Energy relaxation method for chemical non‐equilibrium flow computations

In this paper, an algorithm for chemical non‐equilibrium hypersonic flow is developed based on the concept of energy relaxation method (ERM). The new system of equations obtained are studied using finite volume method with Harten–Lax–van Leer scheme for contact (HLLC). The original HLLC method is modified here to account for additional species and split energy equations. Higher order spatial accuracy is achieved using MUSCL reconstruction of the flow variables with van Albada limiter. The thermal equilibrium is considered for the analysis and the species data are generated using polynomial correlations. The single temperature model of Dunn and Kang is used for chemical relaxation. The computed results for a flow field over a hemispherical cylinder at Mach number of 16.34 obtained using the present solver are found to be promising and computationally (25%) more efficient. The present solver captures physically correct solution as the entropy conditions are satisfied automatically during the computations. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical eigenvalue study of preconditioned non‐equilibrium transport equations

The finite element integration of non‐equilibrium contaminant transport in porous media yields sparse, unsymmetric, real or complex equations, which may be solved by iterative projection methods, such as Bi‐CGSTAB and TFQMR, on condition that they are effectively preconditioned. To ensure a fast convergence, the eigenspectrum of the preconditioned equations has to be very compact around unity. Compactness is generally measured by the spectral condition number. In difficult advection‐dominated problems, however, the condition number may be large and nevertheless, convergence may be good. A numerical study of the preconditioned eigenspectrum of a representative test case is performed using the incomplete triangular factorization. The results show that preconditioning eliminates most of the original complex eigenvalues, and that compactness is not necessarily jeopardized by a large condition number. Quite surprisingly, it is shown that the preconditioned complex problem may have a more compact real eigenspectrum than the equivalent real problem. Copyright © 1999 John Wiley & Sons, Ltd.     

An implicit two‐dimensional non‐hydrostatic model for free‐surface flows

An implicit finite volume model in sigma coordinate system is developed to simulate two‐dimensional (2D) vertical free surface flows, deploying a non‐hydrostatic pressure distribution. The algorithm is based on a projection method which solves the complete 2D Navier–Stokes equations in two steps. First the pressure term in the momentum equations is excluded and the resultant advection–diffusion equations are solved. In the second step the continuity and the momentum equation with only the pressure terms are solved to give a block tri‐diagonal system of equation with pressure as the unknown. This system can be solved by a direct matrix solver without iteration. A new implicit treatment of non‐hydrostatic pressure, similar to the lower layers is applied to the top layer which makes the model free of any hydrostatic pressure assumption all through the water column. This treatment enables the model to evaluate both free surface elevation and wave celerity more accurately. A series of numerical tests including free‐surface flows with significant vertical accelerations and nonlinear behaviour in shoaling zone are performed. Comparison between numerical results, analytical solutions and experimental data demonstrates a satisfactory performance. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

The combined effects of non‐planarity and asymmetry on primary and secondary flows in the small bronchial tubes

The laminar flow in the small bronchial tubes is quite complex due to the presence of vortex‐dominated, secondary flows. In this paper, we report the results of a numerical investigation of the simultaneous effects of asymmetric and non‐planar branching on the primary and secondary flows in the small bronchial tubes, i.e. generations 6–12. We simulate steady‐state inspiratory flow at a Reynolds number of 1000 in three‐generation, asymmetric planar and non‐planar bronchial tube models. The non‐planar model was defined by applying a 90° out‐of‐plane rotation to the third‐generation branches. A detailed mesh refinement study was performed in order to demonstrate mesh independence. Significant differences were observed between flows in the planar and non‐planar models. An uneven mass flow distribution was observed in the non‐planar model in contrast to the evenly distributed mass flow in the planar model. The secondary flows created symmetric vortex patterns in the planar model, whereas vortex symmetry was lost in the non‐planar model. These results illustrate the importance of incorporating asymmetry in addition to non‐planarity in the geometric models. Copyright © 2008 John Wiley & Sons, Ltd.     

Analysis of hypersonic flows using finite elements with Taylor–Galerkin scheme

The aim of the present work is to introduce a formulation for the numerical analysis of three‐dimensional thermochemical non‐equilibrium hypersonic flows, using the finite element method and the Taylor–Galerkin scheme and adopting Park's 2‐temperature, 5‐species (N2, O2, NO, N and O) and 17‐reaction model. Examples using Euler and Navier–Stokes equations are included and compared with experimental and numerical works presented by other authors. The results are close to those analysed by other researches and a good computational performance was obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation and experimental validation of heat transfer within rotating flows for three‐dimensional non‐axisymmetric,turbulent conditions

A control volume type numerical methodology for the analysis of steady three‐dimensional rotating flows with heat transfer, in both laminar and turbulent conditions, is implemented and experimentally tested. Non‐axisymmetric momentum and heat transfer phenomena are allowed for. Turbulent transport is alternatively represented through three existing versions of the k–ε model that were adjusted to take into account the turbulence anisotropy promoted by rotation, streamline curvature and thermal buoyancy. Their relative performance is evaluated by comparison of calculated local and global heat balances with those obtained through measurements in a laboratory device. A modified version of the Lam and Bremhorst, low Reynolds number model is seen to give the best results. A preliminary analysis focused on the flow structure and the transfer of heat is reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

