SHARP INTERFACE IMMERSED BOUNDARY FOR INVISCID FLOWS
4.2 Discrete Conservation
must be remarked that since the IB approach is non–conformal, we choose the contour of integration as the set of grid faces in the vicinity of the body, which leads to an approximated domain stair–step representation as in [111]. The pressure at those faces are simple the cell-centroidal value from the corresponding immersed cell. The errors from this approximation reduces as the grid resolution in the near–
solid regions increases and our studies show that for the grid resolutions considered herein these errors insignificantly influence the force calculations.
IB-FV solver is different from that on the curvilinear body-fitted grid. A closer look at Figures 4.5(b)-(d) also show that the error in shock locations diminish as the grid becomes finer. This is also reflected in Figure 4.6(a) which shows the Cp distributions on the four Cartesian grids, compared with that of the solution on the body-fitted mesh. The IB-FV results on the finest mesh shows a less-diffused shock closer to that predicted by the unstructured FV solver (on the body-fitted mesh) while the results on progressively coarser meshes are more diffused and show a farther shock location.
In contrast, the Cp distributions over the bump in Figure 4.6(b) for the body-fitted meshes show that the “average” shock location is indeed the same, independent of the grid resolution.
10% thick bump
Subsonicinlet Subsonicoutlet
M∝=0.675 Pout=0.737
Figure 4.4: Computational domain for transonic flow past bump along with boundary condition
One can infer from these observations that the IB-FV results approach those ob- tained on conformal meshes using the unstructured FV solver ash−→0 where his the characteristic length scale 2. It must be realised that the unstructured FV approach is discretely conservative on all meshes, which explains why the shock location is cor- rectly predicted on all four curvilinear meshes. The IB-FV solver predicts a different shock location on each mesh and even on the finest mesh, the shock location shows a deviation from that predicted by the unstructured FV solver, although the differences in the predictions are evidently smaller than those on the coarser meshes. It follows therefore that the IB-FV solver is not discretely conservative but the conservation errors do diminish as the grid is refined.
2On uniform Cartesian meshes,h= ∆x= ∆ywhile on unstructured meshes we considerh=1/√ nc wherenc is the total number of cells in the mesh
IB-FV 150 x 50
FV 150 x 50
IB-FV 225 x 75
FV 225 x 75
IB-FV 300 x 100
FV 300 x 100
IB-FV 450 x 150
FV 450 x 150
Figure 4.5: Mach contours depicting normal standing shock for different grid (a) 150
× 50 (b) 225× 75 (c) 300 ×100 (d) 450 ×150 (Min: 0, ∆ : 0.117, Max: 1.52) (Top:- IB-FV solver; Bottom:- FV solver on body fitted mesh)
X -Cp
0 0.25 0.5 0.75 1
-1.5 -1 -0.5 0 0.5 1 1.5 2
2.5 150 x 50, IB-FV
225 x 75, IB-FV 300 x 100, IB-FV 450 x 150, IB-FV 450 x 150, FV
X -Cp
0 0.25 0.5 0.75 1
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
150 x 50 225 x 75 300 x 100 450 x 150
Figure 4.6: Coefficient of pressure distribution along the surface of the body from (a) IB-FV solver on non-conformal grid (b) FV solver on conformal grid
X -Cp
-1 -0.5 0 0.5 1 1.5 2
-1.5 -1 -0.5 0 0.5 1 1.5 2
2.5 450 x 150, IB-FV
450 x 150, FV Luoet al.
