Cats2D Multiphysics > Research Topics > Vector plot verdict: bad but not terrible

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## Vector plot verdict: bad but not terrible

Here is a vector plot that illustrates the evolution of a flow profile at cross sections along a channel. The Reynolds number is 0, 200, and 400 from top to bottom. Cats2D makes it very easy to generate these plots. I think these plots do an excellent job illustrating the rearrangement of a plug flow as it enters a channel, and how adding inertia alters and slows this rearrangement.

This style of vector plot mostly pertains to simple flows that are nearly unidirectional, such as channel or free stream flow. There is a textbook feel to this representation, and it might be particularly useful for developing educational materials, but I don't think I will use it much in my day to day work. Here is one more example before we move on, flow in a diverging channel, just because it looks cool.

Here is a completely different situation, that of a recirculating flow in the lid-driven cavity at Reynolds number equal to 0 (top images) and 5000 (bottom images). The vector plots look pretty good, but they miss out on some of most interesting aspects of this flow. They show that the central vortex shifts downward to a more rotationally symmetric flow at high Reynolds number (closer to the inviscid limit), and that boundary layers develop near the enclosure walls. The extent of the inviscid core is fairly obvious. But they completely miss the flow separations in the lower corners, and on the upper left wall at Re = 5000. The vector plots are useful, but more as a supplement to a good streamline contour plot than in lieu of one.

Here is another example, a reproduction of a slot coater simulation from a paper I wrote in 1992 working with Scriven. First let's look at the streamlines:

A flow recirculation forms under the downstream die face whenever the flow rate from the slot is less than the fully developed Couette flow for a channel of width equal to the gap between slot and web. This critical flow rate yields a film thickness half that of the gap. Here the flow rate is constrained to yield a film thickness one-fifth of the gap. An adverse pressure gradient forms in the gap that modifies the Couette flow to a mixed Poiseuille-Couette flow. This is nicely shown by a vector plot.

The examples I've shown are somewhat contrived in that I've carefully selected cross sections on which to display arrows that are based on the simple structure of these flows. This approach is poorly suited to more general recirculating flows. The usual method in CFD is to distribute arrows more or less uniformly around the flow.

Probably the laziest and commonest way of doing it is to place the arrows at nodes/vertexes/centers of finite elements/volumes. Visually this is a disaster on fine meshes. The plots below show arrows at the element centers on meshes of 30x30, and 100x100, elements of uniform size distribution. Only the coarse mesh gives a reasonable picture of the flow. Selectively reducing the number of arrows on finer meshes to achieve a reasonable density is unappealing, particularly on unstructured or strongly graded meshes.

Next I compare a vector plot to a streamline plot for the EFG problem near the triple phase line. The nodal density in this mesh is about right to make a good vector plot, so I have placed the arrows at the nodes. But the kinks in the streamlines near the corners are a clear signal that the mesh is overly coarse in some areas (with Cats2D, we can rule out poor contouring as an explanation). The vector plot, on the other hand, gives little visual clue of this problem.

A better alternative is to place arrows on a uniform Cartesian grid. This assures a uniform distribution of arrows and uncouples arrow density from details of the discretization. But placement of the arrows is also uncoupled from the general nature of the flow, which can lead to undesirable visual biasing of the flow. The plot shown below appears quite different from the plot above, particularly near the free surface in the upper right corner where the mesh grading is strong. I've also shown the tilted cavity flow to emphasize the effect of a boundary that is misaligned to the Cartesian grid.

In both of these plots the artificial alignment of arrows on straight cross sections is visually distracting in regions of curved streamlines, and near boundaries misaligned to these cross sections. This got me thinking about breaking up the artificial regularity of arrow placement by somehow randomizing the distribution.

To do this I plotted arrows at the centroid of each mesh partition in the nested dissection used by the solver. Since there are several dissection levels, some control over arrow density is possible. Below I show the square cavity flow on a 100x100 element mesh. Also shown is the 10th level of the dissection used to place the arrows. The plot has an obvious visual bias suggesting that the flow meanders under the lid when it is actually quite straight there. The cure might be worse than the disease in this case.

Vector plots just don't show us very much. Only the gross characteristics of a flow can be seen, and these are represented in a way that tends to obscure signs of poor discretization. Also, arrow placement biases our perception of the flow, often to the detriment of our understanding.

Vector plots are more useful for illustrating the flow of heat than the flow of mass. Heat flows tend to not have nearly as much fine detail as mass flows (in particular lacking the complex topologies of flow separation), and often we are only interested in the broad characteristics of these heat flows.