An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 John Wiley & Sons, Ltd.     

A control-volume based finite element method for three-dimensional incompressible turbulent fluid flow,heat transfer,and related phenomena

A control-volume based finite element method of equal-order type for three-dimensional incompressible turbulent fluid flow, heat transfer, and related phenomena is presented. The discretization equations are based mainly on the physics of the phenomena under consideration, more than on mathematical arguments. Special emphasis is devoted to the discretization of the convective terms and the continuity equation, and to the treatment of the boundary conditions imposed by the use of a high Reynolds k-?, type turbulence model. The pressure-velocity coupling in the fluid flow calculation is made from a derivative of the original SIMPLER method, without pressure correction. The discretized equations are solved in a sequential, rather than a coupled, form with significant advantage in the required computer time and storage. The method is an extension of a former version proposed by us for two-dimensional, laminar problems, and is here successfully applied to the following situations: three-dimensional deflected turbulent jet, and flows in 90° and 45° junctions of ducts with rectangular cross sections. The calculated results are in very good agreement with the experimental and numerical (obtained with the well established finite difference method) data available in the literature.     

High-order strand grid methods for shock turbulence interaction

ABSTRACT

In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.     

An immersed boundary/wall modeling method for RANS simulation of compressible turbulent flows

In this paper, an immersed boundary (IB) method is developed to simulate compressible turbulent flows governed by the Reynolds‐averaged Navier‐Stokes equations. The flow variables at the IB nodes (interior nodes in the immediate vicinity of the solid wall) are evaluated via linear interpolation in the normal direction to close the discrete form of the governing equations. An adaptive wall function and a 2‐layer wall model are introduced to reduce the near‐wall mesh density required by the high resolution of the turbulent boundary layers. The wall shear stress modified by the wall modeling technique and the no‐penetration condition are enforced to evaluate the velocity at an IB node. The pressure and temperature at an IB node are obtained via the local simplified momentum equation and the Crocco‐Busemann relation, respectively. The SST k ? ω and S‐A turbulence models are adopted in the framework of the present IB approach. For the Shear‐Stress Transport (SST) k ? ω model, analytical solutions in near‐wall region are utilized to enforce the boundary conditions of the turbulence equations and evaluate the turbulence variables at an IB node. For the S‐A model, the turbulence variable at an IB node is calculated by using the near‐wall profile of the eddy viscosity. In order to validate the present IB approach, numerical experiments for compressible turbulent flows over stationary and moving bodies have been performed. The predictions show good agreements with the referenced experimental data and numerical results.     

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.     

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.     

Turbulent free surface flow simulation using a multilayer model

Based on the steady hydrodynamic equations, a multilayer (ML) model has been formulated for simulating turbulent flow in open channels. The model is imposed on a general curvilinear co-ordinate system with non-staggered finite volume discretization. The turbulent quantities in the model are described by the layer-averaged K-ε turbulence model with standard coefficients. Assuming a vertical hydrostatic pressure distribution, a depth correction scheme, originating in the Rhie and Chow approach for confined flows, is incorporated into the SIMPLE procedure to compute the water surface. Using the multilayer model, flows in a 180° channel bend, near a groin, and in straight open channels are computed. The results are compared with experimental data and with calculations of a depth-averaged model (DAV) having three-dimensional effect corrections. The comparisons show that the predictions of the ML model on mean flow values are in good agreement with the available data and are better than those of the DAV model. The vertical distribution of the turbulent energy dissipation rate is also shown to agree well with the open-channel measurements.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Modelling a recirculating density-driven turbulent flow

A mathematical model of turbulent density-driven flows is presented and is solved numerically. A form of the k–? turbulence model is used to characterize the turbulent transport, and both this non-linear model and a sediment transport equation are coupled with the mean-flow fluid motion equations. A partitioned, Newton–Raphson-based solution scheme is used to effect a solution. The model is applied to the study of flow through a circular secondary sedimentation basin.     

MULTIGRID CONVERGENCE ACCELERATION FOR TURBULENT SUPERSONIC FLOWS

A multigrid convergence acceleration technique has been developed for solving both the Navier–Stokes and turbulence transport equations. For turbulence closure a low-Reynolds-number q–ω turbulence model is employed. To enable convergence, the stiff non-linear turbulent source terms have to be treated in a special way. Further modifications to standard multigrid methods are necessary for the resolution of shock waves in supersonic flows. An implicit LU algorithm is used for numerical time integration. Several ramped duct test cases are presented to demonstrate the improvements in performance of the numerical scheme. Cases with strong shock waves and separation are included. It is shown to be very effective to treat fluid and turbulence equations with the multigrid method. A comparison with experimental data demonstrates the accuracy of the q–ω turbulence closure for the simulation of supersonic flows. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1019–1035, 1997.     

