Modified Galerkin Method for Unsteady Two-Dimensional Viscous Fluid Flow Analysis

The Modified Galerkin Method (MGM) has been proposed as one of the most efficient methods for two-dimensional convection-diffusion equations. In the MGM, the non-symmetric matrices, which are derived from the convection term in the Galerkin formulation, are not used, and an artificial diffusion is introduced through an error analysis approach to improve its discretization accuracy in both time and space directions. In this study, the MGM is applied for two-dimensional viscous fluid flow analysis, and the driven cavity flow problems are solved up to Reynolds number of 10,000 using the vorticity-stream function formulation and non-uniform meshes. The results show the effectiveness of MGM.     

Modified incompressible SPH method for simulating free surface problems

An incompressible smoothed particle hydrodynamics (I-SPH) formulation is presented to simulate free surface incompressible fluid problems. The governing equations are mass and momentum conservation that are solved in a Lagrangian form using a two-step fractional method. In the first step, velocity field is computed without enforcing incompressibility. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. The source term in the Poisson equation for the pressure is approximated, based on the SPH continuity equation, by an interpolation summation involving the relative velocities between a reference particle and its neighboring particles. A new form of source term for the Poisson equation is proposed and also a modified Poisson equation of pressure is used to satisfy incompressibility condition of free surface particles. By employing these corrections, the stability and accuracy of SPH method are improved. In order to show the ability of SPH method to simulate fluid mechanical problems, this method is used to simulate four test problems such as 2-D dam-break and wave propagation.     

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.     

SIMPLE‐type preconditioners for the Oseen problem

In this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.     

A calculation procedure for steady two-dimensional elliptic flows

A calculation procedure is presented for predicting steady two-dimensional elliptic flows. The method introduces a density correction concept and an algebraic equation for the velocity correction instead of the troublesome pressure correction equation in the SIMPLER procedure. Computations show that the method has the same rate of convergence as SIMPLER while saving about 20% computational effort per iteration. Although the method is described for steady two-dimensional situations, its extension to three-dimensional problems is very straightforward.     

Finite element implementation of boundary conditions for the pressure Poisson equation of incompressible flow

In this paper we address the problem of the implementation of boundary conditions for the derived pressure Poisson equation of incompressible flow. It is shown that the direct Galerkin finite element formulation of the pressure Poisson equation automatically satisfies the inhomogeneous Neumann boundary conditions, thus avoiding the difficulty in specifying boundary conditions for pressure. This ensures that only physically meaningful pressure boundary conditions consistent with the Navier-Stokes equations are imposed. Since second derivatives appear in this formulation, the conforming finite element method requires C¹ continuity. However, for many problems of practical interest (i.e. high Reynolds numbers) the second derivatives need not be included, thus allowing the use of more conventional C⁰ elements. Numerical results using this approach for a wall-driven contained flow within a square cavity verify the validity of the approach. Although the results were obtained for a two-dimensional problem using the p-version of the finite element method, the approach presented here is general and remains valid for the conventional h-version as well as three-dimensional problems.     

不可压缩机翼绕流的有限谱法计算

结合有限谱QUICK格式求解不可压缩粘性流问题。这一格式用于模拟不同攻角下的NACA1200机翼绕流问题。利用体积力,提出了将流场速度从0加速到来流速度的方法。区别于传统的压力梯度为零的边界条件,推导出一个更精确的压力边界条件。为使速度散度保持为零,在泊松方程中给速度散度一个特殊的处理。这一成果说明了有限谱法不但具有很高的精度,而且能灵活地和其他格式一起构造出新的格式,从而成功地应用到复杂流场不可压缩流动的数值计算中。     

A fully coupled Navier–Stokes solver for calculation of turbulent incompressible free surface flow past a ship hull

This paper deals with the calculation of free surface flow of viscous incompressible fluid around the hull of a boat moving with rectilinear motion. An original method used to avoid a large part of the theoretical problems connected with free surface boundary conditions in three‐dimensional Navier–Stokes–Reynolds equations is proposed here. The linearised system of convective equations for velocities, pressure and free surface elevation unknowns is discretised by finite differences and two methods to solve the fully coupled resulting matrix are presented here. The non‐linear convergence of fully coupled algorithm is compared with the velocity–pressure weakly coupled algorithm SIMPLER. Turbulence is taken into account through Reynolds decomposition and k–ε or k–ω model to close the equations. These two models are implemented without wall function and numerical calculations are performed up to the viscous sub‐layer. Numerical results and comparisons with experiments are presented on the Series 60 CB=0.60 ship model for a Reynolds number Rn=4.5×10⁶ and a Froude number Fn=0.316. Copyright © 1999 John Wiley & Sons, Ltd.     

Study of the effect of the non‐orthogonality for non‐staggered grids—The results

This article presents the effect of the grid skewness on the ranges of the underrelaxation factors for pressure and velocity. The effect is reflected by the relationship between the numbers of iterations required and the ranges of the underrelaxation factors for a converged solution. Four typical cavity flow problems are solved on non‐staggered grids for this purpose. Two momentum interpolation practices namely, practice A and practice B, together with SIMPLE, SIMPLEC and SIMPLER algorithms are employed. The results show that the ranges of the pressure underrelaxation factor values for convergence exist if the SIMPLE algorithm is used, while no restrictions are observed if the SIMPLEC algorithm is used. From the curves obtained using the SIMPLER algorithm, the ranges of those based on practice B are wider than those based on practice A. Copyright © 1999 John Wiley & Sons, Ltd.     

