A Case-Study for Grid Convergence Study (GCS)

SPICES Workshop, 2013 – Flying wing configuration

836 words / About 4 mins

Welcome to this continuation of the previous article on grid convergence study. For 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 were generated for the configuration ranging from 0.35 million to 28 million. The table below shows the details of the grid family.

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, there by 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.

Top View


Symmetry Plane


Leading Edge


Wing Tip region


Trailing Edge


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 is presented here. The first set of images show the Cp distribution at three span-wise stations for two runs made at angle of attack of 6 and 18 degrees. As can be observed, with refinement the, the pressure distribution moves towards the extra-fine grid result values and the suction peak resolution becomes better and better.

Cp Distribution

The images below shows 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.

Surface Streamlines

Results below show the grid convergence study results of aerodynamic forces and moments. The results labelled 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 y-axis verses the grid size represented by N along 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 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 structured grid at both 6 degree 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.

Grid Convergence Study Results

The plots below show the alpha-sweep for CL, CD and CM on 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 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 are same for all angle of attacks, while moment shows a trend similar to that of lift in the non-linear region of the curve.

Alpha Sweep

Concluding remarks
The above results clearly indicate the fact that a grid-independent result have been achieved and any further refinement will not lead to any significant change in 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 fine grid for routine production runs.

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