Vorticity Variables Research Articles

Stokes flow past an array of circular cylinders through slip-patterned microchannel using boundary element method

Two-dimensional viscous incompressible, pressure-driven, creeping flow at low Reynold's number (Re ≪ 1) around a series of circular cylinders in a slip-patterned rectangular microchannel is investigated numerically by using the boundary element method (BEM) based on a non-primitive variables approach. The non-primitive variables approach refers to the combination of stream function and vorticity variables. The Stokes equations are used to govern the flow of creeping fluid through a microchannel. We consider the alteration of the slip on both the upper and lower surfaces of the microchannel maintain the same phase (i.e., in-phase configuration). Here, the slip boundary condition refers to Navier's slip boundary condition. We considered both small as well as large patterned slip on both surfaces of the microchannel. Moreover, we have assumed that a number of cylinders of equal diameter are present in the in-line configuration in the path of flow. We studied streamlines, velocity profiles, pressure gradients, and the shear stresses with varied slip-length, and the radius of the cylinder, to get a complete comprehension of flow dynamics. We observed that the velocity and shear stress profiles exhibit significant variability in the case of fine slip patterning. Additionally, the proposed investigation holds several potential applications, such as drug capsule delivery systems, hemodynamics, bio-MEMS technology, and so forth.

On the Determination of the 3D Velocity Field in Terms of Conserved Variables in a Compressible Ocean

Explicit expressions of the 3D velocity field in terms of the conserved quantities of ideal fluid thermocline theory, namely the Bernoulli function, density, and potential vorticity, are generalised in this paper to a compressible ocean with a realistic nonlinear equation of state. The most general such expression is the ‘inactive wind’ solution, an exact nonlinear solution of the inviscid compressible Navier–Stokes equation that satisfies the continuity equation as a consequence of Ertel’s potential vorticity theorem. However, due to the non-uniqueness of the choice of the Bernoulli function, such expressions are not unique and primarily differ in the magnitude of their vertical velocity component. Due to the thermobaric nonlinearity of the equation of state, the expression for the 3D velocity field of a compressible ocean is found to resemble its ideal fluid counterpart only if constructed using the available form of the Bernoulli function, the Bernoulli equivalent of Lorenz’s available potential energy (APE). APE theory also naturally defines a quasi-material, approximately neutral density variable known as the Lorenz reference density. This density variable, in turn, defines a potential vorticity variable that is minimally affected by thermobaric production, thus providing all the necessary tools for extending most results of ideal fluid thermocline theory to a compressible ocean.

Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.

Meteorological Conditions That Promote Persistent Contrails

Climate-impacting contrails need ice (super-)saturation to persist longer than a few minutes. However, this simple criterion cannot be easily applied for the prediction of persistent contrails. The current weather forecast models, which lack humidity data for assimilation in the upper troposphere, have difficulties coping with the enormous variability and sharp gradients in the relative humidity field. Thus, ice supersaturation, which is an extremal state of relative humidity, is hard to forecast at a precise location and time to allow contrail-avoiding flight routing. In this paper, we investigate the possibility of using dynamical proxy variables for improved contrail prediction. This idea is guided by the fact that the probability of ice supersaturation differs in different dynamical regimes. Therefore, we determine probability distributions of temperature, water vapour concentration, vertical velocity, divergence, relative and potential vorticity, geopotential height, and lapse rate conditioned on three situations: (a) contrail persistence not possible; (b) contrail persistence possible; and (c) strongly warming persistent contrails possible. While the atmospheric variables are taken from reanalysis data, the conditions (a–c) are based on airborne measurement data and radiation quantities from the reanalysis. It turns out that the vorticity variables, and in particular geopotential and lapse rate, show quite distinct conditional probabilities, suggesting a possibility to base an improved forecast of persistent contrails not only on the traditional quantities of temperature and relative humidity, but on these dynamical proxies as well. Furthermore, we show the existence of long flight tracks with the formation of strongly warming contrails, which are probably embedded in larger ice-supersaturated regions with conditions that foster such contrails. For forecasting purposes, this is a beneficial property since the humidity forecast is easier on large, rather than small, spatial scales.

Numerical investigations of hydrogen and air mixture with vortex tube and duct combinations

Hydrogen gas stands as a renewable energy fuel for the transportation sector. The hydrogen to air ratio is an essential factor in deciding the combustion and therefore the thermal efficiency of an internal combustion engine. In this study, the mixing of hydrogen and air is considered among the four geometrical configurations. These cases are considered from the simple duct and vortex tube which induces an additional swirl inside a cylinder. The approach of the study involves the meshing and computation of mass, momentum and energy equations by finite volume method. After the convergence of results, mixture ratio is analyzed with the velocity components and vorticity variables. The ratio of hydrogen to air is found to vary between 8 and 35 among the cases considered. The desired ratio could be obtained from the combination of vortex tube and plain duct configurations. Stratification of mixture is found to be significant from the gradient of velocity components in this study.

