Large eddy simulation of free surface shallow‐water flow

Shallow‐water flow with free surface frequently occurs in ambient water bodies, in which the horizontal scale of motion is generally two orders of magnitude greater than the water depth. To accurately predict this flow phenomenon in more detail, a three‐dimensional numerical model incorporating the method of large eddy simulation (LES) has been developed and assessed. The governing equations are split into three parts in the finite difference solution: advection, dispersion and propagation. The advection part is solved by the QUICKEST scheme. The dispersion part is solved by the central difference method and the propagation part is solved implicitly using the Gauss–Seidel iteration method. The model has been applied to free surface channel flow for which ample experimental data are available for verification. The inflow boundary condition for turbulence is generated by a spectral line processor. The computed results compare favourably with the experimental data and those results obtained by using a periodic boundary condition. The performance of the model is also assessed for the case in which anisotropic grids and filters with horizontal grid size of the order of the water depth are used for computational efficiency. The coarse horizontal grid was found to cause a significant reduction in the large‐scale turbulent motion generated by the bottom turbulence, and the turbulent motion is predominately described by the sub‐grid scale (SGS) terms. The use of the Smagorinsky model for SGS turbulence in this situation is found inappropriate. A parabolic mixing length model, which accounts for the filtered turbulence, is then proposed. The new model can reproduce more accurately the flow quantities. Copyright © 2000 John Wiley & Sons, Ltd.     

OPERATOR–SPLITTING COMPUTATION OF TURBULENT FLOW IN AN AXISYMMETRIC 180° NARROWING BEND USING SEVERAL k—ϵ MODELS AND WALL FUNCTIONS

This paper describes a finite element implementation of an operator-splitting algorithm for solving transient/steady turbulent flows and presents solutions for the turbulent flow in an axisymmetric 180° narrowing bend, a benchmark problem dealt with at the 1994 WUA-CFD annual meeting. Three k–ϵ based models are used: the standard linear k–ϵ model, a non-linear k–ϵ model and an RNG k–ϵ model. Flow separation after the bend, as observed in the experiment, is predicted by the RNG model and by both the linear and non-linear k–ϵε models with van Driest mixing length wall functions. Good agreement with experimental data of pressure distribution on bending walls is obtained by the present numerical simulation. Results show that there is very little difference between the linear and non-linear k–ϵε models in terms of predicted velocity fields and that the non-linearities mainly affect the distribution of turbulent normal stress and pressure, in analogy to the effect of second-order viscoelastic fluid models on laminar flow. Both the linear and non-linear k–ϵε models fail to predict any flow separation if logarithmic wall functions are used.     

Computations of strongly swirling flows with second‐moment closures

The present study is concerned with simulating turbulent, strongly swirling flows by eddy viscosity model and Reynolds stress transport model variants adopting linear and quadratic form of the pressure–strain models. Flows with different inlet swirl numbers, 2.25 and 0.85, were investigated. Detailed comparisons of the predicted results and measurements were presented to assess the merits of model variants. For the swirl number 2.25 case, due to the inherent capability of the Reynolds stress models to capture the strong swirl and turbulence interaction, both the linear and quadratic form of the pressure–strain models predict the flow adequately. In strong contrast, the k–ϵ model predicts an excessively diffusive flow fields. For the swirl number 0.85 case, both the k–ϵ and Reynolds stress model with linear pressure–strain process, show an excessive diffusive transport of the flow fields. The quadratic pressure–strain model, on the other hand, mimics the correct flow development with the recirculating region being correctly predicted. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical study of turbulent line puffs via the renormalization group (RNG) k–ϵ model

The time evolution of a line puff, a turbulent non-buoyant element with significant momentum, is studied using the renormalization group (RNG) k–ϵ model. The numerical results show that the puff motion is characterized by a vortex pair flow; the computed flow details and scalar mixing characteristics can be described by self-similar relations beyond a dimensionless time of around 30. The added mass coefficient of the puff motion is found to be approximately unity. The predicted puff flow and mixing rate are substantially similar to those obtained from the standard k–ϵ model and are well supported by experimental data. The computed scalar field reveals significant secondary concentration peaks trailing behind in the wake of the puff. The present results suggest that the overall mixing rate of a puff is primarily determined by the large-scale motion and that streamline curvature probably plays a minor role. © 1998 John Wiley & Sons, Ltd.     

