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: • 0 1 0 1 ρ Z 1 @ ρ~u A = @ ~v A 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 · r Q(f ) is the collision operator BGK Model: • @f 1 + v x f = ( [ρ,~u; T ] f ) @t · r 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 j − j (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 · r integrate on a time dependent domain Ω(t) • Z @ Z f + v x f dx = Q(f ) dx: Ω(t) @t · r Ω(t) Stokes formula and Reynolds theorem: • Basic tools (I): integral formulation Boltzmann equation: • @ f + v x f = Q(f ) @t · r integrate on a time dependent domain Ω(t) • Z @ Z f + v x f dx = Q(f ) dx: Ω(t) @t · r 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 · r integrate on a time dependent domain Ω(t) • Z @ Z f + v x f dx = Q(f ) dx: Ω(t) @t · r 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 · r integrate on a time dependent domain Ω(t) • Z @ Z f + v x f dx = Q(f ) dx: Ω(t) @t · r 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 j i j 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): j i j i − j i j i Fij j i j 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): j i j i − j i j i Fij j i j 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 j ij j · 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): j i j i − j i j i Fij j i j 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 j ij j − 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