Non-staggered Grid Research Articles

Measurement of pressure distribution from PIV experiments

In this study, a non-staggered grid SIMPLER pressure solution algorithm, which is able to produce correct pressure distribution directly if correct velocities are given, is proposed to solve the pressure distribution for PIV experiments. The cell face pseudo velocity required in the pressure equation is approximated by a simple linear average of the adjacent nodal pseudo velocities so that the velocity and pressure are collocated without causing the checkerboard pressure distribution problem. In addition, the proposed pressure solution algorithm has the features that upwind effects of the convective terms are considered, boundary conditions are not required, and the pressure distribution obtained can be used to correct the velocity field so that the continuity equation is satisfied. These features make the present algorithm a superior method to calculate the pressure distribution for PIV experiments. The pressure field solved is realistic and accurate. The proposed pressure equation solver is first calibrated with a two-dimensional cavity flow. It is found that the results are almost identical to the exact solution of the test flow. The algorithm is then applied to analyze a uniform flow past two side-by-side circular cylinders in a soap film channel. With the velocity and pressure distributions successfully measured, the structures of the complex shedding flow patterns are clearly manifested.

EFFECT OF OBSTACLE IN RIVER ON THE HEAT DISSIPATION OF HOT WATER INJECTION

Numerical analysis of the cooling process of hot water discharge from a steam power plant into a river has been carried out in the present study. A mathematical model describing the flow field and temperature distribution induced in the river as a result of the cooling process is made based upon the assumptions of steady state, two-dimensional, turbulent flow, in the horizontal plane. The governing equations are the continuity, the momentum, the (К-ε) turbulence model and the energy equation. A proper initial and boundary condition are specified to be used in the solution of the governing equations. A numerical solution of the governing equations is made by using the control volume approach, with non-staggered grid technique and modified SIMPLE algorithm. The numerical solution is capable of calculating the velocity and temperature distributions of the calculation domain. The numerical results show that the developed algorithm is capable of calculating the flow field, properly and accurately. Results are obtained for ten cases of configuration, constant aspect ratio and weather conditions for Baghdad. The results show that the injection velocity effect on the temperature distribution and stream line , the exist of obstacle and its distance from the injection zone but the increase in injection temperature cause a small effect on temperature distribution and stream line.

SUB-NANOSECOND ELECTROMAGNETIC-MICROMAGNETIC DYNAMIC SIMULATIONS USING THE FINITE-DIFFERENCE TIME-DOMAIN METHOD

This paper presents an efficient and simple approach of implementing the Landau-Lifshitz-Gilbert (LLG) equation of mag- netisation motion within the Finite-difference Time-domain (FDTD) method. This combined electromagnetic-micromagnetic simulation technique is particularly important for modeling electromagnetic in- teraction with lossy magnetic material in the presence of current and magnetic sources, particularly at very high frequencies. The efficient implementation involves simple two-point spatial interpolations that are applicable to two and three-dimensional FDTD grids, and uses a stable iterative algorithm for the time integration of the LLG equa- tion. A ferromagnetic resonance numerical experiment on a rectangu- lar Permalloy prism excited through its cross-section by a non-uniform pulse field from a transmission line was carried out for the purpose of verifying the combined FDTD-LLG computations. The numerical results were in good agreement with linearised analytical solutions of the LLG equation for uniform and non-uniform precession modes. This paper also presents a brief investigation on the use of non-staggered FDTD grid schemes to model magnetic material using the LLG equa- tion, and indicates that the classical FDTD staggered scheme offers simplicity in implementation and more accuracy for modeling wave in- teraction with lossy magnetic material than the non-staggered schemes based on Maxwell's equations formulation.

Numerical simulations of ice accumulation under ice cover along a river bend

In this paper, the k-έ two-equation turbulence model has been used to simulate ice accumulation under ice cover along a river bend. A 2D depth-averaged numerical model has been developed in a nonorthogonal coordinate system with nonstaggered curvilinear grids. In this model, the contravariant velocity has been treated as an independent variable. To avoid the pressure oscillation in the nonstaggered grids, the momentum interpolation has been introduced to interpolate variables at the interface. The discretization equations have been solved by using pressure correction algorithms. An equation has been developed for describing the deformation of ice jam bottom. The thickness distribution of ice accumulation (ice jam) along the bend has been simulated. The developed model has been applied to the experimental studies under different conditions carried out at the Hefei University of Technology. Results indicate that all simulated thickness of ice accumulation agrees reasonably well with the measured thickness of ice accumulation in laboratory.

