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Unsteady Analysis of Separated Aerodynamic Flows using an

Unstructured Multigrid Algorithm

Juan Pelaez

Old Dominion University

Hampton, Virginia

Dimitri Mavriplis

ICASE

NASA Langley Research Center, Hampton, Virginia

Osama Kandil

Old Dominion University

Hampton, Virginia

Abstract

An implicit method for the computation of

unsteadyflowson

presented. The resulting nonlinear system of

equations is solved at each time step using an

agglomeration multigrid procedure. The method

allows for arbitrarily large time steps and is

efficient in terms of computational effort and

storage. Validation of the code using a one-

equation turbulence model is performed for the

well-known case of flow over a cylinder. A

Detached Eddy Simulation model [1] is also

implemented and its performance compared to

the one equation Spalart-Allmaras Reynolds-

AveragedNavier-Stokes

model [2]. Validation cases using DES and

RANS include flow over a sphere and flow over

a NACA 0012 wing including massive stall

regimes. The project was driven by the ultimate

goalofcomputing

aerodynamic interest, such as massive stall or

flows over complex non-streamlined geometries.

unstructuredgrids is

(RANS) turbulence

separatedflowsof

1.Introduction

The use of time-dependent simulations for flows

of practical interest is much less widespread than

the use of steady-state flow simulations, due to

limitations related to computational time and

computational resources. However, many flows

are inherently unsteady, particularly when large

amounts of separation are present, and must be

simulated as such in order to obtain meaningful

results.

Copyright c 2001 by AIAA, Inc. All Rights Reserved.

Transient flow simulations can either be based

on explicit or implicit time-stepping schemes.

Stability constraints

allowable time step size of an explicit scheme,

which is proportional to the smallest cell size in

the entire computational mesh. Explicit schemes

are well suited for unsteady applications in

which the time scale of interest is comparable to

the spatial scales. In these cases the mesh should

be clustered only in regions of interest where it is

absolutely necessary or the explicit time step can

become unnecessarily small.

limitthemaximum

Explicit time steps may become too restrictive

for cases characterized by larger time scales or in

cases where there is a wide variation in the grid

resolution. In these cases it is desirable to

develop a fully implicit method in which the

time step is only determined by the physics of

the flow and not by the cell size. At each time

step the unsteady residual must be driven to zero

and this is usually done using inner iterations.

The number of inner iterations needed for each

time step is related to the problem being solved

and the size of the physical time step used.

To develop an unsteady solver that can provide

an accurate description in time of the flow is the

first requirement to address before attempting to

solve flows with massively separated regions

that are inherently unsteady. On the other hand,

it is also important to review the different

techniques available to compute turbulence,

which is at the core of separated flows, and will

be a key parameter to determine how well the

numerical scheme captures all the physical

aspects of the flow. In this sense, numerical

solutions of turbulent flow cases can be achieved

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using different levels of approximation. The

mostwidespread method

Reynoldsaveraged Navier-Stokes

(RANS). In the RANS equations, the turbulent

fluctuations appear in the Reynolds stress term

that must be modeled using any of the turbulence

models available in the literature. However, a

common limitation of these models is their lack

of generality since the model coefficients are

usually set using simple well-documented flows.

In this sense, current RANS solvers are fairly

successful at predicting mostly attached flows,

such as a wing in cruise condition, but fail to

capture a range of different off-design situations

as post-stall regimes,

configurations, and non-streamlined bodies. In

general, in the cases in which the RANS

approach fails, the flow is characterized by large

amounts of separation in which a very wide

range of scales is present in the flow. While the

small scales tend to be universal in nature, the

larger scales are affected by the boundary

conditions. This is the main cause of the lack of

generality of turbulence models, as it is difficult

to model the effect of the large scales in the same

way for many different types of flows.

isto solve

equations

the

certainhigh-lift

The failure to develop a universally valid

turbulence model has led to alternate approaches

such as Direct Numerical Simulation (DNS) and

Large Eddy Simulation (LES). DNS is the most

straightforward approach to the problem. DNS

consists in solving the governing equations on a

mesh fine enough to capture the smallest scales

contained in the flow with a scheme designed to

minimize thenumerical

dissipation. The drawback is extremely high cost

of the DNS computation, which is proportional

to at least Re3. For these reasons, DNS is

generally limited to very simple flows and low

Reynolds Numbers.

