Angular momentum preserving CFD on general grids

Transcription

1 B. Després LJLL-Paris VI Thanks CEA and ANR Chrome Angular momentum preserving CFD on general grids collaboration Emmanuel Labourasse (CEA) B. Després LJLL-Paris VI Thanks CEA and ANR Chrome collaboration Emmanuel Labourasse (CEA) Angular momentum preserving CFD on general grids p. 1 / 29

2 Section 1 Angular momentum preserving CFD on general grids p. 2 / 29

3 Question for conservation laws and Riemann addicts Compressible CFD is based on conservation laws tρ + (ρu) = 0, t(ρu) + (ρu 2 + pi) = 0, t(ρe) + (pu) = 0. 1D Riemann do at the same time excellently and poorly for multid flows. What about using angular momentum numerical conservation law which is a physically sound conservation law? Issue addressed already in astrophysics, but not that often in CFD. See however CFD for helicopters (topic I know almost nothing). Angular momentum preserving CFD on general grids p. 2 / 29

4 Angular momentum in nature, engineering and sport (internet provided) Angular momentum preserving CFD on general grids p. 3 / 29

5 Angular momentum The angular momentum w = u x satisfies an additional conservation laws t(ρw) + (ρu u x) + curl(px) = 0. The vorticity ω = u can be computed from the angular momentum. Let x 0 be a given point in the domain and consider w 0(x) = w(x) u(x) x 0 = u(x) (x x 0). Then w 0(x) = u(x) (x x 0) 2 u(x). Therefore one has the identity ω(x 0) = u(x 0) = 1 2 w0(x) Angular momentum preserving CFD on general grids p. 4 / 29

6 Some references (vorticity/am) : Roe and Morton, Preserving vorticity in finite-volume schemes, Finite volumes for complex applications II, , Hermes Sci. Publ., Paris, $ Caramana and Loubère, Curl-q : A vorticity damping artificial viscosity for essentially irrotational Lagrangian hydrodynamics calculations, $ Dukowicz-Meltz, Vorticity errors in multidimensional lagrangian codes Journal of Computational Physics, Volume 99, Issue 1, March Käppeli, Mishra, Structure preserving schemes. In Seminar for Applied Mathematics, ETH Zürich, Zhang-Zhang-C.W. Shu, Multistage interaction of a shock wave and a strong vortex, Phys. Fluids, Rault-Chiavassa-Donat, Shock-Vortex Interactions at High Mach Numbers, Oort, Angular in the Atmosphere-Ocean-Solid Earth System, Bulletin of the American Meteorological Society, 70 (10), 1, , Angular momentum preserving CFD on general grids p. 5 / 29

7 More references MultiD Riemann - Bourgeade-Le Floch-Raviart : An asymptotic expansion for the solution of the generalized Riemann problem. Part 2 : application to the equations of gas dynamics, Ben-Artzi-Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamic, Brio-Zakharian-Webb, Two-Dimensional Riemann Solver for Euler Equations of Gas Dynamics, Balsara, Multidimensional Riemann Problem with Self-Similar Internal Structure-Part I. Application to Hyperbolic Conservation Laws on Structured Meshes, JCP 277 (2014) Low Mach flows - Guillard-Voizat, On the behaviour of upwind schemes in the low Mach number limit, Compters and Fluids, 28, Dellacherie,... Angular momentum preserving CFD on general grids p. 6 / 29

8 Meshes and computations - Multimaterial computation is courtesy from Delpino (JCP 2011). - 3D is from Chertkov-Levedev on internet. Angular momentum preserving CFD on general grids p. 7 / 29

9 First method We assume the thermodynamics is discretized cell-centered-fv-p 0 ρ j, T j, ε j = C v T j, p j = (γ 1)ρ j ε j R. Probably the velocity will be discretized cell-centered-fv-p 0 : u j R d. What can we do for the angular momentum? First idea : Angular momentum satisfies an independent conservation law, so one can use additional but not fully independent unknowns. Velocity locally in the cell V j made of all solid body displacement v j (x) = a j + b j x Define u j = 1 v j (x) V j V j = a j + b j x j, w j = 1 v j (x) x, V j V j = (a j + b j x j ) x j + 1 (b j (x x j )) (x x j ) V j V j Angular momentum preserving CFD on general grids p. 8 / 29

