On the filling of a glass of water The Reynolds number is about 10 4 - - PowerPoint PPT Presentation

On the filling of a glass of water

The Reynolds number is about 104 − 105. The viscous time scale is about 104s.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 2 / 28

Outline

1

The vanishing viscosity limit

2

The Prandtl 1904 theory

3

Numerical method

4

Results

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 4 / 28

The vanishing viscosity limit

The role of vorticity

In an incompressible flow (ρ = 1), dE dt = d dt u2 2 = −ν

• ω2 = −2νZ

(1) where ωz = ∂xuy − ∂yux , . . . (2) To dissipate energy, vorticity needs to be created and/or amplified, in such a way that Z ∼ ν−1. Examples:

ω ∼ ν−1/2 over O(1) area, ω ∼ ν−1 over O(ν) area.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 5 / 28

The vanishing viscosity limit

The 2d Euler and Navier-Stokes equation

(E) :      ∂tω + u · ∇ω = 0 ω = (∇ × u) · ez, ∇ · u = 0 u|∂Ω · n = 0, ω(0) = ωi (3) (NS) :      ∂tω + u · ∇ω = ν∆ω ω = (∇ × u) · ez, ∇ · u = 0 u|∂Ω = 0, ω(0) = ωi (4) When friction forces are negligible, vorticity is conserved by the flow. This is no longer the case when friction forces are introduced. One can show that there is a Neumann boundary condition for ω: ν∂yω = −∂xp, (5) where p is the pressure field. This is the source of vorticity we were looking for!

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 6 / 28

The vanishing viscosity limit

Vorticity and pressure boundary conditions

Vorticity is produced in reaction to pressure gradient, in the right amount to keep the tangential velocity to zero: ν∂nω = −∂τp, (6) This gives a hope of resolving the paradox. Can we get sufficient vorticity when ν → 0? How do we compute the pressure field? ∆p = −∇((u · ∇)u) inside Ω (7a) ∂np|∂Ω = ν∂τω|∂Ω (Stokes pressure) (7b) Note: this boundary condition for ω is nonlocal and nonlinear ! In fact it is possible to reformulate it in a linear way, but it always remain nonlocal.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 7 / 28

The vanishing viscosity limit

Link with d’Alembert’s paradox

For a solid immersed in a perfect fluid (i.e. satisfying the Euler equations), the only contact force at the boundary is the pressure force. The overall force is thus: F =

• ∂Ω

fdrag =

• ∂Ω

pndS (8) D’Alembert basically showed that in a parallel stationnary flow, the streamwise component Fdrag = 0. In a viscous flow, there is a tangential component to the drag: fviscousdrag = ν(∂xuy + ∂yux)|∂Ω = ν∂yux|∂Ω = −νω|∂Ω (9) This seems to resolve d’Alembert’s paradox.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 8 / 28

The Prandtl 1904 theory

The Prandtl theory

For simplicity we work with Ω = R×]0, +∞[. In the limit ν → 0, and letting y1 = Re

1 2 y, we take the Ansatz :

ω(x, y, t) = ω0(x, y, t) + Re

1 2 ω1(x, y1, t) + R(x, y, t, Re)

(10) By injecting that into the NSE, we get that ω0 should be a solution of (E), and ω1 of the Prandtl equations: (P) :            ∂tω1 + u1,x∂xω1 + u1,y∂y1ω1 = ∂2

y1ω1

u1,x = − y1

0 dy′ 1ω1(x, y′ 1, t)

u1,y = y1

0 dy′ 1

y′

1

0 dy′′ 1 ∂xω1(x, y′′ 1 , t)

∂y1ω1(x, 0, t) = −∂xp0(x, 0, t) , (11) where p0 is the pressure computed only from ω0.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 9 / 28

The Prandtl 1904 theory

Shortcoming of Prandtl’s theory

ω(x, y, t) = ω0(x, y, t) + Re

1 2 ω1(x, y1, t) + R(x, y, t, Re)

(12) By applying the Biot-Savart kernel to (11), we see that as long as the Prandtl boundary layer theory is valid, the flow converges to a solution of the Euler equations. In particular, there can be no injection of vorticity into the bulk flow. Enstrophy production requires breakdown of Prandtl scaling. How does this happen?

