Advances in Computational Fluid Dynamic

8/20/2019 Advances in Computational Fluid Dynamic

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Modelling and Simulation in Engineering

Guest Editors: Guan Heng Yeoh, Chaoqun Liu, Jiyuan Tu,

and Victoria Timchenko

Advances in Computational Fluid

Dynamics and Its Applications


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Advances in Computational Fluid Dynamics andIts Applications


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Modelling and Simulation in Engineering

Advances in Computational Fluid Dynamics andIts Applications

Guest Editors: Guan Heng Yeoh, Chaoqun Liu, Jiyuan Tu,

and Victoria Timchenko


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Copyright © 2011 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in volume 2011 of “Modelling and Simulation in Engineering.” All articles are open access articlesdistributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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Editorial Board

K. Al-Begain, UKAdel M. Alimi, Tunisia

Zeki Ayag, Turkey

Andrzej Bargiela, UK

Philippe Boisse, France

Agostino Bruzzone, Italy

Hing Kai Chan, UK

Andrzej Dzielinski, PolandXiaosheng Gao, USA

F. Gao, UK

Ricardo Goncalves, Portugal

Ratan K. Guha, USA

Chung-Souk Han, USA

Joanna Hartley, UKJing-song Hong, China

MuDer Jeng, Taiwan

Iisakki Kosonen, Finland

Laurent Mevel, France

A. Mohamed, Malaysia

Antonio Munjiza, UK

Petr Musilek, CanadaTomoharu Nakashima, Japan

Gaby Neumann, Germany

Alessandra Orsoni, UK

Javier Otamendi, Spain

Evtim Peytchev, UK

Luis Carlos Rabelo, USA

Ahmed Rachid, FranceSellakkutti Rajendran, Singapore

Waleed Smari, USA

Boris Sokolov, Russia

Rajan Srinivasan, India

S. Taib, Malaysia

Andreas Tolk, USABauke Vries, The NetherlandsFarouk Yalaoui, France

Ivan Zelinka, Czech Republic

Qing-An Zeng, USA


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Contents

Advances in Computational Fluid Dynamics and Its Applications, Guan Heng Yeoh, Chaoqun Liu,

Jiyuan Tu, and Victoria Timchenko

Volume 2011, Article ID 304983, 3 pages

New Findings by High-Order DNS for Late Flow Transition in a Boundary Layer, Chaoqun Liu, Lin Chen,

and Ping Lu


Implicit LES for Supersonic Microramp Vortex Generator: New Discoveries and New Mechanisms,

Qin Li and Chaoqun Liu


An Initial Investigation of Adjoint-Based Unstructured Grid Adaptation for Vortical Flow Simulations,

Li Li


Assessment of Turbulence Models for Flow Simulation around a Wind Turbine Airfoil,

Fernando Villalpando, Marcelo Reggio, and Adrian Ilinca


Computation of Ice Shedding Trajectories Using Cartesian Grids, Penalization, and Level Sets,

Héloı̈se Beaugendre, Franois Morency, Federico Gallizio, and Sophie Laurens


Aerodynamic Optimization of an Over-the-Wing-Nacelle-Mount Configuration, Daisuke Sasaki and

Kazuhiro Nakahashi


New Evaluation Technique for WTG Design Wind Speed Using a CFD-Model-Based Unsteady Flow

Simulation with Wind Direction Changes, Takanori Uchida, Takashi Maruyama, and Yuji Ohya


Comparisons between the Wake of a Wind Turbine Generator Operated at Optimal Tip Speed Ratio andthe Wake of a Stationary Disk , Takanori Uchida, Yuji Ohya, and Kenichiro Sugitani


Experimental and Numerical Simulations Predictions Comparison of Power and Efficiency in HydraulicTurbine, Laura Castro, Gustavo Urquiza, Adam Adamkowski, and Marcelo Reggio


Simulating Smoke Filling in Big Halls by Computational Fluid Dynamics, W. K. Chow, C. L. Chow,

and S. S. Li


Numerical Simulations for a Typical Train Fire in China, W. K. Chow, K. C. Lam, N. K. Fong, S. S. Li,

and Y. Gao


Simulation of Pharyngeal Airway Interaction with Air Flow Using Low-Re Turbulence Model,

M. R. Rasani, K. Inthavong, and J. Y. Tu



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CFD-Guided Development of Test Rigs for Studying Erosion and Large-Particle Damage of Thermal

Barrier Coatings, Maria A. Kuczmarski, Robert A. Miller, and Dongming Zhu


Investigation of Swirling Flows in Mixing Chambers, Jyh Jian Chen and Chun Huei Chen


Numerical Computation and Investigation of the Characteristics of Microscale Synthetic Jets,

Ann Lee, Guan H. Yeoh, Victoria Timchenko, and John Reizes


Recent Eff

orts for Credible CFD Simulations in China, Li Li, Bai Wen, and Liang YihuaVolume 2011, Article ID 861272, 8 pages


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Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2011, Article ID 304983, 3 pagesdoi:10.1155/2011/304983

Editorial Advances in Computational Fluid Dynamics and Its Applications

Guan Heng Yeoh,1, 2 Chaoqun Liu,3 Jiyuan Tu,4 and Victoria Timchenko2

1 Australian Nuclear Science Technology Organisation (ANSTO), Locked Bag 2001, Kirrawee, Sydney, NSW 2233, Australia 2 School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW 2052, Australia3 Center for Numerical Simulation and Modeling, Department of Mathematics, The University of Texas at Arlington, Arlington,

TX 76019-0408, USA4 School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, P.O. Box 71,

Bundoora, Melbourne, VIC 3083, Australia

Correspondence should be addressed to Guan Heng Yeoh, [email protected]

Received 14 October 2011; Accepted 17 October 2011

Copyright © 2011 Guan Heng Yeoh et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Computational fluid dynamics, better known by CFD, iscertainly very prevalent in many fields of engineeringresearch and application. There is no doubt that we arecertainly witnessing a renaissance of computer simulationtechnology where the ever-changing landscape is as a resultof the rapid evolution of CFD techniques and models andthe decreasing computer hardware costs accompanied by faster computing times. In spite of the many significantachievements, there remains much concerted developmentand advancement of CFD to meet the increasing demandsbolstered from various emerging industries such as biomedi-cal and bioengineering, uncharted areas in process, chemical,civil, and environmental engineering as well as traditionally renowned high-technology engineering areas in aeronauticsand astronautics and automotive. In this special issue,we believe that significant coverage on the advances andapplications of CFD has been achieved via the selected

topics and papers which incidentally represent a good paneladdressing the many diff erent aspects of CFD in this specialissue. Of course, we realise that these selected topics andpapers are not an exhaustive representation. Nevertheless,they still represent the rich and many-faceted knowledge thatwe have the great pleasure of sharing with the readers. Wecertainly would like to thank the numerous authors for theirexcellent contributions and patience in assisting us. Moreimportantly, the tiresome work of all referees on reviewingthese papers is very warmly acknowledged.

This special issue contains sixteen papers. The advancesand wide applications of CFD in each of the papers aredetailed in the following.

In the first paper, “ New findings by high-order DNS for late flow transition in a boundary layer ,” C. Liu et al. present asummary of new discoveries by direct numerical simulation(DNS) for late stages of flow transition in a boundary layer.Based on the new findings, ring-like vortex is found to bethe only form existing inside the flow field and the resultof the interaction between two pairs of counter-rotatingprimary and secondary streamwise vortices. Following firstHelmholtz vortex conservation law, the primary vortex tuberolls up and is stretched due to the velocity gradient. In orderto maintain vorticity conservation, a bridge must be formedto link twoΛ-vortex legs. The bridge finally develops as a new ring. This process proceeds to form a multiple ring structure.U-shaped vortices are not new but existing coherent vortex structure. U-shaped vortex, which is a third level vortex,serves as a second neck to supply vorticity to the multiplerings. The small vortices can be found on the bottom of

the boundary layer near the wall surface. It is believed thatthe small vortices, and thus turbulence, are generated by theinteraction of positive spikes and other higher level vorticeswith the solid wall.

