Non-staggered Grid Research Articles

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.

Numerical study of turbulent wake flow behind a three-dimensional steep hill

A numerical investigation on the turbulent flows over a three-dimensional steep hill isrnpresented. The numerical model developed for the present work is based on the finite volume method andrnthe SIMPLE algorithm with a non-staggered grid system. Standard k-e model and Shih's non-linear model are tested for the validation of the prediction accuracy in the 3D separated flow. Comparisons of the mean velocity and turbulence profiles between the numerical predictions and the measurements show good agreement. The Shih's non-linear model is found to predict mean flow and turbulence better than the Standard k-e. Flow patterns have also been examined to explain the difference in the cavity zone between 2D and 3D hills.

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×106 to 1.87×108. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10∘, the average Nusselt number can be correlated into an elegant correlation.

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart–Thomas space and the pressure field in the rotated-Q1Q1nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated-Q1Q1nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

AbstractA recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.

Numerical simulation of fluid flow and heat transfer in a passage with moving boundary

In this paper, a method is presented in detail that can be used to solve the fluid flow and heat transfer in domains with moving boundaries. The primitive variables formulation is adopted and a non-staggered grid, with Cartesian velocity components used as the primary unkowns in the momentum equations, is utilised. Discretisation is carried out using a control-volume method, the simplified QUICK scheme combined with a deferred correction approach is adopted for the convective fluxes and implicit time stepping is used for temporal differencing. The well-known SIMPLE algorithm is employed for handling the velocity?pressure coupling. The computational method is applied for the prediction of fluid flow and heat transfer in a channel with a boundary moving in a prescribed manner. Results show that both the amplitude and Strouhal number have great influences on the characteristics of fluid flow and heat transfer, and in the range studied, the heat transfer rate increases monotonously with the amplitude, whereas the Strouhal number only has a small effect on heat transfer.

Turbulent Flow and Endwall Heat Transfer Analysis in a 90∘ Turning Duct and Comparisons with Measured Data: Part I: Influence of Reynolds Number and Streamline Curvature on Viscous Flow Development

This paper deals with the experimental and computational analysis of flow and heat transfer in a turning duct simulating the overall three-dimensional flow and heat transfer characteristics of gas turbine passages and internal cooling channels. The numerical results obtained for three different Reynolds numbers (790, 40,000, and 342,190) are compared to measured flow characteristics. Extensive measurements of the mean flow structure using a sub-miniature five-hole-probe and a hot-wire provide a quantitative assessment of the computational model used in the analysis. An in-house developed three-dimensional viscous flow solver is the main computational tool of the present study. A well known pressure correction method (SIMPLE) is used for the solution of 3-D incompressible, steady Navier - Stokes equations in a generalized coordinate system. The k - ∈ turbulence model of Launder and Spalding including a curvature correction scheme is utilized to describe the turbulent flow field. A non-staggered grid system is used in an upwind scheme with additional numerical dissipation terms. It is clearly shown that reducing the artificial dissipation terms in a central differencing scheme reduced the numerical dissipation error. Part II of this paper deals with the convective heat transfer aspects of the specific flow near the endwall surface where three-dimensional viscous flow structures are dominant. The measured fluid mechanics data presented in this paper also provide a reliable data set that can be used in the future validation of new computational methods.

CHECKERBOARD PRESSURE PREDICTIONS DUE TO THE UNDERRELAXATION FACTOR AND TIME STEP SIZE FOR A NONSTAGGERED GRID WITH MOMENTUM INTERPOLATION METHOD

(2002). CHECKERBOARD PRESSURE PREDICTIONS DUE TO THE UNDERRELAXATION FACTOR AND TIME STEP SIZE FOR A NONSTAGGERED GRID WITH MOMENTUM INTERPOLATION METHOD. Numerical Heat Transfer, Part B: Fundamentals: Vol. 41, No. 1, pp. 85-94.

Lagrangian numerical simulation of particulate flows

The Lagrangian numerical simulation (LNS) scheme presented in this paper is motivated by the multi-phase particle-in-cell (MP-PIC). In this numerical scheme we solve the fluid phase continuity and momentum equations on an Eulerian grid. The particle motion is governed by Newton's law thus following the Lagrangian approach. Momentum exchange from the particle-to-fluid is modeled in the fluid phase momentum equation. Forces acting on the particle include drag from the fluid, body force and force due to interparticle stress. There is a freedom to use different models for these forces and to introduce other forces. The effect of viscous stresses are included in the fluid phase equations. The volume fraction of the particles appear in the fluid phase continuity and momentum equations. A finite volume method is used to solve for the fluid phase equations on an Eulerian grid. Particle positions are updated using the Runge–Kutta scheme. This numerical scheme can handle a range of particle loadings and particle types. The LNS scheme is implemented using an efficient three-dimensional time-dependent finite volume algorithm. We use a Chorin-type pressure-correction based fractional-step scheme on a non-staggered cartesian grid. In this paper, we consider only incompressible Newtonian suspending fluid. However, the average velocity field of the fluid phase is not divergence-free because its effective density is not constant. Our pressure-correction based fractional-step scheme accounts for varying properties in the fluid phase equations. This method can also account for suspending fluids with non-constant properties. The numerical scheme is verified by comparing results with test cases and experiments.

