that depictsclearly the unsteady nature of the compressible laminar flowover a circular cylinder and thequalitative assessment of applied NRBC.

Table 2: Integral characteristics for 2D unsteady laminar flow over a circular cylinder.

Contributors Year Method Re M Cd C′

d C′

l St −Cpb

Williamson [32] 1996 EXP 140 – 0.180 0.84Muller [21] 2008 HOFDM 150 0.2 1.340 0.026 0.520 0.183Inoue and 2002 DNS 150 0.2 0.026 0.520 0.183

Hatakeyama [12]Isaev et al. [13] 2005 URANS 140 – 1.270 0.011 0.400 0.172

Current results 2010id = 1 URANS 140 – 1.360 0.026 0.533 0.184 0.842id = 2 URANS 140 – 1.342 0.024 0.518 0.184 0.860id = 3 URANS 140 0.2 1.340 0.023 0.480 0.180 1.060

Plane turbulent bluff-body flow - Fujii test case (Re = 17500, M = 0.03)

Experiments [5, 6] were carried out in an open circuit, forced flow type of wind tunnel atambient conditions (Pinf ≈ 100 kPa, Tinf = 280 K). A sketch is presented in Fig 3,(a). Anequilateral (D = 0.025 m) triangular rod was placed inside the channel passage of0.05 m -square cross section. The channel blockage ratio in this test was,B = 0.5. The velocity in afront of the bluff-body was limited touinf = 10 m/s. The turbulence intensity level at the inletof the test section was about2 − 4%. Thus, the corresponding Reynolds number based on thebluff-body base was about Re≈ 17500 and the Mach number, M≈ 0.03.

The two dimensional computational domain (Fig. 3,(b)) was chosen in such a way to replicatelab test conditions. The channel length and height were set to L = 0.305 m andH = 0.05 m,respectively. The bluff-body was located atx = 0.117 m from the channel inlet. Two additionalbuffer domains with length0.05 m and height0.1 m were attached to the channel with thepurpose to simulate acoustic open inlet and outlet boundaries. An additional reason for theinlet buffer domain was to avoid setting of the turbulent velocity and temperature profiles at thechannel inlet since it was not measured in lab tests and to avoid investigating the influence of theboundary layer width on the flow parameters. Two finite-element baseline grids with differentcell types were used in the simulations:

• Unstructured triangular mesh (Fig. 3,(c)). Viscous BLs were attached to the obstacle withthe following parameters:F = 5× 10−5 m, G = 1.25 andJ = 7. The same (exceptG) viscous BL was also applied for the channel walls. Each edge of the bluff-body wasdivided intoN = 80 equal intervals. The computational domain was meshed with the sizeof 0.001 m, which guarantees smooth triangular element distributionfrom the inlet andoutlet to the triangular rod. All these features provided good near-to-wall mesh resolutionand allowed to apply the low-Reynolds-numberk − ǫ turbulence model;

• Baseline unstructured quadrilateral mesh (Fig. 3,(d)) was designed in the same manner;• To check solution mesh independence, two additional grids were built, both with the quad

cells. The type of the attached viscous BLs was the same as in the baseline quad meshand domain was meshing with sizes of0.0015 m and0.0005 m, respectively.

Figure 3: A general view of the experimental [5, 6] test rig (a), computational domain (b) and thefragments of the designed unstructured viscous grids (c-d) at the vicinity of the bluff-body.

The details of the designed grids for this test case are summarized in Tables 3–4.

Time-averaged measured and numerically predicted streamlines of the flow around the triangu-lar bluff-body are presented in Fig. 4,(a). The recirculation zone length (Lr) is defined (here-after) as the distance in streamwise (axial) direction between the bluff-body downstream edgeand the location, where the mean axial velocity in the central-line turns from negative to posi-tive values. The results of the CFD analysis (lower part of theframe Fig. 4,(a)) provided a moreextended length of the reversed zone,Lr = 2.67 compared to the measured one (upper part ofthe frame Fig. 4,(a)), Lr = 2.2. Thus, the difference between them was≈ 18%.

