Non-staggered Grid System Research Articles

Mixed convection in a square enclosure with a rotating flat plate

Mixed convection in a square enclosure arising from thermal buoyancy force and a rotating flat plate is investigated in the paper. The moving boundary problem is solved with the implicit virtual boundary method on a fixed non-staggered Cartesian grid system. Computations are performed for Prandtl number 0.71 at various Rayleigh numbers Ra and rotational Reynolds numbers Re. For a rotor of length d=0.6 rotating at Re=430, the numerical results show that thermal oscillation occurs when the Rayleigh number exceeds 0.55×106. There is a critical Rayleigh number (0.13×106). Below that the rotor enhances the heat transfer. By contrast, the rotor would suppress the heat transfer beyond the critical Rayleigh number. In the case of Ra=106, the Nusselt number oscillates 23 cycles when the rotor makes 29 revolutions. For a longer rotor d=0.9 rotating at Re=430, thermal oscillation exists for all Rayleigh numbers, and the critical Rayleigh number increases to 0.38×106. In the case of Ra=106, the Nusselt number oscillates twice when the rotor makes one revolution.

Implicit Virtual Boundary Method for Moving Boundary Problems on Non-Staggered Cartesian Patch Grids

AbstractA simple numerical method is proposed in the paper for fluid flow around a moving boundary of irregular shape. The unsteady term is discretized with the implicit scheme such that large time step is allowed. All of the computations are performed on non-staggered Cartesian grid system. Fine Cartesian patch grid system covering the moving object is employed to resolve the solution around the solid body. A closed curve defined by connecting the solid grid points adjacent to the solid-liquid interface is referred to as virtual boundary. The narrow irregular strip of solid between the virtual boundary and the actual solid-fluid interface is called pseudo-fluid. The general fluid region consisting of both fluid and pseudo-fluid is a regular domain that can be efficiently solved with conventional numerical method. In this connection, external force is imposed at each fluid grid point adjacent to the solid-fluid interface to compensate for the numerical error arising from the assumption of pseudo-fluid. The solution procedure is iterated until the required external force converges. Accuracy of the new numerical method is validated through three test problems. The numerical method then is used to investigate the flow induced by the flapping wings of a tethered dragonfly in literature. The corresponding CFL numbers of the four examples are infinity, 20, 100, and 3.29.

A parallel VOF IB pressure-correction method for simulation of multiphase flows

This paper develops an improved pressure correction method, coupled with volume-of-fluid (VOF) and immersed boundary (IB) methods, to accelerate multiphase flow computations. A momentum interpolation method (MIM) is used to avoid the issue of a checkerboard pressure field on a non-staggered grid system. A transpose-free quasi-minimal residual (TFQMR) method is applied to reduce the CPU time when solving the Poisson equation for the pressure correction procedure. A suitable initial guess for the TFQMR method is investigated to enhance the stability of the matrix solver. The IB method is applied to treat the solid-wall boundary conditions while OpenMP is used to reduce CPU time. Two physical problems – dam breakage and a water wave impacting on a tall structure – are simulated and compared with known data. The improved numerical scheme has CPU speeds at least 8 times faster than the initial numerical scheme for simulating two-phase flows. With regard to the parallel computational efficiency, using six cores with the OpenMP allows the improved scheme to have 77% parallel efficiency. These modifications greatly enhance overall computational accuracy and efficiency.

A 3-D implicit finite-volume model of shallow water flows

A three-dimensional (3-D) model has been developed to simulate shallow water flows in large water bodies, such as coastal and estuarine waters. The eddy viscosity is determined using a newly modified mixing length model that uses different mixing length functions for the horizontal and vertical shear strain rates. The 3-D shallow water flow equations with the hydrostatic pressure assumption are solved using an implicit finite-volume method based on a quadtree (telescoping) rectangular mesh on the horizontal plane and the sigma coordinate in the vertical direction. The quadtree technique can locally refine the mesh around structures or in high-gradient regions by splitting a coarse cell into four child cells. The grid nodes are numbered with a one-dimensional index system that has unstructured grid feature for better grid flexibility. All the primary variables are arranged in a non-staggered grid system. Fluxes at cell faces are determined using a Rhie and Chow-type momentum interpolation, to avoid the possible spurious checkerboard oscillations caused by linear interpolation. Each of the discretized governing equations is solved iteratively using the flexible GMRES method with ILUT preconditioning, and coupling of water level and velocity among these equations is achieved by using the SIMPLEC algorithm with under-relaxation. The model has been tested in four cases, including steady flow near a spur-dyke, tidal flows in San Francisco Bay and Gironde Estuary, and wind-induced current in a flume. The calculated water levels and velocities are in good agreement with the measured values.