In the second paper, “Implicit LES for supersonic micro-ramp vortex generator: new discoveries and new mechanisms,”Q. Li and C. Liu present the application of an implicitly large eddy simulation (ILES) via fifth-order WENO schemeto study the flow around the microramp vortex genera-tor (MVG) at Mach 2.5 and Reθ = 1440. A numberof new discoveries on the flow around supersonic MVGhave been made including spiral points, surface separationtopology, sourceof the momentum deficit, inflection surface,


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2 Modelling and Simulation in Engineering

Kelvin-Helmholtz instability, vortex ring generation, ring-shock interaction, 3D recompression shock structure, andinfluence of MVG decline angles. A new 5-pair-vortex-tubemodel near the MVG is given based on the ILES observation.The vortex ring-shock interaction is found as the new mechanism of the reduction of the separation zone induced

by the shock-boundary layer interaction.In the third paper, “ An initial investigation of adjoint-based unstructured grid adaptation for vortical flow simula-tions,” L. Li presents a CFD method utilizing unstructuredgrid technology to compute vortical flow around a 65

delta wing with sharp leading edge, specially known as thegeometry of the second international vortex flow experiment(VFE-2). The emphasis of this paper is to investigate theeff ectiveness of an adjoint-base grid adaptation methodfor unstructured grid in capturing concentrated vorticesgenerated at sharp edges or flow separation lines of liftingsurfaces flying at high angles of attack. Earlier vortical flow simulations indicated that the vortex behaviour has beenfound to be highly dependent on the local grid resolutionboth on body surface and space. The basic idea of theadjoint-based adaptation method has been to construct anew adaptive sensor in a grid adaptation process with theintent to allocate where the elements should be smaller orlarger through the introduction of an adjoint formulation torelate the estimated functional error to local residual errorsof both the primal and adjoint solutions.

In the fourth paper, “ Assessment of turbulence models for flow simulation around a wind turbine airfoil,” F. Villalpandoet al. present the investigation of flows over a NACA 63–415 airfoil at various angles of attack. With the aim of selecting the most suitable turbulence model to simulateflow around ice-accreted airfoils, this work concentrates onassessing the prediction capabilities of various turbulencemodels on clean airfoils at the large angles of attack thatcause highly separated flows to occur. CFD simulationshave been performed employing the one-equation Spalart-Allmaras model, the two-equation RNG k- and SST k-ωmodels, and Reynolds stress model.

In the fifth paper, “Computation of ice shedding trajec-tories using cartesian grids, penalization, and level sets,” H.Beaugendre et al. present the modelling of ice sheddingtrajectories by an innovative paradigm based on Cartesiangrids, penalization, and level sets. The use of Cartesiangrids bypasses the meshing issue, and penalization is anefficient alternative to explicitly impose boundary conditions

so that the body-fitted meshes can be avoided, makingmultifluid/multiphysics flows easy to set up and simulate.Level sets describe the geometry in a nonparametric way so that geometrical and topological changes due to physicsand in particular shed ice pieces are straightforward tofollow. The capabilities of the proposed CFD model aredemonstrated on ice trajectories calculations for flow aroundiced cylinder and airfoil.

In the sixth paper, “ Aerodynamic optimization of anover-the-wing-nacelle-mount configuration,” D. Sasaki and K.Nakahashi present the CFD investigation of an over-the-wing-nacelle-mount airplane configuration which is knownto prevent the noise propagation from jet engines toward

ground. Aerodynamic design optimization is conducted toimprove aerodynamic efficiency in order to be equivalent toconventional under-the-wing-nacelle-mount configuration.The nacelle and wing geometry are modified to achieve highlift-to-drag ratio, and the optimal geometry is comparedwith a conventional configuration. Pylon shape is also

modified to reduce aerodynamic interference eff

ect. Thefinal wing-fuselage-nacelle model is compared with the DLR F6 model to discuss the potential of over-the-wing-nacelle-mount geometry for an environmental-friendly future air-craft.

In the seventh paper, “ New evaluation technique for WTGdesign wind speed using a CFD-model-based unsteady flow simulation with wind direction changes,” T. Uchida et al.present the development of a CFD model called RIAM-COMPACT, based on large eddy simulation (LES), that canpredict airflow and gas diff usion over complex terrain withhigh accuracy and simulates a continuous wind directionchange over 360 degrees. The present paper proposes atechnique for evaluating the deployment location of windturbine generators (WTGs) since a significant portion of the topography in Japan is characterized by steep, complex terrain thereby resulting in a complex spatial distribution of wind speed, in which great care is necessary for selecting asite for the construction of WTG.

In the eighth paper, “Comparisons between the wakeof a wind turbine generator operated at optimal tip speed ratio and the wake of a stationary disk,” T. Uchinda et al.present the investigation that the wake of a wind turbinegenerator (WTG) operated at the optimal tip speed ratio iscompared to the wake with its rotor replaced by a stationary disk. Numerical simulations are conducted with a largeeddy simulation (LES) model. The characteristics of thewake of the stationary disk are significantly diff erent fromthose of the WTG. The velocity deficit at a downstreamdistance of 10 D (D being the rotor diameter) behind theWTG is approximately 30 to 40% of the inflow velocity. Incontrast, flow separation is observed immediately behind thestationary disk (≤2 D), and the velocity deficit in the far wake(10 D) of the stationary disk is smaller than that of the WTG.

In the ninth paper, “Experimental and numerical sim-ulations predictions comparison of power and e fficiency inhydraulic turbine,” L. Castro et al. present the CFD simu-lations performed to principal components of a hydraulicturbine: runner and draft tube. On-site power and mass flow rate measurements have been conducted in a hydroelectric

power plant (Mexico). Mass flow rate has been obtainedusing Gibson’s water hammer-based method. Inlet boundary conditions for the runner have been obtained from aprevious simulation conducted in the spiral case. Computedresults at the runner’s outlet are used to conduct thesubsequentdraft tube simulation. Numerical results from therunner’s flow simulation provide data to compute the torqueand the turbine’s power. Power-versus-efficiency curves arebuilt.

In the tenth paper, “Simulating smoke filling in big hallsby computational fluid dynamics,” W. K. Chow et al. presentthe use of CFD in addressing the key hazard due to smokefilling within the enclosure in the event of a fire. This aspect


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primarily concerns the many tall halls of big space volumethat have been and to be built in many construction projectsin the Far East, particularly Mainland China, Hong Kong,and Taiwan. On the basis of the CFD simulations, it can beanticipated that better understanding of the smoke fillingphenomenon and design of appropriate smoke exhaust

systems can be realised.In the eleventh paper, “ Numerical simulations for a typical train fire in china,” W. K. Chow et al. present the useof CFD in addressing the fire safety in passenger trainsdue to big arson and accidental fires since railway is thekey transport means in China including the Mainland,Taiwan, and Hong Kong. The predicted results such as theair flow, temperature distribution, smoke layer height, andsmoke spread patterns inside a train compartment are usefulfor working out appropriate fire safety measures for trainvehicles and determining the design fire for subway stationsand railway tunnels.

In the twelfth paper, “Simulation of pharyngeal airway interaction with air flow using low-Re turbulence model ,”M. R. Rasani et al. present the CFD simulation of theinteraction between a simplified tongue replica with expi-ratory turbulent air flow. A three-dimensional model witha low-Re SST turbulence model is adopted. An Arbitrary Eulerian-Lagrangian description for the fluid governingequation is coupled with the Lagrangian structural solvervia a partitioned approach, allowing deformation of thefluid domain to be captured. Numerical simulations confirmexpected predisposition of apneic patients with narrowerairway opening to flow obstruction and suggest muchsevere tongue collapsibility if the pharyngeal flow regime isturbulent compared to laminar.

In the thirteenth paper, “CFD-guided development of test rigs for studying erosion and large-particle damage of thermal barrier coatings,” M. A. Kuczmarski et al. presentthe use of CFD to accelerate the successful developmentand continuous improvement of combustion burner rigs formeaningful materials testing. Rig development is typically an iterative process of making incremental modificationsto improve the rig performance for testing requirements.Application of CFD allows many of these iterations to bedone computationally before hardware is built or modified,reducing overall testing costs and time, and it can providean improved understanding of how these rigs operate. Thispaper focuses on the study of erosion and large-particledamage of thermal barrier coatings (TBCs) used to protect

turbine blades from high heat fluxes in combustion engines.The steps used in this study—determining the questions thatneed to be answered regarding the test rig performance,developing and validating the model, and using it topredict rig performance—can be applied to the efficientdevelopment of other test rigs.

In the fourteenth paper, “Investigation of swirling flowsin mixing chambers,” J. J. Chen and C. H. Chen presentthe CFD investigation of the three-dimensional momentumand mass transfer characteristics arising from multiple inletsand a single outlet in micromixing chamber. The chamberconsists of a right square prism, an octagonal prism, or acylinder. Numerical results have indicated that the swirling

flows inside the chamber dominate the mixing index. Particletrajectories demonstrate the rotational and extensional localflows which produce steady stirring, and the configuration of coloured particles at the outlet section expressed at diff erentReynolds numbers (Re) represented the mixing performancequalitatively. Eff ects of various geometric parameters and

Re on the mixing characteristics have been investigated.An optimal design of the cylindrical chamber with 4 inletsis found. At Re > 15, more inertia causes more powerfulswirling flows in the chamber, and the damping eff ect ondiff usion is diminished, which subsequently increases themixing performance.

