Stream Function-vorticity Variables Research Articles

Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling

This research investigates laser-induced convection through a stream function-vorticity formulation. Specifically, this paper considers a solution to the steady Boussinesq Navier–Stokes equations in two dimensions with a slip boundary condition on a finite box. A fixed-point algorithm is introduced in stream function-vorticity variables, followed by a proof of the existence of steady solutions for small laser amplitudes. From this analysis, an asymptotic relationship is demonstrated between the nondimensional fluid parameters and least upper bounds for laser amplitudes that guarantee existence, which accords with numerical results implementing the algorithm in a finite difference scheme. The findings indicate that the upper bound for laser amplitude scales by O(Re−2Pe−1Ri−1) when Re≫Pe, and by O(Re−1Pe−2Ri−1) when Pe≫Re. These results suggest that the existence of steady solutions is heavily dependent on the size of the Reynolds (Re) and Peclet (Pe) numbers, as noted in previous studies. The simulations of steady solutions indicate the presence of symmetric vortex rings, which agrees with experimental results described in the literature. From these results, relevant implications to thermal blooming in laser propagation simulations are discussed.

Mixed convection–radiation in lid-driven cavities with nanofluids and time-dependent heat-generating body

The cooling process of electronic devices having heat-generating elements is a major challenge allowing to develop electronics industry. Therefore, a creation of novel cooling techniques is an important task that can be solved numerically taking into account the multiparametric character of this problem. The mixed convection heat transfer combined with thermal radiation in a lid-driven cavity filled with an alumina–water nanofluid under the effect of sinusoidal time-dependent heat-generating solid element is studied numerically. The partial differential equations formulated in stream function–vorticity variables are solved by the finite difference method. Effects of the Rayleigh number, Reynolds number, thermal radiation parameter, heater location, volumetric heat flux oscillation frequency and nanoparticles volume fraction on liquid flow and heat transfer are analyzed. It has been found that an addition of nanoparticles leads to reduction of the heater temperature, while convective flow rate decreases also.

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.

On cat's eyes and multiple disjoint cells natural convection flow in tall tilted cavities

Natural convection fluid flow in air-filled tall tilted cavities is studied numerically with a direct projection method applied on the unsteady Boussinesq approximation in primitive variables. The study is focused on the so called cat's eyes and multiple disjoint cells as the aspect ratio A and the angle of inclination ϕ of the cavity vary. Results have already been reported with primitive and stream function-vorticity variables. The former are validated with the latter ones, which in turn were validated through mesh size and time-step independence studies. The new results complemented with the previous ones lead to find out the fluid motion and heat transfer invariant properties of this thermal phenomenon, which is the novelty here.

From catʼs eyes to multiple disjoint natural convection flow in tall tilted cavities: A direct primitive variables approach

Abstract Natural convection fluid flow in air-filled tall tilted cavities is studied numerically with a new direct projection method on the Boussinesq approximation in primitive variables. The study deals with “catʼs eyes” instabilities and multiple disjoint cells as the aspect ratio A and the angle of inclination ϕ of the cavity vary. The flows are validated with those reported before using the stream function-vorticity variables. New cases, A = 12 and 20 varying ϕ , lead to get more insight on the physical phenomenon.

Cabaret scheme for two-dimensional incompressible fluid in terms of the stream function-vorticity variables

In the present work, the Cabaret method is generalized to the case of two-dimensional incompressible fluid in terms of the stream function-vorticity variables. Using the example of the solitary vortex problem, the high quality of the obtained algorithm in terms of its dissipative and dispersion properties is demonstrated. In the problem about decaying homogeneous isotropic turbulence, the slopes of the energy spectra for all grids (16 × 16, 32 × 32, 64 × 64, 128 × 128) are (−3) up to the highest harmonics, which coincides with Batchelor’s theory.

Numerical simulation of viscous flow around a system of bodies on the basis of the Navier-Stokes equations in stream function-vorticity variables

A method of numerical solution of the Navier-Stokes equations in stream function-vortex-pressure variables is suggested to be used in simulating a nonstationary, nonsymmetric flow of a laminar incompressible viscous stream around a pair of rotating cylinders in a channel. Results of calculations which illustrate application of the method under different conditions of flow around bodies are presented.

On the Boundary Conditions for Navier-Stokes Equations in Stream Function-Vorticity Variables in Simulation of a Flow around a System of Bodies

A method of determining the boundary conditions for the Navier-Stokes equations in stream function-vorticity variables, used for simulation of a nonstationary, asymmetric laminar flow of an incompressible viscous fluid around bodies, has been proposed. Universal relations for desired functions on surfaces around which the stream flows, independent of the method of spatial discretization, have been obtained.