dispersionand

The flow limitations of RANS and the difficulty

of using DNS for realistic applied engineering

problems have generated a great interest in the

Large Eddy Simulation approach (LES) for

computingflowswith

separation. Large Eddy Simulation is a technique

betweenDirect Numerical

Reynolds Averaged Navier-Stokes [3]. In LES

the contribution of the large scales is computed

exactly and only the smallest scales in the flow

are modeled. However, a major difficulty of

LES,particularly for

flows, is that near solid surfaces all the eddies

are small and the “large” and “small” eddies tend

large amountsof

Simulationand

external aerodynamic

to overlap. Therefore the required grid spacing

and time step gradually fall towards DNS as the

solid boundary is approached [4 ].

Using LES to resolve near wall streaks would

bring an immense penalty at industrial Reynolds

numbers. In this line of reasoning the Detached

Eddy Simulation (DES) approach was conceived

with the idea of combining the strengths of

Reynolds Averaged methods near the solid

boundaries and of Large Eddy Simulation

elsewhere [1,4,5].

The ultimate goal of this work is to develop a

large eddy simulation capability based on an

existing unstructured grid Navier-Stokes solver

[6] to be able to perform detached eddy

simulations combing RANS near the walls and

LES in massive separated regions in a non zonal

manner and to compare the solutions obtained

using this approach to the solutions obtained

using a classical RANS approach with a Spalart-

Allmaras one equation turbulence model [6].

2.Unsteady Reynolds Averaged Navier-

Stokes Solver

2.1 Steady solver description

The Reynolds averaged Navier-Stokes equations

are discretized by a finite volume technique on

meshes of mixed elements, including tetrahedra,

pyramids, prisms and hexahedra. In general,

prismatic elements are used in the boundary

layerregions,while

elsewhere.

tetrahedra areused

Flow variables are stored at the grid vertices and

a single unifying edge-based data-structure is

used to handle all elements of the grid.

Convective and viscous fluxes are discretized

along edges using a central difference finite

volumetechniquewith

dissipation.

addedartificial

The non-dimensional steady conservative form

of the full Navier-Stokes equations can be

written as:

0)(

=

wR

(1)

where w represents the solution vector of

conserved variables and R represents the spatial

discretization operator,

vanishes at the steady state.

or residual, which

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An implicit solution procedure for solving

equation (1) begins with the linearization of the

residual about the current time step or iteration

level:

0)(

=∆⋅

∂

∂

++

∂

∂

w

w

R

wR

w

τ

n

(2)

which can be solved as:

{} )(

1

n

wR

w

RI

τ

w

−⋅

ÿÿ

ÿ

ÿÿ

ÿ

∂

∂

+

∆

=∆

−

(3)

where ÿ is a pseudo-time used to advance the

solution, and ∆w represents the new correction to

be applied to the solution. Rather than inverting

the large Jacobian in equation (3), a reduced

Jacobian which is simpler to invert is employed.

In regions of isotropic grid cells, only the block

diagonals of the Jacobian are retained, leading to

a block-Jacobi or point-implicit scheme. In the

boundarylayerregions,

stretching is present, lines are constructed in the

direction normal to the solid wall boundary and

the Jacobians along these lines are inverted,

using a block tridiagonal solution algorithm.

This procedure relieves the stiffness associated

with high grid stretching in these regions, thus

providing more rapid convergence [7 ].

wherehighgrid

This locally implicit scheme is used as the

smoother on all levels of an agglomeration

multigrid algorithm.The multigrid algorithm

constructs coarse level grids automatically by

fusing together neighboring fine grid control

volumes to form a smaller number of large

coarse grid control volumes.

acceleration is achieved by cycling back and

forth between the fine and coarse grid levels of

the multigrid sequence, using the locally implicit

solution procedure as a solver on each grid level

[7 ].

Convergence

The unstructured multigrid solver is parallelized

by partitioning the domain using a standard

graphpartitioner[8]

between the various grid partitions running on

individual processors using the MPI message-

passing library [9]. The solver can be run on

distributedorshared

including clusters of personal computers.

andcommunicating

memorymachines,

2.2 Unsteady term implementation

The unsteady form of the governing equations is

obtained by adding the time derivative to

equation (1):

0)(

=+

∂

∂

wR

t

w

(4)

where R(w) denotes the discretization of the

spatial derivative terms, as previously.

