Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

This paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.     

An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

Lagrange–Galerkin method for unsteady free surface water waves

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.     

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.     

Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

This paper presents a new one-dimensional (1D) second-order Runge–Kutta discontinuous Galerkin (RKDG2) scheme for shallow flow simulations involving wetting and drying over complex domain topography. The shallow water equations that adopt water level (instead of water depth) as a flow variable are solved by an RKDG2 scheme to give piecewise linear approximate solutions, which are locally defined by an average coefficient and a slope coefficient. A wetting and drying technique proposed originally for a finite volume MUSCL scheme is revised and implemented in the RKDG2 solver. Extra numerical enhancements are proposed to amend the local coefficients associated with water level and bed elevation in order to maintain the well-balanced property of the RKDG2 scheme for applications with wetting and drying. Friction source terms are included and evaluated using splitting implicit discretization, implemented with a physical stopping condition to ensure stability. Several steady and unsteady benchmark tests with/without friction effects are considered to demonstrate the performance of the present model.     

An iterative scheme for numerical solution of steady incompressible viscous flows

An iterative solution scheme is proposed for application to steady incompressible viscous flows in simple and complex geometries. The iterative scheme solves the vorticity-stream function form of the Navier-Stokes equations in generalized curvilinear coordinates. The flow system of equations are cast into a Newton's iterative form which are solved using the modified strongly implicit procedure. The solution scheme is benchmarked using two test cases, namely: a shear-driven steady laminar flow in a square cavity; and a simple laminar flow in a complex expanding channel. The iterative process to steady-state convergence in both test cases is highly stable and the convergence rate is without spurious oscillations. At convergence, the flow solutions are second-order accurate.     

Decision support system for water distribution systems based on neural networks and graphs theory for leakage detection

This paper presents an efficient and effective decision support system (DSS) for operational monitoring and control of water distribution systems based on a three layer General Fuzzy Min–Max Neural Network (GFMMNN) and graph theory. The operational monitoring and control involves detection of pipe leakages. The training data for the GFMMNN is obtained through simulation of leakages in a water network for a 24 h operational period. The training data generation scheme includes a simulator algorithm based on loop corrective flows equations, a Least Squares (LS) loop flows state estimator and a Confidence Limit Analysis (CLA) algorithm for uncertainty quantification entitled Error Maximization (EM) algorithm. These three numerical algorithms for modeling and simulation of water networks are based on loop corrective flows equations and graph theory. It is shown that the detection of leakages based on the training and testing of the GFMMNN with patterns of variation of nodal consumptions with or without confidence limits produces better recognition rates in comparison to the training based on patterns of nodal heads and pipe flows state estimates with or without confidence limits. It produces also comparable recognition rates to the original recognition system trained with patterns of data obtained with the LS nodal heads state estimator while being computationally superior by requiring a single architecture of the GFMMNN type and using a small number of pattern recognition hyperbox fuzzy sets built by the same GFMMNN architecture. In this case the GFMMNN relies on the ability of the LS loop flows state estimator of making full use of the pressure/nodal heads measurements existent in a water network.     

Finite difference solution to the flow between two rotating spheres

This paper describes a numerical method for calculating incompressible viscous flows between two concentric rotating spheres. The dependent variables describing the axisymmetric flow field are the azimuthal components of the vorticity, of the velocity vector potential and of the velocity. The coupled set of governing partial differential equations is written as a system of strictly second-order equations by introducing vorticity conditions of an integral character in a meridional plane. Such conditions generalize the one-dimensional integral conditions employed by Dennis and Singh to calculate steady-state solutions of the same problem using Gegenbauer polynomials and finite differences. The basic equations are discretized in space and in time by means of the finite-difference method. A fourth-order accurate centred-difference approximation of the advection terms is employed and a nonlinearly implicit scheme for the discrete time integration is here considered. A general finite-difference algorithm for steady-state and time-dependent problems is obtained which has no relaxation parameter and makes extensive use of fast elliptic solvers. The numerical results obtained by the present method are found to be in good agreement with the literature and confirm the nonuniqueness of the steady-state solution in a narrow spherical gap at certain regimes.     

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem—that standard schemes are not stable on the arbitrarily small cut cells—while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in \(L^1\) and between first- and second-order accurate along the embedded boundary in two and three dimensions.     

Second-order BDF time approximation for Riesz space-fractional diffusion equations

ABSTRACT

Second-order backward difference formula (BDF2) is considered for time approximation of Riesz space-fractional diffusion equations. The Riesz space derivative is approximated by the second-order fractional centre difference formula. To improve the computational efficiency, an alternating directional implicit scheme is also proposed for solving two-dimensional space-fractional diffusion problems. Numerical experiments are provided to verify our theory and to show the effectiveness of numerical algorithms.     

