Efficient deterministic modelling of rarefied gas flows

Transcription

1 Efficient deterministic modelling of rarefied gas flows V.A. Titarev Dorodnicyn Computing Centre of Russian Academy of Sciences 64 IUVSTA Workshop on Practical Applications and Methods of Gas Dynamics for Vacuum Science and Technology, May 16-19, Germany V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 1 / 32

2 Outline of the presentation 1 Motivation: why kinetic equations and what methods we need for industrial applications 2 Model application: gas flow in long micro channels of finite length planar case circular pipe linearised as well as non-linear problems, including flow into vacuum 3 Review of existing methods & results steady-state iteration composite methods 4 Framework of new unstructured mesh solvers mixed-element unstructured spatial solver implicit time evolution efficient HPC version 5 Calculations of flows in long finite-length channels comparison with published data of Loyalka, Graur & Sharipov data for long finite-length planar channel data for long finite-length circular pipe V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 2 / 32

3 Why to use the Boltzmann kinetic equation? At present, the Monte-Carlo statistical simulation method (DSMC) is the computational method of choice. However, Due to statistical fluctuations and 1st order not very suitable for unsteady flows, transitional and near-continuum flows, slow flows computational efficiency is not optimal due to explicit time evolution The Boltzmann equation is free of any limitations of the DSMC: The equation is applicable across all flow regimes, i.e. from free molecular to near-continuum flows Unsteady flows can be treated in a straightforward manner. The deterministic nature of the equation allows the development of efficient high-order accurate methods, including methods with implicit time evolution It is possible to use special properties of the flow problem (e.g. asymptotic solution) in construction of numerical methods V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 3 / 32

4 Requirements for industrial applications Ability to compute flows with Knudsen layers in arbitrary geometries multi-block structured meshes mixed-element unstructured meshes, pure tetras not sufficient High-order of accuracy (at least 2nd order) linear schemes not suitable (e.g. MacCormack) non-linear solution-adaptive methods are required Conservation with respect to collision integral for model equations various approaches available: Rykov et al 1994, Mieussens 2000, Titarev 2006 Rapid convergence to steady state implicit time evolution Good scalability on modern HPC machines MPI model hybrid MPI-OpenMP or MPI-GPU models V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 4 / 32

5 S-model kinetic equation In the non-dimensional variables the kinetic equation takes the form: f M = f t + ξx f x + ξy f y + ξz f ν = 8 5 nt 1 π µ Kn, f + = f M z = ν(f + f ), ( (1 Pr)Sc(c2 52 ) ), n (πt ) exp 3/2 ( c2 ), c = v, v = ξ u, S = 2q T nt. 3/2 Macroscopic quantities defined as ( n, nu, n( 3 ) 2 T + u2 ), q = (1, ξ, ξ 2, 12 ) vv 2 fdξ. Boundary condition on the surface: ξ n<0 n w ( ξ2 f (x, ξ) = f w = exp (πt w ) 3/2 T w n w = N i /N r, N i = ξ nfdξ, N r = + ξ n>0 ), ξ n = (ξ, n) > 0, ) 1 ξ n exp ( ξ2 dξ. (πt w ) 3/2 T w V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 5 / 32

6 Linearised kinetic equation Linearise around the Maxwellian distribution f 0 corresponding to average values of density and temperature n 0, T 0: f 0 = f = f 0(1 + h), h = h(x, ξ), h 1, n 0 (πt exp 0) 3/2 ( ξ2 /T 0) 1 π exp 3/2 ( ξ2 ). Linearised macroscopic quantities: ˆn = n n0 = f 0hdξ, u = ξf 0hdξ, n 0 ˆT = T T0 = 2 ξ 2 f 0hdξ ˆn, q = 5 T u + 1 ξξ 2 f 0hdξ. 2 The evolution equation for the perturbation function h has the following form: h ξ x x + h ξy y + h ξz z = ν0(h(s) h), ν 0 = π Kn, h (S) = ˆn + 2uξ + (ξ ) ˆT (1 Pr)(ξ2 5 2 )qξ. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 6 / 32

