Finite volume/immersed boundary solvers for compressible flows: development and applications

A sharp interface Immersed Boundary (IB) technique based on local reconstruction of the solution has been proposed for inviscid and viscous flows. The IB-FV solver computes weight pressure and skin friction distributions fairly accurately, although the latter requires sufficient fine meshes near the body.

INTRODUCTION

Gradient reconstruction strategy
Cartesian grid based methods
Multi-fidelity framework for optimisation/design problemsproblems
Objectives of the thesis
Outline of the thesis

34] is similar, but the Cartesian grid 'attaches' to a prismatic layer that adapts to the geometry. However, the challenge with IB methods is to 'communicate' the presence of the bodies to the background network in an accurate and cost-efficient manner.

Figure 1.1: (a) Overlapping grid approach (b) Cartesian cut-cell approach Cartesian cut-cell approach (Figure 1.1(b)) represents the body boundary as a sharp-interface by locally altering the cell-topology according to the intersection of the Cartesian gri

GOVERNING EQUATIONS AND MATHEMATICAL PRELIMINARIES

Navier-Stokes equations
Finite volume formulation
Inviscid and viscous flux computations

Inviscid flux discretisation
Viscous flux discretisation

Temporal discretisation
Implementation of boundary conditions

Supersonic inlet & outlet
No-slip walls
Inviscid wall or symmetry boundary

Evaluation of the viscous flux at the face requires calculations of the first derivatives at the faces. The implementation of this boundary condition can be realized by one of the following two approaches.

MODIFIED GREEN-GAUSS RECONSTRUCTION

Overview of Green–Gauss reconstruction
Modified Green Gauss reconstruction
Numerical studies

Supersonic vortex flow
Grashof vortex
Hypersonic flow past compression ramp

Summary

The first integral is a volume integral that precisely defines the cell average value of the vector. 2((∇φ)kc+ (∇φ)knb)·(nf −αrf) (3.29), where it must be recognized that the gradients in the non-orthogonal correction are approximate values determined using MGG reconstruction and that the interpolation accuracy of the face normal derivative defined by Eq.

Figure 3.1: (a) Cell geometry (b) nomenclature for non-orthogonal grid

SHARP INTERFACE IMMERSED BOUNDARY FOR INVISCID FLOWS

Sharp interface immersed boundary method

Classification
Reconstruction

We discuss the details and implementation of the sharp interface immersed boundary (IB) method in the FV framework, discussed in Chapter 2. In the second stage, a local normal is drawn from the centroids of each of the 'I' cells to intersect the discretized geometry at point 'b', which is referred to as the body point. Unlike the scalars, the reconstruction of the velocity field must be treated with more care.

Figure 4.1: Classifications of cells in the immersed boundary finite volume (IB-FV) solver

Discrete Conservation

Transonic flow past bump
Supersonic flow past wedge

In contrast, the Cp distributions over the bump in Figure 4.6(b) for the body-fitted meshes show that the "average" bump location is indeed the same, independent of the grid resolution. It therefore follows that the IB-FV solver is not discretely conservative, but the conservation errors decrease as the grid is refined. It can be seen 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 calculated by Luo et al.

Figure 4.4: Computational domain for transonic flow past bump along with boundary condition

Order of accuracy study

We see from Figure 4.10(a) that the order of accuracy at the L2 norm for both the IB-FV and the body-based solver is approximately 2, which is indicative of the fact that the IB reconstruction does not degrade the nominal second-order global accuracy of finite volume solvers. It should be noted that the error rate L∞ is a strict test of the accuracy of the flow solver and reflects the largest errors in the domain (which are possible due to the boundary treatment) in the case of smooth flows such as those studied here. However, even at the L∞ norm, the IB-FV solver shows an accuracy order between 1 and 2 and hence it can be concluded that the proposed flow solver is second-order accurate.

Figure 4.9: Computational domain for supersonic vortex flow

Numerical investigations

Supersonic flow past a cone
Hypersonic Flow past Sphere
Hypersonic flow past a double ellipse
Hypersonic flow in a scramjet intake
Supersonic flow with moving bodies: Cylinder lift–off
Shape optimisation: Minimum drag geometries in hy- personic flowpersonic flow

This oscillatory nature of the surface pressure distribution found in the IB-FV results appears to decrease as a grid. The solver also clearly resolves the attached oblique shock, which is a manifestation of the high-resolution second-order convective scheme embedded in the IB-FV solver. The importance of the non-uniform lattice in the context of the IB-FV solvent is evident from the surface pressure distribution shown in Figure 4.17 (a).