In the fifteenth paper, “ Numerical computation and investigation of the characteristics of microscale synthetic jets,”A. Lee et al. present the CFD investigation of a synthetic jet generated as a result of the periodic oscillations of amembrane in a cavity. A novel moving mesh algorithm tosimulate the formation of jet is presented. The governingequations are transformed into the curvilinear coordinatesystem in which the grid velocities evaluated are then fedinto the computation of the flow in the cavity domain thusallowing the conservation equations of mass and momentumto be solved within the stationary computational domain.Numerical solution generated using this moving meshapproach is compared with experimental measurement.Comparisons between numerical results and experimentalmeasurement on the streamwise component of velocity profiles at the orifice exit and along the centerline of thepulsating jet in microchannel as well as the location of vortex core indicate that there is good agreement, thereby demonstrating that the moving mesh algorithm developedis valid.

In the sixteenth paper, “Recent e ff orts for credible CFDsimulations in china,” L. Li et al. present the initiation of a series of workshops on credible CFD simulations similarto activities in the West such as ECARP and AIAA DragPrediction Workshops. Another major eff ort in China isthe ongoing project to establish a software platform forstudying the credibility of CFD solvers and performingcredible CFD simulations. The platform, named WiseCFD,has been designed to implement a seamless CFD processand to circumvent tedious repeating manual operations. Aconcerted focus is also given on a powerful job manager forCFD with capabilities to support plug and play (PnP) solverintegration as well as distributed or parallel computations.Some future work on WiseCFD is proposed and also

envisioned on how WiseCFD and the European QNET-CFDKnowledge Base can benefit mutually.

Guan Heng YeohChaoqun Liu

Jiyuan TuVictoria Timchenko


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Research ArticleNew Findings by High-Order DNS for Late Flow Transition ina Boundary Layer

Chaoqun Liu,1 Lin Chen,1, 2 and Ping Lu1

1 Mathematics Department, University of TX at Arlington, Arlington, TX 76019, USA 2 Aerodynamics Department, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Chaoqun Liu, [email protected]

Received 6 October 2010; Accepted 24 November 2010

Academic Editor: Guan Yeoh

Copyright © 2011 Chaoqun Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper serves as a summary of new discoveries by DNS for late stages of flow transition in a boundary layer. The widely spreadconcept “vortex breakdown” is found theoretically impossible and never happened in practice. The ring-like vortex is found theonly form existing inside the flow field. The ring-like vortex formation is the result of the interaction between two pairs of counter-rotating primary and secondary streamwise vortices. Following the first Helmholtz vortex conservation law, the primary vortex tube rolls up and is stretched due to the velocity gradient. In order to maintain vorticity conservation, a bridge must be formed tolink twoΛ-vortex legs. The bridge finally develops as a new ring. This process keeps going on to form a multiple ring structure. TheU-shaped vortices are not new but existing coherent vortex structure. Actually, the U-shaped vortex, which is a third level vortex,serves as a second neck to supply vorticity to the multiple rings. The small vortices can be found on the bottom of the boundary

layer near the wall surface. It is believed that the small vortices, and thus turbulence, are generated by the interaction of positivespikes and other higher level vortices with the solid wall. The mechanism of formation of secondary vortex, second sweep, positivespike, high shear distribution, downdraft and updraft motion, and multiple ring-circle overlapping is also investigated.

1. Introduction

The transition process from laminar to turbulent flow inboundary layers is a basic scientific problem in modern fluidmechanics and has been the subject of study for over acentury. Many diff erent concepts for the explanation of the

mechanisms involved have been developed based on numer-ous experimental, theoretical, and numerical investigations.After over a century study on the turbulence, the linear andearly weaklynonlinear stages of flow transition arepretty wellunderstood. However, for late non-linear transition stages,there are still many questions remaining for research [1–5].

Ring-like vortices play a key role in the flow transition.They generate rapid downward jets (second sweep) whichinduce the positive spike and bring the high energy to theboundary layer and work together with the upward jets(ejection) to mix the boundary layer. In other words, thering-like vortex formation and development are a key topicfor late flow transition in boundary layer. It appears that

there is no turbulence without ring-like vortices. It is naturalto give comprehensive study on the ring formation anddevelopment.

1.1. Ring-Like Vortex Formation. Hama and Nutant [6]described the process of formation of “Ω-shaped vortex”

in the vicinity of the Λ-vortex tip. They found that “. . .asimplified numerical analysis indicates that the hyperbolic vortex filament deforms by its own induction into a milk-bottleshape (the “Ω-vortex ”) and lifts up its tip. . .”. Later, Moin et al.[7] used numerical method with the Biot-Savart law andconcluded that the ring is formed through a mechanism of self-induction and the multiple ring formation is a resultof first ring “pinching off .” They believe that the solutionof Navier-Stokes equation does not change the mechanismbut may delay the ring “pinching off ”. However, they may besomewhat right, but what they discussed is not relevant to theboundary layer transition since the ring-like vortex is formedin a boundary layer and viscosity and full3D time-dependent


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x

z

o

y

Figure 1: Computation domain.

p = 0 1 2 3 n − 1

η

ξ

ζ

Figure 2: The domain decomposition along streamwise directionin the computational space.

Navier-Stokes equations must be considered. In addition,our new DNS results show, the ring is a perfect circle nota deformed shape, the ring stands perpendicularly with 90degree not 45 degree, and, more important, “ring pinch off .”

is never found. The ring-like vortex has its own coherentstructure as we show below, which is quite diff erent fromwhat Moin et al. have shown [7].

1.2. Multiple Ring Formation. Crow theory [8] has beenconsidered as the mechanism of multiple ring formation [2].They pointed out that “the formation of a set of vortex ringsfrom two counter-rotating vortices is usually called “Crow instability.” This instability could be involved at the stage of formation of the ring-like vortices due to interaction of theΛ-vortex legs.” However, there is no evidence to prove such amechanism. We cannot find any links between Crow theory and multiple ring formation. Apparently, Crow theory is not

related to multiple ring formation in a boundary layer.

1.3. Vortex Breakdown. “Vortex Breakdown” has been widely spread in many research papers and textbooks [5, 9–11].However, our analysis found that “Vortex Breakdown” is the-oretically impossible. In practice, “Vortex Breakdown” cannever happen. It is found that “Vortex Breakdown” describedby exiting publications [5, 9, 10] is either misinterpreted by 2D visualization or by using the pressure center as the vortex center.

1.4. U-Shaped Vortex. U-shaped vortex [9] or barrel-shapedwave [5] is still a mystery and thought as a newly formed

Transitional flow (DNS result)

Laminar flow

Turbulent flow (Cousteix ,1989)

1000800600400

x Grids: 960 × 64 × 121

0

0.001

0.002

0.003

0.004

0.005

C f

Figure 3: Streamwise evolution of the time- and spanwise-averagedskin-friction coefficient.

secondary vortex or waves. However, it is found by our new DNS that the U-shaped vortex was an original coherentstructure and it is a tertiary (not secondary) vortex with samevorticity sign as the ring legs. The U-shaped vortices serve as

a second neck to supply vorticity to the rings and keep theleading ring alive.

1.5. High Shear Distribution. The high shear (HS) distribu-tion can be found in the paper by Bake et al. [3], but it isquestionable why the HS disappears near the neck where theflow is very active. However, our new DNS found that there isno normal direction velocity component and then no sweepsand ejections in that area.

Anyway, there are many questions, which are relatedto late flow transition, that have not been answered ormisinterpreted. In order to get deep understanding onthe late flow transition in a boundary layer, we recently conducted a large grid high-order DNS with 1920 × 241 ×128 gird points and about 600,000 time steps to study themechanism of the late stages of flow transition in a boundary layer. Many new findings are made, and new mechanisms arerevealed. Here, we use the λ2 criterion [12] for visualization.

The paper is organized in the following way: Section 2presents the governing Navier-Stokes equations and numer-ical approaches; Section 3 shows the code verification andvalidation to make sure that the DNS code is correct;Section 4 provides the new DNS results and summarizesthe new discoveries and new mechanism of formation andfurther development of ring-like vortices. A number of conclusions are made in Section 5.


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x = 300.79x = 632.33

Linear law

Log law

103102101100

z +

0

10

20

30

40

50

u +

(a) Coarse grids (960 × 64 × 121)

x = 300.79x = 975.14

Linear law

Log law

103102101100

z +

0

5

1015

20

25

30

35

u +

(b) Fine grids (1920 × 128 × 241)

u+ = y +u+ = 1 /κ log y + + C DNS

102101100

y +

0

5

10

15

20

u +

(c) Singer and Joslin [9]

Figure 4: Log-linear plots of the time- and spanwise-averaged velocity profile in wall unit.