Micropolar Blood Flow in a Magnetic Field

In this paper we numerically solve a flow model for the micropolar biomagnetic flow (blood flow) in a magnetic field. In the proposed model we account for both electrical and magnetic properties of the biofluid and we investigate the role of microrotation on the flow regime. The flow domain is in a channel with an unsymmetrical single stenosis, and in a channel with irregular multi-stenoses. The mathematical flow model consists of the Navier–Stokes (N–S) equations expressed in their velocity–vorticity (u–ω) variables including the energy and microrotation transport equation. The governing equations are solved by using the strong form meshless point collocation method. We compute the spatial derivatives of the unknown field functions using the discretization correction particle strength exchange (DC PSE) method. We demonstrate the accuracy of the proposed scheme by comparing the numerical results obtained with those computed using the finite element method.

Unsteady Natural Convection between Two Eccentric Hemispheres

Natural convection heat transfer in open or closed cavities takes place in different engineering areas. The hemispherical cavity is a part of basic geometries although it is not widely studied. The present paper reports the numerical study of natural convection in a closed hemispherical annulus delimited by two vertically eccentric hemispheres filled with Newtonian fluid (air in this case with Pr = 0.7) is conducted. The inner hemisphere is heated by a heat flux of constant density and the outer one is maintained isothermal. Based on the Boussinesq assumptions, the governing equations are numerically studied using unsteady natural convection formulated with vorticity and stream-function variables. These equations are written by using bispherical coordinates system and solved by using a finite difference method. The effect of the control parameters such as the Rayleigh number (103 ≤ Ra ≤ 106) or the eccentricity (e = ±0.2, ±0.5, 0) in the dynamic and thermal behaviours of the fluid is investigated.

Numerical Simulation of Natural Air Convection in Inclined Eccentric Hemispheres Enclosure

The subject of natural convection heat transfer is motivated by a wide range of applications in engineering technology. The hemispherical cavity is a part of basic geometries although it is not widely studied. The effect of inclinaison on natural convection fluid motions in the gap between two eccentric hemispheres is numerically studied. The inner hemisphere is subjected to a heat flux of a constant density and the outer one is maintened isothermal. The walls separating the two hemispheres are thermally adiabatic. Equations are formulated with vorticity and stream-functions variables. It is also assumed the fluid incompressible and obeys the approximation of Boussinesq. These equations are written by using bispherical coordinates system and solved by using a finite difference method. The results show the topology of flow is strongly dependent on the inclinaison because the flow can change from a unicellular regime to a multicellular regime by varying the inclination from 0 to π. By increasing the Rayleigh number (103Ra7), the flow intensifies. The results are shown in terms of streamlines and isotherms during their transient evolution.

Structure of a viscoplastic fluid flow in a pipe with a given obstruction shape

In this paper, we present a study of the axisymmetric steady-state laminar flow of a viscoplastic fluid in a pipe with variable cross section. Mathematical formulation of the problem is written in terms of stream function and vorticity variables in a cylindrical coordinate system. Rheological behavior of the medium is described by the Shvedov-Bingham model. Coordinate transformation is carried out to implement a numerical algorithm. The problem is solved numerically using the finite-difference method and the false transient method. This paper demonstrates the effect of Reynolds number, Bingham number, and geometric parameters of a flow domain on the flow structure.

Thermal performance in transient MHD thermogravitational convection of nanofluid with various heating effects

This work presents numerical simulation of two-dimensional thermogravitational energy transport in a chamber filled with copper–water ( $${\hbox {Cu-H}}_\mathrm{2} \hbox {O}$$ ) nanoliquid under the uniform magnetic impact. In this study, our aim is to analyze the characteristic role of nanoliquid thermal conductivity under the influence of various thermal boundary conditions with constant magnetic effect. The mathematical model of the flow physics consists of the Navier–Stokes (N–S) equations written using streamfunction–vorticity ( $$\psi$$ -- $$\zeta$$ ) variables including the energy transport equation. The governing equations are solved by using a higher-order compact scheme based on finite difference method. The impact of key characteristics including nanoliquid volume fraction ( $$0\le \phi \le 0.04$$ ), Rayleigh number ( $$10^{4}\le {\hbox {Ra}}\le 10^{6}$$ ), Hartmann number ( $$0\le {\hbox {Ha}}\le 60$$ ) and amplitude of heating ( $$0\le I \le 1$$ ) is analyzed in detail. It is found that the energy transport augmentation occurs with nonuniform heating over uniform heating and the rate of thermal transmission rises with a growth of $${\hbox {Ra}}$$ and $$\phi$$ but it decreases with the growth of $${\hbox {Ha}}$$ number. In addition, the transient structures are very useful for understanding the thermo- and magnetohydrodynamic problems.

