Koji Morinishi

A finite difference solution of the Euler equations on non-body-fitted Cartesian grids☆

Abstract A finite difference solution on non-body-fitted Cartesian grids has been developed for the two-dimensional compressible Euler equations. The solution is based on the method of lines. The spatial derivatives of the Euler equations are first discretized by finite difference approximations on stretched grids. The rational Runge-Kutta scheme is used as the time-stepping scheme. Accurate numerical boundary conditions are introduced at the body surfaces where the coordinate lines do not generally fit the boundaries. A series of numerical experiments are carried out to validate the present solution. Numerical results obtained for transonic flows over single-element airfoils agree well with reliable results obtained for the same flows on body-fitted grids. Typical numerical results are also obtained for transonic flows over bi-NACA0012 airfoils. The present solution is confirmed to be easily tractable even for multielement flow fields.

Effective Accuracy and Conservation Consistency of Gridless Type Solver

Gridless type solver can work on structured grids, on unstructured grids, and even on points arbitrarily distributed over computational domains. It is difficult, however, to demonstrate its accuracy order and conservation consistency theoretically except for a few special cases. In this paper, the effective accuracy and conservation consistency of the solver are numerically estimated by grid convergence studies. The effective second order of accuracy and conservation consistency are obtained in the numerical results for incompressible inviscid flows around a circular cylinder and a laminar boundary layer flow over a flat plate. The correct shock strength and position are obtained for a shock reflection problem.

Numerical solution of the kinetic model equations for hypersonic flows

A numerical method for solving the model kinetic equations for hypersonic flows has been developed. The model equations for the distribution function are discretized in phase space using a second order upwind finite difference scheme for the spatial derivatives. The resulting system of ordinary differential equations in time is integrated by using a rational Runge-Kutta scheme. Calculations were carried out for hypersonic flow around a double ellipse under various free stream conditions. Calculated results are compared with the Navier-Stokes solutions and the Direct Simulation Monte Carlo (DSMC) method for the corresponding case. The agreement is quite excellent in general.

Numerical Solution of Two-Dimensional Compressible Navier-Stokes Equations Using Rational Runge-Kutta Method

In this paper, the method of lines approach is proposed for solving viscous compressible flows. In the method of lines, semi-discretization of independent variables reduces the governing partial differential equations to a set of ordinary differential equations (ODEs) in time, which are integrated by using an appropriate time integration scheme. This separation of the space and time discretization assures a steady state solution independent of time step. As a time stepping procedure, we propose to use rational Runge-Kutta (RRK) method. The RRK method proposed by Wambecq [1] is fully explicit, requires no matrix inversion, and is stable at much larger time step than the usual explicit methods. The RRK method has been applied to solve both the Euler [2,3] and the Navier-Stokes equations [4,5]. Local time stepping and implicit residual averaging [6] techniques have been employed to accelerate convergence of solution to steady state.

Convergence acceleration of the rational Runge-Kutta scheme for the Euler and Navier-Stokes equations

Abstract An efficient method-of-lines approach is presented for the Euler and Navier-Stokes equations. The governing equations are spatially discretized by a central finite-difference approximation. The rational Runge-Kutta scheme is used for the time integration. Attention is focused on improving the efficiency and accuracy of the solution. A remarkable improvement in the efficiency is achieved by adopting a combination of the present scheme with the residual averaging and multigrid (M.G.) techniques. The M.G. method and the high suitability of the present scheme to a vector computer partly reduce the computational load imposed on a numerical simulation with a finer grid. The steady-state convergence obtained with the scheme is comparable with those of diagonalized implicit approximate factorization schemes for inviscid and viscous flow equations. The reliability and accuracy of the scheme have also been improved by adopting the artificial dissipation terms scaled down to the minimum level required for stability. The facilities of the scheme are demonstrated in a series of numerical experiments for two- and three-dimensional transonic flows.

Numerical Simulation for Impact of Elastic Deformable Body against Rigid Wall under Fluid Dynamic Force

Numerical simulations of fluid-elastic body interaction problem are presented. In the verification of elastic body model for damped free vibration in stationary fluid, the present solutions agree reasonably well the analytical ones. We can successfully simulate flows around a largely moving and deforming elastic body and collisions of the elastic body with rigid surface under the fluid dynamic force by using the elastic body model. In future work, the elastic body model should be validated through comparison between the numerical result and experiment.

