Figure 1: Blocking strategy to capture physics around an airfoil.
2098 words/ 11 minute Read
CFD 101 starts with airfoils. This geometry is seen as the stepping stone in aerospace / turbomachinery field, before diving deep into CFD. Irrespective of the gridding software and the gridding methodology adopted to mesh, everyone is called to achieve this goal of meshing an airfoil first.
Despite the fact that meshing strategies for an airfoil have come a long way, newer, smarter and more efficient strategies continue to evolve, to capture the subtle physics in the most accurate and optimal way.
This article focusses on the traditional gridding strategies along with improved blocking techniques around airfoils for optimal CFD results.
Figure 2: H-type pattern for a bi-converx airfoil
Figure 3a: H-type pattern for a curved leading edge airfoil, 3b – H-type pattern with a leading edge split
Figure 4: H-type pattern with boundary layer clustering.
The Conventional Ways of Airfoil Meshing!
One of the classical gridding approach is the H-type pattern. It has finely clustered cells at the leading and trailing edges as shown in figure 2. This pattern is simple to construct and holds good for a biconvex airfoil with sharp leading and trailing edges. However, when applied to an airfoil with curved leading edge, the H-pattern creates a singular point (figure 3a). This can partly be alleviated by splitting the singularity into two weaker singularities as shown in figure 3b. But still, it falls short of capturing the leading edge curvature accurately and also propagates the boundary layer in the transverse direction (figure 4).
Alongside the wasted use of point clusters and high aspect ratio cells, the high clustered cells both parallel and perpendicular in regions where the flow accelerates, can result in significant time step reductions due to CFL conditions, leading to slowing of solver convergence.
Figure 5: C-type pattern for a blunt leading edge airfoil with boundary layer clustering.
An improved variant of H-type topology is the C-type pattern, which captures the leading edge curvature without any singularities. Though, the C-type pattern avoids the propagation of boundary layer fineness upstream, it fails to do so in the downstream of the airfoil trailing edge. In a way, this downstream fineness proves beneficial as it helps to capture the shear layer for solver runs at low angles of attack.
In practice however, most applications of the C-type pattern, have not taken advantage of the wonderful alignment of the grid lines along the shear layer. To take full advantage of this pattern, the CFD analysis would have to continually shift the grid curves to dynamically align with the shear layer and this would result in greater conformity to the flow physics. While not doing this alignment results in algorithmic simplicity, it does result in the wastage of grid points downstream and falls short of being an efficient grid type.
Figure 6: O-type pattern for a blunt leading edge airfoil with boundary layer clustering.
The next classical blocking strategy is the O-type, which almost overcomes the disadvantages of the H-type and C-type grids. In O-type the entire grid sheet is wrapped-up around the airfoil without any propagation of the boundary-layer fineness into the field. This helps in getting an optimal cell count, eliminating the redundant propagation of cells as seen in the H-type and C-type patterns.
However, this pattern has its inherent limitations. It creates highly skewed cells for zero thickness trailing edge airfoils. The grid cell on the upper airfoil surface directly connects to the cell on the lower airfoil surface across the horizontal grid line emanating from the trailing edge. For a single cell-to-cell step, it nearly creates a 360 degree turn in one step. Thus, each cell will have an angle of slightly less than 180 degrees which is not exactly of good quality.
The level of skewness created in such cases is acceptable for Euler grids, but is extremely high in viscous grids. Singular points like the trailing edge are critical for accurate CFD computations. High skewness will effect the solver’s robustness and also the quality of the solutions. For this reason, many prefer to go with the C-type grid, and live with the excessive cells in the wake region.
Figure 7a: Cartesian grid. Image source – hanleyinnovations, 7b – Unstructured grid
Unstructured and Cartesian
In the conventional unstructured or cartesian grids, the approach is the same as O-type in the near vicinity of the wall. Few rows of cells are wrapped around the airfoil with rest of the domain filled with unit aspect ratio tria’s or quad cells. Similar to the O-topology approach, the cells in the viscous padding at the trailing edge singularity point are skewed, deviating from the ideal requirement of high quality cells.
In few variants of boundary layer creation, such as seen in figure 8, the trailing edge singularity point can be captured without any skewness or loss of orthogonality. These are ideal grids with optimal cell count. But when the need to capture the wake arises, the Cartesian or the unstructured approach results in humongous increase in cell count, because of the inherent limitation of generating unit aspect ratio cells.
Figure 8: Trailing edge singularity capturing with unit aspect ratio cells.
A Quintessential Mesh
Having gone through the pros and cons of the traditional meshing strategies, you might be wondering what does an ideal grid for an airfoil look like? Let’s have a look at the gridding requirements for an airfoil and and see what the geometric features and the physics demand.
A single element airfoil needs sufficient number of points on the body, say typically in the range of 300-400 points for RANS simulation to capture the geometry. A finite thickness trailing edge demands atleast 20-30 points to maintain adequate grid-density around the singularity region. Point placement is of the order of 0.1 percent of chord length at the leading and trailing edges, with gradual sparsening towards the mid chord region.
At the least 20-40 layers of stretched elements are stacked orthogonally to the wall to capture the boundary layer, with a first cell height placement estimated for a Y-plus of ~1.0 based on the given flow Reynolds Number. Usage of a smaller stretching ratio, say lesser than 1.2 is highly recommended in building the viscous padding. More the orthogonality in these layers, the better the CFD results.
In unstructured and Cartesian gridding approaches, one needs to pay additional attention to the cell height ratio of the last layer of the boundary layer padding and the elements there after into the domain. The size variations need to be as small as possible. This is a critical aspect influencing the solver robustness and solution accuracy. However, this is not an issue in structured grids as the underlying gridding philosophy ensures smooth transitions.
Next, the grid outside of the viscous padding should grow gradually till they reach the farfield. Keeping the farfield as far away as possible is strongly recommended, to ensure that they don’t influence the solution in the near vicinity of the airfoil. For single element airfoil, the farfield needs to be atleast 30 chords away in all direction. NASA’s website for Turbulence Model Verification recommends to keep the farfield as far away as 500 chords. This is too far than actually needed, but they recommend it for sequential grids generated for grid convergence study. For day-today industrial production runs, this is not a must, as the solution one obtains from a 30-100 chord farfield distance is of sufficient quality and accuracy.
The above specifications gives an idea of a basic mesh for an airfoil, fulfilling the need to accurately represent the geometry and capture the gross physics. This basic grid gives reasonably good results and in most instances, this is the grid one settles down for routine production runs.
But, a lot more could be done for capturing the rich flow physics around an airfoil. A C-type, H-type grid can capture the downstream wake for low alpha’s with smooth streamline flow around the airfoil body with no flow separation. Under flow conditions with rich physics like large flow separations and discontinuous flow phenomenons like shock or expansion fans, this basic grid falls short in accurately capturing them.
The “Holy Grail” of Meshing?
Adaptivity looks to be the ideal choice to accurately capture flow phenomenons. By definition, adaptivity refers to local refinement of the flow domain respecting the physics. Refinement can either be solution based or selective local refinement based of the engineers intuitive feel of the flow physics.
Both the adaptive strategies has its pros and cons. Solution based adaptivity on the positive side are excellent in selective refinement of regions rich in flow field physics, ignoring regions with less activity. On the downside, this demands multiple intermediate CFD solver runs before settling down to the final grid. Also, adaptation runs the risk of placing hanging nodes on the body, which hampers the robustness of many CFD solvers.
On the other hand, the user defined local refinement using various field refinement options in the grid-generator helps to refine the important regions with one-to-one cell connectivity. Though the approach is limited by the intuitive feel of the flow physics by the engineer, in many instances it end-up with a few additional cells. Nevertheless, this is still a widely used approach with inherent benefit of one grid catering to the needs of multi-angles of attack simulations, without the intermittent solver runs.
Figure 9: Topology for shock capturing on the upper surface of the airfoil.
New Efficacy of Multi-block Meshing
Major flow phenomenons like the wake and shocks can be effectively captured by smart topology building without the limitations posed by traditional-blocking strategies like the C- type and O-type topologies. Figure 9 shows the grid block layout for shock capturing with a known shock position. As it can be observed, the blocks are placed right at the discontinuity, capturing the shock with a dense cloud of grid points and gradually smoothing out as we move away from the location. The grid density in the central cocoon of blocks can be varied to the users desired level of resolution, without affecting the neighbouring block grid size.
Figure 10: O-Type grid for an Airfoil with Blunt Leading and Trailing Edges with trailing edge and upper surface resolution plus shear layer resolution with a sleeve surface to contain the anticipated shear layer.
Another example is shown in the figure 10, where the wake from the airfoil is accurately captured, by using a loop topology. Stretched high aspect ratio cells are used, which judiciously capture the wake with optimal cell count. By extending the width of the loop, a larger region engulfing the upper part of the airfoil and the region downstream can be captured, extending the benefit of wake capturing and flow separation even at high angles of attack. The pre-eminence of this blocking strategy is that, it avoids the propagation of mesh fineness further downstream, aiding in generating an optimal grid. Unlike the C- type grid, the fineness of the boundary layer does not propagate downstream, the aspect-ratio of the cells inside the loop is under user’s control, it is flexible to meet users varying requirements and hence less prone to code-blow outs.
A more effective technique of wake and flow separation capturing is the topology building concept of “nesting” as shown in figure 11. This, not only has all the benefits of the above discussed looping strategy, but also has the added advantage of rapid cell count reduction in the direction normal to the wake. This serves as a very powerful tool to creatively capture any mid-field flow physics without humongous increase in cell-count.
Figure 11: O-Type topology for an Airfoil with Blunt Leading and Trailing Edges with a sleeve surface to contain the anticipated shear layer plus nested cross-stitching for local downstream cross-sectional enrichment.
Let Bygones be bygones:
In bygone days, if CFD was able to capture the gross flow-physics and make modest prediction of the flow parameters, it was perceived as an accomplishment. CFD now is no more seen as a an aid to Experiments for data cross-verifications, but is widely accepted as a precursor and a reliable design tool.
The innovative strategies discussed here are not out of context. They are very effective and they enhance the CFD results, making them more reliable and accurate. Recent advances in the computer technology has paved the way for transitioning from RANS to higher order CFD methods, LES and even DNS. And they all demand low aspect ratio, high quality quad/hex meshes. These unique blocking strategies help to cater to their needs, without the hidden cost of cell count shoot-up, loss of flow-alignment and orthogonality, poor cell quality as seen by conventional gridding techniques.