Positivity preserving pointwise implicit schemes with application to turbulent compressible flat plate flow

A family of positivity preserving pointwise implicit schemes applicable to source term dominated problems is constructed, where the minimum order of spatial accuracy is one and the maximum is three. It is designed for achieving steady state numerical solutions and is constructed through the analysis of appropriate model problems, where the convective fluxes for the higher‐order members are prescribed by the Chakravarthy–Osher family of total variation diminishing (TVD) schemes. Multidimensionality is facilitated by operator splitting. Numerical experimentation confirms the stability, convergence, accuracy, positivity, and computational efficiency associated with the proposed schemes. These schemes are ideally suited to solving the low‐Reynolds number turbulent k–ϵ equations for which the positivity of k and ϵ and the presence of stiff source terms are critical issues. Hence, using a finite volume formulation of these schemes, the low‐Reynolds number Chien k–ϵ turbulence model is implemented for a flat plate geometry and a series of turbulent flow (steady state) computations are carried out to demonstrate the positivity, robustness, and reliability of the algorithm. The free‐stream and initial k and ϵ values are specified in a very simple manner. Algorithm convergence acceleration is achieved using Multigrid techniques. The k–ϵ model flow predictions are shown to be in agreement with empirical profiles. Copyright © 2001 John Wiley & Sons, Ltd.     

MODELLING OF RELEASE OF GAS FROM HIGH-PRESSURE PIPELINES

The problem investigated is the break of a high-pressure pipeline carrying natural single-phase gas which may condensate (retrograde) when the pressure drops. Single-phase non-ideal gas is assumed using a general- ized equation of state. Taking advantage of the choked massflow condition, the break is split into a pipe flow problem and a dispersion flow problem, both solved using a finite difference control volume scheme. The transient flow field from the pipeline break location is expanded analytically, using an approximation of the governing equations, until ambient pressure is reached and matched to the corresponding gas dispersion flow field using as subgrid model a jet box with a time-varying equivalent nozzle area as an internal boundary of the dispersion domain. The turbulence models used for the pipe and dispersion flow fields are an empirical model of Reichard and the k–ϵ model for buoyant flow respectively. The pipe flow simulations indicate that the flow from the pipeline might include dispersed condensate which will affect quantitatively the mass flow rate from the pipeline and qualitatively the gas dispersion if the condensate rains out. The transient dispersion simulation shows that an entrainment flow field develops and mixes supersaturated gas with ambient warmer air to an unsaturated mixture. Because of the inertia of the ambient air, it takes time to develop the entrainment flow field. As a consequence of this and the decay of the mass flow with time, the lower flammability limit of the gas–air mixture reaches its most remote downstream position relatively early in the simulation (about 15 s) and withdraws closer to the break location.     

AN ADAPTIVE FINITE ELEMENT METHOD FOR A TWO-EQUATION TURBULENCE MODEL IN WALL-BOUNDED FLOWS

This paper presents an adaptive finite element method for solving incompressible turbulent flows using a k–ϵ model of turbulence. Solutions are obtained in primitive variables using a highly accurate quadratic finite element on unstructured grids. A projection error estimator is presented that takes into account the relative importance of the errors in velocity, pressure and turbulence variables. The efficiency and convergence rate of the methodology are evaluated by solving problems with known analytical solutions. The method is then applied to turbulent flow over a backward-facing step and predictions are compared with experimental measurements. © 1997 John Wiley & Sons, Ltd.     

