Stream Function-vorticity Formulation Research Articles

The Influence of Surface Radiation on the Passive Cooling of a Heat-Generating Element

Low-power electronic devices are suitably cooled by thermogravitational convection and radiation. The use of modern methods of computational mechanics makes it possible to develop efficient passive cooling systems. The present work deals with the numerical study of radiative-convective heat transfer in enclosure with a heat-generating source such as an electronic chip. The governing unsteady Reynolds-averaged Navier–Stokes (URANS) equations were solved using the finite difference method. Numerical results for the stream function–vorticity formulation are shown in the form of isotherm and streamline plots and average Nusselt numbers. The influence of the relevant parameters such as the Ostrogradsky number, surface emissivity, and the Rayleigh number on fluid flow characteristics and thermal transmission are investigated in detail. The comparative assessment clearly emphasizes the effect of surface radiation on the overall energy balance and leads to change the mean temperature inside the heat generating element. The results of the present study can be applied to the design of passive cooling systems.

An Iteration-Free Approach to Solving the Navier–Stokes Equations by Implicit Finite Difference Schemes in the Vorticity-Stream Function Formulation

The paper introduces a new algorithm for solving the finite difference equations at the upper time level of an implicit scheme that approximates the Navier–Stokes system in the vorticity-stream function formulation. The algorithm requires no iterations and computes the corresponding discrete solution exactly. It is based on the method of difference potentials and allows one to efficiently address the well-known difficulties typical for this type of formulations—two boundary conditions for the stream function and no boundary conditions for vorticity. Abstract for the translation The paper is translated from the Russian by S. Tsynkov (tsynkov@math.ncsu.edu). The original [12] was published more than 20 years ago (the actual Russian citation is [14]). It has not been previously translated into English and went largely unnoticed by the numerical analysis and scientific computing research community. Yet it presents an important contribution to the discipline as it offers a full answer to the question that has long been outstanding. The incompressible Navier–Stokes equations in the vorticity-stream function formulation require two boundary conditions for the stream function and no boundary conditions for vorticity. A standard approach to addressing numerically the apparent overdetermination in one variable and underdetermination in the other was through the use of iterations. Instead, work [12] shows the unambiguous way of discretizing the Navier–Stokes system implicitly and solving the resulting finite difference equations on the upper time level exactly. It is equivalent to deriving the correct non-local boundary condition for vorticity. Note from the translator My Ph.D. advisor Prof. Ryaben’kii and my postdoc mentor Prof. Abarbanel were friends. They interacted closely and participated in joint projects at ICASE (NASA Langley Research Center) in the late 1990s and early 2000s. Prof. Ryaben’kii visited Tel Aviv several times, and Prof. Abarbanel traveled to Moscow in 2013 to celebrate Prof. Ryaben’kii’s 90th birthday. This paper, which was written by Prof. Ryaben’kii with his Ph.D. student at the time, V. Torgashov, provides a well-deserved tribute to a friend from a friend.

Effects of Microstructure on Three-Dimensional Double-Diffusive Natural Convection Flow of Micropolar Fluid

This work presents a three-dimensional numerical study on double-diffusive natural convection in a cubic cavity filled with a micropolar fluid. The flow is driven by buoyancy forces due to temperature and solutal gradients. Constant temperature and concentration are imposed along the two vertical sidewalls of the cubic enclosure, while the remaining walls are adiabatic and impermeable. The governing equations are expressed in the vorticity-stream function formulation and then solved numerically using the control volume finite method. Numerical results are compared with the case of a Newtonian fluid. The influences of the micro-constituents of the micropolar fluid are depicted. The results show different transitions of the structure of the main flow when varying the vortex viscosity parameter. It is also observed that both the heat and mass transfer rates and the three-dimensional character of the flow are reduced by increasing the vortex viscosity parameter.

New Algorithm for the Lid-driven Cavity Flow Problem with Boussinesq-Stokes Suspension

In the present investigation, a streamfunction-vorticity form for Boussinesq-Stokes liquids (with suspended particles) is suitably used to examine the problem of 2-D unsteady incompressible flow in a square cavity with moving top and bottom wall. A new algorithm is used for this form in order to compute the numerical solutions for high Reynolds numbers up to Re=2500. This algorithm is conducted as a combination of the multi-time-stepping temporal differential transform and the spatial finite difference methods. Convergence of the time-series solution is ensured by multi-time-stepping method. The classical benchmark results of the Newtonian liquid are recovered as a limiting case and the decelerating influence of the suspended particle on the Newtonian liquids’ flow field is clearly elaborated.

A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson's equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells.

