Raghurama Rao, V. Suswaram

A grid-free central scheme for inviscid compressible flows is presented here. The space derivatives in the governing equations are discretized using the least squares minimization of the error in a Taylor series approximation. The discretization is for any arbitrary cloud of points with full stencil support. A new grid-free central scheme is formulated based on convective - pressure splitting using the mid-points of the lines connecting each point to its neighbours for introducing artificial viscosity. The numerical dissipation is reduced due to the modified eigenvalues in the convective - pressure splitting together with a diusion regulator based on the jump in the Mach number. The results obtained for standard test cases in two dimensions with first and second order accuracy are presented.

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied.

Using the framework of a new relaxation system, which converts a nonlinear viscous conservation law into a system of linear convection–diffusion equations with nonlinear source terms, a finite variable difference method is developed for nonlinear hyperbolic–parabolic equations. ...

A novel Lattice Boltzmann Method applicable to compressible fluid flows is developed. This method is based on replacing the govening equations by a relaxation system and the interpretation of the diagonal form of the relaxation system as a discrete velocity Boltzmann system. As a result of this interpretation, the local equilibrium distribution functions are simple algebraic functions of the conserved variables and the fluxes, without the low Mach number expansion present in the equilibrium distribution of the traditional Lattice Boltz-mann Method (LBM). This new Lattice Boltzmann Relaxation Scheme (LBRS) thus overcomes the low Mach number limitation and can successfully simulate compressible flows. While doing so, our algorithm retains all the distinctive features of the traditional LBM. Numerical simulations carried out for inviscid flows in one and two dimensions show that the method can simulate the features of compressible flows like shock waves and expansion waves.

Using the framework of a new relaxation system, which converts a nonlinear viscous conservation law into a system of linear convection–diffusion equations with nonlinear source terms, a finite variable difference method is developed for nonlinear hyperbolic–parabolic equations. ...

An exact discontinuity capturing central solver developed recently, named MOVERS (Method of Optimal Viscosity for Enhanced Resolution of Shocks, JCP, 228 (3) (2009) 770–798), is analyzed and improved further to make it entropy stable. MOVERS, which is designed to capture steady shocks and contact discontinuities exactly by enforcing the Rankine-Hugoniot jump condition directly in the discretization process, is a low diffusive algorithm in a simple central discretization framework, free of complicated Riemann solvers and flux splittings. However, this algorithm needs an entropy fix to avoid nonsmoothness in the expansion regions. The entropy conservation equation is used as a guideline to introduce an optimal numerical diffusion in the smooth regions and a limiter based switchover is introduced for numerical diffusion based on jump conditions at the large gradients. The resulting new scheme is entropy stable, accurate and captures steady discontinuities exactly while avoiding an entropy fix.

This paper introduces a new hybrid shape optimization method using Genetic Algorithm (GA) and Ant Colony Optimization (ACO) coupled with a meshless Computational Fluid Dynamics (CFD) solver. This method converges faster to a better optimum compared to GA-CFD or ...

ABSTRACT Conjugate natural convection in a vertical annulus with a centrally located vertical heat generating rod is studied numerically. The governing equations are discretized on a staggered mesh and are solved using a pressure-correction algorithm. A parametric study is performed by varying the Grashof number, aspect ratio, and the solid-to-fluid thermal conductivity ratio over wide ranges with the Prandtl number fixed at 0.7. Results are presented for the variation of several quantities of interest such as the local Nusselt numbers on the inner and outer boundaries, the axial variation of the centerline and interface temperatures, maximum solid, average solid and average interface temperature variations with Grashof number, and the average Nusselt number variation for the inner and outer boundaries with Grashof number. The average Nusselt number from the conjugate analysis is found to be between the Nusselt numbers of the isothermal and the isoflux cases. The average Nusselt numbers on the inner and outer boundaries show an increasing trend with the Grashof number. Correlations are presented for the Nusselt number and the dimensionless temperatures of interest in terms of the parameters of the problem.

A new upwind method called Peculiar Velocity based Upwind (PVU) method is developed for obtaining numerical solution of Euler equations of gas dynamics.  It is based on the concept of peculiar velocity of Kinetic Theory of Gases.  This method is physically more meaningful and more efficient than its predecessor, the Kinetic Flux Vector Splitting method.  This new method is tested on a variety of test problems on structured meshes and unstructured meshes with mesh refinement.  The results demonstrate robustness of the method and further establish its capability in capturing various flow features.

