Boussinesq Assumption Research Articles

Thermodynamic optimization of conjugate convection from a finned channel using genetic algorithms

For the first time, this study reports the results of numerical investigation of conjugate convection from a finned channel. The computational domain of investigation consists of a horizontal channel with vertical rectangular fins being mounted on outside of the channel. The equations governing two-dimensional, steady, incompressible, constant property laminar flow have been solved for the fluid flowing outside the channel. In doing this, Boussinesq assumption is assumed to be valid for the fluid flowing outside the channel along the fins. For the fluid flowing inside the channel, flow is assumed to be turbulent with forced convection as the mode of heat transfer. From a large volume of numerically generated data correlations have been proposed for (1) Nusselt number and (2) Entropy generated by the system. These correlations are finally used to obtain thermodynamic optimum where in we seek a solution with minimum total entropy generation rate for varying heat duties, by using the state-of-the art Genetic algorithms.

Numerical investigation of flow reversal and instability in mixed laminar vertical tube flow

The problem of the transient laminar mixed convection flow of air in a vertical tube under buoyancy effect and high wall heat flux condition has been numerically investigated by using a full 3D-transient-model and Boussinesq's assumptions. The tube is submitted to a uniform but time-dependent wall heat flux. Results have clearly shown that the flow reversal was first initiated near the tube exit section, on the tube wall and at the level Gr=3×10 5 for opposed-buoyancy case, and on the tube centerline and at the level of Grashof number around 10 6 for assisted-buoyancy one. With further increase in time of the wall heat flux, such reversed flow region has considerably increased in size and intensity and it clearly spreads into the upstream region. The presence of such recirculation cells has drastically perturbed the flow and the thermal field as well as the heat transfer. Although the flow structure appears to conserve its axisymmetrical characters even for cases with very high Grashof numbers, results from this study have shown that the flow seems to remain stable and unique up to the level Gr=5×10 5 and 10 6, respectively, for opposed and assisted buoyancy cases. Beyond these critical Grashof numbers, an extremely slow and rather difficult and tedious convergence behaviors have been experienced, which are believed to be due to a possible flow transition.

Volume Averaged Pressure Interactions for Dispersed Droplet Phase Modeling of Multiphase Flow

This study addresses cross-phase interactions and momentum exchange between dispersed (droplet) and carrier (air) phases of multiphase flow. Alternative formulations are developed for the volume averaged interfacial pressure and apparent mass forces. Algebraic source terms in the momentum equation can accommodate the interfacial shear stresses. Multiphase turbulent stresses in the carrier phase are modeled with a Boussinesq assumption. New interfacial pressure modeling in an Eulerian formulation with volume averaging is developed. Nomenclature ai = surface of the phase interface akw = surface of phase k in contact with the wall surface C D = drag coefficient Dd = droplet diameter g = gravity I = identity tensor jk = flux of the general scalar quantity ˙ mk = mass flux for the particular phase nk, n1, n2 = normal vector for the particular phase nkw = normal vector for the particular phase at the wall surface p = pressure pk = local pressure of phase k � pk� =v olume-averaged pressure of phase k � pki� =v olume-averaged interfacial pressure of phase k Rer = relative (droplet-air) Reynolds number S = surface ˆ S = source term V =v olume v = phase velocity [v, dispersed phase; va,

Flow pattern and heat transfer of swirling flows in cylindrical container with rotating top and stable temperature gradient

The flow pattern and the heat transfer characteristics of confined swirling flows of viscous incompressible fluid in an cylindrical container are numerically investigated in the axisymmetric flow regime under the Boussinesq assumption. The flows are driven by rotating the top cover at a constant angular speed and stable temperature difference is imposed between the top and bottom discs with the side walls thermally insulated. Steady state solutions are obtained for ranges of governing parameters, the Reynolds number Re, the Richardson number Ri in 10 2⩽ Re⩽3×10 3 and 0⩽ Ri⩽1.0 at fixed values of the Prandtl number Pr=1.0 and the cylinder aspect ratio h=1.0. For the flows with small Ri, meridional main circulation resides in the entire container convecting the heat from top to bottom disc. When Ri is increased to O(10 0), horizontally layered structure appears with quiescent lower half and vertically linear distribution of the temperature prevailing in much of the bulk. At intermediate values of Ri, i.e. Ri∼O(10 −1), flow separation occurs on the bottom disc depending on the values of Re and Ri. The flow patterns are classified into several different types on the ( Ri, Re) plane. The average Nusselt number Nu which reflects the change of the flow structure, is a monotonically decreasing function of Ri and an increasing function of Re. The torque coefficient C T is also computed and found to be a mildly decreasing function of Ri for the parameters considered.

