Blasius boundary layer explained

In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow (Falkner–Skan boundary layer), i.e. flows in which the plate is not parallel to the flow.

Prandtl's boundary layer equations

Using scaling arguments, Ludwig Prandtl^[1] argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate). This leads to a reduced set of equations known as the boundary layer equations. For steady incompressible flow with constant viscosity and density, these read:

• Mass Continuity:

\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = 0 x -Momentum: u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} = - \dfrac{ 1 }{\rho} \dfrac{\partial p}{\partial x} + {\nu} \dfrac{\partial ^{2} u}{\partial y ^{2}} y -Momentum: 0 = - \dfrac{\partial p}{\partial y}

Here the coordinate system is chosen with

x pointing parallel to the plate in the direction of the flow and the y coordinate pointing normal to the plate, u and v are the x and y velocity components, p is the pressure, \rho is the density and \nu is the kinematic viscosity.

A number of similarity solutions to this set of equations have been found for various types of flow, including flow on a thin flat-plate. The term similarity refers to the property that the velocity profiles at different positions in the flow are the same apart from scaling factors. Similarity scaling factors reduce the set of partial differential equations to a relatively easily solved set of non-linear ordinary differential equations. Paul Richard Heinrich Blasius,^[2] one of Prandtl's students, developed the similarity model corresponding to the flow for the case where the pressure gradient,

{\partial p} /{\partial x} , along a thin flat-plate is negligible compared to any pressure gradient in the boundary layer region.

Blasius equation - first-order boundary layer

Blasius showed that for the case where

{\partial p}/{\partial x} = 0 , the Prandtl x -momentum equation has a self-similar solution. The self-similar solution exists because the equations and the boundary conditions are invariant under the transformation

$$x\backslash rightarrow\; c^2\; x,\; \backslash quad\; y\backslash rightarrow\; cy,\; \backslash quad\; u\backslash rightarrow\; u,\; \backslash quad\; v\backslash rightarrow\; \backslash frac$$

where

c is any positive constant. He introduced the self-similar variables

$$\backslash eta\; =\; \backslash dfrac\; =\; y\; \backslash sqrt,\; \backslash quad\; \backslash psi\; =\; \backslash sqrt\; f(\backslash eta)$$

where $\backslash delta(x)\; \backslash propto\; \backslash sqrt$ is the boundary layer thickness,

U is the free stream velocity, and \psi is the stream function. The stream function is directly proportional to the normalized function, f( η ) , which is only a function of the similarity thickness variable. This leads directly to the velocity components:^[3]

$$u(x,y)\; =\; \backslash dfrac\; =\; U\; f\text{'}(\backslash eta),\; \backslash quad\; v(x,y)\; =\; -\; \backslash dfrac\; =\; \backslash frac\backslash sqrt\; [\backslash eta\; f\text{'}(\backslash eta)\; -\; f(\backslash eta)]$$

Where the prime denotes derivation with respect to

η .Substitution into the x -momentum equation gives the Blasius equation

$$2f^\; +\; f$$ f = 0

The boundary conditions are the no-slip condition, the impermeability of the wall and the free stream velocity outside the boundary layer

This is a third-order non-linear ordinary differential equation which can be solved numerically, e.g. with the shooting method.

With the solution for

f and its derivatives in hand, the Prandtl y -momentum equation can be non-dimensionalized and rearranged to obtain the y -pressure gradient, {\partial p} /{\partial y} , as^[4]

$$\backslash frac\backslash frac\backslash frac\backslash quad\; =\; \backslash quad\; \backslash frac\; \backslash eta\; f^\backslash ;\; +\; \backslash ;\backslash fracf$$\; - \;\fracff'\; + \;\frac\eta f'^2 \; + \;\frac\eta f f \quad,where

\delta ^{*} is the Blasius displacement thickness.

The Blasius normal velocity

v(x,y) and the y -pressure gradient asymptotes to a value of 0.86 and 0.43, respectively, at large η -values whereas u(x,y) asymptotes to the free stream velocity U . As η goes to zero, the scaled y -pressure gradient goes to 0.16603.