Table

1: Flow parameters. M ∞ Re x in Lx Ly Lz in T w T ∞0.5 1000 300.79δ in 798.03δ in 22δ in 40δ in 273.15 K 273.15 K

2. Governing Equations andNumerical Methods

The governing equations we used are the three-dimensionalcompressible Navier-Stokes equations in generalized curvi-linear coordinates and in a conservative form:

1

J

∂Q

∂t +

∂(E − Ev )∂ξ

+ ∂(F − F v )

∂η +

∂(H − H v )∂ζ =

0. (1)

The vector of conserved quantities Q, inviscid flux vector(E, F , G), and viscous flux vector (Ev , F v , Gv ) are defined as

Q =

ρ

ρu

ρv

ρw

e

, E = 1

J

ρU

ρuU + pξ x

ρvU + pξ y

ρwU + pξ z

U

e + p

,

F = 1 J

ρV

ρuV + pηx

ρvV + pη y

ρwV + pηz

V

e + p

, H = 1

J

ρW

ρuW + pζ x

ρvW + pζ y

ρwW + pζ z

W

e + p

,

Ev = 1 J

0

τ xx ξ x + τ yx ξ y + τ zx ξ z

τ xy ξ x + τ y y ξ y + τ zy ξ z

τ xz ξ x + τ yz ξ y + τ zz ξ z

qx ξ x + q y ξ y + qz ξ z

,

F v = 1 J

0

τ xx ηx + τ yx η y + τ zx ηz

τ xy ηx + τ y y η y + τ zy ηz

τ xz ηx + τ yz η y + τ zz ηz

qx ηx + q y η y + qz ηz

,

H v = 1 J

0

τ xx ζ x + τ yx ζ y + τ zx ζ z

τ xy ζ x + τ y y ζ y + τ zy ζ z

τ xz ζ x + τ yz ζ y + τ zz ζ z

qx ζ x + q y ζ y + qz ζ z

,

(2)

where J is the Jacobian of the coordinate transformationbetween the curvilinear (ξ , η, ζ ) and Cartesian (x , y , z )frames and ξ x , ξ y , ξ z , ηx , η y , ηz , ζ x , ζ y , ζ z are coordinate trans-formation metrics. The contravariant velocity componentsU , V , W are defined as U

= uξ x + vξ y + wξ z , V

= uηx +

vη y + wηz , W = uζ x + vζ y + wζ z ; e is the total energy. Thecomponents of viscous stress and heat flux are denoted by τ xx , τ y y , τ zz , τ xy , τ xz , τ yz , and qx , q y , qz , respectively.

In the dimensionless form of (1), the reference valuesfor length, density, velocities, temperature, and pressure areδ in, ρ∞, U ∞, T ∞, and ρ∞U 2∞, respectively. The Mach numberand Reynolds number are expressed as

M ∞ = U ∞γRT ∞

, Re = ρ∞U ∞δ inµ∞

, (3)

where R is the ideal gas constant, γ the ratio of specific heats,δ in the inflow displacement thickness, and µ∞ the viscosity.


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5101520

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500

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460

440

420

400051015

200

3

6

9

12

(a) t = 6T (b) t = 6.2T

(c) t = 6.4T (d) t = 7.0T

(a)

Where T is period of T-S wave

1 ↑

1 ↑

3 ↑ 1 ↑

1 ↑2 ↑3 ↑4 ↑

(a)

(b)

(c)

(d)

(e)

(f)

(b)

Figure 5: (A) The evolution of vortex structures at the late stage of transition (where T is period of T-S wave). (B) Evolution of type ring-likevortices chain by experiment [13].


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+

−−−−−(a)

+

++−−(b)

Figure 6: Sketch of secondary vortex and third level vorticity formation.

7.576.565.554.543.5

y

0

0.5

1

1.5

2

2.5

3

z

(a) Velocity profile induced by prime vortex

7.576.565.554.543.5

y

0

0.5

1

1.5

2

2.5

3

z

(b) Velocity profile change direction

Figure 7: DNS results of secondary vortex and third level vorticity formation.

Z

X Y

450

445

440435

x

2.842 2.848 2.854 2.86 2.866 p:

Figure 8: Pressure distributions in the evolution of vortex structure(t = 6.2T )

A sixth-order compact scheme [14] is used for the spatialdiscretization in the streamwise and wall normal directions.For internal points j = 3, . . . , N − 2, the sixth-order compactscheme is as follows:

13

f j−1 + f

j + 13

f j+1

= 1h

− 1

36 f j−2 −

7

9 f j−1 +

7

9 f j+1 +

1

36 f j+2

,

(4)

where f j is the derivative at point j. The fourth ordercompact scheme is used at points j = 2, N −1, and the thirdorder one-sided compact scheme is used at the boundary points j = 1, N .

In the spanwise direction where periodical conditionsare applied, the pseudospectral method is used. In orderto eliminate the spurious numerical oscillations caused by central diff erence schemes, a high-order spatial scheme is


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X

Z

Y

480470

460

450

440

430

420

(a)

X

Z

Y

480470

460

450

440

430

420

(b)

Figure 9: Sketch of the relationship of ring-like vortices and streamwise vortices.

X

Z

Y

−2

−1−0.182−0.106−0.02900.0110.0270.0520.070.51.5

ωx

(a)

X

Z

Y

−2

−1−0.182−0.106−0.02900.0110.0270.0520.070.51.5

ωx

(b)

Figure 10: Formation process of ring-like vortices.

Y X

Z Z

X Y X Y

Z

Figure 11: Four-vortex tube and ring-like vortices from three diff erent view angles. Ring like vortex given by our DNS.


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X

Z

Y

10.90.80.70.60.50.40.30.20.10490485480475470

x U/U e

2

4

6

8

10

Z

U = 0.99U e

Figure 12: The position of ring-like vortices in boundary layer (z = 3.56, U = 0.99U e)

(a) (b) (c)

Figure 13: Crow theory about vortex ring chain formation (provided by Dr. Kachanov at FTT09 Short Course).

used instead of artificial dissipation. An implicit sixth-ordercompact scheme for space filtering is applied to the primitivevariables u, v , w, ρ, p after a specified number of time steps.

The governing equations are solved explicitly by a 3rd-order TVD Runge-Kutta scheme [15]:

Q(0)

=Qn,

Q(1) = Q(0) + ∆tR(0),

Q(2) = 34

Q(0) + 1

4Q(1) +

1

4∆tR(1),

Qn+1 = 13

Q(0) + 2

3Q(2) +

2

3∆tR(2),

(5)

CFL ≤ 1 is required to ensure the stability.The adiabatic and the nonslipping conditions are

enforced at the wall boundary on the flat plate. On thefar field and the outflow boundaries, the nonreflectingboundary conditions [16] are applied.

The inflow is given in the form of

q = qlam + A2d q2d ei(α2d x −ωt ) + A3d q3d ei(α3d x ± β y −ωt ), (6)where q represents u, v , w, p, and T , while qlam is the Blasiussolution for a two-dimensional laminar flat plate boundary layer. The streamwise wavenumber, spanwise wavenumber,frequency, and amplitude are

α2d = 0.29919 − i5.09586 × 10−3, β = ±0.5712,

ω = 0.114027, A2d = 0.03, A3d = 0.01,(7)

respectively. The T-S wave parameters are obtained by solv-ing the compressible boundary layer stability equations [17].

The computational domain is displayed in Figure 1. Thegrid level is 1920 × 128 × 241, representing the numberof grids in streamwise ( x ), spanwise ( y ), and wall normal(z ) directions. The grid is stretched in the normal directionand uniform in the streamwise and spanwise directions. Thelength of the first grid interval in the normal direction at theentrance is found to be 0.43 in wall unit (Y + = 0.43).


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Λ-vortex

Ring-like vortex Ring-like vortex

Bridge

Ring-like vortex

Hairpinvortex

New bridgegeneration

(a)

X

Z

Y

560

550540

530

520510

500490

480470460

450440

430

x

(b)

Figure 14: (a) Sketch for mechanism of multi-ring generation. (b) Multiple ring formation.

1

2

3

4Inlinesecondary

hairpin vortex

Side-lobesecondary

hairpin vortex

New U-shaped

vortex

Low pressure

High pressure

Figure 15: U-shaped vortex [9].

The parallel computation is accomplished through theMessage Passing Interface (MPI) together with domaindecomposition in the streamwise direction. The computa-tionaldomainis partitioned into N equally sizedsubdomainsalong the streamwise direction (Figures 1 and 2). N is thenumber of processors used in the parallel computation. Theflow parameters, including Mach number, Reynolds numberare listed in Table 1.

590580570560550

X Z

Y

Figure 16: Our new DNS with a diff erent λ-value.

