Stream Function-vorticity Formulation Research Articles

Buoyancy-aided conjugate mixed convection with surface radiation from a vertical channel with multiple non-identical discrete heat sources

Prominent results of parametric studies on conjugate mixed convection with surface radiation from a vertical channel flush-mounted with multiple non-identical discrete heat sources are presented. The fluid flow and heat transfer equations, without boundary layer approximations and without the assumption of fully-developed flow, are converted to vorticity-stream function form and then solved using finite volume method. A computer code is exclusively written for the purpose. The effects of aspect ratio, surface emissivity and modified Richardson number on salient results concerning the problem are probed into. The gross errors that creep in due to inadvertent omission of radiation are clearly explored.

Numerical simulation of thermal radiation influence on natural convection in a trapezoidal enclosure: Heat flow visualization through energy flux vectors

A theoretical and numerical study of natural convection in two-dimensional laminar incompressible flow in a trapezoidal enclosurein the presence of thermal radiation is conducted, motivated by energy systems applications. Heat flow visualization via the method of energy flux vectors (EFVs) is also included. The trapezoidal cavity has an inclined top wall which in addition to the bottom wall is maintained at constant temperature, whereas the remaining (vertical side) walls are adiabatic. The governing partial differential conservation equations are transformed using a vorticity-stream function formulation and non-dimensional variables and the resulting nonlinear boundary value problem is solved using a finite difference method with incremental time steps. EFVs provide abundant details of the heat flow at the core of the enclosure. The larger energy flux vectors indicate high temperature gradient zones and the sparse EFVs correspond to low temperature gradient zone. Heat flow distribution in the trapezoidal enclosure can be clearly elaborated via energy flux vectors and provides a deeper insight into thermal characteristics. A comprehensive parametric study is performed to evaluate the impact of Rayleigh number (buoyancy parameter) and radiation parameter on transport phenomena. The computations indicate that local Nusselt number and velocity are increasing functions of the Rayleigh number and radiation parameter. Significant changes in streamlines, temperature contours and energy streamlines for high Rayleigh number are observed. The energy flux vectors show that a large eddy is formed within the enclosure which migrates towards the cold wall. Greater thermal buoyancy force accelerates the primary flow whereas it decelerates the secondary flow. The simulations are relevant to solar collector systems, enclosure fire dynamics, electronic cooling and fuel cell systems. Furthermore, the computations furnish a good benchmark for more general computational fluid dynamics (CFD) analysis with commercial software e.g. ANSYS FLUENT.

Computing ill-posed time-reversed 2D Navier–Stokes equations, using a stabilized explicit finite difference scheme marching backward in time

This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier–Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on , with real p>2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier–Stokes initial value problems. Several reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T>0. Successful backward recovery is shown to be possible at parameter values exceeding expectations.

A hybrid [formula omitted]-[formula omitted] HOC approach for surface tension driven flows in level set framework

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function ψ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable ψ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

Computational hemodynamic investigation of a new bileaflet mechanical heart valve

In this paper, we perform a numerical analysis for simulating steady, two-dimensional, laminar blood flow through our proposed design, known as the Butterfly mechanical heart valve, where the leaflets are fully opened. Blood has been assumed to be Newtonian and non-Newtonian fluid using the Casson model for shear-thinning behavior. A non-uniform Cartesian grid generation technique is presented to generate a two-dimensional grid for the irregular geometry of the Butterfly valve. The governing Navier–Stokes equations of flow, written in a stream function–vorticity formulation, are solved by the finite difference method with hybrid differencing of the convective terms. The computed results show that the blood’s non-Newtonian nature significantly affects the flow field with the existence of recirculation and consequently stagnation causing thrombus formation, as well as an increase of the shear stress along the wall, which contributes to hemolytic blood damage. The results demonstrate that the model is capable of predicting the hemodynamic features most interesting to physiologists. It can be used to assess thromboembolic problems occurring with heart valves and in the design of cardiac prostheses.

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids is studied. The left and right walls are maintained at constant temperature and solute while the bottom and top walls are adiabatic and non-diffusive. The finite difference method, together with the successive over-relaxation (SOR) technique, is used to solve the flow governing equations after converted into the vorticity-stream function form. Heat and mass lines visualization techniques are used for the better visualization of energy and solute distribution in the enclosure. The results obtained in this study are compared with experimental and numerical those from literature and found to be in good agreement. The influence of Rayleigh number Ra=103,104,105, Prandtl number Pr=0.015,0.025,0.71, Lewis number Le=1,2,5, and Buoyancy ratio N=−1,0,1 on streamlines, isotherms, isosolutes, heatlines, masslines, total heat and mass transfer are examined. The parameters mentioned above considerably influence the total heat and solute transfer rates; moreover, the heat and mass lines play significant roles in understanding the distribution of energy and solute transfers in the enclosure.

