Stage Runge-Kutta Scheme Research Articles

3D Heat Conduction Solver for Complex Geometries

Abstract: The essential pre-requisites for any solver to tackle real life 3D problems are to handle complex geometries and quick turnaround time. In this direction an Unstructured Finite Volume (FVM) based multi-threaded heat conduction solver has been developed to address both the issues respectively. The solver is object oriented and written in Java language. Unstructured Finite Volume method has been utilized for the space discretization and a four stage Runge-Kutta scheme is used for time integration. This solution methodology is highly parallelizable and extensively used in CFD codes. The present solver can handle non-linear steady state and transient 3D thermal problems with Dirichlet, Neumann and surface radiation boundary conditions. The solver has been validated against bench mark problems. In the present work a 3D complex geometry is discretised into 2.5 lakh hexahedrons has been taken as a test case to demonstrate the efficiency of the solver with multi-threading in dual processor machines. The results obtained are presented.

An efficient parallel algorithm for DNS of buoyancy-driven turbulent flows

In this paper, an explicit low-storage simplified M — stage Runge-Kutta (SRK) scheme for high Reynolds-number incompressible flows is presented. In the SRK scheme, the Poisson equation is solved only once in the final substage of each time step. By taking advantage of the SRK scheme and the advanced hybrid MPI+MPI model, we have developed an efficient parallel solver for buoyancy-driven turbulent flow. The spatial and temporal accuracies of the solver are validated with Taylor-Green vortex flow. Both the RK and SRK schemes are implemented for the simulation of turbulent Rayleigh-Benard convection as well as Rayleigh-Taylor flow. The results show that the SRK scheme can save approximately 20% of the computation time.

Construction, Analysis and Application of Coupled Compact Difference Scheme in Computational Acoustics and Fluid Flow Problems

AbstractIn the present work, a new type of coupled compact difference scheme has been proposed for the solution of computational acoustics and flow problems. The proposed scheme evaluates the first, the second and the fourth derivative terms simultaneously. Derived compact difference scheme has a significant spectral resolution and a physical dispersion relation preserving (DRP) ability over a considerable wavenumber range when a fourth order four stage Runge-Kutta scheme is used for the time integration. Central stencil has been used for the present numerical scheme to evaluate spatial derivative terms. Derived scheme has the capability of adding numerical diffusion adaptively to attenuate spurious high wavenumber oscillations responsible for numerical instabilities. The DRP nature of the proposed scheme across a wider wavenumber range provides accurate results for the model wave equations as well as computational acoustic problems. In addition to the attractive feature of adaptive diffusion, present scheme also helps to control spurious reflections from the domain boundaries and is projected as an alternative to the perfectly matched layer (PML) technique.

Optimization of time integration schemes coupled to spatial discretization for use in CAA applications

In this paper errors arising from spatial and temporal discretization are discussed. A new formulation to optimize time integration schemes is proposed. The new optimization also takes the errors coming from the spatial discretization into account, which leads to a minimization of the total errors. It is applicable to central and upwind type spatial discretizations. A six stage Runge–Kutta scheme is optimized for central and upwind spatial discretizations. Its efficiency is demonstrated on a one dimensional convection equation and on a Linearized Euler problem. The results indicate an important improvement.

A new flux–vector splitting compact finite volume scheme

A new high resolution finite volume method is proposed here that uses flux–vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. [1] and Lele [2] – that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of Burgers’ equation is compared with the analytical results presented in Adams and Shariff [4]. The shock capturing method is based on the proposed technique in [5], that does not require the usage of any nonlinear differencing techniques or limiters. To show that the proposed method can also handle irregular grids, the Burgers’ equation is solved using non-uniform grids. Also to demonstrate the utility of the proposed scheme to solve practical problems, the benchmark problem of nonlinear wave propagation in a one-dimensional shock-tube has been solved with the initial data as given in [23] by solving the Euler equation. The analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method, at the same time providing a neutrally stable scheme when used with four stage Runge–Kutta scheme. Also the same scheme displays extended ranges of wave numbers and circular frequencies over which DRP property is valid. The computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems.

