On two fractional step finite volume and finite element schemes for reactive low Mach number flows

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1 Fourth International Symposium on Finite Volumes for Complex Applications - Problems and Perspectives - July 4-8, 2005 / Marrakech, Morocco On two fractional step finite volume and finite element schemes for reactive low Mach number flows Fabrice Babik *, Thierry Gallouët **, Jean-Claude Latché *, Sylvain Suard *, and Didier Vola * * Institut de Radioprotection et de Sûreté Nucléaire, France ** Laboratoire d Analyse, Topologie, Probabilité, UMR 6632, CMI, France 1/21

2 Outline 1. Introduction 2. Finite Volumes formulation 3. Two numerical schemes 4. Numerical tests Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 2/21

3 Introduction 1. Introduction 1.1 Presentation 1.2 Toward a numerical scheme 2. Finite Volumes formulation 3. Two numerical schemes 4. Numerical tests Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 3/21

4 1.1 Presentation Governing equations: ϱ t + ϱu = 0 ϱu + (ϱu u) = τ p + (ρ ρ )g t ϱz + ϱzu = (κ z) t ϱ = G(z) + boundary and initial conditions Several physical problems enter this abstract framework: Natural convection flows in the low Mach number limit: z stands for the temperature, G(z) is the equation of state of perfect gaz (at fixed pressure in case of open boundaries; otherwise, the preceding system must be supplemented by an evolution equation for the thermodynamic pressure). Non-premixed laminar flame: z is a Zeldovitch s variable called mixture-fraction, G(z) derived from T (z), Y F (z), Y Ox (z) and Y P (z). Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 4/21

5 1.2 Toward a numerical scheme (1/2) Particular features and requirements: the unknown field z must have convergent and stability properties construction of a FV numerical scheme which reproduces these features at the discrete level ϱ = G(z): no independant evolution equation can be stated for the fluid density the mass balance equation acts as a constraint on the velocity - use of fractional step schemes, namely projection methods FEs for an accurate velocity and pressure computation - nonconforming velocity approximations (dof located at the center of the faces) for a coupling with a FV method for the advection-diffusion of z Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 5/21

6 1.2 Toward a numerical scheme (2/2) Overshoots or undershoots (even small) have a dramatic effect in chemically reacting flows From both a physical and numerical viewpoint, all the species concentrations remain positive A maximum principle stands for the unknown field z As the velocity is not divergence free, the conservation equation for z does not imply any maximum principle... But it can be combined with the mass balance equation as follows: ϱz t + ϱzu = ϱ( z t + u z) + z ( ϱ t + ϱu) }{{} =0 = (κ z) For this latter equation, if z(x, 0) [0, 1] and z takes values in [0, 1] on Dirichlet boundaries, then z(x, t) [0, 1], x, t. Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 6/21

7 Mixed FV / FE formulation 1. Introduction 2. Finite Volumes formulation 2.1 A finite volumes scheme for the transport of z 2.2 High order interpolation FV schemes 3. Two numerical schemes 4. Numerical tests Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 7/21

8 2.1 A finite volumes scheme for the transport of z (1/4) Definition: M is a M-matrix if M ii > 0, 1 i N, M ij 0, for i j M ii M ij 0, i j i M ii M ij > 0, for one node j i If M is a M-matrix: if MX 0, = X 0 If M t is an M-matrix, MX 0 = X 0 Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 8/21

9 2.1 A finite volumes scheme for the transport of z (2/4) Scheme: solve in sequence a prediction for the density, then the (now linear) convection-diffusion equation for z, and, finally, update the density using the constitutive equation. m K ϱ K ϱ n K k ϱ K z n+1 K m K k ϱ n+1 K = G(zn+1 K ) + σ E K m σ v σ ϱ σ = 0 ϱn K zn K + m σ v σ ϱ σ zσ n+1 σ E K + σ E K m σ κ (z n+1 K d σ zn+1 L ) = 0 where : v σ stands for the normal velocity to the edge σ ϱ σ and z n+1 σ stand for the standard upwind approximation for ϱ and z n+1. Of course, the prediction step makes sense only if the mass balance holds. Exemple: see the full scheme hereafter. Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 9/21

