REVISITING THE SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS
6.1 Resolution and reconstruction errors
The Cartesian immersed boundary approach discussed in the present study is imple- mented within the framework of an unstructured finite volume (FV) flow solver. This means that one could employ the FV solver with a body-fitted structured mesh in which case the conservation laws are solved in all control volumes. On the contrary, the IB-FV solver uses a Cartesian mesh which does not conform to the geometry with the conserved quantities in the near vicinity of the body obtained using an interpo- lation rather than as a solution to the governing partial differential equations. It is therefore possible to hypothesise that there are two kinds of errors when employing IB-FV solver on non-conformal Cartesian meshes. While the body-fitted grids resolve the boundary layers using large aspect ratio cells aligned with the body, the use of Cartesian mesh would mean that the thin shear layer is being resolved using nearly unit aspect ratio volumes that do not conform to the geometry. The errors that arise due to the use of a non-conformal Cartesian mesh in capturing the viscous layer close to the walls are termed as “resolution errors”. Furthermore, while the finite volume approach is inherent to the IB-FV solver, it is applied almost everywhere except in the close vicinity of the walls which is precisely where the thin boundary layer forms. The use of a simple interpolation for the conserved quantities in the near-wall region using the IB-FV approach would mean that these quantities would not necessarily satisfy discrete conservation that is inherent to the construction of the FV solver. The errors
resulting from this loss of discrete conservation as a consequence of the solution re- construction strategy are referred to as “reconstruction errors”. One must appreciate at this juncture that the immersed boundary approach is merely a philosophy that is employed in conjunction with a suitable discretisation of the conservation laws, which in this study is the finite volume method.
Stair-step boundary Actual boundary
Fluid cell (F) Immersed cell (I)
Solid cell (S) Solid region
Fluid region
Fluid region
Solid region Fluid cells
Immersed cells
Figure 6.1: (a) Stair-step boundary (b) body conformal grid
In order to isolate the role of these errors in estimation of relevant physical pa- rameters, we therefore propose two approaches in addition to the body-fitted finite volume method (referred to as FV herein) and the Cartesian mesh based IB-FV solver (referred to as IB-FV herein). The first is the “stair-step FV” approach which uses a Cartesian mesh similar to that in the IB-FV solver. However, the geometry in this case is represented using a “stair-step approximation” (see Figure 6.1(a)) with the bound- ary conditions directly enforced on the faces of the Cartesian mesh that approximate the true shape of the body. The second approach is the “conformal IB” approach wherein the geometry is immersed into a hybrid mesh (see Figure 6.1(b)) such that the fluid domain is discretised using a structured mesh with the geometry coincident with the grid lines. Even though the mesh now conforms to the geometry, we obtain
the solution in a manner akin to the IB-FV solver with the conserved variables in the near-wall cells computed using the solution reconstruction approach (the near-wall properties corresponds to the cell-centroid values of the ‘I’ cells or cells adjacent to the surface of the body). It must be remarked that while the “stair- step FV” and “conformal IB” are not viable solution methodologies in practice they are valuable tools to analyse the resolution and reconstruction errors. In particular, the “stair-step FV” approach is devoid of reconstruction errors and can thus highlight the role of grid resolution while the “conformal IB” approach may be used to explore the influence of solution reconstruction on a conformal mesh. We also remark that the IB-FV and stair-step FV approaches employ the same Cartesian mesh, the nature of structured mesh resolving the boundary layer and other flow features in the domain is identical between the FV and conformal IB solvers.
θo
Pressure(Pa)
-50 -25 0 25 50
20000 30000 40000 50000 60000 70000 80000
FV Stair-step FV IB-FV Conformal IB
θo Skin-frictioncoefficient(Cf)
-50 -25 0 25 50
-0.00045 -0.0003 -0.00015 0 0.00015 0.0003 0.00045
FV Stair-step FV IB-FV Conformal IB
Figure 6.2: Comparison of (a) pressure (b) skin-friction coefficient Cf
θo
Temperature(K)
-50 -25 0 25 50
200 400 600 800 1000 1200 1400 1600 1800
FV Stair-step FV IB-FV Conformal IB
Figure 6.3: Distribution of near wall temperature
Table 6.1: Comparison of stagnation point heat fluxqo Method qo (W/cm2)
FV 29.3
Stair-step FV 28.3
IB-FV 0.87
Conformal IB 1.05
Figure 6.2(a) shows the pressure distribution for flow past the cylinder using these four approaches and the mutual agreement of the computed pressures is not surprising. The skin friction estimates from the four methods show a fair agreement as well although some differences can be observed with the use of a Cartesian mesh when compared with a body-fitted grid (Figure 6.2(b)). It may be remarked that the grid resolution of the Cartesian and body-fitted meshes are comparable although the meshes employed are not sufficiently fine to ensure grid-independent solutions.
However, we observe that the predictions of near-wall temperature depicted in Figure 6.3 are quite different, with those obtained using FV and stair-step FV approaches exhibiting a good agreement among themselves while also being higher than those computed using conformal IB and IB-FV methods. If the numerical estimates from the FV approach is treated as the benchmark, it is easy to see that the stair-step FV approach is clearly superior to the conformal IB approach. This suggests that the inaccuracies in wall heat flux computations, quantified by the deviation from the FV results, are largely a manifestation of the solution reconstruction. This is because the near-wall temperature distribution on a conformal mesh (also employed for the FV solver) computed using solution reconstruction at the boundaries is significantly lesser than that predicted with a Cartesian mesh and approximated stair-step like geometry
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.