Three-dimensional Steady Flow Research Articles

Effect of surface heat dissipation on thermocapillary convection of low Prandtl number fluid in a shallow annular pool

In order to understand the effect of surface heat dissipation on thermocapillary convection of low Prandtl number in a shallow annular pool, a series of three-dimensional numerical simulations were carried out by using the finite volume method. Results indicate that with the increase of surface heat dissipation, the thermocapillary convection intensity increases at first, and then decreases slightly; furthermore, the center of the thermocapillary convective cell moves gradually to the hot outer wall. With the increase of Marangoni number, the stable axisymmetric flow bifurcates orderly to the three-dimensional steady flow and the three-dimensional oscillatory flow at a weak surface heat dissipation. However, it bifurcates directly to the three-dimensional oscillatory flow at a strong heat dissipation. After thermocapillary convection destabilizes, the hydrothermal waves appear at the weak surface heat dissipation, but the longitudinal rolls near the outer wall are dominant at the strong heat dissipation. The flow pattern transition is always accompanied by the variations of the temperature fluctuation amplitude, the oscillatory frequency and the wave number.

Three-dimensional viscous jet flow with plane free boundaries

The problem of steady three-dimensional viscous flow with plane free boundaries, induced by a linear source or sink, is solved. The nonuniqueness of the solution in the case of a source and its vanishing in the case of a sink, as the Reynolds number reaches a certain critical value, is proved. The problem is investigated within the framework of the known class of the exact solutions of Navier–Stokes equations generalized in this study.

Effect of capillary ratio on thermal-solutal capillary-buoyancy convection in a shallow annular pool with radial temperature and concentration gradients

In order to understand the effect of capillary ratio on the thermal-solutal capillary-buoyancy convection in an annular pool subjected to simultaneous radial thermal and solutal gradients, a series of three-dimensional direct numerical simulations have been carried out. The working fluid was the toluene/n-hexane mixture fluid with the Prandtl number of 5.54 and the Schmidt number of 142.8. The capillary ratio Rσ varied from −0.8 to 0.2. Results show that the flow is axisymmetric and steady at small thermocapillary Reynolds number. Four types of such two-dimensional steady flow, i.e., counter-clockwise unicellular flow, clockwise unicellular flow, flows with two separate cells stratified horizontally and vertically, appear with the variations of capillary ratio and thermocapillary Reynolds number. With the further increase of thermocapillary Reynolds number, two-dimensional steady flow bifurcates firstly to three-dimensional steady flow at the first critical thermocapillary Reynolds number, then to three-dimensional oscillatory flow at the second critical thermocapillary Reynolds number. When the three-dimensional oscillatory flow appears, the temperature and concentration fluctuations on the free surface are indicated as either the coexistence of “hydro-thermal wave” and “hydro-solutal wave”, or the “vibrating straight spokes”. A reverse transition from periodic flow to steady flow is encountered.

An improved heat conduction and mass diffusion models for rotating flow of an Oldroyd-B fluid

An analysis is presented to inspect the steady three-dimensional boundary layer flow of an Oldroyd-B fluid over a stretching surface. Heat and mass transfer analysis are scrutinized by utilizing the non-Fourier’s and non-Fick’s models in rotating frame. The expressions of heat and mass fluxes are based upon the Cattaneo-Christov theory. The Cattaneo-Christov heat and mass flux models are used for the development of energy and concentration equations which account the contribution of thermal and concentration relaxation times. Additionally, temperature dependent thermal conductivity is also considered here. Apposite transformations are betrothed to obtain the ordinary differential equations with high nonlinearity. The analytical solutions are established for the resulting ordinary differential system by utilizing the homotopy analysis method (HAM). Our observations reveal that the rotation parameter serves to diminish the primary velocity f′(η) and associated momentum boundary layer thickness and similar trend is observed for the secondary velocity g(η). It is also observed that the thermal and concentration relaxation time parameters reduce the temperature and concentration profiles. For endorsement of the current relation, the attained consequences are eminent with former exploration in precise cases and splendid agreement has been distinguished.

