Finite difference methods for the Navier-Stokes
equations
Magnus Svärd
School of Mathematics
University of Edinburgh
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A challenging flow problem
[SR-71, shock diamonds]
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Shocks
Turbulence
Sound propagation
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The Navier-Stokes equations
Conservation form:
∂ρ ∂ρuj
+
=0
Mass
∂t
∂xj
∂ρui uj
∂ρui
∂p
1 ∂τij
+
+
=
Momentum
∂t
∂xj
∂xi
Re ∂xj
∂ρ(E + p)uj
∂qj
∂ρE
1 ∂uj τij
+
=
+
Energy
∂t
∂xj
Re
∂xj
∂xj
Equation of state: E =
I
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Re-Reynolds number
Ma-Mach number
T
1
+ uj2
γ(γ − 1) 2
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Objectives
Make a robust and accurate predictive tool for aerodynamics...
Multiblock high-order finite difference method.
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Linear stability for initial-boundary value problem. (Done)
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Non-linear stability and shock capturing. (In progress...)
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A linear model problem
∂u ∂u
+
= 0, t ≥ 0,
∂t
∂x
u(0, t) = g (t)
0≤x ≤1
u(x, 0) = f (x)
Energy method:
Z
1
Z
1
∂u
dx =
∂x
0
0
u(0, t)2 − u(1, t)2 = g (t)2 − u(1, t)2
kuk2t
=2
uut = −2
u
Bounded growth.
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Well-posedness
A bound on the solution is the key to prove well-posedness:
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Uniqueness
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Existence
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Stable to perturbations
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Summation by parts (SBP)
(Kreiss&Scherer -74)
Idea: Mimic the continuous energy estimate. (FEM) Grid: ih,
i = 0...N, h is the grid size.
Semi-discretization: v T = (v0 (t), v1 (t), ...vN (t)).
Derivative approximation: Dv = vx + O(hp ).
I
D = P −1 Qv
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Q + Q T = diag (−1, 0, ..., 0, 1).
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kv k2P = v T Pv , P diagonal positive definite.
Truncation errors are 2p in the interior and p near the boundary.
Convergence rate: p + 1.
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Energy estimate
Semi-discretization of PDE:
vt + P −1 Qv = 0
Energy estimate
(kv k2P )t = v T Pvt + vtT Pv = −v T Qv − v T Q T v = v02 − vN2
How do we impose the boundary condition?
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Weak imposition of boundary conditions
(Carpenter et al -94)
Simultaneous Approximation Term - SAT - Penalty Term
E0 = diag (1, 0, ...., 0)T
vt + P −1 Qv = σP −1 E0 (v − g )
P −1 ∼ 1/h. Dirac mass enforces the boundary condition.
Energy method:
(kv k2P )t = v02 − vN2 + 2σv0 (v0 − g (t))
Stable if σ ≤ −1/2.
stable + accurate ⇒ convergent approximation
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1-D Navier-Stokes equations
In 1-D the system takes the form:
ut + (F I −
1 V
F )x = 0,
Re
0≤x ≤1
with 3 independent variables.
Linearize, symmetrize and freeze coefficients:
wt + Awx =
1
Bwxx ,
Re
0≤x ≤1
(1)
where A, B are symmetric. B does not have full rank.
If (1) is well-posed, then the non-linear problem is well-posed for
small perturbations to smooth solutions.
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Energy estimate
Ignoring the right boundary...
Z 1
Z 1
1
2
wBwxx dx
kw kt + 2
wAwx dx =2
0
0 Re
Z 1
1 T
2
2
kw kt + 2
wx Bwx dx =w0T (Aw0 −
B(wx )0 )
Re
0 Re
With standard assumption on the Lame’ coefficients B is positive
semi-definite and the quadratic term is dissipation.
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Energy stability for...
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SBP-SAT finite difference up to order 5.
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Multi-D.
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Curvilinear grids
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Multi-block grids
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Wall, far-field, pressure outflow boundary conditions.
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SBP artificial diffusion.