The limiting form for small

η \ll 1 is

$$f(\backslash eta)\; =\; \backslash frac\backslash alpha\; \backslash eta^2\; +\; O(\backslash eta^5),\; \backslash qquad\; \backslash alpha\; =\; 0.332057336215196$$

and the limiting form for large

η \gg 1 is^[5]

$$f(\backslash eta)\; =\; \backslash eta\; -\; \backslash beta\; +\; O\backslash left((\backslash eta-\backslash beta)^\; e^\backslash right),\; \backslash qquad\; \backslash beta\; =1.7207876575205$$

The characteristic parameters for boundary layers are the two sigma viscous boundary layer thickness,^[6]

\delta _{v} , the displacement thickness \delta ^{*} , the momentum thickness \theta , the wall shear stress \tau _{w} and the drag force F acting on a length l of the plate. For the Blasius solution, they are given by

The factor

2 in the drag force formula is to account both sides of the plate.

The Von Kármán Momentum integral and the energy integral for Blasius profile reduce to

where

\tau _{w} is the wall shear stress, v _{w} is the wall injection/suction velocity, \varepsilon is the energy dissipation rate, \delta _{2} is the momentum thickness and \delta _{3} is the energy thickness.

Uniqueness of Blasius solution

The Blasius solution is not unique from a mathematical perspective,^[7] as Ludwig Prandtl himself noted it in his transposition theorem and analyzed by series of researchers such as Keith Stewartson, Paul A. Libby.^[8] To this solution, any one of the infinite discrete set of eigenfunctions can be added, each of which satisfies the linearly perturbed equation with homogeneous conditions and exponential decay at infinity. The first of these eigenfunctions turns out to be the

x derivative of the first order Blasius solution, which represents the uncertainty in the effective location of the origin.

Second-order boundary layer

This boundary layer approximation predicts a non-zero vertical velocity far away from the wall, which needs to be accounted in next order outer inviscid layer and the corresponding inner boundary layer solution, which in turn will predict a new vertical velocity and so on. The vertical velocity at infinity for the first order boundary layer problem from the Blasius equation is

$$v\; =\; 0.86\; \backslash sqrt$$

The solution for second order boundary layer is zero. The solution for outer inviscid and inner boundary layer are^[7]

Again as in the first order boundary problem, any one of the infinite set of eigensolution can be added to this solution. In all the solutions

Re = Ux/\nu can be considered as a Reynolds number.

Third-order boundary layer

Since the second order inner problem is zero, the corresponding corrections to third order problem is null i.e., the third order outer problem is same as second order outer problem.^[7] The solution for third-order correction does not have an exact expression, but the inner boundary layer expansion is of the form,

$$\backslash psi(x,y)\; \backslash sim\; \backslash sqrt\; f(\backslash eta)\; +\; 0\; +\; \backslash left(\backslash frac\backslash right)^\; \backslash left[\backslash log\; \backslash left(\backslash frac\{Ux\}\{\backslash nu\}\backslash right)\; \backslash sqrt\{\backslash frac\{x\}\{2\}\}\; f\_\{32\}(\backslash eta)\; +\; \backslash frac\{1\}\{\backslash sqrt\{2x\}\}\; f\_\{31\}(\backslash eta)\; \backslash right]\; +\; \backslash cdot\backslash cdot\backslash cdot$$

where

f _{32} is the first eigensolution of the first order boundary layer solution (which is x derivative of the first order Blasius solution) and solution for f _{31} is nonunique and the problem is left with an undetermined constant.

Blasius boundary layer with suction

Suction is one of the common methods to postpone the boundary layer separation.^[9] Consider a uniform suction velocity at the wall

v( 0 ) =- V . Bryan Thwaites^[10] showed that the solution for this problem is same as the Blasius solution without suction for distances very close to the leading edge. Introducing the transformation

$$\backslash psi\; =\; \backslash sqrt\; f(\backslash xi,\backslash eta),\; \backslash quad\; \backslash xi\; =\; V\backslash sqrt,\; \backslash quad\; \backslash eta\; =\; \backslash sqrty$$

into the boundary layer equations leads to

$$u\; =\; U\; \backslash frac,\; \backslash quad\; v\; =\; -\; \backslash sqrt\; \backslash left(f\; +\; \backslash xi\; \backslash frac\; -\; \backslash eta\; \backslash frac\; \backslash right),$$$$\backslash frac\; +\; f\; \backslash frac\; +\; \backslash xi\; \backslash left(\backslash frac\; \backslash frac\; -\; \backslash frac\; \backslash frac\; \backslash right)\; =0$$

with boundary conditions,

$$f(\backslash xi,0)\; =\backslash xi,\; \backslash quad\; \backslash frac(\backslash xi,0)\; =0,\; \backslash quad\; \backslash frac(\backslash xi,\backslash infty)=0.$$