A numerical simulation on recirculation phenomena of the plume generated by obstacles around a row of cooling towers

The present paper addresses the recirculation phenomena of the plume generated by obstacles around a row of cross-flow type forced draft cooling towers. In order to consider the interaction between external and internal flow of the cooling tower, both outside and inside of the cooling tower are involved in a computational domain. A three-dimensional in-house program based on a non-orthogonal, non-staggered and unstructured grid system is employed. The standard k–ε turbulence model is used for the turbulent effect. In order to analyze flow and heat/mass transfers in the cooling tower, the continuity, momentum, moisture fraction and enthalpy equations have been considered for both air and water. The density and moisture fraction for air and water are obtained by using the function of thermal properties. The geometrical parameters investigated in this research are the height of obstacle, the distance between the inlet region of cooling tower and the obstacle wall, existence of the air-guide and effects of front and rear obstacles. The results show that the mean moisture fractions in the cases with air-guide are, in general, lower than the cases without air-guide under the same conditions. Also, in the cases without air-guide, the mean moisture fraction increases as the obstacle height increases. In addition, it is observed that for the cases without the air-guide with the increase in the distance between the tower and obstacle, the mean moisture fraction is decreased. However, the cases with air-guide do not follow a specific pattern in the same situations.

Development of a Cell-Centered Godunov-Type Finite Volume Model for Shallow Water Flow Based on Unstructured Mesh

Based on the Godunov-type cell-centered finite volume method, this paper presents a two-dimensional well-balanced shallow water model for simulating flows over arbitrary topography with wetting and drying. The central upwind scheme is used for the computation of mass and momentum fluxes on interface. The novel aspect of the present model is a robust and accurate nonnegative water depth reconstruction method which is implemented in the unstructured mesh to achieve second-order accuracy in space and to track the moving wet/dry fronts of the flow over irregular terrain. By defining the bed elevation and primary flow variables at the cell center in the nonstaggered grid system, all computational cells are either fully wet or dry to avoid the problem of being partially wetted. The developed model is capable of being well balanced and preserving the computed water depth to be nonnegative under a certain CFL restriction, which makes it robust and stable. The present model is validated against three benchmark tests and two laboratory dam-break cases. Finally, the good agreement between the numerical results by the established model and measured data of the Malpasset dam break event on a 1/400 scale physical model demonstrates the capability of the model for the real-life applications.

An immersed boundary method with an approximate projection on nonstaggered grids to solve unsteady fluid flow with a submerged moving rigid object

An immersed boundary method is presented, which uses an approximate projection method on nonstaggered grids for computing flows with submerged and moving boundaries. The incompressible Navier–Stokes equations are discretized using a second-order accurate finite difference technique on a nonstaggered grid system, and the new immersed boundary method is proposed based on an approximate projection method with two pressure correction techniques, the Armfield method and the geometrical grid Reynolds modified method on nonstaggered grids. By using this method, the results obtained from (1) flow past a rigid cylinder in two dimensions with different Reynold numbers and (2) flow around an oscillatory circular cylinder in flow at low Keulegan–Carpenter numbers are in agreement with the reported experimental and numerical data, demonstrating that this convenient method of constraining the interface is a reliable and robust numerical approach for solving unsteady fluid flow with a submerged moving rigid object. This method has the advantage of using significantly less computation time and lower computation sources than the traditional immersed boundary methods.

