a coupled finite volume solver for the solution of incompressible flows on unstructured grids this paper reports on a newly developed fully coupled pressure based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids the implicit pressure velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated simple algorithm using the rhie chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix the extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver the performance of the coupled approach as compared to the segregated approach exemplified by simple is tested by solving five laminar flow problems using both methodologies and comparing their computational costs results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent for relatively coarse meshes the cpu time required by the coupled solver on structured grid is lower than the cpu time required on unstructured grid on dense meshes however this is no longer true for low and moderate values of the grid aspect ratio the number of iterations required by the coupled solver remains unchanged while the computational cost slightly increases for structured and unstructured grid systems the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio recorded cpu time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases

an investigation into the characteristics of non linear frequency response functions part 1 understanding the higher dimensional frequency spaces the characteristics of generalized frequency response functions gfrfs of non linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques it is shown how a systematic analysis can be achieved for a wide class of non linear systems in the frequency domain using the proposed methods the paper is divided into two parts in part 1 the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi dimensional frequency spaces the visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time in part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non linear systems described by the narmax class of models case studies are included in part 2 to illustrate the application of the new methods

on the stabilization of finite volume methods with co located variables for incompressible flow finite volume methods with co located variables for incompressible flow suffer from spurious pressure oscillations unless a stabilization method is applied variations of the pressure weighed interpolation pwi method are typically used for this purpose but the pwi method does not only prevent spurious oscillations counter intuitively it also simplifies the approximation of the schur complement pressure matrix which appears in iterative solution methods such as simple

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

optimal sampled data controls with running inequality state constraints pontryagin maximum principle and bouncing trajectory phenomenon sampled data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled data control systems and certain generalizations in this paper we derive a pontryagin maximum principle for general nonlinear finite dimensional optimal sampled data control problems with running inequality state constraints in particular we obtain a nonpositive averaged hamiltonian gradient condition with the adjoint vector being a function of bounded variations our proof is based on the ekeland variational principle in general optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities the jumps and the singular part of the adjoint vector however in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times we call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions

multi level acceleration with manifold mapping of strongly coupled partitioned fluid structure interaction abstract to accelerate the convergence of strongly coupled partitioned fluid structure interaction simulations the manifold mapping algorithm is used which combines a high fidelity model with a low fidelity model a computationally inexpensive low fidelity fluid structure interaction fsi model is combined with a high fidelity fsi model in order to accelerate the convergence of the coupling iterations of the high fidelity fsi model the manifold mapping algorithm is applied to the fluid structure interaction problem in order to minimize the fluid structure interface residual originating from multi fidelity optimization the manifold mapping algorithm is applied for the first time in a simulation context instead of an optimization context the manifold mapping algorithm is applied to an unsteady flow in a one dimensional tube incompressible laminar flow over a fixed cylinder with an attached flexible flap and a wave propagation in a three dimensional elastic tube problem a reduction of approximately 50 in terms of high fidelity iterations is achieved compared to the state of the art interface quasi newton inverse least squares algorithm the convergence of the high fidelity model is accelerated even further when information from previous time steps is reused

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

numerical time step restrictions as a result of capillary waves the propagation of capillary waves on material interfaces between two fluids imposes a strict constraint on the numerical time step applied to solve the equations governing this problem and is directly associated with the stability of interfacial flow simulations the explicit implementation of surface tension is the generally accepted reason for the restrictions on the temporal resolution caused by capillary waves in this article a fully coupled numerical framework with an implicit treatment of surface tension is proposed and applied demonstrating that the capillary time step constraint is in fact a constraint imposed by the temporal sampling of capillary waves irrespective of the type of implementation the presented results show that the capillary time step constraint can be exceeded by several orders of magnitude with the explicit as well as the implicit treatment of surface tension if capillary waves are absent furthermore a revised capillary time step constraint is derived by studying the temporal resolution of capillary waves based on numerical stability and signal processing theory including the doppler shift caused by an underlying fluid motion the revised capillary time step constraint assures a robust aliasing free result as demonstrated by representative numerical experiments and is in the static case less restrictive than previously proposed time step limits associated with capillary waves

finite volume methods for two dimensional incompressible flows with complex boundaries abstract the paper discusses possible ways of developing accurate and efficient finite volume methods for complex incompressible flows the issues of boundary fitted grid generation choice of coupled or uncoupled procedures cartesian or grid aligned velocity components staggered or non staggered variable arrangement discretization methods solution algorithms and convergence acceleration by multigrid methods are addressed an example of a finite volume method for general two dimensional flows is described in some detail and applications of this method to a variety of flows are presented

