Greetings:

I wish to initiate a discussion of how variable viscosity is implmented in available commercial codes, or in the available research codes distributed by government agencies.

In my limited experience, most codes provide for streamlining and economy by having a generic form of the advection-diffusion equation +- additional physics for situations like closure of turbulence equations,enthalpy-temperature, etc. Some also allow for the user to input some form of variable viscosity as a function of variation of some scalar, e.g. temperature or concentration or both.

In my experience this is simply applied to the trasnfer coefficient in front of the diffusive flux term. However, it is a simple matter to show that the governing equations for momentum do not reduce to the usual advection-diffusion form if one starts out with variable viscosity- additional non-linear terms are required that would not be there if one simply had say, a variable thermal diffusivity instead. So the generality of invoking a single form of advection-diffusion equation is lost. Yet I have not yet seen any commercial codes that explicitly addresses this (in my limited experience) with the appropriate discretazation of the momentum equation.

I hand coded the additional terms into a PHOENICS run and found little difference between including them or not, but of course that will be very problem dependent. A recent variable viscosity "benchmark" study run by an engineeering society (I can dig the reference up upon request) did not even mention the issue and few of the governing equations provided for in these studies was correct. This was curious and startling indeed, and a caution about end-users assuming that available codes are really robust in this regard.

George Bergantz Geological Sciences Univ. of Washington

(1). I can only say that turbulence modeling is a 30 year old field, and a lot of it remain to be solved. (2). It is a huge field, and one way to make the results useful is to go through a systematic validation process, to make sure that the model is really useful for the type of problems under study. (3). To make the model useful and practical, it has to be simplified to certain extent. (4). In turbulent flow, the eddy viscosity is always non-constant. So, could you phrase your question more clearly, related to your definition of "variable viscosity" ?

I made no reference to turbulence modeling with regard to variable viscosity because I did not intend that as an application or point of discusion.

I think that you may misunderstand the intent of my message. I completely agree that progress is made incrementally. However code validation and verification are quite different notions, as succinctly stated in the recent monograph by Roache. Solving the wrong equations, which is what I am referring to here, even in laminar flow, is a concern for those who expect "off the shelf" products to be robust in this regard.

I will restate my point, and in the context of laminar flow (e.g. shear viscosity) which high-lights the issues: one cannot treat variable viscosity in the same way as a variable thermo-chemical transport coefficient, such as thermal diffusivity (or conductivity, or species diffusivity) by simply taking the divergence. Yet this seems to be the way in which most codes are structured by default. Perhaps others have had experience with these issues and will care to share their experience.

The applications are not trivial: many chemical engineering and geophysical processes take place at Reynolds numbers of O(10)-O(.01) which may tend to be non-steady and choatic (Lagrangian chaos of Ottino), but not formally turbulent. These applications typically involve, or are driven by thermal-chemical interactions and may involve strong dependence of shear viscosity on some scalar introducing another nonlinear coupling. This is old hat, but what doesn't seem to be widely appreciated is that these couplinsg are not implemented in a rigorous way.

I welcome further comment,

May I seek some clarification. I presume you are not referring to the difference between (a) the constant viscosity form of the momentum transport equation (second derivatives only of solved velocity component in Cartesian coordinates) and (b) the variable viscosity form (additional cross terms involving other velocity components arising from div(mu(symm(grad(u)))). But a difference between (b) and a more physically correct form which arises by dropping stress is proportional to rate of strain?

From my point of view, you have two consider two fields of research:

-mathematics -engineering

For the mathematicians, the goal is two solve as closely as possible the equations modeling the flow. If they have to deal with variable viscosity. They have to retain the additional cross-terms.

For most engineers who have to deal with real industrial problems, the physics is so complicated that they're happy if they can get a solution that is in good agreement with reality. Since most commercial codes are design for engineering, they can make these simplifications. These simplication won't affect the results for most cases. 

Look for the usefulness in a CFD simulation, an innacurate solution can still be a useful one!!

Andy- I'll give you a detailed response later today, but the idea here is that if the dynamic viscosity is an explicit function of say temperature, than any derivatives of that must invoke the chain rule. This will be true regardless of ones constitutive relationships. This then gives rise to additional derivatives. Try it starting with a Newtonian assumption but temperature dependent, and see what you get.

