Blasius boundary layer

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.

Within the boundary layer the usual balance between viscosity and convective inertia is struck, resulting in the scaling argument

: frac{U^{2{L}approx ufrac{U}{delta^{2,

where delta is the boundary-layer thickness and u 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 uoverpartial x}+{partial voverpartial y}=0

: u{partial u over partial x}+v{partial u over partial y}={ u}{partial^2 uover 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{ u x}{U} ight)^{1/2}.

This suggests adopting the similarity variable

: eta=frac{y}{delta(x)}=yleft( frac{U}{ u x} ight)^{1/2}and writing

:u=U f '(eta). It proves convenient to work with the stream function psi , in which case

: psi=( u 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' ightarrow 1 as eta ightarrow 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) ho U^{2}sqrt{ u{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=cx^{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}+{ u}{partial^2 uover partial y^2}.Under these circumstances the appropriate similarity variable becomes

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

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

psi=U(x)delta(x)f(eta) = c x^m delta(x)f(eta)

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).


*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. (English translation)

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