SECOND LAW ANALYSIS OF MIXED CONVECTION OF MAGNETOHYDRODYNAMIC FLOW IN AN INCLINED SQUARE LID-DRIVEN ENCLOSURE

Second law of thermodynamics analysis is formulated for the case of laminar mixed convection in an inclined square lid-driven enclosure in the presence of magnetic field. Vertical sides of the enclosure moves upward when inclination angle is zero. Governing equations of flow and temperature in the form of stream function-vorticity formulation were solved numerically using the differential quadrature method. Governing parameters are: Richardson number (from 0.01 to 100), Prandtl number (from 0.1 to 1.0), inclination angle of enclosure (from 0 to 180), Hartmann number (from 0 to 100) and magnetic field direction (0). It is found that the inclination angle of enclosure is effective parameter on entropy generation especially for higher Richardson number (Ri > 1) due to domination of natural convection. However, Hartmann number is effective on both heat transfer and entropy generation for all values of Richardson and Prandtl numbers and it decreases the convective fluid flow and entropy generation.

A conservative viscous vorticity method for unsteady unidirectional and oscillatory flow past a circular cylinder

A conservative vorticity method for solving the 2-D incompressible unsteady viscous flow in the eulerian frame is proposed. The numerical method implements the vorticity-stream function formulation of the Navier–Stokes (N–S) equations, solved by the cell-centered Finite Volume Method (FVM). A convection limiter, coupling the first order upwind scheme and the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme, is introduced to maintain the accuracy and keep the monotonicity of vorticity interpolations. The vorticity induced stream functions are recovered by a closed-form expression, which also accurately predicts the far-field boundary conditions. The convective flux is evaluated by the difference of stream functions on the neighboring two nodes, which spontaneously eliminates the net mass flux out of a control cell. The conservative FVM and QUICK scheme and the definite evaluations of stream functions and convective flux guarantee the conservation of circulation and mass. The impulsively started flow past a circular cylinder at Reynolds numbers ranging from 100 to 9500 and the inline oscillatory flow around a circular cylinder at Reynolds numbers 100 and 200 are numerically simulated using this method. The current results under the inertial and non-inertial translational frame are well validated by comparisons with other numerical and experimental benchmark tests.

An Efficient Strategy to Deliver Understanding of Both Numerical and Practical Aspects When Using Navier-Stokes Equations to Solve Fluid Mechanics Problems

An efficient and thorough strategy to introduce undergraduate students to a numerical approach of calculating flow is outlined. First, the basic steps, especially discretization, involved when solving Navier-Stokes equations using a finite-volume method for incompressible steady-state flow are developed with the main aim being for the students to follow through from the mathematical description of a given problem to the final solution of the governing equations in a transparent way. The well-known ‘driven-cavity’ problem is used as the problem for testing coding written by the students, and the Navier-Stokes equations are initially cast in the vorticity-streamfunction form. This is followed by moving on to a solution method using the primitive variables and discussion of details such as, closure of the Navier-Stokes equations using turbulence modelling, appropriate meshing within the computation domain, various boundary conditions, properties of fluids, and the important methods for determining that a convergence solution has been reached. Such a course is found to be an efficient and transparent approach for introducing students to computational fluid dynamics.

A modified compact numerical algorithm to solve 2D Navier-Stokes equation

In this paper a modified fourth order compact scheme for solving unsteady 2-D Navier-stokes equation in stream function-vorticity formulation is proposed. The proposed scheme involves domain transformation and the governing cartesian equations are transformed to body fitted coordinate system. Explicit Euler’s forward discretization in time and central difference in space are applied except for few nodes adjacent to the computational boundaries. The scheme is tested on two benchmark problems Taylor vortex flow, Burgraff flow. The results in numerical form are compared with exact solutions and are in excellent agreement.

An Explicit Meshless Point Collocation Solver for Incompressible Navier-Stokes Equations

We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in their stream function-vorticity formulation. We use a uniform Cartesian embedded grid to represent the flow domain. We discretize the governing equations using the Meshless Point Collocation (MPC) method. We compute the spatial derivatives that appear in the governing flow equations, using a novel interpolation meshless scheme, the Discretization Corrected Particle Strength Exchange (DC PSE). We verify the accuracy of the numerical scheme for commonly used benchmark problems including lid-driven cavity flow, flow over a backward-facing step and unbounded flow past a cylinder. We have examined the applicability of the proposed scheme by considering flow cases with complex geometries, such as flow in a duct with cylindrical obstacles, flow in a bifurcated geometry, and flow past complex-shaped obstacles. Our method offers high accuracy and excellent computational efficiency as demonstrated by the verification examples, while maintaining a stable time step comparable to that used in unconditionally stable implicit methods. We estimate the stable time step using the Gershgorin circle theorem. The stable time step can be increased through the increase of the support domain of the weight function used in the DC PSE method.

