Non-uniform Grid System Research Articles

A new implicit finite difference method with a compact correction term for solving unsteady convection diffusion equations

A new implicit finite difference method with a compact correction term is established for solving unsteady convection-diffusion equations. The correction terms by connecting classical and compact finite difference formulas are proposed for improving the accuracies of numerical solutions. This new method with fourth order accuracy can be used for the numerical calculations of convection diffusion equation on both uniform and nonuniform grid systems. It can be extended not only from one-dimensional to multi-dimensional equations, but also from steady to unsteady convection-diffusion equations. Several convection-diffusion problems are stimulated for verifying the flexibility and high accuracy of this method.

Characteristics and analysis of a turbulent offset jet including the effect of density and offset height

This paper presents the results of measurements and numerical predictions of 3D turbulent offset jet flows. Mean velocity and turbulence characteristics of a rectangular offset jet with different offset heights and within variable densities are experimentally and numerically investigated in detail. Four jet gas exit densities, ρj = 1.2, 1.25, 1.3 and 1.4 Kg/m3, and four offset ratios, h/w = 8, 16, 25 and 33, are studied. The considered variation of the jet density is revealed at different Reynolds numbers and the velocity measurements are carried out using a Laser Doppler Velocimetry (LDV) technique. The handled configuration is numerically simulated by solving the Navier-Stokes equations with the finite volume method having a non-uniform grid system. Two different closure models are tested: the standard k–ε model and the Reynolds Stress Model (RSM). Results clearly revealed significant effects of the jet density and the offset height on the flow development in the early region. It is noted that the centerline velocity decay increases as the jet density and the offset height increase. It is also observed that the reattachment length of the jet decreases with the increase of the jet density. However, the reattachment length is found to increase with the increase of the offset height.

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

Numerical investigation of heat transfer and friction characteristics for turbulent flow in various corrugated tubes

Thermal and hydraulic characteristics of turbulent water flow in a transverse corrugated tube with various corrugation direction (inward/outward) and corrugation shape (triangle, curve, rectangle, and trapezoid) are numerically investigated. The axisymmetric model of corrugated tubes with 10 mm inner diameter was investigated by changing the geometrical parameters for Reynolds number ranging from 5000 to 61,000 and constant heat flux boundary condition. Structured, nonuniform grid system is applied. Momentum, continuity, and energy equations were treated by means of a finite volume method using the SIMPLE scheme with the k–ε turbulence model and enhanced wall treatment. The results reveal that corrugation direction and corrugation shape have perceptible effects upon the heat transfer in the form of Nusselt number ( Nu) and pressure drop in the form of friction factor (ƒ). The average Nu for (inward) trapezoidal, rectangular, curved, and triangular corrugation shapes are 52.61%, 50.12%, 47.82%, and 44.96%, respectively, higher than the smooth tube. The average Nu for (outward) trapezoidal, curved, triangular, and rectangular corrugation shapes are 48.31%, 45.72%, 41.23%, and 40.94%, respectively, which are higher than a smooth tube. The results reveal that both inward/outward curved and triangular roughness shape have the superior performance evaluation criterion than rectangular and trapezoidal. Turbulence kinetic energy contour shows the increase in heat transfer performance for all corrugated tubes compared with a smooth tube. Inward corrugated tube provides the highest turbulence kinetic energy along the tube length and, consequently, the highest heat transfer. In addition, inward corrugated tubes provide the highest values and homogeneity of the velocity distribution along the core of tubes.

Efficient surface water flow simulation on static Cartesian grid with local refinement according to key topographic features

Aiming at improving the computational efficiency without accuracy losses for surface water flow simulation, this paper presents a structured but non-uniform grid system incorporated into a Godunov-type finite volume scheme. The proposed grid system can detect the key topographic features in the computational domain where high-resolution mesh is in need for reliably solving the shallow water equations. The mesh refinement is automatically carried out in these areas while the mesh in the rest of the domain remains coarse. The criterion determining the refinement is suggested by a dimensionless number with a fixed value of 0.2 after sensitivity analysis. Three laboratory and field-scale test cases are employed to demonstrate the performance of the model for flow simulations on the new non-uniform grids. In all of the tests, the grid system is shown to successfully generate high-resolution mesh only in those areas with abruptly changing topographic features that dominate the flooding processes. To produce numerical solutions of similar accuracy, the non-uniform grid based model is able to accelerate by about two times comparing with the fine uniform grid based counterpart.

