Stability of the flow of a fluid through a flexible tube at high Reynolds number

The stability of the Hagen-Poiseuille flow of a Newtonian fluid in a tube of radius R surrounded by an incompressible viscoelastic medium of radius R < r < HR is analysed in the high Reynolds number regime. The dimensionless numbers that affect the fluid flow are the Reynolds number Re = (ρVR / η), the ratio of the viscosities of the wall and fluid ηr = (ηs/η), the ratio of radii H and the dimensionless velocity Γ = (ρV2/G)1/2. Here ρ is the density of the fluid, G is the coefficient of elasticity of the wall and Vis the maximum fluid velocity at the centre of the tube. In the high Reynolds number regime, an asymptotic expansion in the small parameter ε = (1/Re) is employed. In the leading approximation, the viscous effects are neglected and there is a balance between the inertial stresses in the fluid and the elastic stresses in the medium. There are multiple solutions for the leading-order growth rate do), all of which are imaginary, indicating that the fluctuations are neutrally stable, since there is no viscous dissipation of energy or transfer of energy from the mean flow to the fluctruations due to the Reynolds strees. There is an O(ε1/2) correction to the growth rate, s(1), due to the presence of a wall layer of thickness ε1/2R where the viscous stresses are O(ε1/2) smaller than the inertial stresses. An energy balance analysis indicates that the transfer of energy from the mean flow to the fluctuations due to the Reynolds stress in the wall layer is exactly cancelled by an opposite transfer of equal magnitude due to the deformation work done at the interface, and there is no net transfer from the mean flow to the fluctuations. Consequently, the fluctuations are stabilized by the viscous dissipation in the wall layer, and the real part of s(1) is negative. However, there are certain values of Γ and wavenumber k where s(l) = 0. At these points, the wail layer amplitude becomes zero because the tangential velocity boundary condition is identically satisfied by the inviscid flow solution. The real part of the O(ε) correction to the growth rate s(2) turns out to be negative at these points, indicating a small stabilizing effect due to the dissipation in the bulk of the fluid and the wall material. It is found that the minimum value of s(2) increases [is proportional to] (H − 1)−2 for (H − 1) [double less-than sign] 1 (thickness of wall much less than the tube radius), and decreases [is proportional to] (H−4 for H [dbl greater-than sign] 1. The damping rate for the inviscid modes is smaller than that for the viscous wall and centre modes in a rigid tube, which have been determined previously using a singular perturbation analysis. Therefore, these are the most unstable modes in the flow through a flexible tube.

Steady compressible flow of cohesionless granular materials through a wedge-shaped bunker

A continuum model based on the critical-state theory of soil mechanics is used to generate stress, density, and velocity profiles, and to compute discharge rates for the flow of granular material in a mass flow bunker. The bin–hopper transition region is idealized as a shock across which all the variables change discontinuously. Comparison with the work of Michalowski (1987) shows that his experimentally determined rupture layer lies between his prediction and that of the present theory. However, it resembles the former more closely. The conventional condition involving a traction-free surface at the hopper exit is abandoned in favour of an exit shock below which the material falls vertically with zero frictional stress. The basic equations, which are not classifiable under any of the standard types, require excessive computational time. This problem is alleviated by the introduction of the Mohr–Coulomb approximation (MCA). The stress, density, and velocity profiles obtained by integration of the MCA converge to asymptotic fields on moving down the hopper. Expressions for these fields are derived by a perturbation method. Computational difficulties are encountered for bunkers with wall angles θw [gt-or-equal, slanted] 15° these are overcome by altering the initial conditions. Predicted discharge rates lie significantly below the measured values of Nguyen et al. (1980), ranging from 38% at θw = 15° to 59% at θw = 32°. The poor prediction appears to be largely due to the exit condition used here. Paradoxically, incompressible discharge rates lie closer to the measured values. An approximate semi-analytical expression for the discharge rate is obtained, which predicts values within 9% of the exact (numerical) ones in the compressible case, and 11% in the incompressible case. The approximate analysis also suggests that inclusion of density variation decreases the discharge rate. This is borne out by the exact (numerical) results – for the parameter values investigated, the compressible discharge rate is about 10% lower than the incompressible value. A preliminary comparison of the predicted density profiles with the measurements of Fickie et al. (1989) shows that the material within the hopper dilates more strongly than predicted. Surprisingly, just below the exit slot, there is good agreement between theory and experiment.

On cavity flow at high Reynolds numbers

The flow in a square cavity is studied by solving the full Navier–Stokes and energy equations numerically, employing finite-difference techniques. Solutions are obtained over a wide range of Reynolds numbers from 0 to 50000. The solutions show that only at very high Reynolds numbers (Re [gt-or-equal, slanted] 30000) does the flow in the cavity completely correspond to that assumed by Batchelor's model for separated flows. The flow and thermal fields at such high Reynolds numbers clearly exhibit a boundary-layer character. For the first time, it is demonstrated that the downstream secondary eddy grows and decays in a manner similar to the upstream one. The upstream and downstream secondary eddies remain completely viscous throughout the range of Reynolds numbers of their existence. It is suggested that the behaviour of the secondary eddies may be characteristic of internal separated flows.

