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 $U$.

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. $u$ 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

$u=Uf'(\eta ).$

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

$\psi =(\nu Ux)^{1/2}f(\eta )$

and on differentiating, to find the velocities, and substituting into the boundary-layer equation we obtain the Blasius equation

$f'''+{\frac {1}{2}}ff''=0$

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:

$u{\partial u \over \partial x}+v{\partial u \over \partial y}=c^{2}mx^{2m-1}+{\nu }{\partial ^{2}u \over \partial y^{2}}.$

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 $-0.0905\leq m\leq 2$. Here, m<0 corresponds to an adverse pressure gradient (often resulting in boundary layer separation) while m > 0 represents a favorable pressure gradient.

References

• 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 Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. vol 56, pp. 1–37. http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf (English translation)

• Liao, S.J. (1999), "An explicit, totally analytic approximation of Blasius' viscous flow problems", International Journal of Non-Linear Mechanics, 34 (4): 759–778, Bibcode:1999IJNLM..34..759L, doi:10.1016/S0020-7462(98)00056-0 (see homotopy analysis method)

• Liao, S.J.; Campo, A. (2002), "Analytic solutions of the temperature distribution in Blasius viscous flow problems", Journal of Fluid Mechanics, 453: 411–425, Bibcode:2002JFM...453..411L (see homotopy analysis method)

• Wilcox, David C. Basic Fluid Mechanics. DCW Industries Inc. 2007