Supersonic steady aerodynamics of a cylinder with a fluid flare

Basing upon the ideal inviscid perfect gas model the aerodynamic characteristics and the gasdynamic function distributions over the surface of a cylinder in a steady supersonic three-dimensional flow are studied in the absence and the presence of combined intense air injection from the flat-ended nose and the lateral surface of the body into the shock layer. The purpose of the study is to investigate the influence of gas injection from different regions of the cylinder surface on supersonic flow past the cylinder at different angles of attack.

Impact of chemical processes on 3D Burgers fluid utilizing Cattaneo-Christov double-diffusion: Applications of non-Fourier's heat and non-Fick's mass flux models

This article communicates the characteristics of Cattaneo-Christov heat and mass flux models on steady three-dimensional flow of Burgers fluid over a bidirectional stretching surface. These models are the modifications of classical Fourier's and Fick's laws through thermal and concentration relaxation times, respectively. Additionally, characteristics of heterogeneous and homogeneous reactions are further investigated. Transformation procedure is adopted to obtain the highly coupled nonlinear ordinary differential equations. The developed nonlinear problem is then solved analytically by utilizing the homotopy analysis method (HAM). The impact of various physical parameters on the temperature and concentration distributions are deliberated and physical characteristics of these parameters are exhibited through graphs and discussed through the reasonable judgment. Our investigation conveys that the temperature and concentration profiles decay as the nondimensional thermal and concentration relaxation time parameters enhances. On the other hand, it is perceived that concentration boundary layer thickness moderates with the growth of Deborah numbers β1 and β2. Additionally, it is fascinating to find that the concentration profiles decline as the Schmidt number escalates. To sum up, the verdict of the present investigation is that the proficiency of thermal and concentration systems can be enriched by introducing the Cattaneo-Christov heat and mass flux models.

Linear instability analysis of low- incompressible flow over a long rectangular finite-span open cavity

TriGlobal linear instability analysis and direct numerical simulations have been performed to unravel the mechanisms ultimately responsible for transition of steady laminar flow over a long rectangular finite-span open cavity with dimensions $L$ : $D$ : $W$$=$ 6 : 1 : 2 to unsteadiness. The steady laminar three-dimensional flow loses stability at $\mathit{Re}_{D,cr}\approx 1080$ as a consequence of linear amplification of a travelling eigenmode that is qualitatively analogous to the shear-layer mode known from analyses of flow in spanwise-periodic cavities, but has a three-dimensional structure which is strongly influenced by the cavity lateral walls. Differences in the eigenspectrum of the present and the spanwise homogeneous flow configuration are documented. Topological changes exerted on the steady laminar flow by linear amplification of the unstable shear-layer mode are reminiscent of observations in experiments at an order of magnitude higher Reynolds number.

Buoyancy effects on the 3D MHD stagnation-point flow of a Newtonian fluid

This work examines the steady three-dimensional stagnation-point flow of an electrically conducting Newtonian fluid in the presence of a uniform external magnetic field H0 under the Oberbeck–Boussinesq approximation. We neglect the induced magnetic field and examine the three possible directions of H0 which coincide with the directions of the axes.In all cases it is shown that the governing nonlinear partial differential equations admit similarity solutions. We find that the flow has to satisfy an ordinary differential problem whose solution depends on the Hartmann number M, the buoyancy parameter λ and the Prandtl number Pr.The skin-friction components along the axes are computed and the stagnation-point is classified. The numerical integration shows the existence of dual solutions and the occurrence of the reverse flow for some values of the parameters.

