A Helmholtz pressure equation method for the calculation of unsteady incompressibl eviscous flows

A time-implicit numerical method for solving unsteady incompressible viscous flow problems is introduced. The method is based on introducing intermediate compressibility into a projection scheme to obtain a Helmholtz equation for a pressure-type variable. The intermediate compressibility increases the diagonal dominance of the discretized pressure equation so that the Helmholtz pressure equation is relatively easy to solve numerically. The Helmholtz pressure equation provides an iterative method for satisfying the continuity equation for time-implicit Navier–Stokes algorithms. An iterative scheme is used to simultaneously satisfy, within a given tolerance, the velocity divergence-free condition and momentum equations at each time step. Collocated primitive variables on a non-staggered finite difference mesh are used. The method is applied to an unsteady Taylor problem and unsteady laminar flow past a circular cylinder.     

Time accurate local time stepping for the unsteady shallow water equations

Two time accurate local time stepping (LTS) strategies originally developed for the Euler equations are presented and applied to the unsteady shallow water equations of open channel flow. Using the techniques presented allows individual cells to be advanced to different points in time, in a time accurate fashion. The methods shown are incorporated into an explicit finite volume version of Roe's scheme which is implemented in conjunction with an upwind treatment for the source terms. A comparison is made between the results obtained using the conventional time stepping approach and the two LTS methods through a series of test cases. The results illustrate a number of benefits of using LTS which included reduced run times and improved solution accuracy. In addition it is shown how using an upwind source term treatment can be beneficial for flows dominated by the geometry. Copyright © 2005 John Wiley & Sons, Ltd.     

UNSTEADY／STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.     

Development of an Artificial Compressibility Methodology with Implicit LU-SGS Method

A numerical method has been developed to solve the steady and unsteady incompressible Navier-Stokes equations in a two-dimensional, curvilinear coordinate system. The solution procedure is based on the method of artificial compressibility and uses a third-order flux-difference splitting upwind differencing scheme for convective terms and second-order center difference for viscous terms. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual-time approach, which introduces pseudo-unsteady terms into both the mass conservation equation and momentum equations. An efficient fully implicit algorithm LU-SGS, which was originally derived for the compressible Eulur and Navier-Stokes equations by Jameson and Toon [1], is developed for the pseudo-compressibility formulation of the two dimensional incompressible Navier-Stokes equations for both steady and unsteady flows. A variety of computed results are presented to validate the present scheme. Numerical solutions for steady flow in a square lid-driven cavity and over a backward facing step and for unsteady flow in a square driven cavity with an oscillating lid and in a circular tube with a smooth expansion are respectively presented and compared with experimental data or other numerical results.     

Anisotropic goal‐oriented error analysis for a third‐order accurate CENO Euler discretization

In this paper, a central essentially non‐oscillatory approximation based on a quadratic polynomial reconstruction is considered for solving the unsteady 2D Euler equations. The scheme is third‐order accurate on irregular unstructured meshes. The paper concentrates on a method for a metric‐based goal‐oriented mesh adaptation. For this purpose, an a priori error analysis for this central essentially non‐oscillatory scheme is proposed. It allows us to get an estimate depending on the polynomial reconstruction error. As a third‐order error is not naturally expressed in terms of a metric, we propose a least‐square method to approach a third‐order error by a quadratic term. Then an optimization problem for the best mesh metric is obtained and analytically solved. The resulting mesh optimality system is discretized and solved using a global unsteady fixed‐point algorithm. The method is applied to an acoustic propagation benchmark.     

Computation of unsteady flow past blunt bodies

In accordance with the recent experimental research for flow visualization,theunsteady behavior of the starting period is investigated numerically for flow past bluntbodies.Finite difference methods are employed to solve the unsteady two-dimensionalincompressible Navier-Stokes equations.A short discussion is presented of explicit,implicit and ADI methods.Finally,the explicit and ADI schemes are used to study the flowfield in the starting period for flow past mountain-shaped and rectangular bodies.     

A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non-uniform cross-sections

The effects of non-Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non-uniform cross-section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd-B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two-step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large-scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non-uniform cross-section.     

ADAPTIVE SOLUTIONS FOR UNSTEADY LAMINAR FLOWS ON UNSTRUCTURED GRIDS

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.     

A new flux-limiting approach–based kinetic scheme for the Euler equations of gas dynamics

This paper proposes a new kinetic-theory-based high-resolution scheme for the Euler equations of gas dynamics. The scheme uses the well-known connection that the Euler equations are suitable moments of the collisionless Boltzmann equation of kinetic theory. The collisionless Boltzmann equation is discretized using Sweby's flux-limited method and the moment of this Boltzmann level formulation gives a Euler level scheme. It is demonstrated how conventional limiters and an extremum-preserving limiter can be adapted for use in the scheme to achieve a desired effect. A simple total variation diminishing criteria relaxing parameter results in improving the resolution of the discontinuities in a significant way. A 1D scheme is formulated first and an extension to 2D on Cartesian meshes is carried out next. Accuracy analysis suggests that the scheme achieves between first- and second-order accuracy as is expected for any second-order flux-limited method. The simplicity and the explicit form of the conservative numerical fluxes add to the efficiency of the scheme. Several standard 1D and 2D test problems are solved to demonstrate the robustness and accuracy.     

