of 56

# Natalini nse slide_giu2013

Published on: Mar 3, 2016
Published in: Technology
Source: www.slideshare.net

#### Transcripts - Natalini nse slide_giu2013

• 1. Finite speed approximations to Navier-Stokes equations Roberto Natalini Istituto per le Applicazioni del Calcolo - CNR INDAM Workshop on Mathematical Paradigms of Climate Science, Rome, June 2013
• 2. click here to open the movie in your browser NASA/JPL’s computational model ”Estimating the Circulation and Climate of the Ocean” a.k.a. ECCO2, a high resolution model of the global ocean and sea-ice
• 3. The incompressible Navier-Stokes equations Find (U, Φ) : IRD × (0, T) → IRD × IR s.t. ∂tU + div(U ⊗ U) + Φ = ν∆U div U = 0 Motivations for a ﬁnite speed approximation ♣ New and more adapted class of estimates ♣ Robust and simple numerical approximations ♣ Natural treatment: upwinding, pressure term and the divergence-free constraint ♣ Possible coupling with other equations
• 4. Plan of the Talk Some methods to solve NS eqs. Relaxation approximations A damped wave equation approximation Boltzmann eq. vs. NS eqs The Vector BGK approximation Comparison with Lattice BGK schemes Some numerical results
• 5. Some Numerical Methods • Finite Element Methods (FEM): variational formulation • + high computational exibility • + rigorous mathematical error analysis → mesh adaptation • - Diﬃcult control of upwinding phenomena and mass conservation • - A lot of theoretical work for implementation • Finite volume methods (FVM): conservation equations • + based on physical conservation properties • - problems on unstructured meshes • - diﬃcult stability and convergence analysis • - heuristic mesh adaptation • Spectral Methods • + high order approximation • - special domains • Finite diﬀerence methods (FDM): direct form • + easy implementation, • - problems along curved boundaries • - diﬃcult stability and convergence analysis • - mesh adaptation diﬃcult
• 6. More on Finite Diﬀerence Schemes Projection methods: Chorin, Temam, Kim & Moin, E & Liu, Bell & Collella & Glaz, .... ⇒ Instability problems for the Pressure MAC methods: Harlow & Welsh, T. Hou & Wetton.... ⇒ Staggered grids: diﬀerent locations for pressure and velocity High order: Strikwerda, Kreiss... ⇒ Implicit methods
• 7. Original projection method (Chorin, Temam) ∂tu + div(u ⊗ u) + φ = ν∆u div u = 0 Splitting method based on Hodge decomposition. First step u∗ = un − ∆t (un · un − ν∆un ) (1) Second Step un+1 = u∗ − ∆t φn+1 (2) where φn+1 is computed from u∗ to force the incompressibility of un+1 div φn+1 = ∆φn+1 = 1 ∆t div u∗ (3)
• 8. The Hyperbolic Relaxation Approach A one-slide presentation (not this one!)
• 9. A simple relaxation model: hyperbolic and diﬀusive scalings • Approximation of ∂tu + ∂x A(u) = 0 Hyperbolic scaling (x , t ), for → 0, and λ > |A (u)| ⇒ u → u ∂tu + ∂x v = 0 ∂tv + λ2∂x u = 1 (A(u ) − v )
• 10. A simple relaxation model: hyperbolic and diﬀusive scalings • Approximation of ∂tu + ∂x A(u) = 0 Hyperbolic scaling (x , t ), for → 0, and λ > |A (u)| ⇒ u → u ∂tu + ∂x v = 0 ∂tv + λ2∂x u = 1 (A(u ) − v ) • Approximation of ∂tu + ∂x A(u) = λ2 ∂xx u Diﬀusive scaling (x , t 2 ), for → 0, u → u ∂tu + ∂x v = 0 ∂tv + λ2 2 ∂x u = 1 2 (A(u ) − v )
• 11. A relaxation approximation of Navier Stokes equations Y. Brenier, R.Natalini, & M. Puel 2004 Let u ∈ IR2 and V ∈ IR4    ∂tu + div V + φ = 0 ∂tV + 1 ν u = 1 (u ⊗ u − V ) · u = 0 → 0 ⇒ ∂tu + div(u ⊗ u) + φ = ν u , · u = 0.
