REVISITING THE SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS
6.2 Studies with local grid refinement
despite the nearly unit aspect ratio quadrilateral cells in the latter not aligned with the boundary layer. We can therefore conclude that the loss in discrete conservation near the boundaries that is a consequence of solution reconstruction in the conformal IB approach is responsible for the under-estimation of the heat flux when compared with the FV benchmark. The good agreement of stagnation point heat flux estimates from the stair-step FV approach with those from the FV approach in Table6.1, even when the grids are different, may therefore be attributed to the fact that both approaches are discretely conservative everywhere in the domain.
θo
Temperature(K)
-50 -25 0 25 50
200 400 600 800 1000 1200 1400 1600 1800
FV (adapted grid) FV (initial grid)
Conformal IB (adapted grid) Conformal IB (initial grid)
θ q/qo
-50 -25 0 25 50
0.4 0.6 0.8 1 1.2 1.4
Weiting (Exp) FV (adapted grid) Conformal IB (adapted grid)
Figure 6.5: Distribution of (a) near wall temperature (b) normalised wall heat flux q/qo
θo Skin-frictioncoefficient(Cf)
-50 -25 0 25 50
-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015
FV (adapted grid) FV (initial grid)
Conformal IB (adapted grid) Conformal IB (initial grid)
Figure 6.6: Comparison of skin-friction coefficientCf along the cylinder on the adapted grid
The lack of discrete conservation is clearly inherent in the immersed boundary ap- proach, but it does lead to reasonably accurate estimates of skin friction (see Figure 6.4(a)) and in some cases, even heat transfer (see Figure 5.18(c), Chapter 5). The lack of mass conservation in sharp-interface IB methods has been shown to be pro- portional to the grid spacing [139] and this motivates the study of adaptive mesh refinement in the context of IB approaches. Specifically, we try to address the issue of whether increasing the near-wall grid resolution can improve the estimates of wall heat flux. Towards this objective, we compare the numerical solutions obtained on the same structured mesh using FV and conformal IB approaches on unadapted as well as locally adapted meshes. The initial unadapted grid consists of 29000 cells with a near-
wall spacing of 2.283×10−4 m and is adapted to five levels in the vicinity of the solid wall as shown in Figure 6.4(b). We carry out an isotropic grid refinement where each quadrilateral cell is divided into four smaller quads. The final adapted grid has 102200 cells with the cell Reynolds number being ∼ 35. The results using the FV approach on the initial and adapted grids clearly show marked improvement with the stagnation heat flux on the adapted mesh being close to the experimental value of 72 W/cm2. It must be noted that the FV simulations on the final adapted mesh were performed with van Leer scheme for the convective computations. The results from the conformal IB approach show improved estimates on grid refinement; however the estimates on the adapted mesh are significantly lower than those from the FV approach (see Table 6.2). This is consistent with the near-wall temperature distributions (Figure 6.5(a)) wherein one can see that the temperatures obtained with the conformal IB approach show a qualitative trend different from those obtained with the FV approach, while also being considerably lesser in magnitude and closer to the wall temperature. Figure 6.5(b) shows the normalised heat flux distribution on the adapted mesh using both approaches. The estimates from FV approach show a reasonable agreement with the experimental data whereas the normalised heat fluxes from the IB-FV approach are anomalous. In fact, the stagnation point heat transfer from the IB-FV approach is not clearly the expected maxima which is likely due to the erroneous near-wall temper- atures which translate into incorrect wall heat fluxes. Interestingly, the skin friction distribution on the adapted mesh using these approaches in Figure 6.6 are in good agreement, indicating that the solution reconstruction does not adversely affect the wall shear stress estimates.
The importance of this study is that it strongly reveals the overwhelming domi- nance of “reconstruction” errors in estimating wall heat fluxes while using an immersed boundary approach. The study demonstrates that the heat flux estimates even on a conformal mesh but using solution reconstruction does not improve significantly with increasing grid resolution. While one could perform further levels of grid refinement, it is not difficult to see that a finitely large number of mesh refinements would be necessary, even with a conformal IB approach, to achieve the accuracy of solutions obtained on an adapted mesh with five levels of refinement using the FV approach.
Consequently, the number of levels of grid refinement on a non-conformal mesh that employs the IB-FV approach would be even higher and therefore not feasible in prac- tice. One must contrast this observation with the studies in [70, 74] where the use of local grid refinement upto 14 levels (leading to a cell Reynolds number of∼1 resulted in quite accurate estimates of skin friction and heat flux. It must however be remarked that these studies were limited to low Reynolds number flows (less than 2000) and do
not consider high Re viscous flows as considered in this study. Our observations for this test case therefore point to the fact that local grid refinement has a weak influence on heat flux estimates obtained with IB-FV solvers and that while it is possible to obtain accurate estimates of wall shear stress with acceptably finite number of grid refinements, the predictions of wall heat fluxes could be practically infeasible with such an approach.