3. Verification and Validation

The skin friction coefficient calculated for the time- andspanwise-averaged profile is displayed in Figure 3. The spatialevolution of skin friction coefficients of laminar flow is alsoplotted out for comparison. It is observed from these figuresthat the sharp growth of the skin-friction coefficient occursafter x ≈ 450δ in, which is defined as the “onset point”. Theskin friction coefficient after transition is in good agreementwith the flat-plate theory of turbulent boundary layer by Cousteix in 1989 [18].


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590580570560550

X Z

Y

(a) Top view

580570560550

X

Z

Y

(b) Side view

580570

560550

X

Z Y

(c) Angle view

Figure 17: View of young turbulence spot head from diff erent directions (t = 8.8T ).

X

Z

Y

550

530

510

490

470

450

430

x

(a) Angle view

x

580570560550540530520510500490480470460450

X Z

Y

(b) Top view

Figure 18: 3D angle view and 2D top view of the young turbulence spot (t

=8.8T ).

Time- and spanwise- averaged streamwise velocity pro-files for various streamwise locations in two diff erent gridlevels are shown in Figure 4. The inflow velocity profiles at

x = 300.79δ in is a typical laminar flow velocity profile. At x =632.33δ in, the mean velocity profile approaches a turbulentflow velocity profile (Log law). Note that there are quite few publications (e.g., Figure 4(c)) which have large discrepancy in velocity profile with Log law.

A vortex identification method introduced by Jeong and

Hussain [12] is applied to visualize the vortex structuresby using an iso-surface of a λ2-eigenvalue. The vortex coresare found by the location of the inflection points of thepressure in a plane perpendicular to the vortex tube axis. The

pressure inflection points surround the pressure minimumthat occurs in the vicinity of the vortex core. By thisvisualization method, the vortex structures shaped by the

nonlinear evolution of T-S waves in the transition process areshown in Figure 5(A). The evolution details are studied in

our previous papers [19–23], and the formation of ring-likevortices chains is consistent with the experimental work (see [13], Figure 5(B)) and previous numerical simulationby Bake et al. [3].

4. New Discoveries and MechanismsRevealed by Our DNS

4.1. Mechanism on Secondary Streamwise Vortex Formation.As shown in Figure 6(a), the primary vortex induces a back-ward velocity which generates negative vorticity by the wallsurface due to the zero velocity on the wall surface. However,negative vorticity generation does not mean formation of thesecondary vortex tube. The secondary vortex tube must begenerated by the flow separation. This means that the flow

direction near the wall must change the sign from backwardto forward (Figure 6(b)). This process is directly shown by the DNS results in Figures 7(a) and 7(b). The separation canonly be done by pressure gradients. The question is where thepressure gradients come from. From Figure 8, we can clearly find that the streamwise vortex center is the low pressurecenter and the pressure outside the primary vortex tube ismuch higher than the center of the primary vortex center.Also from Figure 8, it is shown that the prime vortex changesthe streamwise pressure gradient. These pressure gradientschange the flow direction from backward to forward, and thesecondary vortex tube forms, separates from the wall surface,and rises up.


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X

Z Y

560

540

520

500

480

460

440

420

x

(a) t = 7.0T

X

Z Y

560

540

520

500

480

460

440

420

x

(b) t = 7.5T

X

Z Y

560

540

520

500

480460

440

420

x

(c) t = 8.0T

X

Z Y

560

540520

500

480

460

440

x

(d) t = 8.5

X

Z Y

580

560

540

520

500

480

460

x

(e) t = 8.8T

Figure 19: Small length scale vortex generation at diff erent time steps (view up from bottom): small length scale vortices are generated by the solid wall near the ring necks from the beginning to the end.

4.2. Mechanism of First Ring Formation. Ring-like vorticesplay a vital role in the transition process of wall boundary layer flow. Based on our numerical simulation, the gen-eration of ring-like vortices is the product of the interac-tion between the prime streamwise vortices and secondary streamwise vortices. A ring is the consequence when theinteraction between prime and secondary streamwise vortex becomes strong enough. The formation details are shown inFigure 9. The prime streamwise vortices are inside the ring,while the secondary streamwise vortices are outside the ring.According to the velocity vector trace, the prime streamwise

vortices push flow outward, and the secondary streamwisevortices pull flow in from underneath (Figure 10). Thesemovements bend and stretch the hairpin vortices and thengenerate a narrow neck. Finally a ring is formed at thetip of the Λ-vortices. For comparison, the ring-like vortex shape given by Moin et al. [7] is taken as a reference. Thedetail structure of four vortex tubes and ring-like vortices isdepicted in Figure 11 from three diff erent view angles. They are quite diff erent.

Apparently, our new DNS results show that the ring is aperfect circle (Figure 11(b)) not a deformed shape as shownby Moin et al. [7], the ring stands perpendicularly with 90degrees not 45 degrees as shown byMoin et al. [7], and, more

important, “ring pinch off ” given by Moin et al. [7] is neverfound. The ring-like vortex has its own coherent structure aswe show in Figure 11.

The profile in Figure 12 is a spanwise and time averagestreamwise velocity profile at location x = 486 (the locationof first ring-like vortices). As shown in Figure 12, the ring-like vortices are located outside the boundary layer or inother words located in the inviscid zone (z = 3.56, U =0.99U e). Because the leading rings are in the inviscid zonewhere the flow is almost uniform and isotropic, the shapeof the ring is almost perfectly circular. It is believed that the

velocity in normal direction is almost zero in the inviscidzone and the ring stands almost perpendicularly.

4.3. Mechanism of Multiple Ring Formation. Crow theory [8] has been considered as the mechanism of multiplering formation [2]. Figure 13 shows a typical vortex chainformation by Crow theory. However, there is no evidence toprove such a mechanism for flow transition. According toour DNS, Figure 14(a) gives a schematic about the processof a single ring and multiple ring formation. Following thefirst Helmholtz vorticity conservation law, when the vortex tube is stretched and rolls up, the tube is narrowed, andan additional bridge is developed across the two legs of the


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20151050

Y

0

0.5

1

1.5

2

2.5

3

Z

(a) t = 7.0T and x = 442δ in. Primary z = 1.3δ in, Secondary z = 0.8δ in,Third z = 0.2δ in

20151050

Y

0

0.5

1

1.5

2

2.5

3

Z

(b) t = 7.8T and x = 475δ in. Induced small vortex at z = 0.2δ in

20151050

Y

0

0.5

1

1.5

2

2.5

3

Z

(c) t = 8.2T and x = 515δ in. More small vortices are generatedFigure 20: Small vortices are generated by the wall surface at diff erent time steps and streamwise locations by interaction with secondary vortices and positive spikes.

prime vortex tube to maintain the vorticity conservation.Consequently, the second, third, and so forth. rings areformed. Figure 14(b) shows that the multiple rings (over 8rings) are formed and the first heading ring (right) is skewedand sloped (not perpendicular any more). From our new DNS, the so-called “spike” is nothing but formation of multi-ring (bridge) vortex structure which is quite stable. The mul-tiple rings were generated one by one but not simultaneously like Crow theory, and the rings are perpendicular to thewall not parallel to the wall like Crow theory. Apparently,we cannot find any links between the Crow theory and themultiple ring formation.

4.4. U-Shaped Vortex Development. A description of thedevelopment of the U-shaped vortex can be found in thepaper by Singer and Joslin [9]. Figure 15 has a 2D top view and uses low pressure contour to represent the vortex tube,which could lead to a misinterpretation that the secondary vortex is broken in to smaller pieces. Figure 16 is a 2D topview given by our new DNS with a diff erent λ2-value whichgives a similar structure to Figure 15. However, there is noevidence that the secondary vortex breaks down. Let us look at the head of the so-called “turbulence spot” from diff erentdirections of view (Figure 17). The large vortex structureof the leading head of the turbulence spot is stable and is


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x

900880860840820800780760740720700680660640620600580560540520500480460440

X

Z

Y (a) Side view

X Y

Z

780

770

760

750

740

730720

710

700

x

(b) Angle view (locally enlarged)

x

790780770760750740730720710

X Y

Z

(c) Side view (locally enlarged)

Figure 21: Vortex cycle overlapping and boundary layer thickening.

x

900880860840820800780760740720700680660640620600580560540520500480460440

X

Y

Z

(a) Top view of ring cycle overlapping

X

790780770760750740730720710

X Z

Y

(b) Top view (locally enlarged)

Figure 22: Ring cycle overlapping, but never mixing.

still there. It travels for a long distance and never breaksdown. We found that the head shape looks U-shaped, butthe basic vortex structure really does not change and theheading hairpin vortex never breaks down. The real situationis that the heading primary ring is still there but skewed andsloped, which leads to a disappearance of the second sweep.

