Primitive Variable Formulation Research Articles

Upwind Scheme for Acoustic Disturbances Generated by Low-Speed Flows

Computation of acoustic disturbances generated by unsteady, low-speed flows, such as flows including vortices and shear layers, can be obtained by a recently proposed two-step method. This method requires a hydrodynamic field solution and obtains the acoustic field from the perturbed, inviscid, compressible flow equations. A numerical method for the solution of the equations governing the acoustic field is presented. The primitive variable form of the governing equations is used for the numerical solution. Time integration is performed with a fourth-order, Runge-Kutta method. Discretization of the primitive variables space derivatives is obtained with a high-order, upwind-biased numerical scheme. Upwinding of these convective fluxes is performed according to the eigenvalue sign of the coefficient matrices. Nonreflecting boundary conditions are applied to properly convect outgoing waves away from the computational domain. Solutions are obtained for the acoustic field generated by a pair of corotating point vortices. Computed results are compared with the existing analytic solution for the sound field.

Numerical study of three‐dimensional incompressible thermal flows in complex geometries

In part I uses an iterative point successive over‐relaxation (PSOR) finite difference scheme to solve the coupled unsteady Navier‐Stokes and energy equations for incompressible, viscous and laminar flows in their primitive variable form. Presents the details concerning the derivation of the solution scheme, as well as details on its computer implementation. For validation purposes, includes the results of the two‐dimensional and three‐dimensional benchmark problem of natural convection in a cavity with differentially heated vertical walls. Benchmark computations have been performed for a Prandtl number of 0.71, and different values of the Rayleigh number ranging between 103 and 106 depending on the problem. By comparison with other approaches in the literature, the scheme has been found to be accurate even for large Rayleigh numbers.

TRANSIENT SIMULATION OF INTERNAL SEPARATED FLOWS USING AN INTELLIGENT HIGHER-ORDER SPATIAL DISCRETIZATION SCHEME

SUMMARY This paper summarizes the method-of-lines (MOL) solution of the Navier‐Stokes equations for an impulsively started incompressible laminar flow in a circular pipe with a sudden expansion. An intelligent higher-order spatial discretization scheme, which chooses upwind or downwind discretization in a zone-of-dependence manner when flow reversal occurs, was developed for separated flows. Stability characteristics of a linear advective‐diffusive equation were examined to depict the necessity of such a scheme in the case of flow reversals. The proposed code was applied to predict the time development of an impulsively started flow in a pipe with a sudden expansion. Predictions were found to show the expected trends for both unsteady and steady states. This paper demonstrates the ease with which the Navier‐Stokes equations can be solved in an accurate manner using sophisticated numerical algorithms for the solution of ordinary differential equations (ODEs). Solutions of the Navier‐Stokes equations in primitive variables formulation by using the MOL and intelligent higher-order spatial discretization scheme are not available to date. 1997 by John Wiley & Sons, Ltd.

Spectral analysis of parabolized stability equations

In recent years, the parabolized stability equations (PSE) approach has gained popularity for transition studies in boundary-layer flows. In this paper, the physical origin of ill-posedness of PSE in the primitive-variable form is studied by spectral analysis. In subsonic flows, a continuous spectrum representing the upstream-propagating acoustic waves is shown to be responsible for the ill-posedness of PSE. In supersonic flows, acoustic waves no longer contribute to ill-posedness despite the existence of a subsonic region within the boundary layer. In this case, ill-posedness is caused by some discrete modes with upstream influence. Semi-discretized models of PSE are similarly studied. It is shown that successful implementation of PSE to physical problems can only be achieved through manipulation of the spectra of the PSE operator. A generalized condition for the stable marching solution is devised.

Finite Difference Schemes for Incompressible Flows in the Velocity–Impulse Density Formulation

We consider finite difference schemes based on the impulse density variable. We show that the original velocity—impulse density formulation of Oseledets is marginally ill-posed for the inviscid flow, and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity—impulse density formulation. We present numerical evidence of this instability. We then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation. Finally we give a simplified velocity—impulse density formulation which is free of these complications and yet retains the nice features of the original velocity—impulse density formulation with regard to the treatment of boundary. We present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non-staggered grids.

