An iterative scheme for numerical solution of steady incompressible viscous flows

An iterative solution scheme is proposed for application to steady incompressible viscous flows in simple and complex geometries. The iterative scheme solves the vorticity-stream function form of the Navier-Stokes equations in generalized curvilinear coordinates. The flow system of equations are cast into a Newton's iterative form which are solved using the modified strongly implicit procedure. The solution scheme is benchmarked using two test cases, namely: a shear-driven steady laminar flow in a square cavity; and a simple laminar flow in a complex expanding channel. The iterative process to steady-state convergence in both test cases is highly stable and the convergence rate is without spurious oscillations. At convergence, the flow solutions are second-order accurate.     

An implicit method for three dimensional viscous flow with application to cones at angle of attack

An iteration method for solving the implicit difference equations associated with three nonlinear parabolic differential equations is derived and analyzed. The method is applied to the high Reynolds number viscous flow around a cone at high angle of attack. The requirements which must be met to ensure convergence of the iterations are obtained. In addition, an analysis of the stability of the difference equations is presented and discussed. The numerical results are compared with experimental data for a 10° cone at 12° angle of attack, and a 5·6° cone at 8° angle of attack. The agreement is very good.     

A time-accurate variable property algorithm for calculating flows with large temperature variations

A variable property algorithm is developed for time-dependent resolution of flows with large temperature differences. The non-linear coupled set of equations with property variations is resolved by an implicit iterative procedure at each time step. The momentum and energy equations are integrated in time using an implicit Crank-Nicolson method, and a Helmholtz equation for pressure is solved at each inner iteration. Local density changes are coupled to both changes in local temperature and pressure, whereas other properties are only coupled to changes in temperature. No low Mach number assumption is used in the formulation, and the pressure is calculated directly through the Helmholtz equation. Results are shown to compare well with benchmark calculations and analytical solutions of convection in a differentially heated cavity with very large temperature differences. Unsteady Poiseuille-Bénard flow is used for validation of the time-dependent aspects of the algorithm, which is shown to accurately predict the time-dependent buoyancy driven longitudinal vortices.     

A mathematical model of laminar axisymmetrical natural gas flames

A computational procedure for determining the velocities, temperatures and species concentrations in a laminar Bunsen type flame is presented. The boundary layer form of the Navier-Stokes equations with coupled chemistry are solved for a compressible, viscous and axisymmetric flow. An implicit finite difference method is used in the solutions. Velocities, temperatures and stable species concentrations are compared with experimental data.     

Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

Improved solution methods for inelastic rate problems

The objective of the paper is an assessment of the incremental solution methods for the analysis of inelastic rate problems. In particular, the possibilities of the initial load method are explored to improve the accuracy and stability of the traditional explicit operators by higher-order time expansions and implicit weighting schemes.The convergence limitations are examined for different classes of inelastic growth laws (viscous flow, viscoelasticity, viscoplasticity) which restrict the time step because of the iterative solution of the implicit algorithm. The range and rate of convergence of the initial load method (constant stiffness predictor-corrector iteration) is enlarged by tangential gradient techniques which account for the inelastic response in the structural stiffness matrix. In this way the time step restriction disappears although at a considerable increase of computational expense because of the costly computation and decomposition of structural gradients within each iteration cycle (Newton-Raphson methods).As compared to the linear single-step methods, the cubic Hermitian time expansions furnish far better accuracy than the traditional linear expansions for very little increase of computational cost. Stability and convergence limits correspond to those of the lower-order operators, whereby the implicit midstep of backward weighting schemes are most advantageous. In this context it is worth noting that aging or strain-hardening effects in the inelastic growth law reduce dramatically the time step restrictions of the iterative initial load solution methods (predictor-corrector schemes), as compared to the simplest creep model in which the inelastic growth law depends only on stress, e.g. for viscous flow and viscoplasticity.     

Three-dimensional laminar incompressible flow in straight polar ducts

The flow in the entrance region of long ducts of rectangular and polar cross sections is studied using the three-dimensional parabolized Navier-Stokes equations, together with the energy equation, for an incompressible viscous fluid. Numerical solutions are obtained using an alternating-direction implicit technique for the parabolized equations which are elliptic in the cross-plane of the duct, and the calculations march forward in the axial direction. Computations are also made for polar ducts with a rotating outer-radial wall so as to provide meaningful results for a model problem comprising the simplest form of a turbomachinery cascade. The study of this model problem serves to analyze some of the important features of three-dimensional internal viscous flows. Also of interest is the rapidly converging solution of the Neumann problem for the cross-plane variation of pressure for polar ducts.     

Modified alternating direction-implicit iteration method for linear systems from the incompressible Navier–Stokes equations

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.     

Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir

The development of a capacity to predict the exploitation of structurally complicated and fractured oil reservoirs is essential for the rational use of investment capital. A poor understanding of how the reservoir behaves during production may lead to inept, costly and inefficient development schemes. The mathematical formulation of a three-phase, three-dimensional fluid flow and rock deformation in fractured reservoirs is hence presented. The present formulation, consisting of both the equilibrium and multiphase mass conservation equations, accounts for the significant influence of coupling between the fluid flow and solid deformation, an aspect usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretise the governing equations in space and a finite difference scheme is used to march the solution in time. The final set of equations, which contain the additional cross coupling terms as compared to similar existing models, are highly non-linear and the elements of the coefficient matrices are updated implicitly during each iteration in terms of the independent variables. A field scale example is employed as an alpha case to test the validity and robustness of the currently formulation and numerical scheme. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered.     