A multi-dimensional time-marching method of characteristics for viscous and heat-conducting flows

Summary A numerical scheme is presented which employs the characteristic surfaces in space-time for solving Navier-Stokes equations for compressible fluid flow. We consider the general case of a three-dimensional flow, a simplification of which yields the equations of the two-dimensional case. Emphasis is put on the method itself. We apply it to simulate a laminar hypersonic flow around a circular cylinder of a five-components gas mixture of nitrogen and oxygen with thermally perfect constituents and at chemical nonequilibrium. First, the partial differential equations are transformed into a standard form with directional derivatives, enabling to attain the compatibility conditions, including the viscosity terms. These conditions are discretized by approximating their integrals along the corresponding characteristic surfaces. The result is an explicit time-marching numerical scheme. Using a grid fitted between the shock and the cylinder, and starting from roughly estimated initial conditions, a steady solution is searched. A comparison is made with the solution obtained under the assumption of a perfect gas. Received 6 April 1999; accepted for publication 13 May 1999     

Research on the hysteresis properties of unsteady aerodynamics about the oscillating wings

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Computation of two-dimensional dam-break flood flows

Sudden total collapse of a dam holding back a reservoir of water, whenever it occurs, becomes formidably impressive in the extent of destruction with which it is associated. The movement on a dry bed of a two-dimensional flood wave resulting from the break of a dam has been one of the most important and challenging subjects in rapidly varied unsteady flows from the computational point of view. An implicit time-marching finite volume numerical scheme was developed and subsequently applied for the solution of the two-dimensional unsteady open channel flow equations written in conservation form. In order to avoid the problems associated with a conventional grid system, a body-fitted non-orthogonal local co-ordinate system was utilized. The proposed numerical technique was applied to determine the stage hydrographs, water surface profiles and velocities of flood flows resulting from suddenly breached storage dams. Predictions were compared with an analytical solution, with available numerical solutions using MacCormack's two-step explicit scheme and with experimental measurements. Agreement between predictions and measurements regarding the wave front advancement and stage hydrographs is considered to be satisfactory.     

A time‐accurate pseudo‐compressibility approach based on an unstructured hybrid finite volume technique applied to unsteady turbulent premixed flame propagation

A preconditioning approach based on the artificial compressibility formulation is extended to solve the governing equations for unsteady turbulent reactive flows with heat release, at low Mach numbers, on an unstructured hybrid grid context. Premixed reactants are considered and a flamelet approach for combustion modelling is adopted using a continuous quenched mean reaction rate. An overlapped cell‐vertex finite volume method is adopted as a discretisation scheme. Artificial dissipation terms for hybrid grids are explicitly added to ensure a stable, discretised set of equations. A second‐order, explicit, hybrid Runge–Kutta scheme is applied for the time marching in pseudo‐time. A time derivative of the dependent variable is added to recover the time accuracy of the preconditioned set of equations. This derivative is discretised by an implicit, second‐order scheme. The resulting scheme is applied to the calculation of an infinite planar (one‐dimensional) turbulent premixed flame propagating freely in reactants whose turbulence is supposed to be frozen, homogeneous and isotropic. The accuracy of the results obtained with the proposed method proves to be excellent when compared to the data available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

Numerical computations of internal flows for axisymmetric and two-dimensional nozzles

MacCormack's explicit time-marching scheme is used to solve the full Navier–Stokes unsteady, compressible equations for internal flows. The requirement of a very fine grid to capture shock as well as separated flows is circumvented by employing grid clustering. The numerical scheme is applied for axisymmetric as well as two-dimensional flows. Numerical predictions are compared with experimental data and the qualitative as well as the quantitative agreement is found to be quite satisfactory. © 1997 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

不可压粘流N－S方程的边界积分解法

对原变量的Ｎ－Ｓ方程进行一阶时间离散，采用共轭梯度法解除压强－速度的耦合．对所得的一系列Ｌａｐｌａｃｅ方程、Ｐｏｓｓｉｏｎ方程和Ｈｅｌｍｈｏｔｚ方程均进行边界积分法求解，首次得到了粘性Ｎ－Ｓ方程的边界积分表示式．圆柱的定常、非定常尾迹计算结果表明了本文方法的有效性．     

The influence of source terms on stability,accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes

The two‐dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two‐dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second‐order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case. The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant–Friedrichs–Lewy (CFL) condition. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

Finite volume solution to integrated shallow surface–saturated groundwater flow

This paper describes development of an integrated shallow surface and saturated groundwater model (GSHAW5). The surface flow motion is described by the 2‐D shallow water equations and groundwater movement is described by the 2‐D groundwater equations. The numerical solution of these equations is based on the finite volume method where the surface water fluxes are estimated using the Roe shock‐capturing scheme, and the groundwater fluxes are computed by application of Darcy's law. Use of a shock‐capturing scheme ensures ability to simulate steady and unsteady, continuous and discontinuous, subcritical and supercritical surface water flow conditions. Ground and surface water interaction is achieved by the introduction of source‐sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The numerical solutions and coupling algorithms are explained. The model has been applied to 1‐D and 2‐D test scenarios. The results have shown that the model can produce very accurate results and can be used for simulation of situations involving interaction between shallow surface and saturated groundwater flows. Copyright © 2005 John Wiley & Sons, Ltd.