格子依存性を低減させた実用乱流解析のための新たな高性能LES乱流モデルの構築

九州大学 大学院
工学研究院 航空宇宙工学部門

安倍 賢一

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Background Large eddy simulation (LES) is a useful way to

predict turbulence and turbulent scalar transfer.
 Recent development of computer power allows us

to do LES for several turbulent flows ofengineering interest.
 However, there still remains a serious difficulty in

its application to high Reynolds-number complex
turbulent flows.
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Typical Example (Image)

2D section

3D configuration

Typical 3D grid distribution
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Lack of Spanwise Resolution
High-Re flow aroundairfoil(Away from the wall)
Large scale eddies can beresolved with grid spacing.

y(Close to the wall)
Due to coarse resolution in
z-direction, no eddy can beresolved with grid spacing.
(Large No Eddy Simulation)zTo index(Enough resolution in the wall-normal (y) direction)
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Underprediction of Wall-Shear Stress(Channel Flow,10000)

Nodes:

61 61 61
Domain:6 1.5
Resolution:z   250
Considerable underprediction of wall-shear stress is obtained.
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Subgrid-Scale (SGS) Model Under such a flow condition with coarse grid resolution in near-wallregion, it is very difficult to predict reasonable wall-shear stress(mean friction coefficient) by use of current SGS models.
 Dynamic Smagorinsky SGS model often causes computational

instability in coarse grid-resolution LES.
 Scale-similarity model may have the capability of providing the SGS

stresses more reasonably, but it is not dissipative and thus oftencauses numerical instability.
 Mixed SGS model (linear SGS model + scale-similarity model) may

be a promising way, but even now it cannot return reasonablepredictions for the aforementioned very coarse grid-resolution LES.
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Objectives of the Present Study
Main objectives are to propose an SGS model for representing

reasonable SGS-stress anisotropy.
 to confirm the basic performance of the presently-proposed SGS model.
 to discuss further development of the present SGS

modeling concept for its application to various gridresolutions.
 to elucidate the usefulness of the present

anisotropic SGS model for engineering applications
with coarse grid resolution.
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Governing Equations
Filtered governing equations for incompressible flow

01

Subgrid-scale (SGS) stress tensor

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Representative SGS Models
For, a linear eddy viscosity model (EVM) is most popular.
1
32

1
2

An alternative way may be a scale-similarity model, among
which the Bardina model (1980) is well known.
12122
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The Present SGS Modeling Concept
Basic concept of the present SGS modeling232: extra anisotropic term
(cf: higher-order term of nonlinear eddy-viscosity model)
Desirable features of: Reasonable representation of the SGS-stress anisotropy It doesn’t show serious violence of the realizability.
 Guarantee of computational stability  It doesn’t cause

unnecessary numerical instability.
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Modeling Strategy (1/2)
EVM-type linear approximation for a scale-similarity model.
 An equivalent value of eddy viscosity is evaluated as

22/3,2
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Modeling Strategy (2/2)
This approximation indicates that the following residual term
may be effective for representing the SGS-stress anisotropy.

2 Linear approximation is subtracted from scale-similarity model SGS-stress anisotropy may be successfully reproduced.
 There is no extra energy transfer between GS and SGS parts

 It may not cause undesirable numerical instability.

In this study, this residual term23
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is adopted for:2

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SGS Eddy Viscosity
For the linear SGS eddy viscosity
by Inagaki (IJHFF 2011) is adopted.

, the SGS model proposed⁄

1

Δ0.042,exp

20

⁄

Δ⁄

24,Δ
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Implementation to CFD CodeDU i
1  P    U i U j 




ij xi Dt
 xi x j   x j23

 U i ij  k SGS  ij  SGS  x j

U j  EATij   ij (linear )  EATijxi （Linear term）（Anisotropic term）

  EATij
DU i
1  P    U i U j   ij ( linear )    xi x j   x jDtx jxi 
（Canonical form）（Source term）
Extra anisotropic term is treated as a kind of body force in NS equation.
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Test Case (1: Plane Channel Flow)61 61 61

Grid nodes:Reynolds numberRe  395

 Reb  14000

Domain and resolution：6  2 1.5
Channel flow x  40, y  0.8  36, z  10(Moser, R.D., Kim, J., and Mansour, N.N., Physics of Fluids, 11, 943-945, 1999.) To index
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Test Case (2: Plane Impinging Jet)

Grid nodes:

