Navier-Stokes Equations In Primitive Variables Research Articles

Recent Results from Analysis of Flow Structures and Energy Modes Induced by Viscous Wave around a Surface-Piercing Cylinder

Due to its relevance in ocean engineering, the subject of the flow field generated by water waves around a vertical circular cylinder piercing the free surface has recently started to be considered by several research groups. In particular, we studied this problem starting from the velocity-potential framework, then the implementation of the numerical solution of the Euler equations in their velocity-pressure formulation, and finally the performance of the integration of the Navier-Stokes equations in primitive variables. We also developed and applied methods of extraction of the flow coherent structures and most energetic modes. In this work, we present some new results of our research directed, in particular, toward the clarification of the main nonintuitive character of the phenomenon of interaction between a wave and a surface-piercing cylinder, namely, the fact that the wave exerts its maximum force and exhibits its maximum run-up on the cylindrical obstacle at different instants. The understanding of this phenomenon becomes of crucial importance in the perspective of governing the entity of the wave run-up on the obstacle by means of wave-flow-control techniques.

Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.

The Field of Flow Structures Generated by a Wave of Viscous Fluid Around Vertical Circular Cylinder Piercing the Free Surface

The diffraction of water waves induced by a large-diameter, surface-piercing, vertical circular cylinder is studied numerically. The Navier-Stokes equations in primitive variables are considered for the simulation of a given wave case, and the technique is followed of the Direct Numerical Simulation (DNS). The criterion of the imaginary part of the complex-eigenvalue pair of the velocity-gradient tensor for the extraction of the flow vortical structures is applied to the computed fields, so unveiling a complex configuration of structures at the free surface, and at the cylinder external walls. The most energetic modes of the flow are further extracted from the DNS-simulated fields by using the Karhunen-Loève decomposition (KL). A “reduced” velocity field is reconstructed using the first three most energetic eigenfunctions of the decomposition, and its evolution is followed through a sequence of time steps.

Implicit Solutions of Incompressible Navier-Stokes Equations Using the Pressure Gradient Method

*† A modified pressure gradient method is developed for solving the incompressible twoand three-dimensional Navier-Stokes equations in primitive variables. Velocity and pressure gradient vectors, rather than the velocity and pressure, are considered as dependent variables. This choice of the dependent variables is consistent with the flow physics as the velocity field of incompressible flows is not dependent on the pressure, but rather the pressure gradient. This introduces one and two additional dependent variables for two- and three-dimensions respectively. To close the system of equations, additional equations are obtained from the identity that the curl of the pressure gradient is zero. This choice of dependent variables also leads to Dirichlet boundary conditions for the pressure gradient, which improves numerical stability and convergence rates. The modified pressure gradient method includes a pressure gradient averaging term in the momentum equations to stabilize the solution. This term is extended here to general curvilinear coordinates. The system of equations is discritized using a conservative finite volume formulation. Time integration is achieved with a fully implicit scheme, including implicit boundary conditions. The implicit scheme guarantees that continuity is satisfied to machine zero. Numerical results compare well with solutions available in the literature for the cavity flow problem and flow about a circular cylinder at Reynolds number 40.

Comparative Study of the Accuracy of the Fundamental Mesh Structures for the Numerical Solution of Incompressible Navier-Stokes Equations in the Two-Dimensional Cavity Problem

The accuracies of the staggered, semistaggered, vertex collocated, and cell-center collocated meshes are compared for the incompressible Navier-Stokes equations in primitive variables using the two-dimensional lid-driven cavity test problem in both the traditional form with uniform lid velocity and in the regularized form without corner discontinuities. Central differencing is used in all discretizations. The momentum equations are integrated explicitly after the solution of a Poisson pressure equation that enforces mass conservation. Richardson extrapolation is employed to estimate the correct solutions in cases without precise reference solutions. For both forms of the problem, the staggered and the cell-center collocated meshes are shown to produce comparably accurate results, while the vertex collocated mesh shows poorer accuracy. The almost unknown semistaggered mesh, which is too sensitive to the velocity discontinuity in the traditional cavity problem, appears as the most accurate mesh in the regularized cavity problem.

Comparative Study of UNIFAES and other Finite-Volume Schemes for the Discretization of Advective and Viscous Fluxes in Incompressible Navier-Stokes Equations, Using Various Mesh Structures

The performance of the unified finite approaches exponential-type scheme (UNIFAES) for advective and diffusive transport equations, for the incompressible Navier-Stokes equations in primitive variables, is evaluated by comparison with the central differencing and the exponential scheme. Staggered, semistaggered, vertex collocated, and cell-center collocated meshes are considered. The Cartesian two-dimensional lid-driven cavity test problem is employed both in its traditional form with uniform lid velocity and in the regularized form without corner discontinuities. Richardson extrapolation is employed to produce reference results in the cases without prior precise reference solutions. The UNIFAES shows accuracy generally superior to the other schemes and stability even at the highest Reynolds numbers employed. This article also completes a previous comparative study of the performance of the mesh structures, generalizing the results there observed for higher Reynolds numbers and for schemes other than central differencing.