Time-averaged pressure coefficient distribution downstream the bluff-body inside the recircula-tion bubble is shown in Fig. 4,(b). Numerical results (3-4) obtained at the baseline grids (withid = 1 andid = 3 from Tables 3–4, respectively) demonstrated quite similarbehaviour withexperimental results (2) as well as experimental data [29] that was added for consistency. Min-

Table 3: The description of the designed computational grids for the Fujii labtest simulations: integralparameters.

id Mesh type Mesh size Domain size (m) Bluff-Body size (N )1 triangular 84K 0.0010 3× 802 quad 56K 0.0015 3× 603 quad 110K 0.0010 3× 804 quad 277K 0.0005 3× 160

Table 4: The description of the designed computational grids for the Fujii labtest simulations: viscousBL parameters.

id Mesh type BL channel (F ,G,J) BL obstacle (F ,G,J) y∗ channel y∗ obstacle1 triangular 5× 10−5, 1.3, 7 5× 10−5, 1.25, 7 0.5 0.62 quad 1× 10−4, 1.3, 7 5× 10−5, 1.25, 7 1.6 0.63 quad 1× 10−4, 1.3, 7 5× 10−5, 1.25, 7 1.6 0.64 quad 1× 10−4, 1.3, 7 5× 10−5, 1.25, 6 1.6 0.5

Figure 4: Comparison of time-averaged integral flow characteristics (a – the recirculation zone bubble,calculated by Fujii et al. [5, 6] at the upper side and numerically predicted at the down side;b – axialdistribution of the mean pressure coefficient: 1-2 experimental data [5, 6,29], 3-4 – numerical data, forbaseline triangular (id = 1) and quad (id = 3) grids, from Tables 3–4), respectively;c – axial distributionof normalized turbulent kinetic energy: experimental data (1) of Fujii et al.[5, 6] and numerical results(2-3) for baseline triangular (id = 1) and quad (id = 3) grids) and time history of the instantaneousnormalizedy-velocity obtained numerically (d – all curves ids correlate with Tables 3–4).

imum values were also in a good agreement between numerical (Cp,min = −2.7) and measured(Cp,min = −2.73) data. The normalized turbulence kinetic energy distribution along the centralaxis in the recirculation zone is provided in Fig. 4,(c) where measured (1) and numerical (2-3)results are displayed. Hereafter the following assumptionis used for measured and numericalturbulence kinetic energy definition:

k =3

4

(

u′2x + u′2

y

)

. (4)

The normalisation is defined as:

K =

√

4/3k

uinf

. (5)

One can observe the non-significant shift between numerically predicted data (2-3) that is theinfluence of the grid topology. Overall, the excellent qualitative and quantitative agreement forK distribution between lab test and numerical modelling datawas achieved.

The time history of the normalised instantaneousy-velocity numerically predicted are presentedin Fig. 4,(d). The probe location was in the same position as in the lab test (x = 1.2 andy = 0.6).As expected, periodic sinusoidal type signals that were obtained clearly illustrate self-regularvortex shedding behavior of the wake. All curves (with theirids according to Tables 3–4) arequite close to each other and can confirm mesh independence ofthe numerical solution. Theaveraged calculated Strouhal number was, St= 0.44, with the corresponding main frequency,f = 177 Hz. This is in a quite good agreement with experimental values,where a pronouncedfrequency,f = 160 Hz, and the corresponding Strouhal number St= 0.4, was detected.

Plane turbulent bluff body flow – Volvo test rig (Re = 45000, M = 0.05)

Fig. 5,(a) shows a schematic drawing of the test section. The test set-up consisted of a straightchannel with a rectangular cross-section, divided into an inlet section (with length of0.5 m)and a channel passage section with length of1 m and 0.12 m × 0.24 m cross-section. Theinlet section was used for flow straightening and turbulencecontrol. The air entering the inletsection was distributed over the cross-section by a critical plate that, at the same time, isolatedthe channel acoustically from the air supply system. The channel passage section ended in acircular duct with a large diameter. The triangular bluff-body (with base diameter,D = 0.04 m)was mounted with its reference position0.681 m upstream of the channel exit. The blockageratio for this test case wasB = 1/3.

Figure 5: The sketch of the Volvo test rig (a) taken from [27] and the general view of the computationaldomain (b).