• 12. A relaxation approximation of Navier Stokes equations Y. Brenier, R.Natalini, & M. Puel 2004 The same model as a damped Wave equation.    ∂tu + div(u ⊗ u ) + φ = − ∂ttu + ν∆u · u = 0 → 0 ⇒ ∂tu + div(u ⊗ u) + φ = ν u , · u = 0.
• 13. A convergence result For all ﬁxed T ≥ 0, let U0 be a smooth divergence free vector ﬁeld on T2 . Let (u0, V0 ) be a sequence of smooth initial data for the relaxation approximation. Assume that there exists C s.t. ||u0||H1 + ||∂tu (0, ·)||L2 ≤ C, |u0|H2 < C0 Ks √ |u0(x) − U0 (x)|2 dx ≤ C √ Then, if U is the (smooth) solution of the incompressible Navier Stokes equations with U0 as initial data, we have sup t∈[0,T] |u − U|2 dx ≤ CT √
• 14. A convergence result For all ﬁxed T ≥ 0, let U0 be a smooth divergence free vector ﬁeld on T2 . Let (u0, V0 ) be a sequence of smooth initial data for the relaxation approximation. Assume that there exists C s.t. ||u0||H1 + ||∂tu (0, ·)||L2 ≤ C, |u0|H2 < C0 Ks √ |u0(x) − U0 (x)|2 dx ≤ C √ Then, if U is the (smooth) solution of the incompressible Navier Stokes equations with U0 as initial data, we have sup t∈[0,T] |u − U|2 dx ≤ CT √ Extensions in IR2 and IR3 for less regular initial data in: • R. Natalini, F. Rousset, Proc. AMS, 2006 • M. Paicu and G. Raugel, ESAIM, Proc.,21:6587, 2007 • I. Hachicha, arXiv:1205.5166v1 May 2013
• 15. A Kinetic Approach Goal: a better approximation of the divergence-free constraint
• 16. Hydrodynamic limits The Boltzmann equation in the hyperbolic scaling (x , t ) ∂tf + ξ · x f = 1 Q(f ) → 0 ⇓ If f → f , then f (x, t, ξ) = ρ(x, t) (2πθ(x, t))3/2 exp − |ξ − u(x, t)|2 2θ(x, t) where ρ, u, and θ solve the compressible Euler equations.
• 17. Diﬀusive limits The Boltzmann equation in the parabolic scaling (x , t 2 ) ∂tf + 1 ξ · x f = 1 2 Q(f ) Given the equilibrium conﬁguration M(x, t, ξ) := 1 (2π)3/2 exp −|ξ|2 2 . Then
• 18. Diﬀusive limits The Boltzmann equation in the parabolic scaling (x , t 2 ) ∂tf + 1 ξ · x f = 1 2 Q(f ) Given the equilibrium conﬁguration M(x, t, ξ) := 1 (2π)3/2 exp −|ξ|2 2 . Then f (x, t, ξ) = M(1 + g) + O( 2 ) where g = ρ + ξ · u + (1 2|ξ|2 − 3 2)θ, and div u = 0, (ρ + θ) = 0 ∂tu + div(u ⊗ u) + φ = ν∆u
• 19. Diﬀusive limits The Boltzmann equation in the parabolic scaling (x , t 2 ) ∂tf + 1 ξ · x f = 1 2 Q(f ) Given the equilibrium conﬁguration M(x, t, ξ) := 1 (2π)3/2 exp −|ξ|2 2 . Then f (x, t, ξ) = M(1 + g) + O( 2 ) where g = ρ + ξ · u + (1 2|ξ|2 − 3 2)θ, and div u = 0, (ρ + θ) = 0 ∂tu + div(u ⊗ u) + φ = ν∆u Smooth local solutions: De Masi, Esposito, Lebowitz Renormalized solutions: Golse, Saint-Raymond
• 20. The Vector BGK Approach Relaxation + Kinetic
• 21. The vector BGK approximation First formulation was made in collaboration with F. Bouchut (uncredited) M.F. Carfora & R. Natalini 2008 Y. Jobic, R. Natalini & V. Pavan in preparation Find fi ∈ IRD+1 s.t. ∂tfi + 1 λi · x fi = 1 τ 2 (Mi (ρ , ρu ) − fi ) fi (x, 0) = Mi (ρ, ρu0), i = 1, . . . , N ρ := N i=1 fi,0, ρul := N i=1 f i,l System of semilinear hyperbolic equations Main idea: ρ → ρ, u → U, where U is a solution of the Navier–Stokes eqs.