10 Inverse formulas Define the positive symmetric matrix 0 < H j = H t j R d d b R d, (H j b, b) := 1 V j One can prove H j is non-singular, even if H j = O(h 2 ). V j b (x x j ) 2. Therefore there is a one to one correspondence between the physical unknowns u j, w j and the couple a j, b j { b j = H 1 j (u j x j w j ), a j = u j b j x j. (1) Angular momentum preserving CFD on general grids p. 9 / 29

11 Second method : link with DG Define P n = u = 0 j 1 + +j d n x j x j d d a j1,...,j d a j1,...,j d R d are vectors Rd. The dimension of P n increases with n : dim(p 0 ) = d, dim(p 1 ) = d(d + 1) = d 2 + d and so on. Define a new space P 0 Q = Span {a + Bx} P 1 where a P 0 is arbitrary and the vectorial product is noted with the multiplication by an antisymmetric matrix B t = B. Therefore dim(q) = d + d(d 1) 2 = 1 2 dim(p1 ). The number of extra degrees of freedom with respect to P 0 is N extra = dim(q) dim(p 0 ) = d(d 1). 2 Angular momentum preserving CFD on general grids p. 10 / 29

12 DG procedure Consider the impulse equation DG yields That is d t t(ρu) + M = 0, M = M t = ρu u + pi d. Θ d t Θ ρu ũ + ( M) ũ = 0, ũ Q. Θ ρu ũ M : ũ = (Mũ, n) dσ, ũ Q. Θ Θ Proposition The interior integral vanishes : M : ũ = 0, ũ Q. Θ proof. Indeed ũ = B is an antisymmetric matrix. Since M is symmetric, the contraction vanishes M : B = 0. Angular momentum preserving CFD on general grids p. 11 / 29

13 Equations Take ũ = a. One obtains the conservative inertial momentum equation d t ρu ã = (Mã, n) dσ, ã R d. Θ Θ Take ũ = b x. One obtains the conservative angular momentum equation d t ρu b x = d t ρu x b ( ) = M b x, n dσ, b R d. Θ Θ Θ Both approaches yield the same result. Angular momentum preserving CFD on general grids p. 12 / 29

14 Section 3 Angular momentum preserving CFD on general grids p. 13 / 29

15 Impact is only on Muscl reconstruction Before : most production codes are 2nd order, and based on Muscl techniques. Usually the velocity is reconstructed in the cell V j New method : we decompose v j (x) = u j + u Muscl j (x x j ). u New j = D j + B j, D j = D t j = 1 2 ( ( u Muscl j + u Muscl j ) t ) and B j = B t j provided by the angular momentum degrees of freedom. This is the only place where the AM equation interacts with the usual scheme. Angular momentum preserving CFD on general grids p. 13 / 29

16 Our solver We start from Lagrange CFD ρd tτ u = 0, ρd tu + p = 0, ρd te + (pu) = 0. Our scheme GLACE is a cell-centered Lagrangian scheme, with corner-based Riemann. C jr = xr V j. m j d tτ j (t) = C jr u r, r N (j) m j d tu j (t) = C jr p jr, r N (j) m j d te j (t) = r N (j) C jr u r p jr. where m j, u j and e j are the Lagrangian mass, the velocity and the total energy, and d tϕ ( t + u ) ϕ is the Lagrangian derivative. The nodal unknowns u r (the velocity of the r-th node), and p jr (the nodal pressure) are computed thanks to a nodal Riemann solver for which many different versions exist nowadays. (2) Angular momentum preserving CFD on general grids p. 14 / 29

17 The corner-based Riemann solver It writes where n jr = C jr C jr p jr p j + ρ j c j (u r u j ) n jr = 0, C jr p jr = 0, j C(r) is the normalized corner direction. The first order nodal solver can be rewritten as { Ar u r = j A jr u j + C jr p j, F jr = C jr p j + A jr (u j u r ). In this system, the Glace and Eucclhyd schemes differ ultimately only by the definition of the R d d matrices A jr and of the vector F jr Angular momentum preserving CFD on general grids p. 15 / 29

