COMPLETE PRESSURE CORRECTION ALGORITHM FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID

Abstract

When Navier-Stokes equations for incompressible flow are solved on a nonstaggered grid, the problem of checkerboard prediction of pressure is encountered. So far, this problem has been cured either by evaluating the cell face velocities by the momentum interpolated principle III or by evaluating an effective pressure gradient in the nodal momentum equations [2] In this article it is shown that not only are these practices unnecessary, they can lead to spurious results when the true pressure variation departs considerably from linearity. What is required instead is afresh derivation of the pressure correction equation appropriate for a nonstaggered grid. The pressure correction determined from this equation comprises two components: a mass-conserving component and a smoothing component. The former corresponds to the pressure correction predicted by a staggered grid procedure, whereas the latter simply accounts for the difference between the point value of the pressure and the cell-averaged value of the pressure. The new pressure correction equation facilitates ( in a significant way) computer coding of programs written for three-dimensional geometries employing body-fitted curvilinear coordinate grids.     

A CALCULATION PROCEDURE FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON CURVILINEAR NON STAGGERED GRIDS

A recently proposed pressure-correction algorithm for solution of incompressible Navier-Stokes equations on nonstaggered grids introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure (9). The algorithm was derived for equations in Cartesian coordinates. In this article, the algorithm is extended to solution of Navier-Stokes equations in general curvilinear coordinates. By way of application, two cavity flow problems and two internal flow problems are solved. Comparisons with benchmark solutions or experimental data and (or) previous solutions employing staggered grids are made to validate the calculation procedure.     

NUMERICAL METHOD FOR THE PREDICTION OF INCOMPRESSIBLE FLOW AND HEAT TRANSFER IN DOMAINS WITH SPECIFIED PRESSURE BOUNDARY CONDITIONS

A variety of engineering applications involve incompressible flows in devices for which boundary pressures are known. The purpose of this article is to present a mathematical formulation and a computational method for the prediction of incompressible flow in domains with specified pressure boundaries. The computational treatment of specified pressure boundaries in complex geometries is presented within the framework of a nonstaggered technique based on curvilinear boundary-fitted grids. The construction of the discretization equations for unknown velocities on specified pressure boundaries and the solution of the discretization equations using the SIMPLE algorithm are discussed. The proposed method is applied for predicting incompressible forced flows in branched ducts and in buoyancy-driven flows. These examples illustrate the utility of the proposed method in predicting incompressible flows with specified boundary pressures encountered in practical applications.     

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 interpolation for all state variables, a 6-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor weak statement;(TWS) formulation for dispersion error control. This monograph presents: (I) the formulation of the unsteady evolution of the divergence error, (2) an investigation of the role of nonsmoothness in the discretized continuity-constraint function, (3;) the development of a uniformly H’ Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, and(4;) a derivation of physically and numerically well-posed boundary conditions. In contrast to the general family of ‘pressure-relaxation’ incompressible CFD algorithms, the CCM does not use the pressure as merely a mathematical device to constrain the velocity distribution to conserve mass. Rather, the mathematically smooth and physically motivated genuine pressure is an underlying replacement for the nonsmooth continuity-constraint function to control inherent dispersive-error mechanisms. The genuine pressure is calculated by the diagnostic pressure Poisson equation, evaluated using the verified solenoidal velocity field. This new separation of tasks also produces a genuinely clear view of the totally distinct boundary conditions required for the continuity constraint function and genuine pressure.     

A NEW ANALYTIC SOLUTION OF THE NAVIER-STOKES EQUATIONS FOR MICROCHANNEL FLOWS

In this article we present a new analytic solution of the Navier-Stokes equations for microchannel flows. The solution is based on the concept of the continuum approach using the Chapman-Enskog method, but built upon the proposal to introduce a hyperbolic tangent function of Kn number in the power series of the distribution function and slip boundary condition. The physics behind the mathematical modification are discussed. With the slip boundary condition accurate to O(tanh(Kn)), the solution of the Navier-Stokes equations is extended successfully to the transition flow regime. The analytic solutions are compared with results of DSMC in both slip flow and transition flow regimes. Satisfactory agreements on the velocity profiles and pressure distributions have been achieved. The extension of the upper Knudsen number limits of continuum approach is significant in molecular gas dynamics.     

