Collocated Grid System Research Articles

Numerical Study of Natural Convection for Generalized Second-Grade Fluids Confined in a Square Cavity Subjected to Horizontal Heat Flux

Two-dimensional steady laminar natural convection of a viscoelastic fluid represented by generalized second-grade fluid model in a square enclosure is studied. The cavity is submitted at its vertical sides to a uniform density of heat flux while the horizontal walls are insulated, without slipping conditions at all the solid boundaries. The governing conservation and constitutive equations with the corresponding boundary conditions are solved by finite volume method in a collocated grid system. The contributions of shear rate dependent and elastic characteristics of the viscoelastic fluid are investigated on momentum and heat transport. The effects of elastic number (E) in the range 0 - 1 on heat transfer and fluid motion are interpreted for a power-law index (n) in the range 1.4 - 0.6 and nominal values of Rayleigh number (Ra) range of 103 to 105.

Verification of the numerical analysis using a collocated grid system directly solving the quasi-inviscid incompressible flow around a symmetric airfoil

The present study describes the results of validating a high-quality computational scheme of kinetic energy conservation characteristics under a collocated grid system. In the present study, the pseudo-inviscid incompressible flow around the Kármán-Trefftz airfoil is analyzed. In this analysis, the kinetic energy conservation error is reduced by temporally evolving the governing equation using pressure increments in fractional steps. This sufficient conservation of kinetic energy is validated using an inviscid homogeneous isotropic fluctuation field. The velocity field obtained by this analysis is compared to that of the potential flow obtained using the Kármán-Trefftz airfoil. The numerical analysis is also validated using the lift coefficient profile as a function of a small angle of attack.

Second-order kinetic energy-conservative analysis of incompressible turbulence under a collocated grid system

The study presented in this paper investigates the second-order conservation error of kinetic energy of an incompressible flow under a collocated grid. A collocated grid is often used to numerically analyse the flow around an aerofoil. The kinetic energy conservation error under a collocated grid is often greater than that under a staggered grid. The results of analysis based on coarse computational grids, such as LES analysis, are considerably affected by the kinetic energy conservation error. This study shows the results of numerical analysis in which the kinetic energy has a second-order conservation error under the collocated grid. This higher-order kinetic energy is strictly validated using inviscid flow. Furthermore, the results of conservation errors of higher-order statistics of kinetic energy are discussed in this study. The numerical analysis is then validated using a turbulent channel flow. The statistics obtained by this analysis agree well with those of previous studies in turbulent channel flows.

Solution of lid-driven cavity problems with an improved SIMPLE algorithm at high Reynolds numbers

The SIMPLE algorithm and its variants have been successfully employed to solve the driven-cavity problem at Re<10,000. However, converged steady solutions cannot be achieved at higher Re (15,000–30,000) by SIMPLE-family algorithms owing to its inherent drawbacks. In this work, a new segregated solver, ISTEC-N (incompressible flow solver with two explicit N correction loops), for determining the solution of incompressible flow on structured collocated grid systems is proposed. A pressure equation is deduced from the discretized continuity equation to overcome the “Semi-Implicit” characteristic of the SIMPLE algorithm. An explicit velocity update loop is introduced to improve the consistency between the pressure field and the velocity field in each outer iteration. The performance of the ISTEC-N algorithm is evaluated by solving the driven-cavity problem at high Re number of (15,000<Re<30,000). It is shown that the ISTEC-N algorithm manifests a strong robustness feature under these high convection conditions. The computational results given by the ISTEC-N algorithm are in excellent agreement with the benchmarking data.

