Air entrainment in two-dimensional turbulent shear flows with partially developed inflow conditions

In plunging jet flows and at hydraulic jumps, large quantities of air are entrained at the intersection of the impinging flow and the receiving body of water. The air bubbles are entrained into a turbulent shear layer and strong interactions take place between the air bubble advection/diffusion process and the momentum shear region. New air-water flow experiments were conducted with two free shear layer flows: a vertical supported jet and a horizontal hydraulic jump. The inflows were partially developed boundary layers, characterized by the presence of a velocity potential core next to the entrapment point. In both cases, the distributions of air concentration exhibit a Gaussian distribution profile with an exponential longitudinal decay of the maximum air content. Interestingly, the location of the maximum air content and the half-value band width are identical for both flow situations, i.e. independent of buoyancy effects.     

Turbulent friction factor for two-phase: air-viscoelastic flows through a horizontal tube

The fully developed two-phase turbulent isothermal Fanning friction factors for air-viscoelastic fluid flows through a horizontal tube were measured experimentally. The viscoelastic fluids studied were aqueous solutions of polyacrylamide (100, 200, and 500 ppm by weight). Over the range of the apparent Reynolds number (Re_e) from 10,000 to 100,000, the homogeneous model was found to be accurate enough for engineering prediction of turbulent friction factor for air-viscoelastic flows through horizontal tubes. A new correlation for the turbulent friction factor of airlriscoelastic plug flow is proposed.     

A model for the flow of solid particles in gases

The flow of solid particles in air streams involves a great deal of variables and complex phenomena, difficult to analyse. In practice the flow quantities in gas-solid flows are predicted by the use of empirical correlations of data or semi-empirical methods. The predictive power of these methods varies substantially between different systems. This paper presents an analytical approach to the subject of gas-solid flows, based on a turbulent model. The mixture is modeled as a variable density fluid flowing in a duct; the equations for the Reynolds stress incorporate the variation of velocity and density together, and yield the velocity profile of the flow and average quantities of interest such as the mass flux, the friction factor, the average density and average areas occupied by each phase. The predicted values for the friction factor are compared with known correlations emanating from experimental data. It is found that there is a very good agreement between the predicted values and the experimental correlations.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

Combined forced and free flow in a vertical rectangular duct with prescribed wall heat flux

In this paper, combined forced and free convection is studied in a vertical rectangular duct with a prescribed uniform wall heat flux (H2 boundary condition). A different heat flux value for each plane wall is considered; the condition of a uniform wall heat flux throughout the duct results as a special case. The local momentum and energy balance equations are written in a dimensionless form and solved numerically, by means of a Galerkin finite element method. The numerical solution gives the dimensionless velocity and temperature distributions, together with the values of the Fanning friction factor, of the Nusselt number, of the momentum flux correction factor and of the kinetic energy correction factor. These dimensionless parameters are reported as functions of the aspect ratio and of the ratio between the Grashof number, Gr, and the Reynolds number, Re. The threshold values of Gr/Re for the onset of flow reversal are evaluated.     

A Direct Numerical Study of Transport and Anisotropy in a Stably Stratified Turbulent Flow with Uniform Horizontal Shear

Direct numerical simulations of homogeneous turbulence in stably stratified shear flow have been performed to aid the understanding of turbulence and turbulent mixing in geophysical flow. Two cases are compared. In the first case, which has been studied in the past, the mean velocity has vertical shear and the mean density is vertically stably stratified. In the second case, which has not been studied systematically before, the mean velocity has horizontal shear and the mean density is again vertically stably stratified. The critical value of the gradient Richardson number, for which a constant turbulence level is obtained, is found to be an order of magnitude larger in the horizontal shear case. The turbulent transport coefficients of momentum and vertical mass transfer are also an order of magnitude larger in the horizontal shear case. The anisotropy of the turbulence intensities are found to be in the range expected of flows with mean shear with no major qualitative change in the range of Richardson numbers studied here. However, the anisotropy of the turbulent dissipation rate is strongly affected by stratification with the vertical component dominating the others. This revised version was published online in July 2006 with corrections to the Cover Date.     

FINITE ELEMENT SURFACE MODEL FOR FLOW AROUND VERTICAL WALL ABUTMENTS

A two-dimensional finite element surface model is developed to determine velocities, depths, and turning angles around vertical wall abutments. The model solves the Reynolds-averaged turbulent flow equations along a horizontal plane passing through the average water surface. This approach is an improvement over the depth-averaged flow models where dispersion terms reflecting vertical effects are neglected. In the model, vertical gradient effects are accounted for through the use of power law for the vertical distribution of the longitudinal velocity; a similar treatment is applied to lateral turbulent shear stresses. The model is capable of computing the dynamic pressure distribution, which in turn is converted to water elevation values. The model, being two dimensional, is computationally efficient and practical to use. The numerical model was successfully verified using experimental data from vertical wall abutments and groins with protrusion ratios (ratio of protrusion length perpendicular to direction of flow to total channel width) of 0·1, 0·2 and 0·3. The results show the occurrence of a high intensity velocity zone close to the upstream abutment nose similar to those observed experimentally. The effects of roughness, depth, and energy slope on the intensity of flow field is investigated and an analytical expression is developed. Numerical experiments indicate that grain roughness affects flow field around the abutment nose by controlling the magnitude of the lateral velocity component and by controlling the lateral extent of the affected zone. Velocity amplification at the abutment nose is found to be mainly related to the protrusion ratio and to the friction factor, and can be up to 1·75 times the approach velocities for protrusion ratios of 0·3. For a protrusion ratio of 0·3, for a typical range of roughness values the increase in nose velocities due to friction factor alone was found to be up to 20 percent.     