The cold flow measurements were conducted in the ambient conditions (Tinf = 288 K andPinf = 100 kPa, and corresponding Mach number, M= 0.05). Honeycombs and screens con-trolled the approximate inlet turbulence level of3 − 4%. The test point that was chosen asbaseline for the investigation in a numerical study, had thecorresponding Reynolds number,Re= 45000, based on the bluff-body diameter, which gave the inlet flow velocity in the frontof the bluff-body,uinf = 16.6 m/s. A two-component LDA system was used for thex andy velocity components and its fluctuations measurements. Further detailed description of theLDA system, experimental procedure and sampling techniquecan be found in [27].

The two-dimensional computational domain is presented in Fig. 5,(b) and consisted of an inletbuffer domain (size of0.2 m× 0.24 m) and a channel passage (size of1.5 m× 0.12 m). It wasdecided to attach an inlet buffer domain to the main computational area allowing inlet velocityand temperature profiles to form implicitly in the computations. The integration domain wassplit into three blocks to generate a high-quality unstructured quad/triangular mesh:

1. the inlet buffer and a part of the channel without bluff-body;2. the central part of the channel passage including the obstacle (size of0.2 m × 0.12 m),

with the domain’s central location as described in the sketch at Fig. 5,(a);3. the remaining downstream part of the channel. A total of four unstructured grids were

designed and used in this work, with a detailed description presented in Tables 5–6. Itshould be noticed that the first domain was meshed with quad cells in all cases. Twobaseline grids (quadrilateral and triangular, withid = 1 and id = 3, from Tables 5–6,respectively) were chosen to check the influence of the grid topology on the solution.Two additional triangular grids (withid = 2 andid = 4, from Tables 5–6, respectively)were constructed to check the mesh independence. All grids were built in such a wayto resolve explicitly BLs at the surface of the bluff-body andat the channel walls toguaranteey∗ ≈ 1.0. The boundary layer that was attached to the channel walls was thesame for all grids.

Tables 5–6 give an overview of these grids, meshing details,BLs parameters, correspondingnumbers of the control volumes and averaged values ofy∗.

Table 5: Integral parameters of the designed computational grids for the Volvo test rig.

id Mesh type Domain 1 Domain 2 Domain 3 Bluff-body N cellsSize (m) Size (m) Size (m) N intervals

1 quad 0.0015 0.0007 0.00150 3× 72 282K2 quad\triangular 0.0030 0.0015 0.00300 3× 46 88K3 quad\triangular 0.0015 0.0007 0.00150 3× 72 366K4 quad\triangular 0.0015 0.0004 0.00070 3× 144 754K

Table 6: Viscous BLs parameters of the designed computational grids for the Volvo test rig.

id Mesh type BL obstacle (F ,G,J) y∗ channel y∗ obstacle1 quad 5× 10−5, 1.4, 7 1.5 1.22 quad\triangular 5× 10−5, 1.4, 7 1.4 1.23 quad\triangular 5× 10−5, 1.4, 7 1.5 1.24 quad\triangular 5× 10−5, 1.4, 5 1.4 1.2

Several problem-related articles were found where URANS calculations had been carried outfor the Volvo test rig. But it is important to notice that all ofthem were performed with theincompressible flow assumption. Among them were: 1) the study by Johansson et al. [17]with the results for standard high-Reynolds-numberk − ǫ model, which was also recitatedby Rodi [24]; 2) the paper by Strelets [28] with some limited results available for URANSwith the one-equation Spalart-Allmaras model; 3) the simulation done by Durbin [4] with thek−ǫ−v2 model; 4) the paper by Hasse et al. [9] where brief results forURANS withk−ω SSTmodel were discussed. Thus this work may be considered as thefirst one, where a compressibleURANS approach has been applied to replicate the Volvo test rig.

