Blasius boundary layer

A Blasius boundary layer, in physics and fluid mechanics, describes the steady two-dimensional boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow U.

Within the boundary layer the usual balance between viscosity and convective inertia is struck, resulting in the scaling argument

: frac{U^{2{L}approx ufrac{U}{delta^{2,

where delta is the boundary-layer thickness and u is the kinematic viscosity. However the semi-infinite plate has no natural length scale L and so the steady, two-dimensional boundary-layer equations

: {partial uoverpartial x}+{partial voverpartial y}=0

: u{partial u over partial x}+v{partial u over partial y}={ u}{partial^2 uover 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{ u x}{U} ight)^{1/2}.

This suggests adopting the similarity variable

: eta=frac{y}{delta(x)}=yleft( frac{U}{ u x} ight)^{1/2}and writing

:u=U f '(eta). It proves convenient to work with the stream function psi , in which case

: psi=( u 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' ightarrow 1 as eta ightarrow infty. This non-linear ODE must be solved numerically, with the shooting method proving an effective choice.The shear stress on the plate

: sigma_{xy} = frac{f" (0) ho U^{2}sqrt{ u{sqrt{Ux.

can then be computed. The numerical solution gives f" (0) approx 0.332.

Falkner-Skan boundary layer

A generalisation of the Blasius boundary layer that considers outer flows of the form U=cx^{m} results in a boundary-layer equation of the form: u{partial u over partial x}+v{partial u over partial y}=c^{2}m x^{2m-1}+{ u}{partial^2 uover partial y^2}.Under these circumstances the appropriate similarity variable becomes

eta=frac{y}{delta(x)}=frac{sqrt{c}y}{sqrt{ u}x^{(1-m)/2,

and, as in the Blasius boundary layer, it is convenient to use a stream function

psi=U(x)delta(x)f(eta) = c x^m delta(x)f(eta)

This results in the Falkner-Skan equation

f"'+frac{1}{2}(m+1)f f" - m f'^{2} + m =0

(note that m=0 produces the Blasius equation).

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 Flussigkeiten mit kleiner Reibung", Z. Math. Phys. vol 56, pp. 1-37. http://naca.larc.nasa.gov/reports/1950/naca-tm-1256 (English translation)


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Boundary layer — For the anatomical structure, see Boundary layer of uterus. Boundary layer visualization, showing transition from laminar to turbulent condition In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of… …   Wikipedia

  • Boundary-layer thickness — This page describes some parameters used to measure the properties of boundary layers. Consider a stationary body with a turbulent flow moving around it, like the semi infinite flat plate with air flowing over the top of the plate. At the solid… …   Wikipedia

  • Paul Richard Heinrich Blasius — Infobox Scientist name = Paul Richard Heinrich Blasius image width = caption = birth date = August 9 1883 birth place = Berlin, Germany death date = April 24 1970 death place = Hamburg, Germany residence = citizenship = German nationality =… …   Wikipedia

  • Orr-Sommerfeld-Gleichung — Dieser Artikel befasst sich mit dem Anwachsen kleiner Störungen in einem Strömungsfeld. Für weitere Bedeutungen des Begriffs Lineare Stabilitätstheorie siehe Stabilitätstheorie. Die Lineare Stabilitätstheorie (kurz LST) beschreibt in einer… …   Deutsch Wikipedia

  • Squire-Gleichung — Dieser Artikel befasst sich mit dem Anwachsen kleiner Störungen in einem Strömungsfeld. Für weitere Bedeutungen des Begriffs Lineare Stabilitätstheorie siehe Stabilitätstheorie. Die Lineare Stabilitätstheorie (kurz LST) beschreibt in einer… …   Deutsch Wikipedia

  • Lineare Stabilitätstheorie — Die Lineare Stabilitätstheorie (kurz LST) beschreibt physikalisch in einer Strömung das Anwachsen wellenförmiger Störungen mit kleiner Amplitude. Durch Vorgabe eines stationären Strömungsfeldes lassen sich Anfachungsraten und Form der linearen… …   Deutsch Wikipedia

  • Momentum thickness — In aeronautics and viscous fluid theory, the boundary layer thickness ({delta}) is the distance from a fixed boundary wall where zero flow is considered to occur, and beyond {delta} the fluid is considered to move at a constant velocity. This… …   Wikipedia

  • Direkte Numerische Simulation — Unter Direkter Numerischer Simulation, kurz DNS, versteht man die rechnerische Lösung der vollständigen instationären Navier Stokes Gleichungen. Sie unterscheidet sich von anderen Berechnungsmethoden der Strömungsmechanik dadurch, dass… …   Deutsch Wikipedia

  • Ludwig Prandtl — Infobox Scientist name = Ludwig Prandtl |200px caption = Ludwig Prandtl birth date = birth date|1875|2|4|df=y birth place = Freising, Germany death date = death date and age|1953|8|15|1875|2|4|df=y death place = Göttingen, Germany residence =… …   Wikipedia

  • HIEMENZ — GERMANY (see also List of Individuals) 19.1.1885 Worms/D 2.6.1973 Hamburg/D Karl Hiemenz entered Darmstadt University in 1903 where he won already after two semesters a mathematical competition. After having graduated in mathematics and physics… …   Hydraulicians in Europe 1800-2000


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.