Nobuyuki Satofuka

Overset adaptive-grid method with applications to compressible flows

Abstract A new overset adaptive-grid method which combines a solution-adaptive grid method with an overset-grid method is proposed and applied to transonic flows in this paper. The position on which the subgrid is located is adjusted to the region where the gradient of flow properties is large. The solution-adaptive grid method using an elliptic equation is applied to the overset subgrid. Thus the present overset adaptive-grid method generates the grid adapted twice to the flow solution. Two new ideas are also proposed for the basic adaptive-grid method of the elliptic type in this paper. One of the new ideas is to use a modified distribution of flow property instead of the actual one in order to estimate weighting functions which control grid spacing. The other is a locally-variable relaxation parameter technique which locally changes the relaxation parameter in the Jacobi iteration process according to the value of the coefficient of the first order derivatives in the elliptic equation. The present overset adaptive-grid method is applied to transonic flows, and is shown to be very promising, especially for sharp shock capturing.

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.

A Variable Order Method of Lines: Accuracy, Conservation and Applications

The variable order method of lines is presented for the DNS of incompressible flows. The present method is constructed by the spatial discretization, i.e., the variable order proper convective scheme and modified differential quadrature method, and time integration. The accuracy and conservation property are validated in the 2D Taylor-Green solutions and 3D homogeneous isotropic turbulence. As applications, the flows around a circular cylinder and a sphere are simulated by using Cartesian grid approach with virtual boundary method. Consequently, the present method is very promising for the DNS of the incompressible 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.

A Moving-Mesh Finite-Volume Scheme for Compressible Flows

This paper presents a new scheme for calculating compressible viscous flows with traveling wall boundary on moving mesh system. For the moving-mesh system, it is necessary for the scheme to satisfy the geometric conservation laws[1]. To satisfy the geometric conservation laws, a finite-volume formulation in the complete space-time (x, y, t) domain is adopted in the scheme. This treatment makes it possible that the scheme completely satisfies the physical and geometrical conservation laws for solving unsteady flow equation on moving-mesh system. The resultant fully implicit scheme is solved iteratively at every time step. This “inner” iteration is performed in an explicit manner through the efficient and highly stable Rational Runge-Kutta scheme, so that the algorithm of the scheme can be treated explicitly notwithstanding the implicit formulation. This approach of the inner iteration is similar to so-called pseudo-time approach. This paper also gives some discussion and numerical interpretation between the inner iteration strategy and the pseudo-time approach. The paper gives an application of the present scheme to compressible Navier-Stokes flows with fluid/body-motion coupled interaction.

Numerical Analysis of Particle Behavior Penetrating into Liquid by Level Set Method

A numerical analysis technique using the level set method has been developed to calculate the critical velocity of a particle penetrating into liquid. The contact angle, a dominant factor in this phenomenon, is given by imaginary gas-liquid interfaces inside a solid wall. A Cartesian grid method with immersed boundary is also applied in order to accurately represent a boundary of a moving particle. The current results agree with other calculated and experimental results in contact-angle problems. Finally, calculations of a particle impinging on a liquid surface have been carried out. Current results of the critical velocity agree with experimental and theoretical results.

Parallel Computation of Lattice Boltzmann Equations for Incompressible Flows

Computational Fluid Dynamics (CFD) requires enormous CPU time and huge memory. A convectional single processor computer is very difficult to satisfy these demands. With the advent of reliable, high-performance massively parallel computers, the range of CFD applications is dramatically increasing. This chapter illustrates the lattice BGK (LBGK) method used to simulate decaying 2D homogeneous isotropic turbulence on a massively parallel computer (a Hitachi SR2201). Detailed comparisons of the accuracy, physical fidelity, and efficiency between the LBGK method and traditional Finite Difference Method (FDM) are presented. Speedup and CPU time are also presented for different types of domain decomposition. Parallel computation of decaying 2D homogeneous isotropic turbulence using the LBGK method has shown that the method is accurate when compared with conventional methods using the same lattice size. The LBGK method is able to reproduce the dynamics of decaying turbulence and could be an alternative to solving the Navier-Stokes equation. In parallel computation of the lattice Boltzmann method, the highest speedup is observed in Type A decomposition but it requires the longest CPU time. It is found that the longer the decomposed domain for the horizontal direction the shorter the CPU time.

Numerical simulation of transient turbulent flows by the vorticity-vector potential formulation

Abstract A new eighth-order accurate method is presented for simulating directly turbulent shear flows. In the present method the vorticity-vector potential formulation is adopted in order to satisfy the equation of continuity automatically. The Poisson equations for the vector potentials are discretized also with eighth-order accurate method and the higher-order multi-grid method is employed. The accuracy and efficiency of the present Poisson solver are inspected in detail. The present method is, at the outset, applied to computations of cubic cavity flow and the result shows a striking agreement with that using the pseudo-spectral method. After having shown that the present method predicts accurately the three-dimensional instability of plane Poiseuille flow, the numerical simulation of the transient turbulent flow in a plane channel is dealt with in the 653 grid. The result shows an excellent agreement with that using the pseudo-spectral method; so the present computational method is suitable for the numerical simulation of shear flow turbulences.