Three-Dimensional Stagnation-Point Flow and Heat Transfer of a Dusty Fluid Toward a Stretching Sheet

The steady three-dimensional stagnation-point flow and heat transfer of a dusty fluid toward a stretching sheet is investigated by using similarity solution approach. The freestream along z-direction impinges on the stretching sheet to produce a flow with different velocity components. The governing equations are transformed into ordinary differential equations by introducing appropriate similarity variables and an exact solution is obtained. The nonlinear ordinary differential equations are solved numerically using Runge–Kutta fourth-order method. The effects of the physical parameters like velocity ratio, fluid and thermal particle interaction parameter, ratio of freestream velocity parameter to stretching sheet velocity parameter, Prandtl number, and Eckert number on the flow field and heat transfer characteristics are obtained, illustrated graphically, and discussed. Also, a comparison of the obtained numerical results is made with two-dimensional cases existing in the literature and good agreement is approved. Moreover, it is found that the heat transfer coefficient and shear stress on the surface for axisymmetric case are larger than nonaxisymmetric case. Also, for stationary flat plat case, a similarity solution is presented and a comparison of the obtained results is made with previously published results and full agreement is reported.

Non-linear radiative flow of three-dimensional Burgers nanofluid with new mass flux effect

This article peruses the heat and mass transfer characteristics of three-dimensional steady flow of Burgers nanofluid over a bidirectional stretching surface. The convective boundary and nanoparticles mass flux conditions are considered. Additionally, the impact of non-linear thermal radiation and heat generation/absorption is delved. Further, the most recently proposed model for nanofluid is deliberated that necessitate nanoparticle volume fraction at the wall to be inertly rather than vigorously controlled. A set of similarity transformation is presented to alter the boundary layer equations into self-similar form and then tackled analytically by employing the homotopy analysis method (HAM). The effects of various controlling parameters to the heat and mass transfer characteristics are presented through graphs and scrutinized. The analytical out comes for the wall temperature gradient (Nusselt number) are calculated and presented through tables. It is seen that for enlarging values of the Brownian motion parameter lead to an attenuation in the concentration field as well as corresponding concentration boundary layer thickness. Likewise, it is noticed that the concentration field fall off hastily corresponding to Deborah number (β3) in comparison to Brownian motion parameter. Moreover, in order to perceive the validity of the existing effort, the numerical outcomes are compared with the analytical solutions attained by the HAM and noted an outstanding agreement for the limiting cases.

Thermocapillary convection with bidirectional temperature gradients in a shallow annular pool of silicon melt: Effects of ambient temperature and pool rotation

In order to understand the characteristics and instability of thermocapillary convection of silicon melt with bidirectional temperature gradients, a series of unsteady three-dimensional numerical simulations were carried out in a shallow annular pool. The imposed radial temperature gradient is established by the temperature difference between the outer and inner walls, while the imposed vertical temperature gradient is formed by adding a constant heat flux at the bottom and the thermal radiation on the top free surface. Two kinds of critical heat fluxes are determined. The first type is to ensure the whole silicon in the pool is in the molten state, and the second type is to judge the onset of oscillatory flow. When Marangoni number is small, a three-dimensional steady flow is common without pool rotation, but it becomes oscillatory at a higher ambient temperature or a small rotation Reynolds number. An axisymmetric steady flow can be observed at a large rotation Reynolds number in the case of small or moderate Marangoni number. Furthermore, the transition from the radial temperature gradient dominated flow to the vertical temperature gradient dominated flow is discussed.

The effect of symmetrical perforated holes on the turbulent heat transfer in the static mixer with modified Kenics segments

The numerical simulation on forced convective heat transfer of water in the modified Kenics static mixer (MKSM) was investigated by the three-dimensional turbulent and steady incompressible flow of computational fluid dynamics (CFD). The conservation equations were solved by using finite volume method (FVM) and the SIMPLEC algorithm scheme in ANSYS Fluent 16.1. The simulation results of Nusselt number and the friction factor in MKSM have a good agreement with the literature results. The effects of Reynolds number, perforated diameter, perforated spacing, and segment number on the heat transfer are evaluated in the range of Re=2000–20,000 under uniform heat tube wall temperature conditions. The Nusselt number increases gradually with increasing Re. Furthermore, both the values and deviations of Nusselt number for different perforated diameter accord with an antisymmetrical tendency at the critical Re=11,000. It could be seen that the Nusselt number and the friction factor are not sensitive to the increasing perforated spacing within a given perforated diameter. In view of thermal performance factor, the geometrical diameter and spacing of two symmetrical perforated holes are optimized. The MKSM with d/W=0.3 and s/W=0.6 have the best performance with the consideration of both heat transfer rate and friction loss compared to the others. Thermal performance factor decreases gradually with increasing Re which indicated that the modified Kenics segments are more effective for heat transfer in lower Re. The local and global synergies between secondary flow and temperature fields have been investigated through field synergy principle.