Figure 4.11: Shock wave angle β with grid refinement

Summary

SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS

Hybrid Cartesian Immersed Boundary Method

Reconstruction for velocities
Reconstruction for pressure
Reconstruction for temperature
Reconstruction for density
Calculation of wall pressure, shear stress and heat flux

The values of these velocity components at point f can be calculated using Eq. 5.1 where φ is chosen as|| oru⊥. The temperature value TI at the centers of the immersed cells can then be obtained using eq 5.3. The resulting wall heat flux is related to the temperature gradients at the wall and is defined as, .

Figure 5.1: Reconstruction for obtaining φ at immersed cells along the normal direction as,

Numerical investigations

Inviscid hypersonic flow past a hemisphere
Subsonic flow past NACA0012 airfoil
Transonic flow past biplane NACA0012 airfoil
Low supersonic flow past a 4% thick bump
Supersonic flow past NACA0012 airfoil
Hypersonic flow past a flat plate
Hypersonic flow past a compression ramp
Hypersonic flow past a cylinder
Hypersonic flow past a sphere-cone model

In figure 5.14 we show the Mach contour of the final solution obtained on the adapted grid. In Figure 5.16(a) we compare the pressure distribution along the surface of the flat plate obtained by the IB-FV and FV solvers. In Figure 5.16(b) we compare the results for the Stanton number from both approaches with the experimental data of Lillard and Dries [133].

Figure 5.3: Comparison of normalised pressure coefficient with experimental data [117]

Summary

REVISITING THE SHARP INTERFACE IMMERSED BOUNDARY FOR VISCOUS FLOWS

Resolution and reconstruction errors

Rather, the IB-FV solver uses a Cartesian mesh that does not match the geometry with the conserved quantities in the near vicinity of the body obtained using an interpolation rather than as a solution to the governing partial differential equations not. Using a simple interpolation for the conserved quantities in the near-wall region using the IB-PV approach will mean that these quantities will not necessarily satisfy discrete conservation inherent in the construction of the PV solver not. Taking the numerical estimates of the PV approach as the benchmark, it is easy to see that the stair-step PV approach is clearly superior to the conformal IB approach.

Figure 6.1: (a) Stair-step boundary (b) body conformal grid

Studies with local grid refinement

The results using the PV approach on the original and adapted grids clearly show significant improvement with the stagnation heat flux on the adapted grid being close to the experimental value of 72 W/cm2. Results from the conformal IB approach show improved estimates for network refinement; however the estimates for the adapted grid are significantly lower than those from the PV approach (see Table 6.2). The estimates from the PV approach show a reasonable agreement with the experimental data, while the normalized heat fluxes from the IB-PV approach are anomalous.

Figure 6.5: Distribution of (a) near wall temperature (b) normalised wall heat flux q/q o

Selective solution reconstruction

Realizing that temperature reconstruction led to inaccurate wall heat flux predictions, we consider a scenario where all variables except temperature are reconstructed. Figures 6.7(a) and 6.7(b) show the skin friction and heat flux distributions obtained by the selective reconstruction approach on the coarsest mesh. Surprisingly, the stagnation heat flux estimates obtained by selective reconstruction are much closer to those predicted by the FV approach, as reflected in the ob-wall temperature distributions shown in Figure 6.8.

Figure 6.8: Comparison of near wall temperature along the cylinder

Alternate reconstruction approaches

This non-polynomial interpolant comes from the Walz correlation for compressible boundary layers where we use k = 1 for the case of laminar flows and the recovery factor r is chosen equal to. It can be easily seen that the pressure coefficient and skin friction are not very different even when using the quadratic or non-polynomial interpolants discussed above (see Figures 6.9(a) and (b)). This is despite the fact that these non-polynomial interpolants derive from physical considerations for the temperature distribution in high-velocity boundary layers.

