Finite Volume for Fusion Simulations

Transcription

1 Finite Volume for Fusion Simulations Elise Estibals, Hervé Guillard, Afeintou Sangam To cite this version: Elise Estibals, Hervé Guillard, Afeintou Sangam. Finite Volume for Fusion Simulations. Jorek Meeting 2016, Apr 2016, Sophia Antipolis, France. <hal > HAL Id: hal Submitted on 18 Nov 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Finite Volume for Fusion Simulations E. Estibals H. Guillard A. Sangam Inria Sophia Antipolis Méditerranée November 15, / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

3 Plan MHD equations and issues 1 MHD equations and issues 2 Well-known methods Vector potential Divergence cleaning methods Proposed method 3 2-D system Evolution step 4 2 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

4 Resistive MHD equations Non-conservative equations: E hd = E = B u + ηj p γ ρu2, t ρ + (ρu) = 0, t (ρu) + (ρu u) + p = J B, t E hd + [(E hd + p)u ] = (J B) u + ηj 2, t B + E = 0. Conservative equations: E = E hd B2, p = p B2 t ρ + (ρu) = 0, t (ρu) + (ρu u B B) + p = 0, t E + [(E + p )u (u B)B] = (ηj B), t B + (u B B u) = (ηj). 3 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

5 MHD system MHD equations and issues Hyperbolic part of the system: 3 wave types 2 Alfvén waves, 4 magneto-acoustic waves (2 fast and 2 slow), 2 material waves (moving with u). Involution equation: Then Numerical issues: t ( B) = 0. B(t = 0) = 0 B(t) = 0 t. 1 Shock capturing: need a robust and accurate scheme. 2 constraint. 3 Time scale T >> T alfven = L/v alfven : need implicit scheme. 4 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

6 Well-known methods Vector potential Divergence cleaning methods Proposed method constraint: Well-known methods 2 families of methods: Vector potential A: B = A. Divergence cleaning methods: Enforce B = 0. 5 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

7 Vector potential A MHD equations and issues Well-known methods Vector potential Divergence cleaning methods Proposed method Replace B by A in the equation: t A u ( A) + η ( A) = U. Advantage: Insure B = 0. Drawbacks: One order higher in the spatial derivatives. More complex system of equation. 6 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

8 Well-known methods Vector potential Divergence cleaning methods Proposed method Divergence cleaning methods: Previous works Powell et al. (JCP, 1999): Add a source term proportionnal to B. t B + (u B B u) = ( B)u. Munz et al. (JCP, 2000), Dedner et al. (JCP, 2001): Generalized Lagrange Multiplier (GLM) method. t B + (u B B u) + Ψ = 0, D(Ψ) + B = 0, D(Ψ) = 1 ch 2 t Ψ + 1 Ψ. cp 2 Balsara and Spicer (JCP, 1998): Constrained transport method. d B ds = E dl. dt 7 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation S S

9 Divergence cleaning methods Well-known methods Vector potential Divergence cleaning methods Proposed method Advantages: Easily incorporated with a Riemann type scheme. Drawbacks: Divergence cleaning: B appears in several equations. Not approximated with the same discretization. Divergence cleaning method insure B = 0 for one discrete approximation. Constrained transport: difficult to implement for arbitrary meshes. 8 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

10 Proposed method MHD equations and issues Well-known methods Vector potential Divergence cleaning methods Proposed method A mixture of the two methods. Idea: work with a redundant system. Both vector potential and magnetic fields equations. Based on the relaxation scheme method: 1 Evolution step: FV method for all the variables. 2 Projection step: enforce B = A. 9 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

11 2-D implementation MHD equations and issues 2-D system Evolution step Scalar potential ψ: B = B z e z + e z ψ. Augmented system: t ρ + (ρu) = 0, t (ρu) + (ρu u B B) + p = 0, t E + [(E + p )u (u B)B] = η (J B), t B + (u B B u) = η 2 B, t ψ + u ψ = η 2 ψ. 10 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

12 Evolution step MHD equations and issues 2-D system Evolution step System t U + x F (U) + y G(U) = 0: U n+1 i,j = U n i,j t x (F i+1/2,j F i 1/2,j ) t y (G i,j+1/2 G i,j 1/2 ). F i+1/2,j and G i,j+1/2 computed with a Riemann solver. 11 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

13 Evolution step: HLLD scheme 2-D system Evolution step Miyoshi and Kusano (JCP, 2005). Approximate Riemann solver with 4 intermediate states. S L S L S M S R S R U L U L U R U R U L U R Figure : Riemann fan with four intermediate states. 12 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

14 Evolution step: HLLD scheme 2-D system Evolution step Intermediate fluxes: { F K = F K + S K UK S K U K FK = F K + SK U K (S K S K )UK S, K = L, R. K U K ρψ flux: { ψl, 0 S F (ρψ) = F (ρ) M. ψ R, 0 S M 13 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

15 2-D system Evolution step 1 Evolution step: HLLD scheme { B n+1/2 i,j, ψ n+1/2 i,j. 2 Projection step: { ψ n+1 i,j = ψ n+1/2 i,j, B n+1 i,j = B n+1/2 z,i,j e z + e z ( ψ) n+1 i,j. 14 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

