Numerical Simulation of the Crookes Radiometer
Luc Mieussens, Guillaume Dechristé
Institut de Mathématiques de Bordeaux Université de Bordeaux
Séminaire de mécanique des fluides numériques, Institut Henri Poincaré January 25–26, 2016 Crookes radiometer
invented by W. Crookes (1874) •
rarefied gas effect (Kn 0.1): disappears in vacuum or with • dense gases ≈ Reynolds, Maxwell: light• heated vanes → temperature difference in → white side the gas temperature Tw black side force (radiometric forces) temperature Tb>Tw → Radiometric forces
renewed interest: possible application for microflows (MEMS) • this talk: a numerical tool for 3D simulations of the Crookes • radiometer (based on the Boltzmann equation) other applications: any rarefied flow with moving obstacles • (e.g. vacuum pumps) Outline
1 Origin of the radiometric forces
2 Kinetic model
3 Numerical method
4 2D Simulations
5 Extensions Origin of the radiometric forces
Crookes: radiation pressure ("radiometer”) • Reynolds 1874: pressure difference in the gas • gas
pressure difference
white side temperature Tw pb>pw black side temperature Tb>Tw
“area” effect Origin of the radiometric forces
Reynolds and Maxwell 1879: edge effect •
white side temperature Tw
black side Tb temperature Tb>Tw
Tw temperature gradient Origin of the radiometric forces
Reynolds and Maxwell 1879: edge effect •
white side temperature Tw
black side Tb temperature Tb>Tw
Tw temperature gradient Origin of the radiometric forces
Reynolds and Maxwell 1879: edge effect •
white side temperature Tw
black side Tb temperature Tb>Tw
Tw temperature gradient Origin of the radiometric forces
Reynolds and Maxwell 1879: edge effect •
white side temperature Tw
radiometric force
black side Tb temperature Tb>Tw
Tw temperature gradient Origin of the radiometric forces
Reynolds and Maxwell 1879: edge effect •
white side temperature Tw
radiometric force
black side Tb temperature Tb>Tw
Tw temperature gradient Origin of the radiometric forces
recent works: • • Gimelshein, Muntz, Selden, Ketsdever, Alexeenko, etc. (2009 to 2012): numerical and experimental studies (single vane, steady 2D flow) • Aoki-Taguchi 2012: numerical studies (investigation of various thermally induced flows, single vane, steady 2D flow) • K. Xu 2012 : numerical simulation of a 2D Crookes radiometer only 2D simulations, few experiments • still open questions: • • 3D flow structure not known (lack of experiments) • what is the optimal shape of the vanes? • what is the distribution of forces on the vanes? 1 Origin of the radiometric forces
2 Kinetic model
3 Numerical method
4 2D Simulations
5 Extensions Boltzmann equation
particles: position ~x and velocity ~v • mass density in phase-space f (t,~x,~v) (“Distribution function”): • f (t,~x,~v)dxdv is the mass of particles at x dx with velocity v dv . ± ± Macroscopic quantities: • ρ Z 1 ρ~u = ~v f (t,~x,~v)d~v E R3 1 ~v 2 2 k k 1 3 E = ~u 2 + ρRT 2k k 2 P = ρRT Boltzmann equation
equilibrium distribution ("Maxwellian"): • ρ ~u ~v 2 [ρ,~u, T ] = exp k − k M (2πRT )3/2 2RT
important macroscopic quantity here: stress tensor Σ • Z Σ(t,~x) = (~v ~u) (~v ~u)f (t,~x,~v) dv R3 − ⊗ − force exerted by the gas on a solid surface dS (of normal ~n): •
Fgas→dS = Σ~ndS. − Boltzmann equation
Boltzmann equation: • ∂f + v x f = Q(f ) ∂t · ∇ Q(f ) is the collision operator
BGK Model: • ∂f 1 + v x f = ( [ρ,~u, T ] f ) ∂t · ∇ τ M −
consistency with fluid mechanics • Boltzmann equation
gas-surface interaction: diffuse reflection • for every reflected velocity, i.e v such that (v uw ) n < 0: − · uw 2 n Gas ρw v uw f (t, x, v) := exp Solid 3/2 | − | (2πRTw ) − 2RTw
with ρw such that Z (v uw ) n f (t, x, v) dv = 0 R3 − · (no mass flux across the wall) 1 Origin of the radiometric forces
2 Kinetic model
3 Numerical method
4 2D Simulations
5 Extensions immersed boundary methods e.g. [Filbet Yang 2012], [Perkardan Chigullapalli • Alexeenko 2012], [Bernard Iollo Puppo 2014], [Chen Xu 2015]
. automatic meshing . complex geometries easily handled . low accuracy at solid boundaries : no mass conservation
Numerical Method
CFD with moving obstacles: body fitted methods with moving grids (ALE, moving mesh) • e.g. [Chen, Xu, Lee, Cai, 2012].
