Computations of non-reacting and reacting two-fluid interfaces

Transcription

1 Computations of non-reacting and reacting two-fluid interfaces Kunkun Tang Alberto Beccantini Christophe Corre Lab. of Thermal Hydraulics & Fluid Mechanics (LATF), CEA Saclay Lab. of Geophysical & Industrial Flows (LEGI), Grenoble INP LaMSID, Sep 6, 0

2 Context & motivation Investigation of the effect of premixed combustion of hydrogen-air mixtures in case of nuclear accident in a reactor containment. Difficulties: Characteristic dimensions: EPR (European Pressurized Reactor): (40 m) 3 ; Reaction zone in a laminar deflagration at atmospheric condition: from about mm (X H 0.96) to 0 mm (X H 0.). Direct Numerical Simulation (DNS) for flame acceleration is impossible in such large geometries. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

3 Context & motivation Simplifications: Surface reaction (infinite reaction rate, discontinuous interface); Diffusion phenomena (thermal and species) are neglected, and fundamental flame speed is imposed. Challenges: Accurate and robust computation of interfaces on coarse grids. Extension to multi-d unstructured meshes for complex geometries. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

4 D Chapman-Jouguet deflagration (500 cells) 4 0 State-of-the-art Exact solution Pressure (bar) Position (m) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 3 / 9

5 Numerical strategies: existing tools Interface modelling: Two-fluid model (two pressures & two velocities): [Delhaye, 968, Ishii, 975, Drew and Passman, 998]; Method for interface propagation: Discrete Equations Method (DEM): [Abgrall and Saurel, 003]; Reactive DEM (RDEM): [Le Métayer et al., 005, Beccantini and Studer, 00]; Accurate discretization of interface: Anti-diffusive method: [Després and Lagoutière, 00, Kokh and Lagoutière, 00]. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 4 / 9

6 Present contribution Combine anti-diffusive strategy with DEM/RDEM. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 5 / 9

7 Contents Preliminaries for flow computation Interface & two-fluid model 3 Discrete Equations Method (DEM) 4 Upwind downwind-controlled splitting (UDCS) 5 Numerical results 6 Conclusions & Perspectives Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 6 / 9

8 Compressible Euler equations Euler equations of gas dynamics: ρ t + ρv x = 0, ρv t + (ρv + p) = 0, x ρe t t + (ρet + pv) x = 0. The system is hyperbolic. The solution contains three characteristic waves associated to the eigenvalues below: λ = v c, λ = v, λ 3 = v + c. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 7 / 9

9 Equations of state (EOS) Non-reacting flow: calorically perfect gas, p = (γ )ρe. stiffened gas (liquids under very high pressures), p = (γ )ρe γp. Reacting flow: thermally perfect gas, p = ρrt; ẽ = h 0 + T 0 c v (τ)dτ. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 8 / 9

10 Finite volume method Integrating the Euler equations We note the integral form as U n+ i u t + f(u) x = 0, = U n i t x (F i+ F i ). U U i F i U i F i+ U i+ i i i i + i + Numerical flux F i± to be defined. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 9 / 9

11 Riemann problem (RP) for D Euler equations red: fluid, blue: fluid U l U l U F(U l ) U r U r x rarefaction v c x i+ t v contact shock v + c U l U r U l = U i U r = U i+ x i+ x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 0 / 9

12 Interface model j Σ X = X = 0 Σ X = X = 0 P X D I C i The interface is modelled as a discontinuity of characteristic function X k. Interface evolution [Drew and Passman, 998]: dx k = X k + D I X k = 0. dt t Euler equations inside each phase Σ k : u k t + f(u) k = 0. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

13 Numerical interface model [Abgrall and Saurel, 003] Σ Σ.0 α(volume fraction) Σ Σ Σ Σ Σ x i x i x i+ x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

14 Fully conservative Godunov s Method One-fluid method for two-fluid problem. Σ + Σ F i Σ + Σ i i i x t n+ U l U l * t U r * U r F Averaging here i Σ + Σ Σ + Σ t n Ui = U l Ui+ i i = U i r x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 3 / 9

15 Discrete Equations Method (DEM) [Abgrall and Saurel, 003] α(t = t n ) Σ RP Σ RP Σ α(t = t n + t) Σ F Σ F RP Σ D H C C Σ Σ G B B F x Two-phase Riemann problem 0 i F A i Σ i E x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 4 / 9

16 Local Riemann problem Non-reacting case (we do not need a model for D I ). t Lgnl CD Rgnl U,L U,R i U,L U,R i Reacting case (D I needs to be modelled). Lgnl CD U,L t Taylor RS U,R U,R { i x U,R Rgnl i U,L i U,R x i Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 5 / 9