on the improved finite volume procedure for simulation of turbulent flows over real complex terrains this article presents a new and substantially improved finite volume procedure for simulation of incompressible flows on non orthogonal grids cell centered least squares gradients are obtained in a robust and highly accurate way a new discretization of the diffusive terms is employed which is based on extension of the original cell face gradient interpolation and is more suitable for complex grid distortions a flexible flux limited interpolation of dependent variables on distorted computational grids is introduced an efficient preconditioner for krylov method solution of linear systems is proposed which substantially improves the solution of poisson equation for pressure correction the pressure correction algorithm is adapted for efficient convergence on highly complex grids using a sequence of non orthogonal corrector solutions and its effect on iteration convergence is analyzed the non orthogonalities treated by current procedure are more accustomed to numerical grids generated from a real complex terrain elevation data the main focus is on the simulation of atmospheric micro scale flows pertinent to wind energy application

new bound characteristics of narx model in the frequency domain new results about the bound characteristics of both the generalized frequency response functions gfrfs and the output frequency response for the narx non linear autoregressive model with exogenous input model are established it is shown that the magnitudes of the gfrfs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order gfrf and the coefficients of the polynomial are functions of the narx model parameters these new bound characteristics of the narx model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions reveal the effect of the model parameters on the stability of the narx model to a certain extent and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain for example evaluation of the truncation error in a volterra series expression of non linear systems and the highest order needed in the

performance comparison of the nwf and dc methods for implementing high resolution schemes in a fully coupled incompressible flow solver this paper reports on the use of the normalized weighting factor nwf method and the deferred correction dc approach for the implementation of high resolution hr convective schemes in an implicit fully coupled pressure based flow solver four hr schemes are realized within the framework of the nwf and dc methods and employed to solve the following three laminar flow problems i lid driven flow in a square cavity ii sudden expansion in a square cavity and iii flow in a planar t junction over three grid systems with sizes of 104 5i 104 and 3i 105 control volumes the merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various hr schemes on the different grid systems whereas previous attempts to use the nwf method in a segregated flow solver failed to produce converged solutions current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver in terms of cpu efficiency there is no global and consistent superiority of any method over another even though the dc method outperformed the nwf method in two of the three test problems solved

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

unified formulation of the momentum weighted interpolation for collocated variable arrangements abstract momentum weighted interpolation mwi is a widely used discretisation method to prevent pressure velocity decoupling in simulations of incompressible and low mach number flows on meshes with a collocated variable arrangement despite its popularity a unified and consistent formulation of the mwi is not available at present in this work a discretisation procedure is devised following an in depth analysis of the individual terms of the mwi derived from physically consistent arguments based on which a unified formulation of the mwi for flows on structured and unstructured meshes is proposed including extensions for discontinuous source terms in the momentum equations as well as discontinuous changes of density as shown by the presented analysis and numerical results the mwi enforces a low pass filter on the pressure field which suppresses oscillatory solutions furthermore the numerical dissipation of kinetic energy introduced by the mwi is shown to converge with third order in space and is independent of the time step if the mwi is derived consistently from the momentum equations in the presence of source terms the low pass filter on the pressure field can be shaped by a careful choice of the interpolation coefficients to ensure the filter only acts on the driving pressure gradient that is associated with the fluid motion which is shown to be vitally important for the accuracy of the numerical solution to this end a force balanced discretisation of the source terms is proposed that precisely matches the discretisation of the pressure gradients and preserves the force applied to the flow using representative test cases of incompressible and low mach number flows including flows with discontinuous source terms and two phase flows with large density ratios the newly proposed formulation of the mwi is favourably compared against existing formulations and is shown to significantly reduce or even eliminate solution errors with an increased stability for flows with large density ratios

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

a high resolution pressure based algorithm for fluid flow at all speeds abstract a new collocated finite volume based solution procedure for predicting viscous compressible and incompressible flows is presented the technique is equally applicable in the subsonic transonic and supersonic regimes pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the newly developed algorithm has two new features i the use of the normalized variable and space formulation nvsf methodology to bound the convective fluxes and ii the use of a high resolution scheme in calculating interface density values to enhance the shock capturing property of the algorithm the virtues of the newly developed method are demonstrated by solving a wide range of flows spanning the subsonic transonic and supersonic spectrum results obtained indicate higher accuracy when calculating interface density values using a high resolution scheme