261 61 46

Reynolds number

Re  8500

Domain： 40 D  20 D  x  20 D   2 D  1.5 D(Hattori, H. and Nagano, Y., Int. J. Heat Fluid Flow, 25, 749-758, 2004.)
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Computational Conditions
Simulation code

FrontFlow/Red*

Grid system, Algorithm

Unstructured grid,SIMPLE-baseDiscretization method

Second-order central
difference(for convection terms in)(Second-order upwindscheme)* Muto, M., Tsubokura, M., and Oshima, N., Physics of Fluids, 24, 014102, 2012.
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Comparison of GS and SGS Stresses for
Channel-Flow Case

 A strong stress anisotropy isclearly seen in the SGSstresses.
 The GS and SGS stresses
show a similar trend in thestress anisotropy.
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Comparison of Linear and AnisotropicTerms in the SGS Model The linear EVM part returns

relatively small values of theSGS normal stresses.
 The SGS stress anisotropy isdominated by.
 The linear terms are small, butthey control energy transfer.
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Basic Performance for Plane Impinging Jet(DNS data: Hattori, H. and Nagano, Y., Int. J. Heat Fluid Flow, 25, 749-758, 2004.)

 Fundamental features of this type of flow field are well captured.
 The present LES data are useful enough to investigate thecharacteristics of the present SGS modeling concept for complex
turbulence with flow impingement.
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Comparison of GS and SGS Stresses forPlane Impinging Jet ( ⁄0) Trend of stress components

in the stagnation region iscompletely different from thatin the channel-flow case.
 The GS and SGS stresses
show a similar trend.
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Comparison of GS and SGS Stresses forPlane Impinging Jet ( ⁄3) The GS stresses in wall-jet

region show a trend rather
similar to those in a wall-shearflow.
 The GS and SGS stresses
show a similar trend.
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Characteristics of the Present SGS Model The GS and SGS stresses show a similar trend in thestress anisotropy for both the channel flow and theplane impinging jet, though the flow features are
considerably different between these two test cases.
 Such a common feature that is seen even in a complex

turbulence with flow impingement is thought to be an
important knowledge.

How well does the presently-proposed anisotropic
SGS model works for coarse-grid resolutions?
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Mean-Velocity Distributions for
Various Grid ResolutionsAnisotropic SGS model with no transport equation for∆∆∆∆160, ∆160, ∆80, ∆40, ∆80402010 Although the present anisotropic SGS model provides the SGS stress anisotropy,

its predictive performance is far from satisfactory.
 Underprediction of wall-shear stress is still seen for coarse grid-resolution cases.
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To indexRelation between Unresolved Eddies anda Little Larger Resolved GS Eddies

Dotted lines:
Unresolved eddies
Dashed gray lines: a little larger resolved GS eddies
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25/48Further Development of the Present
SGS Model A conventional SGS model cannot resolve eddies smaller than grid

width (dotted lines), while a scale-similarity model has a possibility ofresolving a little larger GS eddies (dashed gray lines).
 Although the strength of these resolved eddies may be much weaker

than that indeed necessary, it is possible that these eddies, at least
qualitatively, include some fundamental features of turbulent
structures to be resolved.

In order to give an appropriate level of this effect to a flow simulation,
another additional strategy may be necessary such as the introduction.
of solving the transport equation forTo index

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An Advanced Anisotropic One-Equation
SGS Model
is used for representing the degree of stress anisotropy,
while the level of SGS turbulence energy
is evaluatedby its transport equation.

232

2

2
2

1
3
Δ0.1
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Model Functions and Constants
For the linear SGS model, the same model is used as used in
the original model with model constants slightly modified.
⁄Δ

10.05,30exp⁄Δ⁄Δ2,4,0.835
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Contribution of Transport Equation for
to Model Performance

Anisotropic SGS model withno transport equation for

Anisotropic one-equation SGS model The predictive performance is largely improved by one-equation SGS model.