Pure axial flow of viscoelastic fluids in rectangular microchannels under combined effects of electro-osmosis and hydrodynamics

This paper presents an analysis of the combined electro-osmotic and pressure-driven axial flows of viscoelastic fluids in a rectangular microchannel with arbitrary aspect ratios. The rheological behavior of the fluid is described by the complete form of Phan-Thien–Tanner (PTT) model with the Gordon–Schowalter convected derivative which covers the upper convected Maxwell, Johnson–Segalman and FENE-P models. Our numerical simulation is based on the computation of 2D Poisson–Boltzmann, Cauchy momentum and PTT constitutive equations. The solution of these governing nonlinear coupled set of equations is obtained by using the second-order central finite difference method in a non-uniform grid system and is verified against 1D analytical solution of the velocity profile with less than 0.06% relative error. Also, a parametric study is carried out to investigate the effect of channel aspect ratio (width to height), wall zeta potential and the Debye–Huckel parameter on 2D velocity profile, volumetric flow rate and the Poiseuille number in the mixed EO/PD flows of viscoelastic fluids with different Weissenberg numbers. Our results show that, for low channel aspect ratios, the previous 1D analytical models underestimate the velocity profile at the channel half-width centerline in the case of favorable pressure gradients and overestimate it in the case of adverse pressure gradients. The results reveal that the inapplicability of the Debye–Huckel approximation at high zeta potentials is more significant for higher Weissenberg number fluids. Also, it is found that, under the specified values of electrokinetic parameters, there is a threshold for velocity scale ratio in which the Poiseuille number is approximately independent of channel aspect ratio.

Modelling of a non-buoyant vertical jet in waves and currents

A generic numerical model using the large eddy simulation (LES) technique is developed to simulate a non-buoyant vertical jet in wave and/or current environments. The experimental data obtained in five different cases, i.e., one case of the jet in a wave only environment, two cases of the jet in a cross-flow only environment and two cases of the jet in a wave and cross-flow coexisting environment, are used to validate the model. The grid sensitivity tests are conducted based on four different grid systems and the results illustrate that the non-uniform grid system C (205×99×126 nodes with the minimum size of 1/10 jet diameter) is sufficiently fine for the modelling. The comparative study shows that the wave-current non-linear interaction should be taken into account at the inflow boundary while modelling the jet in wave and cross-flow coexisting environments. All numerical results agree well with the experimental data, showing that: (1) the jet under the influence of the wave action has a faster centerline velocity decay and a higher turbulence level than that in the stagnant ambience, meanwhile the “twin peaks” phenomenon exists on the cross-sectional velocity profiles, (2) the jet under a cross-flow scenario is deflected along the cross-flow with the node in the downstream, (3) the jet in wave and cross-flow coexisting environments has a flow structure of “effluent clouds”, which enhances the mixing of the jet with surrounding waters.

3D modeling of reaction-diffusion dynamics in an electrokinetic Y-shaped microreactor

We perform a 3D numerical modeling of reaction-diffusion dynamics in a Y-shaped microreactor, considering a fully developed combined electroosmotic and pressure-driven flow. The governing equations, based on a second-order irreversible reaction, are solved invoking a finite-volume approach for a non-uniform grid system. We demonstrate that the reaction is highly position dependent: more production is observed adjacent to the horizontal walls for a favorable pressure gradient, whereas both the wall and centerline are the regions of highest production when a back pressure is applied. We further show that, to achieve the maximum production rate, the EDL should be thick enough, the pressure gradient should be unfavorable, and both the diffusiveties and the inlet concentrations of the reacting components should be identical. Although the same is true for the production efficiency, defined to be the ratio of the average production concentration to the inlet concentration of the limiting component, there is a main difference: when the inlet concentrations are the same the efficiency is minimal. Moreover, the concentration pick is inclined toward the component with either less diffusivity or less inlet concentration; the inclination is magnified when either the Debye length increases or an unfavorable pressure gradient is employed.

Numerical and experimental study of a jet in a crossflow for different velocity ratio

This paper deals with a flow generated by the interaction between a circular jet and a crossflow for different velocity ratios (R = v 0/u ∞). This particular side of this study is of extremely high interest as it allows a better understanding of the mixing process of different interacting flows. This work presents experimental results obtained by means of the particle image velocimetry technique to track the evolution of the jet among the environment flow. Results showed the dependence of the emerging jet flow structure on its ratio velocity. A three-dimensional numerical model with a second-order turbulent model (RSM) and a non-uniform grid system is used to examine the behavior of the emerging jet in the crossflow. The comparison of the numerical and experimental results gives satisfactory agreement.

Numerical simulation of turbulent heat transfer from perforated plate-fin heat sinks

Excessive heat from microelectronic components is essential to remove to increase the reliability of the system. In this paper, various types of perforations in the form of small channels such as square, circular, triangular and hexagonal cross sections are introduced and thermal performances are compared to improve the cooling performance of heat sink. The governing equations are solved by adopting a control volume based finite element method with an unstructured non-uniform grid system. Flow and heat transfer characteristics are presented for Reynolds numbers from 2 × 104 to 4 × 104 based on the fin length and Prandtl number is taken as Pr = 0.71. RANS based k-e turbulence model is used to predict the turbulent flow parameters. The predicted results are validated by the previously published experimental data and in reasonable agreement with the experiment. Results show that fins having circular perforations have better thermal and fluid dynamic performances than the other types of fins considered here.

