Primitive Variable Formulation Research Articles

Linearized stability analysis of staggered‐grid difference schemes for multidimensional viscous incompressible flows

AbstractThis paper presents a systematic linearized (frozen coefficient) von Neumann stability analysis for various staggered‐grid marker‐and‐cell (MAC) finite difference schemes for solving viscous incompressible fluid flows. These schemes employ the primitive variables formulation and require the velocity field to be divergent free at every time step. It is illustrated that the stability results for staggered‐grid MAC schemes are similar to that for the multidimensional convective‐diffusion equation with constant coefficients. © 1993 John Wiley & Sons, Inc.

THE SOLUTION OF THE TWO-DIMENSIONAL INCOMPRESSIBLE FLOW EQUATIONS ON UNSTRUCTURED TRIANGULAR MESHES

Abstract A numerical method for calculating two-dimensional turbulent incompressible flow on unstructured triangular meshes is developed. A primitive variable formulation is used. The Helmholtz pressure equation algorithm is used to enforce the velocity continuity relation for incompressible flow. A careful treatment of the pressure dissipation model is presented. A standard k-ϵ turbulence model with wall functions is used to provide closure for the governing equations. A backward-facing step turbulent flow is calculated using an unstructured triangular mesh and the results are compared to experimental and computational data.

The computational boundary method for solving the incompressible flows

A new method is described for solving the Navier-Stokes equations in the primitive variable formulation.

The Treatment of Spurious Pressure Modes in Spectral Incompressible Flow Calculations

Algorithms for the transient and steady state simulation of incompressible Newtonian and non-Newtonian flows are described for the primitive variable formulation of the governing equations. Spectral approximations are used for the spatial discretization. Attention is given to the satisfaction of the incompressibility constraint and the determination of the pressure. Spurious pressure modes are removed by means of a singular value decomposition. The corresponding velocity field is divergence free at all the collocation points. Numerical results are presented for Newtonian flow in a regularized driven square cavity and for non-Newtonian flow in a planar channel and in a journal bearing for a realistic range of material parameters.

Numerical modelling of viscoelastic liquids using a finite-volume method

A new staggered-grid, finite-volume method for the numerical simulation of isothermal viscoelastic liquids is presented. The main features of this method are the use of a primitive variable formulation, the location of the velocities, pressures and stresses on different staggered grids and the use of a third-order difference scheme for the discretization of the constitutive equations. The method is applied to a sudden-expansion, viscoelastic-flow problem for a range of Weissenberg numbers. For one case the mesh has been refined to demonstrate that the method is robust and viscoelastic effects are not filtered out as the mesh size decreases. All the computations have been performed on a PC equipped with a floating-point coprocessor. Although the method has been defined for, and applied to, two-dimensional problems, the technique is readily extendable to three dimensions without excessive cost.

Some recent trends and developments in finite element analysis for incompressible thermal flows

AbstractA numerical procedure for solving the time‐dependent, incompressible Navier–Stokes and energy equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a time‐splitting procedure which has unique features in its formulation as well as in its evaluation from the viewpoint of efficiency and accuracy. The applicability of the proposed formulation was verified by computations and comparison with known results.

Preconditioned conjugate gradient methods for the incompressible Navier‐Stokes equations

AbstractA robust technique for solving primitive variable formulations of the incompressible Navier‐Stokes equations is to use Newton iteration for the fully implicit non‐linear equations. A direct sparse matrix method can be used to solve the Jacobian but is costly for large problems; an alternative is to use an iterative matrix method. This paper investigates effective ways of using a conjugate‐gradient‐type method with an incomplete LU factorization preconditioner for two‐dimensional incompressible viscous flow problems. Special attention is paid to the ordering of unknowns, with emphasis on a minimum updating matrix (MUM) ordering. Numerical results are given for several test problems.

A numerical method for incompressible viscous flow simulation

We describe a numerical scheme for computing time-dependent solutions of the incompressible Navier-Stokes equations in the primitive variable formulation. This scheme uses finite elements for the space discretization and operator splitting techniques for the time discretization. The resulting discrete equations are solved using specialized nonlinear optimization algorithms that are computationally efficient and have modest storage requirements. The basic numerical kernel is the preconditioned conjugate gradient method for symmetric, positive-definite, sparse matrix systems, which can be efficiently implemented on the architectures of vector and parallel processing supercomputers.

