Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle‘s governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.     

Computation of the stability derivatives via CFD and the sensitivity equations

The method to calculate the aerodynamic stability derivates of aircrafts by using the sensitivity equations is ex- tended to flows with shock waves in this paper. Using the newly developed second-order cell-centered finite volume scheme on the unstructured-grid, the unsteady Euler equations and sensitivity equations are solved simultaneously in a non-inertial frame of reference, so that the aerodynamic stability derivatives can be calculated for aircrafts with complex geometries. Based on the numerical results, behavior of the aerodynamic sensitivity parameters near the shock wave is discussed. Furthermore, the stability derivatives are analyzed for supersonic and hypersonic flows. The numerical results of the stability derivatives are found in good agree- ment with theoretical results for supersonic flows, and variations of the aerodynamic force and moment predicted by the stability derivatives are very close to those obtained by CFD simulation for both supersonic and hypersonic flows.     

MHD effects on free convective flow over moving semi-infinite vertical cylinder with temperature oscillation

Numerical solutions of magnetodynamics(MHD) effects on the free convective flow of an incompressible viscous fluid past a moving semi-infinite vertical cylinder with temperature oscillation are presented.The dimensionless,unsteady,non-linear,and coupled governing partial differential equations are solved by using an implicit finite difference method of the Crank-Nicolson type.The velocity,temperature,and concentration profiles are studied for various parameters.The local skin-friction,the average skin-fr...     

Generation of linear and nonlinear waves in numerical wave tank using clustering technique-volume of fluid method

A two-dimensional(2D) numerical model is developed for the wave simulation and propagation in a wave flume.The fluid flow is assumed to be viscous and incompressible,and the Navier-Stokes and continuity equations are used as the governing equations.The standard k-ε model is used to model the turbulent flow.The NavierStokes equations are discretized using the staggered grid finite difference method and solved by the simplified marker and cell(SMAC) method.Waves are generated and propagated using a piston type wave maker.An open boundary condition is used at the end of the numerical flume.Some standard tests,such as the lid-driven cavity,the constant unidirectional velocity field,the shearing flow,and the dam-break on the dry bed,are performed to valid the model.To demonstrate the capability and accuracy of the present method,the results of generated waves are compared with available wave theories.Finally,the clustering technique(CT) is used for the mesh generation,and the best condition is suggested.     

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature

An analysis is performed to study the magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The dimensionless governing equations are unsteady, two-dimensional, coupled and non-linear governing equations. It is found that the magnetic field parameter has a retarding effect on the velocities of air and water.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

UNSTEADY／STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.     

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.     

Fully coupled flow-induced vibration of structures under small deformation with GMRES method

Lagrangian-Eulerian formulations based on a generalized variational principle of fluid-solid coupling dynamics are established to describe flow-induced vibration of a structure under small deformation in an incompressible viscous fluid flow. The spatial discretization of the formulations is based on the multi-linear interpolating functions by using the finite element method for both the fluid and solid structures. The generalized trapezoidal rule is used to obtain apparently non-symmetric linear equations in an incremental form for the variables of the flow and vibration. The nonlinear convective term and time factors are contained in the non-symmetric coefficient matrix of the equations. The generalized minimum residual （GMRES） method is used to solve the incremental equations. A new stable algorithm of GMRES-Hughes-Newmark is developed to deal with the flow-induced vibration with dynamical fluid-structure interaction in complex geometries. Good agreement between the simulations and laboratory measurements of the pressure and blade vibration accelerations in a hydro turbine passage was obtained, indicating that the GiViRES-Hughes-Newmark algorithm presented in this paper is suitable for dealing with the flow-induced vibration of structures under small deformation.     

Study of modeling unsteady blade row interaction in a transonic compressor stage part 1: code development and deterministic correlation analysis

The average-passage equation system (APES) provides a rigorous mathematical framework for accounting for the unsteady blade row interaction through multistage compressors in steady state environment by introducing deterministic correlations (DC) that need to be modeled to close the equation system.The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES.In Part 1 of this two-part paper,firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investigate the APES technique.Then steady and unsteady simulations are conducted in a transonic compressor stage.The results from both simulations are compared to highlight the significance of the unsteady interactions.Furthermore,the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial correlations (SC).Lastly,steady and time-averaging (employing APES with DC) simulations for the downstream stator alone are conducted employing DC derived from the unsteady results.The results from steady and time-averaging simulations are compared with the time-averaged unsteady results.The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.     

