A One-Equation Local Correlation-Based Transition Model

This article is available for free download until 15 Oct 2015, h/t ANSYS blog.

Here's the Abstract:

A model for the prediction of laminar-turbulent transition processes was formulated. It is based on the LCTM (‘Local Correlation-based Transition Modelling’) concept, where experimental correlations are being integrated into standard convection-diffusion transport equations using local variables. The starting point for the model was the γ-Re θ model already widely used in aerodynamics and turbomachinery CFD applications. Some of the deficiencies of the γ-Re θ model, like the lack of Galilean invariance were removed. Furthermore, the Re θ equation was avoided and the correlations for transition onset prediction have been significantly simplified. The model has been calibrated against a wide range of Falkner-Skan flows and has been applied to a variety of test cases.

Keywords: Laminar-turbulent transition, Correlation, Local variables
Authors: Florian R. Menter, Pavel E. Smirnov , Tao Liu, Ravikanth Avancha

Transition location, and subsequent turbulence modeling remain the largest source of uncertainty for most engineering flows. Even for chemically reacting flows the source of uncertainty is often less the parameters and reactions for the chemistry, and more the uncertainty in the fluid state driven by shortcomings in turbulence and transition modeling.

Convergence for Falkner-Skan Solutions

About 6 months ago Dan Hughes sent me a link to an interesting paper on "chaotic" behavior in the trajectory of iterates in a numerical Falkner-Skan solution. It struck me that the novel results reported in that paper were an artifact of the numerical method, and had little to do with any "chaotic" physics that might be going on in boundary layers or other systems that might be well described by this equation. This is similar to the point I made in the Fun with Filip post: the choice of numerical method matters. Do not rush to judgment about problems until you have brought the most appropriate methods to bear.

There are some things about the paper that are not novel, and others that seem to be nonsense. It is well-known that there can be multiple solutions at given parameter values (non-uniqueness) for this equation, see White. There is the odd claim that "the flow starts to create shock waves in the medium [above the critical wedge angle], which is a representation of chaotic behavior in the flow field." Weak solutions (solutions with discontinuities/shocks) and chaotic dynamics are two different things. They use the fact that the method they choose does not converge when two solutions are possible as evidence of chaotic dynamics. Perhaps the iterates really do exhibit chaos, but this is purely an artifact of the method (i.e. there is no physical time in this problem, only the pseudo-time of the iterative scheme). By using a different approach you will get different "dynamics", and with proper choice of method, can get convergence (spectral even!) to any of the multiple solutions depending on what initial condition you give your iterative scheme. They introduce a parameter, \(\eta_{\infty}\), for the finite value of the independent variable at "infinity" (i.e. the domain is truncated). There is nothing wrong with this (actually it's a commonly used approach for this problem), but it is not a good idea to solve for this parameter as well as the shear at the wall in your Newton iteration. A more careful approach of mapping the boundary point "to infinity" as the grid resolution is increased (following one of Boyd's suggested mappings) removes the need to solve for this parameter, and gives spectral convergence for this problem even in the presence of non-uniqueness and the not uncommon vexation of a boundary condition defined at infinity (all of external aerodynamics has this helpful feature).

Setting up a project on Github

I've been an SVN user for a while, but it seems like more and more projects are going distributed version control systems like Git so I wanted to learn how to use Git. I found this crash course on Git for SVN users which provides a useful Rosetta stone, and this warning:

SVN is based on an older version control system called CVS, and its designers followed a simple rule: when in doubt, do like CVS. Git also takes a form of inspiration from CVS, and its designer also followed a simple rule: when in doubt, do exactly the opposite of CVS. This approach lead to many technical innovations, but also lead to a lot of extra headscratching among migrators. You have been warned.

This sounds a lot link Linus Torvalds' talk, WWCVSND: What Would CVS Not Do?

Github has a set of steps for setting up on linux. Git comes in the Fedora repos (and probably every other repo), so install is easy. A nice bit of documentation that comes with the install is Everyday GIT With 20 Commands Or So.

Since I already use password-less ssh to hop between the boxes in my little network, I didn't move my old public key as in the instructions. I created a config file in the .ssh directory containing these lines:
Host github.com
User git
Port 22
Hostname github.com
IdentityFile ~/.ssh/id_rsa_git
TCPKeepAlive yes
IdentitiesOnly yes
Where id_rsa_git.pub is the key I uploaded to github. Authenticating to github is then just:
ssh -T git@github.com
Then you accept their RSA key like you would for doing any other ssh login.

The next thing is to create a repo. Clicking through the instructions brings you to a page with several "next steps". Which for my example are:
mkdir FalknerSkan
cd FalknerSkan
git init
touch README
git add README
git commit -m 'first commit'
git remote add origin git@github.com:jstults/FalknerSkan.git
git push -u origin master
Which gives some output ending in something like: Branch master set up to track remote branch master from origin.

If you'd like, you can read up on the Falkner-Skan ODE at the viscous aero course on MIT's OCW:

• Local Scaling. Falkner-Skan Flows.
• Solution of Falkner-Skan Equation by Finite Differences