Showing papers on "Velocity gradient published in 2009"

A Phase Diagram for Turbulent, Transitional, and Laminar Clay Suspension Flows

Abstract: New phase diagrams for the dynamic structure of clay-laden open-channel flows are proposed. These diagrams can be used to distinguish between turbulent Newtonian, transitional, and laminar non-Newtonian flow behavior, on the basis of the balance between turbulent forces (approximated by the horizontal components of flow velocity and turbulence intensity) and cohesive forces (approximated by the suspended clay concentration and rheology). Stability regimes for five different flow types are defined using a comprehensive series of laboratory flume experiments at depth-averaged flow velocities ranging from 0.13 m s−1 to 1.47 m s−1, and at volumetric kaolinite clay concentrations ranging from 0.03% (= 0.8 g L−1) to 16.7% (= 434 g L−1). As clay concentration increases, five flow types can be distinguished: turbulent flow, turbulence-enhanced transitional flow, lower and upper transitional plug flow, and quasi-laminar plug flow. The turbulent properties of transitional flows are shown to be considerably more complex than the common notion of gradual turbulence damping. Turbulence-enhanced transitional flows display higher turbulence intensity than turbulent flows of similar velocity, with such enhancement originating from development of a highly turbulent basal internal shear layer within ~ 0.01 m of the bed. In lower transitional plug flows, the basal internal shear layer separates a lower region of high vertical gradient in horizontal velocity and strong turbulence from an upper region of plug flow with a much gentler velocity gradient and lower turbulence intensity. Kelvin-Helmholtz shear instabilities within the highly turbulent shear layer are expressed as distinct second-scale oscillations in the time series of downstream velocity. Turbulence damping dominates upper transitional plug flows, because strong cohesive forces, inferred to be caused by gelling of the high-concentration clay suspension, start to outbalance turbulent forces. In quasi-laminar plug flows, gelling is pervasive and turbulence is fully suppressed, apart from some minor residual turbulence near the base of these flows. With very few exceptions, all flows pass through the same development stages as clay concentration increases, regardless of their velocity, but the threshold concentrations for turbulence enhancement, gelling, and development of internal shear layers and plug flows are proportional to flow velocity. At flow velocities below ~ 0.5 m s−1, only low concentrations (< 0.75%) of kaolinite are required to induce transitional flow behavior, thus potentially affecting many slow-moving and decelerating clay flows in natural sedimentary environments. However, at flow velocities above 1 m s−1, clay concentrations of at least 6% are required in order for flows to enter the transitional flow phase, but even at these velocities the transitional flow phases make up a significant proportion of the flow phase space. By converting the experimental data to nondimensional Froude number (momentum term) and Reynolds number (cohesive term), it is shown that each boundary between the turbulent, transitional, and laminar flow phases can be described by a specific narrow range of Reynolds numbers. Within the duration of the experiments, settling of clay particles occurred only in plug flows of low flow velocity (and low Froude number), when the flows lacked the strength to support the entire clay suspension load.

Velocity-gradient statistics along particle trajectories in turbulent flows: The refined similarity hypothesis in the Lagrangian frame

Abstract: We present an investigation of the statistics of velocity gradient related quantities, in particular energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The refined similarity hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to Reλ∼400 with 20483 collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time τp of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral time scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial range.

Robust wall gradient estimation using radial basis functions and proper orthogonal decomposition (POD) for particle image velocimetry (PIV) measured fields

Abstract: A robust method for improving the estimation of near-wall velocity gradients from noisy flow data using Gaussian (GA) and generalized multiquadratic (GMQ) radial basis functions (RBFs) that optimizes fitting parameters to minimize the biharmonic equation is introduced. Error analysis of the wall gradient estimation was performed for RBFs, standard finite difference schemes, and polynomial and spline interpolations at various spatial resolutions, interpolation grid sizes and noise levels in synthetically generated Poiseuille and Womersley flow fields. Also, the effectiveness of the methods on digital particle image velocimetry (DPIV) data is tested by processing images generated using velocity fields obtained from direct numerical simulation (DNS) of an open turbulent channel, and the estimated gradients were compared against gradients obtained from DNS data. In the absence of noise, all methods perform well for Poiseuille and Womersley flows yielding a total error under 10% at all resolutions. In the presence of noise, the GMQ performed robustly with a total error under 10–20% even with 10% noise. With DPIV processed data for the turbulent channel flow, the error is on the order of 25–40% using thin plate spline and GMQ interpolations. Optimization of the RBF fitting parameters that minimize the energy functional associated with the analytical surface results in robust velocity gradient estimators but is computationally expensive. This computational expense is reduced and the accuracy of the proposed techniques is further improved by introducing a novel approach that combines the gradient estimators with proper orthogonal decomposition (POD). The implementation of the interpolation schemes on the POD modes results in improving accuracy by 10–15% and reducing the computational cost by approximately 75%.