End-depth in inverted semicircular channels: experimental and theoretical studies

The flow upstream of a free overfall from smooth inverted semicircular channels is theoretically analysed to compute the end-depth ratio (EDR), applying an energy equation based on the Boussinesq assumption. This approach eliminates the need for an experimentally determined pressure coefficient. Experiments were conducted with horizontal channel conditions. The EDR related to the critical depth, which occurs upstream from the end section, is found to be around 0.695 for a critical depth-diameter ratio up to 0.40. A simple method is presented to estimate the discharge from a known end-depth. The theoretical model corresponds closely with the experimental data.

축류형 유체기계에서 익단 누설 유동 해석을 위한 난류 모델 성능 평가

It is experimentally well-known that high anisotropies of the turbulent flow field are dominant inside the tip leakage vortex, which is attributable to a substantial proportion of the total loss and constitutes one of the dominant mechanisms of the noise generation. This anisotropic nature of turbulence invalidates the use of the conventional isotropic eddy viscosity turbulence models based on the Boussinesq assumption. In this study, to check whether an anisotropic turbulence model is superior to the isotropic ones or not, the results obtained from the steady-state Reynolds averaged Navier-Stokes simulations based on the RNG k-εmodel and the Reynolds stress model (RSM) are compared with experimental data for two test cases: a linear compressor cascade and a forward-swept axial-flow fan. Through this comparative study of turbulence models, it is clearly shown that the RSM, which can express the production term and body-force term induced by system rotation without introducing any modeling, should be used to predict quantitatively the complex tip leakage flow, especially in the rotating environment.

Natural convection due to a heat source on a vertical plate

In this work, the flow over a discrete heat source flush mounted on a vertical adiabatic surface is studied. The two- and three-dimensional formulations were adopted in order to compare with the results presented in the literature. The flow was assumed incompressible and laminar with constant fluid properties under Boussinesq assumption. The finite volume control method was used for the discretization of the elliptic governing equations. The numerical results presented confirm in part the theoretical behavior and show the existence of a transition between the two- and three-dimensional plume. Additionally, similarity scales for the thermal boundary layer were proposed for the xy and zy planes. The presence of this region of transition emphasizes the complexity of this kind of flow, where two- or three-dimensional effects must not be studied separately, mainly on electronic packaging where heat sources are in that transition region.

Stationary Transverse Diffusion in a Horizontal Porous Medium Boundary-Layer Flow with Concentration-Dependent Fluid Density and Viscosity

Stable transport of high-concentrated solute is considered in horizontal boundary-layer flows above a wall of constant concentration. Mixing is accomplished by advection and molecular diffusion only. The utilized boundary-layer approximation allows to investigate the exclusive influence of gravity on vertical diffusion. The hydrodynamic dispersion mechanism was disregarded in the present study which confines its applicabilty to flows with small molecular Peclet numbers. A linear variability of both the fluid's density and viscosity with changing concentration is taken into account as well as the complete set of mass-fraction based balance equations. Steady-state concentration and velocity distributions above the horizontal wall have been obtained using the series truncation method which recently had proven successful to solve the corresponding problem using the Boussinesq assumption. The impact of the latter on these distributions is discussed by what has been additionally-facilitated by the existence of an exact analytical solution for the simpler Boussinesq case. Whereas no density variability influence exists with use of the Boussinesq assumption the complete system of mass-fraction based equations predicts opposing effects of density and viscosity differences between oncoming and near-wall fluids on concentration distributions. Larger density differences narrow the transition zone between both fluids, larger viscosity differences widen it. Thus, a compensation of both effects can be observed for individual fluids and for certain regions of the flow field.

Flow and heat transfer in convection-dominated melting in a rectangular cavity heated from below

Melting of a pure phase change material (PCM) in a rectangular container heated from below is simulated using the Streamline Upwind/Petrov Galerkin finite element method in combination with a fixed grid primitive variable method. Boussinesq assumption is invoked and two-dimensionality is assumed. Flow patterns for a wide range of Rayleigh numbers are presented. Instability of free convection flow during the melting process is discovered and discussed.