Analysis of semi-staggered finite-difference method with application to Bingham flows

In this paper, a finite-difference scheme for incompressible flow problems is treated. The scheme uses non-staggered grid for velocity approximation. A special stabilization is introduced to ensure the well-posedness and optimal approximation properties of the scheme. The stability estimate is proved in the form of a mesh-independent bound for the norm of discrete operator inverse. The finite-difference method is particularly suitable for problems in which velocity vector is involved in additional quantities that enter the system of flow equations as, for example, in the Bingham problem. We describe this application in the paper in some detail. Results of numerical experiments are included that confirm the applicability and optimality of the method.

Interaction Effects During Combustion of Linear Arrays of Gaseous Fuel Pockets

The numerical solution for the combustion of an infinite linear array of dense fuel gas pockets in a quiescent ambient is discussed in the present work. Fuel mass, flame, and gas pocket shape behaviors are analyzed in order to quantify the interference effects. The combustion process is considered isobaric. The model is based on mass, momentum, energy, and species conservation equations and considers the simple chemical reaction mechanism and ideal gas behavior. Coupling potentials are used in the solution of the energy and species conservations equations. The thermophysical properties, except the density, are assumed constant. The finite-volume method is used for the numerical solution, using a generalized system of coordinates and a nonstaggered grid. The SIMPLEC algorithm is applied to solve the modified pressure-velocity coupling. The flame and gas pocket morphological evolutions as well as the transient fuel consumption and the flame position behaviors are presented. The effects of the initial fuel and oxidant temperatures, of the interpocket distance, of the stoichiometric condition, of the heat of reaction, and of the mixture thermophysical properties on the combustion process are considered. Results show that the interactive effects on gas pocket linear array combustion are relevant only for small interpocket distances. Results also show flame shape temporal evolutions that indicate an oscillatory behavior dependent on the gas pocket burning conditions.

Development of a dispersively accurate conservative level set scheme for capturing interface in two-phase flows

A two-step interface capturing scheme, implemented within the framework of conservative level set method, is developed in this study to simulate the gas/water two-phase fluid flow. In addition to solving the pure advection equation, which is used to advect the level set function for tracking interface, both nonlinear and stabilized features are taken into account for the level set function so that a sharply varying interface can be stably predicted. To preserve the conservative property, the mapping given by ξ ̲ = x ̲ - u ̲ t is performed between two coordinates x ̲ and ξ ̲ for the advection–diffusion equation in a flow field with velocity u . To capture the interface, the flux term capable of compressing the level set contours is also adopted in the construction of linear inviscid Burgers’ equation, which is indispensable in the level set method. To resolve the physically sharp interface without incurring contact discontinuity oscillations, a damping term which is nonlinear in terms of the level set function and can render an adequate artificial diffusion to stabilize the contact surface is added into the reinitialization step in the modified level set method. For accurately predicting the level set function, the advection scheme for solving the linear inviscid Burgers’ equation in the advection step of the modified level set method is developed to accommodate the true dispersion relation. The solution computed from the resulting two-dimensional dispersion-relation-preserving advection scheme can minimize the phase error. Less artificial damping is needed to damp the oscillations in the vicinity of contact surface and the interface can be less numerically smeared. For the sake of programming simplicity, the incompressible two-phase flow will be discretized in non-staggered grids without incurring checkerboard oscillations by the developed explicit compact scheme for the approximation of pressure gradient terms. For the verification of the proposed two-dimensional dispersion-relation-preserving scheme and the non-staggered incompressible flow solver, three benchmark problems have been chosen in this study. The proposed conservative level set method for capturing the interface in incompressible fluid flows is also verified by solving the dam-break, bubble rising in water, droplet falling in water and Rayleigh–Taylor instability problems. Good agreements with the referenced solutions are demonstrated for all the four investigated problems.