Von Mises transformation

Iglisch obtained the complete numerical solution in 1944.^[11] If further von Mises transformation^[12] is introduced

$$\backslash sigma\; =\; 2\backslash xi,\; \backslash quad\; \backslash psi-Vx\; =\; \backslash frac\; \backslash sigma\; \backslash tau^2,\; \backslash quad\; \backslash phi\; =\; \backslash frac,\; \backslash quad\; \backslash chi\; =\; U^2-u^2\; =\; U^2\backslash left(1-\backslash frac\; \backslash right),$$

then the equations become

$$\backslash sqrt\; \backslash frac\; +\; \backslash left(2\backslash sigma\backslash tau\; +\backslash tau^3\; -\; \backslash frac\; \backslash right)\; \backslash frac\; =\; 2\backslash sigma\; \backslash tau^2\; \backslash frac$$

with boundary conditions,

$$\backslash phi(0,\backslash tau)=4,\; \backslash quad\; \backslash phi(\backslash sigma,0)=0,\; \backslash quad\; \backslash phi(\backslash sigma,\backslash infty)=4.$$

This parabolic partial differential equation can be marched starting from

\sigma =0 numerically.

Asymptotic suction profile

Since the convection due to suction and the diffusion due to the solid wall are acting in the opposite direction, the profile will reach steady solution at large distance, unlike the Blasius profile where boundary layer grows indefinitely. The solution was first obtained by Griffith and F.W. Meredith.^[13] For distances from the leading edge of the plate

x\gg \nu U/V ^{2} , both the boundary layer thickness and the solution are independent of x given by

$$\backslash delta\; =\; \backslash frac,\; \backslash quad\; u\; =\; U(1-e^),\; \backslash quad\; v\; =\; -V.$$

Stewartson^[14] studied matching of full solution to the asymptotic suction profile.

Compressible Blasius boundary layer

h at the wall is studied. The density \rho , viscosity \mu and thermal conductivity \kappa are no longer constant here. The equation for conservation of mass, momentum and energy become

where

Pr = c _{p _{infty}} \mu _{infty/\kappa} _{infty} is the Prandtl number with suffix inf ty representing properties evaluated at infinity. The boundary conditions become

$$u\; =\; v\; =\; h\; -\; h\_w(x)\; =\; 0\; \backslash \; \backslash text\; \backslash \; y=0,$$$$u\; -U\; =\; h\; -\; h\_\backslash infty\; =0\; \backslash \; \backslash text\; \backslash \; y=\backslash infty\; \backslash \; \backslash text\; \backslash \; x=0.$$

Unlike the incompressible boundary layer, similarity solution exists only if the transformation

$$x\backslash rightarrow\; c^2\; x,\; \backslash quad\; y\backslash rightarrow\; cy,\; \backslash quad\; u\backslash rightarrow\; u,\; \backslash quad\; v\backslash rightarrow\; \backslash frac,\; \backslash quad\; h\backslash rightarrow\; h,\; \backslash quad\; \backslash rho\backslash rightarrow\; \backslash rho,\; \backslash quad\; \backslash mu\backslash rightarrow\; \backslash mu$$

holds and this is possible only if

h _{w=constant} .

Howarth transformation

Introducing the self-similar variables using Howarth–Dorodnitsyn transformation

$$\backslash eta\; =\; \backslash sqrt\; \backslash int\_0^y\; \backslash frac\; dy,\; \backslash quad\; f(\backslash eta)\; =\; \backslash frac,\; \backslash quad\; \backslash tilde\; h(\backslash eta)\; =\; \backslash frac,\; \backslash quad\; \backslash tilde\; h\_w\; =\; \backslash frac,\; \backslash quad\; \backslash tilde\; \backslash rho\; =\; \backslash frac,\; \backslash quad\; \backslash tilde\; \backslash mu\; =\; \backslash frac$$

the equations reduce to$$\backslash begin2(\backslash tilde\backslash rho\backslash tilde\backslash mu\; f$$)' + f f =0, \\(\tilde\rho\tilde\mu \tilde h')' + Prf\tilde h' + Pr(\gamma-1)M^2 \tilde\rho\tilde\mu f^2 =0 \endwhere