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.

Effect of Porous Medium and Copper Heat Sink on Cooling of Heat-Generating Element

Cooling of heat-generating elements is a challenging problem in engineering. In this article, the transient free convection of a temperature-dependent viscosity liquid inside the porous cavity with copper radiator and the heat-generating element is studied using mathematical modeling techniques. The vertical and top walls of the chamber are kept at low constant temperature, while the bottom wall is kept adiabatic. The working fluid is a heat-conducting liquid with temperature-dependent viscosity. A mathematical model is developed based on dimensionless stream function, vorticity, and temperature variables. The governing properties are the variable viscosity, geometric parameters of the radiator, and size of thermally insulated strip on vertical surfaces of the cavity. The effect of these parameters on the energy transport and circulation patterns are analyzed numerically. Based on the numerical results obtained, recommendations are given on the optimal values of the governing parameters for the effective operation of the cooling system. It is shown that the optimal number of radiator fins for the cooling system configuration under consideration is 3. In addition, the thermal insulation of the vertical walls and the increased thickness of the radiator fins have a negative effect on the operation of the cooling system.

Differences in Northward Propagation of Convection Over the Arabian Sea and Bay of Bengal During Boreal Summer

AbstractThe governing dynamics that modulate the propagation characteristics of intraseasonal oscillations (ISO) during summer monsoon over the two ocean basins, Bay of Bengal (BoB) and Arabian Sea (AS), are investigated using observational analysis and high‐resolution regional coupled ocean‐atmosphere climate model simulations. ISO features are extracted over the Indian region using a data‐adaptive spectral method called multichannel singular spectrum analysis. ISO exhibits stronger intensity over the BoB than over the AS. But ISO‐filtered rainfall propagates at a faster rate ( 1.25°/day) over AS as compared to BoB ( .74°/day), giving rise to a northwest‐southeast tilted band of rainfall anomalies. However, the composite diagrams of several atmospheric fields associated with northward propagation like vorticity, low‐level convergence, and oceanic variables like sea surface temperature and mixed layer depth do not show this difference in propagation speed and all exhibit a speed of nearly 0.75°/day in both the ocean basins. The difference in speed of ISO‐filtered rainfall is explained through moisture flux convergence. Anomalous horizontal moisture advection plays a major role over AS in preconditioning the atmosphere and making it favorable for convection. Anomalous wind acting on climatological moisture gradient is the dominant term in the moisture advection equation. Easterly wind anomalies associated with a low‐level anticyclone over India helps advect moisture from the eastern side of the domain. The northwest‐southeast tilt of ISO is dictated by the atmospheric processes of moisture advection with the upper ocean playing a more passive role in causing the tilt.

Turbulent natural convection combined with entropy generation in a nanofluid cavity with non-uniformly heated side walls

The two-dimensional turbulent natural convection and entropy generation within an alumina-water nanoliquid chamber in the presence of sinusoidal temperature profiles at vertical walls is calculated. The considered process has been modelled using the RANS (Reynoldsaveraged Navier–Stokes) approach for turbulent regime and Boussinesq approximation for the buoyancy force. The control equations written using the non-dimensional stream function, vorticity and temperature variables combined with parameters of standard κ-ε turbulence model have been solved by the finite difference method with non-uniform computational mesh. The special algebraic transformation has been used for an introduction of the non-uniform mesh in physical domain. The influence of the nano-sized particles concentration and Rayleigh number on the nanofluid circulation and energy transport patterns has been examined.

Numerical Analysis of the Transient Process of Flow and Heat Transfer at High Rayleigh Number in a Partially Open Habitat Heated from Above

In this work, we studied the thermoconvective instabilities in a pentagonal cavity containing a Newtonian fluid. The cavity provided with a side opening is uniformly heated from above by a constant heat flux. The natural ventilation phenomenon in the classic habitat of the hot climate is thus numerically analyzed with unsteady natural convection equations formulated with vorticity and stream-function variables. The finite volume predictions of two-dimensional laminar natural convection at high Rayleigh number are presented. Results show that the incoming fresh air and the hot air discharge begin with the late start of the convection. The phenomenon intensifies with time and the birth of instabilities improves the homogenisation of temperatures which imply the elimination of very cold and very hot areas. However, the competition between the incoming fresh air and the hot air expansion leads to a perpetual displacement of the thermal front. The cross-sections at the opening of the incoming fresh and outgoing hot air are time-varying and the penetration depth of the fresh air is highlighted by the large convective cells originated from the aperture. The non monotonic variation of the Nusselt number reflects not only the multicell nature of the flow but also expresses the heat lost by the active walls due to the fresh air.