Simulation of supercritical flow in crossroads: Confrontation of a 2D and 3D numerical approaches to experimental results

This work deals with the modeling of a flood in an urban environment. Among the various types of urban flood events, it was decided to study specifically the severe surface flooding events, which take place in highly urbanized areas. This work concerns particularly the numerical resolution of the two-dimensional Saint Venant equations for the study of the propagation of flood through the crossroads in the city. A discontinuous finite-element space discretization with a second-order Runge-Kutta time discretization is used to solve the two-dimensional Saint Venant equations. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally more efficient than comparable second-order finite volume methods. The scheme is applied to several supercritical flows in crossroads, which are investigated by Mignot. The experimental results obtained by the author are used to verify the accuracy and the robustness of the method. The results obtained are compared to those obtained by a second-order finite volume method (Rubar20 (2D)) and by FLUENT (3D). A very good agreement between the numerical solution obtained by the Runge-Kutta discontinuous Galerkin (RKDG) method and the experimental measured data were found. The method is then able to simulate the flow patterns observed experimentally and able to predict well the water depths, the discharge distribution in the downstream branches of the crossroad and the location of the hydraulic jumps and other flow characteristics more than the other methods.     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.     

Embedded implicit Runge–Kutta Nyström method for solving second-order differential equations

An embedded diagonally implicit Runge–Kutta Nyström (RKN) method is constructed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. This embedded method is derived using a three-stage diagonally implicit RKN method of order four within which a third-order three stage diagonally implicit RKN method is embedded. We demonstrate how this system can be solved, and by an appropriate choice of free parameters, we obtain an optimized RKN(4,3) embedded algorithm. We also examine the intervals of stability and show that the method is strongly stable within an appropriate region of stability and is thus suitable for oscillatory problems by applying the method to the test equation y″=?ω² y, ω>0. Necessary and sufficient conditions are given for this method to possess non-vanishing intervals of periodicity, for the fourth-order method. Finally, we present the coefficients of the method optimized for small truncation errors. This new scheme is likely to be efficient for the numerical integration of second-order differential equations with periodic solutions, using adaptive step size.     

Comparison of two numerical methods for the stratified flow

The article is devoted to numerical simulation of stratified flows described by the Navier-Stokes equations in Boussinesq approximation. The equations are solved by two high order schemes. The first one using the fifth-order WENO scheme combined with spectral projection to solenoidal field, the second one being based on the second-order AUSM MUSCL scheme with artificial compressibility in dual time.The schemes are used to model a flow around an obstacle moving through the stratified fluid. The setup of the computational case corresponds to the experiment of Chaschechkin and Mitkin [23]. Mutual comparison of results obtained by both schemes as well as of the experimental data is presented.     

An efficient solver for the RANS equations and a one-equation turbulence model

A three-stage Runge-Kutta (RK) scheme with multigrid and an implicit preconditioner has been shown to be an effective solver for the fluid dynamic equations. Using the algebraic turbulence model of Baldwin and Lomax, this scheme has been used to solve the compressible Reynolds-averaged Navier–Stokes (RANS) equations for transonic and low-speed flows. In this paper we focus on the convergence of the RK/Implicit scheme when the effects of turbulence are represented by the one-equation model of Spalart and Allmaras. With the present scheme the RANS equations and the partial differential equation of the turbulence model are solved in a loosely coupled manner. This approach allows the convergence behavior of each system to be examined. Point symmetric Gauss-Seidel supplemented with local line relaxation is used to approximate the inverse of the implicit operator of the RANS solver. To solve the turbulence equation we consider three alternative methods: diagonally dominant alternating direction implicit (DDADI), symmetric line Gauss-Seidel (SLGS), and a two-stage RK scheme with implicit preconditioning. Computational results are presented for airfoil flows, and comparisons are made with experimental data. We demonstrate that the two-dimensional RANS equations and a transport-type equation for turbulence modeling can be efficiently solved with an indirectly coupled algorithm that uses RK/Implicit schemes.     

Accuracy evaluation of unsteady CFD numerical schemes by vortex preservation

Numerical uncertainty is an important but sensitive subject in computational fluid dynamics and there is a need for improved methods to quantify calculation accuracy. A known analytical solution, a Lamb-type vortex unsteady movement in a free stream, is compared to the numerical solutions obtained from different numerical schemes to assess their temporal accuracies. Solving the Navier-Stokes equations and using the standard Linearized Block Implicit ADI scheme, with first order accuracy in time second order in space, a vortex is convected and results show the rapid diffusion of the vortex. These calculations were repeated with the iterative implicit ADI scheme which has second-order time accuracy. A considerable improvement was noticed. The results of a similar calculation using an iterative fifth-order spatial upwind-biased scheme is also considered. The findings of the present paper demonstrate the needs and provide a means for quantification of both distribution and absolute values of numerical error.     

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

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.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

An Essentially Non-oscillatory Crank–Nicolson Procedure for the Simulation of Convection-Dominated Flows

The Crank–Nicolson (CN) time-stepping procedure incorporating the second-order central spatial scheme is unconditionally stable and strictly non-dissipative for linear convection flows; however, its numerical solution in practice can be oscillatory for nonsmooth solutions. This article studies variants of the CN method for the simulation of linear convection-dominated diffusion flows, in which the explicit convection part is approximated by an upwind scheme, to effectively suppress nonphysical oscillations. The second-order essentially non-oscillatory scheme incorporated in the CN procedure (ENO-CN) has been found effective for a non-oscillatory numerical solution of minimum numerical dissipation. A stability analysis is provided for ENO-CN, which turns out to be unconditionally stable for problems of nonzero diffusion. However, for purely convective flows, it is stable only when the CFL condition is satisfied. Numerical results are presented to demonstrate its stability and accuracy.