Fully implicit method for simulation of flows with interfaces

This paper is concerned with simulation of flows with interface between two incompressible and immiscible fluids on a fixed grid using what is called the single fluid formalism. This formalism views flow of two fluids as that of a single fluid whose density and viscosity change abruptly at the interface. The location of the interface is apriori not known but is to be discovered as part of the solution. Problems of these types are typically solved using two main approaches: a) Interface Capturing and b) Interface Tracking. The present contribution is of a former variety. The governing Navier-Stokes equations in primitive variables are solved on collocated grids with a specially derived smoothing pressure correction that satisfies volume conservation whereas superficial density is determined from mass conservation equation. It is shown that the latter equation can be cast in the form of a well known volume fraction equation. The convective terms are represented by a TVD scheme. The overall methodology is tested against few problems for which experimental data and/or numerical predictions are available.

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.

Calculation of impulsively started incompressible viscous flows

AbstractThe paper's focus is the calculation of unsteady incompressible 2D flows past airfoils. In the framework of the primitive variable Navier–Stokes equations, the initial and boundary conditions must be assigned so as to be compatible, to assure the correct prediction of the flow evolution. This requirement, typical of all incompressible flows, viscous or inviscid, is often violated when modelling the flow past immersed bodies impulsively started from rest. Its fulfillment can however be restored by means of a procedure enforcing compatibility, consisting in a pre‐processing of the initial velocity field, here described in detail. Numerical solutions for an impulsively started multiple airfoil have been obtained using a finite element incremental projection method. The spatial discretization chosen for the velocity and pressure are of different order to satisfy the inf–sup condition and obtain a smooth pressure field. Results are provided to illustrate the effect of employing or not the compatibility procedure, and are found in good agreement with those obtained with a non‐primitive variable solver. In addition, we introduce a post‐processing procedure to evaluate an alternative pressure field which is found to be more accurate than the one resulting from the projection method. This is achieved by considering an appropriate ‘unsplit’ version of the momentum equation, where the velocity solution of the projection method is substituted. Copyright © 2004 John Wiley & Sons, Ltd.

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.

PRESSURE BOUNDARY CONDITIONS FOR A SEGREGATED APPROACH TO SOLVING INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

It has been well accepted that Diricklet and Neumann boundary conditions for Ike pressure Poisson equation give the same solution. The purpose of this article is to reveal that the above statement is computationally acceptable but is not theoretically correct. Analytic proof as well as computational evidences are presented through examples in support of our observation. In this work we address that the mixed finite-element formulation for solving incompressible Navier-Stokes equations in primitive variables is equivalent to the formulation that involves solving the pressure Poisson equation, subject to Neumann boundary conditions, Uerativety with the momentum equations provided the velocity field is classified as having divergence-free and conservative properties.

Résolution des équations de Navier-Stokes dans la formulation en variables primitives par approximation diffuse

A diffuse approximation method for solving Navier-Stokes equations in primitive variables is proposed. The results of a numerical example show the accuracy and efficiency of this approach.

A NUMERICAL PROCEDURE FOR PREDICTING MULTIPLE SOLUTIONS OF A SPHERICAL TAYLOR–COUETTE FLOW

SUMMARY A new numerical procedure for predicting multiple solutions of Taylor vortices in a spherical gap is presented. The steady incompressible Navier-Stokes equations in primitive variables are solved by a finitedifference method using a matrix preconditioning technique. Routes leading to multiple flow states are designed heuristically by imposing symmetric propemes. Both symmetric and asymmetric solutions can be predicted in a deterministic way. The current procedure gives very fast convergence rate to the desired flow modes. This procedure provides an alternative way of finding all possible stable steady axisymmetric flow modes.

A NUMERICAL PROCEDURE FOR PREDICTING MULTIPLE SOLUTIONS OF A SPHERICAL TAYLOR-COUETTE FLOW

SUMMARY A new numerical procedure for predicting multiple solutions of Taylor vortices in a spherical gap is presented. The steady incompressible Navier-Stokes equations in primitive variables are solved by a finitedifference method using a matrix preconditioning technique. Routes leading to multiple flow states are designed heuristically by imposing symmetric propemes. Both symmetric and asymmetric solutions can be predicted in a deterministic way. The current procedure gives very fast convergence rate to the desired flow modes. This procedure provides an alternative way of finding all possible stable steady axisymmetric flow modes.

An Analysis of the Fractional Step Method

The fractional step method for solving the incompressible Navier-Stokes equations in primitive variables is analyzed as a block LU decomposition. In this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved. In addition, it is shown that poor temporal accuracy (first-order) is not due to boundary conditions, but due to the method itself. A generalized block LU decomposition that overcomes this difficulty is presented, allowing arbitrarily high temporal order of accuracy. The generalized decomposition is shown to be useful for a wide range of problems including steady problems. Technical issues, such as stability and the appropriate pressure update scheme, are also addressed. Numerical simulations of the unsteady, incompressible Navier-Stokes equations in a square domain confirm the theoretical results.

