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.

This equation cannot be solved analytically and must be solved numerically.

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