17 Upwind & downwind-controlled [Lagoutière, 000] Linear advection (D > 0). α t + D α = 0 αn+ i = α n i x D t x (α i+ α i ). n = 0 upwind everywhere... i i i + i +... n = upwind at i downwind at i +... i i i + i +... i i + exact solution n = downwind controlled upwind... i i i + i +... x x x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 6 / 9

18 Wave propagation point of view α Σ Σ Σ Σ Σ α i α i,i+ 0 Σ i i i i + i + x Detonation is OK, since precursor shock wave travels together with reacting shock. But in general, shock wave can violate the stability condition. Thus, straightforward coupling of DEM/RDEM with downwind-controlled is unstable. Proposition: Upwind downwind-controlled splitting. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 7 / 9

19 New: Upwind downwind-controlled splitting Do Upwind DEM method, let αu = (α,αρ,αρv,αρe t ) T ; α(t n ) (αu) n k,i Upwind method (αu) n+,up k,i. (a) up n i α n i up n i+ (b) α n i+ 0 i 3 x i i + α(t n + t) α n+,up i α n+,up i+ 0 x i 3 i i + Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 8 / 9

20 New: Upwind downwind-controlled splitting Do non-linear downwind phase splitting ; (b) α(t n + t) α n+,up i α n+,up i+ (c) 0 i 3 x i i + α(t n + t) α n+ i re-arrange the volume α n+ i+ 0 x i 3 i i + Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 9 / 9

21 D reacting flow Before combustion, we have stoichiometric mixture of hydrogen-air; Left chamber: the stoichiometric mixture has been completely burnt (red); H + 0.5O H O. Right chamber: the unburnt (blue) stoichiometric mixture of hydrogen-air; T = 800 K v = 0 m/s p =.03 bar ξ = (burnt) T = 90 K v = 0 m/s p =.03 bar ξ = 0 (unburnt) 0 0 m 0 m Fundamental flame speed K 0 = 00 m/s. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 0 / 9

22 Exact solution of reactive Riemann problem Chapman-Jouguet deflagration [Beccantini and Studer, 00] 4 Exact solution 0 Pressure (bar) t contact Position (m) U r Taylor C-J deflagration shock U l U l U r U r precursor shock x Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

23 D result of present work (00 cells) Exact solution -nd minmod UDCS anti α (burnt).5.5 ρ (kg/m 3 ) 0. Exact solution -nd minmod UDCS anti x (m) Exact solution -nd minmod UDCS anti p (bar) x (m) x (m) Exact solution -nd minmod UDCS anti v (m/s) x (m) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 / 9

24 D non-reacting shock bubble interaction 75 mm front side r = 5 mm x = 5 mm y = 44.5 mm R air shock front back side 89 mm 445 mm Location ρ p u x u y γ Air (back) Air (front) R Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 3 / 9

25 Numerical results (DEM+UDCS) Experiment Minmod Minmod Anti-diffusive Anti-diffusive 0 5 quad tri. 0 5 quad tri. (000 00) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 4 / 9

26 Numerical results (DEM+UDCS) 0.9 -nd 500x00 -nd 000x00 Anti 500x α (air) ρ (kg/m 3 ) nd 500x00 -nd 000x00 Anti 500x x (mm) x (mm) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 5 / 9

27 D computation of D cylindrical combustion Cartesian grid. First-order (top) v.s. second-order (bottom)..9.8 p(bar).4. ρ(si) Reference x-axis diag r(m) Reference x-axis diag r(m). p(bar).6.4 ρ(si) Reference x-axis diag r(m) Reference x-axis diag Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 6 / 9 r(m)

28 D computation of D cylindrical combustion Cartesian grid. UDCS method. Computed normal (top) v.s. analytical normal (bottom)..4. p(bar).6.4 ρ(si) Reference x-axis diag r(m) Reference x-axis diag r(m).9.8 p(bar).4. ρ(si) Reference x-axis diag r(m) Reference x-axis diag Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 7 / 9 r(m)

29 Conclusions Accurate and robust results obtained using DEM/RDEM+UDCS for reacting and non-reacting flows. Extra benefit: CPU cost reduction. Internal Riemann problem is no longer required; Less diffused fronts leads to a reduced number of (expensive) two-phase Riemann problems to solve..0 α(volume fraction) α(volume fraction) Σ Σ RP Σ Σ Σ Σ Σ RP RP Σ RP Σ x x i x i x i+ x i x i x i+ x UDCS implemented in fast dynamic fluid-structure interaction code EUROPLEXUS (with multi-d unstructured meshes). Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 8 / 9

30 Perspectives Time discretization: first-order forward Euler method for UDCS anti-diffusive. Extension to -level Runge-Kutta stepping. New approach to compute the normal at the flame interface. Extension of UDCS to simplified multiphase models (e.g. one-pressure and one-velocity two-fluid model). Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 9 / 9

