Modelling and numerical simulation of bi-temperature Euler Equations in toroidal geometry E. Estibals H. Guillard A. Sangam elise.estibals@inria.fr Inria Sophia Antipolis Méditerranée June 9, 2015 1 / 19 E. Estibals, H. Guillard, A. Sangam
Context of this work src : iter.org Hot plasma where ions and electrons may behave in independent way. Ion and electron temperatures can be very different. Complete model of plasma: MHD equations. MHD: MagnetoHydroDynamics. 2 / 19 E. Estibals, H. Guillard, A. Sangam
Goal of this work Introduction Numerical method to approximate a two temperature model: Two temperatures model:. Numerical methods:. Applied to study flows in tokamaks: Spatial discretization: cylindrical coordinates. Adapt the finite volume scheme to this coordinate system. 3 / 19 E. Estibals, H. Guillard, A. Sangam
Table of contents Introduction 1 2 The numerical scheme Numerical test 3 Spatial discretization Volume of control cell Normal and length 4 4 / 19 E. Estibals, H. Guillard, A. Sangam
Non-conservative and conservative systems Non-conservative system, E α = ε α + 1 2 V2, p α = p α (ρ α, ε α ), and c α = ρ α /ρ: t ρ + (ρv) = 0, t (ρv) + (ρv V) + (p e + p i ) = 0, t (ρ e E e ) + ((ρ e E e + p e )V) u(c i p e c e p i ) = ν ei (T i T e ), t (ρ i E i ) + ((ρ i E i + p i )V)+u(c i p e c e p i ) = ν ei (T i T e ). Conservative system, S e = p e ρ γe e et ρe = ρ e E e + ρ i E i : t ρ + (ρv) = 0, t (ρv) + (ρv V) + (p e + p i ) = 0, t (ρe) + ((ρe + p e + p i )V) = 0, t (ρ e S e ) + (ρ e S e V) = (γ e 1)ρe 1 γe ν ei (T i T e ). F. Coquel, and C. Marmignon, Numerical methods for weakly ionized gas, Astrophysics and Space Science, 1998. 5 / 19 E. Estibals, H. Guillard, A. Sangam
Transformation of the system Rotationally invariant system. Can write the system in any direction n, u = n V, v = t V: t ρ + n (ρu) = 0, t (ρu) + n (ρu 2 + p e + p i ) = 0, t (ρv) + n (ρuv) = 0, t (ρe) + n ((ρe + p e + p i )u) = 0, t (ρ e S e ) + n (ρ e S e u) = (γ e 1)ρ 1 γe e ν ei (T i T e ). 1D numerical scheme. 6 / 19 E. Estibals, H. Guillard, A. Sangam
Principle Introduction The numerical scheme Numerical test t U + n F (U, n) = S(U) Relaxation: t [ U V ] + n [ G1 (U, V ) G 2 (U, V ) ] [ = S(U) 1 τ Q(U, V ) If τ 0 then Q(U, V ) 0 and G 1 (U, V ) F (U) Numerically: 1 Transport. 2 Projection. ]. 7 / 19 E. Estibals, H. Guillard, A. Sangam
Relaxed system Introduction The numerical scheme Numerical test Relaxation variables: π e and π i. Relaxed system: t ρ + n (ρu) = 0, t (ρu) + n (ρu 2 + π e + π i ) = 0, t (ρv) + n (ρuv) = 0, t (ρe) + n ((ρe + π e + π i )u) = 0, t (ρ e S e ) + n (ρ e S e u) = (γ e 1)ρ 1 γe e ν ei (T i T e ), t π α + u n π α + a2 ρ α ρ 2 n u = 1 τ (p α π α ). Stability condition on impedance: a 2 ρ 2 max(γ e p e ρ e, γ i p i ρ i ). D. Aregba, J. Breil, S. Brull, B. Dubroca, E. Estibals, in preparation. 8 / 19 E. Estibals, H. Guillard, A. Sangam