Film Cooling Predictions of Simple and Compound Angle Injection from One and Two Staggered Rows

Typical film-cooling configurations of flat plates from one and two staggered rows of simple and compound angle injection holes are investigated using a three-dimensional finite volume method and a multiblock technique, which reduces significantly the core memory needed for the computations and gives more freedom in the generation of the grids. The computational method uses arbitrary curvilinear, body-fitted, multiblock, structured, non-staggered grids. The turbulence is approximated by a standard κ–∊ model with wall functions. The accuracy of the code is improved by using a second-order-bounded scheme for convection terms for all equations including the κ and ∊ turbulence model equations. The influence of number of rows and injection angles as well as the blowing ratio on the film cooling protection has been investigated and compared with experimental data. Comparison between predicted and experimental results indicates that the trends of the streamwise mean velocity and thermal fields are well predicted in most cases. However, the spanwise-averaged film cooling effectiveness is globally underpredicted by the code, probably because of the limited capability of the turbulence model used. Good agreement is obtained between the predictions and measurements made downstream of two rows of compound angle injection.

Natural convection heat transfer by heated partitions within enclosure

In this study, natural convection heat transfer and fluid flow of two heated partitions. within an enclosure have been analysed numerically. The right side wall and the bottom wall of the enclosure were insulated perfectly while the left side wall and top wall were maintained at the same uniform temperature. The partitions were placed on the bottom of the enclosure and their temperatures were kept higher than the non-isolated walls. The effects of position and heights of the partitions on heat transfer and flow field have been investigated. Computations for Rayleigh number in the range of 10 4 and 10 6 have been conducted. Using the control volume approach, finite difference equations are obtained with non-staggered grid arrangement, a computer program based on the SIMPLEM algorithm was developed. The finite difference equations were solved iteratively with a line-by-line Thomas algorithm.

Laminar flow and heat transfer through square duct with twisted tape insert

Numerically predicted characteristics of laminar flow and heat transfer through square duct with twisted tape insert are presented in this paper. The transport equations are solved on a non-staggered non-orthogonal grid using the curvilinear version of the “Complete Pressure Correction” algorithm. Both the flow and heat transfer are in a state of periodic development in the axial direction. The heat transfer characteristics are predicted under axially and peripherally constant wall heat flux conditions. Correlations for friction factor and Nusselt number are developed from the predicted data. The agreement between the correlation and experimental data for friction factor is found to be excellent. Relative thermo-hydraulic performance of square and circular ducts, both fitted with twisted tapes of same twist ratio is also presented in this paper. The study shows significant improvements can be achieved with the square duct, particularly at higher Prandtl numbers and lower twist ratios. Improvement is observed even for air, over certain ranges of Reynolds number.

Simulation of Turbomachinery Flows Using Second Moment Closure on Quasi-Staggered Grid

Introduction W ITH the rapid development of computer technology, it has becomefeasibleto calculateindustrial owswith the secondmoment closure model (RSM). However, few researchers applied RSM to simulate full three-dimensionalindustrial ows. One main difŽ culty is the numericalinstabilitywhenRSM is used.WhenRSM is applied to simulate three-dimensional ows on the nonstaggered grid, the decoupling difŽ culty between the Reynolds stresses and the mean velocities is frequently encountered.Because of this, the solution is often quite unstable. The staggered grid method is the practical way to overcome the checkerboard oscillation of pressure. When it is adopted in RSM, however, seven sets of grids are required for a full three-dimensional  ow. Such simulations have been reported by Huang and Leschziner2 and Lin and Lu.3 Because the Reynolds-stress components are placed at staggered locations, this way is still not so stable. For example, Huang and Leschziner2 had to take some special measures to guarantee the convergenceof solution.Some authors4;5 addedanapparentviscosityin themomentum equations or deŽ ned a new set of pseudoviscositycoefŽ cients to stabilize the solution and got some progresses. However, these ways are quite complex when they are applied to a boundary-Ž tted coordinates system. In thisNote a novelgrid arrangementsuitablefor the computation of turbulent  ows usingRSM is developed.In this grid arrangement gridpointsforpressure,density,temperature,andReynoldsstresses, etc. are located at the centers of the control volumes, while grid points for mean velocities are placed at the corner points of the control volumes. Therefore, only two sets of grids are needed for a full three-dimensional ow. This grid method is called the quasistaggered grid method herein. For a two-dimensional case the grid arrangement is shown in Fig. 1a. Unit 1 is the inner control volume for pressure etc., and unit 2 is the inner control volume for mean velocities. Units 3 and 4 are the corresponding control volumes adjacent to a boundary. As the locations of mean velocities and Reynolds-stresscomponents are offset, the decouplingbetween the mean velocities and the Reynolds stresses can be avoided.Because only two sets of grids are used, the programming becomes simpler. To enhance the coupling between pressure and the mean velocities, a fourth-order pressure gradient formula at the cell face is used. For example, for the gradient in the x direction the formula can be written as

