Vortex Adaptive Grid Refinement

Overview

      The test case being studied is one brought forth in a paper entitled Numerical/Experimental Study of a Wingtip Vortex in the Near Field¹ by Dacles-Mariani, Zilliac, Chow, and Bradshaw. In the paper, a single but very thorough test was run with α = 10 deg and Re = 4.6 x 10⁶, where α is the airfoil's angle of attack. The single test case was performed because the authors determined that a single very complete case was more desireable than several incomplete cases. The main goal of the study is to be able to detect and analyze the vortex being shed from the wingtip in a fully automatic fashion. This will be done by using sensitivity derivatives and a carefully formulated objective function. Presently, we have to manually take slices of the flow solution and determine vortex core points which are then strung together to form a rudimentary vortex core line. From here, the flow solution is run through the program VTAG. VTAG gives information as to the conditions inside the vortex. At some point we would like to incorporate this process into U²NCLE in an automatic fashion. The only "bump in the road" is the automatic vortex core line extraction which is, in fact, in the works and close to completion. The aforementioned paper will be used to verify the results found using numerical solutions.

Process

      The first step was to construct a fairly rough geometry that would still resolve the airfoil's shape enough to have valid results. A very coarse volume grid was then constructed for analysis. Figure 1 below is a cut of the initial volume grid.

Figure 1   Initial Volume Grid

      As shown, in the picture, the grid is extremely coarse. This coarseness is allowed to make the testing easier computationally. From this grid, U²NCLE was used to get a flow solution over the airfoil. After the soulution was found, slices were made at various increments in the flow solution in order to pinpont various locations of the vortex core. The single indepth study done in the paper was done at an x/c = 1.462, therefore the "slices" included this x/c value. The x,y, and z coordinate values were made into a "vortex.in" file which also included radial search length and direction as well as the search area discretization. For more detail on the "vortex.in" file, click here. After an output file is generated by VTAG, we can create a graph of the various condtions captured. Figure 2 and Figure 3 are 3D contour graphs of the velocity profile of the vortex for two different x/c values.

Figure 2  x/c = 1.08


Figure 3  x/c=1.46

      Figure 4 below is a graph of the various parameters involved with a vortex that have been calculated by VTAG. Note the two terms Γ 1 and Γ 2. Γ 1 is the circulation numerically calculated using the radial search length as the representative length, and Γ 2 is the circulation numerically calculated using a_avg, the computed radial vortex core size. The graph shows Γ 1 to be fairly constant, which it should be, but this can oly be attributed to the fact that the representative length has been held constant. On the other hand, Γ 2 is not constant but is inversely proportional to the a_avg term. which is Γ 2's representative length. This means that the value of Γ is dependent upon the representative length used to calculate it which means that using Γ as it stands here is useless in terms of comparing vorticies. Therefore, Γ needs to be modified to make it independent of length. This new strength could then be useful to compare one vortex to the next.

Figure 4   rsl = 0.225

      Since VTAG has computed the tangential velocity, we can compare our computational result to the experimental data from the aforementioned paper¹. Figure 5 below shows this comparison. As shown, the computational data is nowhere close to the actual experimental data. This can be attributed to the courseness of the initial volume grid. This fact brings us to the next step.

Figure 5   x/c = 1.462

NOTE: The "V" shapes in the computational data are due to the fact that the vortex core line is NOT aligned quite properly with the actual vortex center. This is another reason why a reliable vortex core extraction technique is needed.

      After the VTAG analysis has been performed, an SNODE file is then created that will allow the volume grid to be refined in the region of the vortex. Figure 6 below shows the new volume grid constructed with an SNODE file using the locations from the "vortex.in" file with a spacing of 0.01.

Figure 6  Volume Grid w/ SNODE

In all, we hope to make this process completely automatic, where you can begin with a coarse mesh as in Figure 1 and end up with a grid that will resolve the tangential velocity very accurately with no user input in the actual process.

Objective Function

      There are two main factors needed in order to accurately depict vortices computationally. The first is how well one captures the actual creation of the vortex and the second is how well the vortex is preserved. Being able to "create" the vortex lies within how well the activity in the boundary layer is captured. This will be done with boundary layer packing in the volume mesh. The second factor, vortex preservation will be achieved with the aforementioned SNODE file. Naturally, by refining the grid in the region of the vortex, we will be able to more accurately cature the vortex's features.

      Initially, the objective fuction will be simply the magnitude of the difference between the experimental tangential velocity and the computational tangential velocity. Using this function, we will use derivatives to see how sensitive the function is to either boundary layer packing or the refinement of the volume mesh in the region of the vortex. These sensitivity derivatives will be used then to drive the objective function to zero, giving the optimized grid needed to accurately capture the features of a wingtip vortex.