The consequences are that no more energy is transporteddown to the boundary sublayer by the second sweep whichis weakened in the area near the head of the spot. The smalllength-scale-structures located near the laminar bottomdamped, and the originally existing U-shaped vortex is risingand becomes clearly. It was thought [9] that the U-shaped


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876543

y

0

1

2

3

4

z

Reference vector(U = 1)

(a) Vector field

876543

y

0

1

2

3

4

z

(b) Ten contour levels −0.125∼0.323

876543

y

0

1

2

3

4

z

(c) Vector field at x = 462Figure 23:Vector field of (v , w)-velocity disturbance (a) and contours of streamwise velocity disturbanceu (b) inthe ( y , z )-planeat t = 6.6T ,x = 464, (c) t = 6.6T and x = 462. Thick arrows display downward and upward motions associated with sweep 2 and ejection 2 events,respectively.

vortex is a newly formed secondary vortex. However itreally exists for some time. The reason that the U-shaped

vortex is not clearly visible at previous times is that theU-shaped vortex is surrounded by many small length scalestructures. When the heading primary ring is perpendicular,it will generate a strong second sweep which brings a lot of energy from the inviscid area to the bottom of the boundary layer and makes that area very active. However, when theheading primary ring is no longer perpendicular, skewed andsloped, the second sweep disappears. The small length scalestructures rapidly damp and the originally existing U-shapedvortex becomes clear since they are part of the large scalestructure. Diff erent from Singer and Joslin [9], the U-shapedvortex is found not a secondary, but tertiary vortex with samesign of vorticity of the corresponding ring leg. Actually, the

U-shaped vortex is a second neck to supply vorticity to thering (Figures 17(b) and 17(c)).

Due to the increase of the ring (bridge) number andvorticity conservation, the leading rings will become weakerand weaker until they cannot be detected, but they neverbreak down (see Figure 17). The multiring structure is pretty stable and the rings can travel for long distances (Figure 18)because they are located in an inviscid area and the U-shapedneck provides more vorticity.

4.5. Mechanism of Small Length Scale Generation. It is naturalthat one question will be raised. The question is where thesmall vortex length scales come from? We believe that thesesmall length scale vortices are generated by the interactionof secondary and third level vortices with the wall surface.


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876543

y

0

1

2

3

4

z

(a) x = 465. 10 contour levels −0.456∼0.718

876543

y

0

1

2

3

4

z

(b) x = 464. 10 contour levels −0.698∼1.202Figure 24: Contours of spanwise component of the vorticity disturbance in the ( y, z )-plane at t = 6.6T. Light shades of gray correspond to

high values.

470465460455450445

x

0

2

4

6

z

(a) y = 5.5. 10 contour levels −0.364∼1.531

470465460455450445

x

0

2

4

6

z

(b) y = 4.4. 10 contour levels −0.698∼1.202

Figure 25: Contours of spanwise component of the vorticity disturbance in the ( x, z )-plane at t = 6.6T. Light shades of gray correspond tohigh values.

As we know, the vorticity can only be given or generatedin boundaries, but it cannot be generated inside the flow field. Actually, the wall surface is the sole source of thevorticity generation for boundary later. Our new DNS givesthe preliminary answer that the small length scale vorticesare generated by the wall surface and near the wall surfaceas they are generated by second sweep instead of hairpinvortex breakdown. Actually, vortex breakdown is theoreti-cally impossible and practically never found. The animationsof our DNS results give details of flow transition at every

stage and new mechanisms of small length scale vortex generation. We provide some snap shots here (Figure 19).Since we believe that the small vortices are generated by thewall surface and near wall region, we take snap shots in thedirection of view from the bottom to top. It is easily foundthat the small length scale structures first appear on the wallnear the ring neck area in the streamwise direction.

All evidences provided by our new DNS confirm thatthe small length scale vortices are generated near the bottom(Figures 19 and 20) by secondary vortices or third levelvortices, especially by the positive spike caused by strongsecond sweeps which bring high streamwise velocity frominviscid area to sublayer.

4.6. Vortex Cycle Overlapping. As we discussed before, thering head is located in the inviscid area and has much highermoving speed than the ring legs which are located near thebottom of the boundary layer. It is the reason that the hairpinvortex is stretched and multiple rings are generated. Thiswill lead to an overlapping of second ring cycle upside of the first ring cycle. However, no mixing of two cycles isobserved by our new DNS (Figures 21 and 22). The secondring is separated from first ring cycle by a secondary groupof rings which are generated by the wall surface, separated

from wall, and convected to downstream. This is the reasonwhy the transitional boundary layer becomes thicker andthicker.

4.7. High Shear Distribution. The (v , w )-velocity distur-bance vectors and the u-velocity contours are presentedin Figure 23 in the ( y, z )-plane. The projection of theapproximate location of the first vortex ring onto this ( y, z )-plane is indicated in Figure 23 with semitransparent circles.It is seen that there are two strong downdraft jets producedby the first vortex ring behind it at two sides of the structurecenterline. The existence of these jets was called “sweep 2” events [5, 24].


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470465460455450445

x

0

2

4

6

z

(a) y = 5.5. ten contour levels −0.474∼0.150

470465460455450445

x

0

2

4

6

z

(b) y = 4.4. ten contour levels −0.332∼0.340

Figure 26: Contours of streamwise velocity disturbance u in the ( x, z )-plane at t = 6.6T. Light shades of gray correspond to high values.

Formation of a low speed region between the Λ-vortex legs and appearance of an HS-layer slightly above and insideof the Λ-vortex can be found in Figures 24, 25, and 26.It is found (based on comparison of instantaneous fieldscalculated for various time instants) that the low-speedstreaks and high-shear layers associated with the ejection2 events seem to become weaker when the ring-like vortex propagates downstream and finally disappears. This result isconsistent with observations by Bake et al. [3]. They believethat low-speed streaks and high-shear layers become weak and finally disappear because the “tail vortices” connectingthe ring- like vortices move towards the wall. Based on ournew DNS results, it shows that the main reason for the abovephenomena is that in the middle between the two positivespikes near the ring neck, there is an area where the normalvelocity component is almost zero (see Figures 23(a) and23(c)).

5. Conclusions

Based on our new DNS study, the following conclusions canbe made.

(1) Vorticity is generated by the wall. The secondary vortex can be developed only through flow separationwhich is caused by the adverse pressure gradient.

(2) A vortex tube inside the flow field cannot have ends,and thus vortex ring is the only form existing insidethe flow field because the ring has no ends, head, ortail.

(3) There are four streamwise vorticity lines around theΛ-shaped vortex, two primary inside and two sec-

ondary outside. The inside two primary streamwisevorticity lines pull out the Λ-shaped vortex to formthe ring, but the outside two secondary vorticity lines push the Λ-shaped vortex to form the neck.Therefore, the formation of a ring-like vortex isthe interaction between the primary and secondary streamwise vorticity lines.

(4) The leading ring-vortices are almost perfectly circularand stand almostperpendicularly. The so-called “ringpinch off ” is not found.

(5) The stretched vortex tube will be narrowed withstronger vorticity along with faster rotation as a result

of narrowing. According to the first Helmoholtz vor-ticity conservation law, this will lead to developmentof a bridge and further become a second ring, andso on.

(6) U-shaped vortex is a coherent structure and serves assecond neck to supply the rings.

(7) Since the head of the hairpin vortex, that is, rings, islocated in the inviscid area and the U-shaped vortex supplies vorticity to the ring, the consequent multiplering-like vortex structure can travel for long distancein the flow field. The hairpin vortex will become weak until being nondetectable, but never breaks down.

(8) The small length scales are generated by the wallsurface and near the wall region, but not by thehairpin vortex breakdown.

(9) The ring head speed is faster than the ring legs, whichleads to ring cycle overlapping. Multiple ring cycle

overlapping will lead to thickening of the transitionalboundary layer. The multiple rings overlap, but nevermix. They are separated by secondary rings which aregenerated by thewall surface, separated from the wall,and convected to downstream.

(10) The updraft motion leads to the Λ-shaped high-speed streaks and an HS-layer near the wall outsidethe Λ-vortex legs. However, the HS-layer disappearsnear the ring neck because the normal velocity in thenear neck area is almost zero.

Nomenclature

M ∞: Mach numberRe: Reynolds numberδ in: Inflow displacement thicknessT w : Wall temperatureT ∞: Free stream temperatureLz in: Height at inflow boundary Lx : Length of computational domain along x

directionLy : Length of computational domain along y

directionx in: Distance between leading edge of flat plate and

upstream boundary of computational domain A2d : Amplitude of 2D inlet disturbance


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A3d : Amplitude of 3D inlet disturbanceω: Frequency of inlet disturbanceα2d : Streamwise wave number of inlet disturbance β: Spanwise wave number of inlet disturbanceR: Ideal gas constantγ: Ratio of specific heats

µ∞: Viscosity

ωx : Streamwise vorticity λ2: Const representing vortex tube surface.