10 2.1 A finite volumes scheme for the transport of z (3/4) The discrete operator associated to the first equation is the transposed of an M-matrix. The discrete operator associated to the second equation is an M-matrix: Σ= L A K,L = m K ϱ K k + m σ v σ,k ϱ σ = m ϱn K K k > 0 σ E K Let z n and z n be defined by z n = max K zn K, zn = min K zn K. Then: Idea of the proof: K, z n+1 K [zn, z n ] ϱ K (z n+1 K m zn ) K k + σ E K m σ + m σ v σ ϱ σ (zσ n+1 z n ) σ E K κ [(z n+1 K d zn ) (z n+1 L σ z n )] = m K ϱ n K zn K k Σ z n = m K ϱ n K (zn K zn ) k }{{} 0 Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 10/21

11 2.1 A finite volumes scheme for the transport of z (4/4) A L 2 stability property. Multiplying by z the advection-diffusion equation for z and integrating over Ω yields: Ω [ ] ϱz t + (ϱzu) = 1 2 Ω = 1 2 Ω = 1 ϱ z2 2 t Ω z = Ω ϱ z2 t + ϱu (z2 ) ϱ z2 t z2 (ϱu) + z2 ϱ t = 1 2 t [ ϱ t + (ϱu)] z2 + ϱz z Ω t Ω ϱz 2 = Ω + ϱzu z (mass balance equation) (integration by parts) (mass balance equation) κ z 2 The same computation can be followed "line by line" in the discrete case, which provides: a stability result, a tool for proving an error estimate? Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 11/21

12 2.2 High order interpolation FV schemes A second order FV scheme for diffusive cases: Numerical experiments show that the spatial accuracy is limited to first order as soon as an upwind discretization is used for the prediction of the density. Use of a centered deferred correction scheme (2 nd order): ρ ρ n + U (ρv) n+1 + β ( C (ρv) n + U (ρv) n ) = 0 The β parameter is chosen to ensure that ρ remains positive. Use of an hybrid FV interpolation scheme for the transport of z ϱz t + H (ϱzu) = (κ z) with H (φ) = α C (φ)+(1 α) U (φ), α [0, 1] Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 12/21

13 Two numerical schemes 1. Introduction 2. Finite Volumes formulation 3. Two numerical schemes 3.1 A non-conservative scheme 3.2 A conservative scheme 4. Numerical tests Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 13/21

14 3.1 A non-conservative scheme solve for ϱ ϱ ϱ n + U ϱu n = 0 solve for z n+1 ϱz n+1 ϱ n z n + U ϱz n+1 u n C,κ z n+1 = 0 ϱ n+1 given by ϱ n+1 = G(z n+1 ) solve for ũ solve for p n+1, u n+1 ϱũ ϱ n u n + ( ϱu n ũ) τ(ũ) + p n = g n+1 ϱ n+1 u n+1 ϱũ + (p n+1 p n ) = 0 ϱ n+1 u n+1 = ϱn+1 ϱ n the density is updated after the computation of z n+1 this scheme does not conserve the quantity ϱz Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 14/21

15 3.2 A conservative scheme solve for ϱ solve for z ϱ n+1 given by solve for ũ solve for p n+1, u n+1 solve for z n+1 ϱ ϱ n + U ϱu n = 0 ϱ z ϱ n z n + U ϱ zu n C,κ z = 0 ϱ n+1 = G( z) ϱũ ϱ n u n + ( ϱu n ũ) τ(ũ) + p n = g n+1 ϱ n+1 u n+1 ϱũ + (p n+1 p n ) = 0 ϱ n+1 u n+1 = ϱn+1 ϱ n ϱ n+1 z n+1 ϱ n z n + CU ϱ n+1 z n+1 u n+1 C,κ z n+1 = 0 stability of the last step requires: - a centered approximation for the density - an upwind approximation for the variable z Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 15/21