Inertial particle collisions in turbulent synthetic flows: quantifying the sling effect.

Abstract: Turbulent motion increases very significantly the collision rate between particles in dilute suspensions. In the case of heavy inertial particles, the collision rate enhancement results both from the intermittent concentration in the flow, and also from the large relative velocity between colliding particles. The latter effect is a consequence of the ejection of particles out of curved streamlines, denoted here as the ``sling effect.'' Here, we quantitatively study the collision rate between heavy particles in the presence of gravity, with the simplified synthetic model of turbulent flow known as kinematic simulation. Monitoring the velocity of colliding particles and comparing it with the local velocity gradient of the flow of particles allowed us to identify the collision induced by the sling effect and to evaluate their contribution to the total collision rate. Our numerical results are then systematically compared with the estimates based on the properties of particle trajectories in the flow recently proposed by Falkovich and Pumir [G. Falkovich and A. Pumir, J. Atmos. Sci. 64, 4497 (2007)]. At moderate values of the Stokes numbers $(\text{St}\ensuremath{\lesssim}1)$, we demonstrate that the resulting parametrization describes quantitatively correctly the collision rate, and that the sling effect can be responsible for up to $\ensuremath{\sim}50%$ of the total collision rate.

Effects of free-stream turbulence on rough surface turbulent boundary layers

Abstract: Several effects of nearly isotropic free-stream turbulence in transitionally rough turbulent boundary layers are studied using data obtained from laser Doppler anemometry measurements. The free-stream turbulence is generated with the use of an active grid, resulting in free-stream turbulence levels of up to 6.2%. The rough surface is characterized by a roughness parameter k+ ≈ 53, and measurements are performed at Reynolds numbers of up to Reθ = 11300. It is confirmed that the free-stream turbulence significantly alters the mean velocity deficit profiles in the outer region of the boundary layer. Consequently, the previously observed ability of the Zagarola & Smits (J. Fluid Mech., vol. 373, 1998, p. 33) velocity scale U∞δ*/δ to collapse results from both smooth and rough surface boundary layers, no longer applies in this boundary layer subjected to high free-stream turbulence. In inner variables, the wake region is significantly reduced with increasing free-stream turbulence, leading to decreased mean velocity gradient and production of Reynolds stress components. The effects of free-stream turbulence are clearly identifiable and significant augmentation of the streamwise Reynolds stress profiles throughout the entire boundary layer are observed, all the way down to the inner region. In contrast, the Reynolds wall-normal and shear stress profiles increase due to free-stream turbulence only in the outer part of the boundary layer due to the blocking effect of the wall. As a consequence, there is a significant portion of the boundary layer in which the addition of nearly isotropic turbulence in the free-stream, results in significant increases in anisotropy of the turbulence. To quantify which turbulence length scales contribute to this trend, second-order structure functions are examined at various distances from the wall. Results show that the anisotropy created by adding nearly isotropic turbulence in the free-stream resides mostly in the larger scales of the flow. Furthermore, by analysing the streamwise Reynolds stress equation, it can be predicted that it is the wall-normal gradient of 〈u2v〉 term that is responsible for the increase in 〈u2〉 profiles throughout the boundary layer (i.e. an efficient turbulent transport of turbulence away from the wall). Furthermore, a noticeable difference between the triple correlations for smooth and rough surfaces exists in the inner region, but no significant differences are seen due to free-stream turbulence. In addition, the boundary layer parameters δ*/δ95, H and cf are also evaluated from the experimental data. The flow parameters δ*/δ95 and H are found to increase due to roughness, but decrease due to free-stream turbulence, which has significance for flow control, particularly in delaying separation. Increases in cf due to high free-stream turbulence are also observed, associated with increased momentum flux towards the wall.