Buoyant convection in porous annulus with discrete sources-sink pairs and internal heat generation

In the present analysis, numerical investigation of buoyancy-driven convective thermal transport in a vertical annulus has been performed when the interior and exterior cylinders are discretely heated and cooled by heat source-sink pair arrangements. The upper and lower boundaries and unheated parts of inner and outer cylinders are insulated. For the porous annulus, the Brinkman-extended Darcy formulation is adopted for modeling the fluid flow in the porous medium. Also, the effects of internal heat generation are investigated. The governing equations in terms of vorticity-stream function formulation are solved by an implicit finite difference technique based ADI and SLOR methods. In particular, the study is focused on the effects of different sources and sinks arrangements on the convective flow and associated thermal transport features.

Non-equilibrium Model for Nanofluid Free Convection Inside a Porous Cavity Considering Lorentz Forces

In current article, transportation of CuO nanoparticles through a porous enclosure is demonstrated. The enclosure has complex shaped hot wall. Porous media has been simulated via two temperature equations. Magnetic force impact on nanofluid treatment was considered. Control volume based finite element method has been described to solve current article in vorticity stream function form. Single phase model was chosen for nanofluid. Nanofluid characteristics are predicted via KKL model. Roles of solid-nanofluid interface heat transfer parameter (Nhs), porosity, Hartmann and Rayleigh numbers have been illustrated. Outputs illustrated that conduction mode reduces with augment of Ra. Increasing magnetic forces make nanofluid motion to decrease. Temperature gradient of nanofluid decreases with augment of Nhs. Reducing porosity leads to enhance in Nusselt number.

Novel wavelet-homotopy Galerkin technique for analysis of lid-driven cavity flow and heat transfer with non-uniform boundary conditions

In this paper, a brand-new wavelet-homotopy Galerkin technique is developed to solve nonlinear ordinary or partial differential equations. Before this investigation, few studies have been done for handling nonlinear problems with non-uniform boundary conditions by means of the wavelet Galerkin technique, especially in the field of fluid mechanics and heat transfer. The lid-driven cavity flow and heat transfer are illustrated as a typical example to verify the validity and correctness of this proposed technique. The cavity is subject to the upper and lower walls’ motions in the same or opposite directions. The inclined angle of the square cavity is from 0 to π/2. Four different modes including uniform, linear, exponential, and sinusoidal heating are considered on the top and bottom walls, respectively, while the left and right walls are thermally isolated and stationary. A parametric analysis of heating distribution between upper and lower walls including the amplitude ratio from 0 to 1 and the phase deviation from 0 to 2π is conducted. The governing equations are non-dimensionalized in terms of the stream function-vorticity formulation and the temperature distribution function and then solved analytically subject to various boundary conditions. Comparisons with previous publications are given, showing high efficiency and great feasibility of the proposed technique.

Effects of variable viscosity on pulsatile flow of blood in a tapered stenotic flexible artery

The objective of the present study is to investigate the effects of variable viscosity on incompressible laminar pulsatile flow of blood through an overlapping doubly constricted tapered artery. To mimic the realistic situation, wall of the artery is taken to be flexible, and physiologically relevant pulsatile flow is introduced. The governing equations of blood flow are made dimensionless. A coordinate transformation is used to make the overlapping doubly constricted wall geometry of tube to a straight tube. Taking advantage of the Stream function–Vorticity formulation, the system of partial differential equations is then solved numerically by finite difference approximations. Effects of Reynolds number, Strouhal number, degree of contraction, tapering angle, and viscosity parameters are presented graphically and analyzed. The results show that formation of stenosis and tapering disturb the flow field significantly, and degree of stenosis is more important in influencing blood flow compared with tapering.

Analysis of nonequilibrium effects and flow instability in immiscible two-phase flow in porous media

Two-dimensional, high-resolution, numerical solutions for the classical formulation and two widely accepted nonequilibrium models of multiphase flow through porous media are generated and compared with experimental observations from literature. Flow equations for simultaneous flow of two immiscible phases through porous media are written in a vorticity stream-function form. In the resulting system of equations, the vorticity stream-function equation is solved using a spectral method and the transport equation is discretized in space using a central-upwind scheme. The system of equations is solved for a two-dimensional domain using a semi-implicit time-stepper. The solutions reveal behavior that is not apparent in one-dimensional solutions, namely that sharpening of the saturation front caused by the inclusion of a dynamic capillary pressure results in propagation of viscous fingers compared to the classical formulation. The inclusion of nonequilibrium effects in the constitutive relations, in the form of effective saturation, introduces dispersion and smears the otherwise highly resolved viscous fingers in the saturation front. Once developed, the length of the mixing zone in the numerical solutions remains constant with time regardless of the degree of instability. This is contrary to the evolution of the mixing zone observed in unstable flow experiments where, unlike the numerical solutions, the propagation speed of the leading edge of the front appears to increase with time.