Johansson et al. [17] calculated the vortex shedding flow past a triangular flame holder. Theyemployed the standardk − ǫ model with wall functions [20] and the hybrid central/upwinddifferencing scheme for the convective terms discretization. The agreement was fairly good,and profiles of the total fluctuations were also shown to be in good agreement with measureddata of Sjunnesson et al. (1991). Furthermore, the calculated Strouhal number, St= 0.26, wasin good agreement with the experimental value of 0.25. The turbulent separated flow around atriangular cylinder in the same conditions was computed with thek − ǫ − v2 model withoutany wall or damping functions by Durbin [4], and good agreement between experiment andprediction was found. The flow around a triangular bluff-body was investigated in details byHasse et al. [9] by URANS and DES based onk−ω SST model. It was shown that this type ofa flow is essentially more dominated by the eddies created in the shear layers behind the obstaclethan by incoming eddies from the transient turbulent inflow.Statistical turbulent flow quantitiesof URANS were compared to the experiments and it was found thatURANS predictions weregenerally satisfactory.

Fig. 6–7 represents some results. As was reported by Sjunnesson et al. [27], the experimentalprofile of axial velocity component was slightly skewed due to small misalignment in the flangebetween the inlet section and the channel passage. However,fully symmetrical inlet velocityprofile was implicitly formed during flow development in the numerical simulations. In bothcases one could observe the outer regions of BLs at the channelwalls, which corresponded verywell with the ‘1/7 power law profile’ typical for fully developed turbulent channel flows. Fig. 6shows the normalized mean axial velocity (a,c) and normalized turbulence kinetic energy (b,d)at several vertical cross sections in the channel. One can see that there are minimal discrepanciesbetween numerical data obtained at grids with different topology (Fig. 6,(a-b): baseline quad(1) and triangular (2) grids), as well as between numerically predicted (1-2) and measured (3)data. Overall, satisfactory agreement between numerically predicted results using OpenFOAMand FLUENT codes (Fig. 6,(c-d)) and experimental data also was achieved. At the same time,astrong underprediction of numerically predicted turbulence kinetic energy was observed insidethe recirculation zone. Meanwhile the same level of turbulence kinetic energy was containedboth in the downstream of the bluff-body and in the upstream part of channel, after the sepa-ration bubble was vanished. Fig. 7 shows the measured and numerically simulated normalizedmean axial velocity (a) and turbulence kinetic energy (b) along the central-line behind the ob-stacle. It should be noticed a≈ 18% underprediction compared to experimental data of theminimum axial velocity in the recirculation zone independently of the grid type. But, overall,there is a good match between numerical and experimentally measured data. For example, thesame level of underprediction was observed by Hasse et al. [9] for URANS (with k − ω SST)data. Current numerical results showed good prediction for the recirculation lengths: calculatedLr = 1.36 (OpenFOAM) andLr = 1.32 (FLUENT) in comparison with the measured (Sjun-nesson et al., 1991) value ofLr = 1.33. For example, the recirculation zone lengths obtainedfor URANS and DES [9] were only 0.94 and 1.18, respectively. InFig. 7,(b) the comparisonbetween measured and numerically predicted normalized turbulence kinetic energy along thecentral axis is shown. One can observe the same trend betweenexperimentally measured val-ues and the data, obtained by numerical modelling, since thelatter significantly underpredictedthe level of fluctuations inside the recirculation zone. This indicates that the vortex shedding ismuch stronger in the physical experiment and, in spite of that it can be considered as plane, hasdeep three-dimensional nature.

A numerically predicted frequency of von Karman vortex shedding matched quite well to ex-

Figure 6: Comparison of time-averaged normalisedx-velocity (a) and turbulence kinetic energy (b) pro-files in five cross sections with axial coordinates of -2.5, 0.375, 1.525, 3.75 and 9.4 and, which wasobtained experimentally (1) and numerically (OpenFOAM) at baseline quad (2) and triangular (3) grids,with id = 1 andid = 3, from Tables 5–6, respectively. Results comparison of the measured (1) time-averaged normalisedx-velocity (c) and turbulence kinetic energy (d) profiles in five cross sections withaxial coordinates of -2.5, 0.375, 1.525, 3.75 and 9.4 against numerical predictions carried out with Open-FOAM (2) and FLUENT (3) at the baseline triangular grid withid = 3, from Tables 5–6. Experimentaldata were taken from [27].