Effects of Perforations on the Thermal and Fluid Dynamic Performance of a Heat Exchanger

Efficient and optimized heat exchanger design is necessary because of the continuous increase of power consumption rate of electronic equipment and the simultaneous interest in reducing the mass and dimensions of the cooling equipment. This paper presents a numerical study of laminar forced convective heat transfer from an array of solid and different types of perforated fins attached to a rectangular-horizontal base plate. Fluid-flow and heat-transfer characteristics are presented for Reynolds numbers from 100 to 350 based on the fin thickness; Pr and tl number is taken as Pr = 0.71. Finite element approach with an unstructured nonuniform grid system is employed for solving the fluid-flow and heat-transfer equations. The predicted results obtained from the computational fluid dynamics simulations are verified by the previously published data, and the obtained results are in reasonable agreement. This numerical study determines the effects of various types of perforations on the pressure drop and heat-transfer performance of heat exchangers. For each type of perforation, fin effectiveness and performances are determined and compared with those of a solid fin. Results show that perforated fins have better thermal performance than solid fins.

Discussion of Direct Numerical Simulation Method for Supercritical Carbon Dioxide Jet Flow

A kind of direct numerical simulation method suitable for supercritical carbon dioxide jet flow has been discussed in this paper. The form of dimensionless nonconservative compressible Navier-Stokes equations in a two-dimensional cartesian coordinate system is derived in detail. High accurate finite difference compact schemes based on non-uniform grid system are introduced to solve the equations. The simulation results of the three vortex pairing phenomenon of plane mixing layer and a compressible axisymmetric jet flow field show that the discussed numerical simulation method is feasible to calculate the supercritical carbon dioxide jet fluid. And it is found that the difficulties of splitting the convective terms in conservation Navier-Stokes equations, which are brought by the supercritical carbon dioxide fluid pressure state equation, can be avoided by solving the nonconservative compressible Navier-Stokes equations.

THE NEAR FIELD IN THE MIXING OF A THREE-DIMENSIONAL INCLINED POLLUTANT JET WITH A CROSSFLOW

An experimental investigation was carried out to study the structure of the flow field resulting from the interaction of an inclined pollutant jet with a crossflow. The study of the flow field was conducted in a wind tunnel test by means of particle image velocimetry (PIV) system. As the jet was discharged with a variable angle, the resulting flow has been found to be quite complex owing to its three-dimensional nature and the interaction between several flow regions. Results showed the dependence of the emerging jet flow structure on its ratio velocity and the Reynolds number. Extensive wind tunnel experimental results are presented; they concern the Kelvin-Helmholtz vortical structures and the effect of the velocity ratio v0=u1 on the interacting zone. A three-dimensional numerical model with a second-order turbulent model (RSM) and a nonuniform grid system is used to examine the behavior of the emerging jet. The comparison of the numerical and experimental results gives satisfactory agreement.

Transient Radiative Heat Transfer Analysis of Turbid Tissues Subjected to Short-Pulsed Irradiations

Transient radiative heat transfer of turbid tissues with a focused short-pulsed laser is studied. To investigate radiation transfer phenomenon during ultra short time regime, transient radiative transfer equation is adopted and it is solved by TDOM (Transient Discrete Ordinate Method). The nonuniform grid system is considered to capture abrupt change of radiation energy absorption. The hyperbolic conduction model is considered to predict temperature increment of turbid tissues.

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

AbstractIn large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.

Study on the heat transfer and flow characteristics in a spiral-coil tube

In the present study, numerical and experimental results of the heat transfer and flow characteristics of the horizontal spiral-coil tube are investigated. The spiral-coil tube is fabricated by bending a 8.00 mm diameter straight copper tube into a spiral-coil of five turns. The innermost and outermost diameters of the spiral-coil are 270.00 and 406.00 mm, respectively. Hot and cold water are used as working fluids. The k–ε standard two-equation turbulence model is applied to simulate the turbulent flow and heat transfer characteristics. The main governing equations are solved by a finite volume method with an unstructured nonuniform grid system. Experiments are performed to obtain the heat transfer and flow characteristics for verifying the numerical results. Reasonable agreement is obtained from the comparison between the results from the experiment and those obtained from the model. In addition, the Nusselt number and pressure drop per unit length obtained from the spiral-coil tube are 1.49 and 1.50 times higher than those from the straight tube, respectively.

