Velocity-vorticity Formulation Research Articles

A Compact-Difference Scheme for the Navier–Stokes Equations in Vorticity–Velocity Formulation

This paper presents a new numerical method for solving the incompressible, unsteady Navier–Stokes equations in vorticity–velocity formulation. The method is applicable to spatial simulations of transitional and turbulent boundary layer flows. It is based on a compact-difference discretization of the streamwise and wall-normal derivatives in Cartesian coordinates. A Fourier collocation approach is used for the spanwise derivatives. Important new features of the numerical method are the use of nonequidistant differences in the wall-normal direction; the use of split-compact differences in the streamwise direction; a new, fast iteration for a semi-implicit time integration of the wall-normal diffusion terms; and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary. Results of test calculations are presented to verify the improvements obtained by the use of these new techniques.

Solution of the incompressible Navier-Stokes equations on domains with one or several open boundaries

The aim of this paper is to develop a methodology for solving the incompressible Navier–Stokes equations in the presence of one or several open boundaries. A new set of open boundary conditions is first proposed. This has been developed in the context of the velocity–vorticity formulation, but it is also emphasized how it can be formally extended to the equations in primitive variables. The case of a domain involving several independent open boundaries is considered next. An influence matrix technique is applied such that the inlet mass flux is split onto the several outlets in order to enforce the prescribed mean pressure at each outlet. Both approaches are validated by numerical test cases. Copyright © 1999 John Wiley & Sons, Ltd.

Phase change problems with free convection: fixed grid numerical simulation

A numerical and experimental study of unsteady natural convection during freezing of water is presented. The mathematical model for the numerical simulations is based on the enthalpy-porosity method in vorticity-velocity formulation, equations are discretised on a fixed grid by means of a finite volume technique. A fully implicit method has been adopted for the mass and momentum equations. Experiments are performed for water in a differentially heated cube surrounded by air. The experimental data for natural convection with freezing in the cavity are collected to create a reference for comparison with numerical results. The method of simultaneous measurement of the flow and temperature fields using liquid crystal tracers is used. It allows us to collect transient data on the interface position, and the temperature and velocity fields. In order to improve the capability of the numerical method to predict experimental results, a conjugate heat transfer problem was solved, with finite thickness and internal heat conductivity of the non-isothermal walls. These results have been compared with the simulations obtained for the idealised case of perfectly adiabatic side walls, and with our experimental findings. Results obtained for the improved numerical model shown a very good agreement with the experimental data only for pure convection and initial time of freezing process. As time passes the discrepancies between numerical predictions and the experiment became more significant, suggesting a necessity for further improvements of the physical model used for freezing water.

A comparison of the structures of lean and rich axisymmetric laminar Bunsen flames: application of local rectangular refinement solution-adaptive gridding

Axisymmetric laminar methane–air Bunsen flames are computed for two equivalence ratios: lean (Φ=0.776), in which the traditional Bunsen cone forms above the burner; and rich (Φ=1.243), in which the premixed Bunsen cone is accompanied by a diffusion flame halo located further downstream. Because the extremely large gradients at premixed flame fronts greatly exceed those in diffusion flames, their resolution requires a more sophisticated adaptive numerical method than those ordinarily applied to diffusion flames. The local rectangular refinement (LRR) solution-adaptive gridding method produces robust unstructured rectangular grids, utilizes multiple-scale finite-difference discretizations, and incorporates Newton's method to solve elliptic partial differential equation systems simultaneously. The LRR method is applied to the vorticity–velocity formulation of the fully elliptic governing equations, in conjunction with detailed chemistry, multicomponent transport and an optically-thin radiation model. The computed lean flame is lifted above the burner, and this liftoff is verified experimentally. For both lean and rich flames, grid spacing greatly influences the Bunsen cone's position, which only stabilizes with adequate refinement. In the rich configuration, the oxygen-free region above the Bunsen cone inhibits the complete decay of CH4, thus indirectly initiating the diffusion flame halo where CO oxidizes to CO2. In general, the results computed by the LRR method agree quite well with those obtained on equivalently refined conventional grids, yet the former require less than half the computational resources.