Calculation of cell face velocity of non-staggered grid system

In this paper, the cell face velocities in the discretization of the continuity equation, the momentum equation, and the scalar equation of a non-staggered grid system are calculated and discussed. Both the momentum interpolation and the linear interpolation are adopted to evaluate the coefficients in the discretized momentum and scalar equations. Their performances are compared. When the linear interpolation is used to calculate the coefficients, the mass residual term in the coefficients must be dropped to maintain the accuracy and convergence rate of the solution.

Numerical Prediction of Two-Phase Reacting Flow Field and Performance in an Aeroengine Combustor

In a three-dimensional arbitrary curvilinear coordinate system, a FORTRAN program was used to research a condition’s influence on an aeroengine’s two-phase reacting flow field and combustion performance. In this program, an algebraic stress model was adopted to simulate turbulence viscosity. A second-order moment eddy–break up turbulence combustion model was used to determine the chemical reaction rate. In a nonstaggered grid system, the governing equations for the gas phase were solved by the PISO algorithm, and simulation of gas-liquid coupling was achieved by using the particle-source-in-cell method. The aeroengine combustor’s performance was estimated by using an empirical-analytical design methodology. It can be observed that performance at maximum conditions is better than that at idling conditions. The obtained agreement between simulation results and rig-test data is reasonable, which indicates the reliability of the numerical program adopted in this paper. This special-purposed program can be used to predict combustor’s flow field and its performance for the optimization design of a modern aeroengine combustor.

On the performance of linear and nonlinear [formula omitted] turbulence models in various jet flow applications

In this paper, a thorough numerical investigation of the performance of several linear and nonlinear k – ε turbulence model variants in various jet flow applications is carried out. Three k – ε based turbulence models are considered, namely the standard k – ε model, the υ 2 – f model, and the nonlinear k – ε model. The selected turbulence models are applied for the prediction of simple as well as complex jet flow applications to underpin knowledge about the accuracy obtained from the two-equation turbulence models. The numerical code developed by the present authors solves the unsteady RANS equations by using the control volume approach on a non-staggered grid system. Three jet flow applications are selected, namely a turbulent free jet, a turbulent jet impinging on a flat plate, and a turbulent wall jet. In order to validate the numerical results obtained and to investigate the performance of the different turbulence models considered, different experimental measurements from the literature are used. The present work is primarily motivated by the desire to provide a rational way for deciding how complex the turbulence model is required to be for a given application and to find out how the accuracy changes with model complexity. Due to the superior predictive performance of modern turbulence models in a wide range of complex industrial and engineering applications, it was believed that a ‘universal’ turbulence model might exist. In general, that is not true. Simple flows can be analysed using standard two-equation models. The present numerical investigation showed that the linear turbulence model could give good results in simple (non-impinging) jet flows. However, in complicated flows, such as impinging jet problems or wall jet flows, a more elaborate level of modeling is required. In such contexts, nonlinear models are appropriate for predicting the turbulent viscosity structure, namely the inhomogeneous near-wall flow region and the anisotropic Reynolds stresses, which is a vital part of turbulent jet flow prediction.

CFD simulations of wave–current-body interactions including greenwater and wet deck slamming

A numerical method for the solution of unsteady Navier–Stokes equations has been employed in conjunction with an interface-preserving level-set method for the simulation of greenwater effect on offshore structures and ships. In this method, the free surface flows are modeled as immiscible air–water two-phase flows and the free surface itself is represented by the zero level-set function. The Navier–Stokes equations for both the water and air flows are formulated in moving curvilinear coordinate system and discretized using the finite-analytic method on a non-staggered multi-block grid system. Large eddy simulation (LES) approach is used with Smagorinsky model to account for the effects of turbulence induced by violent free surface motions. A chimera domain decomposition approach is implemented using overlapping, embedding, or matching grids to facilitate the simulation of complex flow around practical configurations. The overset grid system also greatly simplified the simulation of arbitrary translational and rotational motions among various computational blocks. Calculations were performed first for dam-breaking flow and free jet problems involving violent free surface motions. The level-set Navier–Stokes method was then employed for the simulation of slamming of a hemisphere, greenwater on offshore structure and ships, and wet deck slamming of an X-Craft in pitch and heave motions. The numerical results clearly demonstrated the capability of the level-set method to deal with violent free surface flows involving breaking waves, water droplets, trapped air bubbles, and wave–current-body interactions.