• 22. Compatibility conditions for the Maxwellian functions N i=1 M0 i (ρ, q) = ρ (4) N i=1 Ml i (ρ, q) = N i=1 λil M0 i (ρ, q) = ql (5) N i=1 λij Ml i (ρ, q) = qj ql ρ + P(ρ)δjl (P(ρ) = Cργ ) (6) τ N i=1 λij λik r ∂qr Ml i (ρ, 0)ur = νδjkul (7)
• 23. Expansion in the D + 1 Conservation Laws Set: ρ := N i=1 fi,0, ρul := N i=1 f i,l ∂tρ + j ∂xj N i=1 λij f 0 i = 0 ∂t( ρul ) + j ∂xj N i=1 λij f l i = 0 l = 1, . . . , D
• 24. Velocity equation To have a the right limit we need P(ρ) − P(ρ) 2 → →0 ρΦ ⇒ ρ = ρ + O( 2 ) and using two compatibility conditions and the Taylor expansion of M M(ρ, ρu) = M(ρ, 0)+∂ρM(ρ, 0)(ρ−ρ)+ qM(ρ, 0)· ρu+O( 2 ), ⇒ ∂tu + div(u ⊗ u) + Φ = ν∆u + O( )
• 25. Incompressibility equation If, in the ﬁrst conservation law, we assume N i=1 λil M0 i (ρ, q) = ql 0 = ∂tρ + j ∂xj N i=1 λij M0 i − τ j,k ∂2 xj xk N i=1 λij λikM0 i + O( ) = j ∂xj (ρuj ) + O( ) ⇒ div u = O( )
• 26. Hyperbolic compatibility conditions (1)–(3) As τ → 0 ( ﬁxed) ; Isentropic Euler Eqs. (A. Sepe in 2011) ∂tρ + div(ρu) = 0 ∂t(ρu) + div(ρu ⊗ u) + 1 2 P(ρ) = 0 Rmk. → 0 in the isentropic Gas-Dynamics yields (formally) the (incompressible) Euler Eqs. ∂tU + div(U ⊗ U) + Φ = 0 div U = 0
• 27. The basic Energy (in)equality H–Theorem ∂tH(f) + Λ · x H(f) ≤ H(M(Uf )) − H(f) ≤ 0 Bouchut’s Theorem (1999): There exist kinetic entropies if each Mi has positive real eigenvalues
• 28. The basic Energy (in)equality H–Theorem ∂tH(f) + Λ · x H(f) ≤ H(M(Uf )) − H(f) ≤ 0 Bouchut’s Theorem (1999): There exist kinetic entropies if each Mi has positive real eigenvalues ⇓ 1 2ρ|u|2 + C (γ−1) 2 ργ − ργ − γργ−1(ρ − ρ) dx + C 4τ |f − M|2 dxdt ≤ 1 2 ρ|u0|2 dx
• 29. A 5 velocities scheme in 2D Orthogonal Velocities Model (D. Aregba-Driollet & R. Natalini 2003). Setting W = (ρ, q) and A1(W ) = q1, q2 1 ρ + P(ρ), q1q2 ρ , A2(W ) = q2, q1q2 ρ , q2 2 ρ + P(ρ) Maxwellian functions in the form Mi (W ) = ai W + 2 j=1 bij Aj (W ) The velocities are λi = λci ,, for some λ > 0, with c1 = (1, 0), c2 = (0, 1), c3 = (−1, 0), c4 = (0, −1), c5 = (0, 0) a1 = · · · = a4 = a, a5 = 1 − 4a; b11 = b22 = −b31 = −b42 = 1 2λ , bij = 0 otherwise.