7 Existing methods: steady-state iteration Typically used in two space dimensions. Let n be the iteration counter. Then: ( ) n+1 ( ) n+1 f f ξ x + ξ y = ν n (fm n f n+1 ) x y Advantages: Low storage requirement Extends to axisymmetric (Shakhov, 1974) and 3D (Arkhipov & Bishaev, 2007) flows directly Allows shock-fitting type difference schemes Disadvantages: Difficult to implement with high order of accuracy Non-conservative with respect to collision integral Poor convergence for small Kn, each iteration adds only a Kn 2 change (Bishaev & Rykov, 1975) V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 7 / 32

8 Composite methods Basic idea: for Kn 1 explicitly use the continuum Navier-Stokes solution in the numerical method 1 Bishaev & Rykov, 1975: one-dimensional non-linear heat transfer problem 2 Sharipov & Subbotin, 1992: multi-dimensional linearised problem, typical of microchannel flow Consider the linearised equation: Now assume Then The use of Chapman-Enskog method gives ξ f = ˆLf + g(x, ξ) f = f 0 + f, f 0 = lim Kn 0 f ξ f = ˆL f + g(x, ξ) + ˆLf 0 ξ f 0 f 0 = 2ξ u 0 where u 0 is the bulk velocity, found from Stoke s equations: µ u 0 = p, divu 0 = 0. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 8 / 32

9 Existing 3D methods & codes for kinetic equations These are time marching methods, in which steady solution is computed as a limit in time: multi-block structured (Z.-H. Li & H.-X. Zhang.J. Comp. Phys.,v.193, 2004) Unified Flow Solver (UFS) on Cartesian semi-structured meshes (V.I. Kolobov et. al. J. Comp. Phys., 223: , 2007) 1st order tetrahedral (Yu.Yu. Kloss et. al. Atomic Physics, 105(4), 2008) However, none of them satisfies all the requriements: spatial meshes not very suitable for certain problems (e.g. long micro channels, Knudsen layer resolution) slow convergence to steady state due to explicit time evolution unclear scalability on HPC More efficient & universal methods of numerical modelling are needed. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 9 / 32

10 2D benchmark problem: flow in long planar micro channel A planar microchannel of length 2l & width a connects large reservoirs filled with the same monatomic gas at pressures p 1,p 2 and temperatures T 1,T 2, respectively. P = p 1 p 2 & T = T 1 T 2 cause the gas movement through channel. The main quantity of interest: mass flow rate M: 2RT0 M = aρ(x, y, z)u(x, y, z)dy, p 0 = 1 p 02a 2 (p1 + p2), T0 = 1 (T1 + T2) 2 a V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 10 / 32

11 3D benchmark problem: flow through a long circular pipe A microchannel (pipe) of length 2l & radius a connects large reservoirs filled with the same monatomic gas at pressures p 1,p 2 and temperatures T 1,T 2, respectively. Due to the spatial symmetry of the problem only quarter of the pipe is considered. P = p 1 p 2 & T = T 1 T 2 cause the gas movement through channel. Non-dimensional mass flow rate: 2RT0 M = p 0 A A ρ(x, y, 0)w(x, y, 0)dxdy V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 11 / 32

12 Hierarchy of solutions The solution to the problem may have the following levels of approximation: 1 Fully non-linear approximation for finite-length channel (arbitrary pressure & temperature jumps) 2 Linearised approximation for finite-length channel (small pressure & temperature jumps) 3 Asymptotic approximation corresponding to l/a gas motion is caused by constant pressure & temperature gradients acting along the channel spatial dimension of the problem is reduced Choice of flow model for the Boltzmann equation: 1 exact collision integral - in theory most accurate, in practice very difficult to achieve good accuracy numerically 2 BGK (or Krook) model - the simplest model collision integral, not accurate for non-isothermal flows 3 S-model collision integral of Shakhov - presently the most accurate model equation for microchannel flows V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 12 / 32

13 Existing results for planar microchannel flows Infinitely long channel - a lot of studies, see e.g. F. Sharipov and V. Seleznev F. Sharipov & G. Bertoldo - comparison of model & exact collision integrals Finite-length channel, linearised flow - only short & moderate tubes, see e.g. C. Cercignani & I. Neudachin 1979 V.D. Akin shin, A.M. Makarov, V.D. Seleznev and F.M. Sharipov, 1988, 1989 E.M. Shakhov 1999,2000 Finite-length channel, nonlinear flow - Larina & Rykov 1996 Sharipov & Seleznev 1994: approximate method to calculate mass flow rate without taking into account end effects. Calculations for really long finite-length channels are missing even in the linearised regime. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 13 / 32