16 Numerical tests 1. Introduction 2. Finite Volumes formulation 3. Two numerical schemes 4. Numerical tests 4.1 Analytical premixed flame propagation 4.2 Planar non-premixed flame propagation 4.3 Natural convection flows Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 16/21

17 4.1 Analytical premixed flame propagation (1/2) Governing equations: ϱ t + ϱu = 0 ϱu + (ϱu u) = τ p t ϱz + ϱzu = (κ z) + f(z) t ϱ = 1/(1 γ + γz) Analytical solution: z(x, t) = 1 2 ( [ ]) x ut 1 tanh δ propagation burnt mixture fresh mixture reaction zone Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 17/21

18 4.1 Analytical premixed flame propagation (2/2) z, N-C S z, C S rho, N-C S, order 2 z, N-C S, order 2 slope order 2 L 2 error norm 0.1 slope order 1 N = 125, N-C S C S N = 500, N-C S C S N = 2000, N-C S C S Time-step L 2 error norm Dx L 2 error norm obtained with the non-conservative (N-C S) and conservative scheme (C S), as a function of: left: time-step for differents grid size; right: grid size for a time-step of Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 18/21

19 4.2 Planar non-premixed flame propagation Infinitely fast chemistry assumption: Burke-Schumann approximation The variable z stands for the mixture fraction A very stiff problem: Large gradients of density and temperature appear near z S 0.05 Mixtures fraction and density mixture fraction, dt = s dt = s dt = s density, dt = s dt = s dt = s Mixtures fraction and density mixture fraction, N = 200 N = 400 N = 800 density, N = 200 N = 400 N = Reduced abscissa Reduced abscissa Distributions of mixture fraction and density along the flow direction obtained with the non-conservative scheme for three time steps (left) and three mesh sizes (right). Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 19/21

20 4.3 Natural convection flows (1/2) A buoyancy-driven flow in a two-dimensional square enclosure [0,L] [0,L] with large temperature differences Two insulated walls and two vertical walls heated to T h respectively cooled down to T c. zero heat flux Flow specification: Dimension 2D T h hot fluid g T c Geometry Temporal Cartesian Steady laminar & Thermal cold fluid Gas law Perfect gas zero heat flux Nusselt number: Nu(y) = L k 0 (T h T c ) k T x Average Nusselt number: y=l Nu = 1 L y=0 Nu h + Nu c = 0 Nu(y) dy, w Governing equations: ϱ t + ϱu = 0 ϱu + (ϱu u) = τ p + (ρ ρ )g t ϱh + ϱhu = (κ h) + dp t dt P = ϱrt Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 20/21

21 4.3 Natural convection flows (2/2) Test case T2 (H. Paillère and P. Le Quéré): Ra = P r g ρ2 0 (T h T c ) L 3 T 0 µ 2 0 = 10 6, Sutherland s Law, ɛ = T h T c T h + T c = 0.6 Initial conditions: The problem is completely defined by the Rayleigh number, the value of ɛ and the following coefficient: P 0 = Pa, T 0 = 600 K, ρ 0 = P 0 /(RT 0 ), P r = 0.71, γ = 1.4, g = 9.81 m/s 2 Boundary conditions: hot wall: T h = T 0 (1 + ɛ), cold wall: T c = T 0 (1 ɛ) horizontal walls: adiabatic conditions, all walls: no-slip condition for the velocity Ref relative error Nu h e := 1 + Nu c / Nu h O(h) for scheme 1 Fourth International Symposium on Finite Volumes for Complex Applications, July 4-8, 2005 / Marrakech 21/21

Convergence of the MAC scheme for incompressible flows

Convergence of the MAC scheme for incompressible flows T. Galloue t?, R. Herbin?, J.-C. Latche??, K. Mallem????????? Aix-Marseille Universite I.R.S.N. Cadarache Universite d Annaba Calanque de Cortiou