Effect of Prandtl number on thermo-fluidic transport characteristics for mixed convection past a sphere

In this paper, we numerically investigate the influence of Prandtl number on the thermo-fluidic transport characteristics for steady laminar mixed convection from a sphere in an assisting flow configuration. The stream function-vorticity formulation with a higher order finite difference compact scheme in spherical polar coordinates has been used to solve the governing transport equation of motion and thermal energy. The fluid flow and heat transfer characteristics are presented in terms of streamline and isotherm contours in close proximity to the sphere together with local Nusselt number along the sphere surface and average Nusselt number for the Reynolds number in the range 5 ≤ Re ≤ 200, Richardson number in the range 0 ≤ Ri ≤ 1.5 and Prandtl number ranging from 0.72 to 40. Critical Richardson number has been identified for different values of Reynolds number and Prandtl number, signifying the strength of the force of streamwise acceleration due to the buoyancy and externally imposed flow field relative to the combined viscous resistance and pressure force, for which the flow separation phenomenon occurring in the downstream region of the sphere gets suppressed. It reveals that for lower values of Reynolds number, the average Nusselt number monotonically increases with Richardson number for any particular value of Prandtl number. At higher values of Reynolds number, contrasting features have been observed. For lower values of Prandtl number, the average Nusselt number increases monotonically with Richardson number, while for higher values of Prandtl number the variation is non-monotonous.

Investigation of dimensionless parameters and geometry effects on heat-transfer characteristics of liquid sodium flowing over a flat plate

To improve the cooling performance of a nuclear fuel element, it is important to appraise the effect of dimensionless parameters and the geometry on heat-transfer characteristic of sodium flowing over a nuclear fuel element. To fulfill this objective, the effects of geometry, Reynolds number (ReH), conductivity ratio ( N cc ), and dimensionless total heat generation parameter ( Qt) on a two-dimensional steady flow of liquid coolant flowing over a nuclear fuel element are studied. For this purpose, the stream function-vorticity formulation method is applied. Full Navier Stokes equations and energy equation for the fluid domain are solved concurrently with conduction equation of fuel element applying finite difference scheme. The pseudotransient form of the vorticity transport and energy equations is solved using the alternating direction implicit method. A line-by-line technique is used for other discretized equations. Isotherms are also plotted and studied in detail. From the obtained results it can be concluded that for fixed values of aspect ratio and Re H there exists a critical value of Qt and N cc beyond and below which peak temperature in the fuel element surpasses its tolerable limit. The results can also be applied to minimize the peak temperature in the nuclear fuel element (hot spots).

Flow past a circular cylinder executing rotary oscillation: Dimensionality of the problem

Computational results for flow past a cylinder, executing rotary oscillation, are presented from the perspective of flow control and drag reduction for bluff bodies. The governing Navier-Stokes equation is solved for Reynolds numbers (Re) of 250–400 using the two-dimensional (2D) stream function-vorticity formulation. Here, we compare the computational results, qualitatively and quantitatively, with experimental results, showing excellent match. Such a successful match between 2D computations and experiments prompted us to investigate the reason with the help of the Taylor-Proudman theorem, which explains the use of the geostrophic assumption used to define atmospheric motion as being 2D. As a consequence, we map the flow domain with equivalent Rossby and Ekman numbers to explain that the flow past a rotationally oscillating cylinder can remain 2D for Re values which are higher than that is known for the stationary cylinder.

Numerical solution of 2D Navier–Stokes equation discretized via boundary elements method and finite difference approximation

In this paper boundary elements method (BEM) is equipped with finite difference approximation (FDA) to solve two-dimensional Navier–Stokes (N–S) equation. The N–S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, forward Euler, Runge–Kutta and Newton’s methods to find a fast ODE solver. Numerical investigations presented in this article show Newton’s method is faster than the others and it is able to solve the N–S equation for Reynolds number up to 40,000 when grid points are at most 141 × 141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM’s two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly agree with that accurate ones available in the literature.

Numerical simulation of physiologically relevant pulsatile flow of blood with shear-rate-dependent viscosity in a stenosed blood vessel

Pulsatile flow of blood in a blood vessel having time-dependent shape (diameter) is investigated numerically in order to understand some important physiological phenomena in arteries. A smooth axi-symmetric cosine shaped constriction is considered. To mimic the realistic situation as far as possible, viscosity of blood is taken to be non-uniform, a shear-thinning viscosity model is considered and a physiologically relevant pulsatile flow is introduced. Taking advantage of axi-symmetry in the proposed problem, the stream function–vorticity formulation is used to solve the governing equations for blood flow. Effect of different parameters associated with the problem on the flow pattern has been investigated and disparities from the Newtonian case are discussed in detail.

Numerical simulation of lid-driven cavity flow of micropolar fluid

Incompressible unsteady laminar flow of a micropolar fluid in a lid driven square cavity has been examined numerically with finite difference scheme. In this paper, we verify the code validation of Newtonian incompressible fluid (K = 0) with standard bench mark problem in the literature and then proceed the present computation. The vorticity-stream function formulation of Navier-Stokes equations and angular momentum equation is solved by the Euler’s explicit method. The effect of various values of Reynolds number (Re) and vortex viscosity parameter (K) of the flow characteristics is studied and discussed through several graphs.

Convergence of a lowest-order finite element method for the transmission eigenvalue problem

The transmission eigenvalue problem arises in scattering theory. The main difficulty in its analysis is the fact that, depending on the chosen formulation, it leads either to a quadratic eigenvalue problem or to a non-classical mixed problem. In this paper we prove the convergence of a mixed finite element approximation. This approach, which is close to the Ciarlet–Raviart discretization of biharmonic problems, is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. The convergence analysis is based on classical abstract spectral approximation result and the theory of mixed finite element methods for solving the stream function–vorticity formulation of the Stokes problem. Numerical experiments are reported in order to assess the efficiency of the method.

Thermal radiation therapy of biomagnetic fluid flow in the presence of localized magnetic field

Present work describes the applications of magnetic field and thermal radiation on the two-dimensional, unsteady flow of bio-magnetic fluid (blood) in a rectangular vessel. The viscosity of the fluid is considered to be an exponential function of temperature. The radiative heat flux is approximated with the help of Stefan Boltzmann law, which is a nonlinear function of temperature. The governing nonlinear, coupled system of partial differential equations for the underlying problem is represented in terms of stream function-vorticity formulation, which is further solved numerically with the help of upwind scheme along with successive over relaxation method. For the validation of the solutions, results are compared with the numerical and experiemental data available in the literature. In order to notice the influence of localized magnetic field and thermal radiation, solutions are presented graphically in terms of streamlines, isotherms, vorticity function contours and velocity component contours and are discussed both qualitatively and quantitatively from the physiological point of view.

Control volume based finite element simulation of magnetic nanofluid flow and heat transport in non-Darcy medium

Magnetic nanofluids are used in cancer therapeutics by utilizing cancer imaging and drug delivery. These are also used to kill cancerous cells without affecting the nearby healthy cells by producing higher temperatures around tumor. This investigation aims to explore the heat transfer in non-Darcy porous medium for magnetic nanofluid flowing due to external magnetic field. Nanoparticle shapes and their properties are also considered. Vorticity stream function formulation is also employed. The solutions of the resulting system of equations are found by CVFEM. Illustrations are outlined pictorially for numerous physical parameters namely, Rayleigh number, radiation parameter, Darcy number, Fe3O4-water based volume concentration and Hartmann numbers. Outcomes indicate that augmenting in Hartmann number results in reduce in velocity of nanofluid and heat transfer rate. The nanoparticle with shape Platelet gives maximum heat transfer rate.

A novel homotopy-wavelet approach for solving stream function-vorticity formulation of Navier–Stokes equations

In this paper, we propose a new homotopy-wavelet approach to solve linear and nonlinear problems with nonhomogeneous boundary conditions. The essence of this technique is to apply the homotopy analysis method (HAM) to transform the governing equations into a set of linear equations and employ the generalized Coiflet-type orthogonal wavelet to express and solve the resulting linear equations. The proposed technique is expected to keep the superiority of the HAM for handling nonlinearities, but with better computational efficiency. The nonhomogeneous boundary conditions including the mixed Dirichlet-Neumann and Robin conditions are reconstructed by introducing the Coiflets on the boundaries, which overcomes the deficiency of the close wavelet method that is difficult to handle the nonhomogeneous boundary conditions. Illustrative examples show very high efficiency of our proposed technique. Furthermore, the classic problem of the incompressible flow in a 2-D lid-driven cavity are investigated. By reconstructing the incompatible boundary conditions with the Coiflets, the singularities of velocity field on rigid points are successfully eliminated so that the vortex on the lid that is difficult to be obtained by previous approaches can be captured clearly. Comparison with previous results is made, excellent agreement is found.