Mixed Convection Heat Transfer From an Equilateral Triangular Cylinder in Cross Flow at Low Reynolds Numbers

A two-dimensional numerical study is undertaken to investigate the influences of cross buoyancy on the vortex shedding phenomena behind a long heated equilateral triangular cylinder for the low-Reynolds-number laminar regime. The flow is considered in an unbounded medium; however, fictitious confining boundaries are chosen on the lateral sides to make the problem computationally feasible. Numerical calculations are performed by using a finite-volume method based on the pressure-implicit with splitting of operators algorithm in a collocated grid system. The range of Reynolds number is chosen to be 10–100 with a fixed Prandtl number, 0.71. The mixed convection effect is studied for the Richardson number range of 0–1. The effects of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as the overall drag and lift coefficients, local and surface average Nusselt numbers, and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers.

Numerical Simulation of Natural Convection in a Parallelogrammic Enclosure Containing Volumetric Heat Source with Non‐uniformly Heated Left Sidewall

The influence of the external Rayleigh number, inclination angle, and internal Rayleigh number on natural convection within an air‐filled parallelogrammic enclosure containing a volumetric source has been investigated numerically. The left sidewall of the enclosure is subjected to a non‐uniformly hot temperature and the right sidewall experiences a uniform cold temperature while the remaining top and bottom walls are kept adiabatic. The physical problems are represented mathematically by various sets of governing equations along with the corresponding boundary conditions. Buoyancy forces are taken into account during the analysis of the present investigation. By using the finite volume method, the dimensionless governing equations are discretized numerically based on a non‐uniform collocated grid system. Results are obtained for a wide range of external Rayleigh numbers varying from 103 to 106 with internal Rayleigh numbers varying from (0) to (108) while the left sidewall from vertical is varied as 0, 30, –30, 60, and −60°, respectively. In the present study, the obtained results are presented in terms of streamlines, isotherms, and average Nusselt number along the hot and cold sidewalls. Two pairs of rotating vortices are observed due to the non‐uniform heating process while the shape of this rotating vortices is sensitive to the inclination angle. Furthermore, the flow field circulation and the average Nusselt number increase remarkably with the increase in the external Rayleigh number. The results of the present work are compared with other published results and give excellent agreement. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(6): 542–560, 2014; Published online 11 November 2013 in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21096

Unsteady mixed convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

A two-dimensional numerical study is carried out to understand the influence of cross buoyancy on the vortex shedding processes behind two equal isothermal square cylinders placed in a tandem arrangement at low Reynolds numbers. The spacing between the cylinders is fixed with five widths of the cylinder dimension. The flow is considered in an unbounded medium, however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a finite volume method based on the PISO algorithm in a collocated grid system. The range of Reynolds number is chosen to be 50–150. The flow is unsteady laminar and two-dimensional in this Reynolds number range. The mixed convection effect is studied for Richardson number range of 0–2 and the Prandtl number is chosen constant as 0.71. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers.

Mixed Convection Heat Transfer from Tandem Square Cylinders for Various Gap to Size Ratios

This article presents a two-dimensional numerical study on the fluid flow and mixed convection heat transfer around two equal isothermal square cylinders placed in a tandem arrangement and subjected to the cross flow of a Newtonian fluid at moderate Reynolds numbers. The spacing between the cylinders is varied by changing the gap to cylinder size ratio as S/d = 1, 2, 3, 4, 5, 7, and 10. The Reynolds number is considered in the range 50 ≤ Re ≤ 150. The mixed convection effect is studied for Richardson number range of 0–2, and the Prandtl number is chosen constant as 0.71. The flow is considered in an unbounded medium; however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a PISO algorithm-based finite volume solver in a collocated grid system. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers, and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers and spacing ratios. The notable contribution is the quantification of the critical spacing ratio which is observed to decrease with increasing thermal buoyancy effect for a specific Reynolds number.

Numerical Prediction of Wind Flow over Sinusoidal Hills Using Multigrid Method

The finite volume method and multigrid method are presented to solve Navier-Stokes equations. MG method is introduced in this work and applied on solving the discrete systems of the governing equations. For turbulent flow, the standard k-ε model is adopted. The non-orthogonal collocated grid systems are generated. The numerical prediction were done for the flow over sinusoidal hills. The numerical results were obtained from the simulations of flow over three types of sinusoidal hills. Comparisons have been made between experimental data and the computational data and agree with each other.

