Far from the boundary, in the free stream, it is valid to use the Bernoulli equation. That's what boundary layer theory is all about.Arjan82 said:
The Prandtl boundary layer equation is a simplified form of the Navier-Stokes equations that describe the flow of fluid near a solid boundary. It specifically addresses the behavior of the flow in the thin region called the boundary layer where viscous forces are significant. For a two-dimensional steady laminar flow, the equation can be written as:
\[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} \]
where \( u \) and \( v \) are the velocity components in the \( x \) and \( y \) directions respectively, and \( \nu \) is the kinematic viscosity of the fluid.
Several key assumptions are made in deriving the Prandtl boundary layer equation:1. The flow is steady and laminar.2. The flow is two-dimensional.3. The boundary layer is thin compared to the characteristic length scale of the problem.4. The pressure gradient in the direction normal to the boundary is negligible.5. The velocity component normal to the boundary is much smaller than the velocity component parallel to the boundary.
The boundary layer thickness \(\delta(x)\) on a flat plate increases with the distance \( x \) from the leading edge of the plate. For a laminar boundary layer, the thickness can be estimated using the Blasius solution, which gives \(\delta(x) \approx 5 \sqrt{\frac{\nu x}{U_\infty}}\), where \( U_\infty \) is the free-stream velocity and \( \nu \) is the kinematic viscosity.
The Reynolds number \( Re \) is a dimensionless quantity that characterizes the relative importance of inertial forces to viscous forces in a fluid flow. In boundary layer theory, it is defined as \( Re = \frac{U_\infty x}{\nu} \). A low Reynolds number indicates laminar flow, while a high Reynolds number indicates turbulent flow. The transition from laminar to turbulent flow in the boundary layer typically occurs at a critical Reynolds number around \( Re \approx 5 \times 10^5 \).
The Prandtl boundary layer equation can be solved using analytical methods for simple cases, such as the Blasius solution for a flat plate. For more complex geometries or flow conditions, numerical
