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 of the
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
PDE. The continuity and Navier–Stokes equations for a two-dimensional steady
incompressible flow in
Cartesian coordinates are given by : {\partial u\over\partial x}+{\partial \upsilon\over\partial y}=0 : 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) : 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) where u and \upsilon are the velocity components, \rho is the density, p is the pressure, and \nu 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 u and \upsilon be streamwise and transverse (wall normal) velocities respectively inside the boundary layer. Using
scale analysis, it can be shown that the above equations of motion reduce within the boundary layer to become : 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} : {1\over \rho} {\partial p \over \partial y}=0 and if the fluid is incompressible (as liquids are under standard conditions): : {\partial u\over\partial x}+{\partial \upsilon\over\partial y}=0 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 \upsilon, the wall normal velocity, is small compared with u the streamwise velocity. Since the static pressure p is independent of y, 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 U be the fluid velocity outside the boundary layer, where u and U are both parallel. This gives upon substituting for p the following result : u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}=U\frac{dU}{dx}+{\nu}{\partial^2 u\over \partial y^2} For a flow in which the static pressure p also does not change in the direction of the flow : \frac{dp}{dx}=0 so U remains constant. Therefore, the equation of motion simplifies to become : u{\partial u \over \partial x}+\upsilon{\partial u \over \partial y}={\nu}{\partial^2 u\over \partial y^2} These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis is for any instantaneous
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 Prandtl observed that from any solution u(x,y,t),\ v(x,y,t) which satisfies the boundary layer equations, further solution u^*(x,y,t),\ v^*(x,y,t) , which is also satisfying the boundary layer equations, can be constructed by writing :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) where f(x) is arbitrary. Since the solution is not unique from mathematical perspective, to the solution can be added any one of an infinite set of eigenfunctions as shown by
Stewartson and
Paul A. Libby.
Von Kármán momentum integral Von Kármán derived the
integral equation by integrating the boundary layer equation across the boundary layer in 1921. The equation is :\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} where :\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 :\tau_w is the wall shear stress, v_w is the suction/injection velocity at the wall, \delta_1 is the displacement thickness and \delta_2 is the momentum thickness.
Kármán–Pohlhausen Approximation is derived from this equation.
Energy integral The energy integral was derived by
Wieghardt. :\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} where :\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 :\varepsilon is the energy dissipation rate due to viscosity across the boundary layer and \delta_3 is the energy thickness.
Von Mises transformation For steady two-dimensional boundary layers,
von Mises introduced a transformation which takes x and \psi(
stream function) as independent variables instead of x and y and uses a dependent variable \chi = U^2-u^2 instead of u. The boundary layer equation then become :\frac{\partial \chi}{\partial x} = \nu \sqrt{U^2-\chi} \, \frac{\partial^2 \chi}{\partial \psi^2} The original variables are recovered from :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. This transformation is later extended to compressible boundary layer by
von Kármán and
HS Tsien.
Crocco's transformation For steady two-dimensional compressible boundary layer,
Luigi Crocco introduced a transformation which takes x and u as independent variables instead of x and y and uses a dependent variable \tau=\mu\partial u/\partial y(shear stress) instead of u. The boundary layer equation then becomes : \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} The original coordinate is recovered from : y = \mu \int \frac{du} \tau . ==Turbulent boundary layers==