Commissariat à l Energie Atomique - Saclay Department of Nuclear Energy Fluid Modeling and Simulation. MIT, November 3rd, 2008

Transcription

1 ON THE NUMERICAL SIMULATION OF GENERAL MULTI-FIELD TWO-PHASE FLOW MODELS Anela KUMBARO Commissariat à l Energie Atomique - Saclay Department of Nuclear Energy Fluid Modeling and Simulation MIT, November 3rd, /25

2 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 2/25

3 CONTEXT Various Riemann solvers such as the Roe-type approximate solver, the VFRoe schemes, the flux vector splitting schemes, or VFFC scheme rely on the spectral decomposition in order to compute the upwind matrices. For efficiency reasons, the eigenstructure information can be obtained analytically by an approximate computation, or an exact computation for a given model. In the case of the multifluid model, an analytical computation of the eigenstructure is always out of question. Moreover, singularities can appear when the phase/field disappears. To surmount this kind of problem, some authors make use of methods, such as the Harten, Lax and van Leer (HLL) solvers or the Rusanov scheme, which need a very limited information on the eigenstructure of the systems, but lack accuracy. We propose either the use of a numerical algorithm or a new intermediary strategy ; the Simplified Eigenstructure DEcomposition Solver (SEDES) aims to have an accuracy close to the one observed by the Roe-type scheme while using an eigenstructure information which is much more restricted. 3/25

4 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 4/25

5 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 5/25

6 SYSTEM OF EQUATIONS A typical form of the conservation equations for the i-field can be written as follows Mass balance equation t (α i ρ i ) + (α i ρ i u i ) = Γ ij + m il (1) j l Momentum balance equation t (α i ρ i u i ) + (α i ρ i u i u i ) + α i p i + (p i p int ij ) α i = (α i [τ + τ T ]) + α i iρ i g + i j M int ij + M w i + j j Γ ij u int ij + l m il u il (2) Energy balance equation t (α i ρ i e i ) + (α i ρ i u i e i ) + p i ( (α i u i ) + t α i ) α i (q i + q T i ) = Q w i + Q int ij + Γ ij h int i + m il h i (3) j j l 6/25

7 CONDENSED FORM The multifield model being non conservative, it can be written in the following condensed form : V d α V i ρ i u i + F + C j = S with F = α i ρ i u i u i + α i pi t x j=1 j α i ρ i u i e i 7/25

8 CONDENSED FORM The multifield model being non conservative, it can be written in the following condensed form : V d α V i ρ i u i + F + C j = S with F = α i ρ i u i u i + α i pi t x j=1 j α i ρ i u i e i The 1D formulation of the above system is where n is a unit vector in R d. V t + F n n + C V n n = S (4) 7/25

9 CONDENSED FORM The multifield model being non conservative, it can be written in the following condensed form : V d α V i ρ i u i + F + C j = S with F = α i ρ i u i u i + α i pi t x j=1 j α i ρ i u i e i The 1D formulation of the above system is where n is a unit vector in R d. V t The quasi-linear form of the 1D problem is V t + F n n + C V n n = S (4) + A n (V) V n = S. (5) 7/25

10 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 8/25

11 REMARKS 1 Energy lines are decoupled in the system matrix and u ni are the corresponding eigenvalues 2 The most convenient vector to obtain the spectral properties of the system (1)-(2) is the so called vector of mixed variables U = ( α 1 ρ 1... α N ρ N u 1... u N ) T. 9/25

12 REMARKS 1 Energy lines are decoupled in the system matrix and u ni are the corresponding eigenvalues 2 The most convenient vector to obtain the spectral properties of the system (1)-(2) is the so called vector of mixed variables U = ( α 1 ρ 1... α N ρ N u 1... u N ) T. 3 P i = j (p i p int ij ) is the pressure default for the i-field at the interface with the j-fields ; 4 H 1 and H 2 are the matrices that express the pressure and volume fractions differentials : dp α 1 dp + P 1 dα 1 α dα 1 ρ 1 1. = H 2. = H 2.H 1 1..d.. α N dp + P N dα N α dα N ρ N N 1 9/25

