The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Non-Periodic Global Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been extended to deal with the case of vertical throughflow.     

The Effects of Combined Horizontal and Vertical Heterogeneity on the Onset of Convection in a Porous Medium with Vertical Throughflow

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, with weak vertical throughflow, are studied analytically for the case of weak heterogeneity. It is found that when the boundary conditions at the upper and lower boundaries are symmetric, the throughflow magnitude and the permeability and conductivity gradients enter the expression for the critical Rayleigh number at second order. The throughflow on its own is stabilizing but the combination of throughflow and heterogeneity may be either stabilizing or destabilizing.     

CFD-based analysis of multistage throughflow surfaces with incidence

A numerical finite-volume solution of Euler equations is performed in the meridional plane of complete axial flow turbomachinery. The throughflow equations contain blade force terms that model the effects of the real blades on the flow and are resolved by further equations. Under the axisymmetric flow assumption, incidence involves a discontinuity through the leading edge, which introduces strong unphysical losses. Incidence is modeled by solving an inverse problem in the front part of the bladed region. The inverse method provides the geometry of the throughflow surface that replaces the discontinuous profile of swirl velocity with a specified, conveniently smooth profile across the leading edge region. The specified velocity profile and computed ideal geometry are used to update the blade force. The Euler solution is compared to a streamline curvature solution in analyzing a three-stage turbine. In the design condition with up to 2° of spanwise-averaged incidence, the method does not significantly affect the prediction of overall performance. In a strong off-design condition with up to 13° of average incidence, performance is predicted with the same accuracy as in the design case.     

The Effect of Vertical Throughflow on the Onset of Convection in a Porous Medium in a Rectangular Box

The effect of vertical throughflow on the onset of convection in a rectangular box occupied by a saturated porous medium uniformly heated from below, is studied using linear stability theory. It is found that, for small values of the throughflow, the stabilizing effect of the throughflow and the stabilizing effect of the confining lateral walls of the box are approximately independent of each other.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Periodic and Localized Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated with the aid of a computer package. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. Previous work on the case of non-periodic global variation is now extended to the case of either periodic variation or localized variation.     

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.     

The Effect of Vertical Throughflow on Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid

The effect of vertical throughflow on the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The dependences of the critical Rayleigh number for the non-oscillatory and oscillatory modes of instability on the thermophoresis and Brownian motion parameters for the cases with and without throughflow are investigated.     

The Effect of Vertical Throughflow on the Onset of Convection Induced by Internal Heating in a Layered Porous Medium

We present an analytical investigation of the effect of vertical throughflow on the onset of convection, induced by internal heating, in a composite porous medium consisting of two horizontal layers. If convection is induced by internal heating, the bulk of the convection occurs in the upper half of the layer where the buoyancy force is destabilizing. For the case of heterogeneous porous medium, if the permeability increases in the upward direction, or if the thermal conductivity decreases in the upward direction, instability is increased. It is also found that upward throughflow is stabilizing but a modest amount of downward throughflow is destabilizing.     

Inlet angle distribution of last stage moving blades for large steam turbines

A simplified analytical method is presented for use in preliminary design studies on the final stages of large steam turbines to derive the required moving blade twist distribution to achieve near-zero incidence. The method incorporates the influence of nozzle twist and of casing flare which is shown to enter the analysis mainly through its effect on the distribution of streamline curvature. Provided flare is properly incorporated into the design, there appears to be no inherent reason why higher flare should limit performance unduly. Optimisation of moving blade twist for variations in load and back pressure is also investigated. Blade skeletons derived analytically are subjected to full throughflow calculations and refined where necessary to improve their likely performance. The results are compared with results of calculations with published commercial blade angle distributions, designed for both low- and high-flare turbines. The results of the throughflow calculations on the blades derived here, and the comparisons with apparently successful commercial blades, suggest that this method can be used as a preliminary design tool, freeing the designer from repetitive trial runs with a full throughflow calculation.     

Throughflow and Quadratic Drag Effects on Thermal Convection in a Rotating Porous Layer

A linear stability analysis is implemented to study thermal convective instability in a horizontal fluid-saturated rotating porous layer with throughflow in the vertical direction. The modified Forchheimer-extended Darcy equation that includes the time-derivative and Coriolis terms is employed as a momentum equation. The criterion for the occurrence of direct and Hopf bifurcations is obtained using the Galerkin method. It is shown that if a Hopf bifurcation is possible it always occurs at a lower value of the Darcy?CRayleigh number than the direct bifurcation. Increase in the throughflow strength and inertia parameter is to decrease the domain of Prandtl number up to which Hopf bifurcation is limited but opposite is the trend with increasing Taylor number. The effect of rotation is found to be stabilizing the system, in general. However, in the presence of both rotation and Forchheimer drag a small amount of vertical throughflow as well as inertia parameter show some destabilizing effect on the onset of direct bifurcation; a result of contrast noticed when they are acting in isolation. The existing results in the literature are obtained as limiting cases from the present study.     

On the throughflow with swirling inflow in annular diffuser

In this paper, a throughflow with swirling inflow in an annular diffuser is calculated. Under the assumption of smallcross-flow, the flow near inner and outer wall surfaces is calculated based on the three-dimensional momentum integral equation of the boundary layer. The potential fiow outside the boundary layer is cornputed by means of the iteration method based on the velocity gradient equation along the quasi-orthogonal direction of the meridional projection of the stream-line on the meridional surface and the constancy of fiux equation. The numerical results agree with the experiments quite well. This method is useful for analyzing the throughfiow with pre-swirl in the annular diffuser.     