Anisotropy of nonaqueous layered silicate suspensions subjected to shear flow

Abstract: Nonaqueous layered silicate suspensions have a complex rheological behavior due to the presence of a microstructure on multiple length scales, which is sensitive to flow and flow history. In the present work, the flow-induced orientation and anisotropy of the nonequilibrium metastable structures in nonaqueous layered silicate suspensions has been studied using a combination of light scattering, scattering dichroism, and advanced rheometric measurements, including two dimensional small amplitude oscillatory shear (2D-SAOS) flow experiments. The nature of the structures during flow was mainly studied by means of small angle light scattering patterns. Linear dichroism measurements in the vorticity and velocity gradient directions were used to assess the microstructural anisotropy. The changes observed in the vorticity plane developed in the same range of shear rate as the shear-thinning behavior of the suspensions. Scattering dichroism was used to demonstrate that the flow-induced anisotropy was locked in up...

Criteria Of Turbulent Transition In Parallel Flows

Abstract: Based on the energy gradient method, criteria for turbulent transition are proposed for pressure driven flow and shear driven flow, respectively. For pressure driven flow, the necessary and sufficient condition for turbulent transition is the presence of the velocity inflection point in the averaged flow. For shear driven flow, the necessary and sufficient condition for turbulent transition is the existence of zero velocity gradient in the averaged flow profile. It is shown that turbulent transition can be effected via a singularity of the energy gradient function which may be associated with the chaotic attractor in dynamic system. The role of disturbance in the transition is also clarified in causing the energy gradient function to approach the singularity. Finally, it is interesting that turbulence can be controlled by modulating the distribution of the energy gradient function.

Homogenized Euler equation: a model for compressible velocity gradient dynamics

Abstract: Along the lines of the restricted Euler equation (REE) for incompressible flows, we develop homogenized Euler equation (HEE) for describing turbulent velocity gradient dynamics of an isentropic compressible calorically perfect gas. Starting from energy and state equations, an evolution equation for pressure Hessian is derived invoking uniform (homogeneous) velocity gradient assumption. Behaviour of principal strain rates, vorticity vector alignment and invariants of the normalized velocity gradient tensor is investigated conditioned on dilatation level. The HEE results agree very well with the known behaviour in the incompressible limit. Indeed, at zero dilatation HEE reproduces the incompressible anisotropic pressure Hessian behaviour very closely. When compared against compressible direct numerical simulation results, the HEE accurately captures the strain rate behaviour at different dilatation levels. The model also recovers the fixed point behaviour of pressure-released (high-Mach-number limit) Burgers turbulence.

Optimization Principle for Variable Viscosity Fluid Flow and Its Application to Heavy Oil Flow Drag Reduction

Abstract: Drag reduction in heavy oil transport systems is a key for high-efficiency oil transfer and, thus, for energy conservation. In this paper, we investigated the influence of viscosity, velocity, and velocity-gradient fields on drag resistance in fluid flow with variable viscosity in terms of the field synergy. The theoretical analysis indicates that the drag during varying viscosity fluid flow processes depends upon not only the synergy between the velocity and its gradient over the entire flow domain but also the viscosity and velocity gradient at the boundary. That is, for a given flow rate or inlet velocity, simultaneously reducing the fluid flow field synergy number over the entire flow domain and decreasing the fluid viscosity and the velocity gradient at the boundary will lead to a smaller flow resistance. In addition, starting from the basic governing equation and via the calculus of variations, we derived Euler’s equation, essentially the momentum equation with a special additional volume force, usi...