On vortex air motions above an axisymmetric source of mass, momentum and heat

We study axisymmetric plume dispersion from a steady source of mass, momentum and/or heat that is subjected to either a time-dependent large-scale external vortex or small-scale turbulent axisymmetric helicity. On the basis of the turbulent boundary layer and Boussinesq assumptions and by assuming similarity profiles with Gaussian distribution in the radial direction the balance equations of mass, momentum, and energy reduce to a system of nonlinear differential equations for amplitude functions of axial velocity, pressure and density differences as well as azimuthal velocity. The system of equations is closed with Taylor's entrainment assumption.The plume radius and the typical radius of the large-scale external vortex are also determined. For a simple density structure of the ambient atmosphere (i.e. adiabatic conditions) analytical results can be obtained, but for more complicated cases, i.e. a layered polytropic atmosphere, the governing equations are examined numerically; computations are reasonably simple and efficient.

On the stream function for variable‐density groundwater flow

When the stream function is used to describe variable‐density ground water flow, it should be defined in terms of mass flux. The historic use of the stream function describes volume flux and implicitly invokes the Boussinesq assumption. A comparison of the flow equations for mass‐based and volume‐based stream functions indicates that this assumption can lead to significant errors in the calculated flow field, especially when flow parallels large density gradients.

Three-dimensional strip-integral method for incompressible turbulentboundary layers

A three-dimensional, strip-integral method was developed to calculate incompressible turbulent boundary layers. The entrainment coefficient is not required and no empirical relation is needed for the skin-friction coefficient. The Boussinesq assumption is used for the Reynolds stresses, and the eddy viscosity is defined using an extended Clauser outer region model. A practical four-parameter velocity profile was formulated: the streamwise velocity is represented by a new law of the wake based on the Johnston law of the wall and the Moses wake function. The crosswise velocity is represented by the Johnston triangular model

Similarity Solution For Convection Near a Vertical Plate Without Using the Boussinesq Approximation

It is well known that the Boussinesq approximation works well in the field of convection flow and heat transfer. However, the basis of this approximation is that temperature in the flow field varies little. Can we extend this assumption to the case in which large temperature difference exists? The engineering answer is definite so far, but from theoretical point of view, it is questionable. What can we do then? This is what we have attempted to do in this paper. In this paper, we propose a new approach to deal with convective heat transfer without using the Boussinesq assumption.

Free thermohaline convection in sediments surrounding a salt column

Complex groundwater convection patterns are possible near salt domes because groundwater is subject to both lateral heat and salinity gradients. In order to assess the mechanisms responsible for driving convection near salt domes we use dimensional analysis and numerical simulations to investigate coupled heat and salt transport in homogeneous sediments surrounding a cylindrical salt column. The dimensional analysis does not require the Boussinesq assumption. The coupled heat, solute, and groundwater transport equations are controlled by three dimensionless parameters: the Rayleigh number, the Lewis number, and the buoyancy ratio. The buoyancy ratio is the ratio of salinity to temperature effects on groundwater density, and it directly affects the groundwater flow equation. A finite difference numerical multigridding algorithm is used to iteratively solve the coupled transport equations. The multigridding technique was about 3 times faster than a point‐wise successive overrelaxation solution. Boundary conditions for the numerical simulations were adjusted to represent different contrasts in the thermal gradient between the salt and the overlying sediments. The contrast in thermal gradient is parameterized by the thermal conductivity ratio and is responsible for isotherms being elevated near the salt. The analysis suggests that a wide range of convective flow patterns are possible, with flow occurring either up or down along the salt flank. The sense of convection is dependent mainly on the value of the buoyancy ratio and how sharply isotherms are pulled up near the salt. These factors in turn depend on the regional salinity variation, the time since diapirism, and the thermal conductivity of water saturated sediments. While this analysis provides useful insight into the mechanisms driving free convection near salt domes, the assumptions about medium and fluid properties may limit the applicability of dimensional analysis in simulating flow in specific geologic settings.

