# Smooth Extrusion for Accurate Viscous Flow Simulation

A mesh that resolves the boundary layer is an absolute necessity for accurate simulation of viscous flow. Resolution in this context implies to

- A sufficient number of cells to capture the gradients across the boundary layer
- A smooth variation in cell-to-cell size ratio across the boundary layer
- A sufficiently small near-wall spacing,
- Orthogonality of the transverse grid lines to the walls, and
- Well-shaped cells.

Here, the resolution largely depends on the applied meshing technique. An optimization-based smoothing technique in Fidelity Pointwise promises to accurately resolve the boundary layers in a viscous flow by extruding a mix of prisms and hexahedra.

**Smoothing in Hybrid Meshes**

Fidelity Pointwise has two techniques for generating hybrid meshes: traditional algebraic extrusion and anisotropic tetrahedral extrusion, also known as T-Rex. Both techniques start from a tri or quad mesh and march outward, creating layers of cells (prisms and hexahedra, respectively). T-Rex is an advancing layer technique that marches each grid point on the extrusion front outward in a nominally orthogonal direction to the wall and with step sizes prescribed to achieve the proper boundary layer resolution.

The anisotropic tetrahedra produced by joining each extruded point back to the extrusion front are combined to form stacks of prisms or hexahedra. T-Rex includes extensive smoothing methods to control the extrusion trajectory, adjust cell shapes, and avoid collisions with other extrusion fronts.

Algebraic extrusion in Fidelity Pointwise consists of defined trajectories for the mesh to follow, including extrusion along a line, rotation around an axis, along a user-prescribed path, and normal to the initial mesh. A variety of smoothing options is necessary to ensure that the algebraic techniques generate a non-folded mesh simply because they lack an elegant mathematical basis like the PDE methods.

**Optimization-based Smoothing Technique**

Introducing mixed-cell grids in Fidelity Pointwise requires smoothing in the extrusion methods to account for cell-to-cell variation. In addition to supporting mixed cell types in the same grid, the new smoothing aims to optimize element shape and size to ensure good boundary layer resolution. To smooth extrusions from a front consisting of both triangles and quads (i.e., extruding prisms and hexahedra side by side), the perturbation of each node on the advanced layer must be smoothed to account for competing effects of different cell types.

**Applying Extrusion and Smoothing**

**Onera M-6**

The ONERA M-6 wing is a slightly swept, low aspect ratio wing with a rounded tip and sharp trailing edge. A surface mesh consisting of zones of quads (for leading and trailing edge resolution) and triangles is shown in Figure 1. Algebraic extrusion using the standard method was applied with a first step height of 0.0001 and a step size growth rate of 10 percent per step.

**Figure 1.** *Close-up view of a hybrid mesh near the tip of the ONERA M-6 wing.*

Forty layers were extruded using 50 smoothing iterations per step for both P=0 and P=2. Views of the symmetry plane regions at the wing root of the extruded grids are shown in Figure 2. The wiggles for the P=0 case (left) result from biasing the smoothing to the worst cost function, which in this case, is the enforcement of the symmetry plane condition. The P=2 case (right) grid is smoother because it is biased toward the average cost function.

**Figure 2.** *40 extrusion layers on the symmetry plane of the ONERA-M6 wing at the leading edge for smoothing exponent P=0 (left) and P=2 (right).*

**The Eglin Pylon**

The finned pylon from the Eglin mutual interference experiment was meshed with all triangles and prisms extruded to test the new smoothing method's ability to handle convex regions such as the fin-body junction. The outer extrusion fronts after 25 steps for 100 smoothing sweeps with smoothing exponent P set to 0, 1, and 2 are shown in figure 3. The P=2 case (bottom) indicated nine prisms with inverted corners in the region near the fin trailing edge, while the P=0 and P=1 cases identified no problem elements. This is due to the lower exponents biasing toward repair of the worst cost function.

**Figure 3.** *25th extrusion layer for prism extrusion off the Eglin pylon/store for smoothing exponent P=0 (left), P=1 (middle), and P=2 (right).*

*To try Fidelity Pointwise to extrude smooth boundary layer meshes for your viscous CFD applications, click the button below –*