Convection Operator Research Articles

A Moving Particle Method with Embedded Pressure Mesh (MPPM) for Incompressible Flow Calculations

An accurate moving particle method for incompressible flow calculations is presented in this article. The major distinctive feature in this proposition is the insertion of a pressure mesh within the particle cloud to handle the continuity constraint. It is motivated by the thought that pressure should be a field variable rather than a material one moving with fluid flow. Both the diffusion and convection operators are executed on the particle locations, while the projection operator to retain a convergence-free velocity field is manipulated on the inserted pressure mesh. It will yield a diagonally dominant and constant-coefficient matrix equation to update the pressure field. Besides the advantages gained to enforce the continuity constraint, the auxiliary mesh can significantly enhance the particle searching efficiency in the particle smoothing process. Numerical verifications on some benchmark problems indicate that the present proposition will provide accurate results for incompressible flow calculations.

Physical Fractional Step Method for Solution of 2D Water Flow around a Spur Dike

A numerical model for simulating of 2D flow around a spur dike was presented by using the physical fractional-step method. The water governing equations were separated into three parts. The first is a convective problem; the second is a diffusive problem; and the third is a source term. The advantage of this method is that in the convective-diffusion equation a more reasonable scheme for the convective and diffusive operator is adopted respectively. Numerical simulation for 2D flow around a spur dike was implemented. The comparisons with other numerical solutions show that the proposed method can be capable of dealing with 2D spur-dike flows.

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

AbstractWe develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L2 norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical application of k-ɛ turbulence model to the flow over a backward-facing step

A model that uses the operator splitting technique based on Eulerian-Lagrangian method and embeds the k-ɛ turbulence mode is developed. The model is used to simulate a two dimensional flow over a backward-facing step with triangular mesh discrete flow field. The convection operator is solved with the characteristic method and the diffusion operator and pressure-Poisson equation are solved by finite element method. The results from the example showed that numerical results are well agreed with the experimental data, and that the method can be adapted to simulate complex turbulent flow with strong non-linear convection. The treatment of boundary conditions is also studied in the paper.

Generalized conservative approximations of split convective derivative operators

We propose a strategy to design locally conservative finite-difference approximations of convective derivatives for shock-free compressible flows with arbitrary order of accuracy, that generalizes the approach of Ducros et al. (2000) [1], and that can be applied as a building block of low-dissipative, hybrid shock-capturing methods. The approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible Euler equations, which guarantee strong numerical stability and (near) energy preservation in the inviscid limit. A convenient implementation of the high-order fluxes is suggested, which guarantees improved computational efficiency over existing methods. Numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten, and effectiveness of convective splitting of Kennedy and Gruber (2008) [2] in providing extra stability in the presence of strong density variations. Numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non-uniform grids, and overall support superior computational efficiency of high-order schemes.

NUR research reactor safety analysis study for long time natural convection (NC) operation mode

The current work represents a part of an overall effort being undertaken to improve NUR nuclear research reactor neutron utilization capabilities using a new core configuration. A RELAP5 model for the reactor NUR under natural convection (NC) operating mode has been developed. The model represents internal pool reactor components with the corresponding geometry, point neutron kinetics, and thermal hydraulics characteristics. An experimental device was designed and implemented in the reactor pool for monitoring the inlet and outlet core temperatures, and other pool temperature positions during NC-operating mode. In this paper, unprotected fast reactivity insertion and total flow blockage of the flapper valve transients have been investigated under NC operating conditions. The achieved steady-state results were found to be in good qualitative agreement with measurements. The results obtained from the transient (FRIA) study were compared to similar approach from recent literature. The second transient herein considered, is an attempt to predict the reactor core thermal-hydraulic behavior under a total flow blockage of the (NC) flapper valve. The latter could be considered as useful contribution for updating the safety analysis report.

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

In this paper we continue the study, which was initiated in (Ben-Artzi et al. in Math. Model. Numer. Anal. 35(2):313---303, 2001; Fishelov et al. in Lecture Notes in Computer Science, vol. 2667, pp. 809---817, 2003; Ben-Artzi et al. in J. Comput. Phys. 205(2):640---664, 2005 and SIAM J. Numer. Anal. 44(5):1997---2024, 2006) of the numerical resolution of the pure streamfunction formulation of the time-dependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in the last three afore-mentioned articles, to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is actually obtained for several test-cases.

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

We address in this paper a parabolic equation used to model the phase mass balance in two-phase flows, which differs from the mass balance for chemical species in compressible multicomponent flows by the addition of a nonlinear term of the form ∇ · ρφ (y)u r , where y is the unknown mass fraction, ρ stands for the density, φ is a regular function such that φ(0) = φ(1) = 0 and u r is a (not necessarily divergence free) velocity field. We propose a finite-volume scheme for the numerical approximation of this equation, with a discretization of the nonlinear term based on monotone flux functions. Under the classical assumption that the discretization of the convection operator must be such that it vanishes for a constant y, we prove the existence and uniqueness of the solution, together with the fact that it remains within its physical bounds, i.e., within the interval [0, 1]. Then this scheme is combined with a pressure correction method to obtain a semi-implicit fractional step scheme for the so-called drift-flux model. To satisfy the above-mentioned assumption a specific time stepping algorithm with particular approximations for the density terms is developed. Numerical tests are performed to assess the convergence and stability properties of this scheme.

