Implicit Large Eddy Simulation of Transitional Flows Over...
Transcript of Implicit Large Eddy Simulation of Transitional Flows Over...
Implicit Large Eddy Simulationof Transitional Flows Over Airfoils and Wings
A. Uranga(*), P.-O. Persson(**), J. Peraire(*), and M. Drela(*)
(*) Department of Aeronautics & AstronauticsMassachusetts Institute of Technology
(**) Department of MathematicsUniversity of California, Berkeley
19th AIAA Computational Fluid Dynamics Conference
June 24, 2009
Outline
1 Introduction
2 MethodologyGoverning Equations and Turbulence ModelingDiscontinuous Galerkin MethodComputational Grids
3 ResultsLaminar Regime: Re = 10,000Transitional Regime: Re = 60,000
4 Conclusions
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 1 / 27
Introduction
Accurate prediction of transition is of crucial importance at low Re
transition location has significant impact on aerodynamic performance
Formation of laminar separation bubble (LSB)
Laminar BL separates in adverse pressure gradientSeparated flow rapidly transitions to turbulenceSubsequent reattachment of turbulent BL
Flow regime encountered in small aircraft and MAVs
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 2 / 27
Introduction
Goal: predict formation of LSB and subsequent transition
Flow around rectangular SD7003 wing at angle of attack of 4
flow exhibits LSB on upper surfaceextensive experimental data available [1]numerical simulations performed by other groups [2]
Laminar regime: Re = 10,000
Transitional regime: Re = 60,000
Implicit Large Eddy Simulations (ILES) using high-orderDiscontinuous Galerkin (DG) method
[1] Ol, M., McAuliffe, B., Hanff, E., Scholz, U., and Kahler, C., “Comparison of laminar separation bubbles measurements on alow Reynolds number airfoil in three facilities”, AIAA-2005-5149, 2005.
[2] Galbraith, M. and Visbal, M., “Implicit Large Eddy Simulaion of low Reynolds number flow past the SD7003 airfoil”,AIAA-2008-225 , 2008.
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 3 / 27
Methodology Governing Equations
Compressible Navier-Stokes equations and ideal gas law
∂ρ
∂t+
∂
∂xi(ρui ) = 0
∂
∂t(ρui ) +
∂
∂xj(ρuiuj) = − ∂p
∂xi+∂τij∂xj
for i ∈ 1, 2, 3
∂
∂t(ρE ) +
∂
∂xj[uj (ρE + p)]− ∂
∂xj(uiτji ) +
∂qj
∂xj= 0
p = (γ − 1)ρ
(E − 1
2ukuk
)
where τij ≡ µ[(
∂ui
∂xj+∂uj
∂xi
)− 2
3
∂uk
∂xkδij
]
qj = − µ
Pr
∂
∂xj
(E +
p
ρ− 1
2ukuk
)
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 4 / 27
Methodology Simulation of Turbulence
Large Eddy Simulation (LES)
large-scale motions are resolvedsmall scales modeled through sub-grid-scale model
Implicit LES (ILES)
unresolved scales accounted for by numerical dissipation(no sub-grid-scale model)solving the full compressible Navier-Stokes equations
ILES approach previously used by Visbal and collaborators [2] withcompact difference method for flow around SD7003
High computational cost ⇒ benefits from high-order methods
[2] Galbraith, M. and Visbal, M., “Implicit Large Eddy Simulaion of low Reynolds number flow past the SD7003 airfoil”,
AIAA-2008-225 , 2008.
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 5 / 27
Methodology Discontinuous Galerkin Method I
High-order Discontinuous Galerkin (DG) method:
High-order, low dissipation
Unstructured meshes
System of conservation laws
∂u
∂t+∇ · Fi (u)−∇ · Fv (u, q) = 0
q −∇u = 0
Domain Ω triangulated into elements K ∈ Th
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 6 / 27
Methodology Discontinuous Galerkin Method II
Seek approximate solutions uh ∈ V ph , qh ∈ Σp
h in spaces ofelement-wise polynomials of order p
V ph = v ∈ [L2(Ω)]m | v |K ∈ [Pp(K )]m, ∀K ∈ Th
Σph = w ∈ [L2(Ω)]dm | r |K ∈ [Pp(K )]dm, ∀K ∈ Th
Multiply by test functions v , w and integrate over element K∫K
qh · w dx = −∫
K
uh∇ · w dx +
∫∂K
uw · n ds , ∀w ∈ [Pp(K )]dm,∫K
∂uh
∂tv dx −
∫K
[Fi (uh)− Fv (uh, qh)] · ∇v dx
= −∫∂K
[Fi − Fv ] · nv ds , ∀v ∈ [Pp(K )]m.
