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Primitive Variables (primitive + variable)
Selected AbstractsA combination of implicit and adaptative upwind tools for the numerical solution of incompressible free surface flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2007V. G. Ferreira Abstract This paper is concerned with the numerical solutions of time dependent two-dimensional incompressible flows. By using the primitive variables of velocity and pressure, the Navier,Stokes and mass conservation equations are solved by a semi-implicit finite difference projection method. A new bounded higher order upwind convection scheme is employed to deal with the non-linear (advective) terms. The procedure is an adaptation of the GENSMAC (J. Comput. Phys. 1994; 110:171,186) methodology for calculating confined and free surface fluid flows at both low and high Reynolds numbers. The calculations were performed by using the 2D version of the Freeflow simulation system (J. Comp. Visual. Science 2000; 2:199,210). In order to demonstrate the capabilities of the numerical method, various test cases are presented. These are the fully developed flow in a channel, the flow over a backward facing step, the die-swell problem, the broken dam flow, and an impinging jet onto a flat plate. The numerical results compare favourably with the experimental data and the analytical solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Hybrid finite element/volume method for shallow water equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010Shahrouz Aliabadi Abstract A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two-fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non-conservation form and solve the non-linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell-center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix-free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd. [source] A Hermite finite element method for incompressible fluid flowINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2010J. T. Holdeman Abstract We describe some Hermite stream function and velocity finite elements and a divergence-free finite element method for the computation of incompressible flow. Divergence-free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence-free flow fields (,·uh,0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid-driven cavity and backward-facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd. [source] Numerical simulation of a single bubble by compressible two-phase fluidsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010Siegfried Müller Abstract The present work deals with the numerical investigation of a collapsing bubble in a liquid,gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non-conservative transport equations of the pressure law coefficients. The original first-order discretization is extended to higher order applying second-order ENO reconstruction to the primitive variables. The derivation of the non-conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale-based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2-D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser-induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd. [source] A streamfunction,velocity approach for 2D transient incompressible viscous flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2010Jiten C. Kalita Abstract We recently proposed (J. Comput. Phys. 2005; 207(1):52,68) a new paradigm for solving the steady-state two-dimensional (2D) Navier,Stokes (N,S) equations using a streamfunction,velocity (,,v) formulation. This formulation was shown to avoid the difficulties associated with the traditional formulations (primitive variables and streamfunction-vorticity formulations). The new formulation was found to be second-order accurate and was found to yield accurate solutions of a number of fluid flow problems. In this paper, we extend the ideas and propose a second-order implicit, unconditionally stable ,,v formulation for the unsteady incompressible N,S equations. The method is used to solve several 2D time-dependent fluid flow problems, including the flow decayed by viscosity problem with analytical solution, the lid-driven square cavity problem, the backward-facing step problem and the flow past a square prism problem. For the problems with known exact solutions, our coarse grid transient solutions are extremely close to the analytical ones even for high Reynolds numbers (Re). For the driven cavity problem, our time-marching steady-state solutions up to Re=7500 provide excellent matches with established numerical results, and for Re=10000, our study concludes that the asymptotic stable solution is periodic as has been found by other authors in recent studies. For the backward step problem, our numerical results are in excellent agreement with established numerical and experimental results. Finally, for the flow past a square prism, we have very successfully simulated the von Kármán vortex street for Re=200. Copyright © 2009 John Wiley & Sons, Ltd. [source] Effect of blockage on free vibration of a circular cylinder at low ReINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2008T. K. Prasanth Abstract The effect of the blockage on vortex-induced vibrations of a circular cylinder of low non-dimensional mass (m*=10) in the laminar flow regime is investigated in detail. A stabilized space,time finite element formulation is utilized to solve the incompressible flow equations in primitive variables form in two dimensions. The transverse response of the cylinder is found to be hysteretic at both ends of synchronization/lock-in region for 5% blockage. However, for the 1% blockage hysteresis occurs only at the higher Re end of synchronization/lock-in region. Computations are carried out at other blockages to understand its effect on the hysteretic behavior. The hysteresis loop at the lower Re end of the synchronization decreases with decrease in blockage and is completely eliminated for blockage of 2.5% and less. On the other hand, hysteresis persists for all values of blockage at the higher Re end of synchronization/lock-in. Although the peak transverse oscillation amplitude is found to be same for all blockage (,0.6D), the peak value of the aerodynamic coefficients vary significantly with blockage. The r.m.s. values show lesser variation with blockage. The effect of streamwise extent of computational domain on hysteretic behavior is also studied. The phase between the lift force and transverse displacement shows a jump of almost 180° at, approximately, the middle of the synchronization region. This jump is not hysteretic and is independent of blockage. Copyright © 2008 John Wiley & Sons, Ltd. [source] High-order ENO and WENO schemes for unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2007W. R. Wolf Abstract This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third- and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge,Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical study of particulate suspension flow through wavy-walled channelsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2006R. Usha Abstract The particulate