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Journal ArticleDOI

A Novel Semi-Explicit Spatially Fourth Order Accurate Projection Method for Unsteady Incompressible Viscous Flows

07 Dec 2009-Numerical Heat Transfer Part A-applications (Taylor & Francis Group)-Vol. 56, Iss: 8, pp 665-684

Abstract: This article describes a simple and elegant compact higher order finite-difference based numerical solution technique to the primitive variable formulation of unsteady incompressible Navier Stokes equations (UINSE) on staggered grids. The method exploits the advantages of the D'yakanov ADI-like scheme and a non-iterative pressure correction based fractional step method. Spatial derivatives are discretized to fourth order accuracy and the time integration is realized through the Euler explicit method. The fast and efficient iterative solution to the discretized momentum and pressure Poisson equations is achieved using a variant of conjugate gradient method. Spatial accuracy and robustness of the solver are tested through its application to relevant benchmark problems.
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Journal ArticleDOI
Tien-Mo Shih1, Martinus Arie1, Derrick I. Ko1Institutions (1)
Abstract: A comprehensive survey of the literature in the area of numerical heat transfer (NHT) published between 2000 and 2009 has been conducted Due to the immenseness of the literature volume, the survey

58 citations

Journal ArticleDOI
Mingang Jin1, Wangda Zuo2, Qingyan Chen1Institutions (2)
Abstract: Fast fluid dynamics (FFD) can potentially be used for real-time indoor air-flow simulations This study developed two-dimensional fast fluid dynamics (2-D FFD) into three-dimensional fast fluid dynamics (3-D FFD) The implementation of boundary conditions at the outlet was improved with a local mass conservation method A near-wall treatment for the semi-Lagrangian scheme was also proposed This study validated the 3-D FFD with five flows that have features of indoor air flow The results show that the 3-D FFD can successfully capture the three dimensionality of air-flow and provide reliable and reasonably accurate simulations for indoor air flows with a speed of about 15 times faster than current computational fluid dynamics (CFD) tools

32 citations

Journal ArticleDOI
Shuying Zhai1, Xinlong Feng1, Demin Liu1Institutions (1)
Abstract: In this article, a new family of fourth-order compact difference schemes for the three-dimensional semilinear convection-diffusion equation with variable coefficients is presented. Like the finite-volume method, a dual partition is introduced. Combining with the Simpson integral formula and parabolic interpolation, fourth-order schemes are derived based on two different types of dual partitions. Moreover, a sixth-order finite-difference discretization strategy is developed, which is based on the fourth-order compact discretization and Richardson extrapolation technique. This extrapolation technique can achieve a sixth-order-accurate solution on fine grids directly, without the need for interpolation. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order schemes and extrapolation formulas.

21 citations

Journal ArticleDOI
Reetesh Ranjan1, Carlos Pantano1Institutions (1)
TL;DR: A new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid, which shows uniform order of accuracy, both in space and time.
Abstract: We present a new finite-difference numerical method to solve the incompressible Navier-Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity-pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier-Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.

11 citations

01 Nov 1997-
TL;DR: The proposed schemes appear to be attractive alternatives to the standard Pade schemes for computations of the Navier?Stokes equations.
Abstract: This paper presents a family of finite difference schemes for the first and second derivatives of smooth functions. The schemes are Hermitian and symmetric and may be considered a more general version of the standard compact (Pade) schemes discussed by Lele. They are different from the standard Pade schemes, in that the first and second derivatives are evaluated simultaneously. For the same stencil width, the proposed schemes are two orders higher in accuracy, and have significantly better spectral representation. Eigenvalue analysis, and numerical solutions of the one-dimensional advection equation are used to demonstrate the numerical stability of the schemes. The computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard Pade schemes. The proposed schemes appear to be attractive alternatives to the standard Pade schemes for computations of the Navier?Stokes equations.

10 citations

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Journal ArticleDOI
Sanjiva K. Lele1Institutions (1)
Abstract: The present finite-difference schemes for the evaluation of first-order, second-order, and higher-order derivatives yield improved representation of a range of scales and may be used on nonuniform meshes. Various boundary conditions may be invoked, and both accurate interpolation and spectral-like filtering can be accomplished by means of schemes for derivatives at mid-cell locations. This family of schemes reduces to the Pade schemes when the maximal formal accuracy constraint is imposed with a specific computational stencil. Attention is given to illustrative applications of these schemes in fluid dynamics.