Flow structure analysis around an oscillating circular cylinder at low KC number: A numerical study

Flow fields around an oscillating circular cylinder are studied by solving the incompressible Navier-Stokes equations in primitive variable formulation. A finite volume method with a pressure predictor-corrector scheme is used and the solution procedure is accelerated by a local time stepping technique. Numerical tests are carried out at the Keulegan-Carpenter number KC < 20 and Reynolds number Re < 4000. For the symmetrical flow (at either very low KC or low Re), the vorticity decaying effect is dominant in the flow field. At higher Re, the vorticity convection becomes stronger, eventually leading to the inception of the asymmetrical flow pattern. It was found that within this critical regime the artificial disturbances imposed to the flow field play an important role in determining the flow patterns: if the flow falls into the symmetrical category, it remains symmetrical even with the artificial disturbances; while if the flow falls into the unsymmetrical category, the artificial disturbances are necessary for the correct predictions. An inception boundary of the asymmetrical flow in the KC — Re plane can thus be defined through a systematic study of the response of the flow field to the small disturbances. For the asymmetrical flow (at higher KC and/or Re), several distinguished flow patterns are identified in the numerical simulations, including quasi-symmetrical flow, diagonal vortex pair shedding, transverse street and double vortex pairs shedding. Agreement between the present results and the flow visualization (particle tracing) is generally good. The forces acting on the cylinder are also predicted for both the symmetrical and the asymmetrical flows. The conventional drag and inertia coefficients are deduced and compared with other numerical and experimental results, also showing good agreement.

A vorticity-velocity variational formulation for the exterior stokes problem in weighted sobolev spaces

In this article, we present a mixed variational formulation for the exterior Stokes problem in terms of the velocity and vorticity variables in the two and three-dimensional cases. The existence and the uniqueness of solutions of the variational problem are proved, as well as the equivalence with the primitive variables formulation.

Simulation Of Wind-Flow Patterns Around Bridge Deck Sections

The finite-difference procedure is used to solve the non-conservative form of the two-dimensional, unsteady, incompressible, Navier-Stokes flow equations, in a body-fitted curvilinear coordinate system. An implicit formulation scheme is employed, in order to avoid excessive computation time, resulting from stringent numerical stability restrictions associated with explicit schemes. The finite-difference mesh is generated using an automatic grid generation technique, based on the boundary-fitted curvilinear coordinate system. Solving the primitive variable form of the Navier-Stokes equations directly yields the desired flow parameters: the two velocity components and the pressure values. The flow velocities are then used in a stream function equation to compute the flow streamlines. Application to the bridge deck section of the Chesapeake and Delaware canal bridge shows good prediction of the wind flow patterns around such a bluff body geometry.

The application of a finite volume multigrid method to three-dimensional flow problems in a highly viscous fluid with a variable viscosity

Abstract In this paper we discuss the application of a finite-volume multigrid method to solve three-dimensional thermally driven convection in a highly viscous, incompressible fluid with a variable viscosity. The conservation laws are solved in the primitive variable formulation, A second-order control volume method is used as discretization. Two schemes are used for time stepping, a second-order implicit-explicit scheme based on the Crank-Nicolson and Adams-Bashforth method, and a fully-implicit θ-method. The implicit system of nonlinear equations are solved using multigrid iteration with the SIMPLER method as smoother. In this paper, we describe the implemented multigrid method and investigate its efficiency and the robustness for different viscosity contrasts. Convergence tests showed that with a small modification of the SIMPLER method, the multigrid method exhibits a satisfactory convergence rate even for viscosity contrasts up to 109. Three cases of time-dependent thermally driven convection with v...

Benchmark results for internal forced convection through integral transformation

The boundary layer equations for steady incompressible laminar forced convection in the entrance region of a parallel plates channel are solved through the integral transform method, by adopting the streamfunction-only formulation, which was preferred over the more commonly used primitive variables formulation. This hybrid numerical-analytical approach allows for the automatic global error control in the final solution and is particularly useful in the validation of purely numerical schemes. A thorough convergence analysis is performed, providing benchmark results for the temperature fields and Nusselt numbers for different values of the Prandtl number.

SIMPLIFIED CONTROL-VOLUME FINITE-ELEMENT METHOD

Localized vector algebra treatment of nonorthogonality is applied to two-dimensional quadrilateral control volumes using Cartesian base vectors in a primitive variable formulation of the Navier-Stokes equations for steady incompressible laminar flow. With optional grid-aligned, locally analytic interpolation, a simplified control-volume finite-element scheme is presented. Discretization of source terms, determination of interface convection-diffusion fluxes, pressure correction factors, and geometric quantities are described briefly. Results of three test cases provide useful initial insights into the performance of the method. The conclusion is reached that a simple finite-volume-based approach to nonorthogonality has been achieved.