Validation of turbulence models in a simulated air-conditioning unit

Details are given of a study to obtain experimental data in an idealized environment for the purpose of evaluating the corresponding computational predictions and which supplement parallel measurements made in actual packaged air-conditioning units. The system consisted of a purpose-built low-speed wind tunnel with a working section constructed to reproduce particular features of the real units. In the experiment, both the mean velocity profiles and turbulence properties of the flow are obtained from triple-hot-wire anemometry measurements. A numerical model, based on finite volume methodology, was used to obtain the solution of the Reynolds-averaged Navier–Stokes equations for incompressible isothermal flow. The Reynolds stress terms in the equations are calculated using the standard k–ϵ model and second-moment closure (Reynolds stress) models. The accuracy of the two models was evaluated against the experimental measurements made 10 mm downstream of a baffle. The results show that the standard k–ϵ model gave the better agreement except in regions of strong recirculation. © 1997 John Wiley & Sons, Ltd.     

An implicit three‐dimensional numerical model to simulate transport processes in coastal water bodies

A three‐dimensional baroclinic numerical model has been developed to compute water levels and water particle velocity distributions in coastal waters. The numerical model consists of hydrodynamic, transport and turbulence model components. In the hydrodynamic model component, the Navier–Stokes equations are solved with the hydrostatic pressure distribution assumption and the Boussinesq approximation. The transport model component consists of the pollutant transport model and the water temperature and salinity transport models. In this component, the three‐dimensional convective diffusion equations are solved for each of the three quantities. In the turbulence model, a two‐equation k–ϵ formulation is solved to calculate the kinetic energy of the turbulence and its rate of dissipation, which provides the variable vertical turbulent eddy viscosity. Horizontal eddy viscosities can be simulated by the Smagorinsky algebraic sub grid scale turbulence model. The solution method is a composite finite difference–finite element method. In the horizontal plane, finite difference approximations, and in the vertical plane, finite element shape functions are used. The governing equations are solved implicitly in the Cartesian co‐ordinate system. The horizontal mesh sizes can be variable. To increase the vertical resolution, grid clustering can be applied. In the treatment of coastal land boundaries, the flooding and drying processes can be considered. The developed numerical model predictions are compared with the analytical solutions of the steady wind driven circulatory flow in a closed basin and of the uni‐nodal standing oscillation. Furthermore, model predictions are verified by the experiments performed on the wind driven turbulent flow of an homogeneous fluid and by the hydraulic model studies conducted on the forced flushing of marinas in enclosed seas. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element simulation of turbulent Couette–Poiseuille flows using a low Reynolds number k–ϵ model

Developing Couette–Poiseuille flows at Re=5000 are studied using a low Reynolds number k–ϵ two‐equation model and a finite element formulation. Mesh‐independent solutions are obtained using a standard Galerkin formulation and a Galerkin/least‐squares stabilized method. The predictions for the velocity and turbulent kinetic energy are compared with available experimental results and to the DNS data. Second moment closure's solutions are also compared with those of the k–ϵ model. The deficiency of eddy viscosity models to predict dissymmetric low Reynolds number channel flows has been demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

Parallel computation of atmospheric pollutant dispersion under unstably stratified atmosphere

Pollutant dispersion under unstably stratified atmosphere was investigated numerically using the finite element method. The effects of atmospheric stability on plume trajectory were studied using a three-dimensional second-order closure dispersion model. The numerical model was implemented using domain decomposition method and carried out using a parallel computer. The computation accelerates significantly and the size of computation can be largely increased as a result of the parallelism. A passive contaminant point source was placed at the middle of the convective boundary layer to simulate the atmospheric dispersion. The requirement of the input of dispersion coefficients in k-theory and Gaussian models was replaced with direct input of turbulence flow data. It was found that the present numerical model can predict several non-Gaussian plume behaviours and the computed results agreed well with findings from experimental observations. © 1998 John Wiley & Sons, Ltd.     