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 7 / 27
Methodology Discontinuous Galerkin Method III
Numerical fluxes Fi , Fv , u are approximations to Fi , Fv , u on boundary ∂Kof element K
Roe scheme for inviscid fluxes Fi
CDG method for viscous fluxes Fv
flux u only function of uh (not of qh)
Resulting system of coupled ODEs solved with incomplete factorizations(ILU) and p-multigrid with Newton-GMRES preconditioning
Code parallelized with block-ILU factorizations
Error O(hp+1) for smooth problems
Time stepping with 3rd order diagonal implicit Runge-Kutta (DIRK)method. Time step ∆t∗ = ∆t U∞/c = 0.01
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 8 / 27
Methodology Computational Grids
SD7003 at 4 AoA, span 0.2c
Domain [-4.3c , 7.4c]×[-6.0c , 5.9c]×[0 , 0.2c]
Span-wise periodic BC
Polynomials of order p = 3→ 4th order method in space
52,800 tetrahedral elements
1,056,000 DOFs
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 9 / 27
Outline
1 Introduction
2 MethodologyGoverning Equations and Turbulence ModelingDiscontinuous Galerkin MethodComputational Grids
3 ResultsLaminar Regime: Re = 10,000Transitional Regime: Re = 60,000
4 Conclusions
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 10 / 27
Results Laminar Regime: Re = 10,000
Flow is essentially 2D
Close-to-periodic vortex shedding
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 11 / 27
Results Laminar Regime: Re = 10,000
Separation at 34%
No transition along airfoil
No reattachment
Velocity correlations
Average streamlines and contours of average velocity magnitude
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Results Laminar Regime: Re = 10,000
Span-wise vorticity ωz on wing’s middle plane
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 13 / 27
Results Laminar Regime: Re = 10,000
Vortical structures: iso-surfaces of q-criterion (∇2p/2ρ)
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 14 / 27
Outline
1 Introduction
2 MethodologyGoverning Equations and Turbulence ModelingDiscontinuous Galerkin MethodComputational Grids
3 ResultsLaminar Regime: Re = 10,000Transitional Regime: Re = 60,000
4 Conclusions
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 15 / 27
Results Transitional Regime: Re = 60,000
3D solution very different from 2D solution
Significant 3D vortical structures present
Non-periodic vortex shedding
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 16 / 27
Results Transitional Regime: Re = 60,000
Separation at 24%
Transition at 51%
Reattachment at 60%
Velocity correlations
Average streamlines and contours of average velocity magnitude
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 17 / 27
Results Transitional Regime: Re = 60,000
Average pressure and skin friction coefficients
Separation and transition well captured
Good agreement with XFoil and previously published ILES [Galbraith& Visbal 2008]
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Results Transitional Regime: Re = 60,000
Span-wise vorticity ωz on wing’s middle plane
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 19 / 27
Results Transitional Regime: Re = 60,000
Vortical structures: iso-surfaces of q-criterion (∇2p/2ρ)
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Results Transitional Regime: Re = 60,000
Boundary layer integral parameters of time-average flow
Pseudo-velocity profile ~u∗(n) =∫ n
0 ~ω × n dn(asymptotes outside BL even with strong curvature)
BL edge ne defined where |~ω|n < ε0|~u∗| and |d~ω/dn|n2 < ε1|~u∗|Streamwise pseudo profile u1(n) = ~u∗(n) · ~u∗e/u∗e
displacement thickness, momentum thickness, shape factor
δ∗1 =∫ ne
0
(1− u1
ue
)dn θ11 =
∫ ne
0
(1− u1
ue
)u1
uedn H11 =
δ∗1θ11
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 21 / 27
Results Transitional Regime: Re = 60,000
Transition Mechanism
Fluctuating stream-wise pseudo-velocity
u′1(~x , t) = u1(~x , t)− u1(x)
Consistent withTollmien-Schlichting (TS)modes
Perturbation amplitudeincreases downstream
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 22 / 27
Results Transitional Regime: Re = 60,000
Transition Mechanism
Amplification of stream-wise TS waves at any chord-wise location x :
A1(x) =1
neue(x)
√∫ ne
0
u′12 dn
Stream-wise amplification factor: N1(x) = ln(
A1(x)A10
)(all waves)
XFoil line: N-factor of thesingle most-amplified waveat given location
Similar slopes
⇒ TS transition
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 23 / 27
Outline
1 Introduction
2 MethodologyGoverning Equations and Turbulence ModelingDiscontinuous Galerkin MethodComputational Grids
3 ResultsLaminar Regime: Re = 10,000Transitional Regime: Re = 60,000
4 Conclusions
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 24 / 27
Conclusions
Use of DG method for ILES of low Reynolds number flows
At Re = 10,000, flow laminar and essentially 2D over wing surface
At Re = 60,000, transition associated with LSB observed
Transition found to be the result of unstable TS waves
Remarkable agreement with XFoil predicitons, finer LES results[Visbal and collaborators], experimental data [Ol and collaborators]→ in spite of use of relatively coarse meshes
Results suggest that DG is particularly suited to simulate these flows
Uranga, Persson, Peraire, Drela (MIT) June 24, 2009 25 / 27
Acknowledgments
Air Force Office of Scientific Research (AFOSR) support underMURI project Biologically Inspired Flight
Scholarship from Mexican’s National Council for Science andTechnology (CONACYT)
NSF through TeraGrid resources provided by TACC
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Questions
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