Total variation diminishing schemes for pollutant transport in porous media

AbstractA bidimensional numerical model has been used in order to simulate the contaminant transport in the coastal groundwater area (Atlantic margin of the Rharb basin, Morocco). This groundwater is materialized by means of the salt contamination derived from several factors: evapotranspiration, lithological series formations, marine intrusion, and processes of interaction between water and rocks. In order to reduce the numerical diffusion and limit the numerical dispersion, we use the Superbee flux limiter as a total variation diminishing scheme to discretize the convective operator. This kind of discretization was applied to the coastal groundwater of the Rharb basin (Morocco). The results show that the Superbee flux limiter is efficient at drawing the path of the contaminant front with high accuracy. Consequently, this scheme could constitute an approach in water management and allows one to prevent the risks of pollution and to manage the groundwater resource from a durable development perspective. Copyright © 2007 John Wiley & Sons, Ltd.

Simulation of vorticity dominated flows using a hybrid approach. II: Numerical method

A family of numerical methods for the solution of the incompressible Navier–Stokes equations in cylindrical coordinates is developed. The formulation for hybrid spectral-finite difference discretizations in domains with cylindrical boundaries presented in Kollman, in press [Kollmann W, Simulation of vorticity dominated flows using a hybrid approach: I formulation, in press] forms the basis of the solvers. The solution methods use the Fourier-spectral approach for the azimuthal and a set of finite-difference operators for the radial and axial directions. The convective terms are discretized with a linear combination of upwind-biased and central difference operators applied to the non-conservative and conservative formulations, respectively. All other spatial derivatives are discretized with central operators. The time integration is specified as a minimum storage, state space, fourth order Runge–Kutta method. The convergence of the solvers as the formal accuracy of the finite-difference operators varies is tested for an axi-symmetric flow for fixed discretization and time integrator. The results show satisfactory convergence with respect to order of accuracy and the convective operators.

Unified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: Numerical investigations

This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn's inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. variations of the Re number on unstructured meshes. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.

A fractional step θ-method for convection–diffusion problems

In this article, we analyze the fractional step θ-method for the time-dependent convection–diffusion equation. In our implementation, we completely separate the convection operator from the diffusion operator, and stabilize the convective problem using a Streamline Upwinded Petrov–Galerkin (SUPG) method. We establish a priori error estimates and show that the optimal value of θ yields a scheme that is second-order in time. Numerical computations are presented which demonstrate the method and support the theoretical results.

Development of a flow analysis code using an unstructured grid with the cell-centered method

A conservative finite-volume numerical method for unstructured grids with the cell-centered method has been developed for computing flow and heat transfer by combining the attractive features of the existing pressure-based procedures with the advances made in unstructured grid techniques. This method uses an integral form of governing equations for arbitrary convex polyhedra. Care is taken in the discretization and solution procedure to avoid formulations that are cell-shape-specific. A collocated variable arrangement formulation is developed, i.e. all dependent variables such as pressure and velocity are stored at cell centers. For both convective and diffusive fluxes the forms superior to both accuracy and stability are particularly adopted and formulated through a systematic study on the existing approximation ones. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are computed by using a linear reconstruction based on the divergence theorem. Momentum interpolation is used to prevent the pressure checkerboarding and a segregated solution strategy is adopted to minimize the storage requirements with the pressure-velocity coupling by the SIMPLE algorithm. An algebraic solver using iterative preconditioned conjugate gradient method is used for the solution of linearized equations. The flow analysis code (PowerCFD) developed by the present method is evaluated for its application to several 2-D structured-mesh benchmark problems using a variety of unstructured quadrilateral and triangular meshes. The present flow analysis code by using unstructured grids with the cell-centered method clearly demonstrate the same accuracy and robustness as that for a typical structured mesh.

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.

Symmetry-Preserving Discretization of Heat Transfer in a Complex Turbulent Flow

Convective and diffusive operators are discretized such that their symmetries are preserved. The resulting discretization inherits all symmetry-related properties of the continuous formulation. It is shown that a symmetry-preserving discretization is unconditionally stable and conservative. A fourth-order, symmetry-preserving discretization method is developed and tested for the numerical simulation of turbulent (flow and) heat transfer in a channel with surface-mounted cubes, where the temperature is treated as a passive scalar. The Reynolds number (based on the channel width and the mean bulk velocity) is Re=13,000. The results of the numerical simulation agree well with available experimental data.

An upwind discretization scheme for the finite volume lattice Boltzmann method

The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys 1992;68:401–7; Peng G, Xi H, Duncan C, Chou SH. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82; Chen H. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys Rev E 1998;58:3955–63] or Finite Element [Lee T, Lin CL. A characteristic Galerkin method for discrete Boltzmann equation. J Comp Phys 2001;171:336–56; Shi X, Lin J, Yu Z. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 2003;42:1249–61] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods. We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.