Special Issue on “Transport Phenomena at the Interface Between Fluid and Porous Domains”

Abstract: Free flow above permeable domains is a common problem in both natural and industrial environments. The classical problem of turbulent flows above forests and urban areas, known as the canopy flow problem (Finnigan 2000), was addressed by hundreds of publications. A library search for the keywords ‘canopy and flow’ in the ISI database returned 1312 results in April 2007 and 1845 results in April 2009, about 40% increase in the last 2years alone. In fact, this search underestimates the actual number of publications since research of flow above permeable domains covers a much wider range of scenarios. Vegetated flows in rivers and wetlands, flow above gravel streambeds, and flow above coral reefs are a few environmental examples; industrial examples include flows in coating processes, chemical reactors, and electronic cooling; and examples of a much smaller length scale include flows above bio-films and inside bio-tissues. Studies of flows over permeable domains include both laminar and turbulent flow regimes. Where most of the porous media studies focus on laminar and creeping flows, terrestrial, ocean, and atmospheric flows are all turbulent. Combinations such as turbulent channel flows above laminar flow inside dense gravel beds are also common. Most of the past and present research activities are often classified by the following four disciplines: porous media flows (e.g., Whitaker 1999), atmospheric canopy flows (e.g., Finnigan 2000), vegetated flow (e.g., Nepf 1999), and fluvial flows. Studies of fluvial flows were recently addressed in a special issue on double averaging applications (Nikora and Rowinski 2008). The following special issue with its 13 publications covers both laminar and turbulent flow regimes while emphasizing on fundamental questions of formulating the transport conditions at the interface between the fluid and the porous domains. The most famous publication on laminar flows above porous interfaces is the study by Beavers and Joseph (1967) (cited 771 times, ISI, April 2009). By investigating the flow above porous domains, they developed a linear relationship between the velocity gradient above the interface and the interface slip velocity. However, numerous studies showed that the Beavers and Joseph condition is not general enough and a coupled solution of both the free flow and

Obtaining reliable transient rheological data on concentrated short fiber suspensions using a rotational rheometer

Abstract: The conventional method for obtaining transient rheological data on short glass fiber-filled polymeric fluids is to use the parallel disk (PP) geometry in a rotational rheometer. Using the PP geometry large transient stress overshoot behavior was observed during the startup of flow measurements on a 30 wt % short glass fiber-filled polybutylene terephthalate. A contributing factor to this behavior is believed to be induced fiber collisions caused by the inhomogeneous velocity field (radial varying velocity gradient). A novel approach was taken in which a “donut” shaped sample was used in a cone-and-plate device (CP-D) to maintain a sufficient gap to fiber length ratio. The magnitude of the first normal stress difference was reduced by 70%, and the time to reach steady state was reduced by 100 strain units. The Lipscomb model coupled with the Folgar–Tucker model for the evolution of fiber orientation was fit to the stress growth behavior measured using both the PP geometry and CP-D resulting in different parameters. In addition, the fitted model parameters were found to depend on the initial fiber orientation. It is believed that the CP-D allows for an accurate determination of the stress growth behavior and eventually will allow one to obtain unambiguous model parameters.

On the spatial resolution of velocity and velocity gradient-based turbulence statistics measured with multi-sensor hot-wire probes

Abstract: A highly resolved turbulent channel flow direct numerical simulation with Reτ = 200 has been used to investigate the ability of 12-sensor hot-wire probes to accurately measure velocity and velocity gradient based turbulence statistics. Various virtual sensor separations have been tested in order to study the effects of spatial resolution on the measurements. First, the effective cooling velocity has been determined for each sensor for (1) an idealized probe where the influence of the velocity component tangential to the sensors and flow blockage by the presence of the prongs and the finite lengths of and thermal cross-talk between the sensors are neglected and, (2) for a real probe, the characteristics of which have been determined experimentally. Then, simulating the response of the virtual probes for these two cases to obtain the effective velocities cooling the sensors, velocity and vorticity component statistics have been calculated by assuming the velocity gradients to be constant over the probe sensing area.

Ultrasound velocity profile (UVP) measurements of pulp suspension flow near the wall.

Abstract: An Ultrasound Velocity profiling (UVP) technique is used in this study to investigate the pipe flow of pulp suspensions in the near wall region. Four flow rates and two consistencies were investigated: 1.9 and 4.8% (w/w) consistency. The mean velocity profiles showed a distinct plug at the centre of the pipe, surrounded by a sharp velocity gradient. The plug size increased with increasing consistency or decreasing bulk velocity. The demodulated echo amplitude (DMEA) profile slowly rises from low values near the wall to a distinct maximum at the plug front before slowly decaying towards the pipe centre. Since only the fibres and fines contribute to the attenuation of ultrasound, the demodulated echo amplitude profiles thus indicate and support the hypothesis of the existence of a consistency profile in the near wall area, with a decreasing amount of fines and fibres close to the pipe wall.

