Editing Boundary layer (section)

==Boundary layer equations==

The deduction of the '''boundary layer equations''' was one of the most important advances in fluid dynamics. Using an [[order of magnitude analysis]], the well-known governing [[Navier–Stokes equations]] of [[viscous]] [[fluid]] flow can be greatly simplified within the boundary layer. Notably, the [[Characteristic polynomial#Characteristic equation|characteristic]] of the [[Partial differential equation|partial differential equations (PDE)]] becomes parabolic, rather than the elliptical form of the full Navier–Stokes equations. This greatly simplifies the solution of the equations. By making the boundary layer approximation, the flow is divided into an inviscid portion (which is easy to solve by a number of methods) and the boundary layer, which is governed by an easier to solve [[Partial differential equation|PDE]]. The continuity and Navier–Stokes equations for a two-dimensional steady [[incompressible flow]] in [[Cartesian coordinates]] are given by

:<math> {\partial u\over\partial x}+{\partial \upsilon\over\partial y}=0 </math>

:<math> u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}=-{1\over \rho} {\partial p \over \partial x}+{\nu}\left({\partial^2 u\over \partial x^2}+{\partial^2 u\over \partial y^2}\right) </math>

:<math> u{\partial \upsilon \over \partial x}+\upsilon{\partial \upsilon \over \partial y}=-{1\over \rho} {\partial p \over \partial y}+{\nu}\left({\partial^2 \upsilon\over \partial x^2}+{\partial^2 \upsilon\over \partial y^2}\right) </math>

where <math>u</math> and <math>\upsilon</math> are the velocity components, <math>\rho</math> is the density, <math>p</math> is the pressure, and <math>\nu</math> is the [[kinematic viscosity]] of the fluid at a point.

The approximation states that, for a sufficiently high [[Reynolds number]] the flow over a surface can be divided into an outer region of inviscid flow unaffected by viscosity (the majority of the flow), and a region close to the surface where viscosity is important (the boundary layer). Let <math>u</math> and <math>\upsilon</math> be streamwise and transverse (wall normal) velocities respectively inside the boundary layer.  Using [[scale analysis (mathematics)|scale analysis]], it can be shown that the above equations of motion reduce within the boundary layer to become

:<math> u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}=-{1\over \rho} {\partial p \over \partial x}+{\nu}{\partial^2 u\over \partial y^2} </math>

:<math> {1\over \rho} {\partial p \over \partial y}=0 </math>

and if the fluid is incompressible (as liquids are under standard conditions):

:<math> {\partial u\over\partial x}+{\partial \upsilon\over\partial y}=0 </math>

The order of magnitude analysis assumes the streamwise length scale significantly larger than the transverse length scale inside the boundary layer.  It follows that variations in properties in the streamwise direction are generally much lower than those in the wall normal direction. Apply this to the continuity equation shows that <math>\upsilon</math>, the wall normal velocity, is small compared with <math>u</math> the streamwise velocity.

Since the static pressure <math>p</math> is independent of <math>y</math>, then pressure at the edge of the boundary layer is the pressure throughout the boundary layer at a given streamwise position. The external pressure may be obtained through an application of [[Bernoulli's equation]].  Let <math> U </math> be the fluid velocity outside the boundary layer, where <math> u </math> and <math> U </math> are both parallel. This gives upon substituting for <math>p</math> the following result

:<math> u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}=U\frac{dU}{dx}+{\nu}{\partial^2 u\over \partial y^2} </math>

For a flow in which the static pressure <math> p </math> also does not change in the direction of the flow

:<math> \frac{dp}{dx}=0 </math>

so <math> U </math> remains constant.

Therefore, the equation of motion simplifies to become

:<math> u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}={\nu}{\partial^2 u\over \partial y^2} </math>

These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis is for any instantaneous [[laminar flow|laminar]] or [[turbulent]] boundary layer, but is used mainly in laminar flow studies since the [[mean]] flow is also the instantaneous flow because there are no velocity fluctuations present. This simplified equation is a parabolic PDE and can be solved using a similarity solution often referred to as the [[Blasius boundary layer]].

