A mixed passive scalar in a differentially rotating flow field.
L. F. Rossi
Associate Professor
T=0T = 90
The image at left is the initial conditions of a passive scalar quantity.  A passive scalar quantity is a field that is convective by a fluid flow and diffuses as well.  The figure at right is the same field after it has been stirred up a bit by a differentially rotating flow field.  A colloquial example might be stirring cream into your coffee.  If you have a high speed connection, you can download and view a video (84 MB) I have prepared of this calculation. 

As you can guess, some of my interests include the mathematical properties of fluid flows, and novel methods for simulating fluid flows.  There is a natural synergy between the mathematical properties of any process (including fluids) and the computation techniques used to approximate solutions to a process.  In one direction, understanding the properties of equations that govern the motion of fluids can lead to better ways to approximate and simulate fluids on computers.  From the other point of view, careful computational experiments can inspire a deeper understanding of fluid properties.

Going back to the images above, A partial differential equation that describes this process is given below:

equation

The concentration of passive scalar c(x,y,t) is the function that must be calculated given some initial distribution and a flow field u(x,y,t).  The initial conditions are the stripe-like pattern that you can see in the figure above at left.  The domain is unbounded, so the only boundary conditions are that the scalar quantity decays to zero as one moves away from the origin.  In this example, u(x,y,t) is a differentially rotating velocity field.  If you were to drop a marker in the fluid, it would travel in a circle.  However, the period varies with radius, so the passive field gets sheared apart since scalars at nearby radii are moving at different speeds.  The differential rotation which is more pronounced at the limbs enhances mixing and diffusion.

Calculating these solutions challenging because the field of interest is concentrated in a small region of the flow field, and there the gradients are large.  The beauty of particle methods is that they represent the field (c, in this case) as a linear combination of moving basis functions.  Instead of calculating the solution on a fixed grid, the basis functions move and deform with the flow field. Unlike more traditional methods like finite difference or finite element, they are naturally adaptive because computational effort is expended only where c is nontrivial.  Here is a picture of the computational domain for the image at the top right.  The computational domain is the spatial representation of the problem on the computer.


pos T=90pos zoom T=90


The left image is the entire computational domain showing all 4431 particles used in the calculation on the [-1,1]X[-1,1] domain.  Notice that computational effort is only expended where c(x,y,t) is significant and nowhere else.  The computer spends no time computing in the white regions.  At right, you can see the particles in the red inset.  I performed this calculation to test a new method that I have developed which uses deforming basis function.  Each particle is a little deformable elliptical Gaussian function, so each particle is represented as a little cross with the same aspect ratio and orientation as the basis function. 

Is 4431 a lot of particles?  Not really.  Consider that this would correspond to solving this problem on a 67X67 fixed mesh with a finite difference method.  With this mesh, one would not be able to resolve the slender limbs on the stripe as it is twisted.  Unfortunately, these fine length scales are crucial in calculating the decay rate of c because the gradients are greatest there.  Naturally adaptive methods are particularly strong here.  I add that nothing was done to tweak the method for this calculation.  Once the initial data is laid down, particles move and evolve with the fluid flow.

If you have a high speed connection, you can download and view a video (84 MB) I have prepared of this calculation.  This will show you the full evolution of c(x,y,t), along with the computational domain and an inset showing the deformation of the particles.

If you would like to read more about these methods please visit our Technical Reports Series.  Reports 2003_6 and 2003_9 are relevant to this work.  A revised versions of these reports will appear in SIAM Journal on Scientific Computing.

Louis Rossi (rossi@math.udel.edu)