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier–Stokes equations

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier–Stokes equations. This proposed method solves a smaller system of nonlinear Navier–Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier–Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection–diffusion–reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier–Stokes solutions.

Numerical simulation of a fountain flow on nonstaggered Cartesian grid system

A liquid–air fountain flow due to the downward motion of a rectangular sleeve over a stationary piston is studied in the paper. Two-dimensional incompressible laminar flows are assumed to prevail in both air and liquid regions. A single set of governing equations over the entire physical domain including the liquid, the air, and the liquid–air interface (free surface) is solved with the extended weighting function scheme and the NAPPLE (nonstaggered APPLE) algorithm on a fixed nonstaggered Cartesian grid system. To ensure the required dynamic contact angle, the liquid meniscus near the sleeve wall is corrected by solving the force balance equation with the geometry method. This is equivalent to introducing a slip condition at the contact line, and thus successfully removes the stress singularity. Steady state solution of the velocity and the pressure as well as the shape of the free surface is obtained. The numerical result evidences the existence of a toroidal-like motion on the free surface postulated by Dussan [E.B. Dussan V., Immiscible liquid displacement in a capillary tube: the moving contact line, AIChE J. 23 (1977) 131–133], although it is quite weak and thin. The resulting free surface profile agrees with the existing experimental observation excellently. Influence of the piston on the flow is discussed.

Large-eddy simulation of two-phase spray combustion for gas turbine combustors

In three-dimensional arbitrary curvilinear coordinates, an Eulerian–Lagrangian formulation is applied to large-eddy simulation (LES) of instantaneous gas–liquid two-phase turbulent combustion flows in gas turbine combustors. Three dimensional block-structured grids are generated by zone method and solving a system of elliptic partial differential equations. The k-equation sub-grid scale model is used to simulate the sub-grid eddy viscosity and the EBU combustion sub-grid scale model is employed to predict the chemical reaction rate. The gas-phase governing equations are solved with SIMPLE algorithm and hybrid scheme in non-staggered grid system. A stochastic separated flow formulation is used to track the droplet trajectories velocities, size and temperature history by Lagrangian equations of motion and thermal balance. Multi-zone coupling method is employed to transport data between interfaces. The influences of two different primary hole positions and three different fuel–air ratios on turbulent two-phase reacting flows are calculated. Predictions are in reasonable agreement with the measured velocity using PIV system and temperature, species concentration measurements at the exit. It is shown that the present approach may be used to study spray combustion flow fields for guiding the design of advanced gas turbine combustors.

A quasi-implicit time advancing scheme for unsteady incompressible flow. Part I: Validation

A quasi-implicit fractional-step method is presented for computing unsteady incompressible flow on unstructured grids. A non-staggered grid system is employed rather than a staggered grid system because of the simplicity and ease of extension to three-dimensional analysis. In this study, the momentum interpolation method, developed by Rhie and Chow [C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532] and further extended by Zang et al. [Y. Zang, R.L. Street, J.R. Koseff, A non-staggered grid, fractional-step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates, J. Comput. Phys. 114 (1994) 18–33], is applied to problems on unstructured grids to resolve the pressure oscillation problem occurring in a non-staggered grid system. An implicit time advancing scheme is used in order to remove the time step restriction and to reduce the required CPU time for problems with complex geometries. The nonlinear equations resulting from this fully implicit scheme are linearized without deteriorating the overall time accuracy. The system matrices are solved using the CG family method, known with P-BiCGSTAB, for momentum equation and P-CG for pressure Poisson equation. The present numerical method is applied to solve four benchmark problems and the results show good agreement with previous experimental and numerical results.