Others:

I agree that less than a "full" solution can still be a very useful one; this is implicit in CFD. My point again is that if one has no notion as to what terms are missing, or that they are indeed missing, then how can one assess the limitations of their model? The Bouss. assumption is a fine example of a robust and widely used and understood assumption.

Useful engineering and science work is done with many simplifications. But one must understand where they arise and when it is useful to make such simplifications. Including the "extra" terms for variable viscosity is not that big a deal and could be routinely added, so it is not simply a matter of convenience on the part of the vendor. I am not referring to "solving the equations right" but "solving the right equations." This is not a matter of math versus engineering.

Another example: the modeling of multiphase flow, whether Eulerian/Eulerian or Lagrangian/Eulerian is full of ad hoc assumptions. But useful work is done because (hopefully) most practitioners understand where the holes are and can defend any decisions in that regard. No problem there.

So, if you are an enduser interested in variable viscosity, my point is simply that you had better look closely at the model equations in light of your design problem, somemthing that has not been discussed previously.

eom

My maths is rusty and my algebra was always abysmal but I get the same answer when expanding using the chain rule or substituting for temperature and differentiating directly. Have I missed the point?

(1). Even in the case of laminar flows, the viscosity is a function of temperature as in the air. (2). So, the viscosity is not treated as a constant. (3). Mathematically, It can be grouped together or expanded out as individual terms. So, it does not matter which form is used in the formulation. (4). If one grouped it together under the divergence form, then, it is up to the discretization process to determine the proper selection of the viscosity value and the location, as long as it is consistent with the original partial differential equations. (5). I normally expand the equation into individual terms, so, I can deal with the effect of viscosity variation explicitly. (6). But, I think, it is up to the person who formualtes the problem to decide whether the viscosity should be explicitly expanded out or grouped under the divergence operation.( it wasn't an issue for me in the past. so, I guess this is my understanding of your question) (7). So, all I can say is "variable viscosity" in cfd code is always treated as non-constant variable. And How this is implemented depends on the algorithm used in the code. For the coarse mesh used, one can see the difference in solution. But if the solution is mesh independent, then, the piece-wise constant treatment inside the cell will not have a major impact on the solution. (8). Once again, I would say that if one is not sure of the cfd solution obtained, then it is important to carry out systematic validation to see whether the error is related to the algorithm used. (9). I have no idea about the commercial cfd code treatment of the viscosity or the algorithms used. But, in the research field, the treatment of temperature dependent viscosity was never a problem. It has been handled properly. (more explicit statement of the question or example would help the readers to understand your question, if my answer is still not quite right.)

I want to start by apologizing if my comments have seemed shrill.

Its hard to write out math in text but here goes...

Consider a cartesian coordinate system, and the usual Newtonian form of the relationship between stress & strain from most advanced texts, for example Convection Heat Transfer by Bejan, chapter 1, equation (17). If we assume that the density is constant ala Bouss., and if the viscosity is temperature dependent say, one gets the usual terms in the standard advection-diffusion equation (as commonly implemented in vendor & reserach codes) that look like (I know these should be partials) where U is the X velocity component, V the Y velocity component, T temperature and mu is viscosity:

d/dx(mu*dU/dx) + d/dy(mu*dU/dy)

but if temp dependent viscosity the additional terms must be also present:

(dmu/dT)*(dT/dx)*(dU/dx) + (dmu/dT)*(dT/dy)*(dV/dx)

Note that now temperature and V velocity appear explicitly in U equation. If the density is not constant (non-Bouss.) then even more terms pop-up. These second set of terms are not present in the consideration of variable diffusivity or conductivity, which holds the same place in the standard template of the advection-diffusion equation.

One can show by invoking the good old similarity solution for boundary layer flow that these terms scale as:

delta(mu)*delta(U)/(delta(X))**2

where 'delta' is some characteristic length scale. It seems that they will only be important vis-a-vis the standard terms if the change in viscosity is of the same order of magnitude as the viscosity itself.

Whew!

I thank you all for your interest. Convection in the earth's mantle, flow of lava, volcanic eruptions, mixing of viscous materials in batch mixers all face these issues. Lastly, I can fax (in US only) the actual derivation to anyone who is seriously interested.

