Can anybody recommend some good references for applying artificial viscosity to an unstructured, finite-volume mesh?

Structured or Unstructured,if you use an explicit artificial viscosity then it would usually be a term of the form epsilon*d^4u/dx^4. So all you need to do is discretize the fouth order derivative using your unstructured discretization.There have been other forms of artificial viscosity which have been applied too of which I do not know much.

Yes, I understand the form of the actual artificial viscosity terms. The trouble I'm having is understanding how to properly apply them--how to calculate the derivatives on a non-cartesian mesh. As the faces of the cells can lie at any angle to the major axes, the dU/dX and dP/dX terms aren't very readily calculated.

Any ideas? Have you read any good references?

There is a book by lohner. It might give you some ideas on implementing the derivatives.

Remember a title?

Applied CFD Techniques: An Introduction Based on Finite Element Methods is the name.

Why don't you let your mesh create the diffusion term properly, then you'll not need to add 'artificial viscosity' at all.

The trick lies in the size of your mesh element - the smaller you go, the more the discrete groups bias towards the diffusion. Of course, in the limit dx->0 , you tend to the correct pde - with dx>>0, you will be convection-dominated based on large mesh size.

desA

It depends on the kind of discretization being used.Central difference schemes have no inbuilt diffusion and the diffusion of the viscous terms can sometimes be insufficient.Also smaller elements restrict your timestepping.Hence you need to evaluate the relevant parameters and choose the one which can help in the solution of the problem more efficiently.

By adding any other terms to your numeric scheme you are violating the rules of the physical phenomena you are trying to study. What answer do you then believe?

Take the Navier-Stokes, Euler - what you will - apply the original definition for a partial-derivative in terms of a forward difference & limit, take away the limit that dx->0 & then study the terms in this 'original' discrete form. This will tell you everything you need to know about dx, dy, dz & their effect on the relative weighting of each group in the original equation. Don't get drawn into the CD vs other debate too early - it tends to cloud the real issues.

desA

Blending CD-UD?

If I were using a flux-split scheme, where flux summations across individual cell faces are applied based on the true propagation of influences--the characteristic directions--I would have no need for artificial viscosity at all. And I've done that in other applications. But this application already exists. It uses a flux summation around cells that essentially decomposes to a central-difference. Thus the need for artificial viscosity terms.

The application is time-critical. I have to get a good, time-accurate approximation of the mass, momentum and energy transport through an enclosed volume, and it has to be calculated fast. So I am VERY limited in the number of cells that can be used. I must use only the minumum number of cells possible to maintain stability and capture the secondary flow features.

I'm already applying artificial viscosity to the system and getting acceptable results. But my technique has its limits, and I'd like to come up with something more robust. So I'm interested in seeing what others have done in this area.

Exactly right. See my post responding to desA, 8 March, 11:00 am.

I'm sorry, I don't understand what you're asking. See my post of 8 March, 11:00 am for more detail.

I mean that you can try to blend a central scheme with an upwind scheme. The level of blend is the level of error of the upwind scheme you will introduce in the solution given by the central scheme. As the upwind scheme tends to be dissipative, you can tune the level of artificial viscosity (which is already proportional to delta_l, your cell-to-cell distance).

Ah, I see. Yes, that's a possibility. The region I'm modelling is typically subsonic, but does have shocks passing through it. It's also largely enclosed by walls, which makes numerical stability a more sensitive issue than in external flows. I'll have to think about your suggestion a bit.

There was a paper at this year's AIAA Aerospace Sciences Meeting that applied a similar philosophy to a fully flux-split approach using second-order-accurate flux summations across all cell faces. But that's major surgery at this point, so I'm looking to see if I can first improve upon the artificial viscosity scheme that's in use already.

Artifical vicsosity methods are so darn old. Why not try any of WENO, CENO, NT schemes? They are pretty good for Euler equations, structured or non-structured grids, all works.

Rich wrote:

If I were using a flux-split scheme, where flux summations across individual cell faces are applied based on the true propagation of influences--the characteristic directions--I would have no need for artificial viscosity at all.

desA's reply:

The characteristic directions for the Euler equations will show up pretty well, but with diffusion present, these directions are suppressed in their action - in many cases. The correct balance between convection & diffusion terms must be maintained if you are to correctly capture the secondary (wave) effects.

If you are forced to use large elements, then you need to be able to capture your physics correctly within your computational cell/element. You may want to look into schemes which could allow use of exponential & sin/cos terms - I've seen something similar in FEM - can't remember offhand their exact element interpolation function (eif), but it was a good idea.

To capture the first 4 terms of a Taylor series of a cos function, you'll need something like a P6 (28 node) element using a conventional polynomial-type eif. This is where a smart element design is useful.

I hope this helps.

desA

Upwinding does its best to suppress some of the 'secondary effects' as you refer to them. I find these 'blurring' schemes to be somewhat like a magical incantation on the way to an incorrect representation of the physics. I see no use at all for stabilisation schemes that no-one can truly understand. Better to understand the underlying physics & to tune your simulation accordingly - then, no stabilisation is necessary.

desA

Unfortunately that cannot be the case all the time. Sometimes the size of your grid is restricted by the memory you have and the time limit. Also sometimes on complex geometries you cannot capture all the physics without resorting to some form of approximations.Why do people do LES when DNS is much more accurate ? In the field of aeroacoustics if you do not filter out the waves due to one sided approximation of boundary conditions or grid non uniformity , you might not be able to get good numerical results for the computations.

Harish wrote:

Unfortunately that cannot be the case all the time...

In the field of aeroacoustics if you do not filter out the waves due to one sided approximation of boundary conditions or grid non uniformity , you might not be able to get good numerical results for the computations.

desA wrote:

How are we, as scientists, able to judge whether these waves are important, or not, if we try to filter them out all the time? Why not try & develop techniques to better capture the bulk + wave mechanisms properly, without trying to tell nature what to do?

Take a good look at Van Dyke's flow gallery - if we cannot see wave activity at almost each & every level of flow activity, then we are modeling the wrong physics. Sometimes this activity is low energy, sometime high-energy. We need to get the simulations to accurately reflect the full situation - not continue to try & exclude intrinsic physics.

Set your solution timing up appropriate to the equations you are using - the 'sweet spot' is different for Euler, creeping & Navier-Stokes. When we stop trying to trick the physics, we will see the root cause of the phenomena we observe.

Hint: Just remember, (1/dt) is the inverse of (dt). What do we use in our discrete simulation space? Is the inverse of a smooth function, another smooth function? Try a line, parabola etc, for instance. For Euler equations, the system is pretty straightforward in (dt) - (dx,i) relationship.

Have fun

desA