Flow and heat transfer characteristics of a nanofluid between a square enclosure and a wavy wall obstacle

A mathematical model for the natural convection flow and heat transfer of a nanofluid in an annulus enclosed by a square cylinder and a wavy wall cylinder is developed. Using vorticity-stream function formulation, we first derive governing equations in the Cartesian coordinates. Then, these equations are transformed utilizing coordinate transformations into a system of equations valid for the present physical domain. The problem is solved using the finite difference method. It is found that for higher values of the volume fraction of nanoparticles, the number of undulations of the wavy wall of the inner cylinder and Rayleigh number, the strength of streamlines significantly increases. However, the amplitude of undulations diminishes the intensity of streamlines. The isotherms are also strongly influenced by these parameters. Contrary to this, the Nusselt number at the inner and outer cylinders is remarkably increased due to the increase of the volume fraction of nanoparticles, amplitude of undulations, and Rayleigh number. For the higher volume fraction of nanoparticles and Rayleigh number, the average Nusselt number at the inner and outer cylinders is higher. The maximum and minimum values of the velocity profile increase with the higher Rayleigh number. Nevertheless, the converse scenario is observed for the larger amplitude of undulation and volume fraction of nanoparticles. The temperature near the inner cylinder noticeably decreases with the increase of the Rayleigh number, whereas it slowly reduces for higher amplitude of undulations. Above all, this investigation might be helpful for the researchers in regard to the approach of making a more complex geometry by using coordinate transformations. Furthermore, the results could provide vital information about the problems in current technological applications.

A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations

A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time dilation of the solid region is approached to infinity. The mathematical and physical properties of the modified equations are investigated and implementation of the no-slip and no-penetration conditions are justified. Moreover, a suitable numerical method is presented for the solution of the modified equations. In the proposed numerical method, time integration is performed via the Crank–Nicolson method, and the semi-discrete equations are spatially discretized via second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, a sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The accuracy of the numerical solutions is evaluated.

Effect of isoflux thermal boundary condition on mixed convective heat transfer from a sphere for liquid metals

Numerical simulation is performed for mixed convective heat transfer past a sphere with uniform heat flux (Isoflux) thermal boundary condition for liquid metals. The steady, incompressible, laminar, axisymmetric and Newtonian fluid is considered in a vertically upward direction. A spherical geometry higher order compact scheme is utilised to solve the governing equations in stream function vorticity formulation together with the energy equation. The fourth order convergence of the current scheme is demonstrated. The velocity, streamline, isotherm contour profiles together with vorticity, pressure, local Nusselt number and average Nusselt number on the surface of the sphere are presented for the Reynolds number in the range , Richardson number in the range and Prandtl number in case of 0.020, 0.0208, 0.065 (liquid Gallium, liquid metal alloy , liquid lithium). The critical Richardson number is identified beyond which the recirculation phenomena of the flow are completely suppressed. The rate of isoflux heat transfer is more when compare to the isothermal case for the same range of parameters.

Structural bifurcation analysis of vortex shedding from shear flow past circular cylinder

In this paper, the unsteady flow separation of two-dimensional (2-D) incompressible shear flow past a circular cylinder is studied using theoretical structural bifurcation analysis based on topological equivalence. The stream function vorticity form of Navier–Stokes (N–S) equations in cylindrical polar coordinates are considered as the governing equations. Numerical simulations are performed, using higher-order compact finite difference scheme (Kalita and Ray J Comput Phys 228:5207–5236 2009), for the Reynolds number (Re) 100, 200 and shear parameter (K) 0.0, 0.05, 0.1 and 0.2. Through this structural bifurcation analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with primary and secondary vortices are studied. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, lift and drag coefficients, phase diagram are also studied to confirm the theoretical results. Unlike the shear flow past a square cylinder, in this case, the structural bifurcation occurs from the downstream surface of the circular cylinder for the same Reynolds number. All the computed results very efficiently and very accurately reproduce the complex flow phenomena. Through this study, many noticeable and interesting results for this problem are reported for the first time.

Analysis of forced convection from a horizontal flat plate for liquid metals

ABSTRACT Numerical simulation of the flow and forced convective heat transfer past a horizontal flat plate for liquid metals has b een performed in elliptical cylindrical coordinates. The two dimensional, steady incompressible Navier–Stokes equations in stream function vorticity formulation together with energy equation are solved using fourth order compact finite difference scheme. A stable and efficient method is introduced to solve the fourth order discretizations of governing partial differential equations through iterations in pseudo time. The fourth order convergence of the present scheme is demonstrated. The drag coefficient, average Nusselt number, variation of skin friction and local Nusselt numbers on the plate surface are shown graphically for Reynolds number up to and Prandtl numbers 0.004, 0.020 and 0.065, respectively for liquid sodium, liquid gallium and liquid lithium. It is found that skin friction coefficient and local Nusselt number approaches to Blasius solution and Pohlhaunsen solution (), respectively as Reynolds number approaches to infinity.

Conjugate Heat Transfer in a Microchannel Simultaneously Developing Gas Flow: A Vorticity Stream Function-Based Numerical Analysis

A simultaneously developing microchannel gas flow is analyzed numerically, using the vorticity–stream function form of the Navier–Stokes equation, together with the fluid energy equation and the solid wall heat conduction equation. Rarefaction, shear work, viscous dissipation, pressure work, axial conduction, and conjugate effects on heat transfer characteristics are investigated. The shear work contribution to the wall heat flux is evaluated in both the developing and the fully developed flow regions and compared with the conductive wall heat flux. The assumption of hydrodynamically fully developed, thermally developing flow—normally used in the analysis of channel heat transfer—is assessed and compared with the simultaneously developing flow case. Analytical expressions for the fluid flow and heat transfer parameters under fully developed conditions are also derived and compared with the numerical results for verification. The analysis presented shows that the shear work and the combined viscous dissipation and pressure work result in extending the thermal entrance length by far. Heat conduction in the wall also contributes to increase the thermal entry length. The results presented also demonstrate the shear work contribution to heat transfer in the slip flow regime, although minor in the very first portion of the thermal entrance length, and it becomes progressively more significant as the flow thermal development conditions are approached and turns out to be exactly equal in magnitude to the conductive wall heat flux in the thermally fully developed region, resulting in a zero Nusselt number, as verified by both the exact and numerical solutions.

Numerical Analyses of Wave Generation and Vortex Formation under the Action of Viscous Fluid Flows over a Depression

Transient free-surface deformations and evolving vortices due to the passage of flows over a submerged cavity are simulated. A two-dimensional stream function–vorticity formulation with a free-surface model is employed. Model results are validated against the limiting case of pure lid-driven cavity flow with comparisons of the vortical flow pattern and velocity profiles. The verification of the free-surface computations are also carried out by comparing results with published potential flow solutions for cases of flows over a depressed bottom topography. The agreements are generally good. Investigations are extended to other viscous flow conditions, where the cavity is set to have the normalized dimension of one by one when scaled by the still water depth. The free-surface elevations and streamline patterns for cases with Froude numbers ranging from 0.5 to 1.1 and different Reynolds numbers (Re = 5000 and 500) are calculated. At the condition of near-critical flow (Fr ≈ 1.0), the phenomenon of upstream advancing solitons is produced. Viscous effects on the free-surface profile reveal that at a lower value of Re (e.g., Re = 500) larger advancing solitary waves are generated. Vortical flow patterns in the cavity are examined for the cases with Fr = 1.0 and various values of Re. When Re = 5000, the vortex pattern includes a primary and a weak, but dominated secondary vortices at the time reaching a nearly quasi-steady motion. For the case of lower Re (e.g., Re = 500), a steady-state vortex pattern can be established with a clockwise primary vortex mostly occupied inside the cavity.

Numerical study of rectangular spectral collocation method on flow over a circular cylinder

Laminar flow over a circular cylinder has been widely used as a classical benchmark test for various numerical methods for partial differential equations. One of the popular ways is by using the vorticity-streamfunction formulation of the Navier-Stokes equations on a bounded numerical domain. The partial differential equations are solved numerically by the method of lines, where space is discretized using the standard collocation method, subject to multiple boundary conditions in the form of Dirichlet at the inlet and Neumann at the outlet. The resulting system of equations are then advanced in time using multistep methods. Fourier-Chebyshev pseudospectral method is used to approximate the solutions in space and Adams-Bashforth third-order backward differentiation method is employed as a time-stepping method. The rectangular spectral collocation method, developed by Driscoll and Hale, is applied to solve the ambiguity in imposing multiple boundary conditions on the same boundary points. The numerical simulations show very good agreement with similar studies for the Strouhal number, drag and lift coefficients over Reynolds numbers ranging from 50 to 150.

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.