Numerical investigation of hydrodynamics and mass transfer for in-line fiber arrays in laminar cross-flow at low Reynolds numbers

In this work, mass transfer at the shell side of an in-line hollow fiber array subjected to cross-flow is simulated by applying the domain decomposition method combined with orthogonal grid generation. Two-dimensional Navier–Stokes equations written in stream function–vorticity variables, were separately solved along with a species conservation equation for different arrays. The main factors influencing the concentration fields, local mass transfer rates and global mass transfer rates in the laminar flow of Re = 10 –200 and Pe = 10 –300 with pitch to tube diameter ratios of 1.45, 1.50, 1.75, 1.85 and 2.00 are discussed in detail. Mass transfer correlations obtained from the numerical simulations show good agreement with typical empirical correlations proposed earlier.

Compact scheme on a nonuniform grid for the ψ–ω-system of Navier-Stokes equations

- This paper deals with the construction and the study of a compact scheme on a nonuniform rectangular grid for the quasilinear system of Navier-Stokes equations in the stream functionvorticity variables. The scheme is written on the 3 × 3 point stencil. In the general case it has a thirdorder error with respect to the steps on a spatial grid, which becomes a fourth-order error on a uniform grid. We obtain a stability criterion based on the requirement for nonnegativeness of the scheme coefficients. It limits the extent to which mesh cells can be elongated and the ratio of neighbouring step sizes in each coordinate direction.

NUMERICAL INVESTIGATION OF TWO-DIMENSIONAL LAMINAR FLOW AND SOLUTE TRANSPORT IN A CHANNEL WITH SOME SYMMETRIC EXPANSIONS AND CONTRACTIONS

Numerical investigation of two-dimensional (2D) laminar flow and solute transport in a channel with some sudden symmetric expansions and contractions has been performed using the fictitious regions method. This method allows us, instead of solving Navier-Stokes equations in a complex domain, to solve equations with suitably continued coefficients in a rectangle. Stream function-vorticity variables are used in the present paper. Dependence of the flow and solute transport from the dimensions of the channel expansions and contractions is numerically investigated for different values of Reynolds and Péclet numbers using a finite differences method on a relatively fine grid.

Steady flow with separation past a thin airfoil at high Reynolds numbers

The results of calculating the steady separated flow of a viscous incompressible fluid past a 12% sinusoidal airfoil with and without a splitter plate are presented. The Navier—Stokes equations written in stream-function—vorticity variables are approximated in accordance with a central difference scheme and solved simultaneously by means of the direct method. The results obtained are analyzed from the standpoint of their consistency with the asymptotic theory based on the well-known three-deck scheme.

A variational finite element method for the two-dimensional Stokes problem

This paper is devoted to the numerical solution of the steady Stokes problem, in stream function-vorticity variables. In two dimensions, this is reduced to a Dirichlet problem for the biharmonic operator. A new formulation for the decomposition of the biharmonic operator using the potential theory is presented. The problem comes down to the calculation of a function defined on the boundary, which is the solution of a variational problem. This kind of decomposition is used in electric potential theory to solve the Laplacian problem outside a bounded domain. For these problems, the function defined on the boundary is called the electric charge of the conductor. The first part of the paper is devoted to the analysis of the continuous problem, and the variational formulation on the boundary is studied. The method may be extended to the Navier-Stokes problem. By using a suitable finite element approximation, similar properties for the discrete problem are found. These properties imply that the discrete function is the solution of a linear system, whose matrix is full, but symmetric and positive-definite. Among the iterative methods, we use the conjugate gradient method, which seems suitable in the context. Numerical results concerning the problem of steady flow in a square cavity are presented.

A numerical case study of a non‐Newtonian flow problem

AbstractThis paper addresses some of the theoretical aspects involved in the numerical study of non‐Newtonian flow problems. We consider the second‐order Rivlin–Erickson constitutive model due to the simple differential form that emerges for the system of equations that govern the flow when expressed in stream function–vorticity variables. This model describes slightly elastic fluids that exhibit a constant viscosity behaviour. A steady two‐dimensional flow is studied through a planar contraction geometry.An auxiliary variable is introduced into the problem formulation producing a non‐linear system of differential equations comprising two elliptic equations and one hyperbolic equation. This system is discretized by finite difference methods and the resulting system of non‐linear algebraic equations is solved iteratively by successive substitutions. The simple structure of this iteration permits a convergence analysis which is presented in Section 2. This analysis is performed prior to the spatial discretization and establishes the dependence of the iteration upon the material parameters. At the discrete linearized equation level a combination of inner iterations for elliptic equations and direct marching for the hyperbolic equation is used. The stability of the marching scheme is considered in Section 4.3 and a discussion on the results is given in Section 5.