二维定常湍流计算中的GMRES算法

在以前工作的基础上将广义极小残差（Generalized Minimum RESidual (GMRES)算法发展到用于求解二维可压Favier平均Navier-Stokes方程组。控制方程经Newton线化处理后构成近似的线性系统,然后采用分别耦合了LUSGS和ILU两种预处理矩阵的GMRES算法求解。Spalart-Allmaras湍流模型被用来封闭流体控制方程组,采用与流体控制方程非耦合的方式,使用LUSGS方法求解。对GMRES算法中矩阵－向量的乘积采用了有限插分方法,从而避免了精确的左端系数矩阵的计算和存储。对预处理矩阵的两种使用方法（左预处理和右预处理）进行了分析和讨论。用两个算例对LUSGS和ILU两各预处理矩阵进行了比较,同时探讨了左预处理和右预处理各自的优缺点。通过对Sajben扩压器和NACA0012有攻角流动的计算,表明带有预处理的GMRES算法在二维定常跨音黏性流动计算中相比于得到广泛应用的DDADI方法具有很大优势,左预处理要优于右预处理。     

A matrix‐free preconditioned Newton/GMRES method for unsteady Navier–Stokes solutions

The unsteady compressible Reynolds‐averaged Navier–Stokes equations are discretized using the Osher approximate Riemann solver with fully implicit time stepping. The resulting non‐linear system at each time step is solved iteratively using a Newton/GMRES method. In the solution process, the Jacobian matrix–vector products are replaced by directional derivatives so that the evaluation and storage of the Jacobian matrix is removed from the procedure. An effective matrix‐free preconditioner is proposed to fully avoid matrix storage. Convergence rates, computational costs and computer memory requirements of the present method are compared with those of a matrix Newton/GMRES method, a four stage Runge–Kutta explicit method, and an approximate factorization sub‐iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and the efficiency of the Newton iteration for the non‐linear system at each time step are analysed for both matrix‐free and matrix methods. Differences in the performance of the matrix‐free method for laminar and turbulent flows are highlighted and analysed. Unsteady turbulent Navier–Stokes solutions of pitching and combined translation–pitching aerofoil oscillations are presented for unsteady shock‐induced separation problems associated with the rotor blade flows of forward flying helicopters. Copyright © 2000 John Wiley & Sons, Ltd.     

用鲁棒Riemann求解器和运动重叠网格计算旋翼粘性绕流

发展了一种基于鲁棒Riemann求解器和运动重叠网格技术计算直升机悬停旋翼流场的方法。基于惯性坐标系,悬停旋翼流场是非定常流场,控制方程为可压缩Reynolds平均Navier-Stoke方程,其对流项采用Roe近似Reimann求解器离散,使用改进的五阶加权基本无振荡格式进行高阶重构,非定常时间推进采用含牛顿型LUSGS子迭代的全隐式双时间步方法。为实施旋转运动和便于捕捉尾迹,计算采用运动重叠网格技术。计算得到的桨叶表面压力分布及桨尖涡涡核位置都与实验结果吻合较好。数值结果表明:所发展方法对桨尖涡具有较高的分辨率,对激波具有较好的捕捉能力,该方法可进一步推广到前飞旋翼粘性绕流的计算。     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

On the Computation of Compressible Turbulent Flows on Unstructured Grids

An accurate, fast, matrix-free implicit method has been developed to solve compressible turbulent How problems using the Spalart and Allmaras one equation turbulence model on unstructured meshes. The mean-flow and turbulence-model equations are decoupled in the time integration in order to facilitate the incorporation of different turbulence models and reduce memory requirements. Both mean flow and turbulent equations are integrated in time using a linearized implicit scheme. A recently developed, fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The spatial discretization is carried out using a hybrid finite volume and finite element method, where the finite volume approximation based on a containment dual control volume rather than the more popular median-dual control volume is used to discretize the inviscid fluxes, and the finite element approximation is used to evaluate the viscous flux terms. The developed method is used to compute a variety of turbulent flow problems in both 2D and 3D. The results obtained are in good agreement with theoretical and experimental data and indicate that the present method provides an accurate, fast, and robust algorithm for computing compressible turbulent flows on unstructured meshes.     

