Problems of Flows Through Short Channels Studied by Means of the Boltzmann Equation

Transcription

1 Problems of Flows Through Short Channels Studied by Means of the Boltzmann Equation Aristov V.V., Frolova A.A. and Zabelok S.A. Dorodnicyn Computing Centre of Russian Academy of Sciences Kolobov V.I. and R.R.Arslanbekov CFD Research Corporation, Huntsville, AL, USA

2 Motivation The kinetic apparatus of the Boltzmann equations allows us to solve the problems of flows through a slit and channels in a wide range of parameters. The solutions of the Boltzmann equation can be the accurate tests which guarantee the reliable basis for the other approaches (e.g. by the model equations), on the other hand the direct Boltzmann approach is reliable for small pressure drop (where DSMC fails). Aristov V.V. Direct Methods for solving the Boltzmann equation and study of nonequilibrium flows, Kluwer Academic Publishers, Dordrecht, 2001.

3 Method of study Unified Flow Solver (UFS) - V.I.Kolobov, R.R.Arslanbekov, V.V.Aristov, A.A.Frolova,S.A.Zabelok, J.Comp.Phys., (2007) is used for numerical study of the problem. The method includes the adaptive mesh refinement, the discrete velocity technique and quasi-monte Carlo procedure for evaluation of the collision integrals. It deals with the direct method of solving the Boltzmann equation (and model equations as the additional instrument). The UFS uses conservative kinetic schemes for approximation of the kinetic equations and special kinetic schemes approximating the NS equations, this hybrid method automatically divides the domain into continuum and kinetic parts, uses appropriate schemes in appropriate parts and couples the continuum and kinetic solutions.

4 UFS applications: supersonic flows 3D problem with conditions of Inflatable Reentry Vehicle Experiment (IRVE).The flow conditions are for 91 km altitude: Kn = 0.01 and Mach number M = 3.94, the flow is at zero angle of attack Streamlines, Mach number, and computational mesh Streamlines and a field of velocities.

5 UFS applications: subsonic flows Kn = 0.01 Kn = 0.06 Kn = 0.3 v max ~5E 4 v max ~0.007 v max ~0.005 Flows induced by the nonuniform temperature on the chamber walls. The kinetic computational domain (red) and the continuum computational domain (blue) are shown in the up left corner of the middle plot. Flow speed of order of 0.01 thermal speed observed, significant flow (~2 3 m/s) of simulated by simple gas radicals flow over semiconductor wafer expected.

6 The benchmark problems Flow through a slit and a short channel is considered. The hard sphere molecular model is used. The diffuse reflection in hard walls is accepted.

7 Flow through a slit. Comparison of a mass flow rate obtained with different methods W (p 2 /p 1 =0) UFS DSMC* BGK** S-mod.** * Sharipov, Kozak, 2009 ** Graur, Polikarpov, Sharipov, 2011

8 Computational resources The most accurate results are obtained with the pure Boltzmann solver with the following parameters: velocity mesh 56x56x28 (~87k cells); ~20k cells in physical space; more than 10 9 cells in phase space; 60 collisions per velocity cell; One step took about 7 seconds on 1500 cores of MVS100k supercomputer.

9 Computational resources The computation for UFS (Boltzmann solver +NS) i.e. the hybrid scheme for small Knudsen numbers: velocity mesh 28x28x14 (~11k cells); 12 collisions per velocity cell; ~15k cells in physical space; The convergence is achieved (p 2 /p 1 =0.5, =10) after 60 hours of computations on P4 3GHz processor.

10 Test variant The convergence of the solutions for the UFS (the Boltzmann equation) have been considered for one variant of the flow, namely for p 2 /p 1 =0, =1. The following range of the computational parameters has been covered The number of collisions per a cell: The velocity meshes: (28, 28, 14) (56, 56, 28). The size of the computation domain: -50H 50H. The adaptive grids with 5 or 6 maximum levels of refinement are used in physical space.

11 Static adapting mesh

12 Dynamically adapting mesh

13 The test computation for a benchmark problem =1, p2/p1=0, L/H=0 Solutions of the Boltzmann equation for differrent numerical parameters =60, (vx,vy,vz)=(56,56,28) W=1.1434, N=60, (vx,vy,vz)=(42,42,21) W=1.1438, N=60, (vx,vy,vz)=(28,28,14) W= The Richardson error estimation: The Richardson correction value: W= = Rough error estimation DW= , DW/W=0.11%.

14 Flow through a slit. Comparison of a mass flow rate obtained with different methods W (p 2 /p 1 =0.5) UFS DSMC* BGK** S-mod.** * Sharipov, Kozak, 2009 ** Graur, Polikarpov, Sharipov, 2011

15 Flow through a slit. Comparison of a mass flow rate obtained with different methods W(p 2 /p 1 =0.5) UFS BGK

16 Density profiles Distribution of density along the symmetry axis for p 2 /p 1 =0.5, =100.

17 Velocity profiles Distribution of macroscopic velocity along the symmetry axis for p 2 /p 1 =0.5, =100.

18 Temperature profiles Distribution of temperature along the symmetry axis for p 2 /p 1 =0.5, =100.

19 Flow through a slit. Comparison of a mass flow rate obtained with different methods W(p 2 /p 1 =0.9) UFS BGK

20 Flow through a slit. Comparison of a mass flow rate obtained with different methods W(p 2 /p 1 =0.99) UFS BGK

21 Reduced flow rate W vs rarefaction parameter

22 CONCLUSIONS We performed a series computations by means of the Boltzmann equations for 2D plane problems. The hybrid scheme of the UFS (Unified Flow Solver) for small Knudsen number is also used. Computations confirm that the differences of the values of the mass flow rates by the Boltzmann direct solutions and by the model equations (and also DSMC) are in the range of the order of 1%..Accurate calculations for the Boltzmann equation allow us to propose that for moderate Knudsen numbers the difference from model equation solutions is 2-3%. For small pressure drops the direct Boltzmann solver produces reliable results. Values of W are different from the linear theory for large values of the rarefaction parameter.