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 Re≫1\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.

Let the half-plane lie in the (x,z)(x,z)-plane (i.e. at y=0y = 0) and exist for all x≥0x \ge 0, such that its edge lies on the zz-axis. A fluid with velocity field v⃗=Ue^x\va{v} = U \vu{e}_x approaches the half-plane’s edge head-on. To describe the fluid’s movements around the plane, we make an ansatz for the so-called slip-flow region’s xx-velocity vx(x,y)v_x(x, y):

vx=Uf′(s)s≡yδ(x)\begin{aligned} v_x = U f'(s) \qquad \qquad s \equiv \frac{y}{\delta(x)} \end{aligned}

Where δ(x)≡νx/U\delta(x) \equiv \sqrt{\nu x / U} is the boundary layer thickness estimate that was used to derive the Prandtl equations. Note that f′(s)f'(s) is the derivative of an unknown f(s)f(s), and that it obeys the boundary conditions f′(0)=0f'(0) = 0 and f′(∞)=1f'(\infty) = 1, i.e. the fluid is stationary at the half-plane’s surface s=0s = 0, and has velocity UU far away at s→∞s \to \infty.

Inserting the ansatz into the incompressibility condition ∇⋅v⃗=0\nabla \cdot \va{v} = 0 yields:

∂vy∂y=−∂vx∂x=−∂vx∂s∂s∂x=Uδ′δsf′′\begin{aligned} \pdv{v_y}{y} = - \pdv{v_x}{x} = - \pdv{v_x}{s} \pdv{s}{x} = U \frac{\delta'}{\delta} s f'' \end{aligned}

Which we integrate by parts to get an expression for the yy-velocity vyv_y, namely:

vy=Uδ′δ∫sf′′dy=Uδ′(sf′−∫f′ds)=Uδ′ (sf′−f)\begin{aligned} v_y = U \frac{\delta'}{\delta} \int s f'' \dd{y} = U \delta' \bigg( s f' - \int f' \dd{s} \bigg) = U \delta' \: (s f' - f) \end{aligned}

Now, consider the main Prandtl equation, assuming the attack velocity UU is constant. Inserting our expressions for vxv_x and vyv_y into it gives:

ν∂2vx∂y2=vx∂vx∂x+vy∂vx∂yν∂2vx∂s2∂2s∂y2=vx∂vx∂s∂s∂x+vy∂vx∂s∂s∂yνU1δ2f′′′=−U2δ′δsf′′f′+U2δ′δf′′(sf′−f)\begin{aligned} \nu \pdvn{2}{v_x}{y} &= v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y} \\ \nu \pdvn{2}{v_x}{s} \pdvn{2}{s}{y} &= v_x \pdv{v_x}{s} \pdv{s}{x} + v_y \pdv{v_x}{s} \pdv{s}{y} \\ \nu U \frac{1}{\delta^2} f''' &= - U^2 \frac{\delta'}{\delta} s f'' f' + U^2 \frac{\delta'}{\delta} f'' (s f' - f) \end{aligned}

We multiply by δ2/U\delta^2 / U, cancel out some terms, and substitute δ(x)≡νx/U\delta(x) \equiv \sqrt{\nu x / U}, leaving:

νf′′′=−Uδ′δf′′f=−Uν2Uf′′f\begin{aligned} \nu f''' &= - U \delta' \delta f'' f = - U \frac{\nu}{2 U} f'' f \end{aligned}

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

2f′′′+f′′f=0\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 δ(x)\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.