Boundary layer equations for incompressible flow

[Felixbro]

Hi.
I am new here so I please let me know if I should post this in another forum. I have been struggling for a while with the following homework problem:

"State the boundary layer equations for incompressible flow over a solid, weakly curved boundary of a Newtonian fluid. What approximations are done compared to the full Navier-Stokes equations and under what conditions are these approximations valid".

I assume I should compare the BL equations with the NS equations for incompressible flow. As I understand it, the boundary layer equations are only valid if the Re-number is sufficiently high (Re>>1). In that case it can be assumed that the viscous effects of the flow are bound only to the boundary layer region whereas outside the boundary layer (the majority of the flow) the flow can be considered inviscous (I think this means that potential flow theory is applied there). This leads to boundary conditions consisting of the no-slip condition at the wall (BL region) and the condition that the velocity gradient drops as we encounter the free stream (u -> U_inf = constant).

The first simplification from the regular NS equations for incompressible flow would be that we are only looking at a twodimensional problem (I assume I don't have to motivate this because I have never seen the BL equations in three dimensions). Thus, we can divide the momentum eq. into the directions x (flow direction). and y (wall normal direction). Then as I understand, we should look at the equation for x- and y-momentum separately. Somehow, by letting Re ->infinity in both equations, the pressure term in the NS equations is simplified to only consider the pressure in the free stream when I look at derivations of the BL equations. I think Re is defined for velocity in x-direction. If this is true, Re-> infinity for an incompressible fluid means that only the velocity in x-direction is increasing and thus we are encountering the free stream where U = constant <-> pressure is constant <-> pressure term in the BL equations can be obtained from potential flow theory (i.e. Bernoulli's equation). I don't know if this is a correct explanation though.

I would really appreciate a good answer on this. I find it very difficult to know if I am covering everything to this question, and whether my thoughts on it so far are correct or not.

 

[KataKoniK]

Hi there,

First of all, welcome to the forum! It's great to see you taking an interest in fluid dynamics and the boundary layer equations. You are correct in your understanding that the boundary layer equations are only valid for high Reynolds numbers (Re >> 1). This is because at high Reynolds numbers, the viscous effects of the flow are confined to a thin layer near the solid boundary, while the majority of the flow can be considered inviscid. This allows us to simplify the Navier-Stokes equations, which describe the motion of a viscous fluid, into the boundary layer equations, which are specific to the boundary layer region.

You are also correct in your understanding that the boundary layer equations are derived by considering the x- and y-momentum equations separately. This is because the flow near a solid boundary is typically two-dimensional, meaning that the flow variables (such as velocity and pressure) only vary in two directions. This allows us to simplify the equations and focus on the important phenomenon happening at the boundary layer.

In terms of the pressure term in the boundary layer equations, it is not exactly correct to say that it is simplified to only consider the pressure in the free stream. The pressure term in the boundary layer equations actually represents the difference between the pressure at the wall and the pressure in the free stream. This is due to the no-slip condition at the wall, which states that the velocity of the fluid at the wall is equal to the velocity of the wall itself. This creates a pressure gradient in the boundary layer, which is represented by the pressure term in the boundary layer equations.

In regards to the Reynolds number, it is defined based on the characteristic length and velocity of the flow. For incompressible flow, it is typically defined using the velocity in the flow direction, as you mentioned. However, it is important to note that the Reynolds number is not a constant value for a given flow. It can vary throughout the flow, and it is only in the limit of Re >> 1 that we can make the simplifications necessary to derive the boundary layer equations.

I hope this helps clarify some of your questions and provides a good answer to your homework problem. Good luck with your studies and feel free to ask any further questions you may have.