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Flow around circular cylinder
y
20
0
-20
-20
0
20
40
x
(b) Ma = 0.1, Re = 100, 5th-order
scheme
Correct prediction of:
I Lift
I Drag
I Strouhal (shedding frequency)
I Separation angle.
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Delicate benchmark case
(Re=5000,Ma=0.5, 3rd-order scheme)
y
0.05
0
-0.05
0.85
Separated but steady flow.
0.9
0.95
1
x
1.05
1.1
1.15
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Further benchmarking...
Linear theory is reasonably complete but turbulence computations
have not been done. (Apart from a low-order FV code.)
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Turbulence in a box.
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Cylinder Re = 3900
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Backward facing step/Cavity at high Reynolds numbers.
High-order finite difference schemes are, by some, considered
ill-suited for turbulence computations.
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HPC challenges.
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Turbulence computations are very, very large.
Explicit time stepping scheme are easiliy parallelized and gives
good load balance. Effective enough?
Scalability on a large number of processors?
Future developments:
I Add openMP to existing MPI code. Easier to construct grids.
I Dynamic load balancing, or at least automated split onto
processors during initialization.
I GPUs?
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Relatively easy (?) to port the main interior computations to
CUDA.
Much logic in the boundary treatment that must be handled
by CPU. Load balance??
Will the CUDA routines run independently or will there be a
lot of communication between GPU and CPU?
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Non-linear stability theory and boundary conditions.
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Linear stability gives a robust scheme for smooth flows.
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When shocks develop the theory breaks down and solution
usually explodes. (It is not the discontinuity per se that is the
problem. Contact discontinuities may be stable.)
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Linear schemes have an inherent ambiguity in the enforcement
of boundary conditions.
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Conservation Laws and Weak Solutions
A system of conservation laws in multi-D.
ut +
d
X
fi (u)xi = 0,
u(x, 0) = u0 (x)
i=1
A weak solution u satisfies:
Z Z
Z
uϕt + f (u)ϕx dxdt +
u0 (x)ϕ(x, 0)dx = 0,
R+
R
R
for all ϕ ∈ Co∞ (R × R+ )
(Scalar, 1-D)
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Entropy pairs and uniqueness
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Entropy pair: (U, Q) such that Q 0 = U 0 f 0 and U is strictly
convex.
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u ∈ L1loc (R × R+ ) is the unique entropy solution if:
Z Z
U(u)ϕt + Q(u)ϕx dxdt ≥ 0.
R+
R
for all 0 ≥ ϕ ∈ Co∞ (R × R+ )
for all (U, Q)
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Entropy stability
1. Cell entropy inequality.
2. A priori bound on global entropy.
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For linear problems, entropy and energy stability are the same.
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For scalar conservation laws, entropy stability implies
convergence to weak entropy solution.
For systems, entropy stability is not sufficient for stability.
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For Navier-Stokes, positivity (of density and pressure) is
required.
Stability does not imply convergence to entropy solution.
(Indeed, the solution has not been proven to exist.)
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Boundary conditions
(mathematical results)
Scalar conservation law in 1-D with entropy pair (U,Q).:
ut + fi (u)x = 0,
u(x, 0) = u 0 (x),
u(0, t) = g (t)
(”if inflow ”)
(Du Bois and Lefloch) With convenient stability assumptions:
Q(u) − Q(g ) − Uu (g )(f (u) − f (g )) ≤ 0.
Entropy inequality: Ut + Qx ≤ 0. Integrate in space:
Z ∞
Ut dx = Q(0, t)
0
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U bounded if Q(0, t) bounded by data.
No boundary condition for Q(0, t).
Requires a strong coupling between flux and entropy flux.
Generally not the case.
Does entropy stability imply boundary inequality?
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Conclusions
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SBP-SAT schemes have been successfully used to compute
linear problems with low Reynolds number
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Turbulent benchmarks are up next...Challenging from HPC
and grid generation perspective.
With entropy stability.
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Boundary conditions clearly defined for conservation laws.
Benchmarking
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Strong shocks. High-speed flow past cylinder.
Shock-turbulence interaction.
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