A novel non‐upwind,interconnected, multi‐grid,overlapping numerical procedure for problems involving fluid flow

A novel numerical procedure for heat, mass and momentum transfer in fluid flow is presented. The new scheme is passed on a non‐upwind, interconnected, multi‐grid, overlapping (NIMO) finite‐difference algorithm. In 2D flows, the NIMO algorithm solves finite‐difference equations for each dependent variable on four overlapping grids. The finite‐difference equations are formulated using the control‐volume approach, such that no interpolations are needed for computing the convective fluxes. For a particular dependent variable, four fields of values are produced. The NIMO numerical procedure is tested against the exact solution of two test problems. The first test problem is an oblique laminar 2D flow with a double step abrupt change in a passive scalar variable for infinite Peclet number. The second test problem is a rotating radial flow in an annular sector with a single step abrupt change in a passive scalar variable for infinite Peclet number. The NIMO scheme produced essentially the exact solution using different uniform and non‐uniform square and rectangular grids for 45 and 30° angle of inclination. All other schemes were unable to capture the exact solution, especially for the rectangular and non‐uniform grids. The NIMO scheme was also successful in predicting the exact solution for the rotating radial flow, using a uniform cylindrical‐polar coordinate grid. Copyright © 2008 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical method for calculation of the incompressible flow in general curvilinear co‐ordinates with double staggered grid

A solution methodology has been developed for incompressible flow in general curvilinear co‐ordinates. Two staggered grids are used to discretize the physical domain. The first grid is a MAC quadrilateral mesh with pressure arranged at the centre and the Cartesian velocity components located at the middle of the sides of the mesh. The second grid is so displaced that its corners correspond to the centre of the first grid. In the second grid the pressure is placed at the corner of the first grid. The discretized mass and momentum conservation equations are derived on a control volume. The two pressure grid functions are coupled explicitly through the boundary conditions and implicitly through the velocity of the field. The introduction of these two grid functions avoids an averaging of pressure and velocity components when calculating terms that are generated in general curvilinear co‐ordinates. The SIMPLE calculation procedure is extended to the present curvilinear co‐ordinates with double grids. Application of the methodology is illustrated by calculation of well‐known external and internal problems: viscous flow over a circular cylinder, with Reynolds numbers ranging from 10 to 40, and lid‐driven flow in a cavity with inclined walls are examined. The numerical results are in close agreement with experimental results and other numerical data. Copyright © 2003 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Calculation of curved open channel flow using physical curvilinear non‐orthogonal co‐ordinates

There are two main difficulties in numerical simulation calculations using FD/FV method for the flows in real rivers. Firstly, the boundaries are very complex and secondly, the generated grid is usually very non‐uniform locally. Some numerical models in this field solve the first difficulty by the use of physical curvilinear orthogonal co‐ordinates. However, it is very difficult to generate an orthogonal grid for real rivers and the orthogonal restriction often forces the grid to be over concentrated where high resolution is not required. Recently, more and more models solve the first difficulty by the use of generalized curvilinear co‐ordinates (ξ,η). The governing equations are expressed in a covariant or contra‐variant form in terms of generalized curvilinearco‐ordinates (ξ,η). However, some studies in real rivers indicate that this kind of method has some undesirable mesh sensitivities. Sharp differences in adjacent mesh size may easily lead to a calculation stability problem oreven a false simulation result. Both approaches used presently have their own disadvantages in solving the two difficulties that exist in real rivers. In this paper, the authors present a method for two‐dimensional shallow water flow calculations to solve both of the main difficulties, by formulating the governing equations in a physical form in terms of physical curvilinear non‐orthogonal co‐ordinates (s,n). Derivation of the governing equations is explained, and two numerical examples are employed to demonstrate that the presented method is applicable to non‐orthogonal and significantly non‐uniform grids. Copyright © 2004 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

A second look at the role of the fast Fourier transform as an elliptic solver

A fast cosine transform (FCT) is coupled with a tridiagonal solver for the purpose of solving the Poisson equation on irregular and non‐uniform rectangular staggered grids. This kind of solution is required for the pressure field during the simulation of the incompressible Navier–Stokes equations when using the projection method. A new technique using the FCT–tridiagonal solver is derived for the cases where the boundaries of the flow regime do not coincide with the boundaries of the computational domain and for non‐uniform grids. The technique is based on an iterative procedure where a defect equation is solved in every iteration, followed by a relaxation procedure. The method is investigated analytically and numerically to show that the solution converges as a geometric series. The method is further investigated for the effects of the relative size of the rigid body, the grid stretching, size and aspect ratio. The new solver is incorporated with the direct numerical simulation (DNS) and large eddy simulation (LES) techniques to simulate the flows around a backward‐facing step and a 3D rectangular obstacle, yielding results that qualitatively compare well with known results. Copyright © 2005 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.     