On the nature of PSE approximation

The recently developed method of parabolized stability equations (PSE) offers a fast and efficient way of analyzing the spatial growth of linear and nonlinear (convective) disturbances in shear layers. For incompressible flows, the governing equations may be represented either in primitive variables or by using other formulations obtained by eliminating the pressure gradient (e.g., vorticity-streamfunction formulation). On the other hand, for compressible flows, primitive variables offer a natural and the only choice. We show that primitive-variable formulation is not well-posed due to the ellipticity introduced by the\(\partial \hat p/\partial x\) term and the marching solution eventually blows up for a sufficiently small step size. However, it is shown that this difficulty can be overcome if the minimum step size is greater than the inverse of the real part of the streamwise wave number, αr. An alternative is to drop the\(\partial \hat p/\partial x\) term, in which case the residual ellipticity is of no consequence for marching computations with much smaller step sizes. However, the ellipticity cannot be completely removed. Results obtained with streamfunction and vorticity-velocity formulations also show that the numerical difficulties arise for a sufficiently small marching step size. This step-size restriction can be overcome by dropping thedα/dx term from the governing equations. The effect of this term on solution accuracy is negligible for Blasius flow but not so for rotating-disk flow.

CALCULATION OF THE TIMING OF VORTEX FORMATION FROM AN OSCILLATING CYLINDER

Vortex shedding from a transversely oscillating circular cylinder in a uniform flow is studied by numerical solutions of the two-dimensional unsteady Navier-Stokes equations with a primitive-variable formulation. As the frequency of excitation of the cylinder is increased relative to the inherent vortex formation frequency, the initially formed concentration of vorticity moves closer to the cylinder until a limiting position is reached; at this point, the vorticity concentration abruptly switches to the opposite side of the cylinder. This process induces distinct changes of the topology of the corresponding streamline patterns. In addition, the influence of this vorticity-switching on forces acting on the cylinder is also investigated.

A NUMERICAL STUDY OF THE FLOW OVER ELLIPSOIDAL OBJECTS INSIDE A CYLINDRICAL TUBE

A numerical scheme is developed to obtain the flow field around one, two and five ellipsoidal objects inside a cylindrical tube. The scheme uses the Galerkin finite element technique and the primitive variable(u–v–p) formulation. The two-dimensional incompressible Navier–Stokes equations are solved numerically by using the direct mixed interpolation method. A Picard iteration scheme is used for the solution of the resulting system of non-linear algebraic equations. The computer code is verified by checking with known analytical solutions for the flow past a sphere. Results for the shear stress distributions along the ellipsoids, forces and drag coefficients are obtained for different geometric ratios and Reynolds numbers. Some of the intermediate computational results on the velocity fields developed are also reported.

Flow in a Pipe with a Ring-Type Obstacle

Turbulent calculations have been carried out to investigate flows in a circular pipe with a ring-type obstacle attached to the wall for Reynolds numbers from 102 to 105. The numerical procedure is based on the solution of the primitive variable formulation of the Reynolds-averaged Navier-Stokes governing equations and the k — e turbulence model in axisymmetric co-ordinate system and with a non-staggered grid. The obstacle opening ratio (d/D) is 0.5, and the obstacle thickness ratio (h/D) is 0.13. The numerical results reveal the effect of the Reynolds number on the flow fields. With the Reynolds number varying from 102 to 105, the reattachment length (Zr/D) increases to a maximum of 3.1 at Re = 5 × 102 and then decreases gradually to a value of 2.1. The velocity profiles of the fully developed flow change from parabolic to power-law curves. The non-dimensional maximum turbulent shear stress (τmax varies in a range from 0.2 to 0.4 and the pressure loss (Ploss ) across the obstacle varies between 8.0 and 11...

A SEGREGATED SOLUTION ALGORITHM FOR INCOMPRESSIBLE FLOWS IN GENERAL CO-ORDINATES

SUMMARY To analyse an incompressible Navier4tokes flow problem in a boundary-fitted curvilinear co-odinatc system is definitely not a trivial task. In the primitive variable formulation, choices between wrhg variables and their storage points have to be made judiciously. The present work engages Contravariant velocity components and scalar pressure which stagger each other in the mesh to prevent even-odd pressure oscillations from emerging. Now that smoothness of the pnssure field is attainable, the remaining task is to ensure a disuete divergence-fire velocity field for an incompmsible flow simulation. Aside hm the flux discretiZations, the indqensable metric tensors, Jacobian and Chistoffel symbols in the transformed equations should be approximated with care. The guidmg idea is to get the property of geometric identity Pertaining to these grid-sensitive discretizatons. In addition, how to maintain the revertible one-toone equivalence at the discrete level between primitive and contravariant velocities is another theme in the present staggered formulation. A semi-implicit segregated solution algorithm felicitous for a large-scale flow simulation was utilized to solve the entire set of basic equations iteratively. Also of note is that the present segregated solution algorithm has the Virtue of reqUiring no userspecified relaxation pammeters for speedmg up the satisfaction of incompmsibility in an optimal sense. hu benchmark problems, including an analytic problem, were investigated to justie the capability of the present formulation in handing problems with complex geometry. The test cases considered and the results obtained herein make a useful contribution in solving problems subsuming cells with arbitrary shapcs in a boundary-fitted grid system.