31 Bibliography I [Abgrall and Saurel, 003] Abgrall, R. and Saurel, R. (003). Discrete equations for physical and numerical compressible multiphase mixtures. Journal of Computational Physics, 86: [Beccantini and Studer, 00] Beccantini, A. and Studer, E. (00). The reactive Riemann problem for thermally perfect gases at all combustion regimes. International Journal for Numerical Methods in Fluids, 64: [Delhaye, 968] Delhaye, J.-M. (968). Équations fondamentales des écoulements diphasiques. Technical report, Commissariat à l énergie atomique, CEA-R-349. [Després and Lagoutière, 00] Després, B. and Lagoutière, F. (00). Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics. Journal of Scientific Computing, 6(4): [Drew and Passman, 998] Drew, D. A. and Passman, S. L. (998). Theory of Multicomponent Fluids, Applied Mathematical Sciences, Vol. 35. Springer, New York. [Ishii, 975] Ishii, M. (975). Thermo-fluid Dynamic Theory of Two-phase Flow. Eyrolles, Paris. [Kokh and Lagoutière, 00] Kokh, S. and Lagoutière, F. (00). An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model. Journal of Computational Physics, 9(8): Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 30 / 9

32 Bibliography II [Lagoutière, 000] Lagoutière, F. (000). Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants. PhD thesis, Université Pierre et Marie CURIE. [Le Métayer et al., 005] Le Métayer, O., Massoni, J., and Saurel, R. (005). Modelling evaporation fronts with reactive Riemann solvers. Journal of Computational Physics, 05: Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 3 / 9

33 Averaged equations for two-fluid model Volume fraction: α k def = X k ; α k t + D I α k = 0; Mass α k ρ k t + α k ρ k v k = ρ k,i ( v k,i D I ) α k ; α k ρ k v k t Momentum ) + α k (ρ k v k v k + p k I = Energy The equation is similar. ) ] [ρ k,i v k,i (v k,i D I + p k,i I α k ; Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 3 / 9

34 Closure laws of two-fluid model Rankine-Hugoniot jump conditions: (f,i u,i D I ) α +(f,i u,i D I ) α = 0. (7.) Reacting interface velocity [Beccantini and Studer, 00]: D flame I = v unburnt + K 0 n. (7.) Equations of state (EOS): Non-reacting problem: stiffened gas EOS, p k = (γ k )ρ k (ẽ k h 0,k ) γ k p,k. Reacting problem: thermally perfect gas, p k = ρ k R k T k ; ẽ k = h 0,k + Tk 0 c v,k (τ)dτ. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 33 / 9

35 Multi-dimensional UDCS algorithm Divide the interfaces as inlet and outlet ones: C i,in, n i,in, C i,in,3 C i,in, C i,in, D n i,in, C i C i,out,4 n i,out,4 C i,out,4 D n i,out,4 C i,out,5 Generated volume at j-th inlet/outlet interface: up n i,in,j = ( α n i,in,j α n ) i Ci,in,j D n i,in,j n i,in,j t, up n i,out,j = ( α n i α n i,out,j) Ci,out,j D n i,out,j n i,out,j t. Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 34 / 9

36 Multi-dimensional UDCS algorithm Upwind scheme: α n+ i = α n i + j up n i,in,j. C i (7.3) Downwind scheme (unconditionally unstable): α n+ i = α n i + j up n i,out,j. C i (7.4) Designed high order scheme: α n+ i = α n i + j ( λ n i,in,j ) upn i,in,j C i + j λ n i,out,j 0 λ ; λ = 0 upwind; λ = downwind. up n i,out,j. C i (7.5) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 35 / 9

37 Two approaches Second order accuracy (use limiter to determine the reconstructed value of α n f,out,j ): λ n i,out,j = αn i αn f,out,j α n i, when α n αn i α n i,out,j. (7.6) i,out,j Downwind-controlled (anti-diffusive) type approach: For simplicity, take a unique value for all outlet interfaces: No Local Extremum of α n+ i : λ n i,out,j = λ n i,out, for any j. m n i αn i + j ( λ n i,in,j ) upn i,in,j C i + λ n up n i,out i,out Mi n C i. (7.7) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 36 / 9

38 Restrictions 0 λ n i,out ; 0 λ n i,in,j, j; λ n i,out up n i,out if up n i,out < 0; up n i,out [ C i (α n i m n i ) + j [ C i (M n i α n i ) j ( λ n i,in,j) up n i,in,j ( λ n i,in,j) up n i,in,j if up n i,out > 0. (7.8) ], ], Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 37 / 9

39 Sub-region of solution 8 >< >: 0 λ n i,out ; 8 h C up n i (α n i m n i ) + X i sign( up ni,in,j ) up n i,in,j, i,out j >< λ n if up n i,out < 0; i,out h C n i (Mi up n α n i ) X i + sign( up ni,in,j i,out ) up n i,in,j, j >: if up n i,out > 0. (7.9) Tang et al. CEA/Grenoble-INP Two-fluid computation Sep 6, 0 38 / 9