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Archive for the ‘Math 597F: topics on boundary layers’ Category

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

We now turn to the delicate case: Orr-Sommerfeld solutions for stable profiles to Rayleigh. The results reported here are in a joint work with E. Grenier and Y. Guo, directing some tedious details of the proof to our paper. We consider the Orr-Sommerfeld problem:

\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi,

with zero boundary conditions on {\phi} and {\phi'}, in which {Ray_\alpha(\cdot)} denotes the Rayleigh operator and {\Delta_\alpha = \partial_z^2 - \alpha^2}.

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(this post was also posted here on my new blog address: http://blog.toannguyen.org/ )

We now return to the Orr-Sommerfeld equations (the linearized Navier-Stokes equations around a boundary layer {U(z)}; see this lecture):

\displaystyle \left \{ \begin{aligned} (U-c) (\partial_z^2 - \alpha^2) \phi - U'' \phi &= \epsilon (\partial_z^2 - \alpha^2)^2 \phi \\ \phi_{\vert_{z=0}} = \phi'_{\vert_{z=0}} &= 0, \qquad \lim_{z\rightarrow \infty} \phi(z) =0. \end{aligned} \right. \ \ \ \ \ (1)

For convenience, we denote {Ray_\alpha(\phi) = (U-c) (\partial_z^2 - \alpha^2) \phi - U'' \phi} and {\Delta_\alpha = \partial_z^2 - \alpha^2}. The Orr-Sommerfeld equation simply reads

\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi.

We shall construct solutions for each fixed pair {(\alpha,c)} and for sufficiently small {\epsilon}.

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We study the linearization of 2D Navier-Stokes around a boundary layer {\vec v_\mathrm{app} =[\bar u, \sqrt \nu \bar v](t', x', y'/\sqrt \nu)}, in which {(t',x',y')} denotes the original coordinates for Navier-Stokes. Materials in this lecture are drawn from the joint paper(s) with Grenier and Guo (here and here).

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Let us give a few examples of boundary layer solutions to the Prandtl problem, derived in the last lecture. In 2D, we recall the Prandtl layer problem:

\displaystyle \left \{ \begin{aligned} u_t + u u_x + v u_z &= \mu u_{zz} - p_x, \qquad v = -\int_0^z u_x(x,\theta)\; d\theta \\ u_{\vert_{t=0}} &= u_0(x,z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,x,z) = u^E(t,x), \end{aligned} \right. \ \ \ \ \ (1)

with the pressure gradient: {p_x = -u_t^E- u^Eu^E_x(t,x)}, where {u^E} denotes the tangential component of Euler flow on the boundary {y=0}. Here, the tangential velocity component {u} is an (only) unknown scalar function, and the normal velocity component {v} is defined through the divergence-free condition. A simplest example: in the case {u^E} is independent of {x} and {t}, any solution to the following heat problem

\displaystyle u_t = \mu u_{zz} , \qquad u_{\vert_{t=0}} = u_0(z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,z) = u^E

gives a boundary layer solution of (1) in the form {[u,0]}. We call a solution of this form to be a shear flow.

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With simple integration by parts, we were able to see in the last two lectures essentially the current “state of art” of the {L^2} convergence of Navier-Stokes to Euler. Embarrassingly, the inviscid limit problem is widely open as discussed. It is noted that the {L^2} energy norm is quite weak, and does not see in the inviscid limit the appearance of thin layers that might (and indeed will) occur near the boundary (for instance, the {L^2} norm of Kato’s layer is of order {\sqrt \nu}\to 0). We will have to work with a different, stronger norm. Regarding the significance of viscosity despite being arbitrarily small (e.g., viscosity of air at zero temperature is about {10^{-4}}, which seems to be neglectable), d’Alembert in the 18th century has already argued out that ideal flows can’t explain well many of the physics, and the viscosity plays a crucial role near the boundary; for instance, one of his conclusions, known as d’Alembert’s paradox, asserts that solid body emerged in stationary ideal flows feels no drag acting on it (in the layman words, birds can’t fly!). Not until the beginning of the 20th century, Prandtl then postulated a solution Ansatz that revolutionized the previous understanding of slightly viscous flows near a boundary, later known as Prandtl boundary layer theory. The theory gave birth to the field of aerodynamics, and is regarded as one of the greatest achievements in fluid dynamics in the last century. Below, I’ll derive the Prandtl boundary layers.