Acknowledgments

This work was supported by AFOSR grant FA9550-08-1-0201 supervised by Dr. John Schmisseur. The authors aregrateful to Texas Advanced Computing Center (TACC) forproviding computation hours. This work is accomplished by using Code DNSUTA which wasreleased by Dr. ChaoqunLiuat University of Texas at Arlington in 2009.

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in wall-bounded shear flows,” Annual Review of Fluid Mechan-ics, vol. 23, no. 1, pp. 495–537, 1991.

[2] V. I. Borodulin, V. R. Gaponenko, Y. S. Kachanov et al., “Late-stage transitional boundary-layer structures. Direct numericalsimulation and experiment,” Theoretical and Computational Fluid Dynamics, vol. 15, no. 5, pp. 317–337, 2002.

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Mechanics, vol. 459, pp. 217–243, 2002.[4] Y. S. Kachanov, “On a universal mechanism of turbulence

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[6] F. R. Hama and J. Nutant, “Detailed flow-field observations inthe transition process in athick boundary layer,” in Proceedingsof the Heat Transfer and Fluid Mechanics Institute, pp. 77–93,Stanford University Press, Palo Alto, Calif, USA, 1963.

[7] P. Moin, A. Leonard, and J. Kim, “Evolution of a curved vortex

filament into a vortex ring,” Physics of Fluids, vol. 29, no. 4, pp.955–963, 1986.

[8] S. C. Crow, “Stability theory for a pair of trailing vortices,” AIAA Journal , vol. 8, no. 12, pp. 2173–2179, 1970.

[9] B. A. Singer and R. D. Joslin, “Metamorphosis of a hairpinvortex into a young turbulent spot,” Physics of Fluids, vol. 6,no. 11, pp. 3724–3736, 1994.

[10] C. B. Lee and J. Z. Wu, “Transition in wall-bounded flows,” Applied Mechanics Reviews, vol. 61, no. 1–6, pp. 030802–03080219, 2008.

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[12] . Jinhee Jeong and F. Hussain, “On the identification of avortex,” Journal of Fluid Mechanics, vol. 285, pp. 69–94, 1995.

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[14] S. K. Lele, “Compact finite diff erence schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no.1, pp. 16–42, 1992.

[15] C. W. Shu and S. Osher, “Efficient implementation of

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Research ArticleImplicit LES for Supersonic Microramp Vortex Generator:New Discoveries and New Mechanisms

Qin Li and Chaoqun Liu

Math Department, University of Texas at Arlington, 411 S. Nedderman Drive, Arlington, TX 76019-0408, USA

Correspondence should be addressed to Chaoqun Liu, [email protected]

Received 6 October 2010; Accepted 12 January 2011

Academic Editor: Guan Yeoh

Copyright © 2011 Q. Li and C. Liu. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper serves as a summary of our recent work on LES for supersonic MVG. An implicitly implemented large eddy simulation(ILES) by using the fifth-order WENO scheme is applied to study the flow around the microramp vortex generator (MVG) atMach 2.5 and Reθ = 1440. A number of new discoveries on the flow around supersonic MVG have been made including spiralpoints, surface separation topology, source of the momentum deficit, inflection surface, Kelvin-Helmholtz instability, vortex ringgeneration, ring-shock interaction, 3D recompression shock structure, and influence of MVG decline angles. Most of the new discoveries, which were made in 2009, were confirmed by experiment conducted by the UTA experimental team in 2010. A new 5-pair-vortex-tube model near the MVG is given based on the ILES observation. The vortex ring-shock interaction is found as thenew mechanism of the reduction of the separation zone induced by the shock-boundary layer interaction.

1. Introduction

It is well known that for the supersonic ramp jet flow,shock boundary layer interaction (SBLI) can significantly degrade the quality of the flow field by triggering large-scale separation, causing total pressure loss, making theflow unsteady and distorted, and can even make an engineunable to start. In order to improve the “health” of theboundary layer, a series of new devices named as micro

vortex generator is designed for flow control, which iswith a height approximately 20–40% (more or less) of theboundary layer thickness. Because of the robust structure,the specific microramp vortex generator (MVG) becomesmore attractive to the inlet designer. Intensive computationaland experimental studies have been made on the MVGrecently.

Lin indicated [1] that the device like MVG could alleviatethe flow distortion in compact ducts to some extent andcontrol boundary layer separation due to the adverse pres-sure gradients. Similar comments were made in the review by Ashill et al. [2]. The formal and systematic studies aboutthe micro VGs including micro ramp VG can be found in

the paper by Anderson et al. [3]. Babinsky et al. [4–7] madea series of experiments on diff erent kinds of mirco VGs andinvestigated their control eff ects in details. The mechanism of MVG flow control from his work was described as that a pairof counter-rotating primary streamwise vortices is generatedby MVG, which is mainly located within the boundary layer and travel downstream for a considerable distance.Secondary vortices are located underneath the primary onesand even more streamwise vortices could be generated under

suitable conditions. Streamwise vortices inside the boundary later will bring low-momentum fluids up from the bottomand high-momentum fluids down to the boundary layer.A striking circular momentum deficit region is observed inthe wake behind the MVG. The vortices will keep lifting upslowly, which is thought to be the consequence of the upwasheff ect of the vortices.

Numerical simulations have been made on MVG forcomparative study and further design purposes. Ghosh et al.[8] made detailed computationsunder the experimental con-ditions given by Babinsky by using RANS, hybrid RANS/LES,and immersed boundary (IB) techniques. Lee et al. [9,10] also made computations on the micro VGs problems


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β = 70

α = 24

8.64464

c s

h

Figure 1: The geometry of MVG.

Fore-regionGrid

transitionGrid

transitionMVGregion

Rampregion

5.6δ 01.875δ 018.63875δ 0

H =

5 δ 0

h

Figure 2: The schematic of the half-grid system of case (1).

by using Monotone Integrated Large Eddy Simulations(MILES). Basic flow structures like momentum deficit andstreamwise vortices were reproduced in the computation.Further studies were also conducted on the improvement of the control eff ect.

Finding physics of MVG for design engineers is definitely needed. RANS, DES, RANS/DES, RANS/LES, and so forthare good engineering tools, but may not be able to reveal themechanism and get deep understanding of MVG. We needhigh-order DNS/LES. A powerful tool is the integration of high-order LES and experiment. Recently, an implicit LESfor MVG to control the shock-boundary layer interaction

around a 24 degree ramp at Mach number of 2.5 andReynolds number of 1440 has been carried out in 2009 [12].In this work, several new discoveries have been made andthe mechanism of MVG is found quite diff erent from thosereported by previous experimental and numerical work inthe literature. The new findings include (1) spiral points andflow topology around MVG, (2) new theory of five pairsof vortices near MVG, (3) origin of momentum deficit, (4)inflection points (surface for 3D) and Kelvin-Helmholtz- (K-H-) type instability, (5) vortex ring generation, (6) ring-shock interaction and separation reduction, (7) 3D re-compressed shock structure, and (8) eff ects of trailing-edgedecline angles.

(a) Computational domain

(b) The grid section intersecting the middleof MVG

(c) The grids at the corner of MVG

Figure 3: Body-fitted grid system.

00.2120.424

0.6360.8491.0611.2731.485

Figure 4: The digital schlieren at central plane.

Most of the new discoveries were made by October2009 and were later confirmed by the UTA experimentalwork in April 2010 [13]. Based on our observation, themechanism of MVG to reduce the flow separation is really caused by interaction of shock and vortex rings generated by MVG.


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Figure 5: The experimental schlieren by Loginov et al. [11].

Faint edge

Figure 6: The experimental schlieren by Babinsky et al. [7].

The paper is arranged as following. The numericalmethods are discussed in Section 2. In Section 3, all new discoveries are summarized and analyzed and the new mechanism is described. Finally, conclusions are made inSection 4.