Computational study on turbulent flows around modern tanker hull forms

To enhance the applicability of CFD techniques to the hull form design of modern commercial ships, an efficient and robust numerical method for turbulent flow calculation is developed. The preprocessor is composed of hull form presentation, surface mesh generation, and field grid generation. The finite‐volume method is utilised to solve the Reynolds‐averaged Navier–Stokes equations. Three k–? turbulence models, i.e., the standard k–? model (SKE), the RNG‐based k–? model (RNG), and the realisable k–? model (RKE), are evaluated to investigate the difference caused by the turbulence model. The developed numerical method is applied to two practical VLCC hull forms with the same forebody and the slightly different afterbodies in order to find out whether CFD can capture the difference of stern flow due to hull form variation. It is found that RKE successfully predict the strength and location of bilge vortex, while SKE and RNG fail. However, all three models can provide the right information on the nominal wake difference between two hull forms. Copyright © 2002 John Wiley & Sons, Ltd.     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

This paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a k-? turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ? is proposed as a numerical treatment for physically meaningless values of k and ?. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

COMPUTATIONAL STUDIES OF IMPINGING JETS USING K-ϵ TURBULENCE MODELS

This paper reports numerical modelling of impinging jet flows using Rodi and Malin corrections to the k–ϵ turbulence model, carried out using the PHOENICS finite volume code. Axisymmetric calculations were performed on single round free jets and impinging jets and the effects of pressure ratio, height and nozzle exit velocity profile were investigated numerically. It was found that both the Rodi and Malin corrections tend to improve the prediction of the hydrodynamic field of free and impinging jets but still leave significant errors in the predicted wall jet growth. These numerical experiments suggest that conditions before impingement significantly affect radial wall jet development, primarily by changing the wall jet's initial thickness.     

NUMERICAL STUDY OF TURBULENT FORCED CONVECTION IN A PERIODICALLY RIBBED CHANNEL WITH OSCILLATORY THROUGHFLOW

Numerical computations are performed on the fully developed flow and heat transfer in a periodically ribbed channel with oscillatory throughflow. A uniform heat flux is imposed at the lower plate of the channel. An externally sustained pressure gradient varies sinusoidally in time. A low-turbulent-Reynolds-number version of the k–ϵ two-equation model of turbulence is invoked, together with a preferential dissipation modification, to predict the complex turbulent flow field. Computed results indicate that much heat transfer enhancement is expected by increasing the Womersley number, which measures the relative strength of the oscillatory motion to the viscous effects.     

NUMERICAL MODELLING OF THREE-DIMENSIONAL FLY-ASH FLOW IN POWER UTILITY BOILERS

A two-fluid Eulerian model in combination with a particle–wall collision model and generalized Eulerian boundary conditions for the particulate phase is employed to predict complex three- dimensional fly-ash flows which often cause severe erosion to boiler tubes located in power utility boilers. Mean momentum and mass conservation equations are solved for each phase using a finite volume scheme with two-way coupling and a modified renormalization group (RNG)-based k –ϵ turbulence model. Comparison of predicted particle concentration with measured data is made and excellent agreement is obtained. The detailed character of the particulate velocity field and concentration just downstream of the 180° bend shows a marked dependence on the Stokes number not previously reported. © 1997 by John Wiley & Sons, Ltd.     

PERFORMANCE PREDICTION OF A BACKSWEPT CENTRIFUGAL IMPELLER AT OFF-DESIGN POINT CONDITIONS

The internal flows inside a backswept centrifugal impeller at off-design point conditions are analysed by solving the fully three- dimensional Navier–Stokes equations coupled with the k–ϵ two-equation turbulence model. Conditions analysed have flow rates of 85 per cent and 113 per cent of the design rate. The calculated results are compared with published experimental data and the comparison shows good agreement. A jet wake structure occurs near the outlet of the impeller for each of the two off-design flow rates and the location and size of the jet wake structure seem to be independent of the flow rate.     

Development of a hybrid k-ε turbulence model for swirling recirculating flows under moderate to strong swirl intensities

A hybrid k-ε turbulence model, based on the concept that the modification of anisotropic effects should not be made in the flow regions inherent to small streamline curvatures, has been developed and examined with the swirling recirculating flows, with the swirl levels ranging from 0·6 to 1·23 in abrupt pipe expansion. A fairly satisfactory agreement of model predictions with the experimental data shows that this hybrid k-ε model can perform better simulation of swirling recirculating flows as compared to the standard k-ε model and the modified k-ε model proposed by Abujelala and Lilley.