Numerical investigation of heat transfer and fluid flow characteristics inside a wavy channel

The periodically fully developed laminar heat transfer and fluid flow characteristics inside a two-dimensional wavy channel in a compact heat exchanger have been numerically investigated. Calculations were performed for Prandtl number 0.7, and Reynolds number ranging from 100 to 1,100 on non-orthogonal non-staggered grid systems, based on SIMPLER algorithm in the curvilinear body-fitted coordinates. Effects of wavy heights, lengths, wavy pitches and channel widths on fluid flow and heat transfer were studied. The results show that overall Nusselt numbers and friction factors increase with the increase of Reynolds numbers. According to the local Nusselt number distribution along channel wall, the heat transfer may be greatly enhanced due to the wavy characteristics. In the geometries parameters considered, friction factors and overall Nusselt number always increase with the increase of wavy heights or channel widths, and with the decrease of wavy lengths or wavy pitches. Especially the overall Nusselt number significantly increase with the increase of wavy heights or channel widths, where the flow may become into transition regime with a penalty of strongly increasing in pressure drop.

An Improved Numerical Scheme for the SIMPLER Method on NonOrthogonal Curvilinear Coordinates: SIMPLERM

In this article, an improved numerical algorithm named SIMPLERM is proposed for incompressible fluid flow computations on the nonstaggered and nonorthogonal curvilinear grid system. In the proposed algorithm, the contravariant velocities are chosen as the cell face velocities and the Cartesian components as the primary variables. The velocity under-relaxation factor is incorporated into the momentum interpolation, and special treatment is adopted to avoid the underrelaxation factor dependence of the velocity solution. In addition, a 1 − δ pressure difference is introduced into the interfacial contravariant velocity determination. Compared with the existing implementation methods of the SIMPLE family on non-staggered and nonorthogonal grids, the SIMPLERM algorithm can guarantee the coupling between velocity and pressure, underrelaxation independence of the solution, and satisfaction of the conservation law, while still possessing sufficient robustness.

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.

Numerical simulation of turbulent flow field and heat transfer in a two-dimensional channel with periodic slit ribs

Spatially periodic turbulent fluid flow and heat transfer in a channel with slit rectangular ribs mounted on one wall have been numerically investigated. A Reynolds stress model with wall function and a modified wall-related pressure-strain model for reducing the overestimation of turbulent kinetic energy at impingement/reattachment region was employed in the two-dimensional simulation. QUICK discretization scheme and nonstaggered grid system were used. The parameters fixed were rib height-to-duct height ratio of 0.17, rib width-to-height ratio of 0.76, rib pitch-to-height ratio of 10, and Reynolds number based on channel hydraulic diameter and bulk mean velocity of 2×10 4, while the main parameter varied was the rib open-area ratio with values of β=0%, 10%, 22%, 32%, and 44%. Two critical ranges of β and three characteristic flow regimes are identified, which is similar to the experimental results reported by previous researchers for the hole-type perforated ribs except a smaller critical β for impermeability. The rationale for the difference in the critical β between the slit-type and hole-type perforated ribs is addressed in terms of effective pore diameters. From the variations of the friction loss, convective mean velocity, turbulent kinetic energy, and negative pressure behind the perforated ribs with β, that β=44% attains the best thermal performance Nu p / Nu s * under the constant pumping power condition is unraveled. A comparison of Nu p / Nu s * distributions between vertical-slit ribs measured by other research groups and horizontal-slit ribs computed presently is also made.

High-Pressure Steam Flow in Turbine Bypass Valve System Part 1: Valve Flow

This paper presents a study of steam e ow behavior through a high-pressure turbine bypass valve when it suffers a high-pressure reduction in an electric-power-plant cogeneratorsystem. Efforts have been mainly directed at investigating the process of steam e ow and property variations in the aforementioned bypass valve as well as at obtaining correlations between the e ow rate and the valve opening ratio. Modeling of the high-pressure turbulent steam e ow was performed on a three-dimensional nonstaggered grid system by employing the e nite volume differencing method and by solving the three-dimensional, turbulent, compressible Navier ‐Stokes and energy equations. Through this research numerous data have been acquired and analyzed. These efforts enable us to obtain a correlation data set for the valve-opening vs the e ow rate coefe cient of the valve. One of the signie cant accomplishments isto use the model presented here to further improvethedesign of a high-pressuresteam turbine bypass e ow valve to reduce a high-velocity spot in the valve e ow pass so that noise can be suppressed.