Making use of a second-order accurate three

point backward approximation for the time

derivative and evaluating R(w) at time level

(n+1), we obtain [10,11,12]:

0)(

2

1

⋅

2

∆

2

3

⋅

111

=+⋅

∆

+⋅−⋅

∆

+−+

nnnn

wRw

t

w

t

w

t

(5)

We may now define and unsteady residual as:

3

)(

+

∆⋅

t

with

2

),(

∆

),()(

2

1*

−

−=

nnwwSwRwR

(6)

11

2

1

⋅

−−

⋅

∆

−⋅=

nnnn

w

t

w

t

wwS

(7)

where wn+1is the approximation to w and the

source term S(wn,wn-1) remains fixed throughout

the solution procedure at each time step. In this

form, the non-linear problem to be solved at each

time-step reduces to R*(w) = 0.

redefinition of the residual, the same iterative

multigrid procedure employed to solve the

steady-state problem can be used to solve the

non-linear unsteady residual at each physical

time step [10,11,12].

Using this

The solution procedure consists of an outer loop

over the physical time steps, used to advance the

problem in time, and an inner (multigrid) loop

over pseudo-time used to drive the unsteady

residual to zero at each time level. While the

maximum size of the physical time step is only

determined by the physics of the problem, the

convergence of the inner sub-iterations for each

time step varies inversely with the size of the

physical time step.

3.Unsteady solver validation

The flow around a circular cylinder is a well-

known case, which has been widely studied

computationally and experimentally. This case is

used as the basis for validation of the unsteady

RANS solver, and for assessing grid resolution

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and

predicting

observed in the cylinder flow. Two different

meshes of 252,000 and 631,000 grid points and

three different time steps of 0.5, 0.25 and 0.1

were used. The time is non-dimensionalized as t

= to/(d/U∞) where d is diameter of the cylinder

and U∞is the freestream velocity.

timestep

the

requirements

vortex

for accurately

frequencyshedding

Figure 1: 3D View of cylinder and lateral walls

The one equation Spalart-Allmaras turbulence

model [2] was used for all calculations in fully

turbulent mode. In all cases the agglomeration

multigrid strategy was used with four levels. The

Mach number is 0.2 and the Reynolds number is

1200 for this case. All runs were performed in

parallel using 16 processors of a Pentium PC

cluster at ICASE.

The computational domain in the plane normal

to the cylinder span has an aspect ratio of 1 and a

side length of 100 cylinder diameters. A span of

two cylinder diameters is employed, and inviscid

(slip velocity) boundary conditions are applied at

theend-walls. The

simulations reported herein were also compared

withtwo-dimensional

aroundacircleusing

dimensional unstructured solver [12], and found

to agree well in terms of force coefficient

histories and shedding frequency.

three-dimensional

simulations

a

offlow

two-validated

Table 1 shows the Strouhal Numbers computed

for each mesh and each time step of the three-

dimensionalsimulations.

achieved as the time step is reduced and the

mesh size increased. A second-order accuracte

convergence behavior is observed as the time-

step is reduced, validating the accuracy of the

Convergenceis

three-point backwards difference scheme used to

discretize the time step. From the smallest time

step results, the solution can be seen to be grid

converged, at least with respect to the prediction

of the vortex shedding frequency. The computed

Strouhal number compares very well to the

experimental value of St = 0.21 given by Dresher

[13,14]. Figure 2 shows the time history of the

lift coefficient, while the oscillatory pattern

corresponding to the vortex shedding is shown in

Figure 3.

Time Step

0.250.50.1

0.252

Million

0.631

Million

0.19249 0.203040.20833

Grids

0.193790.20408 0.20833

Table 1. Predicted Strouhal Number for Various

Grid and Time Step Sizes

Figure 2: Mach contours at two different time

states for flow over circular cylinder. Mach =

0.2, Re = 1200.

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.

Figure 3. Computed LiftCoefficient Time History

for Flow over Circular Cylinder using three

different time steps. Mach = 0.2, Re = 1200

Figure 4. Mean Surface Pressure Coefficient

Distribution for Flow over Circular Cylinder at

Mach = 0.2, Re = 1200, Compared with

Experimental Data at Various Reynolds

Numbers

Figure 4 shows the pressure distribution over the

surface of the cylinder. The pressure distribution

was computed by averaging results at different

times distributed along several oscillations. For

this calculation the finest mesh and the smallest

time step was used. As can be observed from

Figure 4, the computed pressure distribution

compares closely to experimental results at a

higher Reynolds number than the one used for

these computations. This is likely due to the use

of the turbulence model in fully turbulent mode,

in order to avoid the issues of transition

prediction. Similarly, the backpressure obtained

of Cpb= -1.20 compares closely to the Cpb= -

1.20 at Re=27,700 measured by Linke [13,15].