A semi-implicit numerical scheme for transient two-phase flows on unstructured grids

A component-scale two-phase analysis code, CUPID, has been developed for a realistic simulation of transient two-phase flows in light water nuclear reactor components. In the CUPID code, a two-fluid three-field model is adopted and the governing equations are solved on an unstructured grid to make CUPID very useful for flow analysis in complicated geometries. For the numerical solution scheme, the semi-implicit method of the RELAP5 code, which has been proved to be very stable and accurate for most practical applications, was used with some modifications for an application to an unstructured non-staggered grid. This paper presents the modified semi-implicit numerical method for an unstructured grid and the preliminary results of the calculations.

Co-Located Variables Approach Using Implicit Runge-Kutta Methods for Unsteady Incompressible Flow Simulation

A numerical method for transient incompressible viscous flow simulation is presented with a nonstaggered grid (co-located variables) approach. A special class of implicit Runge-Kutta (RK) methods is used for time discretization in conjunction with finite-volume-based semi-implicit pressure linked equations (SIMPLE) algorithm. Owing to the use of implicit RK methods, the proposed method can be used to achieve an arbitrarily high order of accuracy in time-discretization which is otherwise limited to second order in the majority of the currently available simulation techniques. The nonstaggered grid method was tested by solving for velocity field in a lid-driven square cavity. In the test case calculations, the Power Law scheme of Patankar [9] was used in spatial discretization, and time discretization was performed using a second-order implicit Runge-Kutta method. The results from the current method were compared with the results obtained from the staggered grid method of Ijaz and Anand [10-12]. The current method produced results that were nearly equivalent to the ones obtained by using the staggered grid approach. Moreover, the current method was found to have better convergence characteristics compared to the staggered grid method when applied to the lid-driven square cavity problem.

UNSTEADY RESPONSE OF BLOOD FLOW THROUGH A COUPLE OF IRREGULAR ARTERIAL CONSTRICTIONS TO BODY ACCELERATION

A two-dimensional (2D) nonlinear mathematical model to study the response of the pulsatile flow of blood through a couple of irregular stenoses influenced by externally imposed periodic body acceleration is developed. The model is 2D and axisymmetric with an outline of the stenosis obtained from the three-dimensional (3D) casting of a mildly stenosed artery. The combined influence of an asymmetric shape and surface irregularities of the constrictions is explored in a computational study of blood flow through arterial stenoses with 48% areal occlusion. The arterial wall is treated as an elastic (moving wall) cylindrical tube having a couple of stenoses in its lumen, while the streaming blood is considered to be Newtonian. Solutions of the time-dependent nonlinear Navier–Stokes equations in the cylindrical coordinate system are obtained using a finite difference method based on the nonuniform and nonstaggered grids. The finite difference approximation helps to estimate the effects of body acceleration on the doubly constricted flow phenomena through several graphical representations quantitatively in order to validate the applicability of the present, improved mathematical model.

Thermal stratification studies in a side heated water pool for advanced heavy water reactor applications

Strong heat source at the isolation condenser wall of an Advanced Heavy Water Reactor, results in natural convection in gravity driven water pool, which leads to a thermally stratified pool. Governing equations simulating fluid flow and heat distribution are solved numerically by a general purpose Computational Fluid Dynamics solver developed at Indian Institute of Technology, Kanpur. Incompressible finite volume method with non-staggered grid arrangement is used in this exercise. This algorithm is fully implicit and semi-coupled. Turbulent natural convection in a boundary layer for high Rayleigh numbers is analyzed by the Lam–Bremhorst k − e turbulence model. Analysis of unsteady laminar natural convection in a side-heated water cavity is also done for different values of Rayleigh number. Results show a warm fluid layer floating on the top of gradually colder layer (along the vertical direction) that indicates the presence of thermal stratification phenomenon. This fact necessitates additional safety features in such a system so that the detrimental effect such as stratification is minimized.

Consistent Implementation and Comparison of FOU, CD, SOU, and QUICK Convection Schemes on Square, Skew, Trapezoidal, and Triangular Lid-Driven Cavity Flow

This article presents a conservative and consistent implementation of higher-order convection schemes such as the second-order upwind (SOU) and the quadratic upwind interpolation for convective kinematics (QUICK) schemes, and systematic comparison of four convection schemes with the available benchmark solution for various two-dimensional, steady-state, lid-driven cavity flow problems. All these schemes are formulated on Cartesian, nonorthogonal and nonstaggered grid sizes varying from 20 × 20 to 80 × 80 at various Reynolds numbers. The QUICK scheme converges to a grid-independent solution with substantially fewer grid points, requires least computational time for a given level of accuracy, and its accuracy remains almost constant with increasing Reynolds number.

