THREE-DIM ENSIONAL DIAGONAL CARTESIAN METHOD FOR INCOMPRESSIBLE FLOWS INVOLVING COMPLEX BOUNDARIES

A diagonal Cartesian method for the three-dimensional simulation of incompressible fluid flows over complex boundaries is presented in this article. The method is derived utilizing the superposition of the finite analytic solutions of a linearized two-dimensional convection-diffusion equation in Cartesian coordinates. The complex boundary is approximated with a structured grid in a series of calculation planes which are perpendicular to the x, y, and z coordinate axis. In the calculation plane, both Cartesian grid lines and diagonal line segments are used. It is observed that this geometry approximation is more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-vertex nodes on a staggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries. An enlarged control-volume method is introduced for the mass conservation and the pressure boundary condition on complex boundaries. The conservation of momentum on complex boundaries is enforced through the use of three-dimensional 19, 15, 11, or 7-point finite analytic elements. The proposed diagonal Cartesian method is verified by the solution of a rotated lid-driven cavity flow. It is shown that this diagonal Cartesian method predicts the fluid flow very well.     

AN ARTIFICIAL COMPRESSIBILITY METHOD FOR INCOMPRESSIBLE FLOWS

An artificial compressibility method characterized by the pressure-based algorithm is developed on a nonorthogonal collocated grid for incompressible fluid flow problems, using a cell-centered finite-volume approximation. Unlike the traditional pseudo-compressibility concept, the continuity constraint is perturbed by the material derivative of pressure, the physical relevance of which is to invoke matrix preconditionings. The approach provokes density perturbations, assisting the transformation between primitive and conservative variables. To account for the flow directionality in the upwinding, a rotational matrix is introduced to evaluate the convective flux. A rational means of reducing excessive numerical dissipation inherent in the pressure–velocity coupling is contrived which has the expedience of greater flexibility and increased accuracy in a way similar to the MUSCL approach. Numerical experiments in reference to a few laminar flows demonstrate that the overall artifacts expedite enhanced robustness and anticipated oscillation damping properties adhering to the factored pseudo-time integration procedure.     

AN ADAPTIVELY REFINED QUADTREE GRID METHOD FOR INCOMPRESSIBLE FLOWS

An adoptively refined quadtree grid method for the numerical solution of the incompressible Navier-Stokes equations is presented. A pressure-based scheme with allocated primitive variables is used as the solution algorithm. A process of grid refinement and flow solution is repeated until a sufficiently resolved solution is obtained. The present method has been applied to a variety of test cases. The results show that an adoptively refined quadtree grid can yield a better grid distribution over the flow, therefore yielding a more accurate solution as well as an improved convergence rate than a structured grid with a similar number of grid points.     

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 VARIABLE-DENSITY PROJECTION METHOD FOR INTERFACIAL FLOWS

General second-order, variable-density, three-step and four-step projection methods are developed to simulate unsteady incompressible interfacial flows. A high-accuracy, variable-density RKCN projection method is presented, in which the three-stage, low-storage Runge-Kutta technique and second-order semi-implicit Crank-Nicholson technique are employed to temporally update the convective and diffusion terms, respectively. To reduce computation cost, a simplified version of the projection method is also presented, in which the pressure Poisson equation (PPE) is solved only at the last substage. The level set approach is employed to implicitly capture the interface for falling droplet flows. Three-dimensional bubble rising flows and two-dimensional falling droplet flows in a small closed channel are studied numerically via the present method. By the definition of the effective pressure, the flow mechanisms for falling droplet flows with different density ratios, viscosity ratios, Weber numbers, and Reynolds numbers are discussed.     

TEMPORAL SECOND-ORDER ACCURACY OF SIMPLE-TYPE METHODS FOR INCOMPRESSIBLE UNSTEADY FLOWS

SIMPLE-type methods are presented in a more concise formulation. This formulation is used to analyze temporal accuracy for unsteady flows. Detailed error formulations are given. Analysis shows that SIMPLE-type methods have second-order temporal accuracy if a second-order temporal updating technique is employed to update both the convective and diffusion terms. Two algorithms for unsteady flows are presented. Algorithm A is an iteration method, which will cost more time than algorithm B, a noniteration algorithm. Also several second-order updating techniques are presented. A classical validation example is employed to validate the temporal accuracy in this article. A new four-step SIMPLE-type method is presented, in which the pressure Poisson equation, not the pressure difference Poisson equation is solved.     