I respectfully disagree with some of the comments made. I will try to be clear- it is not simply a matter of grouping with regard to the standard form. For example that will not yield the extra physical terms I am speaking of. My point is that one doesn't even start with the standard form regardless of meshing. If one has access to this reference:

"Buoyancy Induced Flows and Transport" by Gebhart et al., go to page 31, equation (2.7.11). One can see that the usual "divergence form" is not simply recovered and an additional dimensionless group will emerge. Other references make the same point.

Of course I do agree that one can write equations in a compact form, the use of the D/Dt notation in the substantial dervative is a good example. And one can find that the stesses can also be written in such a compact way. The initial point of my posting was simply to share my experience that in fact, these terms are rarely included in work I've reviewed, research or otherwise.

Regarding the claim that this has not been an issue in research codes- that may depend on one's field. I have been to many talks involving variable viscosity through the years both at ASME and the geophysical community and rarely have the extra term been added.

I will try to bring closure to this discussion by simply pointing out that (as I said in first posting) available software, as advertised on this web page for example, or commonly used in industrial design, or distributed by government agencies, may not have these extra terms present.

George,

If you look at the first equation in your mail :

d/dx(mu*dU/dx) + d/dy(mu*dU/dy)

This can be written as div(F) where F = (mu*dU/dx,mu*dudy). This can then be solved using a finite volume technique. where this term can be written as a sun of F.n dS on all the faces of a cell.

THe only fact that needs to be remembered is that the mu has to be calculated on the face. Many people do it many ways, they either average the mu's from the cells sharing the face or average the temperature and then compute a mu from this average temperature. Other even do higher order estimates for the temperature on the face.

If the calculation is done this way (as it is usually done in finite volume codes) then the ambiquity you mention does not exist. However, I am not sure how finite difference codes handle this.

Hope this helps Srinivasan

1) All the above is correct. Except for on point, your forgot the cross terms : for incompressible flow, it should have been

d/dx(mu*(dU/dx+dU/dx)) + d/dy(mu*(dU/dy+dV/dx)

As you said, when dealing with FVM, mu is evaluated at the interface (at that point I suggest using an harmonic mean) and the flux for U at the interface is given by

F=( (mu*(dU/dx+dU/dx)) , (mu*(dU/dy+dV/dx) )

2) When the flow is incompressible and mu is constant ( does not depend on T nor space) F simplifies to:

F=mu( (dU/dx) , (dU/dy) )

Thank you Sebastien and Srinivasan

In my coding I deal with these issues, but the advice is welcome and useful. In terms of Sebastien's point my initial comments were simply that it is the "cross terms" that usually seem overlooked. As you both point out it is not a difficult thing to do it right, but that the cross terms make it different from treating it exactly as one would treat say, variable thermal diffusivity. I think this now seems obvious to all of us, but my limited experience with vendor codes suggests that the cross terms are not present because of the convenience of using a single template for all advection-diffusion equations.

For a reference that explicitly address this see the nice book by Gebhart et al., "Buoyancy Induced Flows and Transport" and on page 31 look at equation (2.7.11) where the "extra" or cross terms have been retained.

Your comments have been terrific!

A related issue in CFD of strongly variable viscosity, and typically high Prandtl number, systems is that of convergence of the FVM. It is no surprise that in our experience relaxation of the viscosity is required to achieve convergence. Adding to the problems is that many natural systems where variable viscosity is important are also ones where natural convection provides the potential energy for flow.

This creates a situation where multiple couplings require a lot of trial and error to get convergence, especially if one has double-diffusive convection where there will be a three-way coupling. Interested readers are directed to:

Oldenburg & Spera, 1991, "Numerical modeling of solidification and convection in a viscous pure binary eutectic system", Int. J. Heat Mass Transfer, v. 34, p. 2107-2121.

Ogawa et al., 1991, "Numerical simulations of three-dimensional thermal convection with strongly temperature-dependent viscosity", J. Fluid Mech., v. 233, p. 299-328.

Oldenburg & Spera, 1992, "Hybrid model for solidification", Numer. Heat Transfer, v. B21, p. 217-229.

What others have dealt with this issue? I have probably over-looked key references.