Investigation of two‐dimensional channel flow with a partially compliant wall using finite volume–finite difference approach

This paper presents a numerical simulation of steady two‐dimensional channel flow with a partially compliant wall. Navier–Stokes equation is solved using an unstructured finite volume method (FVM). The deformation of the compliant wall is determined by solving a membrane equation using finite difference method (FDM). The membrane equation and Navier–Stokes equation are coupled iteratively to determine the shape of the membrane and the flow field. A spring analogy smoothing technique is applied to the deformed mesh to ensure good mesh quality throughout the computing procedure. Numerical results obtained in the present simulation match well with that in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

The advantages of using high‐order finite differences schemes in laminar–turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two‐dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second‐order central differences, a fourth‐order central differences, a fourth‐order compact scheme and a sixth‐order compact scheme. A classic fourth‐order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small‐amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high‐order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Analysis of super compact finite difference method and application to simulation of vortex–shock interaction

Turbulence and aeroacoustic noise high‐order accurate schemes are required, and preferred, for solving complex flow fields with multi‐scale structures. In this paper a super compact finite difference method (SCFDM) is presented, the accuracy is analysed and the method is compared with a sixth‐order traditional and compact finite difference approximation. The comparison shows that the sixth‐order accurate super compact method has higher resolving efficiency. The sixth‐order super compact method, with a three‐stage Runge–Kutta method for approximation of the compressible Navier–Stokes equations, is used to solve the complex flow structures induced by vortex–shock interactions. The basic nature of the near‐field sound generated by interaction is studied. Copyright © 2001 John Wiley & Sons, Ltd.     

Implicit–explicit finite‐difference lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator

Difficulties for the conventional computational fluid dynamics and the standard lattice Boltzmann method (LBM) to study the gas oscillating patterns in a resonator have been discussed. In light of the recent progresses in the LBM world, we are now able to deal with the compressibility and non‐linear shock wave effects in the resonator. A lattice Boltzmann model for viscid compressible flows is introduced firstly. Then, the Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite‐difference method with a third‐order implicit–explicit (IMEX) Runge–Kutta scheme for time discretization, and a fifth‐order weighted essentially non‐oscillatory (WENO) scheme for space discretization. Numerical results obtained in this study agree quantitatively with both experimental data available and those using conventional numerical methods. Moreover, with the IMEX finite‐difference LBM (FDLBM), the computational convergence rate can be significantly improved compared with the previous FDLBM and standard LBM. This study can also be applied for simulating some more complex phenomena in a thermoacoustics engine. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

Development of level set method with good area preservation to predict interface in two‐phase flows

A two‐step conservative level set method is proposed in this study to simulate the gas/water two‐phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the coupled compact scheme. For accurately predicting the modified level set function, the dispersion‐relation‐preserving advection scheme is developed to preserve the theoretical dispersion relation for the first‐order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long‐time accurate Casimir functionals and Hamiltonian in the transport equation for the level set function, the time derivative term is discretized by the sixth‐order accurate symplectic Runge–Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second‐step equation of the modified level set method. For the verification of the proposed dispersion‐relation‐preserving scheme applied in non‐staggered grids for solving the incompressible flow equations, three benchmark problems have been chosen in this study. The conservative level set method with area‐preserving property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam‐break, Rayleigh–Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreements with the referenced solutions are demonstrated in all the investigated problems. Copyright © 2010 John Wiley & Sons, Ltd.