13 CHARACTERISTIC POLYNOMIAL The characteristic polynomial can be expressed as follows P(λ) = 1 det(h 1 ) det((v n λi N ) 2 H 1 H 2 ) N (u ni λ) 2, (6) where V n is a N N matrix with diagonal terms equal to u ni. Hence, taking into account also the energy equations, u ni, i = 1,..., N are eigenvalues with multiplicity 3 for a 3D-case. To obtain the 2N remaining non-trivial eigenvalues, we find an expression of the determinant of a sparse Hessenberg matrix : i=1 P 2N (λ) = N α i c 2 i=1 i ( ) ( ) (uni λ) 2 c 2 i ρj (u nj λ) 2 P j. (7) j i 10/25

14 EIGENVALUES STRUCTURE If we sort out the velocities u ni by increasing order 11/25

15 EIGENVALUES STRUCTURE If we sort out the velocities u ni by increasing order 2 non trivial eigenvalues corresponding to pressure waves λ = ũ n a m 11/25

16 EIGENVALUES STRUCTURE If we sort out the velocities u ni by increasing order 2 non trivial eigenvalues corresponding to pressure waves λ = ũ n a m 2(N 1) non trivial eigenvalues corresponding to void waves ( ) λ u ni Pi ρ i, u ni+1 Pi+1 ρ i+1 i = 1,..., N 1 ( ) λ u ni + Pi ρ i, u ni+1 + Pi+1 ρ i+1 i = 1,..., N 1 11/25

17 EIGENVALUES STRUCTURE If we sort out the velocities u ni by increasing order 2 non trivial eigenvalues corresponding to pressure waves λ = ũ n a m 2(N 1) non trivial eigenvalues corresponding to void waves ( ) λ u ni Pi ρ i, u ni+1 Pi+1 ρ i+1 i = 1,..., N 1 ( ) λ u ni + Pi ρ i, u ni+1 + Pi+1 ρ i+1 i = 1,..., N 1 Trivial eigenvalues : 2N shear waves : λ = u ni i = 1,..., N N energy waves : λ = u ni i = 1,..., N 11/25

18 EIGENVALUES STRUCTURE If we sort out the velocities u ni by increasing order 2 non trivial eigenvalues corresponding to pressure waves λ = ũ n a m 2(N 1) non trivial eigenvalues corresponding to void waves ( ) λ u ni Pi ρ i, u ni+1 Pi+1 ρ i+1 i = 1,..., N 1 ( ) λ u ni + Pi ρ i, u ni+1 + Pi+1 ρ i+1 i = 1,..., N 1 Trivial eigenvalues : 2N shear waves : λ = u ni i = 1,..., N N energy waves : λ = u ni i = 1,..., N There are utmost 3N distinct eigenvalues The sound waves, are clearly separate from the others The remaining eigenvalues have more or less the same order of magnitude 11/25

19 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 12/25

20 ROE-TYPE SOLVER : MOTIVATION U n+1 i = U n i t x (A+ Roe (Un i 1, Un i )(Un+1 i U n+1 i 1 ) A Roe (Un i, Un i+1 )(Un+1 i+1 Un+1 i )) with A ± = A± A 2. If P int is a polynomial such that P int (λ i ) = λ i, then P int (A) = A. One can consider the Lagrange interpolating polynomial n d j i(x λ j ) P int (X) = λ i j i(λ i λ j ) No divergence problem, even with complex eigenvalues. i=1 At most (n d 2) matrix-matrix multiplications Need a robust computation of P int (X) 13/25

21 GENERAL FORM OF P int n + eigenvalues 0, n eigenvalues < 0, assume n + n P + = (X λ i ), deg(p + ) = n + λ i 0 P int = X + P + Q +, deg(q + ) < n Determine Q + : interpolation over n support points using Newton divided differences Q + (λ i ) = 2λ i P + (λ i ), λ i < n (n 1) scalar multiplications for Q + At most n + matrix-matrix multiplications to evaluate P int (A) 14/25