. high accuracy at solid boundaries . mesh generation is difficult for complex geometries Numerical Method
CFD with moving obstacles: body fitted methods with moving grids (ALE, moving mesh) • e.g. [Chen, Xu, Lee, Cai, 2012].
. high accuracy at solid boundaries . mesh generation is difficult for complex geometries
immersed boundary methods e.g. [Filbet Yang 2012], [Perkardan Chigullapalli • Alexeenko 2012], [Bernard Iollo Puppo 2014], [Chen Xu 2015]
. automatic meshing . complex geometries easily handled . low accuracy at solid boundaries : no mass conservation Numerical Method
Cut-cell method: •
• Cartesian grid + cut-cells around the solid body • complex geometries easily handled • potentially good accuracy at solid boundaries (body fitted mesh) Numerical Method Numerical Method
the Boltzmann equation must be solved on changing cells Basic tools (I): integral formulation
Boltzmann equation: • ∂ f + v x f = Q(f ) ∂t · ∇ integrate on a time dependent domain Ω(t) • Z ∂ Z f + v x f dx = Q(f ) dx. Ω(t) ∂t · ∇ Ω(t)
Stokes formula and Reynolds theorem: • Basic tools (I): integral formulation
Boltzmann equation: • ∂ f + v x f = Q(f ) ∂t · ∇ integrate on a time dependent domain Ω(t) • Z ∂ Z f + v x f dx = Q(f ) dx. Ω(t) ∂t · ∇ Ω(t)
Stokes formula and Reynolds theorem: • Z ∂ Z Z f dx + v n f dS = Q(f ) dx. Ω(t) ∂t S(t) · Ω(t) Basic tools (I): integral formulation
Boltzmann equation: • ∂ f + v x f = Q(f ) ∂t · ∇ integrate on a time dependent domain Ω(t) • Z ∂ Z f + v x f dx = Q(f ) dx. Ω(t) ∂t · ∇ Ω(t)
Stokes formula and Reynolds theorem: • ∂ Z Z Z Z f dx w n f dS + v n f dS = Q(f ) dx. ∂t Ω(t) − S(t) · S(t) · Ω(t) Basic tools (I): integral formulation
Boltzmann equation: • ∂ f + v x f = Q(f ) ∂t · ∇ integrate on a time dependent domain Ω(t) • Z ∂ Z f + v x f dx = Q(f ) dx. Ω(t) ∂t · ∇ Ω(t)
Stokes formula and Reynolds theorem: • ∂ Z Z Z f dx + (v w) n f dS = Q(f ) dx. ∂t Ω(t) S(t) − · Ω(t) Basic tools (II): finite volume method
cell average of f on a cell Ωn: • i Z n 1 n fi = n f (t , x, v) dx Ω Ωn | i | i integral equation: • ∂ Z Z Z f dx + (v w) n f dS = Q(f ) dx. ∂t Ω(t) S(t) − · Ω(t)
where w is the velocity of the faces of the cell
first order explicit upwind scheme: • X Ωn+1 f n+1 Ωn f n + ∆t n = ∆t Ωn Q(f n). | i | i − | i | i Fij | i | i j Basic tools (II): finite volume method
first order explicit upwind scheme: • X Ωn+1 f n+1 Ωn f n + ∆t n = ∆t Ωn Q(f n). | i | i − | i | i Fij | i | i j numerical flux between cell Ωn and Ωn: • i j Z n n ij (v w) n f (t , x, v)) dS F ≈ n n − · interfaceΩi /Ωj
. for a gas interface: w = 0
n n + n − n = S (v nij ) f + (v nij ) f Fij | ij | · i · j
v n > 0 · ij n n fi n Ωi n Ωj fj v n < 0 · ij Basic tools (II): finite volume method
first order explicit upwind scheme: • X Ωn+1 f n+1 Ωn f n + ∆t n = ∆t Ωn Q(f n). | i | i − | i | i Fij | i | i j numerical flux between cell Ωn and Ωn: • i j Z n n ij (v w) n f (t , x, v)) dS F ≈ n n − · interfaceΩi /Ωj
. for a solid interface: w = uw (velocity of the solid wall)
n n n + n n − = S ((v u ) nij ) f + ((v u ) nij ) M[ρw , uw , Tw ] Fij | ij | − w,ij · i − w,ij ·
(v uw) nij > 0 n− · fi n n Ωj Ωi M[ρw,uw, Tw] (v u ) n < 0 − w · ij Solid-gas coupling