Computational fluid dynamics by boundary-domain integral method

A boundary–domain integral method for the solution of general transport phenomena in incompressible fluid motion given by the Navier–Stokes equation set is presented. Velocity–vorticity formulation of the conservation equations is employed. Different integral representations for conservation field functions based on different fundamental solutions are developed. Special attention is given to the use of subdomain technique and Krylov subspace iterative solvers. The computed solutions of several benchmark problems agree well with available experimental and other computational results. Copyright © 1999 John Wiley & Sons, Ltd.

Fast boundary‐domain integral algorithm for the computation of incompressible fluid flow problems

We deal with the numerical solution of fluid dynamics using the boundary-domain integral method (BDIM). A velocity-vorticity formulation of the Navier-Stokes equations is adopted, where the kinematic equation is written in its parabolic form. Computational aspects of the numerical simulation of two-dimensional flows is described in detail. In order to lower the computational cost, the subdomain technique is applied. A preconditioned Krylov subspace method (PKSM) is used for the solution of systems of linear equations. Level-based fill-in incomplete lower upper decomposition (ILU) preconditioners are developed and their performance is examined. Scaling of stopping criteria is applied to minimize the number of iterations for the PKSM. The effectiveness of the proposed method is tested on several benchmark test problems

A vorticity–velocity formulation for solving the two-dimensional Navier–Stokes equations

This paper describes a vorticity-based integro-differential formulation for the numerical solution of the two-dimensional incompressible Navier–Stokes equations. A finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is evaluated to compute the velocity field from a vorticity distribution over a fluid domain. The Green's scalar identity is employed to solve the total pressure in an integral approach. Global coupling between the vorticity and the pressure boundary conditions is considered when this integro-differential approach is employed. For the early stage development of the flow about an impulsively started circular cylinder, the computational results with our numerical method are compared with known analytical solutions in order to validate the present formulation.

Laminar natural convection heat and mass transfer in vertical rectangular ducts

The present work investigates numerically the laminar natural convection heat and mass transfer in open vertical rectangular ducts with uniform wall temperature/uniform wall concentration (UWT/UWC) or uniform heat flux/uniform mass flux (UHF/UMF) boundary conditions. The vorticity–velocity formulation is applied to solve for the coupled momentum, energy and concentration equations. Results of dimensionless induced volume rate Q, average Nusselt number Nu and Sherwood number Sh are presented in terms of channel length L, buoyancy ratio N, Grashof number Gr, Schmidt number Sc and aspect ratio γ. Analytical solutions for Q, Nu and Sh for the UWT/UWC case are derived under fully developed condition. In addition, the correlation equations of Q, Nu and Sh for both boundary conditions are also presented.

Numerical Simulation of Axisymmetric Viscous Flows by Means of a Particle Method

A vortex particle method for the simulation of axisymmetric viscous flow is presented. The flow is assumed to be laminar and incompressible. The Navier–Stokes equations are expressed in an integral velocity-vorticity formulation. The inviscid scheme is based on Nitsche's method for axisymmetric vortex sheets. Meanwhile, two techniques are proposed for dealing with the viscous term. The first uses an integral Green's function method while the second is based on a diffusion velocity approach. Both are obtained by extension of existing methods for 2D flows. The problem of satisfying boundary conditions along the axis of symmetry is specifically addressed. The problem is solved by using cut-off functions that are derived from the Green's function of the axisymmetric diffusion equation. The scheme is applied to simulate the evolution of vortex rings at intermediate Reynolds number. The processes of entrainment and wake formation are evident in the calculations, as well as the extension of the support of vorticity due to viscous diffusion.

VELOCITY-VORTICITY FORMULATION WITH VORTEX PARTICLE-IN-CELL METHOD FOR INCOMPRESSIBLE VISCOUS FLOW SIMULATION, PART I: FORMULATION AND VALIDATION

A new velocity-vorticity formulation along with a vortex particle-in-cell method (PIC) is developed to solve the incompressible viscous steady and unsteady flow problems for a closed domain. The article demonstrates that the vortex PIC method has some of the advantages of Lagrangian particle methods for computing convection, while the use of a Eulerian grid to compute the diffusion enables viscous flows to be computed. The solution procedure that is described takes advantage of fast Poisson solvers for computational efficiency. Two-dimensional driven cavity flows with impulsively started and oscillating lids are chosen to test the method. The computation results are in good agreement with those of the numerical literature.

Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems

A new solution-adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains. Such a method is required for the numerical solution of realistic combustion problems, in which physical quantities may vary by orders of magnitude over one-tenth of a millimeter at atmospheric pressure, or over micrometers at higher pressures. The local rectangular refinement (LRR) method maintains orthogonality at grid-line intersections but lifts the tensor product restriction common to traditional grids, producing unstructured grids. Governing equations are discretized throughout the domain using newly derived forms, and Newton's method is used to solve the resulting system. On a simple test case with a known solution, the LRR method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature. For the more realistic problem of nonreacting driven square cavity flow, the LRR solution agrees very well with previously published data. When the LRR method is applied to a practical reacting flow (a rich axisymmetric laminar Bunsen flame with complex chemistry, multicomponent transport, and an optically thin radiation submodel), grid spacing highly influences the inner flame's position, which stabilizes only with adequate refinement. The vorticity–velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the LRR gridding technique. Furthermore, each LRR grid is used to form a nonuniform equivalent tensor product (ETP) grid and also, in most cases, an equispaced fully refined (FR) grid; these additional grids are supersets of the LRR grids and thus contain refinement in exactly the same regions. Performance comparisons between the LRR, ETP, and FR grids indicate that the LRR method provides substantial savings in execution time and computer memory requirements, without compromising solution accuracy.

Vorticity-velocity formulations of the Stokes problem in 3D

This work studies the three-dimensional Stokes problem expressed in terms of vorticity and velocity variables. We make general assumptions on the regularity and the topological structure of the flow domain: the boundary is Lipschitz and possibly non-connected and the flow domain may be multiply connected. Upon introducing a new variational space for the vorticity, five weak formulations of the Stokes problem are obtained. All the formulations are shown to lead to well-posed problems and to be equivalent to the primitive variable formulation. The various formulations are discussed by interpreting the test functions for the vorticity (resp. velocity) equation as vector potentials for the velocity (resp. vorticity). Of the five sets of boundary conditions derived in the paper, three are already known, but only for domains with a trivial topological structure, while the remaining two are new. Copyright © 1999 John Wiley & Sons, Ltd.

A Penalty Method for the Vorticity–Velocity Formulation

We present a new vorticity–velocity formulation and implementation for the unsteady three-dimensional Navier–Stokes equations, based on a penalty method. It relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary. This approach is particularly attractive for high-order methods for which the often-used influence matrix method fails to converge for Δt→0. The accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two- and three-dimensional internal and external flows. In particular, the flow past a finite span cylinder attached to end-plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length.

Development of secondary flow and convective heat transfer in isothermal⧹iso-flux rectangular ducts rotating about a parallel axis

The present study investigates rotation-induced secondary flow and mixed convection heat transfer in the entrance region of rectangular ducts rotating about a parallel axis. Boussinesq approximation is invoked to take into account the centrifugal buoyancy. Mechanisms of the secondary vortex development in ducts of isothermal or iso-flux walls are explored by a vorticity transport analysis. Numerical solutions of the parabolized velocity–vorticity formulation of Navier–Stokes⧹Boussinesq systems are used to corroborate the above theoretical analysis and provide evidence for the relevance of axial variation of transport rates to the evolution of the secondary vortices. The present results also reveal that the rotational effect in an iso-flux duct is more significant than that in an isothermal flux. In a fully developed flow region, the iso-flux case can still retain a secondary vortex effect, while the effect vanishes in isothermal ducts.