2. Numerical Methods

2.1. Governing Equations. The governing equations are thenondimensional Navier-Stokes equations in conservativeform as follows:

∂Q∂t

+ ∂E∂x

+ ∂F ∂y

+ ∂G∂z

= ∂Ev ∂x

+ ∂F v

∂y +

∂Gv ∂z

, (1)

where

Q =

ρ

ρu

ρv

ρw

e

, E =

ρu

ρu2 + p

ρuv

ρuwe + p

u

, F =

ρv

ρvu

ρv 2 + p

ρvwe + p

v

, G =

ρw

ρwu

ρwv

ρw2 + pe + p

w

,

Ev = 1Re

0

τ xx τ xy τ xz

uτ xx

+ vτ xy

+ wτ xz

+ qx

, F v = 1Re

0

τ yx τ y y τ yz

uτ yx

+ vτ y y

+ wτ yz

+ q y

, Gv = 1Re

0

τ zx τ zy τ zz

uτ zx

+ vτ zy

+ wτ zz

+ qz

,

e = pγ − 1 +

1

2 ρ

u2 + v 2 + w2, qx =

µγ − 1 M 2∞Pr ∂T ∂x , q y = µγ − 1 M 2∞Pr ∂T ∂y ,

qz = µ

γ − 1 M 2∞Pr ∂T ∂z , p = 1γM 2∞ ρT , Pr = 0.72,

τ = µ

4

3

∂u∂x

− 23

∂v ∂y

+ ∂w∂z

∂u

∂y +

∂v ∂x

∂u∂z

+ ∂w∂x

∂u∂y

+ ∂v ∂x

4

3

∂v ∂y

− 23

∂w∂z

+ ∂u∂x

∂v

∂z +

∂w∂y

∂u∂z

+ ∂w∂x

∂v ∂z

+ ∂w∂y

4

3

∂w∂z

− 23

∂u∂x

+ ∂v ∂y

.

(2)

The viscous coefficient is given by Sutherland’s equation:

µ = T 3 / 2 1 + C T + C

, C = 110.4T ∞

. (3)

The nondimensional variables are defined as follows:

x = x L

, y = y L

, z = z L

,

u =

u

U ∞

, v =

v

U ∞

, w =

w

U ∞

,

T =T

T ∞, µ = µ

µ∞, k =

kk∞

,

ρ = ρ ρ∞

, p = p ρ∞U 2∞

, e = e ρ∞U 2∞

,

(4)

where the variables with “∼” are the dimensional counter-parts.


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0

5

10

−10 0 10

Figure 7: The numerical schlieren at central plane.

Figure 8: Spiral point pairs by ILES in 2009.

Considering the following grid transformation,

ξ

=ξ x , y , z ,η = ηx , y , z ,

ζ = ζ x , y , z ,(5)

the Navier-Stokes equations can be transformed to thesystem using generalized coordinates:

∂ Q∂τ

+ ∂ E∂ξ

+ ∂ F ∂η

+ ∂ G∂ζ

= ∂Ev

∂ξ +

∂ F v ∂η

+ ∂ Gv

∂ζ , (6)

where Q = J −1Q andE = J −1

ξ x E + ξ y F + ξ z G

,

F = J −1ηx E + η y F + ηz G,G = J −1ζ x E + ζ y F + ζ z G,Ev = J −1ξ x Ev + ξ y F v + ξ z Gv ,F v = J −1ηx Ev + η y F v + ηz Gv ,Gv = J −1ζ x Ev + ζ y F v + ζ z Gv ,

(7)

J −1, ξ x , and so forth, are grid metrics, and J −1 = det(∂(x , y , z ) /∂(ξ , η, ζ )), ξ x = J ( y ηz ζ − z η y ζ ), and so forth.

Figure 9: Oil accumulation blocks by UTA experiment in 2010.

(a)

Spiral

points

(b)

Figure 10: Comparison of ILES with UTA experiment.

2.2. Finite Di ff erence Schemes and Boundary Conditions

1. The Fifth-Order WENO Scheme for the Convective Terms[ 14]. In order to decrease the dissipation of the scheme, theless dissipative Steger-Warming flux splitting method is usedin the computation, but not the commonly used dissipativeLax-Friedrich splitting method.


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Primary vortex lifts off

Primary vortex pair

Secondary vortex pair

Characteristic footprintof primary vortex pair

Horseshoevortex

Figure 11: Traditional vortex structure around MVG.

Primary vortex

Horseshoe vortex

Second secondary vortex

First secondary vortices

Figure 12: New 5-pair-vortex-tube model near MVG.

Figure 13: The surface oil flow by Babinsky7.

2. The Di ff erence Scheme for the Viscous Terms. Consideringthe conservative form of the governing equations, the

traditional fourth-order central scheme is used twice tocompute the second-order derivatives for viscous terms.

3. The Time Schems. The basic methodology for the tempo-ral terms in the Navier-Stokes equations adopts the explicitthird-order TVD-type Runge-Kutta scheme:

u(1) = un + ∆tL(un),

u(2) = 34

un + 1

4u(1) +

1

4∆tL

u(1)

,

un+1 = 13

un + 2

3u(2) +

2

3∆tL

u(2)

.

(8)

Figure 14: The surface limiting streamlines by computation.

NP

NP

SPP

SPP

SPP

SDPSDP

Figure 15: The separation pattern near the end of MVG.

4. Boundary Conditions. The adiabatic, zero-gradient of pressure and nonslipping conditions are used for the wall as

∂T ∂n

= 0, ∂p∂n

= 0, −→U = 0. (9)

To enforce the free stream condition, fixed-value boundary condition with the free parameters is used on the upperboundary. The boundary conditions at the front and back boundary surface in the spanwise direction are given asthe mirror-symmetry condition. The outflow boundary conditions are specified as a kind of characteristic-basedcondition, which can handle the outgoing flow without


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(a) SSP on the side of MVG (b) SSP backgrounded using the pressure

contour

Figure 16: The various views of surface separation pattern (SSP).

NP5

NP2

SP

l 1l

(a) Computation: l 1 /l = 0.57

l l

1

(b) Experiment: l 1 /l = 0.545

Figure 17: Comparison of ILES and experiment in surface flow topology.

(a) Secondary vortex lifts up (b) Moving downstream around the primary

Figure 18: Secondary vortex tubes lift up from the bottom plate.

(a) Secondary vortex lifts up (b) Moving downstream around the pri-mary

Figure 19: Secondary vortex tubes lift up from MVG sides.


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Figure 20: New secondary vortex tubes induced by the primary.

0

10 y ( m m )

20

30

−20z ( m m )

−10 0 10 20 −40−20 0

2040

6080

100120

140

x ( m m )

Figure 21: Momentum deficit shown in diff erent streamwisedirections.

reflection. The inflow conditions are generated using thefollowing steps.

(a) A turbulent mean profile is obtained from DNS by Liu and Chen [15] for the streamwise velocity (w -velocity) and the distribution is scaled using the localdisplacement thickness and free stream velocity.

(b) Random fluctuations are added on the primitivevariables, that is, u, v , w , p, ρ . The disturbance has

the form: εdistbe−( y − y w)2 / ∆ y distb × (random − 0.5) / 2,

where the subscript “distb” means the disturbance,“random” is the random function with the value

between 0∼1, εdistb equals to 0.1, and ∆ y distb equalsto 2/3δ 0.

Such inflow conditions are, of course, not the fully developedturbulent flow, but we can consider it as a weakly disturbedinflow while propagating downstream.

5. Body-Fitted Grid Generation. The geometry of MVG isshown in Figure 1. In order to alleviate the difficulty togrid generation caused by original vertical trailing-edge, amodification is made by declining the edge to 70. The othergeometric parameters in the figure are the same as thosegiven by Holden and Babinsky [5], that is, c = 7.2h, α = 24,

and s = 7.5h, where h is the height of MVG and s is thedistance between the center lines of two adjacent MVGs. Sothe distance from the center line to the spanwise boundary of the computation domain is 3.75h.According to experimentsby Holden and Babinsky [5], the ratio h/ δ 0of the modelshas the range from 0.3∼1. The appropriate distance fromthe trailing-edge to the control area is around 19∼56h or 8∼19δ 0. So in this study, the height of MVG h is assumed to beδ 0/2 and the horizontal distance from the apex of MVG to theramp corner is set to be 19.5h or 9.75δ 0. The distance fromthe end of the ramp to the apex is 32.2896h. The distancefrom the starting point of the domain to the apex of MVG is17.7775h. The height of the domain is from 10h to 15h andthe width of the half domain is 3.75h. The geometric relationof the half of the domain can be seen in Figure 2, where thesymmetric plane is the centre plane.The grid number for thewhole system is nspanwise × nnormal × nstreamwise = 128 × 192× 1600. We try to make the grids smooth and orthogonal asmuch as possible (Figure 3).

2.3. Code Validation

1. Supersonic Ramp. Figure 4 gives the instantaneousnumer-ical schlieren image of the central plane, which uses the valueof |∇ ρ|. For qualitative comparison, an experimental pictureof ramp flow at M = 2.9 with large Reynolds number is givenin Figure 5 obtained from the experiment by Loginov et al.[11]. From both pictures, we can observe that the separationshock wave has a declining angle which is a little lager butnearly same to the ramp angle and is almost aligned withthe reflection shock (which is more obv

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