Three-Dimensional Numerical Study of Fluid and Heat Transfer Characteristics of Dimpled Fin Surfaces

In the present study, a code based on the nonorthogonal curvilinear coordinates is developed with a collocated grid system generated by the two-boundary method. After validation of the code, it is used to compare simulated results for a fin-and-tube surface with coupled and decoupled solution methods. The results of the coupled method are more agreeable with the test data. Simulation for dimpled and reference plain plate fin-and-tube surfaces are then conducted by the coupled method within a range of inlet velocity from 1.0 m/s to 5 m/s. Results show that at identical pumping power the dimpled fin can enhance heat transfer by 13.8–30.3%. The results show that relative to the reference plain plate fin-and-tube surface, heat transfer rates and pressure drops of the dimpled fin increase by 13.8%–30.3% and 31.6%–56.5% for identical flow rate constraint. For identical pumping power constraint and identical pressure drop constraint, the heat transfer rates increase by 11.0%–25.3% and 9.2%–22.0%, respectively. By analyzing the predicted flow and temperature fields it is found that the dimples in the fin surface can improve the synergy between velocity and fluid temperature gradient.

Mixed Convection Heat Transfer from an In-Line Row of Square Cylinders in Cross-Flow at Low Reynolds Number

This article presents a two-dimensional numerical study on the unsteady laminar mixed convection heat transfer from a row of five in-line isothermal square cylinders placed in an unconfined medium and subjected to cross-flow of a Newtonian fluid at low Reynolds number (Re = 125). The hydrodynamic and thermal transport phenomena are captured for the separation ratios (spacing to cylinder size ratio, s/d) of 0.5, 1, 1.5, 2, 3, and 4. The mixed convection heat transfer is studied for Richardson numbers (Ri) ranging from 0 to 3 with a fixed Prandtl number (Pr = 0.71). Numerical calculations are performed by using a PISO algorithm-based finite volume solver in a collocated grid system. The instantaneous vorticity fields along with the isotherm patterns are systematically presented and discussed for different separation ratios and Richardson numbers. Depending on the engineering application, the temperature difference between the surface and the free stream could vary to make buoyancy of primary importance, entailing major modification of the flow field. Additionally, the instantaneous and mean drag and lift coefficients, Strouhal numbers, and mean Nusselt numbers are determined and discussed.

3D Numerical Simulation of Flow through a 90 Bending Square Duct

Abstract:A computational model is implemented to investigate hydraulic characteristics of turbulent flow through a 90 bending square duct. In the numerical solution the governing equations are first transformed by using boundary-fitted coordinate to map the physical domain to a uniform transformed space, subsequently discretized by finite-volume method and collocated grid system, and solved via SIMPLEC algorithm.The flow parameters at the control surface are obtained by momentum interpolation method. The modified k-ε turbulence model is applied to close the Reynolds stresses. The 3D hydraulic characteristics of flows in a bend square duct are computed, and the comparison between numerical results and experimental data shows a very good agreement.

Forced Convection Heat Transfer From Tandem Square Cylinders for Various Spacing Ratios

This article presents a two-dimensional numerical study on the fluid flow and forced convection heat transfer around two equal isothermal square cylinders placed in a tandem arrangement and subjected to the cross flow of a Newtonian fluid at low Reynolds numbers. The spacing between the cylinders is varied by changing the gap to cylinder size ratio as S/d = 1, 2, 3, 4, 5, 7, and 10. The flow is considered in an unbounded medium; however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a PISO algorithm based finite volume solver in a collocated grid system. The Reynolds number is considered in the range 50 ≤ Re ≤ 150 and the Prandtl number is chosen constant as 0.71. The instantaneous vorticity and isotherm patterns are presented and discussed at various Reynolds numbers and spacing ratios for the flow and thermal transport visualization. The critical spacing ratios are predicted for the above range of Re. Additionally, the global flow and heat transfer quantities such as the overall drag and lift coefficients, local and surface average Nusselt numbers, and Strouhal number are calculated and discussed for various Reynolds numbers and spacing ratios.