Reasonable predictions of mean velocity are obtained for all grid resolutions.
 This may indicate that the transport equation forsuccessfully works todetermine a reasonable level of the SGS turbulence energy.
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Contribution of Anisotropic Termto Model Performance

Original isotropic one-equationSGS model

Anisotropic one-equation SGS model Original isotropic one-equation SGS model shows strong grid dependency.
 In order to improve the model performance, we must introduce both an extra

anisotropic term and one-equation SGS model. The former is for representing
SGS-stress anisotropy and the latter for determining a reasonable level of.
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Reynolds Shear Stress

∆

40, ∆

10

∆

80, ∆

20

∆

160, ∆40∆

160, ∆80

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Turbulence Intensities

Streamwise direction

Wall-normal direction

All of the components are
generally well predicted,
though a close inspection
indicates some discrepancy
as the grid resolution
becomes coarser.
Spanwise direction
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Mean-Velocity Distributions(Various Grid Resolutions)

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Mean-Velocity Distributions(High Reynolds Number Cases)

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Velocity-vector Plots∆

160, ∆

Original isotropic one-equation
SGS model

80

Anisotropic one-equation SGS model Although both figures generally looks similar, there can be seen

considerable difference between them in near-wall region.
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Velocity-vector Plots (Near-wall Region)∆160, ∆

Original isotropic one-equation
SGS model

80

Anisotropic one-equation SGS model In the near-wall region, smaller structure-like velocity distributions

can be seen for the case of anisotropic one-equation SGS model.
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Second invariant of velocity fluctuations∆160, ∆80

Original isotropic one-equation
SGS model

Iso-surface (7)Anisotropic one-equation SGS model We can see much more vortex-like structures in the near-wallregion for the case of anisotropic one-equation SGS modeleven under such a very coarse-grid resolution.
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Second invariant of velocity fluctuations(Fine, visualized region: 0 < y/δ < 1)
q+=70

∆

Original isotropic one-equation
SGS model
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40, ∆

10

Anisotropic one-equation SGS model

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Second invariant of velocity fluctuations(Fine, visualized region: 0 < y/δ < 0.15)∆q+=70

Original isotropic one-equation
SGS model
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40, ∆10

Anisotropic one-equation SGS model
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Second invariant of velocity fluctuations(Middle, visualized region: 0 < y/δ < 1)
q+=50

∆

Original isotropic one-equation
SGS model
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80, ∆

20

Anisotropic one-equation SGS model

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Second invariant of velocity fluctuations(Middle, visualized region: 0 < y/δ < 0.15)
∆

q+=50

Original isotropic one-equation
SGS model
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80, ∆

20

Anisotropic one-equation SGS model
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Second invariant of velocity fluctuations(Coarse, visualized region: 0 < y/δ < 1)
q+=20

∆

Original isotropic one-equation
SGS model
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160, ∆

40

Anisotropic one-equation SGS model

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Second invariant of velocity fluctuations(Coarse, visualized region: 0 < y/δ < 0.15)
∆

q+=20

160, ∆

Original isotropic one-equation
SGS model
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40

Anisotropic one-equation SGS model
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Transport Equation of Streamwise Vorticity
for GS ComponentD xUUU x y zDtxyz( Production term ) 2  uv  wu   2 2
2 2v  w   2  2 vw
 
y 
x  zy  yz
 z
( SGS-stress term )  2  x( Molecular diffusion term )To index

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Comparison of Terms in Transport Equation ofStreamwise Vorticity (Fine)

Productionterm

SGS-stressterm

Isotropic model
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Anisotropic model

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Comparison of Terms in Transport Equation ofStreamwise Vorticity (Coarse)

Productionterm

SGS-stress
term

Isotropic model
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Anisotropic model

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Concluding Remarks
 An anisotropy-resolving SGS model for large-eddy simulation

was newly proposed by combining a representative isotropic
linear eddy-viscosity SGS model with an extra anisotropic term
based on a scale-similarity model.
 It is found that the GS and SGS stresses obtained by the present

SGS model show a similar trend in the stress anisotropy for boththe channel flow and the plane impinging jet, though the flowfeatures are considerably different between these two test cases.
 By introducing one-equation SGS model, the present SGS model

provides considerable improvement of the predictive performance,
particularly for coarse grid resolution in near-wall region withcomputational stability being maintained.
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ProductsKen-ichi ABE“An Improved Anisotropy-Resolving Subgrid-Scale Model with the Aid of a
Scale-Similarity Modeling Concept”
International Journal of Heat and Fluid Flow, Vol. 39, pp. 42-52, 2013

Tadashi OHTSUKA and Ken-ichi ABE“Performance of an Anisotropy-Resolving Subgrid-Scale Model for Complex
Turbulence with Massive Flow Separation”
IUTAM Symposium on Vortex Dynamics: Formation, Structure and Function,
Fukuoka, pp. 124-125, 2013

Tadashi OHTSUKA and Ken-ichi ABE“On the Role of an Anisotropy-Resolving Extension Term for a Subgrid-Scale
Model in Near-Wall Turbulence”
Journal of Computational Science and Technology, Submitted
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