Wall shear stress in Görtler vortex boundary layer flow

Abstract: The development of wall shear stress in concave surface boundary layer flows in the presence of Gortler vortices was experimentally studied by means of hot-wire measurements. The wavelengths of the vortices were preset by thin vertical perturbation wires so to produce the most amplified wavelengths. Three different vortex wavelengths of 12, 15, and 20 mm were considered, and near-wall velocity measurements were carried out to obtain the “linear” layers of velocity profiles in the boundary layers. The wall shear stress coefficient Cf was estimated from the velocity gradient of the “linear” layer. The streamwise developments of boundary layer displacement and momentum thickness at both upwash and downwash initially follow the Blasius (laminar boundary layer) curve up to a certain streamwise location. Further downstream, they depart from the Blasius curve such that they increase at upwash and decrease at downwash before finally converge to the same value due to the increased mixing as a consequence of transi...

Hydromagnetic flow of viscous incompressible fluid past a wedge with permeable surface

Abstract: Flow of a viscous, electrically conducting fluid past a wedge having permeable surface is analyzed. A constant transpiration through the wedge surface is assumed. The equations governing the flow and the magnetic field being reduced to local non-similarity equations are solved numerically. The implicit finite difference method, as well as the local non-similarity method is being used in finding the solutions of the reduced equations against the transpiration parameter, ξ. Perturbation solutions for small and large ξ values are also obtained. Effect of the physical parameters, such as, the magnetic force parameter, S, the magnetic Prandtl number, Pm and free stream velocity gradient, n, on the local skin-friction coefficient, f'' (0, ξ), and the local current density coefficient, g'' (0, ξ ), are shown graphically. It is found that the perturbation solutions agreed excellently with other solutions at the two extreme ranges of ξ values. From the present investigation we further observe that, incase of withdrawal of fluid both the momentum and magnetic boundary layers decrease with the increase of ξ. On the other hand these layers increase with ξ value when fluid is being injected trough the surface. Further we notice that there is an onset of reverse flow in the down-stream region in case of blowing of fluid and the starting point of this flow, approximately, is ξ = -0.6.

Strain Characteristics Near the Flame Attachment Point in a Swirling Flow

Abstract: Swirling flows are widely used in industrial burners and gas turbine combustors for flame stabilization. In many cases, the flame is stabilized in the shear layer near the centerbody and/or abrupt expansion, where the high speed nozzle flow transitions into the larger combustor. Several prior studies have shown that the flame position becomes increasingly unsteady as it approaches blowoff, due to local extinction/re-attachment of the flame at one or both of these locations. This is apparently due to the local strain rate irregularly oscillating about the extinction strain rate values near the attachment point. In order to characterize these flame strain characteristics, PIV measurements were obtained of several hydrogen/methane mixtures in this spatial region. The fluid mechanic straining of the flame in this region is dominated by two gradients in velocity – that due to the strong shear near the centerbody and to the bulk flow deceleration as it expands from the smaller diameter nozzle into the combustor. These two velocity gradients cause positive and negative stretching of the flame sheet, respectively. The shearing velocity gradient is an order of magnitude larger than the flow deceleration term but, due to the fact that the flame is essentially parallel to the shear layer, actually has a secondary influence relative to the flow deceleration. As a result, the dominant flame straining term near the attachment point is apparently compressive – a somewhat counter-intuitive result, given that the flame is stabilized in the positively straining shear layer.