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Most physicists don’t believe there is such an ideal fluid (i.e., no viscosity). It is clear however that the zero viscosity or infinite Reynolds number limit plays a central role in understanding turbulence, as seen in Kolmogorov’s theory, Onsager’s conjecture, and turbulent boundary layers. Hence, understanding the inviscid limit problem is of great practical and analytical importance. As expected in most singular perturbation problems, new phenomena will arise.

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In this lecture, I will briefly discuss the difficulty of the inviscid limit problem of Navier-Stokes. As it will be clear in the text, the issue is not due to the fact that the million-dollar regularity problem remains unsolved, but rather nature of the singular perturbation problem. Unless otherwise noted, throughout the course the solutions of both Euler and Navier-Stokes are assumed to be sufficiently smooth as one wishes (for instance, one works in the two-dimensional case or with local-in-time solutions, with which smooth solutions are known to exist).

Precisely, consider the incompressible NS equations in {\Omega = \mathbb{R}^n} or {\mathbb{T}^n} (periodic setting), with {n \ge 2},

\displaystyle \left \{ \begin{aligned} v_t + v \cdot \nabla v + \nabla p &= \nu \Delta v \\ \nabla \cdot v &=0 \\ v_{\vert_{t=0}} &= v_0(x), \end{aligned}\right. \ \ \ \ \ (1)

with small viscosity constant {\nu >0}. Unknowns in the equation are velocity field {v} and the pressure {p}. (note here that since {\nabla \cdot v =0}, one gets {\mathrm{div} (v\otimes v) = v \cdot \nabla v =( \sum_j v_j \partial_{x_j} v_k)}, with {v = (v_k)} denoting the vector field {v}). Assume that the initial data {v_0} and the corresponding solutions {v(x,t)} are sufficiently smooth. The most basic quantity associated with (1) is the total energy:
\displaystyle \int_\Omega |v(x,t)|^2 \; dx .

Let us calculate the rate of change of the total energy with respect to time:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx &= \int_{\Omega} v_t \cdot v \; dx \\ &= \int_{\Omega} \Big( \nu \Delta v - \nabla p - v\cdot \nabla v \Big) \cdot v\; dx. \end{aligned}

We now apply the integration by parts (that is, using the divergence theorem and noting that there is no boundary contribution in our case as {\partial \Omega = \emptyset}). The middle term vanishes, since {\nabla \cdot v =0}. The last term is computed as follows:
\displaystyle \int_\Omega (v \cdot \nabla v) \cdot v\; dx = \int_\Omega (v\cdot \nabla ) \frac{|v|^2}{2} \; dx = - \int_{\Omega} \nabla \cdot v \frac{|v|^2}{2}\; dx = 0.

This calculation (we shall refer to it as the standard energy estimate!) yields the energy balance for NS solutions:
\displaystyle \begin{aligned} \frac 12 \frac{d}{dt} \int_{\Omega} |v(x,t)|^2 \; dx = - \nu \int_\Omega |\nabla v (x,t) |^2 \; dx. \end{aligned}

That is, smooth solutions of NS equations dissipate energy, and in particular, smooth solutions of Euler ({\nu =0}) conserve energy (i.e., constant in time).
A side remark: this latter fact turns out to be false for low-regularity Euler solutions. Precisely, Onsager (1949) conjectured that the {C^{\alpha}} Euler weak solutions conserve energy when {\alpha >1/3}, and in fact dissipate energy when {\alpha<1/3}. The first part of the conjecture was proved in 1994 by Eyink, and then by Constantin-E-Titi. The second part is essentially proved by the recent breakthrough of De Lellis and Székelyhidi Jr., and also Isett and Buckmaster (see, for instance this paper, or Isett’s PhD thesis, and the references therein), using convex integration techniques (introduced by Nash in his famous isometric embedding theorem, and later developed by Gromov in his study of h-principle). This {\frac 13} power is closely related to the theory of turbulence of Kolmogorov in 1941. I plan to expand this side remark on another blog post in the near future!

 

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