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

:$u=U\; f\; \text{'}(eta).$ 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

:$f"\text{'}\; +\; frac\{1\}\{2\}f\; f"\; =0$

subject to $f=f\text{'}=0$ on $eta=0$ and $f\text{'}\; ightarrow\; 1$ 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

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

(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)