Non-staggered Grid Research Articles

Flow over a bluff body from moderate to high Reynolds numbers using large eddy simulation

Large eddy simulation (LES) is performed to study the uniform approach flow over a square-section cylinder with different Reynolds numbers, ranging from 10 3 to 5 × 10 6. Two different sub-grid scale models, the Smagorinsky and a dynamic one-equation model, are employed. An incompressible finite-volume code, based on a non-staggered grid arrangement and an implicit fractional step method with second-order accuracy in space and time, is used. The structure of the flow is studied with the instantaneous and the mean quantities such as pressure, turbulent stresses, turbulent kinetic energy, vorticity, the second invariant of velocity gradient and streamlines. The Strouhal number, the mean and RMS values of the lift and drag are computed for various Reynolds numbers, which show a good agreement with the available experimental results. It is found that the effect of Reynolds number on the global quantities, the mean and the large scale instantaneous flow-structures is not much at the higher Reynolds numbers, i.e. Re > 2 × 10 4. In this range of Reynolds numbers, the small scales of the instantaneous structures are more complex and chaotic as they compare with the larger ones.

안정성층류에서 선택취수의 수치해석

3차원 열동수역학 모형을 개발하여 지형학적으로 복잡한 자연 저수지에서 안정한 성층류의 선택 취수를 부정류 모의하였다. 지배방정식은 2차 정확도의 유한체적법을 이용하여 해석하였다. 개발된 수치모형을 3차원 난류, 성층화된 전단층흐름에 적용하여 검정을 하였다. 수치해석결과는 실험실에서 관측된 선택취수시의 속도 및 온도분포의 일반적인 형상을 양호하게 예측하는 것으로 나타났으나, 자연 저수지에서의 흐름에 대한 적용시에는 속도의 크기를 과대모의 하는 것으로 나타났다. 수치모의에서 구해진 선택취수의 물리적 특성을 논하였다. A three-dimensional thermal hydrodynamic model is developed for carrying out unsteady simulation of the selective withdrawal of the stably stratified flow in a geometrically complex, natural reservoir The governing equations are discretized on a non-staggered grid using a second-order accurate, finite-volume scheme. The numerical model is validated by applying it to simulate three-dimensional, turbulent, stratified, shear-layer flow case. The numerical predictions appear to capture reasonably well the general shape of velocity and temperature profiles observed in the laboratory experiments, while significant overestimation of the magnitude of velocity profiles is observed in the application to the flow in a natural reservoir. The physics of selective withdrawal as emerge from the numerical simulations are also discussed.

A comparison of staggered and non-staggered grid Navier--Stokes solutions for the 8:1 cavity natural convection flow

The Navier--Stokes equations may be discretised using finite-volume schemes on non-staggered or on staggered grids. The staggered grid is known to prevent pressure oscillations that may occur on the non-staggered grid; however, this is at the expense of increased code complexity. Non-staggered grid schemes that employ iterative time integration are known to require the use of some form of explicit correction in the construction of the Poisson pressure correction equation. We investigate a fractional-step pressure-correction non-staggered scheme and compare it to a similar fractional-step staggered grid scheme for bifurcated natural convection flow in an 8:1 cavity.

Effect of surface irregularities on unsteady pulsatile flow in a compliant artery

Of concern in the paper is an analytical study of pulsatile blood flow in an irregular stenosed arterial segment through a mathematical model. The model is two-dimensional and axisymmetric with an outline of the stenosis obtained from a three-dimensional casting of a mildly stenosed artery [L. Back, Y. Cho, D. Crawford, R. Cuffel, Effect of mild atherosclerosis on flow resistance in a coronary artery casting of man, J. Biomech. Eng. 106 (1984) 48–53]. The combined influence of an asymmetric shape and surface irregularities of the constriction has been explored in a computational study of blood flow through arterial stenosis with 48% areal occlusion. The moving wall of the artery is included to be anisotropic, linear, viscoelastic, incompressible circular cylindrical membrane shell. The effect of the surrounding connective tissues on the motion of the arterial wall is also paid due attention. Results are also obtained for a smooth stenosis model and also for a stenosis model representative by the cosine curve. An extensive quantitative analysis has been performed in non-uniform non-staggered grids through numerical computations for the effect of surface irregularities on the flow velocity, the flux, the resistive impedance and on the wall shear stress through their graphical representations so as to validate the applicability of such an improved mathematical model.