14 Existing results for 3D microchannel flows Infinitely long channel -most studies for the circular/rectangular cross section. More recent results I. Graur & F. Sharipov, elliptic cross section L. Szalmas & D. Valougeorgis triangular and trapezoidal cross section V. Titarev & E.M. Shakhov arbitrary polygonal cross section Linearised flow through a finite-length circular pipe: Shakhov 2000 Nonlinear flow: Orifice flow Shakhov 1974 Axisymmetric rarefied flow in the pipe caused by evaporation/condensation Shakhov 1996, Larina & Rykov 1998a Flow through short tubes for arbitrary pressure ratios, including into vacuum: Varoutis et al 2008, 2009 (using Monte-Carlo) Anikin, Kloss, Martynov and Tcheremissine 2010: Numerical modelling of Knudsen experiment Accurate calculations for really long finite-length pipes are missing even in the linearised regime. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 14 / 32

15 Nesvetay 2D/3D Framework of 2D/3D one-step implicit methods for deterministic modelling of rarefied gas flows developed by the author from 2007 onwards Flow model - Boltzmann kinetic equation with various model collision integrals (Krook, Shakhov, Rykov etc) The framework consists of the following blocks: high-order TVD method on hybrid unstructured meshes fully conservative procedure to calculate macroscopic quantities one-step implicit time evolution method both OpenMP and MPI parallelisation Due to time restrictions, only 3D version will be presented V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 15 / 32

16 Conservative discrete velocity framework March in time to steady state: g = ξ g + J(g), t J = ν(g (S) g), where g is the distribution function f for the nonlinear case and perturbation h in the linearised case. Replace the infinite domain of integration in the molecular velocity space ξ by a finite computational domain. The kinetic equation is replaced by a system of N ξ advection equations for each of g α = g(x, ξ α ): t g α = ξ α g α + J(g α ), which are connected by the macroscopic parameters in the function g (S) from the model collision integral J. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 16 / 32

17 Approximation of model collision integral on ξ mesh Let ω α be the weights of the second order composite quadrature rule used for integration in ξ space. To compute the vector of primitive variables U = (n, u 1, u 2, u 3, T, q 1, q 2, q 3) T for each spatial cell we have the following system of equations 1 0 ξ ξ 2 (f α + f α)ω α = 0. α vv 2 2 Pr q α Here subscripts i are n are omitted for simplicity. These eight equations are solved using the Newton iterations the initial guess for which is provided by the direct (non-conservative) approximation 3 2 n nu nt + nu2 q = α 1 ξ ξ vv 2 α f αω α Usually, one or two Newton iterations are sufficient for convergence. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 17 / 32

18 Advection scheme Introduce in the physical variables x = (x 1, x 2, x 3) = (x, y, z) a computational mesh consisting of elements (spatial cells) V i. Denote by V i the cell volume, A il area of face l. Omit subscript α for simplicity. Let t = t n+1 t n, g n = g(t n, x, ξ),, One-step explicit method can be written as: g n+1 g n = ξ g n + J n t The implicit one-step method has the following form: (1 + tν n + tξ ) g n+1 g n = ξ g n + J n t V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 18 / 32

19 Fully discrete method Basic idea: integrate over V i them discretize left-hand side with first-order upwind spatial differences the right-hand side L n α is approximated using a TVD method. The result is a system of linear equations for increments of the solution δ n = g n+1 g n : (1 + ν n i t)δ n i + t V i ξ nl F l (δi n, δ n σ l (i)) A il = { (ξ g n ) i + Ji n } t l where σ l (i) the cell index of the cell adjacent to the face l of cell V i. Using divergence theorem sum of face fluxes: (ξ g n ) i = 1 Φ n il, Φ n il = ξ nlg(t n, x, ξ)ds. V i The values at the next time level are given by g n+1 i l A il = g n i + δ n i. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 19 / 32

20 Stencils for piece-wise linear reconstruction method V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 20 / 32

21 Time evolution An approximate factorization of the system is carried out using the approach suggested in Men shov & Nakamura 1995, As a result, the computational cost of one time step of the implicit method is only 25% larger than the computational cost of an explicit method. In calculations, the value of the time step t is evaluated according to the expression t = C min i d i /ξ 0, where C is the prescribed CFL number, d i the characteristic linear size of the cell V i. C 1 3 corresponds to the conventional explicit method V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 21 / 32