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 10 / 28

The Prandtl 1904 theory

Mathematics

Theorem (Kato ’84) In 2D with smooth initial conditions and smooth domain boundaries, the following assertions are equivalent:

1 uν(t) −

− − →

ν→0 u0(t) in L2 uniformly in time for t ∈ [0, T],

2 ∃c > 0, ν

T

• {x∈Ω|d(x,∂Ω)<cν} ∇u2 −

− − →

ν→0 0

For convergence to break down, there has to be energy dissipation in a very thin layer of thickness O(ν) along the wall. We study this (hypothetical) breakdown numerically:

solve the Navier-Stokes equations in a 2D channel. solve the Euler and Prandtl equations for the same initial data.

What happens? How do things scale with Reynolds number?

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 11 / 28

Numerical method

The geometry

π 2π π 2π x y

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 13 / 28

Numerical method

The initial data

π 2π x y

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 14 / 28

Numerical method

Euler solver

2π 2π x y

Play movie Use mirror symmetry around y = 0 to impose boundary condition. Spatial discretization: Fourier pseudo-spectral with hyperdissipation, kmax = 682. Time discretization: third order low storage Runge-Kutta, with exponential propagator for the viscous term.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 15 / 28

Numerical method

Prandtl solver

Artificial boundary condition ω1 = 0 at y1 = 32. Spatial discretization: second order finite differences. Time discretization: second order semi-implicit Runge-Kutta. Neumann boundary condition for ∂tω1 at y1 = 0 appears when inverting Helmholtz problem at each timestep: ∂y1∂tω1 = −∂xtp0(x, 0, t) (13) Time derivative of pressure approximated from Euler solution and using 2nd order Lagrange polynomials in time. Play movie

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 16 / 28

Numerical method

Navier-Stokes solvers

Solver A :

Same as Euler solver, but using standard dissipation, and boundary conditions approximated using volume penalization method.

2π 2π x y

Solver B :

Second order finite volumes in space, Second order time-splitting predictor, Regular 1024 × 4096 grid on subdomain [0 0 25] × [0 0 5], and use

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 17 / 28

Results

Finite-time singularity in Prandtl’s equations

Prandtl equation has well-known finite time singularity (van Dommelen and Shen, 1980):

|∂xω1| and u1,y blows up, ω1 remains bounded.

This is observed in our computations as expected, for t → tD ≃ 55.8:

10 20 30 40 50 60 1 2 3 4 5 6 7 8 x 10

−3

time ||ω1||L

∞

Nx = 512 Nx = 1024 Nx = 2048 Nx = 4096 10 20 30 40 5 10 15 20 25 30 35 40 || ∂x ω||L

∞

time Nx = 512 Nx = 1024 Nx = 2048 Nx = 4096

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 18 / 28

Results

Convergence before the singularity

According to Kato’s theorem, and since ω1 remains bounded uniformly until tD, we expect that uν

L2

− − − →

ν→0 u0 uniformly on [0, tD].

Show convergence!

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 19 / 28

Results

What happens at the singularity?

Play movie

10 20 30 40 50 60 70 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

−7

time −dE/dT/Z NSE/Finite Volumes NSE/Penalization data3 NSE/Finite Volumes NSE/Penalization data6 NSE/Finite Volumes NSE/Penalization data9 NSE/Finite Volumes NSE/Penalization data12

10 20 30 40 50 10 20 30 40 50 60 70 time maximum vorticity Re = 5e+06 Re = 1e+07 Re = 2e+07 Re = 4e+07

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 20 / 28

Results

What happens after the singularity?

Dissipative structures detach from the wall and are advected into the bulk flow1. These structures are likely to play a role in any kind of wall-bounded turbulent flows. The scaling Re−1 for the wall-normal extension of the collapsed boundary layer is roughly consistent with the von K´ arm´ an theory, although it is not clear why or how.

