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.     

NUMERICAL INVESTIGATION OF TURBULENT SHALLOW RECIRCULATING FLOWS BY A QUASI-THREE-DIMENSIONAL k–ϵMODEL

A quasi-three-dimensional multilayer k– ϵ model has been developed to simulate turbulent recirculating flows behind a sudden expansion in shallow waters. The model accounts for the vertical variation in the flow quantities and eliminates the problem of closure for the effective stresses resulting from the depth integration of the non-linear convective accelerations found in the widely used depth- integrated models. The governing equations are split into three parts in the finite difference solution: advection, dispersion and propagation. The advection part is solved using the four-node minimax–characteristics method. The dispersion and propagation parts are treated by the central difference method, the former being solved explicitly and the latter implicitly using the Gauss–Seidel iteration method. The relative effect of bed-generated turbulence and transverse shear-generated turbulence on the recirculating flow has been studied in detail. In comparison with the results computed by the depth-integrated k–ϵ model, the results computed by the present model are found to be closer to the reported data.     

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.     

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.     

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.     

BOUNDARY TREATMENT AND AN EFFICIENT PRESSURE ALGORITHM FOR INTERNAL TURBULENT FLOWS USING THE PDF METHOD

The generalized Langevin model, which is used to model the motion of stochastic particles in the velocity–composition joint probability density function (PDF) method for reacting turbulent flows, has been extended to incorporate solid wall effects. Anisotropy of Reynolds stresses in the near-wall region has been addressed. Numerical experiments have been performed to demonstrate that the forces in the near-wall region of a turbulent flow cause the stochastic particles approachi ng a solid wall to reverse their direction of motion normal to the wall and thereby, leave the near-wall layer. This new boundary treatment has subsequently been implemented in a full-scale problem to prove its validity. The test problem considered here is that of an isothermal, non-reacting turbulent flow in a two-dimensional channel with plug inflow and a fixed back-pressure. An efficient pressure correction method, developed in the spirit of the PISO algorithm, has been implemented. The pressure correction strategy is easy to implement and is completely consistent with the time- marching scheme used for the solution of the Lagrangian momentum equations. The results show remarkable agreement with both k–ϵ and algebraic Reynolds stress model calculations for the primary velocity. The secondary flow velocity and the turbulent moments are in better agreement with the algebraic Reynolds stress model predictions than the k– ϵ predictions. © 1997 by 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.     

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.     

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.     

Computational Modelling of Turbulent Mixing in Confined Swirling Environment Under Constant and Variable Density Conditions

A high-intensity swirling flow in a model combustor subjected to large density variations has been examined computationally. The focus is on the Favre-averaged Navier–Stokes computations of the momentum and scalar transport employing turbulence models based on the differential second-moment closure (SMC) strategy. An updated version of the basic, high-Reynolds number SMC model accounting for a quadratic expansion of both the pressure–strain and dissipation tensors and a near-wall SMC model were used for predicting the mean velocity and turbulence fields. The accompanied mixing between the annular swirling air flow and the central non-swirling helium jet was studied by applying three scalar flux models differing mainly in the model formulation for the pressure-scalar gradient correlation. The computed axial and circumferential velocities agree fairly well with the reference experiment [So et al., NASA Contractor Report 3832, 1984; Ahmed and So, Exp. Fluids 4 (1986) 107], reproducing important features of such a weakly supercritical flow configuration (tendency of the flow core to separate). Although the length at which the mixing was completed was reproduced in reasonable agreement with the experimental results, the mixing activity in terms of the spreading rate of the shear/mixing layer, that is its thickness, was somewhat more intensive. Prior to these investigations, the model applied was validated by computing the transport of the passive scalar in the non-swirling (Johnson and Bennet, Report NASA CR-165574, UTRC Report R81-915540-9, 1981) and swirling (Roback and Johnson, NASA Contractor Report 168252, 1983) flow in a model combustor.     

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.     

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.     

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.     

Evaluation of one‐ and two‐equation low‐Re turbulence models. Part I—Axisymmetric separating and swirling flows

This first segment of the two‐part paper systematically examines several turbulence models in the context of three flows, namely a simple flat‐plate turbulent boundary layer, an axisymmetric separating flow, and a swirling flow. The test cases are chosen on the basis of availability of high‐quality and detailed experimental data. The tested turbulence models are integrated to solid surfaces and consist of: Rodi's two‐layer k–ε model, Chien's low‐Reynolds number k–ε model, Wilcox's k–ω model, Menter's two‐equation shear‐stress‐transport model, and the one‐equation model of Spalart and Allmaras. The objective of the study is to establish the prediction accuracy of these turbulence models with respect to axisymmetric separating flows, and flows of high streamline curvature. At the same time, the study establishes the minimum spatial resolution requirements for each of these turbulence closures, and identifies the proper low‐Mach‐number preconditioning and artificial diffusion settings of a Reynolds‐averaged Navier–Stokes algorithm for optimum rate of convergence and minimum adverse impact on prediction accuracy. Copyright © 2003 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.     