Computational study of desalination by reverse osmosis — Three-dimensional analyses

A computational study has been conducted to examine three-dimensional steady multicomponent fluid flows in the reverse osmosis membrane module. The module contains a net of spacers. The SST k-ω turbulence model is employed to simulate flow and concentration fields at Re=400 and 800, while the laminar model is employed to characterize flow and concentration fields at Re=100. Spacer grids with 30°, 45° and 60° are considered as three different geometries. The membrane is treated as a functional surface where water flux, concentration and local pressure are coupled. The nature of concentration polarization in each membrane module is determined. Characteristics of potential fouling buildup are determined from the wall shear stress distribution. Correlations between potential fouling regions and the concentration distribution are presented. The coefficient of performance for each membrane module is determined at all flow rates considered. It has been illustrated that all membrane modules perform better at higher flow rates. The membrane module containing the net of spacers in the 30° arrangement is shown to be the most efficient membrane module. This study proves that the configuration of spacers is an important optimization parameter for the design of reverse osmosis membrane modules.

Linear stability analysis of thermocapillary flow in a slowly rotating shallow annular pool using spectral element method

The stability of thermocapillary flow in a slowly rotating shallow annular pool was investigated by using the Legendre spectral element method. The silicon melt (Pr=0.011), filling in a pool with adiabatic free surface and bottom, was heated at the outer cylindrical wall and cooled at the inner cylindrical wall. The critical stability conditions for different dimensionless rotation rates ω, ranging from 0 to 2000, were determined by linear stability analysis. Moreover, the energy analysis was applied to further illustrate the underlying mechanism of the flow instability. The results indicate that there is one Hopf bifurcation for ω<940 and ω>1185. Thermocapillary flow is the dominant factor of the instability for ω<940, and the pool rotation becomes the key role for the first instability for ω>1185. Three turning points were observed in the interval 940⩽ω⩽1185, corresponding to three transitions between two-dimensional steady flow and three-dimensional oscillatory flow, owing to the competition of two driving forces with increasing Marangoni number at a fixed ω. With pool rotation, the results exhibit that the flow instability is deduced to occur initially in the zone near the cold wall with the evidence of the extreme velocity gradient and also the distribution of the local kinetic energy.

Numerical simulation on forced convective heat transfer of titanium dioxide/water nanofluid in the cooling stave of blast furnace

In this study, the numerical simulation on forced convective heat transfer of TiO2/water nanofluid in the cooling stave of blast furnace was investigated by the three-dimensional turbulent and steady incompressible flow of computational fluid dynamics (CFD). The conservation equations were solved by using finite volume method (FVM) and the SIMPLEC algorithm scheme to examine the four conditions, namely, the cooling stave materials (Cu, Steel, and Iron); the mass flow rate (0.5–2.5kg/s) of nanofluid; the volume fractions (0–3vol%) of nanoparticles; and the aggregation of nanoparticles (a-4a, where a indicates the average primary particle size), at heat convective boundary condition. In particular, the results show that comparing the 3vol% of nanoparticles dispersed in fluid with base liquid, thermal convection coefficient was raised from 16.3% to 49% by adjusting the velocity in terms of Reynolds number ranging from 56,800 to 75,700; and the Nusselt number was increased with the volume fraction of nanoparticles and mass flow rate of nanofluid because the probability of collision of nanoparticle was increased in the flowing base fluid, but decreased with increasing aggregation of nanoparticles in suspension due to raising the viscosity of nanofluid. Additionally, the ratio of Nusselt number to the friction factor and the correlations of nanofluid have been analyzed and developed in this work

Assessment on characteristics of heterogeneous-homogenous processes in three-dimensional flow of Burgers fluid

