A level set redistancing algorithm for simulation of two-phase flow

Abstract

Incompressible two-phase flow involving interface evolution was studied using a level set method. To maintain the level set function as a signed distance function from the zero level set and meanwhile preserve the mass conservation, a level set redistancing algorithm of level set methods was developed. Important to the above level set redistancing algorithm was a new idea that keeps the zero level set almost unchanged during the redistancing process. Protection of the zero level set is introduced in the redistancing process to ensure that the interface does not change, thus reducing or even avoiding mass errors induced. Accuracy of this algorithm was verified in terms of such benchmark problems as deforming vortex and Zalesak’s disk problems. Numerical examples including 2D and 3D dam-break and 3D single bubble rising simulations are presented to validate the present level set method. Mass conservation and computational costs involving complicated interfacial flow structures are also presented.     

An implicit numerical scheme for the simulation of three-dimensional two-phase flows in light water nuclear reactors

ABSTRACT

An implicit numerical scheme is presented for solving the two-fluid model widely used in the analysis of a gas–liquid two-phase flow in light water nuclear reactors (LWRs). The pressure equation is established by combining the momentum and mass conservation equations. The implicit calculation of each governing equation is separated into phase- and space-link steps. In the phase-link step, the interfacial momentum and heat transfers are implicitly calculated. Then the solution accounting for the convection and diffusion terms is calculated simultaneously in space. The phase- and space-link steps are repeated for convergence. The numerical scheme is implemented in CUPID, which is a multidimensional two-phase flow analysis code for LWRs, and verified against a set of conceptual two-phase flow problems which include typical thermal hydraulic phenomena in LWRs. Calculations are performed using four numerical schemes, semi-implicit ICE and SMAC schemes, an implicit scheme, and an implicit scheme with increased time step size, and the results are discussed.     

Numerical Modeling for Multiphase Incompressible Flow with Phase Change

ABSTRACT

A general formula for the second-order projection method combined with the level set method is developed to simulate unsteady, incompressible multifluid flow with phase change. A subcell conception is introduced in a modified mass transfer model to accurately calculate the mass transfer across the interface. The third-order essentially nonoscillatory (ENO) scheme and second-order semi-implicit Crank-Nicholson scheme is employed to update the convective and diffusion terms, respectively. The projection method has second-order temporal accuracy for variable-density unsteady incompressible flows as well. The level set approach is employed to implicitly capture the interface for multiphase flows. A continuum surface force (CSF) tension model is used in the present cases. Phase change and dynamics associated with single bubble and multibubbles in two and three dimensions during nucleate boiling are studied numerically via the present modeling. The numerical results show that this method can handle complex deformation of the interface and account for the effect of liquid–vapor phase change.     

Topological Shape Optimization of Heat Conduction Problems using Level Set Approach

ABSTRACT

A topological shape optimization method for heat conduction problems is developed using a level set method. The level set function obtained from the “Hamilton-Jacobi type” equation is embedded into a fixed initial domain to implicitly represent thermal boundaries and obtain the finite-element response and adjoint sensitivity. The developed method minimizes the thermal compliance, satisfying the constraint of allowable volume by varying the implicit boundary. During optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition. The newly developed method shows no numerical instability and makes it easy to represent topological shape variations.     

A LEVEL SET METHOD FOR INCOMPRESSIBLE TWO-FLUID FLOWS WITH IMMERSED SOLID BOUNDARIES

ABSTRACT

A numerical method is presented for computing incompressible gas–liquid (or two-fluid) flows with immersed solid boundaries on fixed Cartesian meshes. A level set technique for tracking the gas–liquid interface is modified to treat the contact angle condition at the gas–liquid–solid interline as well as the no-slip condition at the fluid–solid interface. The no-slip condition is imposed by introducing another level set for fluid–solid phases and an effective viscosity formulation. In the immersed solid region where the level set function for gas–liquid phases is not well defined, its zero level set is calculated so that the contact angle condition should be satisfied where the three phases meet. The numerical method is validated through computations of interfacial motion subject to Taylor instability, single-fluid flow past a circular cylinder, and bubbles adhering to a cylindrical solid.     

GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

ABSTRACT

Natural convection in a differentially heated cubic enclosure is studied by solving the velocity–vorticity form of the Navier–Stokes equations by a generalized differential quadrature (GDQ) method. The governing equations in the form of velocity Poisson equations, vorticity transport equations, and energy equation are solved using a coupled numerical scheme via a single global matrix for velocities, vorticities, and temperature. Vorticity and velocity coupling at the solid boundaries is enforced through a higher-order approximation by the GDQ method, thus assuring accurate satisfaction of the continuity equation. Nusselt numbers computed for Ra = 10³, 10⁴, 10⁵, and 10⁶ show good agreement with the benchmark results. A mesh independence study indicates that the present numerical procedure requires much coarse mesh compared to other numerical schemes to produce the benchmark solutions of the flow and heat transfer problems.     

Stress constrained thermo-elastic topology optimization based on stabilizing control schemes

Abstract

This article proposes two effective stabilizing control schemes for addressing the stress constrained thermo-elastic topology optimization in a non-uniform temperature field. Based on the density interpolation scheme, two linear elastic equations for coupling a thermo-elastic problem are considered. For comparison, different topology problem formulations for minimizing compliance or volume subject to stress constraints are solved. By virtue of a stabilization transform method, two stabilizing control schemes combined with the grouped aggregation method are developed to handle the challenging difficulties stemming from the local nature of highly nonlinear stress constraints. Moreover, the adjoint method is adopted to perform the sensitivity analysis. The design variables are updated by utilizing the method of moving asymptotes. The results of several typical numerical examples verify the validity of the proposed methodology, including the present stabilizing control schemes which can be employed to obtain clear topological design and fast convergence rate for thermo-elastic coupling problems. Meanwhile, compliance minimization design with stress constraints is appropriate to achieve balance between stress level and stiffness.     

A Numerical Method for Multiphase Incompressible Thermal Flows with Solid–Liquid and Liquid–Vapor Phase Transformations

ABSTRACT

A numerical method for multiphase incompressible thermal flows with solid–liquid and liquid–vapor phase transformations is presented. The flow is mainly driven by thermocapillary force and vaporization. Based on the level set method and mixture continuum model, a set of governing equations valid for solid, liquid, and vapor phases is derived, considering phase boundary conditions as source terms in the transport equations. The vaporization process is treated as a source term in the continuity equation. The model developed is applied to the laser welding process, where the flow is coupled with optical phenomena. Formation and collapse of a laser-created hole is simulated.     

GDQ METHOD FOR NATURAL CONVECTION IN A SQUARE CAVITY USING VELOCITY–VORTICITY FORMULATION

ABSTRACT

This article describes a compact numerical algorithm based on the generalized differential quadrature (GDQ) method for the numerical analysis of natural convection in a differentially heated square cavity. The velocity–vorticity form of the Navier–Stokes equations and energy equation are used to represent the mass, momentum, and energy conservations of the fluid medium in the cavity. The GDQ form of the governing equations and the vorticity definition at the boundaries are solved by a coupled solution algorithm using a global matrix scheme for all the field variables. The vorticity values at the boundary are correctly imposed using the GDQ method, which approximates a given space derivative with higher-order accuracy compared to the existing schemes based on Taylor's series expansion. This has assured a divergence-free solution for the flow field by satisfying the continuity constraint, though the pressure term is not used directly in the present formulation. The proposed algorithm is validated for a lid-driven cavity flow for Reynolds number Re = 100, 400, and 1,000, and the predicted velocity profiles are in excellent agreement with the benchmark solutions. The algorithm is then used to compute the average Nusselt number and flow parameters for natural convection in a square cavity for Rayleigh number Ra = 10³, 10⁴, 10⁵, and 10⁶. These results are in better agreement with the benchmark solutions than the results obtained by other numerical schemes, which used much finer grids compared to the present scheme.     

A NEWTON-BASED BOUNDARY ELEMENT METHOD FOR NONLINEAR CONVECTIVE DIFFUSION PROBLEMS

Abstract

A Newton-based boundary element method for the solution of nonlinear convective diffusion problems is presented. The problems are formulated through the use of the exponential transformation. The numerical procedures for the boundary element implementation of the formulation are discussed, and the treatment of nonlinear boundary conditions using the Newton-Raphson method is described in detail. The coefficient matrices resulting from the boundary element implementation are so partitioned as to facilitate the construction and efficient computation of the Jacobian matrix. Three numerical examples are provided. The results agree well with the analytical solutions whenever available. The method is free from numerical oscillations even for high Peclet numbers. The Newton-based iterative scheme, when integrated with the BEM, provides an efficient algorithm for the solution of nonlinear convective diffusion problems and is superior to the successive substitution approximation.     