An implicit sub-grid scale model for large eddy simulation is presented by utilizing the concept of a relaxation system for one dimensional Burgers' equation in a novel way. The Burgers' equation is solved for three different unsteady flow situations with  varying the ratio of relaxation parameter to time step.  The coarse mesh results obtained with a relaxation scheme are compared with the filtered DNS solution of the same problem on a fine mesh using a of fourth order CWENO discretization in space and third order TVD Runge-Kutta discretization in time. The numerical solutions obtained through the relaxation system have the same order of accuracy in space and time and they closely match with the filtered DNS solutions.

3-D kinematical conservation laws (KCL) are equations of evolution of a propagating surface in three space dimensions. We start with a brief review of 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an under-determined system to which we add an energy transport equation for a small amplitude disturbance to study the propagation of a three dimensional nonlinear wavefront in a polytropic gas in a uniform state and at rest. We call the enlarged system (3-D KCL and the energy transport equation) equations of weakly nonlinear ray theory-WNLRT. We highlight some interesting properties of the eigenstructure of the equations of WNLRT but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic 7x7 system that is highly nonlinear. Due to a possibility of appearance of  delta-waves and delta-shocks it is a challenging task to develop an appropriate numerical method. Here we use the staggered Lax-Friedrichs and Nessyahu-Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable to solve many complex problems for which there seems to be no other method at present which can give such physically realistic features.

Non-standard finite difference methods (NSFDM) introduced by Mickens are interesting alternatives to the traditional
finite difference and finite volume methods.  When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to non-linear scalar hyperbolic conservation laws. The original non-standard finite difference methods introduced by Mickens were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens recently introduced a non-standard finite difference method in conservative form.  This method captures the shock waves exactly, without any numerical dissipation.  In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the
strategy of composite schemes in which the accurate
non-standard finite difference method is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin are based on relaxation systems which replace the non-linear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter (lambda) is
chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes.

A new Grid-free Upwind Relaxation Scheme for simulating inviscid compressible flows is presented in this paper.  The non-linear conservation equations are converted to linear convection equations with non-linear source terms by using a Relaxation System and its interpretation as a discrete Boltzmann equation.  A splitting method is used to separate the convection and relaxation parts.  Least Squares Upwinding is used for discretizing the convection equations, thus developing a grid-free scheme which can operate on any arbitrary distribution of points.  The scheme is grid-free in the sense that it works on any arbitrary distribution of points and it doesn't require any topological information like elements, faces, edges etc.  This method is tested on some standard test cases.  To explore the power of the grid-free scheme, solution based adaptation of points is done and the results are presented, which demonstrate the efficiency of the new grid-free scheme.

The relation between Lattice Boltzmann method, which has recently become popular, and kinetic schemes, which are routinely used in computational fluid dynamics, is explored.  A new discrete velocity method for the numerical solution of Navier-Stokes equations for incompressible fluid flow is presented by combining both the approaches.  The new scheme can be interpreted as a pseudo-compressiblity method and, for a particular choice of parameters, this interpretation carries over to the Lattice Boltzmann method.

A diffusion regulation parameter, which operates based on the jump in the Mach number, is presented for implementation in Euler solvers.  This diffusion regulation parameter adjusts itself automatically in different regimes of the flow and leads to the exact capturing of steady contact discontinuities which are aligned with the grid-lines.  This diffusion regulator parameter reduces numerical dissipation, is very simple and can be easily incorporated in any Euler solver.  By coupling such a parameter
with a simple numerical method like the Local Lax-Friedrichs (Rusanov) method, an accurate and yet simple numerical method is developed for the numerical simulation of inviscid compressible fluid flows.  To demonstrate the applicability of this approach to any Euler solver, the diffusion regulation parameter is also applied in the framework of a Kinetic Scheme which is very diffusive and the improvements in the accuracy for both the methods are demonstrated through several bench-mark test problems.

A numerical method in which the Rankine-Hugoniot condition is enforced at the discrete level is developed.  The simple format of central discretization in a finite volume method is used together with the jump condition to develop a simple and yet accurate numerical method free of Riemann solvers and complicated flux splittings.  The steady discontinuities are captured accurately by this numerical method.  The basic idea is to fix the coefficient of numerical dissipation based on the Rankine-Hugoniot (jump) condition.  Several numerical examples for scalar and vector hyperbolic conservation laws representing the inviscid Burgers equation, the Euler equations of gas dynamics, shallow water equations and ideal MHD equations in one and two dimensions are presented which demonstrate the efficiency and accuracy of this numerical method in capturing the flow features.