Numerical Simulations of Heat Transfer and Fluid Flow Problems Using an Immersed-Boundary Finite-Volume Method on NonStaggered Grids

ABSTRACT This article describes the application of the immersed boundary technique for simulating fluid flow and heat transfer problems over or inside complex geometries. The methodology is based on a fractional step method to integrate in time. The governing equations are discretized and solved on a regular mesh with a finite-volume nonstaggered grid technique. Implementations of Dirichlet and Neumann types of boundary conditions are developed and completely validated. Several phenomenologically different fluid flow and heat transfer problems are simulated using the technique considered in this study. The accuracy of the method is second-order, and the efficiency is verified by favorable comparison with previous results from numerical simulations and laboratory experiments.

Natural convection in a bottom heated horizontal cylinder

In this work a numerical investigation of two-dimensional steady and unsteady natural convection in a circular enclosure whose lower half is nonuniformly heated and upper half is maintained at a constant lower temperature has been carried out. An explicit finite difference method on a nonstaggered rectangular grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a range of Rayleigh number (Ra) varying between 102 and 106 at a fixed Prandtl number (Pr) taken as 0.71. The numerical experiments reveal that for Ra⩽8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers (Ra<300), it is shown that the velocity field is very weak and the heat transfer is predominantly by conduction. A series solution for the temperature field obtained by neglecting the fluid velocities is shown to agree well with the computed data for Ra<300. The convection takes place in the form of two cells with their interface aligned along the vertical diameter. As Ra is increased further, the isotherms distort to form a plume–like structure of hot fluid rising from the hottest point on the lower half of the cylinder wall. Local Nusselt number distribution over the wall shows that only a portion of the nonuniformly heated bottom half of the cylinder wall is responsible for heating the fluid. For Ra⩾8900, the numerical simulations show that the steady flow looses its stability and the flow undergoes bifurcations to periodic and quasiperiodic states. On the basis of the data on the amplitude of the periodic flows obtained for a set of Ra slightly greater than 8900, it is shown that the steady flow undergoes a supercritical Hopf bifurcation at Ra≈8830. An analysis using the proper orthogonal decomposition shows that the instability is in the form of a standing wave. The structure of the unstable mode is examined via empirical eigenfunctions obtained by the method of snapshots. In the unsteady regime (Ra>8.9×103), the cells start to swing their interface in an oscillatory manner with time. As Ra is increased further, the character of flow changes from periodic to quasiperiodic.

A NUMERICAL PROCEDURE FOR THE LAMINAR HEAT TRANSFER IN CURVED SQUARE DUCTS

ABSTRACT The fully developed forced convective heat transfer flow in a curved square duct is studied, under a constant uniform wall temperature axially and peripherally. The continuity and momentum equations are solved with a new numerical-variational method denoted the CVP method. The method uses a nonstaggered grid for all primitive variables without introducing spurious oscillations in the solutions. Numerical results are presented for the effects of the curvature, the Dean number and the stream pressure gradient, on the velocity, the temperature, the friction factor, and the Nusselt number for fluids with Prandtl numbers 0.7 and 7. The accuracy of the method is discussed, and the results are in close agreement with those obtained by other methods.

Incompressible Fluid Flow Computation in an Arbitrary Two-dimensional Region on Nonstaggered Grids

Abstract A numerical algorithm for solving the Navier-Stokes equations for incom- pressible viscous fluid in an arbitrary two-dimensional region on nonstaggered grids is presented. The idea of the transition to a general curvilinear coordinate system, trans- forming the physical region into a parametrical square is used. For the discrete solution an unconditional a priory estimate has been obtained. The results of the benchmark computations for a driven skewed cavity flow and the results of the fluid flow modeling in a cavity of an arbitrary shape are given.

On a high-order Newton linearization method for solving the incompressible Navier-Stokes equations

The present study aims to accelerate the non-linear convergence to incompressible Navier–Stokes solution by developing a high-order Newton linearization method in non-staggered grids. For the sake of accuracy, the linearized convection–diffusion–reaction finite-difference equation is solved line-by-line using the nodally exact one-dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved owing to the reduction of stencil points. This Newton linearization method is computationally efficient and is demonstrated to outperform the classical Newton method through computational exercises. Copyright © 2005 John Wiley & Sons, Ltd.

