Blasius boundary layer - Wikipedia PDF

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Blasius boundary layer - Wikipedia

Blasius boundary layer 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.

Contents Prandtl's boundary layer equations Blasius equation - First-order boundary layer Uniqueness of Blasius solution Second-order boundary layer Third-order boundary layer Blasius boundary layer with suction[5] Von Mises transformation Asymptotic suction profile Compressible Blasius boundary layer Howarth transformation First-order Blasius boundary layer in Parabolic coordinates See also External links References

Prandtl's boundary layer equations Using scaling arguments, Ludwig Prandtl[1] has 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: Continuity: A schematic diagram of the Blasius flow profile. The streamwise velocity component is shown, as a function of the similarity variable .

-Momentum:

-Momentum: Here the coordinate system is chosen with

pointing parallel to the plate in the direction of the flow and the

towards the free stream,

and

and

are the

velocity components,

is the pressure,

coordinate pointing

is the density and

is the kinematic

viscosity. The -momentum equation implies that the pressure in the boundary layer must be equal to that of the free stream for any given coordinate. Because the velocity profile is uniform in the free stream, there is no vorticity involved, therefore a simple Bernoulli's equation can be applied in this high Reynolds number limit

constant or, after differentiation:

Here

is the velocity of the fluid outside the boundary layer and is solution of Euler equations (fluid dynamics). Von Kármán Momentum integral and the energy integral for Blasius profile reduce to https://en.wikipedia.org/wiki/Blasius_boundary_layer

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where

Blasius boundary layer - Wikipedia

is the wall shear stress,

thickness and

is the wall injection/suction velocity,

is the energy dissipation rate,

is the momentum

is the energy thickness.

A number of similarity solutions to this equation have been found for various types of flow, including flat plate boundary layers. The term similarity refers to the property that the velocity profiles at different positions in the flow are the same apart from a scaling factor. These solutions are often presented in the form of non-linear ordinary differential equations.

Blasius equation - First-order boundary layer Blasius[2]

proposed

a

similarity

solution

for

the

case

in

which

the

free

stream

velocity

is

constant,

, which corresponds to the boundary layer over a flat plate that is oriented parallel to the free flow. Self-similar solution exists because the equations and the boundary conditions are invariant under the transformation

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

where

is the boundary layer thickness and

which the newly introduced normalized stream function,

is the stream function, in , is only a function of the

similarity variable. This leads directly to the velocity components

Developing Blasius boundary layer (not to scale). The velocity profile is shown in red at selected positions along the plate. The blue lines represent, in top to bottom order, the 99% free stream velocity line ( ), the displacement thickness ( ) and ( ). See Boundary layer thickness for a more detailed explanation.

Where the prime denotes derivation with respect to . Substitution into the momentum equation gives the Blasius equation

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

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Blasius boundary layer - Wikipedia

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

is

and the limiting form for large

is

The appropriate parameters to compare with the experimental observations are displacement thickness wall shear stress

The factor

and drag force

, momentum thickness

acting on a length of the plate, which are given for the Blasius profile

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

Uniqueness of Blasius solution The Blasius solution is not unique from a mathematical perspective,[3]:131 as Ludwig Prandtl himself noted it in his transposition theorem and analyzed by series of researchers such as Keith Stewartson, Paul A. Libby.[4] 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 be the

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

The solution for second order boundary layer is zero. The solution for outer inviscid and inner boundary layer are[3]:134

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 solution

can be considered as a Reynolds number.

Third-order boundary layer

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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.[3]:139 The solution for third-order correction don't have an exact expression, but inner boundary layer expansion have the form,

where

is the first eigensolution of the first order boundary layer solution (which is

solution) and solution for

derivative of the first order Blasius

is nonunique and the problem is left with an undetermined constant.