Near‐wall turbulence modelling with enhanced dissipation

A low‐Reynolds number k–ε turbulence model is proposed that incorporates diffusion terms and modified C_ε(1,2) coefficients to amplify the level of dissipation in non‐equilibrium flow regions, thus reducing the kinetic energy and length scale magnitudes to improve prediction of adverse pressure gradient flows, involving flow separation and reattachment. Unlike the conventional k–ε model, it requires no wall function/distance parameter that bridges the near‐wall integration. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation (DNS) and experimental data. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems

A numerical solution for shallow-water flow is developed based on the unsteady Reynolds-averaged Navier–Stokes equations without the conventional assumption of hydrostatic pressure. Instead, the non-hydrostatic pressure component may be added in regions where its influence is significant, notably where bed slope is not small and separation in a vertical plane may occur or where the free-surface slope is not small. The equations are solved in the σ-co-ordinate system with semi-implicit time stepping and the eddy viscosity is calculated using the standard k–ϵ turbulence model. Conventionally, boundary conditions at the bed for shallow-water models only include vertical diffusion terms using wall functions, but here they are extended to include horizontal diffusion terms which can be significant when bed slope is not small. This is consistent with the inclusion of non-hydrostatic pressure. The model is applied to the 2D vertical plane flow of a current over a trench for which experimental data and other numerical results are available for comparison. Computations with and without non-hydrostatic pressure are compared for the same trench and for trenches with smaller side slopes, to test the range of validity of the conventional hydrostatic pressure assumption. The model is then applied to flow over a 2D mound and again the slope of the mound is reduced to assess the validity of the hydrostatic pressure assumption. © 1998 John Wiley & Sons, Ltd.     

Buffeting in transonic flow prediction using time‐dependent turbulence model

In transonic flow conditions, the shock wave/turbulent boundary layer interaction and flow separations on wing upper surface induce flow instabilities, ‘buffet’, and then the buffeting (structure vibrations). This phenomenon can greatly influence the aerodynamic performance. These flow excitations are self‐sustained and lead to a surface effort due to pressure fluctuations. They can produce enough energy to excite the structure. The objective of the present work is to predict this unsteady phenomenon correctly by using unsteady Navier–Stokes‐averaged equations with a time‐dependent turbulence model based on the suitable (k–ε) turbulent eddy viscosity model. The model used is based on the turbulent viscosity concept where the turbulent viscosity coefficient (Cμ) is related to local deformation and rotation rates. To validate this model, flow over a flat plate at Mach number of 0.6 is first computed, then the flow around a NACA0012 airfoil. The comparison with the analytical and experimental results shows a good agreement. The ONERA OAT15A transonic airfoil was chosen to describe buffeting phenomena. Numerical simulations are done by using a Navier–Stokes SUPG (streamline upwind Petrov–Galerkin) finite‐element solver. Computational results show the ability of the present model to predict physical phenomena of the flow oscillations. The unsteady shock wave/boundary layer interaction is described. Copyright © 2005 John Wiley & Sons, Ltd.     

Verification testing in computational fluid dynamics: an example using Reynolds‐averaged Navier–Stokes methods for two‐dimensional flow in the near wake of a circular cylinder

Verification testing was performed for various Reynolds‐averaged Navier–Stokes methods for uniform flow past a circular cylinder at Re= 5232. The standard and renormalized group (RNG) versions of the k–ε method were examined, along with the Boussinesq, Speziale and Launder constitutive relationships. Wind tunnel experiments for flow past a circular cylinder were also performed to obtain a comparative data set. Preliminary studies demonstrate poor convergence for the Speziale relationship. Verification testing with the standard and RNG k–ε models suggests that the simulations exhibit global monotonic convergence for the Boussinesq models. However, the global order of accuracy of the methods was much lower than the expected order of accuracy of 2. For this reason, pointwise convergence ratios and orders of accuracy were computed to show that not all sampling locations had converged (standard k–ε model: 19% failed to converge; RNG k–ε model: 14% failed to converge). When the non‐convergent points were removed from consideration, the average orders of accuracy are closer to the expected value (standard k–ε model: 1.41; RNG k–ε model: 1.27). Poor iterative and global grid convergence was found for the RNG k–ε/Launder model. The standard and RNG k–ε models with the Boussinesq relationship were compared with experimental data and yielded results significantly different from the experiments. Copyright © 2003 John Wiley & Sons, Ltd.