It has been shown in the literature that Euler can predict separation inspite of the fact that it is a viscous phenomenon. I have two queries... 1. How does Euler captures separation 2. Can any body suggest simple testcase to verify the same.

(1). Please do not get this into your head. It was just a joke. (2). I had mentioned this before in this forum, about an engineer who claimed that he was able to compute the flow separation of a flow over a cylinder, using an Euler code from a government lab, more than 15 years ago. (3). The engineer was a yound PhD and had done some inviscid flow using panel code. (4). Anyway, most Euler codes and Navier-Stokes codes have explicit artificial damping terms added or the implicit artificial viscosity effect. (remember that upwind scheme will create artificial viscosity effect?)(5). In other words, when you solve the Euler equation numerically, you are actually solving the viscous equation. And in most cases, it is highly viscous. The boundary conditions used for the Euler equation are slip wall conditions. (6). But all you need to do to capture the viscous solution or flow separation is to use the non-slip wall boundary conditions. (7). Some schemes will have the viscous effect all the time, but some other schemes will have the effect only during iterations or transient conditions. But most of the time, they are there. (8). Unless you are using exact method to solve the Euler equation, most of the time, you are actually solving the viscous equation. (9). This alone tells you that benchmark testing and validation is important. And the viscous solution and flow separation are definitely non-real and are wrong solutions.

Physically, the neccessary conditions for separation from a wall are: (1) surface friction, and (2) adverse pressure gradient. Generally, the both conditions should be exist simultaneously for separation. If the no-slip condition is exactly satisfied on the wall in your computation, it is impossible that the flow is separated naturally unless the flow is enforced by geometric singularity, injection and so on. On the other hand, even without these conditions, the inviscid flow can generate vorticity. If you have a look at the Corroc's vorticity equation, you will find that the vorticity generation occurs when the entropy is producted. In these cases, vortices may be generated due to entropy increasement rather than wall separation.

Zhong

Thanks John and Lei for such prompt response... John if we talk about specific scheme like "Jameson's" which has artificial viscosity in it then can we predict separation ( obviously with fine grid and high fourth order artificial viscosity)

I have certain papers which show that flow on sharp edged delta wings can be predicted by euler method. May be because here point of separation is uniquely defined at leading edges.

My second query still remains unanswered ..

Can we think of a simple 2D/3D test case in which separation occurs and people have solved it with Euler method (Jameson scheme)

(1).I have already given you the test case of the flow over a cylinder. (2). I think, the code used by the engineer in my example from a government lab probably have used the similar method.(such paper will be rejected by technical journals, the example was an in-house case only) (3). Sharp corners and shocks are very special cases due to boundary conditions.(discontinuities in boundary conditions)

Hi,

Euler equations can have discontinuous solutions, which can often be interpreted as separated flows. Importantly, such inviscid flows are not unique. The non-uniqueness is due to two distinctly different reasons. First, the separation point can be prescribed arbitrary to a certain extent. Second, in steady case inside a region of closed streamlines the vorticity, which is constant along streamlines, can be arbitrary distributed across streamlines. (It can be determined from the boundary conditions at infinity but not on closed streamlines.)

Now, in reality, the unique solution is determined due to the action of viscosity. When Euler equations are solved numerically, the unique solution is determined by purely numerical effects, in a sense, by the structure of the approximation error. Therefore, such solutions are of little value.

However, there is an important class of flows when Euler separated solutions are of interest even if obtained numerically. Namely, there should be no closed streamlines, or, more generally, in non-steady case, there should be no regions where vorticity is not swept downstream. Second, if, in reality separation occurs at sharp edges and numerical method is such that in calculations separation occurs also at sharp edges. Many methods ensure this automatically just due to the structure of approximation error. Indeed, otherwise there would be an infinite velocity at a sharp edge resulting in the growth of the approximation error.. In that case, since the Euler solution becomes unique, it can model realistic high-Re flow. Flow past a delta-wing at small incidence is a good example, and it can be the test case you are asking for.

Rgds, Sergei.

There was a discussion about it in some articles in AIAA J. concerning Euler based simulation of flows about airfoil. It was shown that the numerical solution converges to the physically correct result without Cutta condition if the grid used is fine enough. I have not these articles at hand now.

With best regards

Mr. Zhong Lei got the point.

There are two kinds of flow 'seperations': the first one is a viscous phenomenon(such as those due to flow instability in boundary layers) which can never be captured by an inviscid solver. The second kind is an inviscid phenomenon caused by the Corroc effect. The Euler solvers can accuratly predict the second kind seperation (actually, it is not real seperation, it is simply rotational flow with vortexs).

A good yet simple example is flow past a 2-D cylinder. You may use 2 free flow conditions: (1) Free Mach number 0.2. In that case, the flow is fully subsonic, inviscid, and irrotational. If you catch 'seperation' at the back side of the cylinder, your code is a joke. (2) Increase the free Mach number to 0.5 or higher. In this case, there will be a shock wave on top of the cylinder. The shock wave will generate entrop gradient, the flow after the shock becomes inviscid, yet rotational. You should be able to catch a pair of vortexs at the back side of the cylinder. If you can not get the vortexs, your code is another joke.

(1). Regardless of how one look at it, I can only say that flow separation from a smooth surface is result of viscous effect. There is no way one can predict such flow using inviscid Euler equation. (2). The slip flow due to two shock interaction, can be computed by Euler equation and inviscid method. (3). The rotational inviscid flow behind the curved transonic shock, can also be predicted by the inviscid equation. Those are shock related discontinuities. (4). Sip flow, inviscid rotational flow, inviscid jet boundary, etc... are discontinuities. (5). The confusion arises because sometimes it is possible to use inviscid model to "simulate" the global behavior of a viscous rotational flow. (only the global effect but not the real physics)

> result without Cutta condition if the grid used is fine enough. I have not these articles at hand now.

Fine enough but not _too_ fine!

I'm implying, in total agreement with John Chien, that you can't get flow separation with a correctly/accurately implmented Euler solver. If there is no Kutta condition, then there is an infinity at the singularity point. This means, if the solution is "fine enough" to introduce just enough numerical diffusion into the solution, then one can get a "converged" solution. However, with infinite accuracy and extremely fine grids, the solution will blow up, not converge, _unless_ something is done about the singularity; i.e., some viscous model is introduced there.

BTW, this debate of viscous vs. non-viscous vorticity generation has been going on for a long-time now, and some quite famous researchers have argued in favor of the inviscid case (sadly so)

Adrin Gharakhani