Low Reynolds number forced convection in three-dimensional wavy-plate-fin compact channels: fin density effects

Steady forced convection in periodically developed low Reynolds number (10 ⩽ Re ⩽ 1000) air ( Pr = 0.7) flows in three-dimensional wavy-plate-fin cores is considered. Constant property computational solutions are obtained using finite-volume techniques for a non-orthogonal non-staggered grid. Results highlight the effects of wavy-fin density on the velocity and temperature fields, isothermal Fanning friction factor f, and Colburn factor j. The fin waviness is seen to induce the steady and spatially periodic growth and disruption of symmetric pairs of counter-rotating helical vortices in the wall-trough regions of the flow cross-section. The thermal boundary layers on the fin surface are thereby periodically interrupted, resulting in high local heat transfer near the recirculation zones. Increasing fin density, however, tends to dampen the recirculation and confine it. The extent of swirl increases with flow rate, when multiple pairs of helical vortices are formed. This significantly enhances the overall heat transfer coefficient as well as the pressure drop penalty, when compared to that in a straight channel of the same cross-section. The relative surface area compactness as measured by the ( j/ f) performance or the area goodness factor nevertheless increases with fin density.

Numerical study of flow and heat transfer in curvilinear ducts: applications of elliptic grid generation

A numerical study was performed to investigate heat transfer and fluid flow in curvilinear shaped duct geometries. SIMPLEM algorithm was used with a non-staggered grid arrangement to solve Navier–Stokes and energy equations in curvilinear coordinates. To obtain grid generation for curvilinear surfaces elliptic grid generation method was used. Two examples were demonstrated for laminar flow regime in the study: First, asymmetric sinusoidal wavy walled duct and second partially inclined duct geometry were chosen. The results show that SIMPLEM algorithm with non-staggered grid arrangement is a useful technique to solve temperature and flow field in curvilinear ducts by using elliptic grid generation method.

IMPLEMENTATION OF CLEAR ALGORITHM ON COLLOCATED GRID SYSTEM AND APPLICATION EXAMPLES

ABSTRACT Implementation of the CLEAR algorithm on a collocated grid system is conducted. Detailed discussion of the previous momentum interoperation method (MIM) is given to analyze the condition to get a unique converged solution that is independent of relaxation factor for steady flow. Six numerical examples on nonstaggered grids of forced-convective fluid flow and natural convection are provided to compare the convergence performance between CLEAR and SIMPLER. The domain extension method, widely used on staggered grids to deal with mildly irregular domains, is further refined to meet the requirement of collocated grids. It is shown that on a collocated grid the CLEAR algorithm can also greatly enhance the convergence rate, based on iteration number and CPU time consumed, compared with the SIMPLER algorithm with similar robustness.

Solution of transport equations on unstructured meshes with cell-centered colocated variables. Part I: Discretization

This paper describes discretization of transport equations on unstructured meshes with cell-centered colocated variables. The problem of zig-zag pressure prediction is eliminated by introducing smoothing pressure correction derived by Date [A.W. Date, Complete pressure correction algorithm for solution of incompressible Navier–Stokes equations on a non-staggered grid, Numer. Heat Transfer, Part B 29 (1995) 441–458]. The finite-volume discretization is carried out in a structured grid like manner by invoking a special line-structure to evaluate convective–diffusive transport across the cell-faces.

Natural convection flow of air in an inclined open cavity

Consider natural convection flow in a two dimensional inclined open cavity for both transient and steady-state flow. The left hand vertical wall of cavity is heated and facing an opening. The top and bottom boundaries are insulated. Numerical solutions are obtained for the Navier-Stokes equations on a non-staggered grid using a finite volume scheme. Results are considered for a wide range of Rayleigh numbers varying from 10 5 to 10 10 with Prandtl number 0.7 and inclination angles from 10 ? to 90 ? . The flow is steady at the low Rayleigh number for all angles and becomes unsteady at the high Rayleigh number for all angles. The critical Rayleigh numbers for all angles are obtained and we show that the critical Rayleigh number decreases as the inclination angle increases.

Three dimensional simulation of fluid flow in X‐ray CT images of porous media

AbstractA numerical scheme is developed in order to simulate fluid flow in three dimensional (3‐D) microstructures. The governing equations for steady incompressible flow are solved using the semi‐implicit method for pressure‐linked equations (SIMPLE) finite difference scheme within a non‐staggered grid system that represents the 3‐D microstructure. This system allows solving the governing equations using only one computational cell. The numerical scheme is verified through simulating fluid flow in idealized 3‐D microstructures with known closed form solutions for permeability. The numerical factors affecting the solution in terms of convergence and accuracy are also discussed. These factors include the resolution of the analysed microstructure and the truncation criterion. Fluid flow in 2‐D X‐ray computed tomography (CT) images of real porous media microstructure is also simulated using this numerical model. These real microstructures include field cores of asphalt mixes, laboratory linear kneading compactor (LKC) specimens, and laboratory Superpave gyratory compactor (SGC) specimens. The numerical results for the permeability of the real microstructures are compared with the results from closed form solutions. Copyright © 2004 John Wiley & Sons, Ltd.