Numerical simulation of a liquid propellant rocket motor

This work presents a numerical simulation of the flow field in a liquid propellant rocket engine chamber and exit nozzle using techniques to allow the results to be taken as starting points for designing those propulsive systems. This was done using a Finite Volume method simulating the different flow regimes which usually take place in those systems. As the flow field has regions ranging from the low subsonic to the supersonic regimes, the numerical code used, initially developed for compressible flows only, was modified to work proficiently in the whole velocity range. It is well known that codes have been developed in CFD, for either compressible or incompressible flows, the joint treatment of both together being complex even today, given the small number of references available in this area. Here an existing code for compressible flow was used and primitive variables, the pressure, the Cartesian components of the velocity and the temperature instead of the conserved variables were introduced in the Euler and Navier-Stokes equations. This was done to permit the treatment at any Mach number. Unstructured meshes with adaptive refinements were employed here. The convective terms were treated with upwind first and second order methods. The numerical stability was kept with artificial dissipation and in the spatial coverage one used a five stage Runge-Kutta scheme for the Fluid Mechanics and the VODE (Value of Ordinary Differential Equations) scheme along with the Chemkin II in the chemical reacting solution. During the development of this code simulating the flow in a rocket engine, comparison tests were made with several different types of internal and external flows, at different velocities, seeking to establish the confidence level of the techniques being used. These comparisons were done with existing theoretical results and with other codes already validated and well accepted by the CFD community.

Unsteady cascade flow calculations of using dual time stepping and the k-$\omega$ turbulence model

A numerical study on two-dimensional unsteady transonic cascade flow has been performed by adopting dual time stepping and the k-.omega. turbulence model. An explicit 4 stage Runge-Kutta scheme for the compressible Navier-Stokes equations and an implicit Gauss-Seidel iteration scheme for the k-.omega. turbulence model are proposed for fictitious time stepping. This mixed time stepping scheme ensures the stability of numerical computation and exhibits a good convergence property with less computation time. Typical steady-state convergence accelerating schemes such as local time stepping, residual smoothing and multigrid combined with dual time stepping shows good convergence properties. Numerical results are presented for unsteady laminar flow past a cylinder and turbulent shock buffeting problem for bicircular arc cascade flow is discussed.

Computation of three-dimensional transonic viscous flow using the JUMB03D code

A vertex based finite volume method for the solution of the three dimensional Reynolds averaged Navier-Stokes equations has been developed. The computations can be carried out blockwise after dividing the computational domain into smaller blocks to reduce the memory requirement for a single processor computer and also to facilitate parallel computing. A five stage Runge-Kutta scheme has been used to advance the solution in time. Enthalpy damping, implicit residual smoothing, local time stepping, and grid sequencing are used for convergence acceleration. In order to get smooth convergence for transonic, viscous flows, the artificial dissipation has been modified by using the time step for advective and diffusive equations. An algebraic turbulence model has been used to determine the turbulent eddy viscosity. The method has been used to compute transonic flow over a cropped delta wing and the ONERA M-6 wing, and subsonic flow over a launch vehicle configuration. The results obtained show good agreement with available experimental data.

Computation of three-dimensional transonic flow using a cell vertex finite volume method for the Euler equations

A cell vertex finite volume method for the solution of the three dimensional Euler equations has been developed. The computations can be carried out block-wise after dividing the computational domain into smaller blocks to reduce the memory requirement for a single processor computer and also to facilitate parallel computing. A five stage Runge-Kutta scheme has been used to advance the solution in time. Enthalpy damping, implicit residual smoothing, local time stepping, and grid sequencing are used for convergence acceleration. The solution procedure has been studied in detail by computing transonic flow over the ONERA M-6 wing, using both C-H and O-H type structured grids. The effects of changing the artificial viscosity parameters and the distance of the far field boundary are also investigated.

A finite volume nodal point scheme for solving two dimensional Navier-Stokes equations

A finite volume nodal point spatial discretization scheme for the computation of viscous fluxes in two dimensional Navier-Stokes equations has been presented here. The present scheme gives second order accurate first derivatives and at least first order accurate second derivatives even for stretched and skewed grid. It takes almost the same numerical efforts to solve full Navier-Stokes equations as that for using thin layer approximation. The scheme has been implemented to solve laminar viscous flows past NACA0012 aerofoil. To advance the solution in time a five stage Runge-Kutta scheme has been used. To accelerate the rate of convergence to steady state, local time stepping, residual averaging and enthalpy damping have been employed. Only a fourth order artificial dissipation has been used here for global stability of the solution. A comparative study of the results obtained by the present scheme for full Navier-Stokes equations and for thin layer approximation have been made with other numerical methods developed earlier.