Blasius boundary layer

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A Blasius boundary layer, in physics and fluid mechanics, describes the steady two-dimensional boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow U.

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 δ is the boundary-layer thickness and ν is the kinematic viscosity.

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

{\partial u\over\partial x}+{\partial v\over\partial y}=0
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. 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 = Uf'(η).

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

ψ = (νUx)1 / 2f(η)

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

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

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

[edit] Falkner-Skan boundary layer

A generalisation of the Blasius boundary layer that considers outer flows of the form U = cxm results in a boundary-layer equation of the form

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}.

Under these circumstances the appropriate similarity variable becomes

\eta=\frac{y}{\delta(x)}=\frac{\sqrt{c}y}{\sqrt{\nu}x^{(1-m)/2}},

and, as in the Blasius boundary layer, it is convenient to use a stream function

ψ = U(x)δ(x)f(η) = cxmδ(x)f(η)

This results in the Falkner-Skan equation

f'''+\frac{1}{2}(m+1)f f'' - m f'^{2} + m =0

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

[edit] References

  • Pozrikidis, C. (1998), Introduction to Theoretical and Computational Fluid Dynamics, Oxford. ISBN 0-19-509320-8