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, as shown in the above schematic. $u$ 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

$u=U f '(\eta).$

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

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

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 $f'\rightarrow 1$ 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:

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

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

References

• Blasius, H. (1908). "Grenzschichten in Flüssigkeiten mit kleiner Reibung". Z. Math. Phys. 56: 1–37. http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf.  (English translation)
• Parlange, J. Y.; Braddock, R. D.; Sander, G. (1981). "Analytical approximations to the solution of the Blasius equation". Acta Mech. 38: 119-125. doi:10.1007/BF01351467. 
• Pozrikidis, C. (1998). Introduction to Theoretical and Computational Fluid Dynamics. Oxford. ISBN 0-19-509320-8. 
• Lien-Tsai, Yu; Cha'o-Kuang, Chen (1998). "The solution of the blasius equation by the differential transformation method". Math. Comp. Model. 28: 101-111. doi:10.1016/S0895-7177(98)00085-5. 
• He, Jihuan (1998). "Approximate analytical solution of Blasius' equation". Commun. Nonl. Sci. Num. Simul. 3 (4): 260-263. Bibcode 1998CNSNS...3..260H. doi:10.1016/S1007-5704(98)90046-6. 
• 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)
• Ropman-Miller, Lance; Broadbridge, Philip (2000). "Exact integraion of reduced Fisher's equation, Reduced Balsisu equation and the Lorenz model". J. Math. An. Applic. 251 (1): 65-83. doi:10.1006/jmaa.2000.7020. 
• He, Ji-Huan (2002). "A simple perturbation approach to Blasius equation". Appl. Math. Comput. 140 (2-3): 217-222. doi:10.1016/S0096-3003(02)00189-3. 
• 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)
• Schlichting, H. (2004). Boundary-Layer Theory. Springer. ISBN 3-540-66270-7. 
• Wang, Lei (2004). "A new algorithm for solving classical Blasius equation". Appl. Math. Comput. 157 (1): 1-9. doi:10.1016/j.amc.2003.06.011. 
• Wilcox, David C. Basic Fluid Mechanics. DCW Industries Inc. 2007
• Abbasbandy, S. (2007). "A numerical solution of Blasius equation by Adomina's decomposition method and comparison with homotopy perturbation method". Chaos. Solit. Fract. 31 (1): 257-260. doi:/10.1016/j.chaos.2005.10.071. 
• Parand, K; Deghan, Mehdi; Pirkhedri, A. (2009). "Sinc-collocation method for solving the Blasius equation". Phys. Lett. A 373 (44). doi:10.1016/j.physleta.2009.09.005,. 
• Parand, K.; Taghavi, A. (2009). "Rational scaled generalized Laguerre function collocation method for solving the Blasius equation". J. Comp. Appl. Math. 233: 980-989. Bibcode 2009JCoAM.233..980P. doi:10.1016/j.cam.2009.08.106.