On the criteria for reverse transition in a two-dimensional boundary layer flow

Experiments on reverse transition were conducted in two-dimensional accelerated incompressible turbulent boundary layers. Mean velocity profiles, longitudinal velocity fluctuations $\tilde{u}^{\prime}(=(\overline{u^{\prime 2}})^{\frac{1}{2}})$ and the wall-shearing stress (TW) were measured. The mean velocity profiles show that the wall region adjusts itself to laminar conditions earlier than the outer region. During the reverse transition process, increases in the shape parameter (H) are accompanied by a decrease in the skin friction coefficient (Cf). Profiles of turbulent intensity (u’2) exhibit near similarity in the turbulence decay region. The breakdown of the law of the wall is characterized by the parameter \[ \Delta_p (=\nu[dP/dx]/\rho U^{*3}) = - 0.02, \] where U* is the friction velocity. Downstream of this region the decay of $\tilde{u}^{\prime}$ fluctuations occurred when the momentum thickness Reynolds number (R) decreased roughly below 400.

Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis

The tendency of granular materials in rapid shear ow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

Dynamics of pulsed flow in an elastic tube

Internal haemorrhage, often leading to cardio-vascular arrest happens to be one of the prime sources of high fatality rates in mammals. We propose a simplistic model of fluid flow in our attempt to specify the location of the haemorrhagic spot, which, if located accurately, could possibly be operated leading to an instant cure. The model we employ for the purpose is basically fluid mechanical in origin and consists of a viscous fluid, pumped by a periodic force and flowing through an elastic tube. The analogy is with that of blood, pumped from the heart and flowing through an artery or vein. Our results, aided by graphical illustrations, match reasonably well with experimental observations.

Study of gas-fines in decreasing gas velocity in a packed bed

It has been well recognized now that the blast furnace aerodynamics can be represented more accurately under the decreasing gas velocity condition. Therefore, gas-fines study has been carried out in a packed bed under the decreasing gas velocity condition. Gas and fines flow equations have been developed and solved, for two-dimensional case using finite volume method. To take into account the turbulence, k-e turbulent flow model has also been developed in two-dimension. The model's predictions have been validated against the published experimental data for the increasing gas velocity case, as no experimental data are available in open literature for the decreasing gas velocity. This study shows the difference in the results for increasing and decreasing gas velocity cases under various conditions which have been reported here. Implication of the results to the blast furnace condition has also been discussed.

Turbulent flow computations using upwind RANS technique for a flat plate and an airfoil

Reynolds Averaged Navier Stokes (RANS) equations are solved using third order upwind biased Roe's scheme for the inviscid fluxes and second order central difference scheme for the viscous fluxes. The Baldwin & Lomax turbulence model is employed for Reynolds stresses. The governing equations are solved using finite-volume implicit scheme in body fitted curvilinear coordinate O-grid system. Computations axe reported for a flat plate apart from RAE 2822 and NACA 0012 airfoils. Results for the flat plate at M = 0.3, R-c = 4.0 x 10(6) compare favourably with the analytical solution. Results for the two airfoils are compared with experiment. There is a good agreement in C-p distribution between experiment and computation for both the airfoils. Comparison of C-f distribution with experiment for RAE 2822 airfoil is reasonable.

Modeling of nonreacting and reacting turbulent spray jets using a fully stochastic separated flow approach

Numerical modeling of several turbulent nonreacting and reacting spray jets is carried out using a fully stochastic separated flow (FSSF) approach. As is widely used, the carrier-phase is considered in an Eulerian framework, while the dispersed phase is tracked in a Lagrangian framework following the stochastic separated flow (SSF) model. Various interactions between the two phases are taken into account by means of two-way coupling. Spray evaporation is described using a thermal model with an infinite conductivity in the liquid phase. The gas-phase turbulence terms are closed using the k-epsilon model. A novel mixture fraction based approach is used to stochastically model the fluctuating temperature and composition in the gas phase and these are then used to refine the estimates of the heat and mass transfer rates between the droplets and the surrounding gas-phase. In classical SSF (CSSF) methods, stochastic fluctuations of only the gas-phase velocity are modeled. Successful implementation of the FSSF approach to turbulent nonreacting and reacting spray jets is demonstrated. Results are compared against experimental measurements as well as with predictions using the CSSF approach for both nonreacting and reacting spray jets. The FSSF approach shows little difference from the CSSF predictions for nonreacting spray jets but differences are significant for reacting spray jets. In general, the FSSF approach gives good predictions of the flame length and structure but further improvements in modeling may be needed to improve the accuracy of some details of the Predictions. (C) 2011 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

On the Decidability of Model-Checking Information Flow Properties

Current standard security practices do not provide substantial assurance about information flow security: the end-to-end behavior of a computing system. Noninterference is the basic semantical condition used to account for information flow security. In the literature, there are many definitions of noninterference: Non-inference, Separability and so on. Mantel presented a framework of Basic Security Predicates (BSPs) for characterizing the definitions of noninterference in the literature. Model-checking these BSPs for finite state systems was shown to be decidable in [8]. In this paper, we show that verifying these BSPs for the more expressive system model of pushdown systems is undecidable. We also give an example of a simple security property which is undecidable even for finite-state systems: the property is a weak form of non-inference called WNI, which is not expressible in Mantel’s BSP framework.

The Behaviour of Two-Phase Flow of DNAPL and Water through a Fractured Rock under Confining Pressure

This study presents the characterization of DNAPL and water flow in a fracture under confining pressure. A comprehensive mathematical model and the conditions under which DNAPL will enter an initially water-saturated deforming rock fracture are discussed. A numerical model with which to predict the quantity of each phase in terms of their saturations in deforming rock joint is developed. The effect of varying confining stresses on the traverse time of DNAPL across a fractured aquitard is studied. The sensitivity analysis for physical and hydraulic properties like initial fracture apertures, fracture dips, equivalent fracture aperture and confining pressures are performed and discussed.