Equations In Primitive Variables Research Articles

On the Numerical Solution of the Incompressible Navier-Stokes Equations in Primitive Variables Using Grid Generation Techniques

In this paper presents a new model procedure for the solution of the incompressible Navier-Stokes equations in primitive variables, using grid generation techniques. The time dependent momentum equations are solved explicitly for the velocity field using the explicit marching procedure, the continuity equation is implied at each grid point in the solution of pressure equation, while the SOR method is used for the Neumann problem for pressure. Results obtained for the model problem of driven flow in a square cavity demonstrate that the method yields accurate solutions. The results of the numerical computations in a driven cavity, which are presented for the history of the residues at several Reynolds numbers Re=100,1000,4000 and 5000 all the computed results are obtained without any artificial dissipation. This feature of the present procedure demonstrates its excellent convergence and stability characteristics. Numerically results obtained for the steady state static pressure in the driven cavity are presented for the first time at Re=4000 and 5000 using non-staggered grid.

Sloshing in a three-dimensional rectangular tank: Numerical simulation and experimental validation

Pressure variations and three-dimensional effects on liquid sloshing loads in a moving partially filled rectangular tank have been carried out numerically and experimentally. A numerical algorithm based on the volume of fluid (VOF) technique is used to study the non-linear behavior and damping characteristics of liquid sloshing. A moving coordinate system is used to include the non-linearity and avoid the complex boundary conditions of moving walls. The numerical model solves the complete Navier–Stokes equations in primitive variables by using of the finite difference approximations. In order to mitigate a series of discrete impacts, the signal computed is averaged over several time steps. In order to assess the accuracy of the method used, computations are compared with the experimental results. Several configurations of both baffled and unbaffled tanks are studied. Comparisons show good agreement for both impact and non- impact type slosh loads in the cases investigated.

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

Results are presented for the numerical simulation of vortex-induced vibrations (VIVs) of a cylinder at low Reynolds numbers (Re). A stabilized space–time finite-element formulation is utilized to solve the incompressible flow equations in primitive variables. The cylinder, of low nondimensional mass ( m * = 10 ), is free to vibrate in, both, the transverse and in-line directions. To investigate the effect of Re and reduced natural frequency, F n , two sets of computations are carried out. In the first set of computations the Reynolds number is fixed ( = 100 ) and the reduced velocity ( U * = 1 / F n ) is varied. Hysteresis, in the response of the cylinder, is observed at the low- as well as high-end of the range of reduced velocity for synchronization/lock-in. In the second set of computations, the effect of Reynolds number ( 50 ⩽ Re ⩽ 500 ) is investigated for a fixed reduced velocity ( U * = 4.92 ). The effect of the Reynolds number is found to be very significant for VIVs. While the vortex-shedding mode at low Re is 2S (two single vortices shed per cycle), at Re ∼ 300 and larger, the P + S mode of vortex shedding (a single vortex and one pair of counter-rotating vortices are released in each cycle of shedding) is observed. This is the first time that the P + S mode has been observed for a cylinder undergoing free vibrations. This change of vortex-shedding mode is hysteretic in nature and results in a very large increase in the amplitude of in-line oscillations. Since the flow ceases to remain two-dimensional beyond Re ∼ 200 , it remains to be seen whether the P + S mode of shedding can actually be observed in reality for free vibrations.

Numerical Study of Transient and Steady-State Natural Convection and Surface Thermal Radiation in a Horizontal Square Open Cavity

ABSTRACT This article reports a numerical study of the transient and steady-state heat transfer and air flow in a horizontal square open cavity. This study has importance in several thermal engineering problems, for example, in the design of electronic devices and thermal receivers for solar concentrators. The most important assumptions in the mathematical formulation of the horizontal square open cavity are three: The flow is laminar, the fluid is radiatively nonparticipating, and the Boussinesq approximation is valid. The conservation equations in primitive variables are solved for a range of Rayleigh number, Ra, from 103 to 107 using the finite-volume method and the SIMPLEC algorithm. The results show that the radiative exchange between the walls and the aperture increases considerably the total average Nusselt number, from around 94% to 125%, versus the one with no radiative exchange between walls. Also, for Ra = 107, the numerical model predicts periodic formation of thermal plumes in the bottom wall of the cavity, and their further movements explain the oscillation behavior of the Nusselt number with time.

FEM-SIMULATION OF LAMINAR FLAME PROPAGATION II: TWIN AND TRIPLE FLAMES IN COUNTERFLOW

ABSTRACT In a low-Mach-number counterflow geometry, steadily propagating twin and triple flames are investigated. A complete numerical model of such flames is presented including a Galilean-like transformation to achieve steadiness of the flames in a moving frame of reference. The model is based on a similarity transformation of the fundamental conservation Equations in primitive variables, which transforms the inherently three-dimentional problem to two space dimensions. Suitable boundary conditions are derived employing potential theory. The governing Equations are discretized by a finite element method on an unstructured triangulized grid. Employing a local adaptive mesh-refinement procedure, for H2‒O2 systems with both a global one-step model and a detailed mechanism of elementary reactions, numerical solutions have been obtained.

