Hi All ,

I have a question. Is vorticity a conserved variable in all type of viscous flows ?(compressible/incompressible,constant/variable property flows). We can construct a transport equn for vorticity from NS equations. What is the physical interpretation of conservation of vorticity (if it is conserved ) ?

Thanks

Abhijit Tilak

If you do have viscosity, then vorticity can be dissipated. For example the life time of vortices (coherent structure) depends on the viscosity. However, for many type of flows, the viscosity is pretty small actually and the vortices do not dissipate so quickly. So on the time scale studied, the vorticity might be conserved.

On the other side you can also create vorticity due to a baroclinic instability. The baroclinic instability occurs when the lines of constant density are not parallel to the lines of constant pressure, or when the gradient of the pressure and the gradient of density are not parallel. This happens when you solve for an ideal gas equation of state and the pressure is function of the density and also function of the temperature (and the gas is not isothermal). For barotropic flows (when the pressure is a function of the density only), when the gradient of the pressure and the gradient of density are parallel, the baroclinic instability does not occur.

In real life, it is a combination of both. You can produce vorticity with the baroclinic instability and dissipate it with the viscosity, and it can be such that in the flow it looks like the vorticity is conserved (e.g. the atmosphere of Jupiter).

You can also create vorticity in the flow when you have a transition to turbulence (e.g. when you introduce an 'obstacle' in the flow of river, water table, etc..). In this case you tap energy from the laminar flow (non turbulent one) to create vortices in the turbulent region.

Even when you conserve vorticity globally, it is usually not conserved locally. e.g.: two vortices merge together; in 3D a vortex can stretch, etc...

In simple terms, the conservation of vorticity just means that it is a constant of integration of the system of equations, like for example the energy or the momentum. It is an invariant of the system. In compressible flows one rather defines the potential vorticity (vorticity/density), and one then speaks of the conservation of potential vorticity.

There are certainly other cases, ... any other suggestions/examples?

There was a long discussion on this forum about vorticity creation and dissipation. Here is a summary.

Vorticity are mostly created by 3 mechanisms.

i) No-slip condition at walls ii) Baroclinic torque iii) Oblique shocks

i) and iii) are possible only in viscous flows (although discretized Euler equations have been used to generate oblique shocks, it can be shown that no shocks can exist in inviscid flows). It was also arguedearlier that baroclinic torque also can be generated only when there is viscosity.

In the end, some one pointed out that vorticity can generated only if there are entropy gradients (apparently from the most basic form of Crocco's theorem). The person seem fairly well informed on the subject and since no one disagreed with him, I take it that he is right.

However, it wouldn't hurt to go read the posting in that discussion.

An oblique shock on it's own cannot create vorticity. Vorticity can be created by triple point interactions of shocks, which often occur in shock-boundary layer interactions, or shock interactions with obstacles, or detonations.

I'd be interested in knowing why you think shocks cannot exist in inviscid flow. The 1D rankine-hugoniot releations are inviscid and support shocks, the 1d inviscid burgers equations supports shocks, 30 years of simulations of shock tubes with euler codes have shocks, etc....

I'm guess I'm confused by your statement, because it seems inaccurate.

Dan.

I dont think it is correct to say "rankine-hugoniot releations are inviscid". Whether the mathematical model is inviscid or viscous, if you take the flow states far ahead and far behind the shock, the RH conditions must hold. (I am assuming a normal shock in an unbounded flow.)

>>An oblique shock on it's own cannot create vorticity. Vorticity can be created by triple point interactions of shocks..

He probably meant a curved shock, similar to the shockwave forming over a 3D ramp injector for instance, which does create vorticity due to the non-alignment of the pressure and density gradients.

>> I'd be interested in knowing why you think shocks cannot exist in inviscid flow.

Because a shock is purely a viscous phenomenon, the same way as a shear layer or a boundary layer is a viscous phenomenon. You can choose to _model_ a shock as a discontinuity if you so wish, but that does not take away the fact that in reality, the shock has a thickness. You can solve air flowing supersonically over a wedge using a laminar Navier-Stokes code with the domain length set to 80 microns or so. Then solve it again using an ``Euler'' CFD code. The comparison of the two solutions will convince you, I'm certain, that shocks have a thickness function of the viscosity and thermal conductivity, and are indeed viscous.

>> The 1D rankine-hugoniot releations are inviscid and support shocks, the 1d inviscid burgers equations supports shocks

Only the integral form of the inviscid Burgers equations ``supports'' shocks, as the differential form cannot solve the flow through a discontinuity. In the same vein, it could also be stated that the integral form of the Burgers equations ``supports'' shear layers, and would even give quite a good approximation to a shear layer as long as the thickness of the latter is very small compared to the size of the domain.

>> 30 years of simulations of shock tubes with euler codes have shocks, etc....

Contrarily to the popular belief, an inviscid CFD code does _not_ solve the Euler equations as it always introduces artificial dissipation. Artificial dissipation is a necessary evil needed to ``stabilize'' the solution especially in the vicinity of shocks. Without artificial dissipation, a solution to the flowfield inside a shock couldn't be obtained, and shock-capturing methods would not be possible. The ``inviscid'' solutions obtained with shock-capturing CFD methods are effectively solutions lying in between solutions that would be obtained from the Euler and the Navier-Stokes equations.

Dan,

As Bernard pointed out, I meant curved (bow type) shock when I wrote oblique shock.

Just a little addition to Bernard's comments. Euler equations are fully conservative and reversible in time which implies no entropy generation. With out entropy generation, there can be no shocks. As to why discretized Euler equations can capture shocks (as I said in my original posting), Bernard has a very good explanation.

Ah, I just noticed that I did not mention integral or differential form. In fact, as you have already pointed out for the burgers equation, the integral form of the Euler equations also support shocks, and contact discontinuities. That's why I was partly confused by the comments made by kalyan. Sorry for not being specific enough. Entropy generation in this case doesn't matter, shocks exist for inviscid flows as valid solutions of the integral form of the Euler equations.

Funny though, I feel a bit like I've just been "told". I know that shocks are a viscous phenomenon and that we write shock capturing schemes because for the most part we can't practically use grids small enough to resolve a few mean free paths. To get around it we use smart advection schemes to smear shocks out over a few grid cells and supress dispersive wiggles in the solution.

When we start to use grids that can resolve a few mean free paths I would argue that the continuum model breaks down anyways, so even a Navier-Stokes code would be useless. As a result, there's probably not much point in simulating an 80 micron wedge on a fine enough grid. The answer would be wrong with a viscous or inviscid calculation.

I remember sometime ago doing some 1D simulations of detonations with an Euler code. If I refined the grid enough, eventually you would reach a point where there was not enough artificial dissapation to stabilise the solution (with the chemical source terms of course) and the whole thing would blow up. Interesting how once you move beyond the bounds of the model you are using that things just don't work.

Dan.