a novel characteristic parameter approach for analysis and design of linear components in nonlinear systems nonlinearity always brings difficulties to the analysis and design of practical systems this study shows that for a wide class of nonlinear systems described by nonlinear differential equations the nonlinear output spectrum can be expressed as a polynomial function with respect to linear model parameters based on the recently developed concepts of parametric characteristic analysis and nonlinear characteristic output spectrum ncos function the coefficients of the new ncos function are independent of the linear parameters of interest and can be directly determined with proposed procedures this reveals an explicit and analytical relationship between the output spectrum of a nonlinear system and its linear components and therefore provides a convenient tool for the analysis and design of nonlinear systems using only linear components or linear feedback control while simultaneously fully considering inherent nonlinear influence this result is a useful complement to the ncos function based analysis and design recently developed in the literature the effectiveness and usefulness of the results are demonstrated through an application to the analysis of an amplifier with a multi tone input for suppressing intermodulation distortion it is shown that the new results reveal a straightforward insight into the parametric optimization of the amplifier for suppression of intermodulation distortion

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

nonlinear influence in the frequency domain alternating series the nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have volterra series expansion it is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities the magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series sufficient and necessary conditions in which the output spectrum can be cast into an alternating series are studied these results reveal a novel frequency domain insight into the nonlinear influence on a system and provide a new method for the analysis and design of nonlinear systems in the frequency domain examples are given to illustrate the results

frequency response functions for nonlinear convergent systems convergent systems constitute a practically important class of nonlinear systems that extends the class of asymptotically stable linear time invariant systems in this note we extend frequency response functions defined for linear systems to nonlinear convergent systems such nonlinear frequency response functions for convergent systems give rise to nonlinear bode plots which serve as a graphical tool for performance analysis of nonlinear convergent systems in the frequency domain the results are illustrated with an example

feedback bounded stabilisation of certain discrete volterra systems abstractthroughout this paper we present a method for designing feedback laws which stabilise nonlinear discrete volterra systems our method is based on a factorisation algorithm which decomposes the original system as a composition of a polynomial and a linear series

solution for state constrained optimal control problems applied to power split control for hybrid vehicles this paper presents a numerical solution for scalar state constrained optimal control problems the algorithm rewrites the constrained optimal control problem as a sequence of unconstrained optimal control problems which can be solved recursively as a two point boundary value problem the solution is obtained without quantization of the state and control space the approach is applied to the power split control for hybrid vehicles for a predefined power and velocity trajectory and is compared with a dynamic programming solution the computational time is at least one order of magnitude less than that for the dynamic programming algorithm for a superior accuracy

a bound for the magnitude characteristics of nonlinear output frequency response functions part 1 analysis and computation a bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs it is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included the paper is divided into two parts in part 1 an expression for the output magnitude bound is derived properties of the result are discussed and general procedures for the practical computation of the bound are developed in part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed

finite input output representative of a class of volterra polynomial systems n this paper the problem of finite input output representation of a special class of nonlinear volterra polynomial systems is studied via the notion of linear factorization of d series this is an algebraic method based mainly on the star product operation and on a related euclidean type algorithm

efficient solution of 3d electromagnetic eddy current problems within the finite volume framework of openfoam eddy current problems occur in a wide range of industrial and metallurgical applications where conducting material is processed inductively motivated by realising coupled multi physics simulations we present a new method for the solution of such problems in the finite volume framework of foam extend an extended version of the very popular openfoam software the numerical procedure involves a semi coupled multi mesh approach to solve maxwell s equations for non magnetic materials by means of the coulomb gauged magnetic vector potential a and the electric scalar potential the concept is further extended on the basis of the impressed and reduced magnetic vector potential and its usage in accordance with biot savart s law to achieve a very efficient overall modelling even for complex three dimensional geometries moreover we present a special discretisation scheme to account for possible discontinuities in the electrical conductivity to complement our numerical method an extensive validation is completing the paper which provides insight into the behaviour and the potential of our approach

nonlinear system analysis and identification from random data 1991 nonlinear system analysis and identification from random data technometrics vol 33 no 4 pp 482 483

unsteady free surface flow simulation over complex topography with a multidimensional upwind technique in the context of numerical techniques for solving unsteady free surface problems finite element and finite volume approximations are widely used a class of upwind methods which attempts to model the equations in a genuinely multidimensional manner has been recently introduced as an alternative multidimensional upwind schemes mus were developed initially for the approximation of steady state solutions of the two dimensional euler equations on unstructured grids although they can be applicable to any system of hyperbolic conservation laws such as the shallow water equations the formal analogy between the two systems of equations is useful for simple cases however in practical applications of interest in hydraulics complex geometries and bottom slope variation can lead to important numerical errors produced by an inadequate source term discretization this problem has been analyzed and in this work the necessity of a multidimensional upwind discretization of the source terms is justified the basis of the numerical method is stated and the particular adaptation to unsteady shallow water flows over irregular geometry is described as test cases laboratory experimental data are used together with academic tests for validation