A finite element formulation for transient incompressible viscous flows stabilized by local time-steps

Stabilized finite element formulations are well suited for convection dominated flows and for the solution of the incompressible Navier–Stokes equations in primitive variables. In this paper, we present a method where the structure of stabilization terms appear naturally from a least-squares minimization of the time-discretized momentum balance. Local time-steps, chosen according to the time-scales of convection–diffusion of momentum, play the role of stabilization parameters. Numerical solutions of incompressible viscous flows demonstrate the usefulness of the proposed stabilized formulation.

BUOYANCY-AIDED MIXED CONVECTION WITH CONDUCTION AND SURFACE RADIATION FROM A VERTICAL ELECTRONIC BOARD WITH A TRAVERSABLE DISCRETE HEAT SOURCE

This article presents the results of a comprehensive fundamental numerical study of the problem of buoyancy-aided mixed convection with conduction and surface radiation from a vertical electronic board provided with a traversable, flush-mounted, discrete heat source. Air, a radiatively transparent medium, is considered to be the cooling agent. The governing equations in primitive variables for fluid flow and heat transfer are first converted into stream function–vorticity form, and are later converted into algebraic form using the finite-volume method. The resulting finite-difference equations are solved by Gauss-Seidel iterative technique. The governing equation for temperature distribution along the electronic board is obtained by appropriate energy balance. The effects of pertinent parameters, viz., location of the discrete heat source, surface emissivity of the board, and modifiedRichardson number, on various results, including local temperature distribution along the board, maximum board temperature, and contributions of convection and surface radiation to heat dissipation from the board, are studied in great detail. The fact that any design calculation that ignores surface radiation in problems of this kind would be error-prone is clearly highlighted.

FEM-simulation of laminar flame propagation. I: Two-dimensional flames

In this paper, we present a numerical model for two-dimensional low-Mach-number flows of reactive ideal-gas mixtures based on the fundamental conservation equations in primitive variables. Chemical reaction is described by a detailed mechanism of elementary reactions, and detailed models for molecular transport and thermodynamics are taken into account. The equations are discretized by a finite-element method on unstructured grids using the well known Taylor–Hood element. A streamline-diffusion upwinding technique is used to avoid instabilities in convection-dominated regions of the flowfield. A fully operative local adaptive mesh-refinement procedure is used. As numerical examples we consider steadily propagating laminar flames in flat channels, which appear in a variety of shapes depending on the boundary conditions.

An explicit projection method for solving incompressible flows driven by a pressure difference

A finite-difference method for solving the incompressible time-dependent three-dimensional Navier–Stokes equations in open flows where Dirichlet boundary condition (BC) for the pressure are given on part of the boundary is presented. The equations in primitive variables ( v ,p) are solved using a projection method on a non-staggered grid with second-order accuracy in space and time. On the inflow and outflow boundaries the pressure is obtained from its given value at the contour of these surfaces using a two-dimensional form of the pressure Poisson equation, which enforces the incompressibility constraint ∇· v =0 . The obtained pressure in these surfaces is used as Dirichlet BCs for the three-dimensional Poisson equation inside the domain. The solenoidal requirement imposes some restrictions on the choice of the open surfaces. However, these restrictions are usually met in most flows of interest driven by a pressure (or a body force) difference, to which the present numerical method is mainly intended. To check the accuracy of the method, it is applied to several examples including the flow over a backward-facing step, and the three-dimensional pressure driven flow in a circular pipe.

Spectral Element Discretization of the Circular Driven Cavity. Part III: The Stokes Equations in Primitive Variables

This paper deals with the spectral element discretization of the Stokes equations in a disk with discontinuous boundary data. The numerical treatment does not involve any regularization of these data. Based on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of the Stokes problem and derive error estimates in Sobolev spaces with appropriate weights. We propose an algorithm to solve the discrete system and present numerical experiments which confirm the efficiency of the method.

Unsteady flow with heat and mass transfer due to impingement of slot jet on an inclined plate

The unsteady laminar incompressible boundary layer flow due to a two-dimensional slot jet on a flat plate at an angle of attack has been studied. The unsteadiness in the flow field is due to the free stream velocity distribution or wall temperature (concentration) which varies with time. The governing partial differential equations in primitive variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The effect of the variation of the free stream velocity distribution with time is found to be more pronounced on the skin friction than on the heat or mass transfer. The Prandtl number and the variation of the wall temperature with time strongly affect the heat transfer. Similarly, the Schmidt number and the variation of the concentration at the wall with time strongly affect the mass transfer. Beyond a certain critical value of the viscous dissipation parameter, the plate gets heated instead of being cooled.