A three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

Generalized lattice‐BGK concept for thermal and chemically reacting flows at low Mach numbers

The lattice‐BGK method has been extended by introducing additional, free parameters in the original formulation of the lattice‐BGK methods. The relationship between these parameters and the macroscopic moment equations is analysed by Taylor series and Chapman–Enskog expansion. The parameters are determined from the macroscopic moment equations by comparisons with the governing equations to be modelled. Extensions are presented for the Navier–Stokes equations at low Mach numbers in Cartesian or axisymmetric coordinates with constant or variable density, for scalar convection–diffusion equations and for equations of Poisson type. The generalized lattice‐BGK concept is demonstrated by two applications of chemical engineering. These are the computation of chemically reacting flow through an axisymmetric reactor and of the transport and deposition of particles to filters under the action of different forces. Copyright © 2006 John Wiley & Sons, Ltd.     

Study of the effect of the non‐orthogonality for non‐staggered grids—The results

This article presents the effect of the grid skewness on the ranges of the underrelaxation factors for pressure and velocity. The effect is reflected by the relationship between the numbers of iterations required and the ranges of the underrelaxation factors for a converged solution. Four typical cavity flow problems are solved on non‐staggered grids for this purpose. Two momentum interpolation practices namely, practice A and practice B, together with SIMPLE, SIMPLEC and SIMPLER algorithms are employed. The results show that the ranges of the pressure underrelaxation factor values for convergence exist if the SIMPLE algorithm is used, while no restrictions are observed if the SIMPLEC algorithm is used. From the curves obtained using the SIMPLER algorithm, the ranges of those based on practice B are wider than those based on practice A. Copyright © 1999 John Wiley & Sons, Ltd.     

An aeroacoustic hybrid approach for non‐isothermal flows at low Mach number

This paper presents an aeroacoustic hybrid technique for the study of non‐isothermal flows at low Mach number. The flow dynamics and the acoustic production and propagation are computed separately. The fully compressible Navier–Stokes equations are modified through an expansion of the physical quantities using a low Mach number approximation. Compressibility effects are thus removed in the CFD while inhomogeneities of the flow related to heat transfer are preserved. One advantage is a reduction of the time step constraint. Another advantage is that the Mach number does not appear explicitly and a simple rescaling allows a study over a relatively wide band of subsonic Mach number flows with a single dynamic simulation. Compatible acoustic source terms for LEE based propagation have been defined and the procedure is implemented in the case of a temporal mixing layer. Compressible simulations for Mach numbers of 0.2, 0.3 and 0.4 are compared with the numerical results obtained using the proposed method. Very good agreement is obtained even at relatively high subsonic Mach number demonstrating the efficiency of the proposed technique. Copyright © 2004 John Wiley & Sons, Ltd.     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is solved using a finite element method (FEM). A recovery technique has been implemented to improve the FEM solution. The velocity is calculated by a numerical differentiation technique. The corresponding algebraic equations are solved by the conjugate gradient method with a symmetric successive overrelaxation (SSOR) preconditioner. The radiation condition at a truncated boundary is imposed based on the combination of a damping zone and the Sommerfeld condition. This paper (Part 1) focuses on the technical procedure, while Part 2 [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 2. Numerical results and validation. International Journal for Numerical Methods in Fluids 2001] gives detailed numerical results, including validation, for the cases of steep waves interacting with one or two vertical cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

A Riemann solver and upwind methods for a two‐phase flow model in non‐conservative form

We present a theoretical solution for the Riemann problem for the five‐equation two‐phase non‐conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non‐conservative methods to solve the general initial‐boundary value problem for the two‐phase flow model in non‐conservative form. The basic upwind scheme constructed is the non‐conservative analogue of the Godunov first‐order upwind method. Second‐order methods in space and time are then constructed via the MUSCL and ADER approaches. The methods are systematically assessed via a series of test problems with theoretical solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel implementation of a non‐hydrostatic model for free surface flows with semi‐Lagrangian advection treatment

The parallel implementation of an unstructured‐grid, three‐dimensional, semi‐implicit finite difference and finite volume model for the free surface Navier–Stokes equations (UnTRIM ) is presented and discussed. The new developments are aimed to make the code available for high‐performance computing in order to address larger, complex problems in environmental free surface flows. The parallelization is based on the mesh partitioning method and message passing and has been achieved without negatively affecting any of the advantageous properties of the serial code, such as its robustness, accuracy and efficiency. The key issue is a new, autonomous parallel streamline backtracking algorithm, which allows using semi‐Lagrangian methods in decomposed meshes without compromising the scalability of the code. The implementation has been carefully verified not only with simple, abstract test cases illustrating the application domain of the code but also with advanced, high‐resolution models presently applied for research and engineering projects. The scheme performance and accuracy aspects are researched and discussed. Copyright © 2008 John Wiley & Sons, Ltd.