Boundary layer equations for incompressible flow

[Felixbro]

Hello everyone. I posted this question in another forum and got no answers so I'll try and re-post it question here.
I need to deliver a correct answer to the cited question below to my course responsible teacher. Below is also my own solution and thoughts on the problem but I don't know if I am getting it right. Help would be very much appreciated and please let me know if I'm should go to another forum.

"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.

 

[Asim Riaz]

The boundary layer equations for incompressible flow over a solid, weakly curved boundary of a Newtonian fluid are derived from the Navier-Stokes equations. The main difference between the two sets of equations is that in order for the boundary layer equations to be valid, the Reynolds number must be sufficiently high (Re>>1), so that the viscous effects of the flow are bound only to the boundary layer region. In other words, outside of the boundary layer, the flow can be considered inviscous and potential flow theory can be applied.The boundary layer equations consist of two components: the x-momentum equation and the y-momentum equation. In both of these equations, the pressure terms are simplified to only consider the pressure in the free stream when the Reynolds number is taken to infinity. This means that the velocity in the x-direction is increasing, and thus we are encountering the free stream where U = constant, pressure is constant, and the pressure term in the BL equations can be obtained from potential flow theory (i.e. Bernoulli's equation).The boundary layer equations are valid under the conditions that the flow is incompressible, the boundary layer is weakly curved, and the Reynolds number is sufficiently high (Re>>1).