Steady 3D Flow Configurations for the Horizontal Thermal Convection with Thermocapillary Effects

A vast literature exists on the Benard flow, the vertical thermal convection flow, but almost no result is known on the horizontal counterpart. On account of the wide range of applications in geophysics, astrophysics, metereology, and material sciences we think that the horizontal thermal convection flow deserves as much consideration as the Benard problem. The present study is the first step towards the description of the bifurcation pattern of the horizontal thermal convection flow. We present several flow configurations of an incompressible Navier-Stokes fluid contained in a 4 × 1 × 1 parallelepipedic box with open top and vertical transversal walls at different temperature. The flow induced by the buoyancy force and the thermocapillary effects at the fluid/air interface is numerically computed within the Boussinesq approximation. The velocity-vorticity formulation with a fully implicit finite difference method are implemented in a parallel computational code. The strong coupling of the discrete equations ensures the correct mass balance at each time step and provides true-transient numerical flows. For fluids at Prandtl number Pr e 0.015 (semiconductors and liquid metals), we describe the changes in shape of the main vortex and the intensity of the speed of the flow versus variations of the Grashof number. In the case of absence of thermocapillary effects, we observe that as Gr increases, the flow exhibits more and more three-dimensional effects. When we add the thermocapillary effects, we observe an increase of the speed of the flow with the formation of a steep boundary layer around the liquid/air interface.

Computing the velocity of a rotating flow

To compute the time-dependent flow of a rotating incompressible fluid we consider the velocity–vorticity formulation of the Navier–Stokes equations in cylindrical coordinates. In the numerical method employed the velocity field at each time-step is found as the least squares solution of an overdetermined system of linear equations, Ax= b. We consider how to compute x using the preconditioned conjugate gradient algorithm for least squares (PCGLS) on a distributed parallel computer. The various aspects of using a parallel computer are discussed, and results for a wide range of parallel computers are presented. The parallel speed-up depends on the architecture but is typically about 80% of the number of processors used.

Three‐dimensional incompressible Navier–Stokes equations on non‐orthogonal staggered grids using the velocity–vorticity formulation

Three‐dimensional incompressible Navier–Stokes equations on non‐orthogonal staggered grids using the velocity–vorticity formulation

Three-dimensional incompressible Navier-Stokes equations on non-orthogonal staggered grids using the velocity-vorticity formulation

This paper is concerned with the numerical resolution of the incompressible Navier-Stokes equations in the velocity-vorticity form on non-orthogonal structured grids. The discretization is performed in such a way, that the discrete operators mimic the properties of the continuous ones. This allows the discrete equivalence between the primitive and velocity-vorticity formulations to be proved. This last formulation can thus be seen as a particular technique for solving the primitive equations. The difficulty associated with non-simply connected computational domains and with the implementation of the boundary conditions are discussed. One of the main drawback of the velocity-vorticity formulation, relative to the additional computational work required for solving the additional unknowns, is alleviated. Two- and three-dimensional numerical test cases validate the proposed method

On the Velocity-Vorticity Formulation of the Two-Dimensional Stokes System and its Finite Element Approximation

This paper deals with the numerical study of the two-dimensional Stokes equations in terms of the velocity and the vorticity. We take one of the possible sets of boundary conditions, for which the equivalence between the standard velocity-pressure Stokes system and such velocity-vorticity formulation holds. We consider the corresponding rigorous variational framework and an uncoupled solution scheme. In this context, the authors proposed several families of Galerkin finite element approximations, for which they derived optimal or suboptimal convergence results. Here their numerical approach is illustrated by several numerical examples. As a byproduct, hints are given on the best way to interpolate the different fields involved in the analogous uncoupling scheme, for solving the incompressible Navier-Stokes equations.

Pressure boundary conditions for the vorticity–velocity formulation of Navier–Stokes equations

The pressure boundary conditions (BC) for the vorticity–velocity formulation of the NS equations are considered. The aim of the paper is to formulate the BC problem on the strict basis of the theory of potential. Two aspects of the pressure BC are discussed. The first one is pressure BC at open boundaries (for example, at the inlet and outlet of the pipe). The problem is reduced finally to the system of Fredholm equations of the second type relative to unknown vorticity and sources (sinks) distributions over the boundaries. The second aspect is the determination of the pressure field on the basis of the known velocity field. The approach alternative to the pressure Poisson equation is proposed. It is based on the Helmholz decomposition of the vector field, and it is free from the ambiguities inherent to the previous considerations.