Blasius boundary layer

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

File:Blasius1.PNG

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\[ {\partial u\over\partial x}+{\partial v\over\partial y}=0 \]

x-Momentum\[ u{\partial u \over \partial x}+v{\partial u \over \partial y}={\nu}{\partial^2 u\over \partial y^2} \]

(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).

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

(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

• (English translation)
• (see homotopy analysis method)
• (see homotopy analysis method)
• Wilcox, David C. Basic Fluid Mechanics. DCW Industries Inc. 2007