18 Workflow 1 From the mesh, compute the geometrical features C jr, x j, x r, H j, 2 Compute b j = H, 1 j (u j x j w j ) from which B j is deduced. 3 Limit the reconstructed velocity, and deduce B # j and eventually D # j. 4 Compute v # j (x r ) = u j + u # j (x r x j ) 5 Construct 2nd order nodal pressure p # j (x r ) = p j + p # j (x r x j ). 6 Solve the system with the reconstructed quantities : { A r u # r = j A jr v # j (x r ) + ( C jr p # j (x r ), ) = C jr p # j (x r ) + A jr v # j (x r ) u # r. F # jr 7 Update u # j, w # j and e # j u # j = u j t F # jr m, j r w # j = w j t F # jr m x r. j r e # j = ej t F # jr u# r. m j r (3) 8... Angular momentum preserving CFD on general grids p. 16 / 29

19 Stability/entropy Consider the new AMC scheme with D j = 0 and no limitation on B j. The entropy balance of the Lagrangian scheme writes with (similar to first order scheme) Q j = r m j T j d ts j = Q j + R j (u r v j (x r ) A jr (u r v j (x r )) 0 and R j = m j b j u j d tx j + m j b j (d t(h j )b j /2). Since R j make take any sign, it is in practice controlled by a limiter technique. Angular momentum preserving CFD on general grids p. 17 / 29

20 Section 4 Angular momentum preserving CFD on general grids p. 18 / 29

21 Rotating ring Note : for central forces, the AMC is a local d t(ru θ ) = 0. Angular momentum preserving CFD on general grids p. 18 / 29

22 Rotating ring : standard 2nd Muscl scheme Angular momentum preserving CFD on general grids p. 19 / 29

23 Rotating ring : final mesh with the new method Angular momentum preserving CFD on general grids p. 20 / 29

24 Taylor vortex The mesh is fixed Eulerian, and the scheme is run Lagrange+remapp ρ(x) = (1 ) 1 (γ 1)β2 e 1 r 2 γ 1, p(x) = ρ(x) γ, u(x) = β 8γπ 2 2π e 1 r2 2 x x where β = 5 corresponds to the strength of the vortex, and r = x is the distance to the center. The analytic solution is stationnary 1D plot of the density (left) and the velocity (right) for the Taylor vortex. Angular momentum preserving CFD on general grids p. 21 / 29

25 Convergence table for the Taylor Vortex with amc no amc number of cells L 2 error order L 2 error order The order of our Muscl procedure is not degraded by the introduction of the angular momentum variable. The are even slightly better. Muscl shows more smearing of b = β e 1 r 2 2π 2 than new AMC technique. Angular momentum preserving CFD on general grids p. 22 / 29

26 Unexpected advantage of angular Claim : Angular momentum conservation lowers the mesh imprint on implosion problems. Example of a initial skewed mesh. The center of implosion is at M. Relevant for computational inertial fusion : NIF, ICF,.... Angular momentum preserving CFD on general grids p. 23 / 29

27 Principle Consider I = Ω ρu and W = Ω ρu x Assume for simplicity the boundary conditions are such that I and W are constant. Assume M be the center of implosion : t f u (x M) = 0. Consequently W = I M. such that x, one has This is a linear system with unknown M which cannot be fully recovered. Additional symmetry considerations yield M. The previous example corresponds to W = 0 0 w, I = 0 i 0, M = x M y M 0 = w = ix M = x M = w i. The symmetry assumption is y M = 0 since the center is on the main axis. Angular momentum preserving CFD on general grids p. 24 / 29

28 Results no AMC with AMC O1 O2 Our best result is O2-AMC. Angular momentum preserving CFD on general grids p. 25 / 29

29 Shock with rotation Set up Angular momentum preserving CFD on general grids p. 26 / 29

30 Reference solution AMC yields d dt (ru θ) = 0, d dt = t + u Angular momentum preserving CFD on general grids p. 27 / 29

31 Results no AMC with AMC O1 O2 Our best result is O2-AMC. Angular momentum preserving CFD on general grids p. 28 / 29

32 Conclusions Angular momentum is a natural conservation law. It can be easily incorporated in any standard 2nd order Muscl-FV code. It increases the accuracy. An issue is stability : the decomposition of the velocity gradient does the job in our case. My feeling is it could be also incorporated in most high order Eulerian CFD codes. D. Labourasse : see JCP online Angular momentum preserving CFD on general grids p. 29 / 29