Categories: Fluid dynamics, Fluid mechanics, Physics.

Prandtl equations

In fluid dynamics, the Prandtl equations or boundary layer equations describe the movement of a viscous fluid with a large Reynolds number \(\mathrm{Re} \gg 1\) close to a solid surface.

Fluids with a large Reynolds number are often approximated as having zero viscosity, since the simpler Euler equations can then be used instead of the Navier-Stokes equations.

However, in reality, a viscous fluid obeys the no-slip boundary condition: at every solid surface the local velocity must be zero. This implies the existence of a boundary layer: a thin layer of fluid “stuck” to solid objects in the flow, where viscosity plays an important role. This is in contrast to the ideal flow far away from the surface.

We consider a simple theoretical case in 2D: a large flat surface located at \(y = 0\) for all \(x \in \mathbb{R}\), with a fluid trying to flow parallel to it at \(U\). The 2D treatment can be justified by assuming that everything is constant in the \(z\)-direction. We will not solve this case, but instead derive general equations to describe the flow close to a flat surface.

At the wall, there is a very thin boundary layer of thickness \(\delta\), where the fluid is assumed to be completely stationary \(\va{v} = 0\). We are mainly interested in the region \(\delta < y \ll L\), where \(L\) is the distance at which the fluid becomes practically ideal. This the so-called slip-flow region, in which the fluid is not stationary, but still viscosity-dominated.

In 2D, the steady Navier-Stokes equations are as follows, where the flow \(\va{v} = (v_x, v_y)\):

\[\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} &= - \frac{1}{\rho} \pdv{p}{x} + \nu \Big( \pdv[2]{v_x}{x} + \pdv[2]{v_x}{y} \Big) \\ v_x \pdv{v_y}{x} + v_y \pdv{v_y}{y} &= - \frac{1}{\rho} \pdv{p}{y} + \nu \Big( \pdv[2]{v_y}{x} + \pdv[2]{v_y}{y} \Big) \\ \pdv{v_x}{x} + \pdv{v_y}{y} &= 0 \end{aligned}\]

The latter represents the fluid’s incompressibility. We non-dimensionalize these equations, and assume that changes along the \(y\)-axis happen on a short scale (say, \(\delta\)), and along the \(x\)-axis on a longer scale (say, \(L\)). Let \(\tilde{x}\) and \(\tilde{y}\) be dimenionless variables of order \(1\):

\[\begin{aligned} x = L \tilde{x} \qquad \quad y = \delta \tilde{x} \qquad \quad \pdv{x} = \frac{1}{L} \pdv{\tilde{x}} \qquad \quad \pdv{y} = \frac{1}{\delta} \pdv{\tilde{y}} \end{aligned}\]

Furthermore, we choose velocity scales to be consistent with the incompressibility condition, and a pressure scale inspired by Bernoulli’s theorem:

\[\begin{aligned} v_x = U \tilde{v}_x \qquad \quad v_y = \frac{U \delta}{L} \tilde{v}_y \qquad \quad p = \rho U^2 \tilde{p} \end{aligned}\]

We insert these scalings into the Navier-Stokes equations, yielding:

\[\begin{aligned} \frac{U^2}{L} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \frac{U^2}{L} \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} &= - \frac{U^2}{L} \pdv{\tilde{p}}{\tilde{x}} + \nu \Big( \frac{U}{L^2} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{U}{\delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) \\ \frac{U^2 \delta}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{U^2 \delta}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} &= - \frac{U^2}{\delta} \pdv{\tilde{p}}{\tilde{y}} + \nu \Big( \frac{U \delta}{L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{U}{L \delta} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) \end{aligned}\]

For future convenience, we multiply the former equation by \(L / U^2\), and the latter by \(\delta / U^2\):

\[\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{x}} + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{L}{U \delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) \\ \frac{\delta^2}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\delta^2}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{y}} + \nu \Big( \frac{\delta^2}{U L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) \end{aligned}\]

We would like to estimate \(\delta\). Intuitively, we expect that higher viscosities \(\nu\) give thicker layers, and that faster velocities \(U\) give thinner layers. Furthermore, we expect downstream thickening: with distance \(x\), viscous stresses slow down the slip-flow, leading to a gradual increase of \(\delta(x)\). Some dimensional analysis thus yields the following estimate:

\[\begin{aligned} \delta \approx \sqrt{\frac{\nu x}{U}} \sim \sqrt{\frac{\nu L}{U}} \end{aligned}\]

We thus insert \(\delta = \sqrt{\nu L / U}\) into the Navier-Stokes equations, giving us:

\[\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{x}} + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{1}{\nu} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big) \\ \frac{\nu}{U L} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\nu}{U L} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{y}} + \nu \Big( \frac{\nu}{U^2 L^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big) \end{aligned}\]

Here, we recognize the definition of the Reynolds number \(\mathrm{Re} = U L / \nu\):

\[\begin{aligned} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \pdv[2]{\tilde{v}_x}{\tilde{y}} \\ \frac{1}{\mathrm{Re}} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}} &= - \pdv{\tilde{p}}{\tilde{y}} + \frac{1}{\mathrm{Re}^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_y}{\tilde{y}} \end{aligned}\]

Recall that we are only considering large Reynolds numbers \(\mathrm{Re} \gg 1\), in which case \(\mathrm{Re}^{-1} \ll 1\), so we can drop many terms, leaving us with these redimensionalized equations:

\[\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdv[2]{v_x}{y} \qquad \quad \pdv{p}{y} = 0 \end{aligned}\]

The second one tells us that for a given \(x\)-value, the pressure is the same at the surface as in the main flow \(y > L\), where the fluid is ideal. In the latter regime, we apply Bernoulli’s theorem to rewrite \(p\), using the Bernoulli head \(H\) and the mainstream velocity \(U(x)\):

\[\begin{aligned} p = \rho H - \frac{1}{2} \rho U^2 = p_0 - \frac{1}{2} \rho U^2 \end{aligned}\]

Inserting this into the reduced Navier-Stokes equations, we arrive at the Prandtl equations:

\[\begin{aligned} \boxed{ v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = U \dv{U}{x} + \nu \pdv[2]{v_x}{y} \qquad \quad \pdv{v_x}{x} + \pdv{v_y}{y} = 0 } \end{aligned}\]

A notable application of these equations is the Blasius boundary layer, where the surface in question is a semi-infinite plane.

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.

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