22 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 15/25

23 GENERAL PRESENTATION The solution of a one-dimensional Riemann problem at cell interfaces shows how to define backward and forward differences to approximate the spatial derivatives F n. The solver proposed is : n F Φ n+1 K,L = n (V n+1 K ) + F n (V n+1 L ) ASED 2 n (V n K, Vn L ) 2 (V n+1 L V n+1 K ) with the upwinding matrix evaluated at an average state on the interface between K and L. We sort out the eigenvalues of the linearised system matrix A n (V n K, Vn L ) by increasing order : λ 1 λ p < 0 < λ M, M = N(d + 2). 16/25

24 CONSTRUCTION OF THE UPWIND MATRIX Initial matrix Complete eigen-decomposition N A n = λ k l k r k k=1 17/25

25 CONSTRUCTION OF THE UPWIND MATRIX Initial matrix Complete eigen-decomposition N A n = λ k l k r k k=1 Hypothesis We state that : - it is essential to take account of the contributions of the fastest eigenvalues, - the remaining eigenvalues can be represented by a unique candidate, for instance by the fastest one of this group : λ = max( λ 2,..., λ M 1 ) 17/25

26 CONSTRUCTION OF THE UPWIND MATRIX Initial matrix Complete eigen-decomposition N A n = λ k l k r k k=1 Hypothesis We state that : - it is essential to take account of the contributions of the fastest eigenvalues, - the remaining eigenvalues can be represented by a unique candidate, for instance by the fastest one of this group : λ = max( λ 2,..., λ M 1 ) SEDES upwind matrix A n A SED n M 1 = λ 1 l 1 r 1 + λ M l M r M + λ k=2 l k r k A SED n = ( λ 1 λ )l 1 r 1 + ( λ M λ )l M r M + λ I 17/25

27 REMARKS ABOUT SEDES MATRIX SEDES needs only the eigenvalues of the system, or an estimation, together with the eigenvectors associated to the sound waves 18/25

28 REMARKS ABOUT SEDES MATRIX SEDES needs only the eigenvalues of the system, or an estimation, together with the eigenvectors associated to the sound waves eigenvalues/vectors degeneration during phase disappearing is not an issue 18/25

29 REMARKS ABOUT SEDES MATRIX SEDES needs only the eigenvalues of the system, or an estimation, together with the eigenvectors associated to the sound waves eigenvalues/vectors degeneration during phase disappearing is not an issue The definition of λ leads to Sp( A SED n A ROE n ) R +. The difference on the numerical diffusion between Roe and SEDES solvers will depend on A SED n A ROE n = N 1 ( λ λ k )l k r k k=2 18/25

30 REMARKS ABOUT SEDES MATRIX SEDES needs only the eigenvalues of the system, or an estimation, together with the eigenvectors associated to the sound waves eigenvalues/vectors degeneration during phase disappearing is not an issue The definition of λ leads to Sp( A SED n A ROE n ) R +. The difference on the numerical diffusion between Roe and SEDES solvers will depend on A SED n A ROE n = N 1 ( λ λ k )l k r k Comparing the numerical diffusion of SEDES solver with Rusanov solver yields : N 1 A SED (s λ )l k r k A Rus n n k=2 k=2 18/25

31 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 19/25

32 THREE-FIELD TWO-PHASE FLOW RANSOM FAUCET PROBLEM The problem consists of a water jet contained within a channel, 12 m long. 20/25

33 THREE-FIELD TWO-PHASE FLOW RANSOM FAUCET PROBLEM The problem consists of a water jet contained within a channel, 12 m long. Boundary and initial conditions Liquid velocity of 10 m/s and stagnant gas Gas volume fraction, α g = 0.2, is divided between the two vapor fields into equal parts Pressure is 10 5 Pa and flow temperature is 50 C Gravity term is the only one source term 20/25