Throughflow Effects on Penetrative Convection in Superposed Fluid and Porous Layers

The effect of vertical throughflow on the onset of penetrative convection simulated via internal heating in a two-layer system in which a layer of fluid overlies and saturates a layer of porous medium is studied. Flow in the porous medium is governed by Forchheimer-extended Darcy equation, and Beavers?CJoseph slip condition is applied at the interface between the fluid and the porous layers. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The eigenvalue problem is solved using a regular perturbation technique with wave number as a perturbation parameter. The ratio of fluid layer thickness to porous layer thickness, ??, the direction of throughflow, and the presence of volumetric internal heat source in fluid and/or porous layer play a decisive role on the stability characteristics of the system. In addition, the influence of Prandtl number arising due to throughflow is also emphasized on the stability of the system. It is observed that both stabilizing and destabilizing factors can be enhanced because of the simultaneous presence of a volumetric heat source and vertical throughflow so that a more precise control (suppress or augment) of thermal convective instability in a layer of fluid or porous medium is possible.     

The Onset of Convection in a Suspension of Gyrotactic Microorganisms in Superimposed Fluid and Porous Layers: Effect of Vertical Throughflow

The purpose of this paper is to investigate the effect of vertical throughflow on the onset of bioconvection in a suspension of gyrotactic microorganisms. A dilute suspension of gyrotactic microorganisms in a shallow system that consists of superimposed fluid and porous layers is considered. A linear instability analysis of this problem is performed and the Galerkin method is utilized to solve the eigenvalue problem. The analysis leads to an equation for the critical Rayleigh number. It is shown that the vertical throughflow stabilizes the system.     

Throughflow and g-jitter effects on binary fluid saturated porous medium

A non-autonomous complex Ginzburg-Landau equation (CGLE) for the finite amplitude of convection is derived, and a method is presented here to determine the amplitude of this convection with a weakly nonlinear thermal instability for an oscillatory mode under throughflow and gravity modulation. Only infinitesimal disturbances are considered. The disturbances in velocity, temperature, and solutal fields are treated by a perturbation expansion in powers of the amplitude of the applied gravity field. Throughflow can stabilize or destabilize the system for stress free and isothermal boundary conditions. The Nusselt and Sherwood numbers are obtained numerically to present the results of heat and mass transfer. It is found that throughflow and gravity modulation can be used alternately to heat and mass transfer. Further, oscillatory flow, rather than stationary flow, enhances heat and mass transfer.     

Effect of an axial throughflow on buoyancy-induced flow in a rotating cavity

In this paper large-eddy simulation is used to study buoyancy-induced flow in a rotating cavity with an axial throughflow of cooling air. This configuration is relevant in the context of secondary air systems of modern gas turbines, where cooling air is used to extract heat from compressor disks. Although global flow features of these flows are well understood, other aspects such as flow statistics, especially in terms of the disk and shroud boundary layers, have not been studied. Here, previous work for a sealed rotating cavity is extended to investigate the effect of an axial throughflow on flow statistics and heat transfer. Time- and circumferentially-averaged results reveal that the thickness of the boundary layers forming near the upstream and downstream disks is consistent with that of a laminar Ekman layer, although it is shown that the boundary layer thickness distribution along the radial direction presents greater variations than in the sealed cavity case. Instantaneous profiles of the radial and azimuthal velocities near the disks show good qualitative agreement with an Ekman-type analytical solution, especially in terms of the boundary layer thickness. The shroud heat transfer is shown to be governed by the local centrifugal acceleration and by a core temperature, which has a weak dependence on the value of the axial Reynolds number. Spectral analyses of time signals obtained at selected locations indicate that, even though the disk boundary layers behave as unsteady laminar Ekman layers, the flow inside the cavity is turbulent and highly intermittent. In comparison with a sealed cavity, cases with an axial throughflow are characterised by a broader range of frequencies, which arise from the interaction between the laminar jet and the buoyant flow inside the cavity.     

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Transport in Porous Media - Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous...     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Convective Instability in Superposed Fluid and Porous Layers with Vertical Throughflow

A closed form solution to the convective instability in a composite system of fluid and porous layers with vertical throughflow is presented. The boundaries are considered to be rigid-permeable and insulating to temperature perturbations. Flow in the porous layer is governed by Darcy–Forchheimer equation and the Beavers–Joseph condition is applied at the interface between the fluid and the porous layer. In contrast to the single-layer system, it is found that destabilization due to throughflow arises, and the ratio of fluid layer thickness to porous layer thickness, , too, plays a crucial role in deciding the stability of the system depending on the Prandtl number.     

Linear Stability of Horizontal Throughflow in a Brinkman Porous Medium with Viscous Dissipation and Soret Effect

Transport in Porous Media - The onset of double-diffusive convective instability of a horizontal throughflow induced by viscous dissipation in a fluid-saturated porous layer of high permeability is...     

The Onset of Double-Diffusive Convection in a Vertical Cylinder Occupied by a Heterogeneous Porous Medium with Vertical Throughflow

This paper investigates the onset of convection in a vertical cylinder occupied by a saturated porous medium of vertically heterogeneous permeability. The flow is induced by an applied vertical temperature gradient and an imposed solute concentration gradient. The main interest of this paper is studying the effect of vertical throughflow on the onset of instability in this system. The study is performed using linear stability theory. The problem is of considerable interest for hydrological and geophysical situations.