Two‐dimensional modeling for stability analysis of two‐phase stratified flow

The effect of wavelength and relative velocity on the disturbed interface of two‐phase stratified regime is modeled and discussed. To analyze the stability, a small perturbation is imposed on the interface. Growth or decline of the disturbed wave, relative velocity, and surface tension with respect to time will be discussed numerically. Newly developed scheme applied to a two‐dimensional flow field and the governing Navier–Stokes equations in laminar regime are solved. Finite volume method together with non‐staggered curvilinear grid is a very effective approach to capture interface shape with time. Because of the interface shape, for any time advancement, a new grid is performed separately on each stratified field, liquid, and gas regime. The results are compared with the analytical characteristics method and one‐dimensional modeling. This comparison shows that solving the momentum equation including viscosity term leads to physically more realistic results. In addition, the newly developed method is capable of predicting two‐phase stratified flow behavior more precisely than one‐dimensional modeling. It was perceived that the surface tension has an inevitable role in dissipation of interface instability and convergence of the two‐phase flow model. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐Newtonian blood flow dynamics in a right internal carotid artery with a saccular aneurysm

Flow dynamics plays an important role in the pathogenesis and treatment of cerebral aneurysms. The temporal and spatial variations of wall shear stress in the aneurysm are hypothesized to be correlated with its growth and rupture. In addition, the assessment of the velocity field in the aneurysm dome and neck is important for the correct placement of endovascular coils. This work describes the flow dynamics in a patient‐specific model of carotid artery with a saccular aneurysm under Newtonian and non‐Newtonian fluid assumptions. The model was obtained from three‐dimensional rotational angiography image data and blood flow dynamics was studied under physiologically representative waveform of inflow. The three‐dimensional continuity and momentum equations for incompressible and unsteady laminar flow were solved with a commercial software using non‐structured fine grid with 283 115 tetrahedral elements. The intra‐aneurysmal flow shows complex vortex structure that change during one pulsatile cycle. The effect of the non‐Newtonian properties of blood on the wall shear stress was important only in the arterial regions with high velocity gradients, on the aneurysmal wall the predictions with the Newtonian and non‐Newtonian blood models were similar. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

The Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

A higher order compact (HOC) finite difference solution procedure has been proposed for the steady two‐dimensional (2D) convection–diffusion equation on non‐uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. Effectiveness of the method is seen from the fact that for the first time, an HOC algorithm on non‐uniform grid has been extended to the Navier–Stokes (N–S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for difficult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non‐uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex flows such as that in the lid‐driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen. Copyright © 2004 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.     

A multigrid finite difference approach to steady flow between eccentric rotating cylinders

A simple finite difference scheme over a non‐uniform grid is proposed to solve the two‐dimensional, steady Navier–Stokes equations. Instead of the Newton method, a more straightforward line search algorithm is used to solve the resultant system of non‐linear equations. By adopting the multigrid methodology, a fast convergence is achieved, at least for low‐Reynolds number flow. This scheme is applied, in particular, to flow between eccentric rotating cylinders. The computed results are shown in good agreement with some analytic findings. Copyright © 2000 John Wiley & Sons, Ltd.     

非正交同位网格下压力与速度耦合算法

发展了一种在非正交同位网格下以笛卡儿速度分量作为动量方程的独立变量、压力与速度耦合的S IM-PLER算法。该算法的特点是显式处理界面速度中的压力交叉导数项,得出压力与压力修正方程,使得压力及压力修正值与界面逆变速度直接耦合。通过对分汊通道内的流动问题进行验证计算,结果表明该算法可以有效而准确地模拟复杂区域内的流动与换热问题。     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.