Finite element modelling of natural‐convection‐controlled change of phase

AbstractThe problem of phase change in the presence of natural convection has been investigated. A model has been proposed based on the treatment of the release/absorption of latent heat as a heat source/sink in combination with the standard Galerkin finite element method with a primitive variable formulation on a fixed grid. To demonstrate the capabilities of the model, three cases of phase change of an aluminium alloy in the presence of natural convection arc considered, i.e. solidification, melting and combined solidification and melting. The solidification of water in a square cavity is modelled as another example, taking into account the density extremum, and the results are compared with a previously published work.

Numerical Solution of Incompressible Flow of Power-Law Fluid through a Pipe with Abrupt Contraction.

A numerical solution for an axisymmetric non-Newtonian flow of an incompressible inelastic power-law fluid using a primitive variable formulation is presented. The present method is based on the method of lines using the rational Runge-Kutta time integration scheme combined with the central finite difference method for the spatial discretization. The Poisson equation is solved by means of the SOR method using the concept of the SMAC method. As a test problem, numerical results for a circular pipe flow with uniform inlet velocity are shown and compared with the analytic fully developed velocity profile to test the validity of the present method. Next, numerical results for the flow through a pipe with abrupt contraction are shown. The difference between Newtonian (n=1) and non-Newtonian fluids (n=0.75 and 1.25) for the Reynolds number (Re) from 0.01 to 100 is discussed concerning the flow pattern. It is evident that the size of the secondary vortex increases with increasing volumes of n. The entry length and the excess pressure drop in the contraction increase with decreasing values of n. It is confirmed that the present method is suitable for the numerical solution of a power-law fluid.

Numerical Solution of Incompressible Flow of Power-Law Fluid Using Boundary-Fitted Curvilinear Coordinates.

A numerical method using boundary-fitted curvilinear coordinates (BFC) is applied to compute non-Newtonian flow equations in primitive variable formulations for an incompressible inelastic power-law fluid. The flow equations are solved on a special staggered grid by the method of lines using the rational Runge-Kutta time integration scheme and the central finite difference method for spatial discretization, while the Poisson equation is solved by means of the successive overrelaxation (SOR) method using the modified simplified marker and cell (MSMAC) method. It is concluded that the results using BFC are comparable to those using Cartesian coordinates, and the proposed method can be applied to the numerical solution of flow problems of a power-law fluid in arbitrary computational domains.

An efficient finite element incompressible Navier-Stokes solver that approximates nonlinear terms with finite differences

A new method for solving the steady-state incompressible Navier-Stokes equations in primitive variable form is presented. The approach uses finite difference operators for the convective terms within the framework of a penalty finite element method. The technique is shown to be computationally very efficient. Indeed, not only does the penalty function method remove one degree of liberty per node, but also the special treatment given to the nonlinear terms yields a global coefficient matrix that is both symmetric and banded. Both storage needs and calculation time are reduced tremendously. Furthermore, for all flow cases tested, the authors have been able to reach a convergent and stable solution up to Reynolds number of the order of those normally attained by “conventional” finite element models using an identical space discretisation. Results are given for the popular benchmark problem of the driven cavity, as well as for a practical application in the Port of Montreal.

Finite element solutions of the Euler equations for transonic external flows

We present a finite element Newton-Galerkin scheme for the solution of the steady subsonic and transonic Euler equations, in primitive variable form. The scheme is based on the explicit introduction of an artificial viscosity in the governing equations to provide the necessary dissipation for numerical stability. The system of equations is linearized by a Newton method and a direct solver is used for the resulting fully coupled system of algebraic equations

Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms.

Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.

Three‐dimensional simulations of natural convection in a sidewall‐heated cube

AbstractA high‐resolution, finite difference numerical study is reported on three‐dimensional natural convection of air in a differentially heated cubical enclosure over an extensive range of Rayleigh number from 103 to 1010. The maximum number of grid points is 122 × 62 × 62. Solutions to the primitive variable formulation of the incompressible Navier‐Stokes and energy equations are acquired by a control‐volume‐based procedure together with a higher‐order upwind‐differencing technique. The field characteristics at large‐time limits are examined in detail by state‐of‐the‐art numerical visualizations of the three‐dimensional results. The emergence of the well‐defined boundary layers and the interior core at high Rayleigh numbers is captured by elaborate numerical visualizations. Both the similarities and discrepancies between the three‐ and two‐dimensional computations are pointed out. These emphasize the need for three‐dimensional calculations to accurately determine the flow characteristics and heat transfer properties in realistic, high‐Rayleigh‐number situations.

