Applicability of potential flow theory (fluid dynamics)

[Chuckstabler]

To start this question off, I'll just state what I do understand about potential flow theory

1.) I understand that the navier stokes equations are really difficult to solve analytically except in degenerate cases where the assumptions about the flow cancel out the non-linear terms.

2.) I understand that potential flow theory makes three simplifying assumptions to deal with the problems of finding exact solutions to the navier stokes equations. The first assumption is that the fluid has no viscosity, and hence has no term that acts to diffuse momentum. The second assumption is that the fluid is not compressible, which is to say that the density throughout the fluid is constant. The third assumption is that the fluid is irrotational, which is to say that the curl of the velocity field of the fluid is zero everywhere. Because the curl of the velocity field is zero everywhere, we can express the velocity field as being the gradient of a scalar potential function (owing to the fact that the curl of the gradient of a scalar is zero).

3.) I understand that the continuity equation for an incompressible fluid when the velocity field is the gradient of a scalar potential function reduces to the homogeneous laplace equation. This is because the continuity equation for an in-compressible fluid demands that the divergence of the velocity vector field be zero everywhere. Replace the velocity vector with the gradient of a scalar potential and we find that continuity demands that the divergence of the gradient of the scalar potential be zero. The divergence of the gradient of a scalar is just the laplacian operator. So the continuity equation simply demands that the laplacian operator acting on the scalar potential return 0 everywhere. This is a linear second order PDE in the scalar potential.

4.) I think I understand that the momentum equations for such a potential flow end up reducing to an equation for the pressure, such that the kinematics/momentum balance for the flow is only needed to determine the pressure distribution afterwards. In this way the pressure and velocity are decoupled entirely, and the only thing that the scalar potential needs to do is satisfy laplaces equation.

5.) Because of the linearity of laplaces equation, any two potential functions that solve laplaces equation on their own can be linearly combined to form a potential function that also satisfies laplaces equation. In other words if F(x,y,z) solves laplaces equation and G(x,y,z) solves laplaces equation then A*F(x,y,z) + B*G(x,y,z) also solves laplaces equation. In this way more complex flows can be constructed by adding together elementary potential functions. This, I believe, is the idea behind potential flow around a cylinder (if I remember correctly it's a linear superposition of a doublet flow and a flow with a constant velocity in some direction). Now ; here's my actual question. In what domains is potential flow theory a good approximation to more realistic flows with non zero viscosity and non zero curl? Let's say for example that we run a time dependent CFD simulation of a two dimensional flow around a disc, with a constant free stream velocity and the boundary condition that the velocity must be tangental to the surface of the disc at the surface of the disc, with zero viscosity. In what regions will the potential flow solution be applicable? I've run a few simulations with very very small viscosity myself, with a free stream velocity going in the positive x direction (flow moving from negative x to positive x), and I've found that what typically happens is that the flow matches potential flow theory quite well when when x is negative (in the area leading up the disc), but that when we get to the area behind the disc vortexes form and shed off the back of the disc, breaking the assumption of zero curl.

So what is the mechanism responsible for forcing the velocity field to obey potential flow theory quite well in the area before the flow goes around the disc, but completely diverge from potential flow theory in the area behind the disc (the wake)?

 

[boneh3ad]

The inviscid assumption is what is breaking down in your example. As soon as the flow encounters that disc (or any object), a boundary layer will form at its surface. The boundary layer is a thin region of flow near a surface in which viscosity is important to the flow behavior, and potential flow does a poor job in modeling flows in the boundary layer region. Eventually, the vorticity generated by a boundary layer (or by boundary-layer separation) can continue on into the wake.

So when is potential flow theory useful? Any time you aren't in a boundary layer or a wake, and sometimes even in those situations it can still be helpful. For example, you can solve the flow field using potential theory and then use the resulting pressure field as inputs to a boundary-layer solver, which are more complicated than potential flow solvers but considerably simpler than solving the Navier-Stokes equations directly. Then you can adjust the effective size of the object, re-solve the potential flow problem, and then re-solve the boundary layer based on this new solution and get a more accurate answer. You can iterate on that until the solutions stop changing and you would then have a fairly accurate solution to the problem. This still doesn't account for a wake with shed vortices, but it can handle flows where the wake remains smooth just fine and is subject mainly to the accuracy of the boundary-layer solver.

This is typically how the initial stops of airfoil design are performed (with several other assumptions used to mathematically enforce effects that would not otherwise arise without viscosity). For more information on that, look up panel methods.

 

[Chuckstabler]

Thanks for your response. That makes sense. You also say that several other assumptions are used to mathematically enforce effects that would not otherwise arise without viscosity. Is the kutta condition such an assumption?

 

[boneh3ad]

Right. Viscosity would ordinarily dictate that there is a stagnation point at the trailing edge (assuming no earlier separation). Potential flow won't handle that due to its inviscid nature, so enforcing the Kutta condition essentially forces the potential flow solution to abide by that particular consequence of the real world being viscous.