Investigations on the validity of Boussinesq approximation on free convection in vertical porous annulus

Validity of the Boussinesq approximation on free convective heat transfer in fluid filled porous annulus has been examined considering fluid density to be variable in all the conservation equations. The variable fluid density is characterized by a density difference ratio. Numerical solutions have been obtained using successive accelerated replacement scheme. Numerical results obtained for a wide range of Rayleigh numbers, aspect ratio, radius ratio and density difference ratio show that, relaxing the Boussinesq approximation, leads to insignificant changes in the average Nusselt number when the porous medium is filled with a liquid. However, for a gas filled porous medium, owing to a relatively large change in the density, change in the average Nusselt number is significant. For example, the change in the average Nusselt number is as high as 17% at a Rayleigh number of 300, when the aspect ratio is 5, radius ratio is 5 and the density difference ratio is 0.5.     

Entropy generation analysis of tangent hyperbolic fluid in quadratic Boussinesq approximation using spectral quasi-linearization method

Applied Mathematics and Mechanics - In many industrial applications, heat transfer and tangent hyperbolic fluid flow processes have been garnering increasing attention, owing to their immense...     

Accuracy of the Boussinesq approximation for an incompressible fluid

The corrections of first order to the eigenvalues and critical Rayleigh numbers obtained in the Boussinesq approximation are determined for convection in a fluid with zero compressibility. The ratio of the equilibrium difference of the densities to a mean density of the fluid is taken as the small parameter. The corrections are found by the methods of perturbation theory for self-adjoint operators. It is shown that in the class of problems with symmetry with respect to a horizontal plane the first-order corrections vanish. The restrictions on the system needed if the Boussinesq approximation is to be meaningful in the problem of the occurrence of convective instability are established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 2, pp. 19–26, March–April, 1981.     

Heat transfer by rotational flow in an annulus with porous lining

The flow and heat transfer in an annulus between rotating coaxial cylinders, with non-erodible porous lining, is investigated. The flow in the porous lining is obtained by using Brinkman equation. At the boundary between the porous lining and the free flow (the so called nominal surface), the velocity slip and the temperature slip are used. A quasi-numerical technique developed by the authors is employed in obtaining the solution of the energy equation. The effect of the thickness of the porous lining and the permeability on the velocity and the Nusselt numbers at the walls is studied.

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Wärmeübergang bei rotierender Strömung in einem Ring mit poröser Wand

Zusammenfassung In dieser Arbeit wird die Strömung und Wärmeübertragung zwischen rotierenden koaxialen Zylindern mit unauswaschbarem porösem Überzug untersucht. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmanschen Gleichung berechnet. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung wurde die Geschwindigkeitsgleitung und Temperaturgleitung benutzt. Die Energiegleichung ist mit Hilfe eines von den Autoren entwickelten quasi-numerischen Verfahrens gelöst. Der Einfluß der Dicke und der Durchlässigkeit des porösen Überzugs auf die Strömung und die Nusseltschen Zahlen an den Wänden wird untersucht.

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Nomenclature R ₂ radius of the outer cylinder forming the annulus - ₂ angular velocity of the outer cylinder - T ₂ temperature of the outer cylinder - R _l radius of the inner cylinder forming the annulus - ₁ angular velocity of the inner cylinder - T ₁ temperature of the inner cylinder - h thickness of the porous lining - R radial distance of any point in the annulus - V azimuthal component of velocity in zone 1 (of Fig. 1) - V part of velocity in zone 2 (of Fig. 1) due to transfer of momentum from the main flow - V _p velocity in zone 2 (of Fig. 1) - Q Darcy velocity in the porous medium (zone 2 of Fig. 1) - velocity slip parameter - k absolute permeability of the material used for lining - ₀ nondimensional shearing stress at the outer cylinder - _i nondimensional shearing stress at the inner cylinder - K thermal conductivity in zones 1 and 2 (of Fig. 1) - coefficient of viscosity of the fluid - T temperature in zone 1 (of Fig. 1) - T temperature in zone 2 (of Fig. 1) - temperature slip parameter - (Nu) _o nondimensional Nusselt number at the outer cylinder - (Nu) _i nondimensional Nusselt number at the inner cylinder - radii ratio - nondimensional rotational parameter - nondimensional thickness of the porous lining     

Boussinesq approximation in the Rayleigh-Benard problem

Instability of mechanical equilibrium and initiation of plane steady-state convective flows in an infinite horizontal fluid layer heated from below (Rayleigh-Benard problem) are investigated. The convection model for an isothermal incompressible fluid is not assumed to have small thermal expansion (contrary to the Oberbeck-Boussinesq approximation). The influence of a supplementary thermal expansion parameter on the convection process is numerically investigated. The results are compared with the known results for the Oberbeck-Boussinesq approximation. It is shown that subcritical instability is possible if the thermal expansion parameter increases. The linearization and Lyapunov-Schmidt methods are applied.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–10, September–October, 1995.     

