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In many cases the flow about an airfoil is calculated by solving the Laplace equation, (for example in the Hess-Smith panel method). If the velocity field is irrotational and its divergence its zero, that field will satisfy the Laplace equation. My question is: how do you tell when these two conditions (zero curl and zero divergence) are still a good approximation of the actual flow in the reality?

If the flow is incompressible, its divergence will be zero, right?. But in what conditions you can treat air as a incompressible fluid?

And how can I tell if the flow can be treated as irrotational? I thought that this was related to viscosity, but I've read that in some cases there can be irrotational flows with viscous fluids and also rotational flows with inviscid fluids. And the Kutta condition, that is fundamental in the Hess-Smith method, is based on the fact that the fluid has viscosity.

And I've also read that it's important, to apply the Hess-Smith method, that the airspeed is subsonic everywhere. What are the reasons of this condition?

I'd be happy to receive explanations or also useful links. Thank you.

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1 Answer 1

You have lots of questions here, so I'll try to address them all but I may miss/gloss over some.

A fluid that is considered incompressible is divergence-free. This comes from the conservation of mass equation:

$$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_i}{\partial x_i} = 0$$

where if we say the density does not change, the time derivative is zero and the $\rho$ can be pulled out of the spatial derivative and divided from both sides, leaving:

$$ \frac{\partial u_i}{\partial x_i} = 0 $$

which is the divergence free condition.

Under which conditions can this be assumed? The typical rule of thumb is that any flow where the Mach number is less than 0.3 can be assumed incompressible. This typically corresponds to a change in density due to compression of less than 5%.

Of course, potential flow can exist in compressible cases as well so divergence free is not a requirement for potential flow.

Potential flow is defined as irrotational flow. This means that there is no vorticity in the flow, or the curl of the velocity is zero.

Under which conditions does that occur? Well, flow without turbulence (so no separation from the body) and flow outside the boundary layer (which is why we need the slip boundary condition on the body) are the two most common examples of potential flow.

You are correct that inviscid flows can have rotation, and there are books on viscous potential flow as well. The criterion for being potential is just the irrotationality, or vorticity free, condition.

As for the Kutta-condition... this is a mathematical requirement to make up for the fact that when we drop viscosity from the equations, we lose a boundary condition. For more information on that, and for more information on the whole "lift needs viscosity but inviscid potential flow has lift" issue, see my answer to Does a wing in a potential flow have lift?

Conclusion

If you use a panel method solver like XFoil, you will get warnings if your setup has a velocity that is too high or will cause separation of the flow from the body. The important point about speed is keeping the Mach number under 0.3 at all points on the body. A very thick airfoil may reach higher velocities at the thickest point even for small freestream velocities.

Also, a thick airfoil or an airfoil at a large angle of attack will cause separation, violating the irrotational assumption. Well-established panel method codes will warn you in both of these cases; if writing your own, you must ensure that you don't just blindly take an answer you get without making sure you are in a reasonable range.

Knowing the range takes some experience. For divergence-free, it's relatively easy. If you get answers from your solver that put the velocity near Mach 0.3 or greater, you need to be careful. For irrotational flow, you will get some experience and intuition about how thick is too thick, or how large an angle of attack is too much. Until then, you can look for similar shapes in canonical texts such as Theory of Wing Sections which has $C_L$, $C_D$ and $C_M$ data for almost all of the known NACA airfoils. If you look at the $C_L$ plots for a shape of similar thickness, you can see the angle of attack where the curve is no longer linear and this is the point that separation occurs. You can't be near that point if you want your panel method to work.

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Very good answer. –  OSE Jul 22 '13 at 15:11

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