5,460 citations

Journal ArticleDOI
U Ghia1, K.N Ghia1, C. T. Shin1Institutions (1)
TL;DR: The vorticity-stream function formulation of the two-dimensional incompressible NavierStokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions.
Abstract: The vorticity-stream function formulation of the two-dimensional incompressible NavierStokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10.000 and meshes consisting of as many as 257 x 257 points. For Re = 1000, the (129 x 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional gridclustering coordinate transformations. The past decade has witnessed a great deal of progress in the area of computational fluid dynamics. Developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems. For some of these applications, the use of simple iterative techniques to solve the Navier-Stokes equations leads to a rather slow convergence rate for the solutions. The solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries. The rate of convergence is also generally strongly dependent on such problem parameters as the Reynolds number, the mesh size, and the total number of computational points. This has led several researchers to examine carefully the recently emerging multigrid (MG) technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously.

3,728 citations

"A Novel Semi-Explicit Spatially Fou..." refers background or methods in this paper

  • ...[35], are found to be negligible for the grid spacing beyond 1 84....


  • ...[35] when the fourth order accurate spatial discretization is replaced with the second order counterpart....


  • ...The streamlines and vorticity contours for this case, shown in Figures 10a and 10b, respectively, are found to have captured all essential patterns that are reported in an earlier work [35] for this Re, but with a grid size of 256 256....


Journal ArticleDOI
Alexandre J. Chorin1Institutions (1)
Abstract: A numerical method for solving incompressible viscous flow problems is introduced. This method uses the velocities and the pressure as variables and is equally applicable to problems in two and three space dimensions. The principle of the method lies in the introduction of an artificial compressibility ? into the equations of motion, in such a way that the final results do not depend on ?. An application to thermal convection problems is presented.

2,649 citations

"A Novel Semi-Explicit Spatially Fou..." refers methods in this paper

  • ...This approach is feasible through the usage of the fractional step based projection method which was first proposed in reference [13]; wherein, the pressure gradient term is eliminated while solving momentum equations to compute pseudovelocity field....


Journal ArticleDOI
Abstract: Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass, momentum, and kinetic energy. This deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems. Finite difference schemes in a collocated grid system are also analyzed, and a violation of kinetic energy conservation is revealed. The predicted conservation properties are demonstrated numerically in simulations of inviscid white noise, performed in a two-dimensional periodic domain. The proposed fourth order schemes in a staggered grid system are generalized for the case of a non-uniform mesh, and the resulting scheme is used to perform large eddy simulations of turbulent channel flow.

908 citations

"A Novel Semi-Explicit Spatially Fou..." refers background in this paper

  • ...The conservation properties of these forms in terms of mass, momentum, and kinetic energy have been discussed in reference [23]....


  • ...Previous works [23, 25], signify that the higher order spatial accuracy pays penalty in terms of shrinking the CFL limit....


  • ...In addition, it is pointed out in references [23, 24], when the skew symmetric form of NOMENCLATURE...


Journal ArticleDOI
Abstract: This paper considers the accuracy of projection method approximations to the initial–boundary-value problem for the incompressible Navier–Stokes equations. The issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second-order methodology a decade and a half ago. It has been observed that while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the L ∞ -norm. This paper identifies the source of this problem in the interplay of the global pressure-update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second-order accurate, as demonstrated by a normal mode analysis and numerical experiments. In addition, a numerical method based on a gauge variable formulation of the incompressible Navier–Stokes equations, which provides another option for obtaining fully second-order convergence in both velocity and pressure, is discussed. The connection between the boundary conditions for projection methods and the gauge method is explained in detail.

776 citations

"A Novel Semi-Explicit Spatially Fou..." refers methods in this paper

  • ...The fractional step method recommended by reference [31] is responsible for generating these high order solutions....


  • ...The purpose of this work is to test the utility of a semi-explicit fractional step method, suggested by reference [31], when combined with advantages of matrix form of compact finite-difference schemes....


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