Blasius boundary layer with suction[5] Suction is one of the common methods employed to postpone the boundary layer separation. Consider a uniform suction velocity at the wall

. Bryan Thwaites[6] 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

into the boundary layer equations leads to

with boundary conditions,

Von Mises transformation Iglisch obtained the complete numerical solution in 1944.[7] If further von Mises transformation[8] is introduced

then the equations become

with boundary conditions,

This parabolic partial differential equation can be marched starting from

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.[9] For distances from the leading edge of the plate and the solution are independent of

, both the boundary layer thickness

given by

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Blasius boundary layer - Wikipedia

Stewartson[10] studied matching of full solution to the asymptotic suction profile.

Compressible Blasius boundary layer Here Blasius boundary layer with a specified specific enthalpy conductivity

at the wall is studied. The density , viscosity

and thermal

are no longer constant here. The equation for conservation of mass, momentum and energy become

where

is the Prandtl number with suffix

representing properties evaluated at infinity. The boundary

conditions become

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

holds and this is possible only if

.

Howarth transformation Introducing the self-similar variables using Howarth–Dorodnitsyn transformation

Compressible Blasius boundary layer

the equations reduce to

where once

is the specific heat ratio and

is the Mach number, where

is the speed of sound. The equation can be solved

are specified. The boundary conditions are

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Blasius boundary layer - Wikipedia

The commonly used expressions for air are

. If

is constant, then

. It

should be noted that 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

is large.

First-order Blasius boundary layer in Parabolic coordinates Since the boundary layer equations are Parabolic partial differential equation, the natural coordinates for the problem is parabolic coordinates.[3]:142 The transformation from Cartesian coordinates

to parabolic coordinates

is given by

.

See also Falkner–Skan boundary layer Emmons problem

External links [1] (http://naca.central.cranfield.ac.uk/reports/1950/naca-tm-1256.pdf) - English translation of Blasius' original paper - NACA Technical Memorandum 1256.

References 1. Prandtl, L. (1904). "Über Flüssigkeitsbewegung bei sehr kleiner Reibung". Verhandlinger 3. Int. Math. Kongr. Heidelberg: 484– 491. 2. Blasius, H. (1908). "Grenzschichten in Flüssigkeiten mit kleiner Reibung". Z. Angew. Math. Phys. 56: 1–37. 3. Van Dyke, Milton (1975). Perturbation methods in fluid mechanics. Parabolic Press. ISBN 9780915760015. 4. Libby, Paul A., and Herbert Fox. "Some perturbation solutions in laminar boundary-layer theory." Journal of Fluid Mechanics 17.3 (1963): 433-449. 5. Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963. 6. Thwaites, Bryan. On certain types of boundary-layer flow with continuous surface suction. HM Stationery Office, 1946. 7. Iglisch, Rudolf. Exakte Berechnung der laminaren Grenzschicht an der längsangeströmten ebenen Platte mit homogener Absaugung. Oldenbourg, 1944. 8. Von Mises, Richard. "Bemerkungen zur hydrodynamik." Z. Angew. Math. Mech 7 (1927): 425-429. 9. 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. 10. Stewartson, K. "On asymptotic expansions in the theory of boundary layers." Studies in Applied Mathematics 36.1-4 (1957): 173-191. 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 (https://doi.org/10.1007%2FBF01351467). Pozrikidis, C. (1998). Introduction to Theoretical and Computational Fluid Dynamics. Oxford. ISBN 978-0-19-509320-9. Schlichting, H. (2004). Boundary-Layer Theory. Springer. ISBN 978-3-540-66270-9. Wilcox, David C. Basic Fluid Mechanics. DCW Industries Inc. 2007 Boyd, John P. (1999), "The Blasius function in the complex plane" (http://projecteuclid.org/getRecord?id=euclid.em/104726235 9), Experimental Mathematics, 8 (4): 381–394, doi:10.1080/10586458.1999.10504626 (https://doi.org/10.1080%2F10586458. 1999.10504626), ISSN 1058-6458 (https://www.worldcat.org/issn/1058-6458), MR 1737233 (https://www.ams.org/mathscinetgetitem?mr=1737233) Retrieved from "https://en.wikipedia.org/w/index.php?title=Blasius_boundary_layer&oldid=863178375"

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