22 Results 1 Calculations are run for both 2D and 3D channels 2 Both linearised & non-linear formulations are considered 3 Efficiency of implicit time evolution is assessed 4 Parallel scalability tests are carried out 5 All calculations are run on the HPC Facility Astral of the Cranfield university, which is a Hewlett Packard HPC, comprising 856 Intel Woodcrest cores (3.0GHz). V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 22 / 32

23 Results: pressure driven & creep flow Calculations have been carried out for l/a = 10, 100 and 1000 in 2D l/a = 1, 10 and 100 in 3D Main calculated characteristic of the flow - mass flow rate as function of ν 0, l/a and ratios of temperature and pressure in reservoirs Pressure driven flow: movement is caused by pressure difference only Linearised problem: solution is independent of pressure difference Non-linear problem: pressure ratio p 1/p 2 defines the flow Creep flow movement is caused by temperature gradient; pressure difference is zero. Only linearised problem is considered To compare results for different l/a and p 1/p 2 consider mass flow rate scaled with the average pressure gradient: M p = 2l P M T =0, M T = 2l T M P=0 V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 23 / 32

24 Mass flow rate in linearised 2D problem V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 24 / 32

25 Efficiency of implicit time evolution in 3D The solution of the nonlinear problem with pressure ratio p 1/p 2 = 2 is computed for ν 0 = 1 and hexa mesh with cells V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 25 / 32

26 Scalability studies using 16 3 velocity mesh in 3D Spatial mesh of hexa cells and 16 3 velocity mesh are used, performance normalised by the result on 16 cores Weak scaling (fixed size per core) Strong scaling (fixed problem size) V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 26 / 32

27 Asymptotic case l/a = & circular pipe Flow is described by spatially two-dimensional kinetic equation with a source term Solid line - Lo & Loyalka, 1982, semi-analytic method Crosses Graur & Sharipov, 2009, 10 6 spatial cells, composite scheme Red circles - Titarev & Shakhov, 2010, spatial cells, direct solution V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 27 / 32

28 Mass flow rate in linearised 3D problem V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 28 / 32

29 Flow in a 3D composite tube Channel: circular and pentagonal sections smoothly connected B.C.: z = 0 - evaporation, z = 5 complete condensation. Mixed element mesh with Knudsen layer: spatial cells intotal, of which 6401 tetras, 880 hexa, 2344 prisms and 426 pyramids. Velocity mesh: 16 3 cells. Calculations run on a PC V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 29 / 32

30 Mass flow ρw for Kn = 0.1 Cuts along the axis (left) and symmetry plane (right) V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 30 / 32

31 Conclusions 1 A brief summary of existing numerical methods for kinetic equations has been presented 2 Review of existing results for 2D and 3D microchannels shows that there are virtually no numerical results for long finite-length channels 3 A new framework for analyzing rarefied flows in complex 2D/3D geometries has been presented, which is fully unstructured, efficient and scalable up to 512 cores 4 Using the proposed approach, for the first time an accurate calculation of the mass flow rate through long microchannels has been performed for various flow regimes V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 31 / 32

32 Publications 1 V.A. Titarev. Conservative numerical methods for model kinetic equations // Computers and Fluids, 36(9): , V.A. Titarev. Numerical method for computing two-dimensional unsteady rarefied gas flows in arbitrarily shaped domains // Computational Mathematics and Mathematical Physics, 49(7): , V.A. Titarev. Implicit numerical method for computing three-dimensional rarefied gas flows using unstructured meshes // Computational Mathematics and Mathematical Physics, 50(10): , V.A. Titarev. Implicit unstructured-mesh method for calculating Poiseuille flows of rarefied gas // Communications in Computational Physics, 8(2): , V.A. Titarev and E.M. Shakhov. Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic S-model // Computational Mathematics and Mathematical Physics, 50(12): , V.A. Titarev. Efficient deterministic modelling of three-dimensional rarefied gas flows, submitted to Communications in Computational Physics. 7 V.A. Titarev. Linearised problem of rarefied gas flow through a long circular pipe of finite length, in preparation. V.A. Titarev (Computing Center of RAS) Deterministic modelling 64 IUVSTA Workshop 32 / 32