INCOMPRESSIBLE COMPUTATIONAL FLUID DYNAMICS AND THE CONTINUITY CONSTRAINT METHOD FOR THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

Abstract As the field of computational fluid dynamics (CFD;) continues to mature, algorithms are required to exploit the most recent advances in approximation theory, numerical mathematics, computing architectures, and hardware. Meeting this requirement is particularly challenging in incompressible fluid mechanics, where primitive-variable CFD formulations that are robust, while also accurate and efficient in three dimensions, remain an elusive goal. This monograph asserts that one key to accomplishing this goal is recognition of the dual role assumed by the pressure, i.e., a mechanism for instantaneously enforcing conservation of mass and a force in the mechanical balance law for conservation of momentum. Proving this assertion has motivated the development of a new, primitive-variable, incompressible, CFD algorithm called the continuity constraint method (CCM;). The theoretical basis for the CCM consists of a finite-element spatial semidiscretization of a Galerkin weak statement, equal-order interpolati...

Investigation of three-dimensional thermocapillary convection in a cubic container by a multi-grid method

Three-dimensional thermocapillary convection of a Boussinesq fluid in an open cubic container is considered in the absence of buoyancy. The steady motion is computed with a modified Full-Approximation Scheme, Full Multi-Grid (FAS-FMG) method based on a finite difference, primitive variable formulation. Solutions on a staggered grid of 32 × 32 × 32 points for each unknown are obtained by a line coupled, block implicit relaxation method for the Navier-Stokes equation, and an alternating zebra-relaxation method for the energy equation. Three-dimensional (3D) flow structures and temperature fields are presented for Reynolds numbers up to 5 × 10 4 and Prandtl numbers Pr = 0.01, 1, 10 and 50. Apart from the main vortex known from 2D calculations pronounced secondary vortices appear. The mechanisms driving this secondary flow are discussed in terms of well-known properties of the main vortex.

Local and Overall Condensation Heat Transfer Behavior in Horizontal Tube Bundles

A numerical investigation has been carried out to determine the effects of condensation inundation, noncondensable gases, and cooling water velocity upon the local and overall condensation heat transfer in horizontal tube bundles. Five inundation correction factors are tested against the experimental data. Comparisons between the predicted results and the experimental data are made in terms of total condensation rates and local heat flux distributions. The effects of noncondensable gases and cooling water velocity are assessed by comparing local heat flux distributions, total condensation rates, and average steam-side and average overall heat transfer coefficients under different inlet air mass fraction and inlet cooling water velocity conditions. A quasi-three-dimensional approach is used in the numerical analysis, in which three-dimensional effects due to the rise of the cooling water temperature along the cooling water flow direction have been included in the simulations. The governing equations are solved in primitive variable form using a semiimplicit consistent control-volume formulation in which a segregated pressure correction-linked algorithm is employed. The modeling is carried out based on porous media concepts using flow, heat, and mass transfer resistances.

Numerical Solutions of Incompressible Navier-Stokes Equations Using Non-staggered Finite Difference Method.

A new method for solving the primitive variable formulation of the incompressible Navier-Stokes equations on the non-staggered grid is presented. The spatial derivatives of the momentum equations are discretized by the second-order centered finite difference approximation on the cell interfaces, and the fluxes on the non-staggered grid are estimated from the simple average of the cell interfaces. The resulting system of ordinary differential equations in time is integrated by the forward Euler scheme. The Poisson equation for the pressure is discretized in the same manner as the momentum equations and solved by the successive-overrelaxation (SOR) method. The present method satisfies the divergence-free condition, i.e., the continuity equation, at each time step. As the test problem, square driven cavity flows are considered. The results obtained by the present method are compared with those of the marker-and-cell (MAC) method. The results show that the present method gives solutions comparable to the MAC solutions, and that the allowable time step is larger than in the MAC method.