Heat and brine transport in porous media: the Oberbeck-Boussinesq approximation revisited

This paper discusses the Oberbeck-Boussinesq approximation for heat and solute transport in porous media. In this commonly used approximation all density variations are neglected except for the gravity term in Darcy’s law. However, in the limit of vanishing density differences this gravity term disappears as well. The main purpose of this paper is to give the correct limits in which the gravity term is retained, while other density effects can be neglected. We show that for isothermal brine transport, fluid volume changes can be neglected when a condition is fulfilled for a dimensionless number, which is independent of the density difference and specific discharge. For heat transfer an additional condition is required. One-dimensional examples of simultaneous heat and brine transport are given for which similarity solutions are constructed. These examples are included to elucidate the volume effects and the corresponding induced specific discharge variations. Finally, a two-dimensional example illustrates the relative effects of volume changes and gravity.     

The quadratic approximation in rotational dynamical equations

Summary For many problems of interest in rotational dynamics, linearized equations obscure essential features of system response, and non-linear formulations are required. Meaningful progress in dynamic analysis may nonetheless require some level of approximation, and frequently the retention of second degree terms in the variational coordinates and their derivatives is sufficient to reveal the inherently nonlinear characteristics of the response. The formulation of equations in this quadratic approximation is typically fraught with difficulties of detail, with consequent likelihood of computational error. The present note offers a rationale for performing the desired quadratic approximation at the matrix level, and indeed explicit formulas are developed which provide in general terms the required expressions for the direction cosine matrix and the angular velocity matrix. These formulas are illustrated in application to the gravity-gradient satellite, which is known to exhibit nonlinear resonances which can be detected in the quadratic approximation.

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Übersicht Bei der Untersuchung von Drehbewegungen starrer Körper erweist es sich oft als notwendig, nichtlineare Glieder zu berücksichtigen, da durch Linearisierung wesentliche Eigenschaften verloren gehen können. Vielfach können die nichtlinearen Effekte bereits durch Berücksichtigung der quadratischen Glieder erfaßt werden. Die Aufstellung der quadratischen Näherungen ist im einzelnen schwierig, so daß leicht Rechenfehler gemacht werden können. Deshalb wird in der vorliegenden Arbeit ein zweckmäßiges Verfahren zur Aufstellung der quadratischen Näherungen als Matrizen-Gleichungen angegeben. Für den allgemeinen Fall werden explizite Formeln für die Matrizen der Richtungscosinus und der Winkelgeschwindigkeiten angegeben. Diese Formeln werden auf das Beispiel eines schwerestabilisierten Satelliten angewendet, bei dem nichtlineare Resonanzen festgestellt wurden. Durch quadratische Näherungen lassen sich diese rechnerisch erfassen.

     

An improved depth-averaged nonhydrostatic shallow water model with quadratic pressure approximation

Phase-resolved information is necessary for many coastal wave problems, for example, for the wave conditions in the vicinity of harbor structures. Two-dimensional (2D) depth-averaging shallow water models are commonly used to obtain a phase-resolved solution near the coast. These models are in general more computationally effective compared with computational fluid dynamics software and will be even more capable if equipped with a parallelized code. In the current article, a 2D wave model solving the depth-averaged continuity equation and the Euler equations is implemented in the open-source hydrodynamic code REEF3D. The model is based on a nonhydrostatic extension and a quadratic vertical pressure profile assumption, which provides a better approximation of the frequency dispersion. It is the first model of its kind to employ high-order discretization schemes and to be fully parallelized following the domain decomposition strategy. Wave generation and absorption are achieved with a relaxation method. The simulations of nonlinear long wave propagations and transformations over nonconstant bathymetries are presented. The results are compared with benchmark wave propagation cases. A large-scale wave propagation simulation over realistic irregular topography is shown to demonstrate the model's capability of solving operational large-scale problems.     

Peristaltic flow and heat transfer in a vertical porous annulus, with long wave approximation

In this paper, we study the interaction of peristalsis with heat transfer for the flow of a viscous fluid in a vertical porous annular region between two concentric tubes. Long wavelength approximation (that is, the wavelength of the peristaltic wave is large in comparison with the radius of the tube) is used to linearise the governing equations. Using the perturbation method, the solutions are obtained for the velocity and the temperature fields. Also, the closed form expressions are derived for the pressure-flow relationship and the heat transfer at the wall. The effect of pressure drop on flux is observed to be almost negligible for peristaltic waves of large amplitude; however, the mean flux is found to increase by 10-12% as the free convection parameter increases from 1 to 2. Also, the heat transfer at the wall is affected significantly by the amplitude of the peristaltic wave. This warrants further study on the effects of peristalsis on the flow and heat transfer characteristics.     

Optimization of heat transfer in the thermal Marangoni convective flow of a hybrid nanomaterial with sensitivity analysis

Applied Mathematics and Mechanics - The heat transfer rate of the thermal Marangoni convective flow of a hybrid nanomaterial is optimized by using the response surface methodology (RSM). The...     

