Ganesh Natarajan

Performance comparison of flux schemes for numerical simulation of high-speed inviscid flows

Numerical investigations to assess the performance of different flux schemes for the spatial discretisation of the Euler equations have been performed. The schemes employed in the study include flux vector and flux difference splitting schemes as well as hybrid schemes. The schemes are cast in an unstructured high-order finite volume framework and are studied on typical engineering problems in the supersonic and hypersonic regimes. While all the flux schemes perform well in the supersonic and low hypersonic range, their performance show a marked change in the high Mach number regimes. Numerical experiments suggest that the AUSM family of schemes is the most accurate while the Rusanov scheme is the most robust for the range of Mach numbers considered in the study. These studies indicate that a robust and accurate numerical solver for high-speed compressible flows necessitate the use of blended schemes that provide a right balance between accuracy and numerical dissipation.

Effect of Fineness Ratio on Minimum-Drag Shapes in Hypersonic Flows

The importance of fineness ratio in determining the aerodynamic shapes of minimum-drag, zero-lift axisymmetric bodies in inviscid hypersonic flows is investigated using a shape optimization framework. The framework employs modified Newtonian theory for surface pressure computation, Bezier curves for geometric parameterization and a steepest-descent approach for minimization of the wave-drag coefficient. Studies are performed on axisymmetric bodies of a given length for fineness ratios varying from 1 to 6 and Mach numbers ranging from 6 to 12. It is shown that the optimal axisymmetric body for a given freestream Mach number is blunt-nosed for smaller fineness ratios but becomes sharp-nosed at larger fineness ratios. The fineness ratio at which the optimal body of revolution transitions from blunt-nosed to sharp-nosed is found to be nearly constant at hypersonic speeds and is around 3. Results indicate that the optimal bodies derived from the framework are superior in terms of wave drag to the von Karman og...

Defect correction based velocity reconstruction for physically consistent simulations of non-Newtonian flows on unstructured grids

Abstract A new algorithm to recover centroidal velocities from face-normal data on two-dimensional unstructured staggered meshes is presented. The proposed approach uses iterative defect correction in conjunction with a lower-order accurate Gauss reconstruction to obtain second-order accurate centroidal velocities. We derive the conditions that guarantee the second-order accuracy of the velocity reconstruction and demonstrate its efficacy on arbitrary polygonal mesh topologies. The necessity of the proposed algorithm for non-Newtonian flow simulations is elucidated through numerical simulations of channel flow, driven cavity and backward facing step problems with power-law and Carreau fluids. Numerical investigations show that second-order accuracy of the reconstructed velocity field is critical to obtaining physically consistent solutions of vorticity-dominated flows on non-orthogonal meshes. It is demonstrated that the spurious solutions are not linked to discrete conservation and arise solely due to the lower order accuracy of velocity reconstruction. The importance of the proposed algorithm for hemodynamic simulations is highlighted through studies of laminar flow in an idealized stenosed artery using different blood models.

A novel consistent and well-balanced algorithm for simulations of multiphase flows on unstructured grids

Abstract We discuss the development and assessment of a robust numerical algorithm for simulating multiphase flows with complex interfaces and high density ratios on arbitrary polygonal meshes. The algorithm combines the volume-of-fluid method with an incremental projection approach for incompressible multiphase flows in a novel hybrid staggered/non-staggered framework. The key principles that characterise the algorithm are the consistent treatment of discrete mass and momentum transport and the similar discretisation of force terms appearing in the momentum equation. The former is achieved by invoking identical schemes for convective transport of volume fraction and momentum in the respective discrete equations while the latter is realised by representing the gravity and surface tension terms as gradients of suitable scalars which are then discretised in identical fashion resulting in a balanced formulation. The hybrid staggered/non-staggered framework employed herein solves for the scalar normal momentum at the cell faces, while the volume fraction is computed at the cell centroids. This is shown to naturally lead to similar terms for pressure and its correction in the momentum and pressure correction equations respectively, which are again treated discretely in a similar manner. We show that spurious currents that corrupt the solution may arise both from an unbalanced formulation where forces (gravity and surface tension) are discretised in dissimilar manner and from an inconsistent approach where different schemes are used to convect the mass and momentum, with the latter prominent in flows which are convection-dominant with high density ratios. Interestingly, the inconsistent approach is shown to perform as well as the consistent approach even for high density ratio flows in some cases while it exhibits anomalous behaviour for other scenarios, even at low density ratios. Using a plethora of test problems of increasing complexity, we conclusively demonstrate that the consistent transport and balanced force treatment results in a numerically stable solution procedure and physically consistent results. The algorithm proposed in this study qualifies as a robust approach to simulate multiphase flows with high density ratios on unstructured meshes and may be realised in existing flow solvers with relative ease.

A Hybrid Grid Based Algebraic Volume of Fluid Method for Interfacial Flows

In the present work for numerically investigating the interfacial flows, an algebraic Volume of Fluid technique has been implemented over hybrid unstructured meshes. Following the work of Dalal et al. (Numer Heat Transf Part B 54(2):238–259, 2008 [2]), the governing equations are discretised by cell centered finite volume method wherein pressure-velocity coupling has been achieved by momentum interpolation due to Rhie and Chow (AIAA J 21:1525–1532, 1983 [12]). The binary fluid problem is represented by a single fluid formulation with a fluid property jump at the interface. Two schemes namely NVD based GAMMA scheme (Jasak, Int J Numer Meth Fluids 31:431–449, 1999 [7]) and Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA) (Alves et al., Numer Heat Transf 49:19–42, 2006 [1]) has been incorporated into an in-house fully coupled Navier-Stokes solver. These schemes are validated with the published results of collapse of water column also known as dam break problem by Martin and Moyce (Math Phys Sci 244:312–324, 1952 [9]) and Rayleigh-Taylor instability (Tryggvason, J Comput Phys 75:253–282, 1988 [13]). The results are found to be in good agreement.