Buoyant convection driven by an encapsuled spinning disk with axial suction

An analysis is made of the flow and thermal structures of a viscous fluid confined in a vertically mounted cylindrical container. The bottom endwall disk and the sidewall are rotating steadily about the cylinder axis, and the top endwall disk is stationary. A gravitationally stable temperature difference is applied on the container boundaries. In order to formulate a flow model of greater relevance to the Czochralski growth of crystals from melt, an axial suction through the rotating disk and a concomitant radial inflow through the sidewall are imposed. The main motivation of the study is to examine the effects of suction in a finite configuration and of the fluid stratification. Numerical solutions to the axisymmetric Navier-Stokes equations with the Boussinesq assumption have been obtained. The parameter ranges are that the rotational Reynolds number is large and the cylindrical aspect ratio, the Prandtl number, the suction parameter, and the stratification number are all of order unity. Comprehensive flow details have been acquired. For a stratified flow, the principal balance in the interior core is characterized by a relationship between the radial temperature gradient and the vertical shear in the azimuthal flow. The magnitudes of azimuthal flows in the core region increase as the suction increases. Significant differences are seen in the meridional flow patterns as the effects of suction and stratification are incorporated in the analysis. As the stratification effect becomes appreciable, larger portions of the meridional fluid transport are short-circuited directly from the sidewall to the rotating disk. The structure of the thermal field for a stratified fluid is plotted to illustrate the dominant modes of heat transfer. It is shown that the convective heat transfer associated with the meridional fluid transport is dominant in the main body of the flowfield.

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow.

A two-dimensional time-dependent model for surface shear and buoyancy-driven flows in domains with large aspect ratio

A two-dimensional numerical model has been developed for studying flows in enclosures having a large length-to-depth ratio. The model solves the two-dimensional Navier-Stokes and energy equations subject to hydrostatic, Boussinesq, and rigid-lid assumptions. Turbulent momentum and heat diffusion are incorporated using variable eddy coefficients. The flow is driven by surface wind shear and heat transfer. Two cases were run for a rectangular domain, subject to a suddenly imposed surface shear stress while the surface and bottom were mainained at two different but constant temperatures. The difference between the cases was the formulation used for predicting the eddy viscosities and diffusivities. The variations of velocities and temperatures with time were studied. Both velocities and temperatures were found to undergo a sudden rapid change, after an initial period of slow change, before attaining steady state. This has been explained in light of the differences in the convective and diffusive time scales and the nature of coupling between the governing equations.

Prediction of the Natural Convective Heat Transfer From a Horizontal Heated Disk

A thin-layer approximation is applied to the laminar momentum and energy equations governing the natural convection above an isothermal heated disk in air. Using the Boussinesq assumption the equations are nondimensionalized in terms of a stream function, pressure and temperature difference. The variables are expanded in a series solution and the resulting set of equations are solved numerically. The solution is cast in terms of the nondimensional radial position and the disk Grashof number. These two parameters are shown to define the outer boundary conditions which are uniquely determined from a point source solution. The outer velocity boundary condition is shown to decrease in relative magnitude as the disk Rayleigh number increases. Beyond a Rayleigh number of approximately 106 the outer flow may be ignored in calculating the disk heat transfer rate. The radial variation of the outer flow is to the – 1/5 power measured inward from the leading edge. This is a result of the scaling difference of the thin layer flow, and the outer, plume entrainment flow. The local heat transfer rate is increased by including the entrainment effects on the outer flow and varies as the Grashof number to the power (1/5–ε), where ε is a decreasing function of inward radial distance.

A two-dimensional time-dependent model for surface shear and buoyancy-driven flows in domains with large aspect ratio

A two-dimensional numerical model has been developed for studying flows in enclosures having a large length-to-depth ratio. The model solves the two-dimensional Navier-Stokes and energy equations subject to hydrostatic, Boussinesq, and rigid-lid assumptions. Turbulent momentum and heat diffusion are incorporated using variable eddy coefficients. The flow is driven by surface wind shear and heat transfer. Two cases were run for a rectangular domain, subject to a suddenly imposed surface shear stress while the surface and bottom were maintained at two different but constant temperatures. The difference between the cases was the formulation used for predicting the eddy viscosities and diffusivities. The variations of velocities and temperatures with time were studied. Both velocities and temperatures were found to undergo a sudden rapid change, after an initial period of slow change, before attaining steady stage. This has been explained in light of the differences in the convective and diffusive time scales and the nature of coupling between the governing equations.

Modeling subsurface stormflow on steeply sloping forested watersheds

Five mathematical models for predicting subsurface flow were compared to discharge measurements made by Hewlett and Hibbert (1963) on a uniform sloping soil trough at the Coweeta Hydrologic Laboratory. The models included one‐ and two‐dimensional finite element models based on the Richards equation, a kinematic wave model, and two simple storage‐discharge models based on the kinematic wave and Boussinesq assumptions. The simple models simulated the subsurface response and water table positions as well as the more complex models based on the Richards equation and were much more economical to use from the point of view of computational costs. Such models have features that would allow them to be incorporated into more complex watershed models, thus placing hydrologic prediction on a more physically correct and less empirical footing.