Turbulence in a three-dimensional wall-bounded shear flow

Abstract: A new turbulent flow with distinct three-dimensional characteristics has been designed in order to study the impact of mean-flow skewing on the turbulent coherent vortices and Reynolds-averaged statistics. The skewing of a unidirectional plane Couette flow was achieved by means of a spanwise pressure gradient. Direct numerical simulations of the statistically steady Couette–Poiseuille flow enabled in-depth explorations of the turbulence field in the skewed flow. The imposition of a modest spanwise gradient turned the mean flow about 8° away from the original Couette flow direction and this turning angle remained nearly the same over the entire cross section. Nevertheless, a substantial non-alignment between the turbulent shear stress angle and the mean velocity gradient angle was observed. The structure parameter turned out to slightly exceed that in the pure Couette flow, contrary to the observations made in some other three-dimensional shear flows. Coherent flow structures, which are known to be associated with the Reynolds shear stress in near-wall regions, were identified by the λ2-criterion. Instantaneous and ensemble-averaged vortices resembled those found in the unidirectional Couette flow. In the skewed flow, however, the vortex structures were turned to align with the local mean-flow direction. The conventional symmetry between Case 1 and Case 2 vortices was broken due to the mean-flow three-dimensionality. The turning of the coherent vortices and the accompanying symmetry-breaking gave rise to secondary and tertiary turbulent shear stress components. By averaging the already ensemble-averaged shear stresses associated with Case 1 and Case 2 vortices in the homogeneous directions, a direct link between the educed near-wall structures and the Reynolds-averaged turbulent stresses was established. These observations provide evidence in support of the hypothesis that the structural model proposed for two-dimensional turbulent boundary layers remains valid also in flows with moderate mean three-dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.

PIV Investigation of Flow Behind Surface Mounted Detached Square Cylinder

Abstract: The flow field behind surface mounted detached square ribs under the approaching flat plate turbulent boundary layer has been experimentally studied using the particle image velocimetry (PIV) (two-component and stereo) technique in both streamwise and cross stream measurement planes. An oil film visualization study has been carried out for correlating the surface flow patterns to the flow structures. The Reynolds number based on the rib height is equal to 11,075. The ratio of the gap height to the square rib size is set equal to 0.2, 0.37, 0.57, and 1.0. The ratio of approaching boundary layer thickness to rib height is equal to 0.2. The mean and rms velocity fields, streamwise and spanwise vorticity fields, velocity gradient and velocity vector fields, turbulent kinetic energy budgets, and stream trace results are reported. The second invariant of the velocity gradient tensor results are presented to distinguish between the rotational and shear contribution of the vorticity field. The recirculation bubbles with a focilike structure are observed behind the detached ribs. These structures are displaced upward, i.e., away from the wall surface with an increase in gap size of the detached cylinder. The size of the recirculation bubble also drops with an increase in the gap size. The stream traces in the cross stream plane show node-saddle patterns, whose near wall concentration is high for a lower gap size detached cylinder. The oil film visualization images show saddle patterns at the meeting point between the flow through the gap and the reattaching shear layer for the lower gap size detached cylinder. The v-velocity magnitude distribution shows greater wall-normal motion across the wake for the detached cylinder of lower gap size. There is a significant near wall velocity fluctuation for the lower gap size detached cylinder. The higher velocity fluctuation due to the near wall flow structures contributes toward an increase in the near wall mixing of a detached cylinder geometry. Overall, the present study clearly demonstrates the flow structures behind detached ribs, which are responsible for effective near wall mixing. The results from this study provide useful understanding for the design of turbulators in various practical applications.

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

Abstract: The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.

Relativistic Variable Eddington Factor in a Relativistic Plane-Parallel Flow

Abstract: We examine the behavior of the variable Eddington factor for a relativistically moving radiative flow in the vertical direction. We adopt the “one-tau photo-oval” approximation in the comoving frame. Namely, the comoving observer sees radiation coming from a closed surface where the optical depth measured from the observer is unity; such a surface is called a one-tau photo-oval. In general, the radiative intensity emitted by the photo-oval is nonuniform and anisotropic. Furthermore, the photo-oval surface has a relative velocity with respect to the comoving observer, and therefore the observed intensity suffers from the Doppler effect and aberration. In addition, the background intensity usually depends on the optical depth. All of these introduce anisotropy to the radiation field observed by the comoving observer. As a result, the relativistic Eddington factor, f , generally depends on the optical depth � , the four velocity u, and the velocity gradient du=d� . In the case of a plane-parallel vertical flow, we found that the relativistic variable Eddington factor, f , generally decreases as the velocity gradient increases, but it increases as the velocity increases for some cases. When the comoving radiation field is uniform, it is well approximated by 3f � 1=[ 1 + (16=15)(� du=�d� ) + (� du=�d� ) 1:6� 2 ]. When the radiation field in the inertial frame is uniform, on the other hand, it is expressed as f = (1 + 3ˇ 2 )=(3 + ˇ 2 ). These relativistic variable Eddington