UNSTEADY HYDRODYNAMIC AND THERMAL BOUNDARY LAYERS OVER A FLAT PLATE WITH UNIFORM INJECTION OR SUCTION

A computational algorithm for calculating the unsteady compressible and turbulent hydrodynamic and thermal boundary layers over a porous flat plate with uniform suction or injection is developed in the present work. The mathematical modeling involves the derivation of the governing partial differential equations of the problem. These are the continuity, the momentum, the ( κ −ε ) turbulence model and the energy equations. Besides, the perfect gas law and the Sutherland' s law of molecular viscosity are also used. A proper initial and boundary conditions are specified to be used in the solution of the governing equations. A numerical solution of the governing equations is made by using the control volume approach, with non-staggered grid technique and modified SIMPLE algorithm. The numerical solution is capable of calculating the velocity and temperature distributions of the calculation domain, the kinetic energy and dissipation of turbulence, the local and average skin-friction and heat transfer coefficients and Nusselt number, and the hydrodynamic and thermal boundary layers thicknesses. All these parameters are calculated at the transient and steady states. The numerical results show that the developed algorithm is capable of calculating the flow field, properly and accurately. The results show that injection causess light decrease in the temperature inside the thermal boundary layer, a decrease in skin – friction coefficient, a slight increase in the Nusselt number, a decrease in hydrodynamic boundary layer thickness, an increase in thermal boundary layer thickness and an increase in the time required to reach the steady state. Suction almost causes reverse effects.

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.

A semi-implicit numerical scheme for a transient two-fluid three-field model on an unstructured grid

A three-dimensional (3D) unstructured hydrodynamic solver for transient two-phase flows has been developed for a 3D component of a nuclear system code and a component-scale analysis tool. A two-fluid three-field model is used for the two-phase flows. The three fields represent a continuous liquid, an entrained liquid, and a vapor field. An unstructured grid is adopted for realistic simulations of the flows in a complicated geometry. The semi-implicit ICE (Implicit Continuous-fluid Eulerian) numerical scheme has been adapted for an unstructured non-staggered grid. This paper presents the numerical method and the preliminary results of the calculations. The results show that the modified ICE scheme is robust and predicts the phase changes and the flow transitions due to a boiling and a flashing very well.

909 しゅう動面に凹凸を有する非接触シールの力学的特性 : 集中格子を用いた数値解析(OS11-2 機械等の動的解析,オーガナイズドセッション:11 機械と車輌のダイナミクス)

909 しゅう動面に凹凸を有する非接触シールの力学的特性 : 集中格子を用いた数値解析(OS11-2 機械等の動的解析,オーガナイズドセッション:11 機械と車輌のダイナミクス)

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.

Comparison of trust-region-based and evolutionary methods for optimization of flow geometries†

Optimization has become increasingly important in computer-aided engineering, and applications of engineering optimization to design, control, operations, and planning already exist. Since in the context of numerical flow simulation no information on the gradient information of the objective function is available, or is very difficult to obtain, in such optimization cases it is advantageous to use an optimization technique which does not directly depend on the derivative. The aim of this study is to investigate two optimization techniques, the trust-region-based method and the evolutionary algorithm technique, and compare them quantitatively with respect to efficiency, quality, and working strategy. A tool based on free-form deformation (FFD) is employed for the variation of flow geometry. This simulation tool is the parallel multigrid flow solver FASTEST, which uses a fully conservative finite-volume method for solving the incompressible Navier–Stokes equations on a non-staggered cell-centred grid arrangement. The optimization tools are investigated by considering the optimization of the connection of two pipes with respect to the minimization of the pressure drop. This problem can be considered as a representative test case for a practical three-dimensional flow configuration. †This is an extended and enhanced version of work presented at the mini-symposium on Evolutionary Algorithms: Recent Applications in Engineering and Science organized by Dr William Annicchiarico at the 7th World Congress on Computational Mechanics, Los Angeles, July 2006.