Finite speed approximations to
Navier-Stokes equations
Roberto Natalini
Istituto per le Applicazioni del Calcolo - CNR
IND...
click here to open the movie in your browser
NASA/JPL’s computational model ”Estimating the Circulation
and Climate of the...
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...
Plan of the Talk
Some methods to solve NS eqs.
Relaxation approximations
A damped wave equation approximation
Boltzmann eq...
Some Numerical Methods
• Finite Element Methods (FEM): variational formulation
• + high computational exibility
• + rigoro...
More on Finite Difference Schemes
Projection methods: Chorin,
Temam, Kim & Moin, E &
Liu, Bell & Collella & Glaz,
....
⇒
In...
Original projection method (Chorin, Temam)
∂tu + div(u ⊗ u) + φ = ν∆u
div u = 0
Splitting method based on Hodge decomposit...
The Hyperbolic Relaxation Approach
A one-slide presentation (not this one!)
A simple relaxation model: hyperbolic and diffusive scalings
• Approximation of
∂tu + ∂x A(u) = 0
Hyperbolic scaling (x
, t...
A simple relaxation model: hyperbolic and diffusive scalings
• Approximation of
∂tu + ∂x A(u) = 0
Hyperbolic scaling (x
, t...
A relaxation approximation of Navier Stokes equations
Y. Brenier, R.Natalini, & M. Puel 2004
Let u ∈ IR2
and V ∈ IR4


...
A relaxation approximation of Navier Stokes equations
Y. Brenier, R.Natalini, & M. Puel 2004
The same model as a damped Wa...
A convergence result For all fixed T ≥ 0, let U0
be a smooth divergence
free vector field on T2
. Let (u0, V0 ) be a sequenc...
A convergence result For all fixed T ≥ 0, let U0
be a smooth divergence
free vector field on T2
. Let (u0, V0 ) be a sequenc...
A Kinetic Approach
Goal: a better approximation of the divergence-free constraint
Hydrodynamic limits
The Boltzmann equation in the hyperbolic scaling (x
, t
)
∂tf + ξ · x f =
1
Q(f )
→ 0
⇓
If f → f , the...
Diffusive limits
The Boltzmann equation in the parabolic scaling (x
, t
2 )
∂tf +
1
ξ · x f =
1
2
Q(f )
Given the equilibri...
Diffusive limits
The Boltzmann equation in the parabolic scaling (x
, t
2 )
∂tf +
1
ξ · x f =
1
2
Q(f )
Given the equilibri...
Diffusive limits
The Boltzmann equation in the parabolic scaling (x
, t
2 )
∂tf +
1
ξ · x f =
1
2
Q(f )
Given the equilibri...
The Vector BGK Approach
Relaxation + Kinetic
The vector BGK approximation
First formulation was made in collaboration with F. Bouchut
(uncredited)
M.F. Carfora & R. Na...
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) ...
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 ...
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 con...
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
λi...
Hyperbolic compatibility conditions (1)–(3)
As τ → 0 ( fixed) ; Isentropic Euler Eqs. (A. Sepe in 2011)
∂tρ + div(ρu) = 0
∂...
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 k...
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 k...
A 5 velocities scheme in 2D
Orthogonal Velocities Model (D. Aregba-Driollet & R. Natalini
2003). Setting W = (ρ, q) and
A1...
Consistency, Stability, and Global Existence
♣ The continuous model is consistent if τ = ν
2λ2a
♣ The Maxwellian functions...
The fully discrete scheme
Solve (using the upwind scheme) a discrete version of



∂tfi + 1
λi · x fi = 0 tn ≤ t < tn+1...
The fully discrete scheme
Solve (using the upwind scheme) a discrete version of



∂tfi + 1
λi · x fi = 0 tn ≤ t < tn+1...
Comparison with the Lattice BGK models:
(McNamara & Zanetti, Higuera & Jimenez, Succi, Benzi, H. Chen,
S. Chen, Doolen,......
Consistency
To reach consistency with the Navier-Stokes equations, fix ∆x and
ω = ∆t
τ ∈ (0, 2)
⇓
|c| =
3ν
∆x
2ω
2 − ω
∆t =...
Consistency
To reach consistency with the Navier-Stokes equations, fix ∆x and
ω = ∆t
τ ∈ (0, 2)
⇓
|c| =
3ν
∆x
2ω
2 − ω
∆t =...
Numerical Validation
in collaboration with V. Pavan and Y. Jobic
(IUSTI, Aix-Marseille Universit´e)
Lid-driven cavity : Computational domain
u = U, v = 0
wall
wall
wall
primary
vortex
top left
vortex
(T)
bottom
left vortex...
Lid-driven cavity : results 1
Figure: streamlines at Re 400, Nx = Ny
= 400
Figure: streamlines at Re 7500, Nx =
Ny = 7500
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...
Transient couette flow : Computational domain
Wall
UmPeriodic
Periodic
H
Figure: boundary conditions
analytical solution
ux...
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....
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 2...
Backward-Facing Step : Computational domain
H
Um
h
hi
L
x1
x2
x3
Figure: Conditions limites
Backward-Facing Step : results 1
0 5 10 150
1
2
Figure: streamlines at Re 100, Nx = 1000, Ny = 250, ε = 0.016
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 20...
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
Biswa...
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
Biswa...
Figure: Landsat 7 image of clouds off the Chilean coast near the Juan
Fernandez Islands (also known as the Robinson Crusoe ...
Figure: Von Karman vortices off the coast of Rishiri Island in Japan
Von Karman streets
click here to open the movie in your browser
The instabilities of the steady flow make the Von karman st...
Conclusions (?) and possible directions
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to study
NSE
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to study
NSE
• Some analytica...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to study
NSE
• Some analytica...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to study
NSE
• Some analytica...
Conclusions (?) and possible directions
• Hyperbolic approximations furnish a nice framework to study
NSE
• Some analytica...
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...

Related Documents