How to avoid very large discrete velocity grids in deterministic simulations of rarefied gas flows? Luc Mieussens Institut de Mathématiques de Bordeaux, Université de Bordeaux, France Issues in Solving the Boltzmann Equation for Aerospace Applications (June 3-7, 2013) collaborators: C. Baranger - J. Claudel - N. Hérouard (CEA) S. Brull - L. Forestier-Costes (IMB) Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 1 / 49
Introduction flows of rarefied gases Deterministic simulations based on a discrete velocity approximation of the Boltzmann (BGK) equation (discrete ordinate method) A global discrete velocity grid, generally Cartesian, is commonly used for the whole computational domain (Kyoto group, Aristov et al., etc.) For practical applications in aerodynamics (atmospheric re-entry problems), the grid is so large that the computational ressources (memory storage and CPU time) required by the simulation are huge two projects: automatic generation of a locally refined velocity grid (steady flows), and construction of local velocity grids (unsteady flows) Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 2 / 49
Outline of the talk Basics for discrete velocity approximations of the models of RGD 1 Steady flows: local refinement of the velocity grid 2 Unsteady flows: a method to use local velocity grids 3 Conclusions 4 Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 3 / 49
Outline Basics for discrete velocity approximations of the models of RGD 1 Steady flows: local refinement of the velocity grid 2 Unsteady flows: a method to use local velocity grids 3 Conclusions 4 Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 4 / 49
Discrete velocity approximation of a kinetic equation of RGD Physical model : steady equation v · ∇ x f = Q ( f ) + BCs Velocity space R 3 � V = � v k , k ∈ N 3 � discrete velocity set Discrete kinetic equation: v k · ∇ x f k = Q k ( f ) + BCs Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 5 / 49
How to design a discrete velocity set? Generally: V is a Cartesian grid l ∆ v x l 0 : l max , = V = v l , m , n = c + m ∆ v y m 0 : m max n ∆ v z n 0 : n max Constraints: bounds and grid step : the grid must be large enough to contain the main part of the distribution functions for every position in the computational domain fine enough to capture the core of the distribution functions for every position in the computational domain Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 6 / 49
How to design a Cartesian velocity grid? At each space position x , we have information on the support of the distribution function f ( x , . ) : assume f is close to its local Maxwellian, then � it is centered on u ( x ) with standard deviation RT ( x ) � RT ( x ) � u ( x ) u ( x ) + c � RT ( x ) u ( x ) − c RT ( x ) The grid contains all the distributions if the bounds are at least � � v min = min x ∈ Ω { u ( x ) − c RT ( x ) } v max = max x ∈ Ω { u ( x ) + c RT ( x ) } (c around 4 ) There should be at least 3 points into the “core” of each distribution. The grid step should be: � ∆ v ≤ min x ∈ Ω { RT ( x ) } Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 7 / 49
Estimation of the bounds and the step of the grid Temperature and velocity fields are a priori unknown! Several solutions : Use the boundary values: upstream flow ( u ∞ , T ∞ ) and velocity and temperature of the body ( u wall , T wall ) More information with flow parameters inside the shock: u shock and T shock can be estimated by Rankine-Hugoniot relations. Even better: a compressible Navier-Stokes pre-simulation: gives u NS and T NS in Ω set x ∈ Ω { u NS ( x ) − c � x ∈ Ω { u NS ( x ) + c � v min = min RT NS ( x ) } v max = max RT NS ( x ) } � ∆ v = a min x ∈ Ω { RT NS ( x ) } simple, fast and efficient Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 8 / 49
Drawbacks of a Cartesian grid For re-entry problems: the velocity is very large ( ≈ 6.000 m/s), the temperature is very large in the shock ( ∼ 10 5 K), and very small in the upstream and at the boundary ( ∼ 10 2 K) Consequently: very large grid bounds, and very small step, so very large number of discrete velocities Example: Mach 20, altitude 90 km: V contains 52 × 41 × 41 points. Around 350 GB memory requirements with a coarse 3D mesh in space!... Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 9 / 49
What can be done? existing works for unsteady flows: variable velocity grids (all the grids have the same bounds, and the grids are refined or coarsened by AMR techniques), see Kolobov et al. (2011) and K. Xu et al. (2012) for steady flows: use the unsteady solution, but expensive this talk: for steady flows: estimation of an “optimal” locally refined global grid for unsteady flows: variable velocity grids, with variable bounds and steps Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 10 / 49
Outline Basics for discrete velocity approximations of the models of RGD 1 Steady flows: local refinement of the velocity grid 2 Unsteady flows: a method to use local velocity grids 3 Conclusions 4 Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 11 / 49
Idea in 1D 1D view: several distributions to be represented on the same grid Natural idea: refine the grid in the main parts of the distributions: Main contribution of this work: automatic generation of a locally refined velocity grid based on a CNS pre-simulation Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 12 / 49
Automatic generation of a locally refined velocity grid Main “ingredient”: construction of a “support” function Assume we have an estimation of the macroscopic velocity and temperature fields (compressible Navier-Stokes simulation) to build a Cartesian velocity grid V At each point x ∈ Ω , the “main” support of the distribution function � f ( x , . ) in the velocity space is centered at u ( x ) with a radius 4 RT ( x ) We define the support function φ ( v ) by �� � � φ ( v ) = min RT ( x ) , where x is such that || v − u ( x ) || ≤ 4 RT ( x ) . Take a velocity v : among all the distribution functions whose support contains v , one of them has a smallest support, and φ ( v ) is the standard deviation of this distribution. φ computed on the thin Cartesian grid defined by u and T . Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 13 / 49
Automatic generation of a l. r. velocity grid: algorithm Example : flow around a cylinder Tells us how the Cartesian grid around v l , m , n should be refined, or coarsened Algorithm to generate an AMR velocity grid: recursive algorithm that cuts a cell if it is larger than the minimum of φ in the velocity cell property of the final grid: every cell has a size smaller than the minimum of the support function in the cell. Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 14 / 49
Automatic generation of a l. r. v. grid: application Computational code 2D steady rarefied flows, monoatomic gases, BGK model finite volume scheme on curvilinear structured grids entropic and conservative velocity discretization Why BGK? simpler than Boltzmann (parametric model) often sufficiently accurate in transition regime and slow microflows (see "A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases" Hung et al. Commun. Comput. Phys. 12 (3) 2012) more physics might be included (polyatomic, multispecies, reactive) Code of the CEA 3D flows polyatomic gases 2D test case: steady flow over a cylinder of radius 0 . 1m, Mach 20, altitude 90 km, space mesh of 50 × 50 cells Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 15 / 49
Automatic generation of a l. r. v. grid: application Compressible Navier-Stokes pre-simulation gives a uniform Cartesian grid of 44 × 45 = 1980 points, total memory required: 1 Gb Support function and induced AMR velocity grid: AMR grid of 316 points, memory required: 140 Mb Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 16 / 49
Automatic generation of a l. r. v. grid: application Gain in memory: factor 7 Gain in CPU time: factor 7 Accuracy: average error less than 2% for macroscopic quantities maximum error on the normal heat flux along the boundary smaller than 3% Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 17 / 49
Automatic generation of a l. r. v. grid: application - 2D axisymmetric air flow around a sphere (radius = 1 cm ) , Ma =20 , altitude =90km gain: factor 10 (CPU and memory) accuracy: less than 1% error Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 18 / 49
Automatic generation of a l. r. v. grid: application - 2D axisymmetric Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 19 / 49
Automatic generation of a l. r. v. grid: application - full 3D air flow, cone with spherical nose, (radius = 1 cm), Ma = 20, altitude = 90km gain: factor 27 (CPU and memory) Luc Mieussens (Bordeaux, France) Variable velocity grids for DVM ICERM 2013 20 / 49
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