16 MHD equations and issues Shock capturing tests:,, Kelvin-Helmholtz. Plasma fusion tests:. 15 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

17 1-D shock tube Magnetic field: B = B x e x + x ψe y. 1-D equation for the potential ψ: Initial conditions: t (ρψ) + x (ρψu) = B x ρv. ρ u p B x B y = x ψ ψ x < x x > x 1-D mesh: 100 points Final time: / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

18 1-D : Magnetic component B y Figure : HLLD O(1). Figure : HLLD O(2). 17 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

19 2-D shock tube Potential: ψ(x, y) = 2-D mesh: cells. { x 0.75y, x < x 0.75y, x > / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

20 2-D shock tube Figure : B y HLLD O(2). Figure : ψ, HLLD O(2). 19 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

21 MHD equations and issues Initial conditions: γ = 5/3 ρ u(x, y) v(x, y) p(x, y) B x (x, y) B y (x, y) ψ(x, y) γ 2 sin(2πy) sin(2πx) γ sin(2πy) sin(4πx) 1 2π cos(2πy) 1 4π cos(4πx) Mesh: cells. Final time: / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

22 : Pressure field Figure : Without projection. Figure : With projection. 21 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

23 MHD equations and issues Figure : Pressure at y = Red: without projection, Blue: with projection. 22 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

24 MHD equations and issues Figure : B. 23 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

25 Initial data ρ p u(x, y) v(x, y) B x (x, y) B y (x, y) B z (x, y) ψ(x, y) 1 1 γ 1 2 tanh( y y 0 ) cos( π 3 ) ρ sin( π 3 ) ρ 0.1 cos( π 3 ) ρy Single mode pertubation v(x, y) = 0.01 sin(2πx) exp( y 2 Mesh: points B pol B 2 B tor = x +By 2 B z. ), σ = σ2 24 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

26 Kelvin-Helmholtz: t = 5.0 Figure : Without projection. Figure : With projection. 25 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

27 Kelvin-Helmholtz: t = 8.0 Figure : Without projection. Figure : With projection. 26 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

28 Kelvin-Helmholtz: t = 12.0 Figure : Without projection. Figure : With projection. 27 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

29 Kelvin-Helmholtz: t = 20.0 Figure : Without projection. Figure : With projection. 28 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

30 Kelvin-Helmholtz MHD equations and issues Figure : B. 29 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

31 Preliminary test for fusion plasma Equilibrium computation: p = J B. Initial conditions: R 0 = 10, B r = 0, r ρ = 1, B θ (r) = R 0 (3r 2 +1), u = 0, B z = 1, 1 p(r) = 6R0 2(3r, ψ(r) = ) 2 6R 0 ln(3r 2 + 1). Aligned mesh: cylindrical coordinates ( cells). Non aligned mesh: Cartesian coordinates ( cells). 30 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

32 Cylindrical coordinates Figure : B θ. Figure : p. 31 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

33 Cylindrical coordinates: steady state Figure : Residual on r-momentum equation. Figure : Relative error of B θ. 32 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

34 Cartesian coordinates Figure : Relative error of p in function of Alfvén time. Non aligned meshes: with projection simulations maintains the equilibrium on 1000 Alfvén times (around 0.5ms and 10 6 time steps). However, we want to perform the simulation on several milliseconds. 33 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

35 Remedies MHD equations and issues A simple trick: remove the truncation error evaluated at time t = 0. t (ρu) + (ρu u) + p J B ( h p J h B h ) eq = 0. Well-balanced scheme: work in progress. Figure : Relative error of p. 34 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

36 Conclusions and perspectives Conclusions: Shock capturing tests: gives satisfactory results. Plasma fusion test: Work in progress on well-balanced scheme. Perspectives: Perform more tests for plasma fusion (kink instability). Adapt to 3-D geometry. Test on resistive problems. 35 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

37 Cylindrical equations MHD equations and issues t (rρ) + r (rρu [ r ) + θ (ρu θ ) ] = 0, t (rρu r ) + r r(ρur 2 + p Br 2 ) + θ (ρu r u θ B r B θ ) = ρuθ 2 + p B 2 ] ] θ, t (r 2 ρu θ ) + r [r 2 (ρu r u θ B r B θ ) + θ [r(ρuθ 2 + p Bθ 2 ) = 0, t (rρu z ) + [ r [r(ρu r u z B r B z )] + θ (ρu θ u z B θ B z ) = 0, t (re) + r r[(e + p )ur (u B)B r ] ] [ + θ (E + p ] )uθ (u B)B θ = 0, t (rb r ) + θ (u θ B r u r B θ ) = 0, t B θ + r (u r B θ u θ B r ) = 0, t (rb z ) + r [r(u r B z u z B r )] + θ (u θ B z u z B θ ) = 0, t (rρψ) + r (rρψu r ) + θ (ρψu θ ) = / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation

38 1 Evolution step: HLLD scheme 2 Projection step: p n+1/2 i,j, B n+1/2 i,j, ψ n+1/2 i,j. ψ n+1 i,j = ψ n+1/2 i,j, B n+1 i,j = B n+1/2 z,i,j e z + e z ( ψ) n+1 i,j, E n+1 i,j = pn+1/2 i,j γ ρn+1/2 u (Bn+1 i,j ) / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation