Plume flows in porous media driven by horizontal differential heating

2012 ◽  
Vol 696 ◽  
pp. 263-284 ◽  
Author(s):  
P. Adamou-Graham ◽  
P. G. Daniels
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Constant Pressure ◽  
Lower Surface ◽  
Differential Heating ◽  
Vertical Heat ◽  
Theoretical Predictions ◽  
Flow Through ◽  
Plume Flows ◽  
Darcy Rayleigh Number

AbstractIn this paper we describe flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the impermeable lower surface. The upper surface is held at constant pressure and at a constant temperature equal to the minimum temperature of the lower surface, while the sidewalls are impermeable and thermally insulated. Numerical results for general values of the Darcy–Rayleigh number $R$ and the cavity aspect ratio $A$ are compared with theoretical predictions for the small Darcy–Rayleigh number limit $(R\ensuremath{\rightarrow} 0)$ where the temperature field is conduction-dominated, and with a boundary-layer theory for the large Darcy–Rayleigh number limit $(R\ensuremath{\rightarrow} \infty )$ where convection is significant. In the latter case a horizontal boundary layer near the lower surface conveys fluid to the hot end of the cavity where it rises to the upper surface in a narrow plume. Predictions are made of the vertical heat transfer through the cavity.

The Prediction of Axisymmetric Turbulent Boundary Layer in Conical Nozzles

1974 ◽  
Vol 41 (1) ◽  
pp. 20-24
Author(s):  
S. B. Au
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Digital Computer ◽  
Computation Method ◽  
State Equations ◽  
Favorable Pressure Gradient ◽  
Theoretical Predictions ◽  
Flow Through ◽  
Throat Section

A theoretical axisymmetric model of flow through conical Venturimeters, capable of solution by digital computer, has been developed to predict the development of the turbulent boundary layer along the convergent and throat section where the flow is subjected to a favorable pressure gradient. A computation method was developed to solve the momentum, auxiliary, continuity, and state equations simultaneously step-by-step. For a given inlet condition the throat condition can be predicted. Theoretical predictions were compared with experimental results obtained from two 10-in-dia Venturimeters of area ratios 0.25 and 0.5. The agreement between the theory and detailed boundary-layer surveys is good.

Rayleigh instability of a thermal boundary layer in flow through a porous medium

1960 ◽  
Vol 9 (2) ◽  
pp. 183-192 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Steady State ◽  
Rayleigh Number ◽  
Thermal Boundary Layer ◽  
Wave Number ◽  
Rayleigh Instability ◽  
Exponential Form ◽  
Flow Through ◽  
The Stability

It is supposed that a heated liquid is rising very slowly through a semi-infinite porous medium towards the permeable horizontal surface, where it mixes with a layer of cool overlying fluid. In the steady state a thermal boundary layer of exponential form exists in the medium. It is shown that the layer is stable provided that the Rayleigh number for the system does not exceed a critical positive value, and that the wave-number of the critical neutral disturbance is finite. The stability properties of the layer are explained qualitatively from physical considerations.

scholarly journals On the boundary layer structure of differentially heated cavity flow in a stably stratified porous medium

2007 ◽  
Vol 586 ◽  
pp. 347-370 ◽  
Author(s):  
P. G. DANIELS
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Rayleigh Number ◽  
Numerical Solutions ◽  
Layer Structure ◽  
Cavity Flow ◽  
Asymptotic Results ◽  
Boundary Layer Structure ◽  
Horizontal Boundary ◽  
Darcy Rayleigh Number

This paper considers two-dimensional flow generated in a stably stratified porous medium by monotonic differential heating of the upper surface. For a rectangular cavity with thermally insulated sides and a constant-temperature base, the flow near the upper surface in the high-Darcy–Rayleigh-number limit is shown to consist of a double horizontal boundary layer structure with descending motion confined to the vicinity of the colder sidewall. Here there is a vertical boundary layer structure that terminates at a finite depth on the scale of the outer horizontal layer. Below the horizontal boundary layers the motion consists of a series of weak, uniformly stratified counter-rotating convection cells. Asymptotic results are compared with numerical solutions for the cavity flow at finite values of the Darcy–Rayleigh number.