A COMPARATIVE ASSESSMENT OF THE PERFORMANCE OF MASS CONSERVATION-BASED ALGORITHMS FOR INCOMPRESSIBLE MULTIPHASE FLOWS

This work is concerned with the implementation and testing, within a structured collocated finite-volume framework, of seven incompressible-segregated multiphase flow algorithms that belong to the mass conservation-based algorithms (MCBA) group in which the pressure-correction equation is derived from overall mass conservation. The pressure-correction schemes in these algorithms are based on SIMPLE, SIMPLEC, SIMPLEX, SIMPLEM, SIMPLEST, PISO, and PRIME. The performance and accuracy of the multiphase algorithms are assessed by solving eight one-dimensional two-phase flow problems spanning the spectrum from dilute bubbly to dense gas-solid flows. The main outcome of this study is a clear demonstration of the capability of all MCBA algorithms to deal with multiphase flow situations. Moreover, results displayed in terms of convergence history plots and CPU times indicate that the performance of the MCBA versions of SIMPLE, SIMPLEC, and SIMPLEX are very close. In general, the performance of SIMPLEST approaches that of SIMPLE for diffusion-dominated flows. As expected, the PRIME algorithm is found to be the most expensive, due to its explicit treatment of the phasic momentum equations. The PISO algorithm is generally more expensive than SIMPLE, and its performance depends on the type of flow and solution method used. The behavior of SIMPLEM is consistent, and in terms of CPU effort it stands between PRIME and SIMPLE.     

AN ADAPTIVE LEVEL SET METHOD FOR MOVING-BOUNDARY PROBLEMS: APPLICATION TO DROPLET SPREADING AND SOLIDIFICATION

A three-dimensional adaptive level set method has been developed for deformable free surface problems with or without solidification. In the new scheme, a three-dimensional multizone adaptive grid generation (MAGG) scheme is employed to track the moving boundaries and a level set method is used to capture the free surface deformation. The effectiveness and robustness of the algorithm are demonstrated by solving the droplet spreading and solidification problem, in which both free surface and solidification interface movements are important.     

HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS,PART II: APPLICATIONS

Within the framework of the SIMPLE algorithm, a dual-dissipation scheme is proposed on nonorthogonal collocated grids for incompressible fluid flow problems, using a cell-centered finite-volume approximation. The dissipative mechanism employs dynamic limiters to control the amount of dissipation, preserving expediences of greater flexibility and increased accuracy in a way similar to the MUSCL approach. The artificial density concept is combined with the pressure Poisson equation, facilitating an avoidance of pressure underrelaxation. To account for the flow directionality in the upwinding, a rotational matrix is evoked to evaluate the convective flux.     

NONSTAGGERED BOUNDARY-FITTED COORDINATE METHOD FOR FREE SURFACE FLOWS

A new approach based on the finite-difference technique has been developed to study the steady incompressible Navier-Stokes equations in the laminar region, where the domain is partially bounded by a free surface. The nonstaggered fractional step method is used to solve the flow equations written in terms of primitive variables. The physical domain is transformed to a rectangle by means of a numerical mapping technique. The location of the phase boundary is accomplished by means of two methods depending on the surface tension effect: the normal-stress boundary condition or the kinematic boundary condition. We have tested the accuracy and efficiency of the numerical method by solving four different test problems: lid-driven flow in an inclined cavity, film in the absence of gravity, the "stick-slip" problem, and the Newtonian jet swell problem.     

ADVANCED THREE-DIMENSIONAL TWO-FLUID POROUS MEDIA METHOD FOR TRANSIENT TWO-PHASE FLOW THERMAL-HYDRAULICS IN COMPLEX GEOMETRIES

A three-dimensional two-fluid model for two-phase flow within tube or rod bundles is presented. The porous media concept is applied in the model statement. Closure laws for interfacial mass, momentum and energy transfer, bundle flow resistance and heat transfer are presented. Correlations for the interfacial drag force are proposed. Some illustrative examples of application (including also verification) to two-phase flows in complex geometries and across vertical and horizontal tube/rod bundles are presented below. The developed model is suitable for the simulation and analyses of complex multidimensional thermal-hydraulics of nuclear fuel rod bundles or shell-and-tube heat exchangers with vapor generation within a tube bundle on the shell side.     

HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS,PART I: THEORETICAL DEVELOPMENT

A higher-order-accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth- and sixth-order compact finite-difference schemes for spatial discretization. New insights are presented on the elimination of the odd-even decoupling problem in the solution of the pressure Poisson equation. For consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Accuracy and robustness issues are addressed by application to several pertinent benchmark problems in Part II.     

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.     

AN ACCURATE METHOD FOR THE SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE HEATED PIPE FLOW

A numerical scheme using Fourier expansions in the streamwise and azimuthal directions and Jacobi polynomials in the radial direction for the direct numerical simulation of three-dimensional incompressible pipe flow with heating is presented. The proposed basis and test functions for the thermal field in conjunction with those for the velocity field set forth by Leonard and Wary offer an accurate representation of flow variables in transitional flow. A simple test to a linear stability problem demonstrates that this method yields very accurate results with relatively few radial modes and is well suited for the simulation of nonisothermal pipe flow transition.     

COMPUTATIONS OF MULTIPHASE FLOWS WITH SURFACE TENSION USING AN ADAPTIVE FINITE ELEMENT METHOD

An adaptive finite element method for solving incompressible steady-state axisymmetric free surface flow problems is presented. While the methodology is applicable to most types of multiphase flows, we are particularly interested in modeling laminar free surface flows such as free and impinging jets, which are usually modeled in a Lagrangian framework. An Eulerian strategy is used to capture the interface. Surface tension is included in the model in order to study its influence on the topology of the free surface. A stabilized finite element discretization is used to help solve the coupled system of partial differential equations. An adaptive methodology helps optimize the accuracy of the computed solution. The proposed adaptive free surface capturing methodology is competitive with interface tracking techniques, while being more flexible. The verification and validation of the methodology completes the article.     

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 NONLINEAR INDIRECT MEASUREMENT PROBLEM FOR A MULTITHERMOCOUPLE PROBE IMMERSED IN A LIQUID

The problem described herein concerns the processing of the time-dependent, internal temperatures within a multithermocouple probe. These are used to compute the temperature of the surrounding fluid, as part of an inverse heat conduction problem (IHCP). The novel achievement in this work is that the exchange coefficients do not have to be supplied a priori, but instead are an additional solution output. Consequently the IHCP is nonlinear and requires significant stabilization. Four methods are applied successively, until a satisfactory solution is found: the parameterization of spatial variations in fluid temperatures and exchange coefficients; a functional specification method (using future time data) to address the noncausal nature of the solution; a lower bound on the exchange coefficient; and a maximum number of iterations at each time step (in accordance with the discrepancy principle).     

CO-LOCATED EQUAL-ORDER CONTROL-VOLUME FINITE-ELEMENT METHOD FOR MULTIDIMENSIONAL,INCOMPRESSIBLE. FLUID FLOW -PART I: FORMULATION

A co-located equal-order control-volume-based finite-element method (CVFEW) for two- and three-dimensional, incompressible, viscous fluid flow is presented. The method works directly with the primitive variables. Triangular elements and polygonal control volumes, and tetrahedral elements and polyhedral control volumes are used to discretize the calculation domains in two- and three-dimensional problems, respectively. Two available flow-oriented upwind schemes (FLO and FLOS) and a novel mass-weighted skew upwind scheme (MAW) are investigated. In each dement, the velocity components in the mass flux terms are interpolated by special functions that prevent the generation of spurious pressure oscillations. The discretized equations are solved using an iterative sequential variable adjustment algorithm. Verification of the proposed CVFEM is presented in a companion article.     

A NEW WALL FUNCTION STRATEGY FOR COMPLEX TURBULENT FLOWS

Wall functions are widely used and offer significant computational savings compared with low-Reynolds-number formulations. However, existing schemes are based on assumed near-wall profiles of velocity, turbulence parameters, and temperature which are inapplicable in complex, nonequilibrium flows. A new wall function has therefore been developed which solves boundary-layer-type transport equations across a locally defined subgrid. This approach has been applied to a plane channel flow, an axisymmetric impinging jet, and flow near a spinning disc using linear and nonlinear k–ε turbulence models. Computational costs are an order of magnitude less than low-Reynolds-number calculations, while a clear improvement is shown in reproducing low-Re predictions over standard wall functions.