A NEW APPROACH TO NONLINEAR EQUATIONS FOR A SHALLOW UNSYMMETRICAL HEATED SANDWICH SHELL OF DOUBLE CURVATURE

A completely new set of uncoupled differential equations is derived that governs the behavior of an elastic, unsymmetrical, doubly curved, heated sandwich shell whose face sheets are of unequal thickness and of different materials. The equations include nonlinear effects and can be used for the analysis of the movable as well as immovable edge conditions. However, a restriction that the element's radii of curvature be large compared with the overall thickness of the sandwich is imposed. Reducing the present set of equations to the symmetrical case, the equations can be easily identified with those of Grigotyuk. Finally, the numerical results of a rectangular unsymmetrical sandwich cylindrical shell under thermal loading have been computed and compared with other available results.     

SOLUTION OF THE STEADY NAVIER-STOKES EQUATIONS BY THE PRESSURE CORRECTION METHOD IN THE ALE GRID

The arbitrary Lagrangian-Eulerian (ALE) grid system possesses some computational merits over the fully staggered and collocated grids in solving the fluid flow problem with primitive variables. However, severe pressure wiggle problems will occur when a standard centered difference scheme is adopted in this grid. In this article, a split velocity concept is introduced to eliminate the checkerboard pressure field in the ALE grid. In this scheme, the cell face velocities are corrected to account for the true pressure gradients by a simple algebraic arrangement of adjacent velocities. The consequence of cell face velocities is analogous to that from momentum interpolation in the collocated grid; however, the split velocities do not produce deterioration of the total mass flux in the original velocity field. Computed results are compared with other numerical predictions and available experimental data for developing channel flow, lip-driven cavity flow, sudden expansion flow, and flow over a backward-facing step. It is shown that this method successfully simulates complex flow situations with different flow characteristics. Finally, comparison of convergent rate with other grid systems is also discussed in this article.     

A FINITE DIFFERENCE METHOD FOR SOLVING 3-D HEAT TRANSPORT EQUATIONS IN A DOUBLE-LAYERED THIN FILM WITH MICROSCALE THICKNESS AND NONLINEAR INTERFACIAL CONDITIONS

We develop a finite difference method for solving 3-D heat transport equations in a double-layered thin film with microscale thickness and nonlinear interfacial conditions. The scheme is solved by using a preconditioned Richardson iteration, so that only two tridiagonal linear systems with nonlinear interfacial conditions are solved at each iteration. Applying a parallel Gaussian elimination coupled with Newton's iteration to solve these two linear systems with nonlinear interfacial conditions, we develop a domain decomposition algorithm for thermal analysis of the double-layered thin film. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained.     

A DOMAIN DECOMPOSITION METHOD FOR SOLVING 3-D HEAT TRANSPORT EQUATIONS IN A DOUBLE-LAYERED CYLINDRICAL THIN FILM WITH SUBMICROSCALE THICKNESS AND NONLINEAR INTERFACIAL CONDITIONS

A domain decomposition method is developed for solving 3-D heat transport equations in a double-layered cylindrical thin film with submicroscale thickness and nonlinear interfacial conditions. The heat transport equations are discretized based on the hybrid finite elementfinite difference (FE-FD) method. The schemes are solved by using an iteration, so that only two-block tridiagonal linear systems with nonlinear interfacial conditions are solved. Finally, a parallel Gaussian elimination coupled with the Newton iteration is applied to solve these two systems with nonlinear interfacial conditions. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained.     