Transient, three-dimensional simulation of particle dispersion in flows around a circular cylinder Re = 140–260)

To understand three-dimensional dispersion of particles in the flow around a bluff body, direct numerical simulations of particle-laden wakes of a circular cylinder with Reynolds number ranging from 140 to 260 were performed. The domain decomposition method and high-order finite-difference schemes were applied to solve the fluid flow. A Lagrangian tracking solver was developed to trace the trajectories of each particle in the non-uniform grid system. It is observed that the predicted coherent structures and vortex dislocation frequency agree well with those of previous experimental studies. Particles at low Stokes numbers follow the vortex motion and can disperse into the core regions of the vortex-street and the recirculating region. Particles at larger Stokes numbers try to maintain their own motion, but take on a non-uniform distribution in the wake due to the strong folding effect of shed vortices. But for particles at intermediate Stokes numbers, higher concentration occurs in the outer boundary regions of the vortices as well as the thin band regions connecting adjacent vortex structures. Besides the spanwise vortex structures, the streamwise vortex tubes also have a remarkable effect on particle dispersion, which results in a ‘mushroom’ shape particle distribution in the wake. Furthermore, non-uniform forcing at inlet has additional effect on particle dispersion and mixing.

Wind Tunnel Investigation and Numerical Simulation of the Near Wake Dynamics for Rectangular Obstacles

An experimental investigation was carried out to study the structure of the flow field around three-dimensional rectangular obstacles. The principal objective was to analyze three-dimensional flow past a single rectangular obstacle and two obstacles placed in tandem by combining the application of a PIV experimental technique and an RSM turbulence model. The study of the flow field around an obstacle was performed in a wind tunnel using a particle image velocimetry (PIV) system. The flow of a fluid around a rectangular obstacle with different attack angles was examined. The results showed the dependence of the flow structure developing around the obstacle on the corresponding Reynolds number, and on the spacing between a pair of tandem aligned obstacles. Detailed quantitative information of turbulence parameters in the vicinity of the obstacle was obtained. Extensive wind tunnel experimental results are presented and compared with numerically simulated ones. A three-dimensional numerical model with the turbulent Reynolds Stress Model (RSM) and a nonuniform grid system was used to examine the effects of a single rectangular obstacle and a double tandem obstacle on the development of the incoming flow. For the case of two rectangular tandem obstacles, the flow patterns were characterized in the gap region as a function of the distance between the obstacles. Good agreement was found between the experimental results of flow and the numerical simulation data.

Experimental and numerical modelling of the three-dimensional incompressible flow behaviour in the near wake of circular cylinders

An experimental investigation was carried out to study the structure of the flow field around three-dimensional circular cylinders. The study of the flow field around an obstacle was performed in a wind tunnel using a particle image velocimetry (PIV) system. The flow of a fluid around an obstacle with a different velocity to the oncoming flow was examined. The results showed the dependence of the flow structure around the obstacle on its Reynolds number, and the spacing between a pair of obstacles. Detailed quantitative information of turbulence parameters in the vicinity of the obstacle was attained. Extensive wind tunnel experimental results are presented and compared with numerical simulation. A three-dimensional numerical model with Reynolds stress model (RSM) turbulence and a non-uniform grid system were used to examine the effects of a single cylinder and two cylinders in tandem on the flow. The principal objective was to analyse three-dimensional flow past a single cylinder and two circular cylinders placed in tandem by combining the application of a PIV experimental technique and an RSM turbulence model. For the case of two cylinders in tandem, the flow patterns are characterized in the gap region as a function of the distance between the cylinders. A good level of agreement was found between the experimental results of flow and numerical simulation.

A high-order compact difference scheme for 2D Laplace and Poisson equations in non-uniform grid systems

In this study, a high-order compact scheme for 2D Laplace and Poisson equations under a non-uniform grid setting is developed. Based on the optimal difference method, a nine-point compact difference scheme is generated. Difference coefficients at each grid point and source term are derived. This is accomplished through the consideration of compatibility between the partial differential equation and its difference discretization. Theoretically, the proposed scheme has third- to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Two examples are provided to examine performance of the proposed scheme. Compared with the traditional five-point difference scheme, the proposed scheme can produce more accurate results with faster convergence. Another reference scheme with the same nine-point grid stencil is derived based on the five-point scheme. The two nine-point schemes have the same coefficients for each grid points; however, their coefficients for the source term are different. The overall accuracy level of the solution resulting from the proposed scheme is higher than that of the nine-point reference scheme. It is also indicated that the smoothness of grids has significant effects on accuracy and convergence of the solutions; efforts in optimizing the grid configuration and allocation can improve solution accuracy and efficiency. Consequently, with the proposed method, solution under the non-uniform grid setting with appropriate grid allocation would be more accurate than that under the uniform-grid manipulation, with the same number of grid points.