Comparative Studies of Three Approaches for GDQ Computation of Incompressible Navier–Stokes Equations in Primitive Variable Form

Three numerical approaches for solving the incompressible Navier–Stokes equations in primitive variable form are proposed and compared in this work. All these approaches are based on the SIMPLE strategy with GDQ discretization on a non-staggered grid. It was found that the satisfying of the continuity equation on the boundary is critical to obtain an accurate numerical solution. The proposed three approaches are to make sure that the continuity equation is satisfied on the boundary, and in the meantime, recommend the boundary condition for pressure correction equation. Through a test problem of two-dimensional driven cavity flow, the performance of three approaches is comparatively studied in terms of the efficiency and accuracy. For all three approaches, accurate numerical results can be obtained by using just a few grid points.

An Improved Three-Dimensional Level Set Method for Gas-Liquid Two-Phase Flows

For the present study, we developed a three-dimensional numerical method based on the level set method that is applicable to two-phase systems with high-density ratio. The present solver for the Navier-Stokes equations was based on the projection method with a non-staggered grid. We improved the treatment of the convection terms and the interpolation method that was used to obtain the intermediate volume flux defined on the cell faces. We also improved the solver for the pressure Poisson equations and the reinitialization procedure of the level set function. It was shown that the present solver worked very well even for a density ratio of the two fluids of 1:1000. We simulated the coalescence of two rising bubbles under gravity, and a gas bubble bursting at a free surface to evaluate mass conservation for the present method. It was also shown that the volume conservation (i.e., mass conservation) of bubbles was very good even after bubble coalescence.

가상 경계 방법을 이용한 유동 해석 기법에 관한 기초 연구

In most industrial applications, the geometrical complexity is combined with the moving boundaries. These problems considerably increase the computational difficulties since they require, respectively, regeneration and deformation of the grid. As a result, engineering flow simulation is restricted. In order to solve this kind of problems the immersed boundary method was developed. In this study, the immersed boundary method is applied to the numerical simulation of stationary, rotating and oscillating cylinders in the 2-dimensional square cavity. No-slip velocity boundary conditions are given by imposing feedback forcing term to the momentum equation. Besides, this technique is used with a second-order accurate interpolation scheme in order to improve the accuracy of flow near the immersed boundaries. The governing equations for the mass and momentum using the immersed boundary method are discretized on the non-staggered grid by using the finite volume method. The results agree well with previous numerical and experimental results. This study presents the possibility of the immersed boundary method to apply to the complex flow experienced in the industrial applications. The usefulness of this method will be confirmed when we solve the complex geometries and moving bodies.

On Nonstaggered Rectangular Grids Using Streamfunction and Velocity Potential or Vorticity and Divergence

Abstract Dispersion characteristics of the shallow water gravity–inertia waves are discussed for two nonstaggered rectangular grids that use the velocity potential and streamfunction, and the divergence and vorticity, respectively. It is shown that the simplest second-order accuracy schemes on the two grids produce identical frequencies on an infinite rotating plane, and that, therefore, the two grids are equivalent. Stability analysis of the linearized shallow water equations using leapfrog and semi-implicit time-differencing techniques indicates that viable numerical schemes can be designed on these grids.

CFD STUDIO: AN EDUCATIONAL SOFTWARE FOR CFD ANALYSIS

The main goal of this paper is to demonstrate the general characteristics of the educational user-friendly CFD Studio package for CFD teaching. The package was designed for teaching 2D fluid mechanics and heat transfer process, including conduction, coupled conduction/convection, natural and forced convection, external and internal flows, among other phenomena. The finite volume methodology and its related topics can also be taught using the software. Therefore, general aspects of the three main modules, pre-processor, solver and post-processor are discussed aiming to show the generality of the tool. These modules are integrated in the application by a so-called “numerical problem project” which guide the student through the steps to obtain the solution. To approximate the partial differential equations the finite volume approach is employed using a fully-implicit formulation with the interpolation schemes CDS, UDS and WUDS. Mesh editing and nonorthogonal boundary-fitted mesh generation, using algebraic interpolation and elliptic equations, are important features of the package. Coupled heat transfer problems are handled using the “solid-block” formulation and the pressure-velocity coupling uses the SIMPLE and SIMPLEC methods with non-staggered grids. To demonstrate the capabilities two fluid flow and heat transfer “problem projects” are presented.