Numerical Analysis of Thermal Stratification in a Circular Pipe

This paper presents an effective numerical method for predicting the stratified flow in a horizontal circular pipe. The method employs a body-fitted, nonorthogonal grid system to accommodate the pipe wall of circular geometry and the interface of the two fluids at different temperatures, of which the level is variable. The transient behaviors of fluid flow and temperature distribution in the piping are simulated using the finite volume approach. The convection term is approximated by a higher-order bounded scheme named HLPA, which is known as a high-resolution and bounded discretization scheme. The cell-centered, nonstaggered grid arrangement is adopted and the resulting checkerboard pressure oscillation is prevented by the application of modified momentum interpolation scheme. The SIMPLE algorithm is employed for the pressure and velocity coupling. The new way of treating the unsteady conjugate heat transfer problem is presented. The present method has been applied to the stratified flow in the pressurizer surge line of the nuclear reactor, and the results have been discussed. In addition, this study has investigated the effects of level of the interface between the two stratified fluids on the transient evolution of temperature distributions in the piping wall.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

The solution of viscous incompressible jet flows using non-staggered boundary fitted co-ordinate methods

A new approach for the solution of the steady incompressible Navier–Stokes equations in a domain bounded in part by a free surface is presented. The procedure is based on the finite difference technique, with the non-staggered grid fractional step method used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. In order to design an effective free solution scheme, we distinguish between flows dominated by surface tension and those dominated by inertia and viscosity. When the surface tension effect is insignificant we used the kinematic condition to update the surface; whereas, in the opposite case, we used the normal stress condition to obtain the free surface boundary. Results obtained with the improved boundary conditions for a plane Newtonian jet are found to compare well with the available two-dimensional numerical solutions for Reynolds numbers, up to Re=100, and Capillary numbers in the range of 0≤Ca<1000. Copyright © 2001 John Wiley & Sons, Ltd.

Fractional step methods for the Navier-Stokes equations on non-staggered grids

The Navier-Stokes equations are solved on a non-staggered grid using a semi-implicit fractional step method in both iterative and non-iterative form. It is shown that the iterative scheme in standard form is second-order accurate in timei, but is very slow to integrate as a result of the non-elliptic pressure coupling at the grid scale. Inclusion of additional terms into the pressure correction equation for the iterative scheme ensures an elliptic pressure coupling at the grid scale, but introduces a first order in time error into the scheme, leading to a reduction in solution accuracy. The non-iterative scheme is shown to be second order accurate in time in standard form and to be considerably more efficient than the iterative scheme.

Computational modeling of pulverized coal combustion processes in tangentially fired furnaces

In this work, an Eulerian/Lagrangian approach has been employed to investigate numerically flow characteristics, heat transfer and combustion processes in a tangentially fired furnace. A new method of cell face velocity interpolation for non-staggered grid system is employed to avoid pressure oscillation. Grid-independence tests have been conducted. To avoid pseudo-diffusion that is significant in modeling tangentially fired furnaces, some attempts have been made at improving the finite-difference scheme. The standard k-ε model performs well in predicting flows without swirling or without sharp change within the calculated region. But for tangentially fired boiler furnaces, where swirling flow is very marked, we must resort to other more valid, more efficient turbulent models to gain accuracy. In this paper, we try to use RNG k-ε model as an alternative to the standard k-ε model. Comparisons have been made between standard k-ε and RNG k-ε models. Some new developments on turbulent diffusion of particles are taken into account for improving computational accuracy, and probability error is also discussed. Finally, temperature deviation is studied numerically so as to gain deeper insight into tangentially fired furnaces.

Numerical modeling of helical flow of viscoplastic fluids in eccentric annuli

AbstractHelical flow of viscoplastic Herschel‐Bulkley fluids in concentric and eccentric annuli with rotating inner cylinder was investigated numerically. Similar flows occur in drilling operations of oil and gas wells. A finite volume algorithm with a nonstaggered grid system and a nonorthogonal curvilinear coordinate system to handle irregular geometry of an eccentric annulus was used to analyze the problem. Papanastasiou's modification of the Herschel‐Bulkley (Yield‐Power‐Law) constitutive equation was used to model the shear rate‐dependent viscosity of a viscoplastic fluid. For a fixed axial pressure gradient, the axial flow rate increased with increasing rotational speed of the inner cylinder. The discharge, as well as torque required to rotate the inner pipe, increased with increasing eccentricity for a fixed axial pressure gradient and inner cylinder rotational speed. Discharge also increased with increasing axial pressure gradient at a fixed eccentricity and rotational speed of the inner pipe. The flow field in an eccentric annulus is complex because vigorous secondary flow is produced in addition to the primary axial helical flow. Blockage at the narrow part of the eccentric annulus, when present, intensifies this secondary flow, with the discharge decreasing initially, then increasing, and decreasing again with increasing height of the blockage.