An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: II. Three-Dimensional Cases

An efficient and accurate numerical scheme is presented for the three-dimensional Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a spectral-Galerkin approximation for the space variables and a second-order projection scheme for time. The new spectral-projection scheme is implemented to simulate unsteady incompressible flows in a cylinder.

Nonlinear modeling of liquid sloshing in a moving rectangular tank

A nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated. The fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility. The tank is forced to move harmonically along a vertical curve with rolling motion to simulate the actual tank excitation. The volume of fluid technique is used to track the free surface. The model solves the complete Navier–Stokes equations in primitive variables by use of the finite difference approximations. At each time step, a donor–acceptor method is used to transport the volume of fluid function and hence the locations of the free surface. In order to assess the accuracy of the method used, computations are verified through convergence tests and compared with the theoretical solutions and experimental results.

Space and time error estimates for a first order, pressure stabilized finite element method for the incompressible Navier–Stokes equations

In this paper we analyze a pressure stabilized, finite element method for the unsteady, incompressible Navier–Stokes equations in primitive variables; for the time discretization we focus on a fully implicit, monolithic scheme. We provide some error estimates for the fully discrete solution which show that the velocity is first order accurate in the time step and attains optimal order accuracy in the mesh size for the given spatial interpolation, both in the spaces L 2(Ω) and H 1 0(Ω) ; the pressure solution is shown to be order 1 2 accurate in the time step and also optimal in the mesh size. These estimates are proved assuming only a weak compatibility condition on the approximating spaces of velocity and pressure, which is satisfied by equal order interpolations.

Simulation of laminar vortex shedding flow past cylinders using a coupled BEM and FEM model

This paper describes a novel computational model developed to simulate laminar vortex-shedding flows past circular cylinders in 2D incompressible viscous flows in external flow fields. The model based on unsteady 2D Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The external flow field is simulated using the boundary element method (BEM) by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three-step finite element method (FEM). The present model is applied to simulate vortex-shedding flow past a single circular cylinder and flow past two cylinders in which one act as a control cylinder, for low Reynolds number flows. For both the cases, the time development of the flow towards periodic vortex shedding, are illustrated by streamline pictures. The simulation results are compared with experimental data and other numerical models and found to be very reasonable and satisfactory.

A CONTROL-VOLUME FINITE-ELEMENT METHOD (CVFEM) FOR UNSTEADY, INCOMPRESSIBLE, VISCOUS FLUID FLOWS

A control-volume finite-element method (CVFEM), to simulate unsteady, incompressible, and viscous fluid flows, using nine-noded quadrilateral elements, has been developed. The Navier-Stokes equations in primitive variables u-v-p are the mathematical model of the flows. A technique of upwind, known as MAW, Mass-Weighted Interpolation, was extended for the finite element employed in this work. The set of nonlinear partial differential equations was integrated, and after using interpolation functions and time discretization, the algebraic system of equations was solved by using the frontal method of solution. Results obtained for some benchmark problems compared favorably with available results from the literature.

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

AbstractThis paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.

Incremental unknowns for solving the incompressible Navier–Stokes equations

Incremental unknowns, earlier designed for the long-term integration of dissipative evolutionary equations, are introduced here for the incompressible Navier–Stokes equations in primitive variables when multilevel finite-difference discretizations on a staggered grid are used for the spatial discretization; extending to this practical case, the notion of small and large wavelengths that stems naturally from spectral methods when Fourier series expansions are considered. Furthermore, for the temporal discretization, we use the θ-scheme, which allows to decouple the nonlinearity and the incompressibility in these equations; then we have to solve a generalized Stokes equation — we consider here a leading preconditioned outer/inner iteration strategy — and a nonlinear elliptic equation — we linearize its nonlinear terms to get an iterative process. Roughly, at each iterative stage several Poisson equations must be solved, incremental unknowns appear there as an efficient preconditioner. The incremental unknowns methodology appears well suited to capture the turbulent behavior of the flow whose small eddies, even for moderate Reynolds numbers, never go to a steady state; instead they converge to a strange attractor, and keep always bringing imperceptible kinetic energy to the flow. First, they wander around, then they converge to the strange attractor.

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.

Numerical Simulation of Unsteady Incompressible Flow (\em Re \protect\boldmath $\leq$ 9500) on the Curvilinear Half-Staggered Mesh

In this paper, we continue our efforts toward numerical simulation of high $Re$ ($1000 \leq Re \leq 9500$) unsteady incompressible flows with the finite difference method on the curvilinear half-staggered mesh. The finite difference scheme and the projection method for the numerical solution of the incompressible Navier--Stokes equations in primitive variables was proposed and verified in [L. C. Huang, J. Comput. Math, to appear]. Here we derive the discrete Poisson equation on the curvilinear half-staggered mesh and discuss its properties. For unsteady complex flow simulation, rapid solution of this equation is crucial; the multilevel one-way dissection methods offer effective solutions. Numerical simulation of high $Re$ flows around circular and aerofoil-shaped cylinders show our method to be efficient and robust.