Unsteady natural convection in a partially porous cavity having a heat-generating source using local thermal non-equilibrium model

PurposeThe purpose of this study is a numerical analysis of transient natural convection in a square partially porous cavity with a heat-generating and heat-conducting element using the local thermal non-equilibrium model under the effect of cooling from the vertical walls. It should be noted that this research deals with a development of passive cooling system for the electronic devices.Design/methodology/approachThe domain of interest is a square cavity with a porous layer and a heat-generating element. The vertical walls of the cavity are kept at constant cooling temperature, while the horizontal walls are adiabatic. The heat-generating solid element is located on the bottom wall. A porous layer is placed under the clear fluid layer. The governing equations, formulated in dimensionless stream function, vorticity and temperature variables with corresponding initial and boundary conditions, are solved using implicit finite difference schemes of the second order accuracy. The governing parameters are the Darcy number, viscosity variation parameter, porous layer height and dimensionless time. The effects of varying these parameters on the average total Nusselt number along the heat source surface, the average temperature of the heater, the fluid flow rate inside the cavity and on the streamlines and isotherms are analyzed.FindingsThe results show that in the case of local thermal non-equilibrium the total average Nusselt number is an increasing function of the interphase heat transfer coefficient and the porous layer thickness, while the average heat source temperature decreases with the Darcy number and viscosity variation parameter.Originality/valueAn efficient numerical technique has been developed to solve this problem. The originality of this work is to analyze unsteady natural convection within a partially porous cavity using the local thermal non-equilibrium model in the presence of a local heat-generating solid element. The results would benefit scientists and engineers to become familiar with the analysis of convective heat transfer in enclosures with local heat-generating heaters and porous layers, and the way to predict the heat transfer rate in advanced technical systems, in industrial sectors including transportation, power generation, chemical sectors and electronics.

Natural convection of Al2O3/H2O nanofluid in a cavity with a heat-generating element. Heatline visualization

Cooling of electronic devices is a very important topic. The present paper deals with numerical simulation of natural convection cooling of heat-conducting and heat-generating source using an alumina-water nanofluid under the effect of cold vertical walls. Governing equations formulated in dimensionless stream function, vorticity and temperature variables have been solved by the finite difference method. The effects of nanoparticles concentration, heat source location, internal heat generation flux and heat source material on fluid flow and heat transfer have been studied. It has been found that an addition of nanoparticles can intensify the cooling process when the heat is located near the vertical cold wall.

Magneto-hydrodynamic natural convection of CuO-water nanofluid in complex shaped enclosure considering various nanoparticle shapes

PurposeThe purpose of this study is to peruse natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method.Design/methodology/approachGoverning equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with the Koo–Kleinstreuer–Li correlation for the effective dynamic viscosity and the effective thermal conductivity have been solved numerically by control volume finite element method.FindingsEffects of various pertinent parameters such as Rayleigh number, Hartmann number, volume fraction of nanofluid and shape factor of nanoparticle on the convective heat transfer characteristics are analysed. It was observed that local and average heat transfer rates increase for higher value of Rayleigh number and lower value of Hartmann number. Among various nanoparticle shapes, platelets were found to be best in terms of heat transfer performance. The amount of average Nusselt number reductions was found to be different when nanofluids with different solid particle volume fractions were considered due to thermal and electrical conductivity enhancement of fluid with nanoparticle addition.Originality/valueA comprehensive study of the natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method is addressed.

MHD natural convection of Cu/H2O nanofluid in a horizontal semi-cylinder with a local triangular heater

Purpose The purpose of this study is to investigate free convection of copper-water nanofluid in an upper half of circular horizontal cylinder with a local triangular heater under the effects of uniform magnetic field and cold cylinder shell using control volume finite element method (CVFEM). Design/methodology/approach Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with Brinkman correlation for the effective dynamic viscosity and Hamilton and Crosser model for the effective thermal conductivity have been solved numerically by CVFEM. Findings The impacts of control parameters such as the Rayleigh number, Hartmann number, nanoparticles volume fraction, local triangular heater size, shape factor on streamlines and isotherms as well as local and average Nusselt numbers have been examined. The outcomes indicate that the average Nusselt number is an increasing function of the Rayleigh number, shape factor and nanoparticles volume fraction, while it is a decreasing function of the Hartmann number. Originality/value A complete study of the free convection of copper-water nanofluid in an upper half of circular horizontal cylinder with a local triangular heater under the effects of uniform magnetic field and cold cylinder shell using CVFEM is addressed.

Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

We describe an energy–enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy–enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy–enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere.