Categories: Fluid dynamics, Fluid mechanics, Physics.

# Blasius boundary layer

In fluid dynamics, the Blasius boundary layer is an application of the Prandtl equations, which govern the flow of a fluid at large Reynolds number $$\mathrm{Re} \gg 1$$ close to a surface. Specifically, the Blasius layer is the solution for a half-plane approached from the edge by a fluid.

A fluid with velocity field $$\va{v} = U \vu{e}_x$$ flows to the plane, which starts at $$y = 0$$ and exists for $$x \ge 0$$. To describe this, we make an ansatz for the slip-flow region’s $$x$$-velocity $$v_x(x, y)$$:

\begin{aligned} v_x = U f'(s) \qquad \quad s \equiv \frac{y}{\delta(x)} \end{aligned}

Note that $$f'(s)$$ is the derivative of an unknown $$f(s)$$, and that it obeys the boundary conditions $$f'(0) = 0$$ and $$f'(\infty) = 1$$. Furthermore, $$\delta(x)$$ is the thickness of the stationary boundary layer at the surface. To derive the Prandtl equations, the estimate $$\delta(x) = \sqrt{\nu x / U}$$ was used, which we will stick with. For later use, it is worth writing the derivatives of $$s$$:

\begin{aligned} \pdv{s}{x} = - y \frac{\delta'}{\delta^2} = - s \frac{\delta'}{\delta} \qquad \quad \pdv{s}{y} = \frac{1}{\delta} \end{aligned}

Inserting the ansatz for $$v_x$$ into the incompressibility condition then yields:

\begin{aligned} \pdv{v_y}{y} = - \pdv{v_x}{x} = U s f'' \frac{\delta'}{\delta} \end{aligned}

Which we integrate to get an expression for the $$y$$-velocity $$v_y$$, namely:

\begin{aligned} v_y = U \frac{\delta'}{\delta} \int s f'' \dd{y} = U \delta' \: (s f' - f) \end{aligned}

Now, consider the main Prandtl equation, assuming that the attack velocity $$U$$ is constant:

\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = \nu \pdv{v_x}{y} \end{aligned}

Inserting our expressions for $$v_x$$ and $$v_y$$ into this leads us to:

\begin{aligned} - U^2 \frac{\delta'}{\delta} s f'' f' + U^2 \frac{\delta'}{\delta} f'' (s f' - f) = \nu U \frac{1}{\delta^2} f''' \end{aligned}

After multiplying it by $$\delta^2 / U$$ and cancelling out some terms, it reduces to:

\begin{aligned} \nu f''' + U \delta' \delta f'' f = 0 \end{aligned}

Then, substituting $$\delta(x) = \sqrt{\nu x / U}$$ and $$\delta'(x) = (1/2) \sqrt{\nu / (U x)}$$ yields:

\begin{aligned} \nu f''' + U \frac{\nu}{2 U} f'' f = 0 \end{aligned}

Simplifying this leads us to the Blasius equation, which is a nonlinear ODE for $$f(s)$$:

\begin{aligned} \boxed{ 2 f''' + f'' f = 0 } \end{aligned}

Unfortunately, this cannot be solved analytically, only numerically. Nevertheless, the result shows a boundary layer $$\delta(x)$$ exhibiting the expected downstream thickening.

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