Three-stage Runge-Kutta Method Research Articles

Numerical simulation of high speed reacting shear layers using AUSM+- up scheme-based unstructured finite volume method solver

Development of an Advection Upstream Splitting Method (AUSM[Formula: see text]-up) scheme-based Unstructured Finite Volume (UFVM) solver for the simulation of two-dimensional axisymmetric/planar high speed compressible turbulent reacting shear layers is presented. The inviscid numerical flux is evaluated using AUSM[Formula: see text]-up upwind scheme. An eight-step hydrogen–oxygen finite rate chemistry model is used to model the development of chemical species in a supersonic reacting flow field. The chemical species terms are alone solved implicitly in this explicit flow solver by rescaling the equation in time. The turbulence modeling has been done using RNG-based [Formula: see text]–[Formula: see text] model. Three-stage Runge–Kutta method has been used for explicit time integration. The nonreacting two-dimensional Cartesian version of the same solver has been successfully validated against experimental and numerical results reported for the wall static pressure data in sonic slot injection to supersonic stream. Detailed validation studies for reacting flow solver has been done using experimental results reported for a coaxial supersonic combustor, in which species profile at various axial locations has been compared. Present numerical solver is useful in simulating combustors of high speed air-breathing propulsion devices.

Investigation of compressibility effects on dynamic stall of pitching airfoil

In the present work, effects of compressibility on the dynamic stall of NACA 0012 airfoil, pitching sinusoidally from 5.03° to 24.79°, are investigated computationally using implicit large eddy simulations in a finite difference framework. Simulations of two-dimensional (2D), high Reynolds number, compressible flows are carried out without any transition or turbulence model to capture the physics of the dynamic stall process. The problem is formulated in a body-fixed, rotating, non-inertial frame. High accuracy, dispersion relation preserving optimized upwind compact scheme is used to compute convective flux derivatives, and an optimized three-stage Runge-Kutta method is used for time integration. Results are presented for free stream Mach number M∞ = 0.283, 0.4, and 0.5, where the Mach number is varied independent of the Reynolds number. The computations have been quite successful in capturing the essential features of the dynamic stall mechanism. It is observed that dynamic moment and lift stalls occur at smaller angles of attack as the Mach number increases. Reduction in the size of airload hysteresis loops and maximum attainable load coefficients are observed with increasing Mach number. Weak shock waves are observed near the leading edge (LE) at M∞ = 0.4, and lambda-shock is formed near the LE for M∞ = 0.5. It is observed that with increasing Mach number, the impact of dynamic stall on the aerodynamic loads (Cl, Cd, and Cm) becomes less dramatic as the maximum value attained by these aerodynamic loads decreases with an increase in the Mach number. An increase in positive damping area in the hysteresis loop is observed with an increase in the Mach number, inhibiting possible vulnerability to stall flutter.

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.

Multigrid acceleration of a block structured compressible flow solver

We study a multiblock method for compressible turbulent flow simulations and present results obtained from calculations on a two-element airfoil. A cell-vertex or vertex-based spatial discretization method and explicit multistage Runge-Kutta time stepping are used. The vertex-based method is found to give better results than the cell-vertex method. In the latter method a larger amount of artificial dissipation is required since different control volumes are used for the discretization of the viscous and convective fluxes. The slow convergence of the time stepping method makes a multigrid acceleration technique indispensable. This technique leads to an acceleration by about a factor of 10. The numerical predictions are in good agreement with experimental results. It is shown that the convergence of the multigrid process depends considerably on the ordering of the various loops. If the block loop is put inside the stage loop the process converges more rapidly than if the block loop is situated outside the stage loop in case a three-stage Runge-Kutta method is used. If a five-stage scheme is adopted the process does not converge in the latter block ordering. Finally, the process based on the five-stage scheme is about 60% more efficient than with the three-stage scheme, if the block loop is inside the stage loop.

Euler Solver Using Unstructured Upwind Method.

The unstructured upwind method is proposed to solve compressible flow with complicated boundaries or an arbitrarily shaped mesh. The flux difference splitting (FDS) scheme and flux vector splitting (FVS) scheme, which are extended to the unstructured grid system, are compared. The Euler equation is discretized by the control volume method. All conserved variables are stored at the cell center. The MUSCL approach for the cell-centered unstructured grid is proposed to obtain higher-order accuracy. A three-stage Runge-Kutta method is used for the time stepping scheme. The present calculation methods are applied to two-dimensional front-facing step flows. The results demonstrate that the present MUSCL approach is effective to obtain high resolution, and the triangular mesh solver is comparable to the uniform orthogonal mesh solver. FDS gives slightly better resolution than FVS.

Boundary modifications of the dissipation operators for the three-dimensional Euler equations

Explicit methods for the solution of fluid flow problems are of considerable interest in supercomputing. These methods parallelize well. The treatment of the boundaries is of particular interest with respect to both the numeric behavior of the solution and the computational efficiency. We have solved the three-dimensional Euler equations for a twisted channel using second-order, centered difference operators, and a three-stage Runge-Kutta method for the integration. Three different fourth-order dissipation operators were studied for numeric stabilization: one positive definite, one positive semidefinite, and one indefinite. The operators only differ in the treatment of the boundary. For computational efficiency all dissipation operators were designed with a constant bandwidth in matrix representation, with the bandwidth determined by the operator in the interior. The positive definite dissipation operator results in a significant growth in entropy close to the channel walls. The other operators maintain constant entropy. Several different implementations of the semidefinite operator obtained through factoring of the operator were also studied. We show the difference both in convergence rate and robustness for the different dissipation operators, and the factorizations of the operator due to Eriksson. For the simulations in this study one of the factorizations of the semidefinite operator required 70%–90% of the number of iterations required by the positive definite operator. The indefinite operator was sensitive to perturbations in the inflow boundary conditions. The simulations were performed on a 8,192 processor Connection Machine system CM-2. Full processor utilization was achieved, and a performance of 135 Mflops/sec in single precision was obtained. A performance of 1.1 Gflops/sec for a fully configured system with 65,536 processors was demonstrated.