• 30. Consistency, Stability, and Global Existence ♣ The continuous model is consistent if τ = ν 2λ2a ♣ The Maxwellian functions are positive and the model has a kinetic entropy if the following conditions are veriﬁed: 1 4 > a > 1 2λ P (ρ) + um So it is enough to take λ > 2 P (ρ) + um ♣ Under these conditions, for ﬁxed and τ and small initial data, the smooth solution is global in time (Kawashima conditions → Hanouzet-Natalini ARMA 2003)
• 31. The fully discrete scheme Solve (using the upwind scheme) a discrete version of    ∂tfi + 1 λi · x fi = 0 tn ≤ t < tn+1 fi (x, tn) = f n i (x), (8) ρn+1 ρn+1un+1 = N i=1 fi (tn+1−) and fn+1 = M ρn+1 , ρn+1 un+1 Consistent with the Navier-Stokes equations (order 2 in space) if = aλ∆x ν , ∆t ≤ a(∆x)2 ν
• 32. The fully discrete scheme Solve (using the upwind scheme) a discrete version of    ∂tfi + 1 λi · x fi = 0 tn ≤ t < tn+1 fi (x, tn) = f n i (x), (8) ρn+1 ρn+1un+1 = N i=1 fi (tn+1−) and fn+1 = M ρn+1 , ρn+1 un+1 Consistent with the Navier-Stokes equations (order 2 in space) if = aλ∆x ν , ∆t ≤ a(∆x)2 ν Main idea: the artiﬁcial viscosity is used to reconstruct the Navier-Stokes viscosity
• 33. Comparison with the Lattice BGK models: (McNamara & Zanetti, Higuera & Jimenez, Succi, Benzi, H. Chen, S. Chen, Doolen,.....) Give a set of velocities ci , and a grid such that ∆x = ∆tci fi (x + ∆tci , t + ∆t) = fi (x, t) + 1 τ (Mi − fi ) The D2Q9 lattice ρ := N i=1 fi , q := N i=1 ci fi Mi (ρ, q) = Wi ρ ρ + 3ci · q − 3 2|q|2 + 9 2(ci · q)2
• 34. Consistency To reach consistency with the Navier-Stokes equations, ﬁx ∆x and ω = ∆t τ ∈ (0, 2) ⇓ |c| = 3ν ∆x 2ω 2 − ω ∆t = ωτ = ∆x2 3ν 2 − ω 2ω
• 35. Consistency To reach consistency with the Navier-Stokes equations, ﬁx ∆x and ω = ∆t τ ∈ (0, 2) ⇓ |c| = 3ν ∆x 2ω 2 − ω ∆t = ωτ = ∆x2 3ν 2 − ω 2ω BGK Lattice Boltzmann models vs. Kinetic schemes • Lattice grids (µ = 1); • Scalar distribution function (for ﬁxed i); • No nonlinear stability criteria; • boundary conditions ; Junk & Klar (2000): ﬁnite diﬀerence version
• 36. Numerical Validation in collaboration with V. Pavan and Y. Jobic (IUSTI, Aix-Marseille Universit´e)
• 37. Lid-driven cavity : Computational domain u = U, v = 0 wall wall wall primary vortex top left vortex (T) bottom left vortex (BL1) bottom right vortex (BR1) BL2 BR2 Figure: Setting of the problem
• 38. Lid-driven cavity : results 1 Figure: streamlines at Re 400, Nx = Ny = 400 Figure: streamlines at Re 7500, Nx = Ny = 7500
• 39. Lid-driven cavity : results 2 y/N −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 u/U −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 a Re = 100 present work b Re = 400 present work c Re = 1000 present work d Re = 3200 present work e Re = 5000 present work f Re = 7500 present work Re 100 Ghia&al Re 400 Ghia&al Re 1000 Ghia&al Re 3200 Ghia&al Re 5000 Ghia&al Re 7500 Ghia&al a b c d e f Figure: u values at the centerline v/U −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0 −0 −0 0 0.2 0.4 0.6 x/N −0.2 0 0.2 0.4 0.6 0.8 1 1.2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 a Re = 100 present work b Re = 400 present work c Re = 1000 present work d Re = 3200 present work e Re = 5000 present work f Re = 7500 present work Re 100 Ghia&al Re 400 Ghia&al Re 1000 Ghia&al Re 3200 Ghia&al Re 5000 Ghia&al Re 7500 Ghia&al a b c d e f Figure: v values at the centerline