A Conservative Pressure-Correction Method on Collocated Grid for Low Mach Number Flows

A novel extension to SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor—corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. The proposed algorithm has second order centrally differenced convective fluxes with upwinding based on Cell Peclet number while diffusive flux are viscous fourth order accurate. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. The algorithm is kinetic energy preserving. Further in this paper robustness and accuracy are demonstrated by performing test on channel flow with non-Boussinesq condition on different temperature ratios.

Numerical Prediction of Turbulent Flow over 3D Sinusoidal Hill Using Non-orthogonal Hexahedron Grid and AMG Method

Numerical predictions of the wind energy distribution over terrains are essential for the appropriate selection of a suitable site for a wind power plant. This paper presents three-dimension numerical simulations of turbulence flow over sinusoidal hills. AMG in HYPRE (High Performance Preconditioner Library) is introduced in this work and applied on solving the FVM-based discrete systems of the governing equations for turbulent flow. The three-dimensional non-orthogonal collocated grid systems are generated and the AMG-based algorithm is established. Turbulence is modelled with the standard model and flow parameters of Governing equations are solving by AMG method. The numerical results were obtained from the simulations of flow over three types of three-dimensional sinusoidal hills. Comparisons have been made by virtue of experimental data with the computational data. The numerical results agree with the experimental results, the numerical methods are effective.

Two-dimensional unsteady aerodynamics analysis based on a multiphase perspective

Compressible aerodynamics is analyzed by the CIP/CCUP (constraint interpolation profile/CIP-combined unified procedure) method. The CIP method is feasible for an analysis of various phase change problems, those associated with compressible and incompressible flow areas. For aeroelastic problems based on CFD (computational fluid dynamics), the use of Lagrangian body-fitted grids is problematic because these grids increase the skewness of a mesh, or can be broken – even during structural motions. Therefore, a method embedding a physical boundary in a fixed Eulerian grid is appropriate compared to Lagrangian grids. In this paper, the CCUP method based on a pressure-based algorithm in which the pressure Poisson equation is modified to deal with compressible flows. A collocated grid system with the velocity components on the cell face obtained from the CIP interpolations is applied. Boundary condition for arbitrary surfaces are newly derived using a simple algebraic relationship based on the immersed boundary method. Far-field boundary condition is replaced with sponge layers. Several numerical benchmarking problems, in this case a wedge or a bump under transonic and supersonic flows, are tested to verify the code. Unsteady motions are directly applied to embedded solid areas. Finally, the steady and unsteady aerodynamics of airfoil sections are calculated and compared to reference data to validate the proposed method.

Effect of thermal buoyancy on vortex shedding behind a square cylinder in cross flow at low Reynolds numbers

Two-dimensional numerical simulation is performed to understand the influence of cross buoyancy on the vortex shedding process behind a stationary heated square cylinder at low Reynolds numbers. The flow is considered in an unbounded medium, however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are carried out by using a finite volume method based on the PISO algorithm in a collocated grid system. The range of Reynolds number is chosen to be 5–40. In this range the flow and thermal fields are found to be steady and separated without the superimposed thermal buoyancy (i.e. for pure forced convection). However, as the buoyancy parameter (Richardson number) increases flow becomes unstable and subsequently, at some critical value of Richardson number, periodic vortex shedding is observed to characterize the flow and thermal fields. The global flow and heat transfer quantities such as the recirculation length, overall drag coefficient and surface average Nusselt number are computed for the steady separated flow (without buoyancy effect) and are found in good agreement with the available results in the literature. The effect of superimposed thermal buoyancy for the range of Reynolds number is studied for various Richardson numbers. The frequency of vortex shedding (Strouhal number) is calculated for the unsteady periodic flow. The critical Richardson number for the onset of vortex shedding decreases and the corresponding critical Strouhal number increases with increasing Reynolds number in the chosen range.