A new stabilized method for quasi-Newtonian flows

For a generalized quasi-Newtonian flow, a new stabilized method focused on the low-order velocity-pressure pairs, （bi）linear/（bi）linear and （bi）linear/constant element, is presented. The pressure projection stabilized method is extended from Stokes problems to quasi-Newtonian flow problems. The theoretical framework developed here yields an estimate bound, which measures error in the approximate velocity in the W 1,r（Ω） norm and that of the pressure in the L r＇ （Ω） （1/r ＋ 1/r＇ = 1）. The power law model and the Carreau model are special ones of the quasi-Newtonian flow problem discussed in this paper. Moreover, a residual-based posterior bound is given. Numerical experiments are presented to confirm the theoretical results.     

Moving least squares reconstruction for sharp interface immersed boundary methods

We propose a new approach for reconstructing velocity boundary conditions in sharp-inerface immersed boundary (IB) methods based on the moving least squares (MLS) interpolation method. The MLS is employed to not only reconstruct velocity boundary conditions but also to calculate the pressure and velocity gradients in the vicinity of the immersed body, which are required in fluid structure interaction problems to obtain the force exerted by the fluid on the structure. To extend the method to arbitrarily complex geometries with nonconvex shaped boundaries, the visibility method is combined with the MLS method. The performance of the proposed curvilinear IB MLS (CURVIB-MLS) is demonstrated by systematic grid-refinement studies for two- and three-dimensional tests and compared with the standard CURVIB method employing standard wall-normal interpolation for reconstructing boundary conditions. The test problems are flow in a lid-driven cavity with a sphere, uniform flow over a sphere, flow on a NACA0018 airfoil at incidence, and vortex-induced vibration of an elastically-mounted cylinder. We show that the CURVIB-MLS formulation yields a method that is easier to implement in complex geometries and exhibits higher accuracy and rate of convergence relative to the standard CURVIB method. The MLS approach is also shown to dramatically improve the accuracy of calculating the pressure and viscous forces imparted by the flow on the body and improve the overall accuracy of FSI simulations. Finally, the CURVIB-MLS approach is able to qualitatively capture on relatively coarse grids important features of complex separated flows that the standard CURVIB method is able to capture only on finer grids.     

Direct simulation of flows with suspended paramagnetic particles using one-stage smoothed profile method

The so-called smoothed profile method, originally suggested by Nakayama and Yamamoto and further improved by Luo et al. in 2005 and 2009, respectively, is an efficient numerical solver for fluid-structure interaction problems, which represents the particles by a certain smoothed profile on a fixed grid and constructs some form of body force added into the momentum (Navier-Stokes) equation by ensuring the rigidity of particles. For numerical simulations, the method first advances the flow and pressure fields by integrating the momentum equation except the body-force (momentum impulse) term in time and next updates them by separately taking temporal integration of the body-force term, thus requiring one more Poisson-equation solver for the extra pressure field due to the rigidity of particles to ensure the divergence-free constraint of the total velocity field. In the present study, we propose a simplified version of the smoothed profile method or the one-stage method, which combines the two stages of velocity update (temporal integration) into one to eliminate the necessity for the additional solver and, thus, significantly save the computational cost. To validate the proposed one-stage method, we perform the so-called direct numerical simulations on the two-dimensional motion of multiple inertialess paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and compare their results with the existing benchmark solutions. For the validation, we develop the finite-volume version of the direct simulation method by employing the proposed one-stage method. Comparison shows that the proposed one-stage method is very accurate and efficient in direct simulations of such magnetic particulate flows.     

Performance analysis of IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems

Recently, an efficient segregated algorithm for incompressible fluid flow and heat transfer problems, called inner doubly iterative efficient algorithm for linked equations (IDEAL), has been proposed by the present authors. In the algorithm there exist inner doubly iterative processes for pressure equation at each iteration level, which almost completely overcome two approximations in SIMPLE algorithm. Thus, the coupling between velocity and pressure is fully guaranteed, greatly enhancing the convergence rate and stability of solution process. However, validations have only been conducted for two‐dimensional cases. In the present paper the performance of the IDEAL algorithm for three‐dimensional incompressible fluid flow and heat transfer problems is analyzed and a systemic comparison is made between the algorithm and three other most widely used algorithms (SIMPLER, SIMPLEC and PISO). By the comparison of five application examples, it is found that the IDEAL algorithm is the most robust and the most efficient one among the four algorithms compared. For the five three‐dimensional cases studied, when each algorithm works at its own optimal under‐relaxation factor, the IDEAL algorithm can reduce the computation time by 12.9–52.7% over SIMPLER algorithm, by 45.3–73.4% over SIMPLEC algorithm and by 10.7–53.1% over PISO algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Implementation of CLEAR algorithm on non‐orthogonal curvilinear co‐ordinates for solution of incompressible flow and heat transfer