The mean value of the drag coefficient computed

is Cd=1.3 compared to Cd=1.2 as measured by

Wieselsberger [13,16] for Re=30,000.

4. Detached Eddy Simulation.

Detached Eddy Simulation (DES) is a hybrid

technique that combines RANS and LES in a

non-zonal manner. DES is based on the Spalart-

Allmaras one equation RANS model [2] in

which the length scale d, which is traditionally

taken as the nearest distance at any given point to

the closest wall, is replaced as the minimum

between the distance to the wall and a length

proportional to the local grid spacing:

dDES= min (d, CDES∆x)

where CDESrepresents a model constant which

has been taken as 0.65 in previous work [1,17].

Traditionally, on structured grids [1,17,18], ∆x is

taken as the maximum grid spacing over all three

directions. In our particular case, the definition

of ∆x has been modified for unstructured grids

by taking it as the maximum edge length

touching a given vertex. In boundary layer

regions, ∆x far exceeds the distance to the wall

d, and the standard Spalart-Allmaras RANS

turbulence model is recovered. However, away

from the boundaries d exceeds CDES∆x and the

models turns into a simple one equation sub-

grid-scale (SGS) model with the mixing length

proportional to the grid spacing. This effect is

illustrated by plotting contours of the distance or

length scale function for both the RANS and

DES models in Figure 5, where it is observed

that both models employ the same length scales

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near the wall, but use vastly differing length

scales in the regions far removed from the wall,

where the DES model reverts to an LES mode

and a Smagorisnky–like expression for the eddy

viscosity is obtained.

URANSDES

Figure 5. Distance function/Length Scale

comparison between URANS and DES.

5.Validation.

5.1 Flow around a Sphere.

DES is applied to predict the flow around a

sphere. Similarly to the flow around a circular

cylinder, this case has been widely studied using

experimental and computational approaches. It

has been shown that over a wide range of

Reynolds numbers (280< Re < 3.7x105) the flow

is characterized by the vortex shedding with

large-scale vorticity emanating from the shear

layer which separates from the surface of the

sphere. In our case a Reynolds number of 104is

used, corresponding to the sub-critical regime

(laminar boundary layer separation). This is

similartothecomputationsperformedby

Constantinescu

comparative basis for our results.

[17],whichprovidesa

The main goal of this case is to compare the

results obtained using traditional URANS and

DES. The comparison includes the time history

of integral parameters such as drag coefficient

and mean distribution of pressure around the

sphere.

5.2.1 Numerical approach.

The flow over a sphere is calculated for a

Reynolds number of 104and a Mach number of

0.2 on an unstructured mesh of 767,000 vertices.

The computational domain is a cube of 100

sphere diameter lengths in each direction. This

large domain is chosen to ensure that the

downstream boundary condition does not lead to

spurious oscillations within the domain. Near the

surface of the sphere, the normal grid spacing is

10-4× d, where d represents the diameter of the

sphere.

For the unsteady calculations two different time

steps of 0.1 × d/U∞and 0.05 × d/U∞were used.

In all cases the multigrid strategy was used with

four grid levels. All runs were performed on 32

processors of a Pentium PC cluster at ICASE.

For the DES runs, the CDESvalue used was 0.65.

This value was chosen based on previous studies

by Shur et al. [1], where the value was calibrated

in isotropic turbulence.

Constantinescu et al. [17]

CDES

asthemost

calculations of the flow over a sphere at Re =

104. A parallel study is being conducted to

examine the effects of variations in the value of

CDES using the same test case of decaying

homogeneous turbulence in a box as done by

Shur et al. [1].

Moreover,

prove this value of

appropriate fortheir

5.2.2 Code validation.

Following the procedure used by Constantinescu

et al. [17], the accuracy of the numerical

approach was established by comparing the

results obtained in the steady regime at Reynolds

number 250 with previous computational and

experimental results.

The drag coefficient is compared in Table 2 to

other simulation results and experimental data.

