Boundary layer equations

Formation of a laminar boundary layer on a flat surface (bottom line)

The boundary layer equations appear in boundary layer theory as simplifications of the Navier-Stokes equations . For a two-dimensional steady flow with constant density they read: ${\displaystyle \rho }$

${\displaystyle {\begin{aligned}u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}&=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}+\nu {\frac {\partial ^{2}u}{\partial y^{2}}}\\{\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}&=0\\0&={\frac {\partial p}{\partial y}}\end{aligned}}}$

With

• ${\displaystyle x}$: Coordinate in the direction of flow
• ${\displaystyle y}$: Coordinate perpendicular to the direction of flow (wall distance)
• ${\displaystyle u={\frac {dx}{dt}}}$: Velocity component in the direction of flow ( in the figure) ${\displaystyle v_{x}}$
• ${\displaystyle v={\frac {dy}{dt}}}$: Velocity component perpendicular to the direction of flow
• ${\displaystyle p}$the pressure
• ${\displaystyle \nu }$the kinematic viscosity
• ${\displaystyle {\frac {\partial }{\partial }}}$the partial derivative .

The second equation is the continuity equation for incompressible flows.

The third equation says that the pressure does not change over the height under consideration; H. the pressure on the body surface corresponds to the pressure in the frictionless external flow.

Pressure gradient and external flow

In the outer flow , the Euler equation applies : ${\displaystyle \left({\tfrac {\partial u}{\partial y}}=0\right)}$

${\displaystyle \Rightarrow u_{\delta }{\frac {\partial u_{\delta }}{\partial x}}=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}}$

With

• ${\displaystyle u_{\delta }}$: Velocity of the external flow.

It says: the pressure gradient, i. H. is the course of the pressure in the direction of flow

• negative with accelerated flow ${\displaystyle \left({\frac {\partial p}{\partial x}}<0\quad \Rightarrow \quad {\frac {\partial u_{\delta }}{\partial x}}>0\right)}$
• Zero with plate flow
• positive with delayed flow.

Initial and boundary conditions

The following initial and boundary conditions are required to calculate the speed distribution :

${\displaystyle {\begin{aligned}u(x,y=0)&=0\\v(x,y=0)&=0\\u(x,y\rightarrow \infty )&=u_{\infty }(x)\end{aligned}}}$

The first two equations describe the adhesion condition on the body surface, the third condition is the speed of the external flow ( is the thickness of the boundary layer). ${\displaystyle \delta }$

The following equation can be derived from the adhesion condition:

${\displaystyle \Rightarrow \eta \left({\frac {\partial ^{2}u}{\partial y^{2}}}\right)_{y=0}={\frac {\mathrm {d} \,p(x)}{\mathrm {d} \,x}},}$

which relates the curvature of the velocity profile on the wall to the pressure gradient imposed by the external flow ( is the dynamic viscosity ). ${\displaystyle \eta =\rho \cdot \nu }$

A boundary layer detachment can only occur with a delayed external flow, i.e. H. with a positive pressure gradient. The boundary layer becomes detached from the body contour when the wall shear stress disappears:

${\displaystyle \tau _{\text{wand}}=\eta \left({\frac {\partial u}{\partial y}}\right)_{y=0}=0.}$

solution

In contrast to the elliptical Navier-Stokes equations, the boundary layer equations form a parabolic system of equations. As a result, there is no upstream flow of information, so that a numerical solution with an upstream method is possible.

An analytical solution of the boundary layer equations is only possible in a few special cases. The simplest solution is the boundary layer flow along an infinitely thin, flat plate (Blasius solution). In this case, the solutions are similar at different points along the plate and can be converted into one another by suitable scaling of the coordinate normal to the wall. This gives an expression for the boundary layer thickness:

${\displaystyle {\frac {\delta (x)}{x}}={\frac {5}{\sqrt {\mathrm {Re} _{x}}}},}$

with the Reynolds number ${\displaystyle \mathrm {Re} _{x}={\frac {u_{\infty }\cdot x}{\nu }}.}$

The thickness of the boundary layer is defined as the thickness at which the speed has reached 99% of the speed of the free external flow: ${\displaystyle \delta (x)}$

${\displaystyle u_{\delta }=u(y=\delta ):=0{,}99\cdot u_{\infty }\approx u_{\infty }}$

In addition to this definition of the boundary layer thickness, the displacement thickness or the pulse loss thickness is often used as a physically more sensible measure . ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$

swell

• Hermann Schlichting (et al.): Boundary Layer Theory. Springer, Berlin 2006, ISBN 3-540-23004-1

literature

• Spurk, Aksel: Fluid Mechanics, Springer, Berlin 2007, ISBN 978-3-540-38439-7