• 40. Transient couette ﬂow : Computational domain Wall UmPeriodic Periodic H Figure: boundary conditions analytical solution ux (y, t) = Um 1 − y H − 2 π ∞ k=1 1 k exp − k2π2 H2 νt sin kπ H y (9)
• 41. Transient couette ﬂow : results 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 FluidvelocityUx y / H t = 0.125 t = 0.5 t = 1 t = 2 t = 3 Figure: Diﬀerent time solutions at Re 10 0 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 FluidvelocityUx y / H t = 0.1 t = 1 t = 2 t = 3 t = 4 t = 6 Figure: Diﬀerent time solutions at Re 800
• 42. Transient couette ﬂow : results 2 L1relativeerror 1e−07 1e−06 1e−05 0.0001 1e−07 1e−06 1e−05 0.0001 Number of points 100 200 300 400 500 600 700 100 200 300 400 500 600 700 slope : 2 Figure: Order 2 in space
• 43. Backward-Facing Step : Computational domain H Um h hi L x1 x2 x3 Figure: Conditions limites
• 44. Backward-Facing Step : results 1 0 5 10 150 1 2 Figure: streamlines at Re 100, Nx = 1000, Ny = 250, ε = 0.016
• 45. Backward-Facing Step : results 1 0 5 10 150 1 2 Figure: streamlines at Re 100, Nx = 1000, Ny = 250, ε = 0.016 0 5 10 15 200 1 2 Figure: streamlines at Re 800, Nx = 20000, Ny = 1000, ε = 0.25
• 46. Backward-Facing Step : results 2 2 4 6 8 10 12 14 100 200 300 400 500 600 700 800 x1/h Re present work Armaly Erturk Biswas Figure: attachment point for the ﬁrst vortex
• 47. Backward-Facing Step : results 2 2 4 6 8 10 12 14 100 200 300 400 500 600 700 800 x1/h Re present work Armaly Erturk Biswas Figure: attachment point for the ﬁrst vortex 6 8 10 12 14 16 18 20 22 24 450 500 550 600 650 700 750 800 850 x/h Re X2 X3 present work Armaly Erturk Biswas present work Armaly Erturk Biswas Figure: detachment/attachment point for the second vortex
• 48. Figure: Landsat 7 image of clouds oﬀ the Chilean coast near the Juan Fernandez Islands (also known as the Robinson Crusoe Islands)
• 49. Figure: Von Karman vortices oﬀ the coast of Rishiri Island in Japan
• 50. Von Karman streets click here to open the movie in your browser The instabilities of the steady ﬂow make the Von karman streets appear
• 51. Conclusions (?) and possible directions
• 52. Conclusions (?) and possible directions • Hyperbolic approximations furnish a nice framework to study NSE
• 53. Conclusions (?) and possible directions • Hyperbolic approximations furnish a nice framework to study NSE • Some analytical problems are still open.
• 54. Conclusions (?) and possible directions • Hyperbolic approximations furnish a nice framework to study NSE • Some analytical problems are still open. • It is possible to derive simple and eﬀective schemes for NSE.
• 55. Conclusions (?) and possible directions • Hyperbolic approximations furnish a nice framework to study NSE • Some analytical problems are still open. • It is possible to derive simple and eﬀective schemes for NSE. • 3D dimensions, complex geometries (coming soon...)
• 56. Conclusions (?) and possible directions • Hyperbolic approximations furnish a nice framework to study NSE • Some analytical problems are still open. • It is possible to derive simple and eﬀective schemes for NSE. • 3D dimensions, complex geometries (coming soon...) • coupling with other equations, multidomains...