Because previousresultswerebased on

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incompressible simulations, and the current

solveris adensity

formulation, the importance of compressible

effects was also investigated by running the

simulation at Mach numbers of 0.2 and 0.1, both

withandwithouta

preconditioner [18,19].

satisfactory for all the cases tested.

basedcompressible

low Machnumber

The agreement is

Cd

Constantinescu et

al.

(2000)

Johnson and Patel

(1999)

0.70

0.70

Experimental

0.70 - 0.72

M = 0.1

0.7141

M = 0.2

0.7015

M = 0.1

Low Mach

Number pre-

conditioner.

M = 0.2

Low Mach

Number pre-

conditioner.

0.6961

0.6950

Table 2. Computed Steady Drag Coefficient for

Flow over Sphere at Re = 250 compared with

Experimental and Previous Computational

Values

5.2.3 Unsteady Results.

For the unsteady runs, the flow around a sphere

is computed at a Mach number of 0.2, without

any additionallow

preconditioning, and a Reynolds number of 104.

At this Reynolds number the detached vortex

sheet from the sphere is fully turbulent while the

boundary layer on the sphere remains laminar.

The Strouhal number associated with the vortex

shedding at this Reynolds number is in the range

of 0.185-0.200 depending on the investigation.

The large scatter of the data is mainly due to the

influencingparameters

methods of the different investigations.

Mach number

and themeasuring

Figure 6. DES Time history of Drag Coefficient

for flow over Sphere at Mach = 0.2, Re = 10,000

using two different time steps

Figure 7. URANS Time history of the drag

coefficient for flow over Sphere at Mach = 0.2,

Re = 10,000 using two different time steps

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Figure 9.. URANS and DES Average Surface

Pressure Distributions Compared with

Experimental Data at Various Reynolds

Numbers for flow over Sphere

Mach = 0.2, Re = 10,000s

The time history of the drag coefficient is shown

in Figures 6 and 7 and reveals important

differences between URANS and DES. The

mean value of the drag coefficient in both cases

is close to the experimentally reported value of

0.40from Schlicting

frequency content in each case

different. The URANS simulation appears to

damp out most of the oscillations present in the

DES run, while the DES runs show a very

[20].However,

is completely

the

chaotic oscillatory pattern quite similar to the

solutions obtained by Constantinescu et al [17].

Spectral analysis of the time-dependent drag

coefficient history reveals a peak corresponding

to a Strouhal number of 0.1 which is not in

agreement with the values 0.18 to 0.2 reported

experimentally [21].This may be due to an

insufficiently long time history sample, since

less than three full periods of this frequency are

present in our sample.

The Mach number contours depicted in Figure 8

corroborate the difference in the predicted flow

using regular Spalart-Allmaras RANS turbulence

model and detached eddy simulation (DES).

DES shows a wider range of scales present in the

flow while regular RANS models tend to smooth

out the smaller scales. This is the effect expected

from DES since the length scale redefinition

increases the magnitude of the destruction term

in the Spalart-Allmaras model, drawing down

the eddy viscosity and allowing instabilities to

develop. Predictions of the mean pressure

distribution over the surface of the sphere are

shown in Figure 9. The surface pressure is seen

to match well with experimental results at

Re=165,000 reported by Achenbach [21], which

is in agreement with the results reported by

Constantinescu [17]. The average computed

separation angle of 81ocompares resonably to

experimental vale of 82.5o.

5.3 Flow over a NACA0012 airfoil.

RANS and DES were used to compute the flow

over a NACA0012 airfoil at a Reynolds number

of 105and a Mach number of 025. Runs were

performed for 10 different angles of attack

ranging from 0 to 16. Our main interest was to

study and compare RANS and DES in the stall

regime. Other studies (Shur et al. [1]) have

focused their interest in higher angles of attack,

such as 45, 60 or even 90 degrees, showing very

good and promising results in which DES is

within 10% of the experimental values, while

traditional RANS predict results with 40%

errors. All runs are computed as fully turbulent

to avoid having to trigger the transition point. A

mesh of 966,00 node points is used with a first

grid spacing normal to the solid boundary of 10-5

chords. The spanwise boundary conditions are

slip (inviscid)walls, and the span of the

geometry is two chords lengths.

In this case, due to time and computing

constraints, only one time step of 0.25 × c/U∞

Figure 8. Comparison of URANS and DES

Mach Contours for flow over sphere.

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was used, where c is the airfoil chord. The

calculations were carried out for a maximum of

50 time units for the post-stall cases, where a

time unit represents the time it takes for the

undisturbed far-field flow to travel one chord

length.

