Blasius boundary layer

In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow U.

A schematic diagram of the Blasius flow profile. The streamwise velocity component u(\eta)/U(x) is shown, as a function of the stretched co-ordinate \eta.

The solution to the Navier–Stokes equation for this flow begins with an order-of-magnitude analysis to determine what terms are important. Within the boundary layer the usual balance between viscosity and convective inertia is struck, resulting in the scaling argument

\frac{U^{2}}{L}\approx \nu\frac{U}{\delta^{2}},

where \delta is the boundary-layer thickness and \nu is the kinematic viscosity.

However the semi-infinite plate has no natural length scale L and so the steady, incompressible, two-dimensional boundary-layer equations for continuity and momentum are

Continuity: \dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}=0

x-Momentum: u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}={\nu}\dfrac{\partial^2 u}{\partial y^2}

(note that the x-independence of U has been accounted for in the boundary-layer equations) admit a similarity solution. In the system of partial differential equations written above it is assumed that a fixed solid body wall is parallel to the x-direction whereas the y-direction is normal with respect to the fixed wall, as shown in the above schematic. u and v denote here the x- and y-components of the fluid velocity vector. Furthermore, from the scaling argument it is apparent that the boundary layer grows with the downstream coordinate x, e.g.

\delta(x)\approx \left( \frac{\nu x}{U} \right)^{1/2}.

This suggests adopting the similarity variable

\eta=\frac{y}{\delta(x)}=y\left( \frac{U}{\nu x} \right)^{1/2}

and writing

u=U f '(\eta).

It proves convenient to work with the stream function \psi , in which case

\psi=(\nu U x)^{1/2} f(\eta)

and on differentiating, to find the velocities, and substituting into the boundary-layer equation we obtain the Blasius equation

f''' + \frac{1}{2}f f'' =0

subject to f=f'=0 on \eta=0 and f'\rightarrow 1 as \eta\rightarrow \infty. This non-linear ODE can be solved numerically, with the shooting method proving an effective choice. The shear stress on the plate

\tau_{xy} = \frac{f'' (0) \rho U^{2}\sqrt{\nu}}{\sqrt{Ux}}.

can then be computed. The numerical solution gives f'' (0) \approx 0.332.

Falkner–Skan boundary layer

We can generalize the Blasius boundary layer by considering a wedge at an angle of attack {\beta} from some uniform velocity field U_{0} . We then estimate the outer flow to be of the form:

u_{e}(x)= U_{0} \left( x/L \right) ^{m}

Where L is a characteristic length and m is a dimensionless constant. In the Blasius solution, m = 0 corresponding to an angle of attack of zero radians. Thus we can write:

{\beta} = \frac{2m}{m + 1}

As in the Blasius solution, we use a similarity variable {\eta} to solve the Navier-Stokes Equations.

{\eta} = y \sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{2}}

It becomes easier to describe this in terms of its stream function which we write as

\psi=U(x)\delta(x)f(\eta) = y \sqrt{\frac{2{\nu} U_{0}L}{m+1}}\left(\frac{x}{L}\right)^\frac{m+1}{2}f(\eta)

Thus the initial differential equation which was written as follows:

u{\partial u \over \partial x} + v{\partial u \over \partial y} = c^{2}m x^{2m-1} + {\nu}{\partial^2 u\over \partial y^2}.

Can now be expressed in terms of the non-linear ODE known as the Falkner–Skan equation (named after V. M. Falkner and Sylvia W. Skan[1]).

\frac{\partial^3 f}{\partial \eta ^3}+f\frac{\partial^2 f}{\partial \eta^2}+ \beta \left[1-\left(\frac{\mathrm{d}f}{\mathrm{d}\eta}\right)^2 \right]=0

(note that m=0 produces the Blasius equation). See Wilcox 2007.

In 1937 Douglas Hartree revealed that physical solutions exist only in the range -0.0905 \le m \le 2 . Here, m < 0 corresponds to an adverse pressure gradient (often resulting in boundary layer separation) while m > 0 represents a favorable pressure gradient.

References

  1. ↑ V. M. Falkner and S. W. Skan, Aero. Res. Coun. Rep. and Mem. no 1314, 1930.