at tn: f n is known for every cell Ωn, velocity un of solid • i i w,i nodes as well new position of grid nodes at tn+1: • n+1 n n xi = xi + ∆t uw,i this gives new cut cells • average values f n+1 on new cells Ωn+1 are computed with the • i i finite volume scheme force and torque exerted by the gas on the solid wall are • computed with stress tensor + boundary conditions new node velocity un+1 are computed with Newton laws • w,i 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
time integration in the control • volume distribute the gas of the control • volume to its cells :
Technical details (I): small cell problem
very small cut cells: very small ∆t • 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
time integration in the control • volume distribute the gas of the control • volume to its cells :
Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
time integration in the control • volume distribute the gas of the control • volume to its cells :
Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
distribute the gas of the control • volume to its cells :
Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → time integration in the control • volume 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → time integration in the control • volume distribute the gas of the control • volume to its cells : 1 Z f dS = , n+1 n+1 S1 S1 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S
Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → time integration in the control • volume distribute the gas of the control • volume to its cells : 1 Z n+1 f dS S Sn+1 Technical details (I): small cell problem
very small cut cells: very small ∆t • cells merged in a control volume → time integration in the control • volume distribute the gas of the control • volume to its cells : 1 Z 1 Z n+1 f dS = n+1 f dS, n+1 S n+1 S1 S1 S 1 Z 1 Z n+1 f dS = n+1 f dS. n+1 S n+1 S2 S2 S Technical details (II): disappearing cut-cells
disappearing cut cell: • Iteration n Iteration n + 1
Boundary at tn Boundary at tn+1
Merging Scheme Update Technical details (II): appearing cut-cells
appearing cut-cell: • Iteration n Iteration n + 1
Boundary at tn Boundary at tn+1
Merging Scheme Update Properties
stable: standard CFL condition • exact mass conservation • 1 Origin of the radiometric forces
2 Kinetic model
3 Numerical method
4 2D Simulations
5 Extensions Numerical results
Simulations of Taguchi and Aoki [2012] : • T0
L0/10 T0 2T0
L0 4L0
4L0
T0 Comparison with Taguchi and Aoki (steady case): • -0.02 0.06 Kn = 0.1 Taguchi Kn = 0.3 Maille coupÂee Kn = 0.5 0.05 1/2
-0.03 1/2 ) ) 0 0 0.04 /(2RT /(2RT ∞ w
-0.04 u u 0.03
-0.05 0.02 0 40 80 120 160 0 0.2 Kn 0.4 0.6 t/t0 Numerical results
Simulations of Taguchi and Aoki [2012] : • T0
L0/10 T0 2T0
L0 4L0
4L0
T0 Unsteady cut-cell simulation: • Numerical results
Simulations of Taguchi and Aoki [2012] : • T0
L0/10 T0 2T0
L0 4L0
4L0
T0 Unsteady cut-cell simulation: • Numerical results
Simulations of Taguchi and Aoki [2012] : • T0
L0/10 T0 2T0
L0 4L0
4L0
T0 Comparison with a body fitted unstructured mesh code • (steady case):
1.10 1.64 1.06 1.58 1.01 1.52 0.97 1.46 1.39 0.92 1.33 0.88 1.27 0.83 1.21 0.79 1.15 0.74 1.09 0.70 1.02 Numerical results
2D radiometer: comparison with Chen et al. (2012, moving • mesh method)