In this paper, the CLEAR (coupled and linked equations algorithm revised) algorithm is extended to non‐orthogonal curvilinear collocated grids. The CLEAR algorithm does not introduce pressure correction in order to obtain an incompressible flow field which satisfies the mass conservation law. Rather, it improves the intermediate velocity by solving an improved pressure equation to make the algorithm fully implicit since there is no term omitted in the derivation process. In the extension of CLEAR algorithm from a staggered grid system in Cartesian coordinates to collocated grids in non‐orthogonal curvilinear coordinates, three important issues are appropriately treated so that the extended CLEAR can lead to a unique solution without oscillation of pressure field and with high robustness. These three issues are (1) solution independency on the under‐relaxation factor; (2) strong coupling between velocity and pressure; and (3) treatment of the cross pressure gradient terms. The flow and heat transfer problems in a rectangular enclosure with an internal eccentric circle and the flow in a lid‐driven inclined cavity are computed by using the extended CLEAR. The results show that the extended CLEAR can guarantee the solution independency on the under‐relaxation factor, the smoothness of pressure profile even at very small under‐relaxation factor and good robustness which leads to a converged solution for the small inclined angle of 5^° only with 5‐point computational molecule while the extended SIMPLE‐series algorithm usually can get a converged solution for the inclined angle larger than 30^° under the same condition. Copyright © 2006 John Wiley & Sons, Ltd.     

On the Reynolds-number independence of mixed convection in a vertical channel subjected to asymmetric wall temperatures with and without flow reversal

The problem of mixed convection in a vertical channel with asymmetric wall temperatures including situations of flow reversal is studied numerically. The SIMPLER algorithm with a staggered grid system is employed to solve the corresponding numerical equations formulated by the finite-volume method. A second-order upwind scheme is used to model the convective term, and a suitable grid distribution is introduced. The ranges of the parameters studied are 0 r_t 1, 1 Re 1000, and 0 Gr/Re 500.

The numerical results, with the streamwise coordinate scaled by the Reynolds number (Re), show that solutions for the velocity and temperature fields are independent of the Reynolds number when Re 50, even in the presence of flow reversal. These solutions, however, are dependent on r_t and Gr/Re. Subsequently, correlations are proposed for the bulk temperature distribution and the local Nusselt numbers along the hot wall and the cold wall.     

A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations

A new stable unstructured finite volume method is presented for parallel large-scale simulation of viscoelastic fluid flows. The numerical method is based on the side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure–velocity–stress coupling. The log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix–logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in order improve the limiting Weissenberg numbers in the proposed finite volume method. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present numerical algorithm is validated for the Kovasznay flow, the flow of an Oldroyd-B fluid past a confined circular cylinder in a channel and the three-dimensional flow of an Oldroyd-B fluid around a rigid sphere falling in a cylindrical tube. Parallel large-scale calculations are presented up to 523,094 quadrilateral elements in two-dimension and 1,190,376 hexahedral elements in three-dimension.     

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.     

PEGASE: A NAVIER–STOKES SOLVER FOR DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS

A hybrid conservative finite difference/finite element scheme is proposed for the solution of the unsteady incompressible Navier–Stokes equations. Using velocity–pressure variables on a non-staggeredgrid system, the solution is obtained with a projection method basedon the resolution of a pressure Poisson equation. The new proposed scheme is derived from the finite element spatial discretization using the Galerkin method with piecewise bilinear polynomial basis functions defined on quadrilateral elements. It is applied to the pressure gradient term and to the non-linear convection term as in the so-called group finite element method. It ensures strong coupling between spatial directions, inhibiting the development of oscillations during long-term computations, as demonstrated by the validation studies. Two- and three-dimensional unsteady separated flows with open boundaries have been simulated with the proposed method using Cartesian uniform mesh grids. Several examples of calculations on the backward-facing step configuration are reported and the results obtained are compared with those given by other methods. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 833–861, 1997.     

Wave resistance of a viscous liquid

In this paper we examine the resistance encountered by a system of normal stresses during its rectilinear motion along the surface of a viscous liquid of infinite depth. The problem is solved in the linear formulation, i.e., it is assumed that amplitudes of the waves which arise are small and the waves are shallow. The solution for the two-and three-dimensional problems is obtained in the general case in closed form. In the two-dimensional case a detailed study is made of the case when a constant pressure p₀, moving with the constant velocity U, is given on a segment of length 2l. In the three-dimen-sional problem the case is studied when the normal stress is concentrated on a segment of a straight line of length 2l, which can replace a ship moving along a straight course with the constant velocity U. The integrals obtained in both cases are studied using the stationary phase method, the application of which for the three-dimensional integrals with respect to a volume with boundaries is justified in §1 of the paper. As a result we obtain equations for the wave resistance in the two- (§2) and three-dimensional (§3) cases.     

Conservation relation of generalized growth rate in boundary layers

The elementary task is to calculate the growth rates of disturbances when the e^N method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the e^N method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90° which can be easily avoided in practical application.