Turbulent flow calculations using unstructured and adaptive meshes

A method of efficiently computing turbulent compressible flow over complex two-dimensional configurations is presented. The method makes use of fully unstructured meshes throughout the entire flow field, thus enabling the treatment of arbitrarily complex geometries and the use of adaptive meshing techniques throughout both viscous and inviscid regions of the flow field. Mesh generation is based on a locally mapped Delaunay technique in order to generate unstructured meshes with highly stretched elements in the viscous regions. The flow equations are discretized using a finite element Navier-Stokes solver, and rapid convergence to steady state is achieved using an unstructured multigrid algorithm. Turbulence modelling is performed using an inexpensive algebraic model, implemented for use on unstructured and adaptive meshes. Compressible turbulent flow solutions about multiple-element aerofoil geometries are computed and compared with experimental data.     

非稳态流动的隐式最小二乘等几何方法

对非稳态粘性不可压流动问题提出了一种隐式最小二乘等几何计算方法。该方法先用隐式的向后多步差分格式对Navier-Stokes方程进行时间离散,再用Newton法线性化对流项,最后在每个时间步上用最小二乘等几何方法进行求解。根据该算法编制了计算程序,通过构造解析解的方法验证了程序的正确性,用该程序求解了雷诺数为5000时的非稳态二维顶盖驱动流问题,计算结果捕捉到了流动过程中涡的演化过程,表明本文方法可用于非稳态流动的求解。     

A Fast,Matrix-free Implicit Method for Computing Low Mach Number Flows on Unstructured Grids

A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and Navier-Stokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES + LU?SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both in viscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU?SGS method. The numerical results obtained indicate that the use of the GMRES + LU?SGS method leads to a significant increase in performance over the LU?SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.     

An implicit meshless method for application in computational fluid dynamics

An implicit meshless scheme is developed for solving the Euler equations, as well as the laminar and Reynolds‐averaged Navier–Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The system of equations is linearised, and solved implicitly using approximate, analytical Jacobian matrices and a preconditioned Krylov subspace iterative method. The details of the spatial discretisation, linear solver and construction of the Jacobian matrix are discussed; and results that demonstrate the performance of the scheme are presented for steady and unsteady two dimensional fluid flows. Copyright © 2012 John Wiley & Sons, Ltd.     

GMRES physics-based preconditioner for all Reynolds and Mach numbers: numerical examples

This paper presents several numerical results using a vectorized version of a 3D finite element compressible and nearly incompressible Euler and Navier–Stokes code. The assumptions were set on laminar flows and Newtonian fluids. The goal of this research is to show the capabilities of the present code to treat a wide range of problems appearing in laminar fluid dynamics towards the unification from incompressible to compressible and from inviscid to viscous flow codes. Several authors with different approaches have tried to attain this target in CFD with relative success. At the beginning the methods based on operator splitting and perturbation were preferred, but lately, with the wide usage of time-marching algorithms, the preconditioning mass matrix (PMM) has become very popular. With this kind of relaxation scheme it is possible to accelerate the rate of convergence to steady state solutions with the modification of the mass matrix under certain restrictions. The selection of the mass matrix is not an easy task, but we have certain freedom to define it in order to improve the condition number of the system. In this paper we have used a physics-based preconditioner for the GMRES implicit solver developed previously by us and an SUPG formulation for the semidiscretization of the spatial operator. In sections 2 and 3 we present some theoretical aspects related to the physical problem and the mathematical model, showing the inviscid and viscous flow equations to be solved and the variational formulation involved in the finite element analysis. Section 4 deals with the numerical solution of non-linear systems of equations, with some emphasis on the preconditioned matrix-free GMRES solver. Section 5 shows how boundary conditions were treated for both Euler and Navier–Stokes problems. Section 6 contains some aspects about vectorization on the Cray C90. The performance reached by this implementation is close to 1 Gflop using multitasking. Section 7 presents several numerical examples for both models covering a wide range of interesting problems, such as inviscid low subsonic, transonic and supersonic regimes and viscous problems with interaction between boundary layers and shock waves in either attached or separated flows. © 1997 John Wiley & Sons, Ltd.