X
Entropy,S
0 0.2 0.4 0.6 0.8 1
1 1.02 1.04 1.06 1.08
Figure 4.7: (a) Coefficient of pressure distribution along the surface of the body on conformal and non-conformal mesh (b) Numerical entropy generation along immersed cells
This test case also serves to study the efficacy of the IB approaches for problems with a non-conformal geometry and strong compression. One can see from Figure 4.7(a) that the pressure distribution on the surface of the bump using the IB-FV solver on the 450 × 150 grid agrees well with those computed by Luo et al. [114] as well as that obtained using the unstructured FV solver on body-fitted mesh of equivalent resolution. The distribution of numerical entropy S =P/ργ in Figure 4.7(b), on the surface of the bump clearly indicates a jump at the location of the standing shock, which makes it a good indicator of discontinuities in compressible flows. This test problem effectively achieves the twin-objectives of investigating discrete conservation and validating the solver on flow past curved geometries with strong compression. Our results show that discrete conservation can only be achieved in the limit as the grid size tends to zero and on sufficiently fine meshes the IB-FV solver does compute the numerical solution to reasonable accuracy.
4.2.2 Supersonic flow past wedge
As a second test case in analysing the issue of discrete conservation, we simulate the supersonic flow past a wedge of semi–vertex angle of 20◦ at freestream Mach number M∞=2. While the geometry for this test case is quite simple, the use of IB-FV solver with a uniform Cartesian mesh means that the body is not aligned with the background mesh. Moreover, the flow creates an attached shock at the leading edge which is also not aligned with the grid. We carry out simulations on four different meshes on a
domain of size 0.1 × 0.15 maintaining unit aspect ratio cells on all grids. The details of all grids are shown in Table 4.1 along with their respective characteristic lengths.
For all these simulations, we also compute the mass defect defined by
∆m=Xρf U⊥,f ∆Sf (4.8)
where the summation is over all the boundary faces of the domain as well as the interior faces shared by S and I cells. In other words, Eq. 4.8 is a summation of the mass fluxes through the inlet and outlet as well as the body surface, which is ap- proximated by the stair-step representation in Figure4.1. On body-conforming grids, the mass flux through the body surface would be zero and the mass fluxes through the inlet and outlet would differ by an amount which is of the order of residual at steady-state, which we also observed in our numerical studies (not reported here for sake of brevity). This is a consequence of the discrete mass preservation of FV solvers.
In the case of IB-FV solvers, the solutions at I cells are reconstructed as opposed to F cells where they are “solved” and there would be a finite mass defect on any mesh even when steady-state solutions are obtained. Table 4.1 shows this mass defect on the four meshes at steady-state convergence, defined by the relative residual (in en- ergy) becoming 10−6 or lesser. On the backdrop of observations in Section4.2.1, it is not surprising to see mass defects that are three orders of magnitude larger than the residual errors. This clearly shows that the IB-FV solver does not discretely conserve mass on any mesh but the errors do diminish as the grid becomes finer. Figure 4.8 shows that the discrete conservation errors, which are quantified by the mass defect at steady-state in this case, indeed decrease linearly with grid refinement. A similar result was obtained for a class of meshfree methods with regard to mass conservation [115]. This study reinforces the fact that there is a finite O(h) conservation error for the IB-FV approach and it is strictly conservative only in the limit as the grid spacing tends to zero.
Table 4.1: Mass defect ∆m on different grids
Grid h ∆m
100 ×150 1/100 0.058 150 ×225 1/150 0.039 200 ×300 1/200 0.028 250 ×375 1/250 0.022
The lack of machine-zero conservation errors on any mesh using the IB-FV ap- proach means that it is necessary that the Cartesian/curvilinear meshes employed for compressible flows are “sufficiently” fine in the vicinity of the bodies to keep con-
servation errors to acceptably low values. This may be achieved using local grid refinement. We emphasise that the studies on discrete conservation (and its errors) conducted herein for sharp interface IB methods is likely the first comprehensive ex- position of this issue, unlike other studies which often ignore the problem or even implicitly assume that the use of fine meshes guarantee conservation.
Log (h)
Log(∆m)
-2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9
-1.7 -1.6 -1.5 -1.4 -1.3 -1.2
1.057
Figure 4.8: Variation of mass defect ∆m with grid refinement