Figure 7: Comparison of time-averaged normalisedx-velocity (a) and turbulence kinetic energy (b) pro-files near the bluff-body and throughout the computation domain in the channel midsection, which wasobtained experimentally (1) and numerically using OpenFOAM (2-3) and FLUENT (4). Simulationswere done for baseline quad (2) and triangular (3,4) grids withid = 1 andid = 3, from Tables 5–6,respectively. Time history of normalised instantaneousy-velocity numerically predicted at all designedgrids (c), all curves correlate with gridids according to Tables 5–6, curve 5 – FLUENT run with gridid = 3. The reference point was located at the same position as in the lab test of Sjunnesson et al. [27]with the following coordinates:x = 1, y = 0.

perimental data:f = 117 Hz (OpenFOAM) andf = 118 Hz (FLUENT) vs.f = 105 Hz inexperimental data, and St= 0.28 vs. St= 0.25, respectively, which are differences of about10%. These results correspond well also with those obtained by Hasse et al. [9]:f = 122 Hzandf = 117 Hz peak frequencies for URANS and DES, respectively.

Although the accuracy of the pressure loss measurements wasnot too high, as it was mentionedin [27], it is still of interest in many engineering applications. The experimental value for

the normalized pressure drop was measured (between axial statesx = −5 andx = 10) to∆P = 0.6 while the numerically predicted value was∆P ≈ 0.7 (both for OpenFOAM andFLUENT), which correspond quite well between each other.

It should be noticed that unstructured grids with triangular and quadrilateral cell types wereused, but with the same viscous BLs. The comparison provided in Fig. 6 for the mean axialvelocity and turbulence kinetic energy show very small (about 1 − 2%) errors between thenumerical solutions. Fig. 7(c) time history of the normalisedy-velocity measured at the sameprobe location and obtained for all the meshes in this work ispresented. The distribution of thecurves confirms the mesh independence of the solutions sincethe variations between them arequite small (about±5% both for the frequencies and amplitudes).

Discussion

The turbulent fully developed bluff-body flows replicated Fujii and Volvo lab test conditionshave been analyzed using the conventional URANS approach, but without the incompressibil-ity assumption. Since a limited number of problem-related articles were found in literature, andall cited numerical analysis were carried out with incompressible flow assumption, we can re-gard that these data can be considered as the first one where a compressible URANS approachhas been applied for these test cases. The influence of ‘discretized’ errors on solutions were val-idated and confirmed by an independence from grids topology and applied numerical schemesand factors (CFD codes). Two different implementation ofk − ǫ turbulence model was used tocheck the ‘modeling’ or ‘systematic’ error influence.

Integral measured and numerically predicted flow parameters were summarized in Table 7.Predicted integral flow parameters are in a satisfactory agreement with experimental values.The compliance between them are in the range of10 − 20%. Moreover, these discrepanciesare in approximately the same range as, for example the data predicted and reported recentlyby Hasse et al. [9] using DES. Strelets [28] reported resultscomputed at the basis of URANSand DES for a triangular cylinder in a plane channel. Strelets concluded that the time-averagedforces and other mean-flow characteristics between URANS andDES were not highly differentand explained it possibly due to the geometrical restrictions caused by the channel walls.

Table 7: Integral parameters of Fujii and Volvo lab test simulations.

Contributors Year Method B Re M −Cp,min Lr StSullerey [29] 1975 Exp 0.25 11500 0.03 2.87 2.3 0.18Fujii [5, 6] 1981 Exp 0.5 17500 0.03 2.73 2.2 0.40

Current work 2010 URANS-LSKE 0.5 17500 0.03 2.7 2.67 0.45

Sjunnesson et al. 1991 Exp 1/3 45000 0.05 – 1.33 0.250Johansson et al. 1993 URANS-SKE 1/3 45000 – – 1.30 0.270

Durbin 1995 URANS-KEV2 1/3 45000 – – 1.30 0.285Strelets 2001 URANS-SA 1/3 45000 – – 0.90 –

Hasse et al. 2009 URANS-SST 1/3 45000 – – 0.94 0.296

Current work 2010 URANS-LSKE 1/3 45000 0.05 – 1.36 0.28Current work 2010 URANS-RKE 1/3 45000 0.05 – 1.3 0.28

Conclusion

1. The generalized numerical setup for solving compressible URANS equations (based ona factorised FVM with global second-order accuracy both in space and in time, modi-fied Launder and Sharma low-Reynolds-numberk − ǫ turbulence model, PISO pressure-velocity coupling algorithm, GAMMA differencing scheme for convective terms approx-imation, BDF2 formula with dynamic time stepping technique for time integration andNRBC) was established.