===Prandtl's transposition theorem===
[[Ludwig Prandtl|Prandtl]] observed that from any solution <math>u(x,y,t),\ v(x,y,t)</math> which satisfies the boundary layer equations, further solution <math>u^*(x,y,t),\ v^*(x,y,t) </math>, which is also satisfying the boundary layer equations, can be constructed by writing<ref>{{Cite journal|doi = 10.1002/zamm.19380180111|title = Zur Berechnung der Grenzschichten|year = 1938|last1 = Prandtl|first1 = L.|journal = Zeitschrift für Angewandte Mathematik und Mechanik|volume = 18|issue = 1|pages = 77–82|bibcode = 1938ZaMM...18...77P}}</ref>

:<math>u^*(x,y,t) = u(x,y+f(x),t), \quad v^*(x,y,t) = v(x,y+f(x),t) - f'(x) u(x,y+f(x),t)</math>

where <math>f(x)</math> is arbitrary. Since the solution is not unique from mathematical perspective,<ref>Van Dyke, Milton. Perturbation methods in fluid mechanics. Parabolic Press, Incorporated, 1975.</ref> to the solution can be added any one of an infinite set of eigenfunctions as shown by [[Keith Stewartson|Stewartson]]<ref>{{Cite journal|doi=10.1002/sapm1957361173|title=On Asymptotic Expansions in the Theory of Boundary Layers|year=1957|last1=Stewartson|first1=K.|journal=Journal of Mathematics and Physics|volume=36|issue=1–4|pages=173–191}}</ref> and [[Paul A. Libby]].<ref>{{Cite journal|doi=10.1017/S0022112063001439|title=Some perturbation solutions in laminar boundary-layer theory|year=1963|last1=Libby|first1=Paul A.|last2=Fox|first2=Herbert|journal=Journal of Fluid Mechanics|volume=17|issue=3|page=433|doi-broken-date=14 January 2025 |s2cid=123824364 }}</ref><ref>{{Cite journal|doi=10.1017/S0022112064000830|title=Some perturbation solutions in laminar boundary layer theory Part 2. The energy equation|year=1964|last1=Fox|first1=Herbert|last2=Libby|first2=Paul A.|journal=Journal of Fluid Mechanics|volume=19|issue=3|pages=433–451|bibcode=1964JFM....19..433F|s2cid=120911442 }}</ref>

===Von Kármán momentum integral===
[[Theodore von Kármán|Von Kármán]] derived the integral equation by integrating the boundary layer equation across the boundary layer in 1921.<ref>{{Cite journal|doi = 10.1002/zamm.19210010401|title = Über laminare und turbulente Reibung|year = 1921|last1 = von Kármán|first1 = T.|journal = Zeitschrift für Angewandte Mathematik und Mechanik|volume = 1|issue = 4|pages = 233–252|bibcode = 1921ZaMM....1..233K|url = https://zenodo.org/record/1447403}}</ref> The equation is

:<math>\frac{\tau_w}{\rho U^2 } = \frac{1}{U^2}\frac{\partial }{\partial t}(U\delta_1) + \frac{\partial \delta_2}{\partial x} +\frac{2\delta_2+\delta_1}{U} \frac{\partial U}{\partial x} + \frac{v_w}{U} </math>

where

:<math>\tau_w = \mu \left( \frac{\partial u}{\partial y}\right)_{y=0}, \quad v_w = v(x,0,t), \quad \delta_1 = \int_0^\infty \left(1- \frac{u}{U} \right) \, dy, \quad \delta_2 = \int_0^\infty \frac{u}{U} \left(1- \frac{u}{U}\right) \, dy</math>

:<math>\tau_w</math> is the wall shear stress, <math>v_w</math> is the suction/injection velocity at the wall, <math>\delta_1</math> is the displacement thickness and <math>\delta_2</math> is the momentum thickness. [[Kármán–Pohlhausen Approximation]] is derived from this equation.

