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Blasius boundary layer

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 $U$ .

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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 = U f'(η).

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

ψ = (ν U x) 1 / 2 f (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 $f'' (0) \approx 0.332$ .

Falkner-Skan boundary layer

A generalisation of the Blasius boundary layer that considers outer flows of the form $U = c x m$ 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

$ψ = U (x)δ(x) f (η) = c x m δ(x) f (η)$

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).

References

Schlichting, H. (2004), Boundary-Layer Theory, Springer. ISBN 3-540-66270-7

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

Blasius, H. (1908), Grenzschichten in Flussigkeiten mit kleiner Reibung, Z. Math. Phys. vol 56, pp. 1-37. http://naca.larc.nasa.gov/reports/1950/naca-tm-1256 (English translation)

Category: Boundary layers

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Blasius_boundary_layer". A list of authors is available in Wikipedia.