A Case-Study for Grid Convergence Study (GCS)

Figure 1: SPICES Workshop, 2013 – Flying wing configuration used for grid convergence study

836 words / About 4 mins

Welcome to this continuation of the previous article on grid convergence study. For a better insight on the grids generated for GCS, this article presents a case study showcasing the GCS grid family generated for a workshop, similar to DPW, called SPICES-2013. The configuration chosen was a Flying Wing Configuration, and the meshing software used was GridPro.

The Grid Family

The geometry for the workshop was a delta wing type of configuration called the flying wing configuration with a sting base. A grid family with 5 grids was generated for the configuration ranging from 0.35 million to 28 million. The table below shows the details of the grid family.

Figure 2: Mesh details.

The surface mesh elements are increased by a factor of 2, while that of volume by a factor of 3. The first spacing in the viscous padding becomes smaller and smaller as we move from tiny (yplus ~ 1) to extra-fine grid (yplus ~ 0.2). Also, the number of layers in the viscous padding is increased from 15 to 45, thereby resolving the laminar sub-layer and the turbulent regions of the boundary layer in a better way.

Below is a gallery of images showing the variation of grid resolution with each grid level at various regions of the computational domain.

Grid convergence study: Top view
Figure 3: Grid convergence study: Top view
Grid convergence study: Symmetry plane
Figure 4: Grid convergence study: Symmetry plane
Grid convergence study: Leading edge
Figure 5: Grid convergence study: Leading edge.
Grid convergence study: Wing Tip Region
Figure 6: Grid convergence study: Wing Tip Region
Grid convergence study: Trailing Edge
Figure 7: Grid convergence study: Trailing Edge
Grid convergence study: Base Region
Figure 8: Grid convergence study: Base Region

Meshing in GridPro

The initial meshing for the configuration took about a couple of days for topology building and density readjustments to suit the specifications. Once the first grid was generated in GridPro, the subsequent grids were generated by programming the grid generator. The video gives an overview of the process of generating a family of grids from a single topology.

CFD Flow Solver: HiFUN

The code HiFUN the primary product of SandI, is a general-purpose flow solver employing unstructured data-based algorithms. It is fine-tuned to solve typical aerospace applications and certain flow problems encountered in automotive industries. The code has been extensively used for solving a number of problems, over a wide range of Mach numbers, ranging from airship aerodynamics to aerodynamics of hypersonic vehicles.

Analysis of CFD Results

The participants made use of the GridPro grids for their computations. Only the results using the code HiFUN are presented here. The first set of images shows the Cp distribution at three span-wise stations for two runs made at an angle of attack of 6 and 18 degrees. As can be observed, with refinement, the pressure distribution moves towards the extra-fine grid result values and the suction peak resolution becomes better and better.

Figure 9: Cp Distribution

The images below show the surface streamlines. With grid refinement, the flow features become clearer and crisper. The bubble in the wing-sting junction region is better captured and is observed to grow in size with higher grid resolution.

Figure 10: Grid convergence study: Surface streamlines

The results below show the grid convergence study results of aerodynamic forces and moments. The results labeled as Str-grid represented by red color lines are the results obtained using GridPro grids, while the Uns-grid is the organizer’s unstructured grid. The plots show the variation of aerodynamic quantities CL, CD, and CM along the y-axis versus the grid size represented by N along the x-axis. As the grid size has a power of -2/3, the smallest of the grids is on the right side of the figure and with every level of refinement, the data point moves towards the x=0 value.

With refinement, the delta change in the aerodynamic quantity reduces significantly, clearly showing a tendency towards a grid-independent solution. The change in solution from coarse to medium is large, which progressively becomes smaller with higher levels of mesh refinement. Ideally, all the plots should have been smooth without any kinks as seen for the CD data for the structured grid at both 6 degrees and 18-degree angle of attack. The CL and CM curves show non-monotonous variation with an amplitude of less than 1 count. This subtle variation could be attributed to the lack of solver convergence for fine or extra-fine grids.

Figure 11: Grid Convergence Study Results

The plots below show the alpha-sweep for CL, CD, and CM on the medium grid. The match between the structured and the unstructured grids is excellent in the linear region of the curve. Differences in the predicted results are seen in the non-linear region of the curve, especially near CL-max.  CL-max prediction by the structured grid is more than the unstructured grid. This is a commonly observed phenomenon and is attributed to the fact that structured grids are more flow-aligned, and the flow separation is not as prominent as in the unstructured grid near stall, resulting in a slightly larger lift-prediction. However, the drag prediction by both the grids is the same for all angles of attacks, while the moment shows a trend similar to that of lift in the non-linear region of the curve.

Figure 12: Alpha Sweep

Concluding Remarks

The above results clearly indicate the fact that a grid-independent result has been achieved and any further refinement will not lead to any significant change in the solution. The variation of CL between fine and extra-fine structured grids is about 0.5 counts at alpha 6 degrees, while at alpha 18 degrees, it is about 2 counts. Also, the CD variation between fine and extra-fine is 8 counts at alpha 6 degrees, and this variation increases to 20 counts at 18 degrees.

The variations between fine and extra-fine results are small and within acceptable limits, and hence it may be safe to use the grid sizing of the fine grid for routine production runs.

Further Reading:

  1. Grid Convergence Study ! – Is it Necessary?
  2. Grid Convergence Study for Francis Turbine- From a Meshing Perspective
  3. Do Mesh Still Play a Critical Role in CFD ?
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