Boundary Layer Theory
Although the steady boundary-layer equations \(\boxed{\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,\quad
u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}
=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu\,\frac{\partial^2 u}{\partial y^2},\quad
0=-\frac{\partial p}{\partial y}}\) do represent a significant simplification of the full equations, they are still nonlinear second-order partial differential equations that can only be solved when appropriate boundary and matching conditions are specified
If the exterior flow is presumed to be known and irrotational (and the fluid density is constant), the pressure gradient at the edge of the boundary layer can be found by differentiating the steady constant-density Bernoulli equation (without the body force term)
\(\boxed{p+\tfrac{1}{2}\rho U_e^2=\text{const.}}\) to find
\[
-\,\frac{1}{\rho}\,\frac{dp}{dx}
=
\underbrace{U_e(x)}_{\substack{\text{velocity at the edge} \\ \text{of the boundary layer}}}\,
\frac{dU_e}{dx}
\]
It represents a matching condition between the outer ideal-flow solution and the inner boundary-layer solution in the region where both solutions must be valid
The (usual) remaining boundary conditions on the fluid velocities of the inner solution are
- $$
u(x,0)=v(x,0)=0 \quad \text{(no slip and no through flow at the wall)}
$$
- $$
u(x, y\to\infty)=U_e(x) \quad \text{(matching of inner and outer solutions)}
$$
- $$
u(x_0,y)=u_{\text{in}}(y) \quad \text{(inlet boundary condition at } x_0\text{)}
$$
- \(\boxed{u(x, y\to\infty)=U_e(x) \quad \text{(matching of inner and outer solutions)}
}\) merely means that the boundary layer must join smoothly with the outer flow; for the inner solution, points outside the boundary layer are represented by $y\to\infty$, although we mean this strictly in terms of the dimensionless distance $y/\bar\delta=(y/L)\mathrm{Re}^{1/2}\to\infty$
- \(\boxed{u(x_0,y)=u_{\text{in}}(y) \quad \text{(inlet boundary condition at } x_0\text{)}}\) implies that an initial velocity profile $u_{\text{in}}(y)$ at
some location $x_0$ is required for solving the problem. Such a condition is needed because the terms $u\,\partial u/\partial x$ and $v\,\partial^2 u/\partial y^2$ give the boundary-layer equations a parabolic character, with $x$ playing the role of a time-like variable
The complete Navier–Stokes equations require boundary conditions on the velocity (or its derivative normal to the boundary) upstream, downstream, and on the top and bottom boundaries, that is, all around. (The upstream influence of the downstream boundary condition is a common concern in CFD)
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