40 T0
30 Zoom 10 L ) -1 20 (s .
l θ 5 0 0.001 R 10
Th Tc Results of Chen et al. Cut cell method 0 0 0.005 0.01 0.015 0.02 time (s) Numerical results
Other application: vacuum pump (Roots pump) • Numerical results
Other application: vacuum pump (Roots pump) • 1 Origin of the radiometric forces
2 Kinetic model
3 Numerical method
4 2D Simulations
5 Extensions High performance computing (I): adaptive mesh refinement
Implementation of a Quadtree • algorithm in space. Refinement criterion based on the • distance between the cell and the nearest solid boundary. Coupling AMR - and the cut cell • method is simple. High performance computing (I): adaptive mesh refinement
Implementation of a Quadtree • algorithm in space. Refinement criterion based on the • distance between the cell and the nearest solid boundary. Coupling AMR - and the cut cell • method is simple. High performance computing (I): adaptive mesh refinement
Implementation of a Quadtree • algorithm in space. Refinement criterion based on the • distance between the cell and the nearest solid boundary. Coupling AMR - and the cut cell • method is simple. High performance computing (I): adaptive mesh refinement
Implementation of a Quadtree 30 • algorithm in space.
) 20 Refinement criterion based on the -1 • (s . θ distance between the cell and the 10 CarteÂsien nearest solid boundary. AMR 0 Coupling AMR - and the cut cell 0 0.005 0.01 0.015 0.02 • temps (s) method is simple. High performance computing (II): parallel implementation
parallelization with MPI • load balancing is difficult with moving boundaries • hybrid domain decomposition in space and velocity : • • a small number of subdomains in space • in each subdomain, the velocity grid is distributed to several processors this allows a good load balancing • 3D implementation
How to treat all the different cut cells in a simple and single way?
a normal cell is a cube (6 faces) • a cut cell is a polyhedron with 4 to 7 faces • n n n L 1 L L i,j+ i,j+1 i,j+1 | 2 | 2 | 2 | | |
|L n i | ,j L i n |
| Ωi j + , L y y 2 1 i n 0 + , j | 1 2 , =
L j n = ∆ n Ωi j
| , |
= ∆ L 1 | i | | , j | j i j+ Ω = , j i j j , | , , n 2 L , i
j | | 2 1 1 2
| ,
i n L 2 1 + 0 − − | n i n i 2 1 − n i , L L j | | | L n | L i,j 1 | −2 |
Ln = ∆x Ln = 0 Ln i,j 1 i,j 1 i,j 1 | −2 | | −2 | | −2 | notion of “virtual cell”: • • a virtual cell is a polyhedron with 7 faces, possibly degenerated • a virtual cell is associated to each cells generic treatment of every kind of cut cell • passing from 2D to 3D is easy • Numerical Results
3D Radiometer: • . 213 discrete velocities. . 240 000 cells (AMR). . 10 000 time steps. CPU time: 12h on 128 processors. → 3D Simulation Conclusion
Summary: • . a general method to simulate rarefied flows with moving obstacles, . validation in 1 et 2 dimensions . general treatment of cut cells allows complex 3D simulations Perspectives: • . might be useful for exploring radiometric effects, 3D flow structure, force magnitude and location . validation of the 3D code (experiments ?) . 2D free version available soon ("CAKE")