34 THREE-FIELD TWO-PHASE FLOW RANSOM FAUCET PROBLEM The problem consists of a water jet contained within a channel, 12 m long. Boundary and initial conditions Liquid velocity of 10 m/s and stagnant gas Gas volume fraction, α g = 0.2, is divided between the two vapor fields into equal parts Pressure is 10 5 Pa and flow temperature is 50 C Gravity term is the only one source term Tabulated equations of state for thermodynamic properties 20/25

35 THREE-FIELD TWO-PHASE FLOW RANSOM FAUCET PROBLEM The problem consists of a water jet contained within a channel, 12 m long. Boundary and initial conditions Liquid velocity of 10 m/s and stagnant gas Gas volume fraction, α g = 0.2, is divided between the two vapor fields into equal parts Pressure is 10 5 Pa and flow temperature is 50 C Gravity term is the only one source term Tabulated equations of state for thermodynamic properties The computations are realized using a 100 and 1000 cells grid with a CFL number of 1 Comparison of numerical results obtained using Sedes, Roe and Rusanov scheme, respectively 20/25

36 THREE-FIELD TWO-PHASE FLOW RANSOM FAUCET PROBLEM : RESULTS Total vapor volume fraction Roe 1000 cells Roe 100 cells Sedes 100 cells Sedes 1000 cells Rusanov 100 cells Rusanov 1000 cells analytical Pressure [Pa] 1.005e e e e e X [m] 9.950e X [m] Vapor velocity [m/s] Liquid velocity [m/s] X [m] X [m] FIG.: Water faucet problem. From left to right, from top to bottom : volume fraction, pressure, vapor velocity, and liquid velocity profiles. 21/25

37 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit 22/25

38 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit Calculations were performed including four groups of bubbles of different sizes 22/25

39 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit Calculations were performed including four groups of bubbles of different sizes Use of coalescence, breakup, and drag models 22/25

40 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit Calculations were performed including four groups of bubbles of different sizes Use of coalescence, breakup, and drag models We hypothesize that all of the heating goes to the first group bubbles 22/25

41 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit Calculations were performed including four groups of bubbles of different sizes Use of coalescence, breakup, and drag models We hypothesize that all of the heating goes to the first group bubbles Tabulated equations of state for thermodynamic properties 22/25

42 BOILING IN A VERTICAL HEATED CHANNEL The incipience of boiling occurs at channel inlet, and the heating rate is such that a slug flow regime occurred at channel exit Calculations were performed including four groups of bubbles of different sizes Use of coalescence, breakup, and drag models We hypothesize that all of the heating goes to the first group bubbles Tabulated equations of state for thermodynamic properties The computations are realized using a 200 cells grid. Comparison of numerical results obtained using Sedes and Roe solvers 22/25

43 0.3 BOILING IN A VERTICAL HEATED CHANNEL field 1 field 2 field 3 field 4 Vapor volume fraction x (m) FIG.: Boiling in a vertical heated channel. Five group model. Vapor volume fraction profiles. 23/25

44 OUTLINE 1 MULTIFIELD TWO-PHASE FLOW MODEL 2 EIGENSTRUCTURE OF THE MULTIFIELD MODEL 3 ROE-TYPE APPROXIMATE RIEMANN SOLVER 4 SIMPLIFIED EIGENSTRUCTURE DECOMPOSITION SOLVER (SEDES) 5 NUMERICAL RESULTS 6 CONCLUSION 24/25

45 CONCLUSION We presented a method which is able to treat general multifield two-phase flow models with realistic state equations. Our overall evaluation methodology seeks to prove three hypothesis : that three informations coming from the eigenvalues are to take into account to obtain an accurate solver ; that field/phase appearing/disappearing does not affect the computation of the upwinding matrix ; that results are quite satisfactory for reasonable relative velocities, i.e. when the hypotheses for deriving the scheme fully apply. 25/25