2. The proposed methodology was preliminary tested for the plane laminar compressibleflow over circular cylinder. Predicted main flow parameters were very well in compliance(the deviations less than5%) with the experimental data and other numerical results.

3. The numerical method was validated in detail for two selected plane turbulent bluff-bodyflows with excessive measured data of Fujii et al. [5, 6] and Sjunnesson et al. [27]. Thelatter is quite popular among researchers, and several articles were found in literaturewhere this test was used for a verification of quite close numerical procedures. But it isimportant to notice that all the studies found were done in anincompressible formulation.No results were found for the Fujii lab test. We can consider that this paper is the firstone where the conventional URANS approach has been utilized for numerical modellingand replication these lab tests in a compressible formulation (with weak compressibility,since the Mach numbers were limited as M≤ 0.05 in the mentioned experiments).

4. All numerical results presented and discussed in this paper showed a non-significant (lessthan5%) sensitivity from the different grid topologies. The mesh independence study wascarried out to demonstrate mesh independence solutions forall test cases. Furthermore,both OpenFOAM and FLUENT technolgies demonstrated similarresults of analysed testproblems.

5. Overall, satisfactory agreement (with deviations of10 − 20%) between numerical andmeasured data for integral and local flow characteristics was achieved for the selectedbenchmarks, which indicates on the adequacy and accuracy ofthe established general-ized numerical setup.The described numerical methods can be used for calculating un-steady separation flows and the simplified quasi-two-dimensional approach can be usedfor numerical representation of a three-dimensional vortex flow in a wake.

Acknowledgment

This work was conducted as a part of the CenBio Center for environmentally-friendly energy.We are very appreciated to the Norwegian Meta center for Computational Science (NOTUR)for providing the uninterrupted HPC computational resources and the useful technical support.

Notation

∆P Static pressure drop,∆P = 2∆P/(

ρu2

inf

)

.

ǫ Dissipation rate of turbulence kinetic energy, [m2/s3].

γ NVD GAMMA differencing scheme parameter.

κ von Karman constant.

µ, µτ Dynamic molecular and turbulence viscosity, [kg/m/s].

ω Specific rate of turbulence energy dissipation, [Hz].

ρ Mass density, [kg/m3].

−Cpb Mean base suction coefficient.

B Channel blockage ratio,B = D/H.

Cµ Constant,Cµ = 0.09.

Cd Drag coefficient.

Cl Lift coefficient.

Cp Mean (time-averaged) pressure coefficient,Cp = 2(P − Pinf )/(

ρu2

inf

)

.

Cǫ1, Cǫ3 Constants in production and sink terms ofǫ equation.

D Bluff-body diameter or base, [m].

F First row size in the boundary layer, [m].

G Growth factor of the cells in the boundary layer.

H Channel height, [m].

J The number of rows in the BL.

K Normalised turbulence kinetic energy,K =√

4/3k/uinf .

L Channel length, [m].

Lr Recirculation zone length, normalized byD.

N The number of cells (intervals).

P Static pressure, [Pa].

R Bluff-body half-diameter or base(R = D/2), [m].

T Temperature, [K].

U Velocity, normalized byuinf .

f von Karman vortex shedding frequency, [Hz].

f2 Function in sink term ofǫ equation.

id Mesh identification number.

k Turbulence kinetic energy, [m2/s2].

t∗ Dimensionless time,t∗ = tuinf/D.

u Velocity, [m/s].

x,y Axial and vertical direction, normalized byD.

y∗ Dimensionless distance from the wall.

z Axial direction, normalized by recirculation zone length.

CFL Courant number

M Mach number

Re Reynolds number

St Strouhal number

φ Mean (or time-averaged) value ofφ.

φ′

Fluctuation component of a valueφ.

inf Value in an incoming flow.

min Minimum value.

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