A three-dimensional hydrodynamic model in curvilinear co-ordinates with collocated grid

A three-dimensional hydrodynamic model has been developed for turbulent flows with free surface. In the horizontal x–y-plane, a boundary-fitted curvilinear co-ordinate system is adopted, while in the vertical direction, a σ-co-ordinate transformation is used to represent the free surface and bed topography or lower boundary. Using the finite volume method, the convection terms are discretized using Roe's second-order-accurate scheme. The governing equations are solved in a collocated grid system by a fractional three-step implicit algorithm that has been developed to handle the velocity–pressure–depth coupling problem of free surface incompressible fluid flows. The present study is the extension of previous work to three-dimensional turbulent flows. The model has been applied to three test cases. Comparison with available data shows that the model developed is successful, and is valuable to engineering application. © 1998 John Wiley & Sons, Ltd.     

Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids

This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulation of unsteady turbulent flows around moving aerofoils using the pseudo‐time method

The pseudo‐time formulation of Jameson has facilitated the use of numerical methods for unsteady flows, these methods have proved successful for steady flows. The formulation uses iterations through pseudo‐time to arrive at the next real time approximation. This iteration can be used in a straightforward manner to remove sequencing errors introduced when solving mean flow equations together with another set of differential equations (e.g. two‐equation turbulence models or structural equations). The current paper discusses the accuracy and efficiency advantages of removing the sequencing error and the effect that building extra equations into the pseudo‐time iteration has on its convergence characteristics. Test cases used are for the turbulent flow around pitching and ramping aerofoils. The performance of an implicit method for solving the pseudo‐steady state problem is also assessed. Copyright © 2000 John Wiley & Sons, Ltd.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

Numerical simulation of the unsteady behaviour of cavitating flows

A 2D numerical model is proposed to simulate unsteady cavitating flows. The Reynolds‐averaged Navier–Stokes equations are solved for the mixture of liquid and vapour, which is considered as a single fluid with variable density. The vapourization and condensation processes are controlled by a barotropic state law that relates the fluid density to the pressure variations. The numerical resolution is a pressure‐correction method derived from the SIMPLE algorithm, with a finite volume discretization. The standard scheme is slightly modified to take into account the cavitation phenomenon. That numerical model is used to calculate unsteady cavitating flows in two Venturi type sections. The choice of the turbulence model is discussed, and the standard RNG k–εmodel is found to lead to non‐physical stable cavities. A modified k–εmodel is proposed to improve the simulation. The influence of numerical and physical parameters is presented, and the numerical results are compared to previous experimental observations and measurements. The proposed model seems to describe the unsteady cavitation behaviour in 2D geometries well. Copyright © 2003 John Wiley & Sons, Ltd.     

A Fully—Implicit 2‐D Navier-Stokes Algorithm for Unstructured Grids

A fully-implicit algorithm is developed for the two-dimensional, compressible, Favre-averaged Navier-Stokes equations. It incorporates the standard k-? turbulence model of Launder and Spalding and the low Reynolds number correction of Chien. The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. A generalization of wall functions including pressure gradient effects is implemented to solve the near-wall region for turbulent flows using a separate algorithm and a hybrid grid. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, Trapezoidal or 3-Point Backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. The accuracy of the code and the efficiency of the solution strategy are presented for three test cases: a supersonic turbulent mixing layer, a supersonic laminar compression corner and a supersonic turbulent compression corner.     

基于滑移速度壁模型的复杂边界湍流大涡模拟

采用滑移速度壁模型实现了浸入边界方法与壁模型相结合的大涡模拟.本文首先分别采用平衡层模型和非平衡壁模型对周期山状流进行数值模拟,以考查在壁模型中考虑切向压力梯度的作用.数值结果表明,流场的压力对本文所采用的壁模型形式并不敏感,但是考虑切向压力梯度可以显著改进壁面摩擦力的计算结果,并且能够准确的预测强压力梯度区以及分离区内的流动平均统计特性.不考虑压力梯度效应的平衡层模型显著低估了壁面摩擦力的分布,同时无法准确预测分离区内的平均速度剖面.非平衡模型的修正项正比于切向压力梯度和壁面法向距离,因此在强压力梯度区或者网格较粗时,计算得到的平均压力和摩擦力分布以及流动的低阶统计量均与参考的实验和计算结果吻合.在此基础上,通过回转体绕流的大涡模拟考查了该方法用于模拟高雷诺数壁湍流的适用性,非平衡壁模型可以准确地捕捉流动的物理结构并较准确地预测其水动力学特性.结果表明,将浸入边界方法与非平衡滑移速度壁模型相结合的大涡模拟,有望成为数值模拟复杂边界高雷诺数壁湍流的工具.      

Finite volume computation of the turbulent flow over a hill employing 2D or 3D non-orthogonal collocated grid systems

A general numerical method for the solution of the complete Reynolds-averaged Navier-Stokes equations for 2D or 3D flows is described. The method uses non-orthogonal co-ordinates, Cartesian velocity components and a pressure-velocity-coupling algorithm adequate for non-staggered grid systems. The capability of the method and the overall performance of the κ–? eddy viscosity model are demonstrated by calculations of 2D and 3D flow over a hill. Solution error estimations based on fine grids, e.g. 320 × 192 control volumes, together with comparisons with standard turbulence model modifications, low-Reynoldsnumber or streamline curvature effects, have allowed the investigation of model drawbacks in predicting turbulent flows over surface-mounted hills.     

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.