\gamma is the specific heat ratio and M = U/c _{infty} is the Mach number, where c _{infty} is the speed of sound. The equation can be solved once \tilde \rho = \tilde \rho(\tilde h), \tilde \mu = \tilde \mu(\tilde h) are specified. The boundary conditions are

$$f(0)=f\text{'}(0)=\backslash theta(0)-\backslash tilde\; h\_w=f\text{'}(\backslash infty)-1=\backslash tilde\; h(\backslash infty)-1=0.$$

The commonly used expressions for air are

\gamma = 1 . 4 , Pr = 0 . 7 , \tilde\rho = \tilde h ^{-1} , \tilde\mu = \tilde h ^{2/3} . If c _{p} is constant, then \tilde h = \tilde \theta = T/T _{infty} . The temperature inside the boundary layer will increase even though the plate temperature is maintained at the same temperature as ambient, due to dissipative heating and of course, these dissipation effects are only pronounced when the Mach number M is large.

First-order Blasius boundary layer in parabolic coordinates

(x,y) to parabolic coordinates (\xi, η ) is given by

$$x\; +\; i\; y\; =\; \backslash tfrac\; (\backslash xi\; +\; i\; \backslash eta)^2,\; \backslash quad\; x\; =\; \backslash tfrac\; (\backslash xi^2-\backslash eta^2),\; \backslash quad\; y\; =\; \backslash xi\; \backslash eta.$$

See also

• Falkner–Skan boundary layer
• Emmons problem

External links

• https://ntrs.nasa.gov/citations/20050028493 - English translation of Blasius' original paper - NACA Technical Memorandum 1256.

References

• J. Y.. Parlange. R. D.. Braddock. G.. Sander. Analytical approximations to the solution of the Blasius equation. Acta Mech.. 1981. 38. 1–2. 119–125. 10.1007/BF01351467. 1981AcMec..38..119P. 122114667.
• Book: C. . Pozrikidis. 1998. Introduction to Theoretical and Computational Fluid Dynamics. Oxford. 978-0-19-509320-9.
• Book: Schlichting. H.. 2004. Boundary-Layer Theory. Springer. 978-3-540-66270-9.
• Wilcox, David C. Basic Fluid Mechanics DCW Industries Inc. 2007

Notes and References

1. Prandtl. L.. Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlinger 3. Int. Math. Kongr. Heidelberg. 1904. 484–491.
2. Blasius. H.. Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. Angew. Math. Phys.. 1908. 56. 1–37.
3. Schlichting, H., (1979). Boundary-Layer Theory, 7th ed., McGraw-Hill, New York.
4. Book: Weyburne, David . Aspects of Boundary Layer Theory . 2022 . 978-0-578-98334-9 .
5. Boyd. J.. The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems . SIAM Rev.. 2008 . 50 . 4 . 791–804. 10.1137/070681594 . 2008SIAMR..50..791B .
6. Weyburne. D.. New thickness and shape parameters for the boundary layer velocity profile . Experimental Thermal and Fluid Science. 2014. 54. 22–28. 10.1016/j.expthermflusci.2014.01.008 .
7. Book: Van Dyke, Milton . Milton Van Dyke . Perturbation methods in fluid mechanics . Parabolic Press . 1975 . 9780915760015 .
8. Libby, Paul A., and Herbert Fox. "Some perturbation solutions in laminar boundary-layer theory." Journal of Fluid Mechanics 17.3 (1963): 433-449.
9. Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
10. Thwaites, Bryan. On certain types of boundary-layer flow with continuous surface suction. HM Stationery Office, 1946.
11. Iglisch, Rudolf. Exakte Berechnung der laminaren Grenzschicht an der längsangeströmten ebenen Platte mit homogener Absaugung. Oldenbourg, 1944.
12. Von Mises, Richard. "Bemerkungen zur hydrodynamik." Z. Angew. Math. Mech 7 (1927): 425-429.
13. Griffith, A. A., and F. W. Meredith. "The possible improvement in aircraft performance due to the use of boundary layer suction." Royal Aircraft Establishment Report No. E 3501 (1936): 12.
14. Stewartson, K. "On asymptotic expansions in the theory of boundary layers." Studies in Applied Mathematics 36.1-4 (1957): 173-191.