numerical solution of constrained optimal control problems with parameters this article presents a numerical solution technique for constrained optimal control problems that contain parameters here the state control and parameter inequality constraints are accommodated via an extended penalty function this penalty function takes on large values when the constraints are violated and small values when the constraints are satisfied using the calculus of variation it is shown that the first order necessary conditions for optimality are in the form of a two point boundary value problem involving differential and algebraic equations bvp dae a multiple shooting continuation method is developed for solving this bvp dae two examples are presented to demonstrate the effectiveness of the solution approach developed in the paper

a coupled pressure based computational method for incompressible compressible flows pressure based flow solvers couple continuity and linearized truncated momentum equations to derive a poisson type pressure correction equation and use the well known simple algorithm momentum equations and the pressure correction equation are typically solved sequentially in many cases this method results in slow and often difficult convergence the current paper proposes a novel computational algorithm solving for pressure and velocity simultaneously within a pressure correction coupled solution approach using finite volume method on structured and unstructured meshes the method can be applied to both incompressible and subsonic compressible flows for subsonic compressible flows the energy equation is also coupled with flow field and the density of fluid is obtained by equation of state the procedure eliminates the pressure correction step the most expensive component of the simple like algorithms the proposed coupled continuity momentum energy equation method can be used to simulate steady state or transient flow problems the method has been tested on several cfd benchmark cases with excellent results showing dramatically improved numerical convergence and significant reduction in computational time

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

an investigation into the characteristics of non linear frequency response functions part 2 new analysis methods based on symbolic expansions and graphical techniques in part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi dimensional frequency spaces the visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented in this the second part symbolic expansion techniques are introduced and together with the results achieved in part 1 yield new methods for analysing the properties of generalized frequency response functions case studies are included to illustrate the application of the new methods

interpretation of non linear frequency response functions analytical and graphical methods of interpreting generalised frequency response functions for non linear systems are derived it is shown that non linear phenomena can be classified into intra kernel and inter kernel interference and that worst case responses can be computed the results are illustrated using several discrete and continuous time non linear systems

frequency domain truncation of volterra series for weakly non linear systems subject to harmonic excitation new results for model order reduction for weakly non linear systems in the frequency domain are derived based on a parametric modelling approach

a pressure based algorithm for multi phase flow at all speeds a new finite volume based numerical algorithm for predicting incompressible and compressible multi phase flow phenomena is presented the technique is equally applicable in the subsonic transonic and supersonic regimes the method is formulated on a non orthogonal coordinate system in collocated primitive variables pressure is selected as a dependent variable in preference to density because changes in pressure are significant at all speeds as opposed to variations in density which become very small at low mach numbers the pressure equation is derived from overall mass conservation the performance of the new method is assessed by solving the following two dimensional two phase flow problems i incompressible turbulent bubbly flow in a pipe ii incompressible turbulent air particle flow in a pipe iii compressible dilute gas solid flow over a flat plate and iv compressible dusty flow in a converging diverging nozzle predictions are shown to be in excellent agreement with published numerical and or experimental data

a finite volume formulation for transient convection and diffusion equations with unstructured distorted grids and its applications in fluid flow simulations with a collocated variable arrangement abstract this paper describes a finite volume method for simulating transport processes governed by convection diffusion type equations the formulation is based on a cell centred unstructured grid with an edge based data structure discretisation is independent of control volume or cell shape by using a surface vector decomposition at the midpoint of the interface between cells along with a deferred correction approach any cross diffusion due to grid skewness can be readily accounted for when discretising the diffusive flux for modelling fluid flow processes a collocated arrangement of variables is employed so that a single coefficient matrix applies for the momentum equations of each velocity component to avoid checkerboard oscillation arising from pressure velocity decoupling occurring under the collocated variable arrangement when a pressure based solution algorithm is employed a novel pseudo flux interpolation method is proposed for unstructured grids ensuring that the solution is both under relaxation factor and time step for transient calculation independent the methodology can be formulated within a framework whereby either a coupled or a decoupled solution algorithm can be employed the features and advantages of the method are demonstrated by solution of the navier stokes equations for two benchmark flow problems

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation