A new adaptative numerical method for kinetic equation St ephane - PowerPoint PPT Presentation

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions A new adaptative numerical method for kinetic equation St´ ephane Brull, Louis Forestier-Coste, Luc Mieussens MNMCFF2014 - Beijing May 21-27, 2014 Forestier-Coste local velocity grid 1/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Introduction rarefied gas flow Deterministic simulations (DVM) of Boltzmann equation (BGK) A global discrete velocity grid in space and time For practical applications in aerodynamics (atmospheric re-entry problems), the grid is so large that the computational ressources (memory storage and CPU time) require by the simulation are huge Project : construction of local velocity grids (unsteady flows) Forestier-Coste local velocity grid 2/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions Forestier-Coste local velocity grid 3/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions Forestier-Coste local velocity grid 4/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . � mass density ρ ( t , x ) = R D f ( t , x , v ) d v Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . � mass density ρ ( t , x ) = R D f ( t , x , v ) d v 1 � velocity u ( t , x ) = R D v f ( t , x , v ) d v ρ ( t , x ) Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . � mass density ρ ( t , x ) = R D f ( t , x , v ) d v 1 � velocity u ( t , x ) = R D v f ( t , x , v ) d v ρ ( t , x ) � v � 2 � energy E ( t , x ) = f ( t , x , v ) d v 2 R D Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . � mass density ρ ( t , x ) = R D f ( t , x , v ) d v 1 � velocity u ( t , x ) = R D v f ( t , x , v ) d v ρ ( t , x ) � v � 2 � energy E ( t , x ) = f ( t , x , v ) d v 2 R D 1 � 2 E ( t , x ) � ρ ( t , x ) − � u ( t , x ) � 2 temperature T ( t , x ) = DR Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions f ( t , x , v ) distribution function : mass of particles at time t , at position x ∈ Ω ⊂ R D with a velocity of v ∈ R D . � mass density ρ ( t , x ) = R D f ( t , x , v ) d v 1 � velocity u ( t , x ) = R D v f ( t , x , v ) d v ρ ( t , x ) � v � 2 � energy E ( t , x ) = f ( t , x , v ) d v 2 R D 1 � R D � v − u ( t , x ) � 2 f ( t , x , v ) d v temperature T ( t , x ) = DR ρ ( t , x ) Forestier-Coste local velocity grid 5/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Notations 2 � v � 2 � T 1 , v , 1 � collisional invariants : m ( v ) = moments : ρ = ( ρ, ρ u , E ) T � < g > = R D g d v ρ = < m f > Forestier-Coste local velocity grid 6/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions BGK model ∂ t f + v . ∇ x f = Q ( f ) Q ( f ) = 1 τ ( M [ ρ, u , T ] − f ) + boundaries conditions Forestier-Coste local velocity grid 7/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Maxwellian Maxwellian M [ ρ, u , T ] ( t , x , v ) : solution of an entropy minimisation problem ( H ( f ) = < f ln( f ) > ). H ( M [ ρ, u , T ] ) = min { H ( g ) , g ≥ 0 s.t. < m g > = ρ } − | v − u ( t , x ) | 2 ρ ( t , x ) � � M [ ρ, u , T ] ( t , x , v ) = exp D 2 RT ( t , x ) (2 π RT ( t , x )) 2 Forestier-Coste local velocity grid 8/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions 1 BGK Model 2 Standard method 3 Presentation of the method (1D) 4 2D method 5 Conclusions Forestier-Coste local velocity grid 9/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Discretization Physical model: ∂ t f + v · ∇ x f = Q ( f ) + BCs Velocity space R D � V = v k , k ∈ N D � � set of discrete velocities Discrete kinetic equation: ∂ t f k + v k ·∇ x f k = Q k ( f ) + BCs Forestier-Coste local velocity grid 10/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Velocity grid? Cartesian grid V : 8 9 0 l ∆ v x 1 0 l 1 0 0 : l max 1 < = A = V = : v l , m , n = v min + m ∆ v y m 0 : m max A , @ @ @ A n ∆ v z n 0 : n max ; Constraints: bounds and grid step : the grid must be large enough fine enough Forestier-Coste local velocity grid 11/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions How to design a cartesian velocity grid? 2 � RT ( x ) Information on the support of f ( x , . ): assuming f close to its local Maxwellian, then centered on u ( x ) � with standard deviation RT ( x ) u ( x ) − c � RT ( x ) u ( 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 ) At least 3 points into the ”core” of each distribution. The grid step should be: � ∆ v ≤ min x ∈ Ω { 2 RT ( x ) } Forestier-Coste local velocity grid 12/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Estimation of the bounds and the step of the grid Temperature and velocity fields are a priori unknown! several solutions : use the boundaries values: upstream flow ( u ∞ , T ∞ ) and velocity and temperature of the body ( u wall , T wall ) Flow parameter inside the shock: u shock and T shock estimated by Rankine-Hugoniot relations. Even better: A compressible Navier-Stokes pre-simulation : u NS and T NS in Ω let x ∈ Ω { u NS ( x ) − c p x ∈ Ω { u NS ( x ) + c p v min = min RT NS ( x ) } v max = max RT NS ( x ) } p ∆ v = a min x ∈ Ω { 2 RT NS ( x ) } Forestier-Coste local velocity grid 13/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Drawback of a global grid For reentry problem : 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!... Forestier-Coste local velocity grid 14/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Need of dynamics grids : example two interacting blastwaves: ` a t = 0 ρ = 1 , u = 0 , T = [1000 , 0 . 01 , 100] √ max ( u + 4 RT ) = 126 √ at t = 0 . 02 (after the interaction): max ( u + 4 RT ) = 236: larger bounds! the optimal velocity grid get 2551 points! Forestier-Coste local velocity grid 15/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea 1 BGK Model 2 Standard method 3 Presentation of the method (1D) Main Idea Illustration Extension of the LVG 4 2D method 5 Conclusions Forestier-Coste local velocity grid 16/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea Local velocity grids idea: define a velocity grid for each t and x Forestier-Coste local velocity grid 17/ 49

BGK Model Standard method Presentation of the method (1D) 2D method Conclusions Main Idea Local velocity grid: problems 1 how to define the Local Velocity Grid: bounds and step? 2 how to exchange informations between two grids? Example: ∂ x f ( t , x , v ) ≈ f ( t , x + ∆ x , v ) − f ( t , x , v ) ∆ x but f ( t , x , . ) and f ( t , x + ∆ x , . ) are not defined on the same velocity grid... Forestier-Coste local velocity grid 18/ 49