SOLUTIONS TO THE EXTENDED GRAETZ PROBLEM FOR A POWER-MODEL FLUID WITH VISCOUS DISSIPATION AND DIFFERENT ENTRANCE BOUNDARY CONDITIONS

The transport phenomena of the extended Graetz problem with three different entrance boundary conditions are discussed. The expansion coefficients of the solution corresponding to the different conditions play an important role in effecting the solution form. The solution, assuming that the entrance boundary conditions for both temperature and energy flux (TFBC) are continuous, is the same as that for the problem in which the downstream region was considered infinite. Among all the procedures used, the computational procedures for TFBC are the simplest. The TFBC condition is recommended for use in analyzing the problem. Results show that temperature profiles and local Nussell number are influenced by Piclet number and different entrance boundary conditions. In addition, it is also shown that the asymptotic Nussell numbers for the three different conditions are the same.     

A COMPARISON OF OUTFLOW BOUNDARY CONDITIONS FOR THE MULTIDOMAIN STAGGERED-GRID SPECTRAL METHOD

This article investigates four outflow boundary conditions (BCs) for use with the multidomain spectral method (D. A. Kopriva, J. Comput. Phys. , vol. 244, pp. 142-158, 1998). The objective is to find low-reflective, stable BCs suitable for use in direct numerical simulation (DNS) of turbulent flows in complex geometries. This investigation is focused on outflow BCs, which are critical in DNS. The methods were tested on a two-dimensional Poissueille flow, a vortex propagating through an outflow boundary, and a transitional flow over a rectangular cylinder. The best results were found by specifying the exit pressure and using a linear Riemann solver to compute the velocity and density. Temporal damping in the subdomains bordering the outflow boundary improved the results in specific cases.     

EVOLUTION TO CHAOTIC NATURAL CONVECTION IN A RECTANGULAR ENCLOSURE WITH MIXED BOUNDARY CONDITIONS

The purpose of this numerical study is to analyze the characteristics of transition from laminar to chaotic natural convection in a fluid-filled two-dimensional, unity aspect ratio rectangular cavity with mixed thermal boundary conditions. For a medium Prandtl number fluid ( Pr ) the numerical solution of the two-dimensional Navier-Stokes momentum and energy equation with Bousinessq approximation, it is found that there are finite Rayleigh numbers Ral Ra2, and Ra3 for the onset of single-, double-, and multiple-frequency oscillatory motion at different spatial locations in the enclosure. As Ra increases, the flow exhibits a change from steady convection to periodic to quasi-periodic flow, while no period doubting is observed. The onset of strong chaos appears when Ra = 57,000 Rac. This system does not revert to steady state convection for Rac as high as 285,000. As Ra increases, various measures of chaos, such as power spectrum, Poincare sections, phase portrait, and time series of various dynamical variable signals, all show an increasing degree of characteristics of chaos,     

A THREE-LEVEL FINITE-DIFFERENCE SCHEME FOR SOLVING MICRO HEAT TRANSPORT EQUATIONS WITH TEMPERATURE-DEPENDENT THERMAL PROPERTIES

Heat transport at the microscale is important for the processing of materials with a pulsed laser. In this study, we develop a three-level finite-difference scheme for solving micro heat transport equations with temperature-dependent thermal properties obtained based on the parabolic two-step model. It is shown by the discrete energy method that for constant thermal properties the scheme is unconditionally stable. Numerical results for thermal analysis of a gold film are obtained.     

HYGROTHERMAL STRESS IN A PLATE SUBJECTED TO ANTISYMMETRIC TIME-DEPENDENT MOISTURE AND TEMPERATURE BOUNDARY CONDITIONS

When the temperature and/or moisture at the surfaces of a composite change suddenly, stresses will arise in the composite owing to the nonuniform diffusion of heat and moisture. Recent investigations have shown that under certain conditions the classical uncoupled theory of diffusion can significantly underestimate the coefficient of diffusion. The coupling between heat and moisture is an inherent part of the diffusion process that cannot be neglected on intuitive grounds. This investigation is an inquiry into the influence of antisymmetric boundary conditions on the magnitude of the hygrothermal stresses in a plate made of T300/5208 epoxy material, commonly used in graphite fiber-reinforced composites. Both moisture and temperature boundary conditions are considered. Because of the nonlinear character of the coupled equations, a finite-difference scheme is adopted. Numerical results involving time-dependent moisture, temperature, and stress distributions in the plate are displayed graphically; they show that the stresses derived from the coupled theory differ appreciably from the uncoupled results, both qualitatively and quantitatively. The hygrothermal stresses with coupling taken into account acquire an oscillatory character when the temperature on the plate is raised suddenly; this factor could contribute to material damage. In addition, antisymmetric boundary conditions can either raise or lower the stress levels, depending on time and the transient nature of the applied temperature.     