Convection in a Porous Circular Cavity

Volume 1 ◽  
2004 ◽  
Author(s):  
Kamyar Mansour
Keyword(s):  
Boundary Layer ◽  
Rayleigh Number ◽  
Porous Material ◽  
Convective Flow ◽  
Dimensionless Parameter ◽  
Cross Flow ◽  
Boundary Layer Theory ◽  
Circular Cavity ◽  
The Core ◽  
Darcy Rayleigh Number

This investigation in a porous circular cavity driven by heating in the horizontal direction is analyzed by a self-consistence boundary layer theory. We use Darcy’s law for this cavity filled with porous material. The solution is governed by dimensionless parameter Darcy-Rayleigh number. There are a lot of ways to approach boundary layer limit see Mansour [1]. Our model is based on the fact that the convective flow of fluid driven by buoyancy is carried entirely by the cross-flow in the core. In other words, we assume a thermally stratified core. This model is based on analyses of a rectangular cavity containing a viscous fluid (Gill [2]) carried over to porous material (Weber [3], Walker et al [4]). The global heat flux, the Nusselt number, is found asymptotically as the 1/2 power of Darcy-Rayleigh number, which is in consistence with other Results.

scholarly journals The convection of a Bingham fluid in a differentially-heated porous cavity

2016 ◽  
Vol 26 (3/4) ◽  
pp. 879-896 ◽  
Author(s):  
D. Andrew S. Rees
Keyword(s):  
Rayleigh Number ◽  
Yield Stress ◽  
Piecewise Linear ◽  
Rectangular Cavity ◽  
Content Type ◽  
Multigrid Algorithm ◽  
Bingham Number ◽  
Steady Solutions ◽  
Darcy Flux ◽  
Darcy Rayleigh Number

Purpose – The purpose of this paper is to determine the manner in which a yield stress fluid begins convecting when it saturates a porous medium. A sidewall-heated rectangular cavity is selected as the testbed for this pioneering work. Design/methodology/approach – Steady solutions are obtained using a second order accurate finite difference method, line relaxation based on the Gauss-Seidel smoother, a Full Approximation Scheme multigrid algorithm with V-cycling and a regularization of the Darcy-Bingham model to smooth the piecewise linear relation between the Darcy flux and the applied body forces. Findings – While Newtonian fluids always convect whenever the Darcy-Rayleigh number is nonzero, Bingham fluids are found to convect only when the Darcy-Rayleigh number exceeds a value which is linearly dependent on both the Rees-Bingham number and the overall perimeter of the rectangular cavity. Stagnation is always found in the centre of the cavity and in regions close to the four corners. Care must be taken over the selection of the regularization constant. Research limitations/implications – The Darcy-Rayleigh number is restricted to values which are at or below 200. Originality/value – This is the first investigation of the effect of yield stress on nonlinear convection in porous media.

Plume formation and resonant bifurcations in porous-media convection

1994 ◽  
Vol 272 ◽  
pp. 67-90 ◽  
Author(s):  
Michael D. Graham ◽  
Paul H. Steen
Keyword(s):  
Boundary Layer ◽  
Porous Media ◽  
Rayleigh Number ◽  
Scaling Laws ◽  
Thermal Plumes ◽  
Scaling Behaviour ◽  
Parametric Resonances ◽  
Classical Boundary ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The classical boundary-layer scaling laws proposed by Howard for Rayleigh–Bénard convection at high Rayleigh number extend to the analogous case of convection in saturated porous media. We computationally study two-dimensional porous-media convection near the onset of this scaling behaviour. The main result of the paper is the observation and study of instabilities that lead to deviations from the scaling relations.At Rayleigh numbers below the scaling regime, boundary-layer fluctuations born at a Hopf bifurcation strengthen and eventually develop into thermal plumes. The appearance of plumes corresponds to the onset of the boundary-layer scaling behaviour of the oscillation frequency and mean Nusselt number, in agreement with the classical theory. As the Rayleigh number increases further, the flow undergoes instabilities that lead to ‘bubbles’ in parameter space of quasi-periodic flow, and eventually to weakly chaotic flow. The instabilities disturb the plume formation process, effectively leading to a phase modulation of the process and to deviations from the scaling laws. We argue that these instabilities correspond to parametric resonances between the timescale for plume formation and the characteristic convection timescale of the flow.

Wall Boundary Layers in Turbomachines

1972 ◽  
Vol 14 (6) ◽  
pp. 411-423 ◽  
Author(s):  
H. Marsh ◽  
J. H. Horlock
Keyword(s):  
Boundary Layer ◽  
Integral Equations ◽  
Cross Flow ◽  
Inlet Guide Vanes ◽  
Wall Boundary Layer ◽  
Wall Boundary ◽  
Alternative Approach ◽  
Flow Through ◽  
The Many ◽  
Momentum Integral

Equations for the passage-averaged flow in a cascade are used to derive the momentum integral equations governing the development of the wall boundary layer in turbomachines. Several existing methods of analysis are discussed and an alternative approach is given which is based on the passage-averaged momentum integral equations. The analysis leads to an anomaly in the prediction of the cross flow and to avoid this it is suggested that for the many-bladed cascade there should be a variation of the blade force through the boundary layer. This variation of the blade force can be included in the analysis as a force deficit integral. The growth of the wall boundary layer has been calculated by four methods and the predictions are compared with two sets of published experimental results for flow through inlet guide vanes.

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