A block-centered finite-difference method for the time-fractional diffusion equation on nonuniform grids

ABSTRACT

In this article, a block-centered finite-difference scheme is introduced to solve the time-fractional diffusion equation with a Caputo derivative of order α ∈ (0, 1) on nonuniform grids. The resulting scheme is second-order-accurate in space and (2 ? α)-order-accurate in time, and the unconditional stability and convergence are proved theoretically. Moreover, numerical solutions of the unknown variable along with its first derivatives are obtained. Finally, numerical experiments, including boundary-layer and high-gradient problems, are carried out to support our theoretical analysis and indicate the efficiency of this method.     

APPLICATIONS OF AN EXPONENTIAL FINITE-DIFFERENCE TECHNIQUE

ABSTRACT

In this paper an expontial finite-difference scheme, first presented by Bhattacharya for one-dimensional, unsteady heat conduction problems in a plane wail, is used to solve various partial differential equations. Solutions of the unsteady diffusion equation in three dimensions and of the viscous form of Burgers’ equation ere used to illustrate the method. Predicted results are compared with exact solutions or with results obtained by other numerical methods.     

FINITE-VOLUME APPROACH TO THERMOVISCOELASTICITY

ABSTRACT

This article presents a development of the finite-volume method for solving linear thermoviscoelastic deformation problems. Hereditary continuum problems represented by spatially elliptic second-order partial differential equations with memory are considered. This is motivated by the need to develop numerical algorithms for the solution of thermoviscoelastic stress analysis problems, although it is expected that results presented will generalize to other Volterra problems.

Assuming that the hydrostatic and deviatoric responses are uncoupled, and using the temperature–time equivalence hypothesis, the constitutive equations are expressed in an incremental form. Procedures for analyzing linear viscoelastic deformation are described, and numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms. The accuracy of the method is demonstrated through comparison with analytical and experimental results as well as with numerical solutions obtained elsewhere.     

Extending a low-order upwind-biased scheme to solve turbulent flames using detailed chemistry model

Abstract

Achieving more accurate reacting flow numerical solutions apparently demand employing higher-order schemes, utilizing finer grids, and benefiting from more advanced chemistry models. One major objective of this work is to extend an inclusive low-order upwind-biased scheme in the context of finite-volume-element method to predict turbulent reacting flows on coarse grid resolutions very reliably. In this regard, a low-order upwind-biased scheme is suitably extended to approximate the mixture fraction variances at the cell-faces. This scheme implements the reacting flow physics explicitly in deriving the proposed mixture fraction variance expressions. These physical implementations enhance the derived expressions to result in superior turbulent reacting flow solutions even on coarse grid resolutions. To assess the accuracy of new expressions, we simulate a sample turbulent non-premixed flame with strong non-equilibrium effects of turbulence on chemistry. The comparisons show that the current low-order scheme is robust enough to predict the complex structure of non-premixed flames very reliably even on coarse grids.     

Wave Instability of Mixed Convection Flow Along a Vertical Flat Plate

Abstract

An analysis is performed to investigate the linear wave instability of laminar mixed convection flow over an isothermal vertical flat plate, in which the buoyancy force arises solely from the temperature gradients in the fluid. In the stability analysis, the main flow and thermal fields are treated as nonparallel, and are obtained by the local nonsimilarity solution method The eigenvalue problem consisting of the linearized system of coupled differential equations for the velocity and temperature disturbances are solved by a direct Runge-Kutta numerical integration scheme along with a filtering technique to remove the “parasitic errors” inherent in the numerical integration of the disturbance equations. Neutral stability curves and critical Reynolds numbers are presented for a range of buoyancy parameters covering both assisting and opposing flow situations for two representative Prandtl numbers of 0.7 and 7. It is found that the flow becomes more stable as the buoyancy force increases for assisting flow and less stable as the buoyancy force increases for opposing flow. The curve of Grashof number versus Reynolds number that separates the unstable flow region from the stable one is also presented.     