Petrov-Galerkin solutions of the incompressible Navier-Stokes equations in primitive variables with adaptive remeshing

The solution of the incompressible Navier-Stokes equations in primitive variables is addressed in this paper. A Petrov—Galerkin formulation which automatically introduces Streamline upwinding and allows equal order interpolation for all flow variables is presented. The solution algorithm is segregated, involving a series of updatings of the velocity and pressure fields. Error estimation and adaptive remeshing strategies for both steady and unsteady problems are demonstrated in some representative numerical examples.

Finite element solution of the incompressible Navier-Stokes equations by a Helmholtz velocity decomposition

Finite element solution methods for the incompressible Navier-Stokes equations in primitive variables form are presented. To provide the necessary coupling and enhance stability, a dissipation in the form of a pressure Laplacian is introduced into the continuity equation. The recasting of the problem in terms of pressure and an auxiliary velocity demonstrates how the error introduced by the pressure dissipation can be totally eliminated while retaining its stabilizing properties. The method can also be formally interpreted as a Helmholtz decomposition of the velocity vector. The governing equations are discretized by a Galerkin weighted residual method and, because of the modification to the continuity equation, equal interpolations for all the unknowns are permitted. Newton linearization is used and at each iteration the linear algebraic system is solved by a direct solver. Convergence of the algorithm is shown to be very rapid. Results are presented for two-dimensional flows in various geometries.

Coupled space-marching method for the Navier-Stokes equations for subsonic flows

An implicit space-marching finite-difference procedure is described for solving the compressible form of the steady, two-dimensional Navier-Stokes equations in body-fitted curvilinear coordinates. The coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space-marching procedure. A new pressure correction method has been developed that significantly accelerates convergence of the iterative process. The scheme is used to compute incompressible flows by taking the low Mach number limit of the compressible formulation. Computed results are compared with other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and external flow over a cylinder. NUMBER of different methods have been developed to solve the Navier-Stokes equations in primitive variables for steady, subsonic flows. Many of these algorithms are described in Anderson et al. 1 The present methods are not entirely adequate for all problems, and advanced techniques continue to be developed. New procedures and recent im- provements to existing methods are briefly reviewed in the following paragraphs. The most frequently used primitive variable algorithms for incompressible flows solve the momentum equations for the velocity components in an uncoupled (segregated) manner, holding pressure fixed. Although different in detail, the segre- gated solution schemes (e.g., Refs. 2-4) use the continuity equation indirectly in the formulation of a separate Poisson equation that is solved for the pressure. Since the velocity components and pressure are computed from different algo- rithms, rather than in a coupled manner, convergence is slowed. Furthermore, the solution of the elliptic Poisson equa- tion consumes a large part of the total computaion time for each global iteration. Van Doormaal et al.5 recently have eval- uated improved algorithms for solving the pressure Poisson equation, and Rhie4 has applied a multigrid method to acceler- ate the solution of the momentum equations and the pressure equation. Several strategies have been advanced for solving the cou- pled momentum and continuity equations for the primitive variables, including the pressure. The schemes all share the advantage that a separate procedure for imposing the continu- ity constraint and determining the pressure is not required. The choice of variables and the algebraic approach used to solve the system of nonlinear equations distinguish the direct, time- marching, and space-marching algorithms. Vanka and Leaf6 and recently Patankar et al. 7 have com- pared solutions for two-dimensional flows obtained by direct and segregated methods. A large sparse matrix solver is used to invert a system of equations spanning the entire flow domain.

Finite element method for computing turbulent propeller flow

A numerical procedure based on the primitive variable Navier-Stokes equations is applied to the simulation of the three-dimensional axisymmetric flow near a propeller in a uniform flow. The time-averaged Navier-Stokes equations are solved using both a mixed and a penalty function finite element method. A new model is developed to compute axisymmetric viscous throughflow past propellers. This is achieved by forcing the flow to be tangent to the mean stream surface between the blades. This condition is enforced by introducing a Lagrange multiplier that can be interpreted as the force applied by the blades on the fluid. Turbulence is modelled by a simple mixing length equation. Detailed comparison with wind tunnel measurements shows good prediction of velocity and pressure. The present approach represents a first step toward the development of a viscous quasi-three-dimensional capability.

Coupled fully implicit solution procedure for the steady incompressible Navier-Stokes equations

This paper presents a new fully implicit procedure for the solution of the steady incompressible Navier-Stokes equations in primitive variables. The momentum equations are coupled with a Poisson-type equation for the pressure and solved using the Beam and Warming approximate factorization method. The present formulation does not require the iterative solution of the pressure equation at each time step. Thus, the major drawback of the pressure-Poisson approach, which made it prohibitively expensive for complex three-dimensional applications, is eliminated. Numerical solutions for the problem of the two-dimensional driven cavity are obtained using a non-staggered grid at Re = 100, 400, and 1000. All the computed results are obtained without any artificial dissipation. This feature of the present procedure demonstrates its excellent convergence and stability characteristics. Those characteristics result from the coupling of the pressure equation, which is elliptic in space, with the momentum equations.