===Energy integral===
The energy integral was derived by [[Karl Wieghardt (physicist)|Wieghardt]].<ref>Wieghardt, K. On an energy equation for the calculation of laminar boundary layers. Joint Intelligence Objectives Agency, 1946.</ref><ref>{{Cite journal|doi = 10.1007/BF00548007|title = Über einen Energiesatz zur Berechnung laminarer Grenzschichten|year = 1948|last1 = Wieghardt|first1 = K.|journal = Ingenieur-Archiv|volume = 16|issue = 3–4|pages = 231–242| bibcode=1948AAM....16..231W |s2cid = 119750449}}</ref>

:<math>\frac{2 \varepsilon}{\rho U^3 } = \frac{1}{U}\frac{\partial }{\partial t}(\delta_1 + \delta_2) + \frac{2 \delta_2}{U^2}\frac{\partial U}{\partial t}  +\frac{1}{U^3} \frac{\partial }{\partial x}(U^3\delta_3) + \frac{v_w}{U} </math>

where

:<math>\varepsilon = \int_0^\infty \mu \left( \frac{\partial u}{\partial y}\right)^2 dy, \quad \delta_3 = \int_0^\infty \frac{u}{U}\left(1- \frac{u^2}{U^2}\right) \, dy</math>

:<math>\varepsilon</math> is the energy dissipation rate due to viscosity across the boundary layer and <math>\delta_3</math> is the energy thickness.<ref>Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.</ref>

===Von Mises transformation===
For steady two-dimensional boundary layers, [[Richard von Mises|von Mises]]<ref>{{Cite book|doi=10.1007/978-3-662-11836-8_49|chapter=Bemerkungen zur Hydrodynamik|title=Ludwig Prandtl Gesammelte Abhandlungen|year=1961|last1=Tollmien|first1=Walter|last2=Schlichting|first2=Hermann|last3=Görtler|first3=Henry|last4=Riegels|first4=F. W.|pages=627–631|isbn=978-3-662-11837-5}}</ref> introduced a transformation which takes <math>x</math> and <math>\psi</math>([[stream function]]) as independent variables instead of <math>x</math> and <math>y</math> and uses a dependent variable <math>\chi = U^2-u^2</math> instead of <math>u</math>. The boundary layer equation then become

:<math>\frac{\partial \chi}{\partial x} = \nu \sqrt{U^2-\chi} \, \frac{\partial^2 \chi}{\partial \psi^2}</math>

The original variables are recovered from

:<math>y = \int \sqrt{U^2-\chi} \, d\psi, \quad u = \sqrt{U^2-\chi}, \quad v = u\int \frac{\partial}{\partial x} \left(\frac{1}{u}\right) \, d\psi.</math>

This transformation is later extended to compressible boundary layer by [[Theodore von Kármán|von Kármán]] and [[Qian Xuesen|HS Tsien]].<ref>{{Cite journal|doi=10.2514/8.591|title=Boundary Layer in Compressible Fluids|year=1938|last1=von Kármán|first1=T.|last2=Tsien|first2=H. S.|journal=Journal of the Aeronautical Sciences|volume=5|issue=6|pages=227–232}}</ref>

===Crocco's transformation===
For steady two-dimensional compressible boundary layer, [[Luigi Crocco]]<ref>Crocco, L. "A characteristic transformation of the equations of the boundary layer in gases." ARC 4582 (1939): 1940.</ref> introduced a transformation which takes <math>x</math> and <math>u</math> as independent variables instead of <math>x</math> and <math>y</math> and uses a dependent variable <math>\tau=\mu\partial u/\partial y</math>(shear stress) instead of <math>u</math>. The boundary layer equation then becomes

: <math>
\begin{align}
& \mu \rho u \frac{\partial}{\partial x}\left(\frac{1}{\tau}\right)  + \frac{\partial^2 \tau}{\partial u^2} -\mu \frac{dp}{dx} \frac{\partial }{\partial u}\left(\frac{1}{\tau}\right) =0, \\[5pt]
& \text{if } \frac{dp}{dx}=0, \text{ then } \frac{\mu\rho}{\tau^2} \frac{\partial \tau}{\partial x} = \frac{1}{u}\frac{\partial^2 \tau}{\partial u^2}.
\end{align}
</math>

The original coordinate is recovered from

:<math> y = \mu \int \frac{du} \tau .</math>