Alemania

When computing flows which have boundary layers growing over walls or flat surfaces, require the grid to be clustered normal to the wall or flat surface in order to resolve the large gradients present through the boundary layer. As a consequence, it is important to define a set of variables that can used to quantify grid ”quality” so that it is possible to measure the ability of a selected grid in being able to give an accurate solution to the flow problem.

One such factor is the ”Reynolds number per cell”, (Reh) defined as,

Re_h= ρ∞U∞h

µ∞

(3.50)

where (h) is the height of the cell and (∞) denote free stream conditions. The (Reh)

coefficient is a measure of the number of grid points across the boundary layer, and is representative of the grids ability to capture the inner deck of the ”triple deck theory”2. Generally, the smaller the (Reh) coefficient, the better the grid is in being able to capture

the low deck of the boundary layer. The minimum (Reh) is located in cells next to the

blunt-fin and the leading edge.

Another important parameter is the cell dimension ratio (4x/4y), which due to mesh stretching reaches its maximum next to the plate. Very large or small ratios increase the oscillatory effects in second order schemes. Best results are obtained with values close to unity. Unfortunately, there are no analytic forms to determine the optimum ratios and empirical tests should be done. The coefficients for different grids evaluated at the leading edge are shown in Table 3.1.

The recommended grid parameters of Katzer [41] for Shock-Boundary Layer Interactions, near separation, (4x/4y≤20) and (Reh<<100) are followed to accurately capture the

low deck displacement.

2_{- Triple deck theory states that there are three decks within the boundary layer: the outer deck is largely}

potential flow, the middle deck covers most of the boundary layer profile and is comprised of rotational, inviscid, disturbance flow, and finally the inner deck close to the surface contains viscous disturbance flow.

3.8 Numerical Grid Generation S.J.VITHANA The quality of the grid is also assessed against an additional set of criteria usually used on flows with turbulent boundary layers. The parameter (y+) is a non-dimensional distance measured normal to the wall measured across the turbulent boundary layer. This variable is defined as:

y+ =y

√

ρwτw

µw (3.51)

where the variables with the (w) subscript represent the values at the wall. The general rule is that the first grid point off the wall should have (y+< 1) to accurately define a turbulent velocity profile. It can go up to (y+ = 5) for less accuracy. Resolution of heat fluxes requires (y+) to be about 0.1 [48].

The grids used in the following simulations will use all three parameters defined to measure grid quality, therefore assuring that the grids are effective in determining the most accurate solution.

H-Type vs C-Type

To assess the merits or shortcomings of both the grid types to yield an accurate solution to the problem in terms of numerical accuracy and time dependency, two test grids with identical grid densities (43×40×36) were run, and the solutions compared.

From the resulting simulations the following observations were made. When using the explicit method under a first-order scheme, it was observed that when using the H-type

grid it was possible to run the scheme using higher explicitCFLnumbers when compared to using theC-type grid. As a direct consequence the H-type grid gave a faster converged solution to the flowfield. However, due to the grid mapping technique the grid cells near the curved blunt section of the fin were highly skewed, even though an elliptic solver was used to smooth out these regions. This had a direct adverse affect on the numerical solution. This was expressed as numerical irregularities in the final solution. These irregularities were localised near the line of symmetry close to the blunt fin section, and manifested as ”kinks” in the contour results of the solution parameters. In contrast theC-type grid did

3.8 Numerical Grid Generation S.J.VITHANA not appear to suffer from this problem.

As a result of these observations, and taking into account the competing factors; it was decided that all subsequent numerical simulations of the blunt-fin were to use theC-type

grid.

To better mimic the leading edge properties of the flat plate a modifiedC-type Hybrid was also tested. It was found that the new grid gave improved resolutions in the leading edge shock and the subsequent boundary layer.

Grid Studies

For this case two baseline grids (labelled A1 and A2) were generated, (43×40 ×36) and (58×51×45) respectively. To obtain finer grids these two grids were doubled and redoubled to obtain grids typeB andC. Properties of these grids are shown in Table 3.1:

Grid Type ∆x/∆y ∆x of smallest cell Reh y+

(×f in diameter) 43×40×36 A1 15 0.008D 88 6.61 58×51×45 A2 15 0.008D 82 5.35 86×80×72 B1 10 0.004D 65 3.67 116×102×90 B2 8 0.004D 44 1.16 172×160×144 C1 6 0.002D 17 0.92 232×204×180 C2 4 0.002D 8 0.73

Table 3.1: Grid properties for 5mm diameter blunt-fin

From preliminary tests, it was found that a higher resolution of grid points was necessary in the boundary layer region in order to obtain a suitably well converged surface heat transfer rate solution. As a result of the grid clustering process, the shorter cell lengths close to the surface necessitate a shorter time step hence requiring a significantly higher number of time steps to reach a prescribed time level. Therefore using very fine grids for all three-dimensional simulations was not practical due to excessive computational and

3.8 Numerical Grid Generation S.J.VITHANA time demands. As a result of these restrictions very fine grid-dependent studies were only undertaken for the unswept 5mmdiameter fin case.

It was found that grid type B2 was the minimum requirement as a standard to give reasonably well defined surface heat transfer profiles. Hence from Table 3.1 it is clear that grid typesB2,C1 andC2 would give the best solutions to the flow problem.

In a similar fashion, for the 10mm diameter fin two baseline grids were generated, (58× 51×45) and (78×72×60). To obtain finer grids these two grids were doubled to obtain grid typeB. Properties of these grids are shown in Table 3.2:

Grid Type ∆x/∆y ∆xof smallest cell Reh y+

(×f in diameter)

58×51×45 A1 15 0.008D 82 5.35

78×72×60 A2 10 0.004D 71 3.89

116×102×90 B1 8 0.004D 44 1.16

156×144×1204 B2 5 0.002D 22 0.98

Table 3.2: Grid properties for 10mmdiameter blunt-fin

The simulations carried out for the 5mm diameter fin with 30◦ _{sweep are mainly for}

comparison purposes, hence fine grid studies were not necessary. The properties of this grid are shown in Table 3.3:

Grid Type ∆x/∆y ∆x of smallest cell Reh y+

(×f in diameter)

58×51×45 A1 15 0.008D 82 5.35

116×102×90 B1 8 0.004D 44 1.16