Parallel Computation of Higher Order Gridless Type Solver

This paper describes higher order gridless type solvers for hyperbolic partial differential equations and their performance of parallel computing. At each point distributed over the computational domain, the spatial derivatives of the partial differential equations are evaluated using cloud of neighboring points. In addition to the previous second order upwind solver, the third and fifth order upwind gridless type solvers are presented. Reliability of the solver is demonstrated for scalar and vector hyperbolic partial differential equations. Almost no numerical dissipation is found in the results of the fifth order method. Parallel computing performance of the solvers is also examined using domain decomposition method and MPI library on the Hitachi SR2201 parallel computer at Kyoto Institute of Technology. The linear speedups are found up to 16 PUs.

Solution of Compressible Euler Flows Using Rational Runge-Kutta Time Stepping Scheme

A Method of lines approach has been applied for solving compressible flows governed by the Euler Equations. The method is based on a central difference approximation to spatial derivatives and subsequent time integration using the rational Runge-Kutta scheme. Numerical results are presented for several test cases of GAMM Workshop on the Numerical Simulation of Compressible Euler Flows.

Influence of Vortices in the Sinus of Valsalva on Local Wall Shear Stress Distribution

Transposition of the great arteries (TGA) is one of the most severe congenital heart diseases. The arterial switch operation (ASO) is the procedure of preference for treatment of TGA. After ASO, some patients suffer from circulatory system problems such as neo-aortic root dilatation and neo-aortic valve regurgitation, and supravalvar pulmonary stenosis. The neo-aortic root dilatation is often explained by the structural vascular difference between normal great arteries and the neo-aorta after ASO. Since the aortic and pulmonary roots generally remain in situ after ASO, i.e., the original pulmonary artery is connected to the left ventricle (LV), whereas the original aorta is connected to the right ventricle, the neo-aorta has no sinus of Valsalva after ASO. The influence of these morphological changes on the blood flow field at the aortic root should be investigated in detail as well as the structural vascular difference to consider the circular system problems. In this study, we apply the virtual flux method (VFM), which is a tool to describe stationary or moving body shapes in a Cartesian grid, to the 2D aortic valves and reproduce the blood flow fields around the aortic valves and the sinus of Valsalva by regularized lattice Boltzmann method (RLBM), and consider the influence of longitudinal length of sinus of Valsalva on blood flow fields around the aortic valves. As a result, we found that the longitudinal length of the sinus affects development of vortices around the aortic valves strongly. We also assessed the wall shear stress (WSS) distribution on the aortic valves and sinus wall and showed the effect of vortices in the sinus of Valsalva on local WSS distribution.

Influence of Geometric Changes in the Thoracic Aorta due to Arterial Switch Operations on the Wall Shear Stress Distribution

Background: The transposition of the great arteries (TGA) is one of the most severe congenital heart diseases. The arterial switch operation (ASO) is the preferred procedure to treat TGA. Although numerous reports have shown good results after ASOs, some patients suffer from circulatory system problems following the procedure. One reason for problems post-ASO is the local changes in the curvature and torsion of the thoracic aorta. Objective: The influence of these geometric changes on the blood flow field needs to be investigated in detail to consider possible cardiovascular problems after an ASO. Method: In this study, we conduct blood flow simulations in the thoracic aorta post-ASO, evaluate geometric changes in the aorta due to the ASO in terms of curvature and torsion, and consider the effect of geometric changes on blood flow in the aorta. Results: It was found that a large curvature near the aortic root causes an increase in the maximal wall shear stress value in the middle systole. Moreover, a large torsion results in a circumferential change in the maximal wall shear stress region. It was also found that the maximal wall shear stress in the post-ASO models is significantly higher than that in the normal models. This indicates that the aortic aneurysm initiation risk for a post-ASO artery may be higher than that of a normal artery. Conclusion: To reduce the risk of initiating an aneurism, it is suggested that the curvature near the aortic root should be decreased during the ASO.