Post on 23-Mar-2018
transcript
Compressible Navier-Stokes (Euler) Solver based on Deal.II Library
Lei Qiao
Northwestern Polytechnical University
Xi’an, China
Texas A&M University
College Station, Texas
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Fifth deal.II Users and Developers Workshop Texas A&M University, College Station, TX, USA
August 5, 2015
Outline
Motivation and Goal
Current work
Conclusion and TODOs
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Motivation
This is what I typically need to compute
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The pain pointMesh generation:
Time consuming Boring Experience depending
Solution: Mesh adaptation — let solver tell what is good mesh
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Deal.II the library
• Mesh adaptation with tree data structure • FEM discretization • Parallelization and excellent scalability
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GoalA N-S solver based on deal.II that: • Starts from initial value on coarse mesh • Converges to solution on a reasonable fine
mesh adaptively • With High-order capability
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Current workGoverning Equation
Solving technique
Solver implementation
Solver verification
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Governing Equation
Compressible Navier-Stokes Equation
@Q(w)
@t+r · F(w) = S(w)
Use primitive variables as working var.
w = (uj , ⇢, p)T
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Governing Equation
Fc(w) =
0
@⇢u⌦ u+ Ip
⇢u(E + p)u
1
A =
0
@⇢uiuj + �ijp
⇢ui
(E + p)ui
1
A
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The Flux
F = Fc � Fv
Governing EquationThe Flux
F = Fc � Fv
WithT =
p
�R⇢
⌧ij = µ(@ui
@xj+
@uj
@xi) + ��ij
@uk
@xk
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Fv
(w) =1
Re⇤1
0
@⌧ij
0⌧ij
ui
+ @T
@xj
1
A
Governing Equation
Weak form:
w = (uj , ⇢, p)T
Z
⌦vl@Ql(w)
@t
+
Z
⌦vl@Fl,i(w)
@xi=
Z
⌦vlSl(w)
8v = vl 2 T
, Find w 2 S that satisfy On domain ⌦
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Governing Equation
Weak form:
Integrate by part to get boundary flux
Z
⌦vl@Ql(w)
@t
+
Z
@⌦vlFl,i(w)ni
�Z
⌦
@vl
@xiFl,i(w) =
Z
⌦vlSl(w)
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Governing EquationTreat boundary conditions and discontinuities on hanging node with this boundary flux
Compute numerical flux with Roe scheme[1]
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Governing Equation
Boundary conditions:
Far field: Riemann invariant
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Slip wall: Non-penetration
u · n = 0⇢out
= ⇢in
pout
= pin
Solving technique
BDF-1 (Implicit Euler) time integration
Newton linearization
Direct linear solver from Trilinos** Iterative solver doesn’t stable enough, only used for scalability test
For steady run, time step size is determined according to norm of Newton update[2]
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Solver implementation• Starts from step-33 of deal.II tutorial
• Change working vars. from conservative ones to primitive ones • viscous flux • Riemann boundary condition • Roe flux[2] to replace the too viscous Lax flux • Parallelize: learned from step-40
• Version control with Git • CMake project • Regression test suit
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Solver verification• Scalability • Manufactured solution[3]
• Subsonic • Supersonic
• Adaptive simulation over circle • Flow over foil NACA2412
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Scalability22,288 cells 91,312 DoFs
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GMRES solver from Trilinos with DD+ILUT precond.
Solver verification• Manufactured solution[3]
@Q(w)
@t+r · F(w) = S(w)
w = J(x, y) =
8>><
>>:
u(x, y)= u0+u
x
sin(a
ux
⇡x)+u
y
cos(a
uy
⇡y)+u
xy
cos(a
uxy
⇡xy)
v(x, y)= v0+v
x
cos(a
vx
⇡x)+v
y
sin(a
vy
⇡y) +v
xy
cos(a
vxy
⇡xy)
⇢(x, y)= ⇢0+⇢
x
sin(a
⇢x
⇡x)+⇢
y
cos(a
⇢y
⇡y)+⇢
xy
cos(a
⇢xy
⇡xy)
p(x, y)= p0+p
x
cos(a
px
⇡x)+p
y
sin(a
py
⇡y) +p
xy
sin(a
pxy
⇡xy)
S(x, y) =@Q(J)
@t
+r · F(J)
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SolverJh(x, y)
Solver verification• Manufactured solution[3]
w = J(x, y) =
8>><
>>:
u(x, y)= u0+u
x
sin(a
ux
⇡x)+u
y
cos(a
uy
⇡y)+u
xy
cos(a
uxy
⇡xy)
v(x, y)= v0+v
x
cos(a
vx
⇡x)+v
y
sin(a
vy
⇡y) +v
xy
cos(a
vxy
⇡xy)
⇢(x, y)= ⇢0+⇢
x
sin(a
⇢x
⇡x)+⇢
y
cos(a
⇢y
⇡y)+⇢
xy
cos(a
⇢xy
⇡xy)
p(x, y)= p0+p
x
cos(a
px
⇡x)+p
y
sin(a
py
⇡y) +p
xy
sin(a
pxy
⇡xy)
Choosing different Constants to get subsonic
or supersonic
case
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Solver verification• Manufactured solution
• Subsonic • convergence
order
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Solver verification• Manufactured solution
• Supersonic • convergence
order
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Solver verification• Adaptive simulation over circle
• C1 mapping on wall boundary • Subsonic: Mach = 0.1 • Almost inviscid: Cv=1e-6 just for stabilization • Converged Cd=0.00029 which ideal value is zero
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Solver verification• Adaptive simulation over cylinder
Converged density contour
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Solver verification• Adaptive simulation over cylinder
Converge history of density contour and mesh adaptation
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Solver verification• Flow over NACA2412 foil
• Subsonic: Mach=0.3
Converged pressure contour
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Solver verification• Flow over NACA2412 foil
• Subsonic: Mach=0.3
Oscillation may be caused by geometry non-smoothness
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Solver verification
• Almost inviscid: Cv=3e-6 just for stabilization • Converged Cd=0.00025 which ideal value is zero • Converged Cl=0.2576, Cm=0.05235
• As a reference, xFoil[4] gives • Cd=-0.00076 • Cl=0.2704 • Cm=0.0585
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• Flow over NACA2412 foil • Subsonic: Mach=0.3
Conclusion• A prototype NS solver based on deal.II is constructed • The solver could run in parallel but the solver doesn’t
scale perfectly • High order convergence is confirmed by MMS • Solving process could start from very coarsen mesh
and go on with mesh adaptation • The solver can give out reasonable aerodynamics data
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ToDos• Stable and efficient preconditioner for iterative solver
(Now my hope is on MDF ordering of ILU) • Describe wall boundary with NURBS geometry • Non-slip boundary condition • Anisotropic adaptation for boundary layer
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Reference1. J. Blazek. Computational Fluid Dynamics: Principles and
Applications (Second Edition). Elsevier Science, Oxford, second edition edition, 2005. ISBN 978-0-08- 044506-9.
2. J. Gatsis. Preconditioning Techniques for a Newton–Krylov Algorithm for the Compressible Navier–Stokes Equations. PhD thesis, University of Toronto, 2013.
3. C. J. Roy, C. C. Nelson, T. M. Smith, and C. C. Ober. Verification of Euler/Navier- stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44(6):599--620, 2004.
4. M. Drela, H. Youngren. http://web.mit.edu/drela/Public/web/xfoil/
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Thanks
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