An analysis is performed to explore the characteristics of heterogeneous-homogeneous processes for the steady three-dimensional flow of Burgers fluid over a bidirectional stretching surface. In this paper, we utilized the advanced model of a homogeneous-heterogeneous reactions with equal diffusivities for reactant and autocatalysis. Additionally, heat transfer analysis is carried out in the presence of nonlinear thermal radiation and convective boundary conditions. The basic governing non-linear problem is presented and reduced into self-similar form with the aid of suitable similarity approach. The advanced non-linear problem is then tackled analytically by employing the homotpy analysis method (HAM). The effectiveness of relevant physical parameters on temperature and concentration profiles are taken into consideration. It is evident from the graphs that the concentration distribution diminishes by uplifting the homogeneous process parameter while it enhances for strength of Schmidt number. Moreover, it is observed that the surface heat transfer rate enhances for larger values of the Prandtl number.

Impact of thermophoresis particle deposition on three-dimensional radiative flow of Burgers fluid

This paper explores the analytical solution to heat and mass transfer of a three-dimensional steady flow of Burgers fluid over a bidirectional stretching surface. Additionally, analysis is carried out in the presence of thermal radiation, heat generation/absorption and thermophoretic effects. The governing non-linear problem is developed and transformed into self-similar form by utilizing similarity approach. The resulting non-linear problem is solved analytically by employing the homotpy analysis method (HAM). The obtained results are plotted and discussed in detail for interesting physical parameters. It is seen that increasing values of the thermophoretic parameter leads to a decrease in the concentration field and the corresponding concentration boundary layer thickness. Also, it is noticed that the concentration field decays quickly corresponding with thermophoretic parameter in comparison to Schmidt number.

A pressure-based velocity decomposition procedure for the incompressible Navier–Stokes equations

An alternative formulation is presented for the incompressible form of the Navier–Stokes equations. This is done through the introduction of a new vector variable which represents the sum of the velocity vector and the pressure gradient vector. The introduction of the new variable modifies the continuity equation to a Poisson equation for pressure, with the forcing function on the right-hand side representing the divergence of the new vector variable. Hence, the proposed formulation utilizes pressure to satisfy the divergence-free condition on the differential level in a simple and straightforward manner. Moreover, with pressure introduced to the continuity equation on the differential level, the present formulation allows direct computation of the pressure and velocity fields from the continuity and momentum equations, respectively, and eliminates the need for computing pressure and velocity corrections to satisfy mass conservation as in SIMPLE-type pressure correction methods. The proposed procedure is numerically tested against several two-dimensional and three-dimensional steady flow problems. Discretization is carried out using a second-order accurate central difference scheme on a staggered grid and the discretized equations are solved in a segregated manner. Numerical results from the present formulation are in good agreement with conventional incompressible flow solvers and with experimentation.

Modeling and analysis for three-dimensional flow with homogeneous-heterogeneous reactions

This work addresses the steady three-dimensional boundary layer flow of Maxwell fluid over a bidirectional stretching surface with homogeneous-heterogeneous reactions. A system of ordinary differential equations is obtained by using suitable transformations. Convergent series solutions are derived by homotopic procedure. Impact of various pertinent parameters on the velocity and concentration is discussed. It is noted that an increase in the Deborah number decreases both the velocity components. Also concentration distribution decreases for larger values of strength of homogeneous reaction parameter while it increases for strength of heterogeneous reaction parameter.

Numerical Simulation of Turbulent Thermal Fluid in Tee Water Duct Cooled Nuclear Reactor

The turbulent and thermal mixing in a vertically oriented T-junction is investigated numerically using ANSYS FLUENT software. By taking account the buoyancy forces, a steady state three-dimensional turbulent flow is considered with a Reynolds number of 0.4×105 at the cold inlet and 3.3×105 at the hot entrance. The k-ε standard model with standard wall function is chosen to provide closure for the Reynolds stress tensor. The numerical results presented in the form of velocity vectors field and contours of temperature distribution gave a good prediction of the dynamic and the thermal fields namely in the mixing region where a reversed flow is captured.