A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh

The consistent and conservative scheme developed on a rectangular collocated mesh [M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system, Journal of Computational Physics 227 (2007) 174–204] and on an arbitrary collocated mesh [M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh, Journal of Computational Physics 227 (2007) 205–228] has been extended and specially designed for calculation of the Lorentz force on a staggered grid system (Part III) by solving the electrical potential equation for magnetohydrodynamics (MHD) at a low magnetic Reynolds number. In a staggered mesh, pressure ( p) and electrical potential ( φ) are located in the cell center, while velocities and current fluxes are located on the cell faces of a main control volume. The scheme numerically meets the physical conservation laws, charge conservation law and momentum conservation law. Physically, the Lorentz force conserves the momentum when the magnetic field is constant or spatial coordinate independent. The calculation of current density fluxes on cell faces is conducted using a scheme consistent with the discretization for solution of the electrical potential Poisson equation, which can ensure the calculated current density conserves the charge. A divergence formula of the Lorentz force is used to calculate the Lorentz force at the cell center of a main control volume, which can numerically conserve the momentum at constant or spatial coordinate independent magnetic field. The calculated cell-center Lorentz forces are then interpolated to the cell faces, which are used to obtain the corresponding velocity fluxes by solving the momentum equations. The “conservative” is an important property of the scheme, which can guarantee computational accuracy of MHD flows at high Hartmann number with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers. 2D fully developed MHD flows with analytical solutions available have been conducted to validate the scheme at a staggered mesh. 3D MHD flows, with the experimental data available, at a constant magnetic field in a rectangular duct with sudden expansion and at a varying magnetic field in a rectangular duct are conducted on a staggered mesh to verify the computational accuracy of the scheme. It is expected that the scheme for the Lorentz force can be employed together with a fully conservative scheme for the convective term and the pressure term [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, Journal of Computational Physics 143 (1998) 90–124] for direct simulation of MHD turbulence and MHD instability with good accuracy at a staggered mesh.

3D Open Channel Flow Modeling by Applying 1D Adjustment

A three-dimensional (3D) finite volume model with a novel adjustment scheme was developed to solve shallow water equations in open channels. An explicit finite volume method was used to discretize the governing equations in a boundary-fitted structured and collocated grid system. Because a simple second-order central scheme was used for spatial discretization and due to the occurrence of high Peclet numbers in open channel flows, some treatments were needed to reduce oscillation. Thus, a special adjustment scheme designed to minimize differences in the averaged free surface elevation and flow discharge in a 3D model and 1D flow data was applied to some cross-sections. The model was applied to simulate shallow water flow in a backwardfacing step, a meandering channel with 90° bends and a 180° bend channel. A comparison of the model results with available experimental and numerical data illustrated that the proposed numerical procedure decreases the numerical oscillations and increases the stability of the 3D numerical model in open channel flow modeling.

The Effects of Reynolds and Prandtl Numbers on Flow and Heat Transfer Across Tandem Square Cylinders in the Steady Flow Regime

The effects of Reynolds and Prandtl numbers on the fluid flow and heat transfer characteristics over two equal isothermal square cylinders placed in a tandem arrangement in cross flow have been investigated in this article. The spacing between the cylinders is fixed with four widths of the cylinder. The numerical results are presented for the range of conditions as 1 ≤ Re ≤ 30 and 0.7 ≤ Pr ≤ 1000 for three different blockage parameters B = 0.05, 0.25, and 0.5. Numerical simulations are performed with a finite volume code based on the PISO algorithm in a collocated grid system. The representative streamlines and isotherm patterns are presented and discussed. In addition, the overall drag coefficient and average Nusselt number are determined to elucidate the role of Reynolds and Prandtl numbers on flow and heat transfer. It is found that the flow is completely steady for the chosen ranges of the parameters.