suspension flow in a channel whose walls describe a travelling wave motion is examined numerically. A perturbation method is employed and the primitive variables are expanded in a series with the wall amplitude as the perturbation parameter. The boundary conditions are applied at the mean surface of the channel and the first-order perturbation quantities are numerically determined by solving the governing system of ordinary differential equations by shooting technique. The present approach does not impose any restriction on the Reynolds number of the flow and the wave number and frequency of the wavy-walled channel, although it is limited by the linear analysis. The wall shear stress and the positions of flow separation and reattachment points are computed and the influence of the volume fraction density of the particles is examined. The variations of velocity and pressure of the particulate suspension flow with frequency of excitation are also presented. Copyright © 2005 John Wiley & Sons, Ltd. [source] Rotating incompressible flow with a pressure Neumann conditionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2006Julio R. Claeyssen Abstract This work considers the internal flow of an incompressible viscous fluid contained in a rectangular duct subject to a rotation. A direct velocity,pressure algorithm in primitive variables with a Neumann condition for the pressure is employed. The spatial discretization is made with finite central differences on a staggered grid. The pressure and velocity fields are directly updated without any iteration. Numerical simulations with several Reynolds numbers and rotation rates were performed for ducts of aspect ratios 2:1 and 8:1. Copyright © 2005 John Wiley & Sons, Ltd. [source] Dendritic solidification of binary alloys with free and forced convectionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005P. Zhao Abstract Dendritic solidification with forced convection and free convection driven by contraction and thermo- solutal buoyancy is simulated in two-dimensional space using a sharp-interface model. Both pure substances and alloys are considered. The model is formulated using the finite element method and works directly with primitive variables. The coupled energy- and solutal concentration-equations, along with the Navier,Stokes equations for incompressible flow, are solved using different meshes. Temperature is solved in a fixed mesh that covers the whole domain (solid + liquid) where the solid,liquid interface is explicitly tracked using marker points. The concentration and momentum equations are solved in the liquid region using an adaptive mesh of triangular elements that conforms to the interface. The velocity boundary conditions are applied directly on the interface. The model is validated using a series of problems that have analytical, experimental and numerical results. Four simulations are presented: (1) crystal growth of succinonitrile with thermal convection under two small undercoolings; (2) dendritic growth into an undercooled pure melt with a uniform forced flow; (3) equiaxial dendritic growth of a pure substance and an alloy with contraction-induced convection; and (4) directional solidification of Pb,0.2 wt% Sb alloy with convection driven by the combined action of contraction, thermal and solutal buoyancy. Some of the simulation results are compared to those reported using other methods including the phase-field method; others are new. In each case, the effects of convection on dendritic solidification are analysed. Copyright © 2005 John Wiley & Sons, Ltd. [source] Effect of leading edge cut on the aerodynamics of ram-air parachutesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005R. Balaji Abstract The effect of the configuration of leading edge cut on the aerodynamic performance of ram-air parachutes is studied via two-dimensional flow simulations. The incompressible Reynolds-averaged Navier,Stokes equations, in primitive variables, are solved using a stabilized finite-element formulation. The Baldwin,Lomax model is employed for turbulence closure. Flow past an LS(1) 0417 airfoil is investigated for various configurations of the leading edge cut and results are compared with those from a Clark-Y airfoil section. It is found that the configuration of the leading edge cut affects the lift-to-drag ratio (L/D) of the parachute very significantly. The L/D value has strong implications on the flight performance of the parachute. One particular configuration results in a L/D value that is in excess of 25 at 7.5° angle of attack. Results are presented for other angles of attack for this configuration. Copyright © 2004 John Wiley & Sons, Ltd. [source] A block-implicit numerical procedure for simulation of buoyant swirling flows in a model furnaceINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003Marcelo J. S. de Lemos Abstract This work reports numerical results for the case of incompressible laminar heated flow with a swirl in a vertical cylindrical chamber. Computations are obtained with a point-wise block-implicit scheme. Flow governing equations are written in terms of the so-called primitive variables and are recast into a general form. The discretized momentum equations are applied to each cell face and then, together with the mass-continuity, tangential velocity and energy equations, are solved directly in each computational node. The effects of Rayleigh, Reynolds and Swirl numbers on the temperature field are discussed. Flow pattern and scalar residual history are reported. Further, it is expected that more advanced parallel computer architectures can benefit from the error smoothing operator here described. Copyright © 2003 John Wiley & Sons, Ltd. [source] Local block refinement with a multigrid flow solverINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2002C. F. Lange Abstract A local block refinement procedure for the efficient computation of transient incompressible flows with heat transfer is presented. The procedure uses patched structured grids for the blockwise refinement and a parallel multigrid finite volume method with colocated primitive variables to solve the Navier-Stokes equations. No restriction is imposed on the value of the refinement rate and non-integer rates may also be used. The procedure is analysed with respect to its sensitivity to the refinement rate and to the corresponding accuracy. Several applications exemplify the advantages of the method in comparison with a common block structured grid approach. The results show that it is possible to achieve an improvement in accuracy with simultaneous significant savings in computing time and memory requirements. Copyright © 2002 John Wiley & Sons, Ltd. [source] Boundary element analysis of driven cavity flow for low and moderate Reynolds numbersINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2001M. Aydin Abstract A boundary element method for steady two-dimensional low-to-moderate-Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier,Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non-linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data. Copyright © 2001 John Wiley & Sons, Ltd. [source] | ||||||||||