On the numerical solution of generalized convection heat transfer problems via the method of proper closure equations – part I: Description of the method

ABSTRACT

The purpose of this paper is to introduce a new physical-based computational approach for the solution of convection heat transfer problems on co-located non-orthogonal grids in the context of an element-based finite volume method. The approach has already been presented in the context of two-dimensional incompressible flow problems without heat transfer. It has been shown that the pressure–velocity coupling on co-located grids can be correctly modeled via the so-called method of proper closure equations (MPCE). Here, MPCE is extended to the numerical simulation of natural, forced, and mixed convection heat transfer problems. It is shown that the couplings between pressure, velocity, and temperature can be conveniently handled on co-located grids by resorting again to the modified forms of the governing equations, i.e., the proper closure equations. The set of discrete equations is solved in a fully coupled manner in this study. Here, in part I of the paper, only the basic methodology is described; in part II, the results of application of the method to some test problems are presented.     

A Numerical Study of Weight Functions,Scaling, and Penalty Parameters for Heat Transfer Applications

ABSTRACT

This article describes a numerical study of weight functions, scaling, and penalty parameters for heat transfer problems. The numerical analysis is carried out using a meshless element-free Galerkin (EFG) method, which utilizes moving least-square (MLS) approximants to approximate the unknown function of temperature. These MLS approximants are constructed by using a weight function, a basis function, and a set of coefficients that depend on position. Lagrange multiplier and penalty methods are used to enforce the essential boundary conditions. MATLAB software is developed to obtain the EFG results. A new rational weight function is proposed. Comparisons are made among the results obtained using cubic spline, quartic spline, Gaussian, quadratic, hyperbolic, rational, exponential and, cosine weight functions in one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) heat transfer problems. The L₂ error norm and rate of convergence are evaluated for different EFG weight functions and the finite-element method (FEM). The effect of scaling and penalty parameters on EFG results is discussed in detail. The results obtained by the EFG method are compared with those obtained by finite-element and analytical methods.     

A coupled volume of fluid and level set method based on analytic PLIC for unstructured quadrilateral grids

We develop a coupled volume of fluid and level set (VOSET) method for unstructured quadrilateral grids to simulate incompressible two-phase interfacial flows in irregular domains. In the method, an analytic piecewise linear interface calculation (PLIC) is first proposed and its solution speed is much faster (about five times) than that of Brent's iterative method; then an iterative geometric operation by which the level set function near interfaces can be calculated, is extended to unstructured quadrilateral grids; moreover, the volume fraction advection is solved by a Lagrangian-Eulerian advection scheme. Finally, the present method is validated by the single vortex flow problem, bubble rising problem and falling droplet problem. Our simulation results show good agreement with those in previous studies.     

NUMERICAL SIMULATIONS ON THE HYDRODYNAMICS OF A TURBULENT SLOT JET IMPINGING ON A SEMICYLINDRICAL CONVEX SURFACE

This study presents the numerical simulations of flow characteristics of a turbulent slot jet impinging on a semicylindrical convex surface. The turbulent-governing equations are solved by a control-volume-based finite difference method with power-law scheme, and the well-known k  ? ε turbulence model associated with the wall function is used to describe the turbulent behavior and structure.

While the width of the slot nozzle is fixed at 9.38 mm, the diameter of the semicylinder is at 150 mm, and air is the working medium, the adopted modifying parameters here include the Reynolds number of the inlet flow (Re = 6000 ~ 20000), jet to impingement surface spacing (y / w = 7 ~ 13), and the entrainment or wall boundary is employed nearby the convex surface. The numerical simulations of flow fields indicate that the velocity distribution of the free jet region departs from the center with increasing y / w. When we increase Reynolds number Re, the variation of the velocity on the convex surface becomes rapid, and the turbulent kinetic energy increases.     

A time discretization scheme based on integrated radial basis functions for heat transfer and fluid flow problems

Abstract

This article reports a new numerical procedure, which is based on integrated radial basis functions (IRBFs) and Cartesian grids, for solving time-dependent differential problems that can be defined on non-rectangular domains. For space discretizations, compact five-point IRBF stencils are utilized. For time discretizations, a two-point IRBF scheme is proposed, where the time derivative is approximated in terms of not only nodal function values at the current and previous time levels but also nodal derivative values at the previous time level. This allows functions other than a linear one to also be captured well on a time step. The use of the RBF width as an additional parameter to enhance the approximation quality with respect to time is also explored. Various kinds of test problems of heat transfer and fluid flows are conducted to demonstrate the the attractiveness of the present compact approximations.