5.3.1 Results.

The time history of the lift and drag coefficient

show good agreement between URANS and

DES for angles of attack bellow 11.5 degrees.

This was expected since for pre-stall conditions

the DES model operates primarily in the RANS

mode. However, for the post-stall condition, i.e.

angles of attack over 11.5, the time history of the

force coefficients obtained using URANS and

DES showed differences similar to the ones

observed for the sphere case. The time history of

the DES results shows higher unsteadiness that

URANS, indicating that more scales are being

captured in the separated region. The DES

simulation predicts a more severe stall than the

URANS results, i.e. lower post-stall lift and

higher drag. However, both computations stall at

thesameincidence

experimental data [22] is not sufficiently close to

favor agreement with one method over the other.

This is in contrast to previous work [1] where

DES was reported to provide closer agreement

with experiement for very high angles of attack,

such as 45 to 90 degrees.

attention was devoted to the prediction of the

onset of stall in [1].

andcomparisonwith

However, little

Figure 10: Sample URANS Drag Coefficient

Time History for Flow over NACA0012 Wing at

Various Angles of Attack

Figure 11. Sample DES Drag Coefficient Time

History for Flow Over NACA0012 Wing at

Various Angles of Attack

Figure 12.

Coefficient versus Angle of Attack for URANS

and DES versus Experimental Data [22] at two

Different Reynolds Numbers

Comparison of Computed Lift

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Figure 13..

Coefficient versus Angle of Attack for URANS

and DES versus Experimental Data [22] at two

Different Reynolds Numbers

Comparison of Computed Drag

6.Conclusions.

A steady-state unstructured multigrid solver was

extended to unsteady flows and validated by

computing the flow over a circular cylinder.

Results show good agreement with experimental

results in Strouhal number, pressure distribution ,

and flow patterns over the cylinder.

A Detached Eddy Simulation (DES) technique

was implemented in the code and compared with

regular URANS by computing the flow over a

sphere and the flow over a NACA0012 airfoil.

The results for the sphere show that DES

provides a better description of the flow in the

separated regions. This was concluded from the

time history of the force coeffcients and from the

pressure distribution over the surface of the

sphere. The time history of the drag coefficient

computed using DES showed small scales

present in the flow that URANS did not capture.

The Strouhal number obtained does not match

the experimental value but longer time histories

are required for a more definite determination of

this values.

Prediction of the onset of stall for a NACA0012

wing was not substantially improved using the

DES approach over the URANS approach.

However, DES has been reported to provide

better agreement with experiement at very high

angles of attack, and this will be investigated in

future work. Additional work is also underway

to simulate the decay of homogeneous isotropic

turbulence in a periodic domain, in order to

calibrate the DES model for unstructured grids.

6.

References..

1.Shur M., Spalart P. R., Strelets M., Travin

A. “Detached-eddy Simulation of an Airfoil

at High Angle of Attack”, 4thInt. Symp.

Eng. Turb. Modeling and Measurements,

May 24- 26, 1999, Corsica.

2. Spalart, P. R. and Allmaras S. R., “A one

equation turbulence model for aerodynamic

flows”, La Recherche Aerospatiale, 1, pp 5-

21., 1994

3. Piomelli, U. “Large –eddy Simulation of

Turbulent Flows’’ TAN Report No. 767.

UILU-ENG-94-6023, September 1994.

4. Spalart, P. R., Jou, W. H., Strelets, M. and

Allmaras, S. R., “Comments on the

Feasibility of LES Wings and on Hybrid

RANS/LES Approach”,First AFOSR

InternationalConference on DNS/LES,

Rouston, Louisiana, USA, 1997.

5.

Shur, M., Spalart P. R., Strelets, M., and

Travin, A. K., “Navier-Stokes simulation of

shedding turbulent flow past a circular

cylinder with a backward splitter plate” 4th

Int. Symp. Eng. Turb. Modeling and

Measurements, May 24- 26,1999,Corsica .

6. Mavriplis, D.J. , ``Viscous Fow Analysis

Using a Parallel Unstructured Multigrid

Solver’’, AIAA Journal, Vol 38, No. 11

pp.2067-2076, Nov 2000.

7.Mavriplis, D. J. ``Directional Agglomeration

Multigrid Techniques for High-Reynolds

Number Viscous Flows’’, AIAA paper 98-

0612, Jan 1998.

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