Finite volume computation of incompressible turbulent flows in general co-ordinates on staggered grids

A brief review of the computation of incompressible turbulent flow in complex geometries is given. A 2D finite volume method for the calculation of turbulent flow in general curvilinear co-ordinates is described. This method is based on a staggered grid arrangement and the contravariant flux componets are chosen as primitive variables. Turbulence is modelled either by the standard k–ε model or by a k–ε model based on RNG theory. Convection is approximated with central differences for the mean flow quantities and a TVD-type MUSCL scheme for the turbulence equations. The sensitivity of the method to the grid properties is investigated. An application of this method to a complex turbulent flow is presented. The results of computations are compared with experimental data and other numerical solutions and are found to be satisfactory.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Anisotropic adaptive stabilized finite element solver for RANS models

Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.     

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.     

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

An unsteady incompressible Navier–Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high‐order‐accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.     

Turbulent boundary-layer analysis using finite elements

The use of finite element methods for turbulent boundary-layer flow is relatively recent and of limited extent.¹ In the present study, we extend the group variable approach of Fletcher and Fleer^2,3 to treat turbulent boundary layer flows with heat transfer using a two-equation turbulence model. The main concepts in the formulations include a Dorodnitsyn-type transformation which uses a velocity component as the transverse variable, a ‘variational’ formulation for the transformed equations using special test functions and development of a two-equation turbulence model in terms of the turbulent kinetic energy and turbulence dissipation rate as additional field variables. Several numerical test cases have been examined comparing the results with finite difference calculations and comparing the two-equation turbulence model with an algebraic turbulence model.     

Parallel large eddy simulation of turbulent flow around MIRA model using linear equal‐order finite element method

A parallel large eddy simulation code that adopts domain decomposition method has been developed for large‐scale computation of turbulent flows around an arbitrarily shaped body. For the temporal integration of the unsteady incompressible Navier–Stokes equation, fractional 4‐step splitting algorithm is adopted, and for the modelling of small eddies in turbulent flows, the Smagorinsky model is used. For the parallelization of the code, METIS and Message Passing Interface Libraries are used, respectively, to partition the computational domain and to communicate data between processors. To validate the parallel architecture and to estimate its performance, a three‐dimensional laminar driven cavity flow inside a cubical enclosure has been solved. To validate the turbulence calculation, the turbulent channel flows at Re_τ = 180 and 1050 are simulated and compared with previous results. Then, a backward facing step flow is solved and compared with a DNS result for overall code validation. Finally, the turbulent flow around MIRA model at Re = 2.6 × 10⁶ is simulated by using approximately 6.7 million nodes. Scalability curve obtained from this simulation shows that scalable results are obtained. The calculated drag coefficient agrees better with the experimental result than those previously obtained by using two‐equation turbulence models. Copyright © 2007 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.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

A non‐linear k–ε model with realizability for prediction of flows around bluff bodies

The incompressible flow around bluff bodies (a square cylinder and a cube) is investigated numerically using turbulence models. A non‐linear k–ε model, which can take into account the anisotropy of turbulence with less CPU time and computer memory then RSM or LES, is adopted as a turbulence model. In tuning of the model coefficients of the non‐linear terms are adjusted through the examination of previous experimental studies in simple shear flows. For the tuning of the coefficient in the eddy viscosity (=C_μ), the realizability constraints are derived in three types of basic 2D flow patterns, namely, a simple shear flow, flow around a saddle and a focal point. C_μ is then determined as a function of the strain and rotation parameters to satisfy the realizability. The turbulence model is first applied to a 2D flow around a square cylinder and the model performance for unsteady flows is examined focussing on the period and the amplitude of the flow oscillation induced by Karman vortex shedding. The applicability of the model to 3D flows is examined through the computation of the flow around a surface‐mounted cubic obstacle. The numerical results show that the present model performs satisfactorily to reproduce complex turbulent flows around bluff bodies. Copyright © 2003 John Wiley & Sons, Ltd.     

Computation of internal high-speed separated flow with modified B–L and J–K models

The existence of shock–turbulent boundary layer interactions lead to very complicated flow phenomena and pose a challenge for numerical simulation. In this paper, two turbulence models, the Baldwin–Lomax (B–L) model and the Johnson–King (J–K) model, which were originally developed for simple external flow simulation, are modified to model complex high-speed internal separated flows. The full Navier–Stokes solver used in this paper is based on a cell-centered finite volume method and multistepping time marching scheme. Both implicit residual smoothing and local time stepping techniques are incorporated to accelerate the convergence rate. To ensure the numerical stability with the present explicit scheme, a point-implicit treatment to the source term in the ordinary differential equation (ODE) of the J–K model has been developed and has proved to be very effective in modeling such a complex flow. An arc-bump channel flow case has been studied. Comparisons of computed results with experimental data show that the present solver, with the modified turbulence models, predicts the shock and the flow separation very well. The J–K model is found to predict the size of the separation bubble with a higher accuracy. © 1998 John Wiley & Sons, Ltd.     

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.     

The calculation of incompressible separated turbulent boundary layers

The algebraic turbulent model of Baldwin and Lomax was incorporated into the incompressible full Navier–Stokes code FIDAP. This model was extensively tested in the past in finite difference codes. We believe that the incorporation of the model also into the finite element code has resulted in a practical method to compute a variety of separated turbulent 2D flows. Firstly, we use the model to compute the attached flow about an aerofoil. Next, the application of the model to separated flows is presented by computing the flows at high angles of attack up to maximum lift. It is shown that the model is capable of predicting separation, steady stall and C_Lmax. As a difficult test of the model we compute the laminar separation bubble development directly using the full Navier–Stokes finite element code. As far as we know, this approach has not yet been reported. The importance of using an appropriate upwinding is discussed. When possible, comparison of computed results with experiments is presented and the agreement is good.     

A ghost-cell immersed boundary method for large eddy simulation of flows in complex geometries

An efficient ghost-cell immersed boundary (IB) method is proposed for large eddy simulations of three-dimensional incompressible flow in complex geometries. In the framework of finite volume method, the Navier–Stokes equations are integrated using an explicit time advancement scheme on a collocated mesh. Since the IB method is known to generate an unphysical velocity field inside the IB that violates the mass conservation of the cells near the IB, a new IB treatment is devised to eliminate the unphysical velocity generated near the IB and to improve the pressure distribution on the body surface. To validate the proposed method, both laminar and turbulent flow cases are presented. In particular, large eddy simulations were performed to simulate the turbulent flows over a circular cylinder and a sphere at subcritical Reynolds numbers. The computed results show good agreements with the published numerical and experimental data.     

基于S-A湍流模型的Runge-Kutta有限元算法

采用一方程S-A模型(Spalart-Allmaras模型)封闭雷诺时均N-S方程(RANS方程)进行湍流数值计算,可以减少方程求解数量,节约计算时间。本文对其进行了有限元数值算法研究,首先通过沿流线坐标变换,得到无对流项RANS方程,并引入三阶Runge-Kutta法对其进行时间离散;然后利用沿流线的Taylor展开解决坐标变换带来的网格更新的困难;最后采用Galerkin法进行空间离散,得到湍流模型的有限元算法。基于方柱绕流和覆冰输电线绕流模型,与试验结果进行对比,验证了该算法的有效性,与一阶数值算法相比,该算法在精度和收敛性方面更具优势。