비정렬격자계를 사용하는 3차원 유동해석코드 개발 (I) - 수치해석방법 -

A conservative pressure-based finite-volume numerical method has been developed for computing flow and heat transfer by using an unstructured grid system. The method admits arbitrary convex polyhedra. Care is taken in the discretization and solution procedures to avoid formulations that are cell-shape-specific. A collocated variable arrangement formulation is developed, i.e. all dependent variables such as pressure and velocity are stored at cell centers. Gradients required for the evaluation of diffusion fluxes and for second-order-accurate convective operators are found by a novel second-order accurate spatial discretization. Momentum interpolation is used to prevent pressure checkerboarding and the SIMPLE algorithm is used for pressure-velocity coupling. The resulting set of coupled nonlinear algebraic equations is solved by employing a segregated approach, leading to a decoupled set of linear algebraic equations for each dependent variable, with a sparse diagonally dominant coefficient matrix. These equations are solved by an iterative preconditioned conjugate gradient solver which retains the sparsity of the coefficient matrix, thus achieving a very efficient use of computer resources.

Spectral element formulations on non-conforming grids: A comparative study of pointwise matching and integral projection methods

Pointwise matching (PM) and integral projection (IP) methods are two widely used techniques to extend the classical weak formulations to include non-conforming grids. We present spectral element formulations on polynomial (p-type) and geometric (h-type) non-conforming grids using both the PM (also known as the Constrained Approximation) and IP (also known as the Mortar Element) methods. We systematically compare the convergence characteristics of PM and IP methods for diffusion, convection, and convection–diffusion equations. Consistency errors due to the non-conforming formulations of the diffusion equation result in convergence problems for the PM method using the maximum rule. Both non-conforming formulations for the unsteady convection operator result in eigenvalue spectrum with positive real values, causing convergence problems due to the consistency errors. However, small “physical” diffusion in the convection–diffusion equation eliminates these problems, resulting in spectral convergence for both methods. Encouraged by this, we present spectral element formulations for incompressible Navier–Stokes equations using PM and IP methods on p-type and h-type non-conforming grids, and demonstrate spectral convergence for unsteady and steady test cases. Results for two-dimensional lid-driven cavity flow at Re = 1000 are also presented.

A shift transformation for fully conservative methods: turbulence simulation on complex, unstructured grids

Operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries. These “shift” transformations permit conservative, skew-symmetric convective operators and symmetric, positive-definite diffusive operators to be obtained for staggered variables using collocated operators. Shift transformations are not limited to uniform or structured meshes, and this formulation leads to a generalization of the works of Perot (J. Comput. Phys. 159 (2000) 58) and Verstappen and Veldman (J. Comput. Phys. 187 (2003) 343). A set of shift operators have been developed for, and applied to, a time-adaptive Cartesian mesh method with a fractional step algorithm. The resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time. A mass conserving interpolation is used for the variables during mesh adaptation; the interpolation conserves momentum and energy to second order in space. Turbulent channel flow simulations were conducted at Re τ ≈ 180 using direct numerical simulation (DNS). The DNS results from the adaptive method compare favourably with spectral DNS results despite the use of a (formally) second-order accurate scheme.

Towards the ultimate variance-conserving convection scheme

In the past various arguments have been used for applying kinetic energy-conserving advection schemes in numerical simulations of incompressible fluid flows. One argument is obeying the programmed dissipation by viscous stresses or by sub-grid stresses in Direct Numerical Simulation and Large Eddy Simulation, see e.g. [Phys. Fluids A 3 (7) (1991) 1766]. Another argument is that, according to e.g. [J. Comput. Phys. 6 (1970) 392; 1 (1966) 119], energy-conserving convection schemes are more stable i.e. by prohibiting a spurious blow-up of volume-integrated energy in a closed volume without external energy sources. In the above-mentioned references it is stated that nonlinear instability is due to spatial truncation rather than to time truncation and therefore these papers are mainly concerned with the spatial integration. In this paper we demonstrate that discretized temporal integration of a spatially variance-conserving convection scheme can induce non-energy conserving solutions. In this paper the conservation of the variance of a scalar property is taken as a simple model for the conservation of kinetic energy. In addition, the derivation and testing of a variance-conserving scheme allows for a clear definition of kinetic energy-conserving advection schemes for solving the Navier–Stokes equations. Consequently, we first derive and test a strictly variance-conserving space–time discretization for the convection term in the convection–diffusion equation. Our starting point is the variance-conserving spatial discretization of the convection operator presented by Piacsek and Williams [J. Comput. Phys. 6 (1970) 392]. In terms of its conservation properties, our variance-conserving scheme is compared to other spatially variance-conserving schemes as well as with the non-variance-conserving schemes applied in our shallow-water solver, see e.g. [Direct and Large-eddy Simulation Workshop IV, ERCOFTAC Series, Kluwer Academic Publishers, 2001, pp. 409–287].