Variable Eddington Factor in a Relativistic Plane-Parallel Flow

Abstract: We examine the Eddington factor in an optically thick, relativistic flow accelerating in the vertical direction % When the gaseous flow is radiatively accelerated and there is a velocity gradient, there also exists a density gradient The comoving observer sees radiation coming from a closed surface where the optical depth measured from the observer is unity Such a surface, called a {\it one-tau photo-oval}, is elongated in the flow direction In general, the radiation intensity emitted by the photo-oval is non-uniform, and the photo-oval surface has a relative velocity with respect to the position of the comoving observer Both effects introduce some degree of anisotropy in the radiation field observed in the comoving frame As a result, the radiation field observed by the comoving observer becomes {\it anisotropic}, and the Eddington factor must deviate from the usual value of 1/3 Thus, the relativistic Eddington factor generally depends on the optical depth $\tau$ and the velocity gradient $du/d\tau$, $u$ being the four velocity % In the case of a plane-parallel vertical flow, we obtain the shape of the photo-oval and calculate the Eddington factor in the optically thick regime We found that the Eddington factor $f$ is well approximated by $f(\tau, \frac{du}{d\tau}) = {1/3} \exp (\frac{1}{u} \frac{du}{d\tau}) $ % This relativistic variable Eddington factor can be used in various relativistic radiatively-driven flows

Review of Compressible Pulsating Flow Effects on System Performance

Abstract: Pulsating flow is as old as the human being. It appears in different forms from a compressible to incompressible, one phase, two or multi phase flows. Some forms of pulsating flow are favorable like that increase the combustion efficiency in combustors. Other forms of pulsating flow are harmful such as pulsation associated with compressor surge leading to increase of the noise level. Understanding the behavior of the pulsating flow in different forms, its effects on the flow system performance and on the gas turbine engine cooling system are reviewed. Reynolds number, velocity gradient, pressure gradient and frequency are the main parameters that affecting the pulsating flow behavior. The measurement of pulsating flow in its forms is one of the major tasks of the researchers. From the review it appears that number of researches tried to cover the whole sides of pulsating flow phenomena and the different ways to measure it. However, a lot of works is still needed to be understood thoroughly the effect of pulsating flow on the flow systems and how it could be used to developed the performance.

Internal instability of thin liquid sheets

Abstract: Linear stability analysis of an inviscid liquid sheet with different velocity profiles across its thickness is reported. The velocity profiles for which there is a progressive increase or decrease in velocities between the two interfaces are demonstrated to be inherently unstable even in the absence of the destabilizing aerodynamic shear at the liquid-gas interfaces. Compared to a flat velocity profile, a linear or a parabolic profile, symmetric at the center line of the sheet reduced both the maximum growth rate and the wavelength range over which the waves grow. The convective acceleration from the velocity gradient is found to stabilize longer waves while the growth of shorter waves is hampered by the combined effect of the surface tension and a decrease in the interface velocity between gas and liquid media. The wave forms are dominantly sinuous for symmetric velocity profiles; however, with larger velocity gradients the dilatational modes are observed. The inherent instability of liquid sheets with a progressive change in velocities between the interfaces is seen to arise from the differential convective acceleration at the two interfaces in the plane of reference of the liquid sheets.

Lyapunov exponent for small particles in smooth one-dimensional flows

Abstract: This paper discusses the Lyapunov exponent for small particles in a spatially and temporally smooth flow in one dimension. Using a plausible model for the statistics of the velocity gradient in the vicinity of a particle, the Lyapunov exponent is obtained as a series expansion in the Stokes number, St, which is a dimensionless measure of the importance of inertial effects. The approach described here can be extended to calculations of the Lyapunov exponents and of the correlation dimension for inertial particles in higher dimensions. It is also shown that there is correction to this theory which arises because the particles do not sample the velocity field ergodically. Using this non-ergodic correction, it is found that (contrary to expectations) the first order term in the expansion does not vanish.