Resonance study of spring pendulum based on asymptotic solutions with polynomial approximation in quadratic means

Meccanica - The nonlinearities of geometric nature that is characteristic for pendulum-type systems are expressed by the trigonometric functions. In order to apply the method of multiple scales in...     

Experimental investigation of turbulent entrainment in an annulus with moving sidewalls

The establishment of a turbulent mixed layer in a two-layer stratified shear flow, and the rate of entrainment into that layer were studied experimentally in a modified annulus. The modification of the conventional annulus was made by replacing the upper rotating screen with inner rotating sidewalls, extending over the upper half of the channel, so that the flow in the upper layer was nearly uniform and almost laminar, while the bottom layer was quiescent. Vertical density profile measurements were conducted using single electrode conductivity probes. The flow was visualized during the various stages of the experiment using the hydrogen bubble technique.After the start of the sidewalls rotation, the upper layer accelerates from rest, and consequently a transition process is taking place during which the initial density interface between the two layers is developed into a turbulent mixed layer. This turbulent layer is bounded by two sharp interfaces, each separating it from an outer non-turbulent zone. The generation of this five-layer structure seemed to be dominated by instabilities activated by the velocity difference between the upper and lower layer.Once a turbulent mixed layer is formed, entrainment of nonturbulent fluid into that layer is taking place causing its thickness to increase continuously. Depending on the overall Richardson number, based on the channel width, the slope of the entrainment law curve was found to have two different values, each indicating the dominance of a different source of turbulent energy production. For relatively low Richardson numbers, the slope is close to -1.8, implying that the velocity shear across each interface contributes significantly to the entrainment. On the other hand, for larger Richardson numbers the slope is about -1.25, in agreement with previous results of shear-free entrainment experiments.The measured velocity profiles indicate that as long as the mixed layer is not too thick, the radial inhomogeneities are small and the flow may be considered as nearly one-dimensional. It seems, therefore, that for the understanding of entrainment processes occurring in realistic stratified flows, the modified annulus is a more reliable tool than the conventional one.     

Sensitivity analyses in a buoyancy-driven closed system with high resolution CFD using Boussinesq approximation and variable density models

The influences of the Schmidt number and fluid models of the Boussinesq approximation and variable density (low-Mach formulation) are investigated using large eddy simulation (LES) where fluid naturally advects between two tanks connected by a pipe. The geometry resembles a reactor-like vessel which comprises of a cold leg-downcomer region. The closed system consists of two miscible fluids with different densities and viscosities separated by a valve. Analysis with Reynolds-Averaged Navier–Stokes (RANS) is used for calculating Taylor length scales to establish minimum grid resolution for LES. Using particle image velocimetry (PIV) data, validation is conducted for four different magnitudes of Schmidt numbers. Evidence indicates that capturing the effects from the highest Schmidt number is essential for properly predicting fluctuations and instabilities in the flow. Although downcomer comparisons with PIV consist of larger errors, they are explained through additional sensitivity analyses involving initial conditions, geometry, and residual energy. Lastly, the influence of larger density differences is demonstrated using simulations with a higher Atwood number, temporally scaled using the fluid front velocity. Recommendations are made regarding future experiments and simulations for design of similar buoyant mixing problems.     

Heat transfer and pressure drop of a transversely finned concentric annulus with longitudinal flow

The characteristics of heat transfer and pressure drop have been experimentally studied for the fully developed concentric annular flow with transverse fins normal to the flow direction by the naphthalene sublimation technique. Correlations for calculating the heat transfer coefficient with different inner diametersD ₀ of the outer tube are presented. A characteristic Reynolds number has been proposed, by which the predominant role of the transverse fins can be evaluated. It has been indicated that the inner diameterD ₀ has much more effect on pressure drop than on heat transfer. The effect ofD ₀ on the overall performance is also compared under the same flow velocity or flow rate. It has been found that the effect of developing flow on heat transfer is significant and should be taken into account during experiment.

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Wärmeübergang und Druckverlust in einem querberippten konzentrischen Ringkanal bei Längsströmung

Zusammenfassung Mit Hilfe der Naphthalin-Sublimationstechnik werden die Wärmeübergangs- und Druckverlustcharakteristiken in quer zur voll ausgebildeten Strömung bei berippten Ringkanälen experimentell ermittelt und Korrelationen zur Berechnung des Wärmeübergangskoeffizienten bei variablen InnendurchmesserD ₀ des umschließenden Rohres angegeben. Ferner wird eine charakteristische Reynolds-Zahl vorgeschlagen, über die sich der dominierende Einfluß der Querrippen erfassen läßt. Es zeigte sich, daß der InnendurchmesserD ₀ den Druckverlust wesentlich mehr beeinflußt als den Wärmeübergang. Auch wurde die Abhängigkeit des Gesamt-Übertragungsverhaltens vonD ₀ bei gleicher Strömungsgeschwindigkeit bzw. Volumenstromdichte ermittelt. Es zeigte sich, daß die Ausbildung des Strömungsprofils bei Einlaufströmung den Wärmeübergang signifikant beeinflußt und deshalb im Experiment zu berücksichtigen ist.