A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver

In this paper, we improve the Navier–Stokes flow solver developed by Sun et al. based on the spectral volume method (SV) in the following two aspects: the development of a more efficient implicit/p-multigrid solution approach, and the use of a new viscous flux formula. An implicit preconditioned LU-SGS p-multigrid method developed for the spectral difference (SD) Euler solver by Liang is adopted here. In the original SV solver, the viscous flux was computed with a local discontinuous Galerkin (LDG) type approach. In this study, an interior penalty approach is developed and tested for both the Laplace and Navier–Stokes equations. In addition, the second method of Bassi and Rebay (also known as BR2 approach) is also implemented in the SV context, and also tested. Their convergence properties are studied with the implicit BLU-SGS approach. Fourier analysis revealed some interesting advantages for the penalty method over the LDG method. A convergence speedup of up to 2-3 orders is obtained with the implicit method. The convergence was further enhanced by employing a p-multigrid algorithm. Numerical simulations were performed using all the three viscous flux formulations and were compared with existing high order simulations (or in some cases, analytical solutions). The penalty and the BR2 approaches displayed higher accuracy than the LDG approach. In general, the numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

Implicit treatment of molar mass equations in secondary oil migration

The hydrocarbon migration can be described by a coupled set of partial differential equations describing the dynamics of the temperature, component flow, pressure and velocity. A sequential solution procedure where the component flow is solved explicitly, gives severe restrictions on the time step given by the CFL condition. In this paper an implicit solution procedure is given and results from numerical tests are presented. The results are compared with the explicit solutions. As expected the implicit algorithm allows for substantially larger time steps. Received: 31 January 2001 / Accepted: 30 September 2001     

三维Euler方程的隐式间断有限元算法

为了求解三维欧拉方程,对隐式时间离散格式间断有限元方法进行了研究。根据间断Galerkin有限元方法思想,构造内迭代SOR-LU-SGS隐式时间离散格式,结合当地时间步长技术、多重网格方法,实现了三维流场的计算。数值计算了ONERAM6机翼、大攻角尖前缘三角翼以及DLR-F4翼身组合体的亚声速绕流问题。结果表明,加入SOR内迭代步的LU-SGS隐式算法具有较大的优势,相较于GMRES算法所占用的内存少且收敛速度相当,是LU-SGS算法的三倍以上。针对三维算例,具有较好的稳定性和较高的收敛速度,能够给出准确的流场信息。与原方法相比,SOR-LU-SGS方法无论是在迭代步数上还是在CPU时间上,效率均有明显提高,适合于三维复杂流场计算。     

On the accuracy and efficiency of a finite element tensor product algorithm for fluid dynamics applications

A finite element numerical solution algorithm is derived for application to problems in computational fluid dynamics. Through extension of the basic error extremization constraints offered using the method of weighted residuals, a selective dissipation mechanism is introduced to control phase error and/or non-linearly induced error. Using an implicit integration algorithm, the Jacobian of a Newton iteration procedure is replaced by a tensor matrix product construction for solution economy. Numerical results for multi-dimensional equations, modeling the substantial time derivative within the Navier-Stokes system, are utilized to assess solution accuracy, stability and error control.     

Computation of MHD flow due to moving boundaries

A non-iterative numerical scheme is presented which computes in a single iteration the steady, laminar flow of a viscous, incompressible, electrically conducting fluid caused by moving boundaries in the presence of a transverse magnetic field. It also eliminates the possible error induced by taking the value of numerical infinity (representing the unbounded domain of the flow) as a finite number. The scheme is based on implicit use of infinite series of exponentials for velocity components. The issue of convergence of these series is also discussed. An asymptotic solution valid for large values of M, the Hartmann number, and an approximate solution valid for any value of M are further developed. In particular, the case of axisymmetric magnetohydrodynamic (MHD) flow due to a stretching sheet has been dealt with in some detail. A comparison has been made of the merits of various techniques used in the paper and appropriate conclusions are drawn.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Numerical simulation of 3-D wing flutter with fully coupled fluid-structural interaction

A numerical methodology coupling Navier-Stokes equations and structural modal equations for predicting 3-D transonic wing flutter is developed in this paper. A dual-time step implicit unfactored Gauss-Seidel iteration with the Roe scheme is employed for the flow solver. A modal approach is used for the structural response. The flow and structural solvers are fully coupled via successive iterations within each physical time step. The mesh-deformation strategy is described. The accuracy of the modal approach is validated with ANSYS. The results indicate that the first five modes are sufficient to accurately model the wing-structure response for the studied case of this paper. The computed flutter boundary of AGARD wing 445.6 at free stream Mach numbers ranging from 0.499 to 1.141 agrees well with the experiment.     

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.