Diffuse interface immersed boundary method for multi-fluid flows with arbitrarily moving rigid bodies

Abstract We present an interpolation-free diffuse interface immersed boundary method for multiphase flows with moving bodies. A single fluid formalism using the volume-of-fluid approach is adopted to handle multiple immiscible fluids which are distinguished using the volume fractions, while the rigid bodies are tracked using an analogous volume-of-solid approach that solves for the solid fractions. The solution to the fluid flow equations are carried out using a finite volume-immersed boundary method, with the latter based on a diffuse interface philosophy. In the present work, we assume that the solids are filled with a “virtual” fluid with density and viscosity equal to the largest among all fluids in the domain. The solids are assumed to be rigid and their motion is solved using Newtons second law of motion. The immersed boundary methodology constructs a modified momentum equation that reduces to the Navier–Stokes equations in the fully fluid region and recovers the no-slip boundary condition inside the solids. An implicit incremental fractional-step methodology in conjunction with a novel hybrid staggered/non-staggered approach is employed, wherein a single equation for normal momentum at the cell faces is solved everywhere in the domain, independent of the number of spatial dimensions. The scalars are all solved for at the cell centres, with the transport equations for solid and fluid volume fractions solved using a high-resolution scheme. The pressure is determined everywhere in the domain (including inside the solids) using a variable coefficient Poisson equation. The solution to momentum, pressure, solid and fluid volume fraction equations everywhere in the domain circumvents the issue of pressure and velocity interpolation, which is a source of spurious oscillations in sharp interface immersed boundary methods. A well-balanced algorithm with consistent mass/momentum transport ensures robust simulations of high density ratio flows with strong body forces. The proposed diffuse interface immersed boundary method is shown to be discretely mass-preserving while being temporally second-order accurate and exhibits nominal second-order accuracy in space. We examine the efficacy of the proposed approach through extensive numerical experiments involving one or more fluids and solids, that include two-particle sedimentation in homogeneous and stratified environment. The results from the numerical simulations show that the proposed methodology results in reduced spurious force oscillations in case of moving bodies while accurately resolving complex flow phenomena in multiphase flows with moving solids. These studies demonstrate that the proposed diffuse interface immersed boundary method, which could be related to a class of penalisation approaches, is a robust and promising alternative to computationally expensive conformal moving mesh algorithms as well as the class of sharp interface immersed boundary methods for multibody problems in multi-phase flows.

Towards an improved conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid

Abstract In [1] , the authors proposed a charge-conservative numerical framework for simulating electrohydrodynamic two-phase flows where the electric force was discretely treated as the divergence of Maxwell stress tensor because the use of volume-averaged electric force was found to be inaccurate. In this letter, we show that this framework still suffers from inaccuracies, particularly at high permittivity ratios and propose a simple solution that involves reconstruction of electric displacement rather than the electric field. We demonstrate the efficacy of the new remedial approach through simple numerical experiments for different electrical behaviour of the fluids over a range of permittivity ratios.

NUMERICAL INVESTIGATION OF HIGH TEMPERATURE GRADIENT THERMOBUOYANT FLOWS WITH MAGNETIC FIELD

We propose an algorithm to solve thermobuoyant flows in the presence of magnetic field in enclosures with large temperature differences. The method is based on a staggered/non–staggered finite volume framework for incompressible flows which is modified to solve quasi-incompressible flows with heat transfer in the presence of applied magnetic field. The present framework is solves a single equation for normal momentum in the domain. This equation is discretized using a second–order convection scheme while central differencing is adopted for the viscous terms. An implicit solution approach which is first order accurate in time is employed and the resulting non–linear system of equations is solved using the Newton–Krylov approach. The momentum field at the cell centroids is then reconstructed using a defect–correction algorithm. The energy conservation equation is discretized similar to a collocated framework, with a first–order upwind scheme for the convective terms and implicit Euler time stepping. These equations are linearized by considering the velocity field at the latest available time step, and the resulting linear system of equations are solved using an ILU preconditioned GMRES solver using the LiS library. The approach is however still pressure–based with the energy equation employed to derive the divergence constraint to be used for solution of the pressure correction equation. Studies in enclosures over a range of temperature differences indicate that proposed approach can successfully simulate magneto-hydrodynamics convective flows with large density variation.

Unified Solver for Thermobuoyant Flows on Unstructured Meshes

We propose an unified formulation for thermobuoyant flows an arbitrary mesh topologies. Unlike incompressible flow, the pressure correction equation is derived from the energy equation. The resulting Poisson’s equation reduces to continuity constraint \( \nabla \cdot \varvec{u} = 0 \), only in absence of thermal gradient and compressibility effects. Investigations are carried out for flows with small and large temperature differences in a differentially heated square enclosure. Studies using Cartesian and triangular grids show that the proposed approach can successfully simulate non-Boussinesq convection with extreme density variation.

Computation of Variable Density Flows on Hybrid Unstructured Grids

Variable density flows are basically those in which density variations cannot be neglected and hence the density is treated as a variable. This formidable change in density may be caused by either of three variables of the ideal gas equation (when dealing with gases) namely pressure, temperature and molar-mass. When pressure causes considerable change in density, the flow is no longer incompressible and they are dealt separately under compressible flows. But when density variation is caused by the other two aforementioned variables then the flow is still incompressible but the density will vary. Presence of varying density in governing equations poses a formidable challenge for numerical simulation as there is a very strong coupling between momentum and scalar transport equations. This work involves the development of three-dimensional Navier-Stokes solver to compute variable density incompressible flows on hybrid unstructured grids. The density variation due to temperature and species concentration has been considered.