Figure 6.9: Comparison of (a) pressure coefficient C p (b) skin-friction coefficient C f , along the cylinder

Dependence on freestream and wall conditions

Therefore, one can conclude that the heat flux estimates for the conformal IB solver tend to be further away from those predicted using the conservative FV approach as the freestream Mach number increases. While it can be noted that the discrepancy between the FV and conformal IB solutions for skin friction distribution is not significant at either Reynolds number, near-wall temperatures differ between the two methods, the differences being significantly higher for the higher Reynolds number case (see Figure 6.13 ). It can be seen from Figure 6.14(a) that the wall pressure distribution from the conformal IB approach using either temperature reconstructions is in excellent agreement with those calculated using the FV solver.

Figure 6.11: Comparison of near wall temperature distribution for freestream Mach number (a) 2.0 (b) 3.5 (c) 5.0

Discussions and remedial approaches

Quasi-conservative immersed boundary approach: The idea behind this strategy is to use the conservation laws even in the near-wall regions, but to ensure that the boundary conditions are properly accounted for. Hidden Grid/Immersed Boundary Approach: In this approach, a curvilinear mesh can be generated in the vicinity of the geometry that resolves the boundary layer quite accurately. The third approach is therefore the only methodology that is in the true spirit of the sharp-interface immersed boundary approximations.

Summary

While these three strategies generally fall within the submerged Cartesian boundary framework, they remain significantly unexplored for hypersonic flows. Moreover, it is also important to investigate the existing class of IB approaches with sharp interfaces (and these possible correction strategies) in a wide range of canonical flows crossed by generic configurations such as the HB2 geometry [144] and the cone configuration of dual [145] among others, to gain greater insight into the strengths and limitations of different variants for complex hypersonic flow problems.

APPLICATIONS TOWARDS DESIGN AND OPTIMISATION

Aerodynamic shape optimisation of nose cone

We briefly describe the salient features of the proposed multi-fidelity aerodynamic shape optimization framework (MFF) below. One of the most important aspects in aerodynamic design for hypersonic flow is the heat transfer to the body. The heat flux distribution over the length of the maximum β-body atl/d=6 obtained from laminar and turbulent simulations is depicted in Figure 7.9.

Figure 7.1: Flowchart describing the pro- pro-posed multi–fidelity optimisation frame-work

Design of scramjet inlets

At the start of the insulator placement, there is a sudden jump in βp and DI due to the shock wave. This is not entirely surprising since the HF framework is based on the invisible dynamics of the flow. In addition, the deflection angle correction enables uniform flow in the isolator to the scramjet inlet.

Figure 7.11: Scramjet inlet schematic representation where β and θ represent the shock and flow-deflection angle respectively

Design of optimal nozzle for supersonic flows

The calculations performed using the IB-FV solver clearly highlight the usefulness of the design framework based on the one-dimensional approach. The convergence diagram in Figure 7.18 obtained using LFF shows monotonically decreasing cost function (ie vr/Vt). Figure 7.20 shows the Mach contour, and Figure 7.21 shows the Mach number distribution over the nozzle wall, where small differences can be seen between the low-fidelity approach and the IB-FV.

Figure 7.16: Variable area nozzle Table 7.5: Low-fidelity flow solver Algorithm: LF Flow solver

Summary

However, the exit Mach number obtained from low-fidelity flow solver is 3.19 while the IB-FV flow solver gives an average exit Mach number of 3.35. One can see that the quantitative agreement for the mean exit Mach number is good, similar to what was also observed in the scramjet intake studies (see Table 7.4). The importance of IB-FV strategy as a flow solver in a high-fidelity framework is that, in contrast to conventional CFD solvers such as ANSYS Fluent, it allows for an automatic, robust and more flexible multi-fidelity strategy as it avoids issues with grid generation and the shapes generated are simply immersed in an underlying grid.

CONCLUSIONS AND FUTURE SCOPE

Conclusions

We have devised a sharp interface immersed boundary approach that employs a non-conservative near-body reconstruction. The use of the IDW reconstruction approach preserves the nominal second-order accuracy of the finite volume flow solver, despite a lack of near-body conservation. This clearly indicates that the reconstruction approach does not degrade the accuracy of the solution in the computational domain.

Scope of future work

Diffuse Interface Immersed Boundary Method for Convective Heat-Fluid Flow. International Journal of Heat and Mass Transfer. A Diffuse Interface Immersed Boundary Method for Multifluid Flows with Arbitrarily Moving Rigid Bodies. Journal of Computational Physics. Sources of spurious force fluctuations from the immersed boundary method for moving-body problems. Journal of Computational Physics.