[Mike_K]

I am using Cart3D to solve for the flow around streamlined bodies.

As far as I understand Cart3D uses a completely inviscid Euler solver with no automatic viscous corrections. (Perhaps someone here knows more about the solver?)

I would appreciate some insight regarding which drag values are actually calculated by Cart3D (and I would assume Euler solvers in general).

I have experimented with a few body shapes, the surface pressure results are attached. Mach nr is 0.78 in all cases and all bodies have a length of about 37 units (As set in OpenVSP, in which I created the geometry).

The three bodies are:

• Symmetric body of revolution (sharp nose and tail)
• Non-symmetric body of revolution (with rounded nose and sharp tail))
• Primitive airliner fuselage shaped body

The drag coefficients for the three cases, based on a reference are of 122.4 units^2 are:

Symm: 0.0007147
Non-symm: 0.000906
CSR: 0.00108

I am curious as to what physical drag mechanisms these results represent? I know that they don't contain contributions from skin-friction, but shouldn't inviscid flow result in zero drag for any shape, due to D'Lamberts paradox?

Pressure drag contains contributions from the pressure distribution integrated over the surface of a body which should be zero for inviscid flow, yet all these solutions are not nearly zero.

I understand that for thorough understanding I need to read more details about the solver, but I don't think the documentation is freely available.

I would appreciate any insight into the correct drag bookkeeping methodology when working with an inviscid Euler solver.

[sbaffini]

I don't know the specific details of Cart3D but, depending on the specific formulation adopted, an inviscid FV solver might or not give you exact 0 for drag. And besides this, you will still get results within the truncation error of the scheme.

For example, a first case might be a solver that solves for the incompressible potential flow equation directly (say, the same equation you use for steady temperature distributions in isotropic materials). Here you have the inviscid/irrotational feature of the flow embedded in the continuous form of the equations. I have no experience in using this for flows (as panel methods apply under the same circumstances and I use them instead), but I suspect that if you compute such field, than its gradient to obtain the velocity, and finally use Bernoulli to get the pressure, the resulting drag will be close to 0 within the discretization accuracy. Also, there should be no way to obtain lift (your streamlines will close even around sharp edges).

A very different case is that of a solver which actually discretize the full Navier-Stokes equations, and for inviscid cases only some terms are dropped. In this case, even if you are not directly including any viscosity in your equations, that doesn't mean you don't have some numerical viscosity. This, I think, is the specific case of Cart3D (and any other NS solver I can think of), and the numerical viscosity comes from the discretization of the convective terms, because of their upwinding or because it is explicitly added to an otherwise central scheme. Such a solver is, in practice, capable to predict lift to a certain extent because the numerical viscosity will prevent streamlines to close around sharp edges.

Thus, most probably, the drag you are seeing comes from the pressure integral all over the body. The numerical viscosity has a role here in reducing the total pressure reachable at the rear stagnation point, so that the frontal pressure is always higher than the rear one and you get drag.

So, it is not physical, but will help you get "close to physical" results for an aerofoil at, say, an AOA of 5 degrees.

Actually, if this fortuitous mechanism (still, not as much fortuitous as the Kutta condition at TE is for panel methods) was not present, inviscid solvers would have been pretty much useless in most fields (aerodynamics for sure).

Edit: it's interesting how you managed to get the code but not the manual. Also, are you visualizing both a flow section and the surface pressure distribution?

[Mike_K]

Thank you for the explanation Paolo.

So, to make sure I understand: The numerical error introduces a viscous term into the Euler equation. This artificial viscosity allows the flow to separate from the body, causing a pressure imbalance on the fore and aft sections of the body, leading to some drag force.

If I break viscous drag down, I can say it has components from skin-friction drag, boundary layer interactions with the 'inviscid' free stream, and drag due to separation.

Is it fair to say that the drag reported by an Euler solver with artificial viscosity will only give some (probably inaccurate) estimate of the drag due to separation?

PS: The screenshots attached are pressure values on the surface of the body and on a cutplane through the axis of symmetry of the bodies.

Regards
Michael

[rmcdo]

I've done similar experiments with OpenVSP and Cart3D years ago.

In conventional terms, Cart3D can model wave drag and induced drag.

If you have shocks on your body, flow will likely separate behind them -- contributing to drag. Numerical viscosity prevents flow from staying attached to sharp corners (a good thing), so a aft-step or a wing trailing edge will exhibit reasonable separation behavior. This lets an Euler code model lift, induced drag, and blunt base separation drag.

The zero-drag paradox really only applies to potential flow, even without shocks or separation, a compressible flow can have drag.

I once generated a very-gentle body that exhibited no symmetry (a potato) and had no sharp edges or corners. I cranked up the surface resolution and then set Cart3D's adjoint adaptation to refine extensively on a metric of combined forces (I think). I ran the case with an odd value of aoa/sideslip just to be sure. And of course, I set Mach very low.

Once done, the forces were very near zero (but finite moment). When I visualized the result, sure enough, the forward and aft stagnation points were at somewhat unpredictable locations -- they 'find' the position that results in zero force.

Rob

BTW, There is a Cart3D Google Group if you have specialized questions. Your first post will be moderated (it may take a week), but after that you can post on-demand.

http://groups.google.com/group/cart3d

[Mike_K]

Thank you very much Rob.

I have indeed posted a question there, but have not been moderated yet.

Have you perhaps also experimented with the viscousDrag program that was written to estimate viscous drag a priori for Cart3D results?

In the linked paper, it seems to be quite promising.

Regards
Michael


https://www.nas.nasa.gov/publication..._2006-0652.pdf

[sbaffini]

As also stated by Rob, a full Euler code should give you accurate values for all the drag forces except the viscous one. Not really an expert here in using inviscid codes for general predicitons but:

- I would certainly not trust them in cases far from the basic inviscid assumption, not even for the lift
- I would probably never count on them for the drag prediction, probably just on some qualitative behaviour

[FMDenaro]

As I understand from the discussion, the key is on the possibility that numerical solutions of Euler equation can show some behaviour similar to the viscous solution.
I think that:
1) For incompressible (M=0) inviscid (potential) flow we must have for symmetric body and zero angle of attack no drag, no lift, no separation as the theory says. The only way to produce such effects are by fixing a proper Kutta condition. The presence of residuals in such values are garbage due to the numerical viscosity (discretization of the convective terms over non-symmetrical stencil)
2) In the compressible case, the Euler equation can produce a singularity in the solution. But, as Euler equation are a only a mathematical model for the Navier-Stokes equations for vanishing viscosity, this mathematical discontinuity can or cannot be called shock wave in the thermodynamic sense. We must ensure that physically the entropy increases across the discontinuity. Theoretically, yet no flow separation exists on a body for the presence of an inviscid discontinuity. Numerical solution with proper entropy properties produces physically relevant discontinuity. The other effects are garbage from the numerical errors.

[rmcdo]

I don't have any experience with that boundary layer model -- or the one that Aerion has been developing (perhaps they're related). I also believe the Cart3D team is working on a more traditional RANS capability for Cart3D. I am excited about any viscous model tied to Cart3D as I think they have promise to develop very productive tools for practical use.

Mike and Marion produce top notch work -- they are very patient about waiting for work to be 'done' before making it available.

Rob