A PRESSURE CORRECTION SCHEME USING COUPLED MOMENTUM EQUATIONS

Abstract

This article presents a method for the numerical solution of the incompressible 2D Navier-Stokes equations, based on the coupled solution of the momentum equations and a fully compatible pressure correction equation. With in each iteration, the linearized momentum equations are simultaneously, though inexactly, solved through a two-step, noniterative scheme. Their solution employs the appropriate factorization of the diagonal coefficient matrices into upper and lower triangular ones. The continuity equation is satisfied by means of the SIMPLE concept in a particular form that arises from the coupled solution of the momentum equations. The algorithm is more efficient in terms of convergence rates when compared to a segregated algorithm, given that identical discretization schemes are used.     

水域边界上存在点源流动的相似性解的一种求解方法

从流体力学复位势出发，对水域边界上存在点源流动问题给上似变量及相似性解，导出了相似函数所满足的常微分方程及边界条件，该方法物理概念清晰，结果简明，对水利资源 的扩散问题有理论参考及工程应用意义 。     

NUMERICAL MODELLING OF MICROSCALE EFFECTS IN CONDUCTION FOR DIFFERENT THERMAL BOUNDARY CONDITIONS

Natural convection in a two-dimensional, rectangular enclosure with sinusoidal temperature profile on the upper wall and adiabatic conditions on the bottom and sidewalls is numerically investigated. The applied sinusoidal temperature is symmetric with respect to the midplane of the enclosure. Numerical calculations are produced for Rayleigh numbers in the range 10 2 to 10 8 , and results are presented in the form of streamlines, isotherm contours, and distributions of local Nusselt number. The circulation patterns are shown to increase in intensity, and their centers to move toward the upper wall corners with increasing Rayleigh number. As a result, the thermal boundary layer is confined near the upper wall regions. The values of the maximum and the minimum local Nusselt number at the upper wall are shown to increase with increasing Rayleigh number. Finally, an increase in the enclosure aspect ratio produces an analogous increase of the fluid circulation intensity.     

A HYBRID FINITE ELEMENT-FINITE DIFFERENCE METHOD FOR SOLVING THREE-DIMENSIONAL HEAT TRANSPORT EQUATIONS IN A DOUBLE-LAYERED THIN FILM WITH MICROSCALE THICKNESS

W e develop a finite element?finite difference method for solving three-dimensional heat transport equations in a double-layered thin film with microscale thickness. The implicit scheme is solved by using a preconditioned Richardson iteration, so that only two block tridiagonal linear systems with unknowns at the interface are solved for each iteration. W e then apply a parallel Gaussian elimination procedure to solve these two block tridiagonal linear systems and develop a domain decomposition algorithm for thermal analysis of the double-layered thin film. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained.     

APPLICATION OF THE PARTIAL ELIMINATION ALGORITHM FOR SOLVING THE COUPLED ENERGY EQUATIONS IN POROUS MEDIA

This article discusses the solution of coupled energy equations in local thermal nonequilibrium models for porous media. The decoupled solution approach, in which the interaction between the solid and the fluid temperature fields is treated in an explicit manner, converges very slowly when the interface heat transfer coefficient and/or the specific surface area of the porous medium are large (large Biot number). An attractive alternative to the decoupled approach is the partial elimination algorithm, proposed by D. B. Spalding. In this algorithm, the discretization equations are rearranged so that the resulting equations are more implicit and take directly into account the coupling between the two phases. The convergence rates of these two solution procedures are studied with reference to convective heat transfer in a two-dimensional channel filled with a porous medium. The partial elimination algorithm converges much more quickly than the decoupled procedure, with the number of iterations required for convergence becoming constant for large Biot numbers.