1Romain Nguyen van yen, Marie Farge, and Kai Schneider. “Energy Dissipating

Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity”. In:

• Phys. Rev. Lett. 106.18 (2011), p. 184502.
• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 21 / 28

Results

Main messages of this talk

The production of dissipative structures is the key feature of unstationnary BL detachment at vanishing viscosity. The viscous solution converges uniformly to the inviscid solution for t ≤ tD, and ceases to converge for t > tD. The detachment process involves spatial and temporal scales as fine as Re−1. The Stokes pressure seems to play a key role in the dynamics of detachment.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 22 / 28

Results

References I

[1] Ludwig Prandtl. “¨ Uber Fl¨ ussigkeitsbewegung bei sehr kleiner Reibung”. In: Proc. 3rd Inter. Math. Congr. Heidelberg. 1904,

• pp. 484–491.

[2] O.A. Oleinik. “On the mathematical theory of boundary layer for an unsteady flow of incompressible fluid”. In: Journal of Applied Mathematics and Mechanics 30.5 (1966), pp. 951 –974. [3] LL Van Dommelen and SF Shen. “The spontaneous generation of the singularity in a separating laminar boundary layer”. In: J. Comp.

• Phys. 38.2 (1980), pp. 125–140.

[4] Tosio Kato. “Remarks on Zero Viscosity Limit for Nonstationary Navier-Stokes Flows with Boundary”. In: Seminar on nonlinear partial differential equations. MSRI, Berkeley, 1984, pp. 85–98.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 24 / 28

Results

References II

[5] L.L. Van Dommelen and S.J. Cowley. “On the Lagrangian description of unsteady boundary-layer separation. Part 1. General theory”. In: J. Fluid Mech. 210.-1 (1990), pp. 593–626. [6] Weinan E and Bjorn Engquist. “Blowup of solutions of the unsteady Prandtl’s equation”. In: Communications on Pure and Applied Mathematics 50.12 (1997), pp. 1287–1293. [7] Marco Sammartino and Russel E. Caflisch. “Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space”. In: Comm. Math. Phys. 192 (1998), pp. 433–491. [8] Kevin W. Cassel. “A Comparison of Navier-Stokes Solutions with the Theoretical Description of Unsteady Separation”. English. In: Philosophical Transactions: Mathematical, Physical and Engineering Sciences 358.1777 (2000), pp. 3207–3227.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 25 / 28

Results

References III

[9]

• H. Clercx and G. J. van Heijst. “Dissipation of kinetic energy in

two-dimensional bounded flows”. In: Phys. Rev. E 65 (2002),

• p. 066305.

[10] AV Obabko and KW Cassel. “Navier-Stokes solutions of unsteady separation induced by a vortex”. In: Journal of Fluid Mechanics 465 (2002), pp. 99–130. [11] Zhouping Xin and Liqun Zhang. “On the global existence of solutions to the Prandtl’s system”. In: Advances in Mathematics 181.1 (2004), pp. 88 –133. [12]

• H. Clercx and C.-H. Bruneau. “The normal and oblique collision of a

dipole with a no-slip boundary”. In: Comput. Fluids 35.3 (2006),

• pp. 245–279.
• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 26 / 28

Results

References IV

[13]

• W. Kramer, H. Clercx, and G. J. van Heijst. “Vorticity dynamics of

a dipole colliding with a no-slip wall”. In: Phys. Fluids 19 (2007),

• p. 126603.

[14]

• F. Gargano, M. Sammartino, and V. Sciacca. “Singularity formation

for Prandtl’s equations”. In: Physica D 238.19 (2009),

• pp. 1975–1991.

[15]

• D. G´

erard-Varet and E. Dormy. “On the ill-posedness of the Prandtl equation”. In: Journal of the American Mathematical Society 23.2 (2010), p. 591. [16]

• F. Gargano, M. Sammartino, and V. Sciacca. “High Reynolds

number NavierStokes solutions and boundary layer separation induced by a rectilinear vortex”. In: Computers &amp; Fluids 52.0 (2011), pp. 73 –91.

• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 27 / 28

Results

References V

[17] Romain Nguyen van yen, Marie Farge, and Kai Schneider. “Energy Dissipating Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity”. In: Phys. Rev. Lett. 106.18 (2011),

• p. 184502.
• R. Nguyen van yen (FU Berlin)

Boundary layers and dissipation October 16, 2012 28 / 28