Numerical Simulation of Different Types of Steam Surface Condensers

A numerical procedure is developed to simulate the fluid flow and heat transfer processes in the shell-side of steam surface condensers. The governing equations are solved in primitive variable form using a semi-implicit consistent control-volume formulation in which a segregated pressure correction linked algorithm is employed. The procedure is applied to three different types of surface condenser. The numerical predictions are critically assessed by comparison to available experimental data for condensers, and in general, the solutions are in good agreement with the experimental data.

Some current CFD issues relevant to the incompressible Navier-Stokes equations

The goals of this paper are to carefully define a particular class of well-set incompressible Navier-Stokes problems in the continuum (partial differential equation/PDE) setting and to discuss some relevant and sometimes poorly understood issues related to these well-posed PDE problems, both in the continuum world and in its computer counterpart. Emphasized are three common formulations used to generate computer codes: the primitive variable (velocity-pressure) formulation and two derived variable formulations (pressure Poisson equation and vorticity transport equation).

A fractional step solution method for the unsteady incompressible navier-stokes equations in generalized coordinate systems

The time-dependent, three-dimensional incompressible Navier-Stokes equations are presently solved in generalized coordinate systems by means of a fractional-step method whose primitive variable formulation uses as dependent variables, in place of the Cartesian components of the velocity: (1) pressure (defined at the center of the computational cell), and (2) volume fluxes across the faces of the cells. The momentum equations are solved by means of an approximate factorization method. A novel 'ZEBRA' scheme incorporating four-color ordering efficiently solves the Poisson equation. Illustrative two- and three-dimensional laminar flow test cases are computed and evaluated relative to extant numerical and experimental results, and good agreement is obtained.

Numerical Solutions of the Incompressible Navier-Stokes Equations with Boundary Conditions Switching

The boundary condition switching method for solving stream function-vorticity equations is applied to solve the incompressible Navier-Stokes equations in primitive variable form. The key idea of this method is that on boundaries, where both Dirichlet and Neumann boundary conditions are known, the Neumann boundary condition is satisfied naturally while the Dirichlet boundary condition is used to substitute the unknown boundary condition of the other variable, for instance, vorticity in stream function-vorticity equations and pressure in Navier-Stokes equations. When solving the Navier-Stokes equations with primitive variables for incompressible laminar flows, the Dirichlet boundary condition is the velocity boundary condition by giving zero velocity at wall boundaries. The Neumann boundary condition are given by the continuity condition. In the methods solving Navier-Stokes equations with primitive variables, the velocity boundary conditions are implemented with the Neumann boundary condition satisfied by continuity equation. It is well known that these methods result in pressure oscillations using equal-order interpolation in finite element method (FEM) and using non-staggered grid in finite difference method (FDM). This difficulty could be removed by using the present method. Application of the method is tested by both FEM and FDM and the results show that the method does not result in any pressure oscillations. Thus, pressure solutions do not need to be filtered or smoothed.

Simulation of dynamic stall phenomenon using unsteady Navier-Stokes equations

A non-inertial body-fixed generalized coordinate frame is employed to develop an unsteady Navier-Stokes (NS) analysis for arbitrary maneuvering bodies. A fully implicit direct numerical simulation (DNS) methodology is implemented for the solution of the governing equations. A velocity-vorticity (disturbance stream function - vorticity in 2-D) formulation is shown to simplify both the theoretical and numerical analysis of maneuvering flight when compared with the primitive-variable formulation. The resulting method is highly vectorizable and currently achieves a computational index of 6 micro-seconds per time step per mesh point using a single processor on the CRAY Y-MP 8 864 at the Ohio Supercomputer Center. Vectorization has been achieved satisfactorily, with the code currently performing in the range of 130–180 Mflops. The rapid pitch-up maneuver of a NACA 0015 airfoil is wxamined at Re=1000 and Re=10 000, with particular emphasis being placed on the development and evolution of the dynamic stall vortex.