Introduction The numerical scheme Numerical test Transport step: Godunov scheme t ρ + n (ρu) = 0, t (ρu) + n (ρu 2 + π e + π i ) = 0, t (ρv) + n (ρuv) = 0, t (ρe) + n ((ρe + π e + π i )u) = 0, t (ρ e S e ) + n (ρ e S e u) = 0, t (ρπ α ) + n (ρπ α u + a 2 ρα ρ u) = 0. Projection step: dρ dt = du dt = dv dt = 0, π α = p α, α = e, i, dt C e v,e dt = ν ei (T i T e ), dt C i v,i dt = ν ei (T i T e ). 9 / 19 E. Estibals, H. Guillard, A. Sangam
Implosion, Initial data The numerical scheme Numerical test ρ L, V L T L i, T L e ρ R, V R 10µm Ti R, Te R ρ L = ρ R = 1, V L = V R = 0, T L T R i = 150eV = 1.7406.10 6 K, i = 1500eV = 1.7406.10 7 K, Te L = 198, 2eV = 2, 3.10 6 K, Te R = 1982eV = 2, 3.10 7 K. 10 / 19 E. Estibals, H. Guillard, A. Sangam
The numerical scheme Numerical test Density and Temperatures at t = 5ps 11 / 19 E. Estibals, H. Guillard, A. Sangam
Velocity at t = 5ps Introduction The numerical scheme Numerical test 12 / 19 E. Estibals, H. Guillard, A. Sangam
Torus model Introduction Spatial discretization Volume of control cell Normal and length Coordinate system: cylindrical. Input: 2D mesh. Simulation: 2D axisymetric and 3D. 2D mesh: plan (R, Z) i.e. poloidal plan. 3 rd direction: φ, toroidal direction. 13 / 19 E. Estibals, H. Guillard, A. Sangam
Numerical tools Introduction Spatial discretization Volume of control cell Normal and length Time step: U n+1 i = U n i t Ω i j V (i) (F (U ij, n ij )n ij l ij ). Numerical approach: Vertex Centered. Numerical tools: 1 Ω i : volume of control cell. 2 n ij : unit normal. 2 cases: Normal between 2 points of the same poloidal plan. Normal between 2 points which are not inside the poloidal plan 3 l ij : length of segments. 14 / 19 E. Estibals, H. Guillard, A. Sangam
Computing of Ω is Introduction Spatial discretization Volume of control cell Normal and length φ iplan = φ iplan+1/2 φ iplan 1/2. is Ω 2D is Ω is = Ω is RdRdZdφ, = φ iplan+1/2 φ iplan 1/2 dφ Ω RdRdZ, 2D is = φ iplan Ω 2D = φ iplan is, τ Ω 2D is R τ τ 6. A. Bonnement, T. Fajraoui, H. Guillard, M. Martin, A. Mouton, B. Nkonga, and A. Sangam, Finite Volume Method in Curvilinear Coordinates, ESAIM Proc., Vol. 32, 163-176, 2011. 15 / 19 E. Estibals, H. Guillard, A. Sangam
l ij n ij in poloidal plane Spatial discretization Volume of control cell Normal and length is1 is3 τ1 G 1 M N 1 N 2 is2 N = N 1 + N 2 = = n R cos(φ) n R sin(φ) n Z [ nr n Z ] (x,y,z), (R,Z), G 2 is3 τ2 l ij n ij = Ω is NRdRdZ, [ ] 2n = n R sin( φ iplan 2 ) n Z φ iplan 16 / 19 E. Estibals, H. Guillard, A. Sangam (R,Z).
l ij and n ij in toroidal direction Spatial discretization Volume of control cell Normal and length n ij : easy to compute, the points are not in the same poloidal plan, n ij = e φiplan+1/2. l ij computing: l ij = Ω ± is RdRdZ = τ Ω 2D is R τ τ 6. 17 / 19 E. Estibals, H. Guillard, A. Sangam
Summary and perpectives Summary: Bi-temperature model tested in 2D geometry. Spatial discretezation of a torus. Perpective: Need a numeric test with the toroidal geometry. Extension to a bi-temperature MHD model. 18 / 19 E. Estibals, H. Guillard, A. Sangam
Merci de votre attention! 19 / 19 E. Estibals, H. Guillard, A. Sangam
Density and Temperatures at t = 5ps 20 / 19 E. Estibals, H. Guillard, A. Sangam
Velocity at t = 5ps Introduction 21 / 19 E. Estibals, H. Guillard, A. Sangam