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:

\[\begin{aligned} v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} = \nu \pdv[2]{v_x}{y} \end{aligned}\]

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.

References

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

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