(1). In early days, when people started using CFD to solve 2-D transonic flows (inviscid), they were surprised that inviscid solution agreed better with the test data of pressure distribution including shock wave than viscous solutions. (2). But eventually, when the mesh refinement was carried out later with more powerful computer, it was realized that the pressure distribution across the shock of inviscid solution became less and less like the test data.(high pressure gradients were observed at the shock location, while the test data of the pressure distribution exibited smooth pressure rise.) (3). With nearly 30 years in solving Naavier-Stokes equations, my feeling is: from inviscid equation to Navier_Stokes equation, there is an invisible barrier, similar to the Mach one for earlier aircraft. (4). In the inviscid world, everything is nice and clean. On the other hand, you get all kind of troubles in the viscous Navier-Stokes world. Without the proper training and experience, it is not easy to move from the inviscid world to the more realistic viscous world. (5). All I can say is: inviscid equation and world is not real. It is a simplified model. But sometimes, by clever superposition of inviscid models, one can display the results "similar" to that exists in the viscous world.

>> (1) do you regard the weak solution (to the integral equation) to be somehow inferior to the the differential solution?

No. However, I would say that the integral form of the Euler equations can lead to some misunderstandings. While the properties after the shock can be predicted by the integral Euler equations, a solution to the shock cannot be obtained. The integral equations model the shock as a discontinuity, but do not solve it, the same way as a turbulence model approximates turbulent eddies on a grand scale through the use of an eddie viscosity without solving the eddies.

From a physical standpoint, a shock is not a discontinuity and has a thickness, and the entropy gain through the shock is due to the viscous effects. A solution that does not show this behaviour can only be approximate, and entails assumptions and some amount of ``modelling''.

>> (2) if I were to predict a curved shock using the method of characteristics in 2 would I still not see the production of vorticity?

I'd say you can predict where the shock will lie using the MOC, but you cannot obtain a solution to the shock itself.

John,

I would say that the viscous systems are much better understood than the inviscid systems. Viscous systems provide mechanisms like Kutta condition and separation naturally (as Adrin pointed out). Viscous systems also do not admit discontinuous solutions like the hyperbolic inviscid equations. As Adrin pointed out, it it very hard to obtain purely inviscid solutions except for those problems where an analytical solution is known.

Inviscid continuum systems often tend to have infinite dimensions. I am not sure what the mathematical requirements for this are but ergodicity (on which almost all theories, models on turbulence are based) is a sufficient condition. Dissipation does not make the system finite-dimensional, but you can capture the behavior of the system using finite dimensions (which is what closure theories and DNS rely on). Hence I would say viscous systems are more tractable.

An open question : Is the second law of thermodynamics the "invisible barrier" John is refering to.

(1). It is exactly like what you have just said, that you think viscous flow is easier to handle. (2). The person on the inviscid world would say that inviscid flow is easier to handle. (3). This is the reason why even for a well-known professor, it is difficult to switch over from one side to the other. (4). If you are on the inviscid world, you don't need: (a). boundary layer mesh stretching, which is not easy to handle for 3-D problem, (b). boundary layer theory which include both the laminar, transitional and turbulent flows. (c). turbulence flows and turbulence modelling. This can easily take several years to study, not to mention the wall function black art, the low Reynolds number models, etc. (5). It will take five to ten years for a person specialized in the inviscid world to switch over to the viscous world. To get rid of the old habit and to get hands on experience. (6). This is the reason why a PhD is a minimum requirement in doing CFD research or applications. (well if you don't care about the accuracy of the results, then I guess, nothing is required beyond one week training to use the code). (7). For example, if you need the accurate loss prediction to improve your blade design in the turbomachinery field, then the currently available commercial codes are all useless, because it is not possible to predict the loss accurately. (you can still use these codes though. but then it depends on who is using it and for what) (8). If a professor spent a lot of time doing inviscid flow research, then it will take ten years for him to switch over to viscous world if he works very hard to learn the items I just mentioned above. (you can't say you are now an expert by just taking these courses alone) (9). Yes, you will have to generate enough entropy before you can become an expert again. (inviscid theory used to predict no drag for flow over a body, at the begining)