Modelling of particle-wall collisions in confined gas-particle flows

This paper demonstrates that numerical simulations of confined particulate two-phase flows require a detailed modelling of particle—wall collisions which includes the wall surface structure and the particle shape. These effects are taken into account by “irregular bouncing” models which are based on the statistical treatment of the collision process. In this study, results obtained using various “irregular bouncing” models based on the impulse equations for a particle—wall collision are considered and compared with experimental observations. The wall roughness is simulated by assuming that the particle collides with a virtual wall which has a randomly distributed inclination with respect to the plane, smooth wall. A Gaussian distribution for this random inclination showed the best agreement with experimental results. Numerical predictions of a turbulent two—phase flow in a vertical channel, where the particle phase is treated using a Lagrangian approach, showed that the different models applied for a particle-wall collision have a strong effect on the particle velocity fluctuations and the mass flux profiles in the region of fully developed flow. The numerical simulations using the irregular bouncing models yielded considerably higher values for the particle velocity fluctuations, which also agreed better with the experimental values. This effect was most pronounced for large particles, where the distance they need to respond to the fluid flow is larger than the characteristic dimension of the confinement. On the other hand, the motion of small particles is less affected by the choice of the wall-collision model. These effects of the wall roughness on the velocity fluctuations of the dispersed phase have not been considered in previous studies using irregular bouncing models.     

Analytical solutions for transverse distributions of stream-wise velocity in turbulent flow in rectangular channel with partial vegetation

The theory of poroelasticity is introduced to study the hydraulic properties of the steady uniform turbulent flow in a partially vegetated rectangular channel. Plants are assumed as immovable media. The resistance caused by vegetation is expressed by the theory of poroelasticity. Considering the influence of a secondary flow, the momentum equation can be simplified. The momentum equation is nondimensionalized to obtain a smooth solution for the lateral distribution of the longitudinal velocity. To verify the model, an acoustic Doppler velocimeter (ADV) is used to measure the velocity field in a rectangular open channel partially with emergent artificial rigid vegetation. Comparisons between the measured data and the computed results show that the method can predict the transverse distributions of stream-wise velocities in turbulent flows in a rectangular channel with partial vegetation.     

Brinkman Flow Past a Stretching Sheet

The problem of steady viscous flow of an incompressible fluid over a flat deformable sheet in a porous medium, when the sheet is stretched in its own plane is revisited. An exact solution is recovered for the two-dimensional case and a totally analytic approximate solution is developed for the axisymmetric case. Stretching rate of two-dimensional case is assumed as double the stretching rate of axisymmetric case. The analytical expressions of residual errors, horizontal, vertical velocity distributions, stream lines, vorticity lines, pressure distributions have been obtained and plotted. The values of skin friction, entrainment velocity, boundary layer thickness, momentum thickness and energy thickness have been tabulated. For the first time, two-dimensional and axisymmetric cases are compared by means of a unified scale.     

A geometric approach for predicting vertical stationary profiles of weakly inertial advecting-diffusing particles in closed incompressible flows

Mixing of weakly inertial particles in closed flows is often addressed by considering individual particles as passive advecting-diffusing tracers, subjected to an additional settling velocity resulting from body forces (e.g. gravity). We show that the qualitative and quantitative features of the vertical particle distribution (i.e. the horizontal cross-sectional averages of particle concentration) can be predicted from the structure of the flow resulting from the superposition of the stirring field and the settling velocity. The prediction is based upon the observation that the resulting flow can be divided into two nonoverlapping regions, namely trajectories that are confined within the mixing space (recirculation loops), and trajectories that cross the mixing space. The spatial extent of these regions is exploited to define an effective vertical convective velocity entering the one-dimensional lumped model. Model two-dimensional flows possessing different flow patterns are used to illustrate the proposed estimate for effective velocity. A CFD-computed three-dimensional turbulent flow inside a baffled stirred vessel is used as a benchmark test to assess the model performance in typical industrial flows.     

Modeling study of three-phase low liquid loading flow in horizontal pipes

A theoretical study is conducted to model the flow characteristics of three-phase stratified wavy flow in horizontal pipelines with a focus on the low liquid loading condition, which is commonly observed in wet gas pipelines. The model predictions are compared to the experimental data of Karami et al. (2016a, b). These experiments were conducted with water or 51 wt% of MEG in the aqueous phase, and inlet aqueous phase fraction values from 0 to100%.Modeling of three-phase flow can be described as a combination of two-phase gas-liquid flow modeling, and a liquid phase oil-water mixing modeling. A mechanistic model is proposed to predict flow characteristics of three-phase stratified wavy flow in pipeline. For the gas-liquid interactions, Watson's (1989) combined momentum balance equation derivation was applied. However, the calculation procedure was reversed, and the wave celerity was assumed as an input, while interfacial friction factor was one of the model's outputs. The liquid-liquid interactions were modeled using a simple energy balance equation and shift in liquid phase center of gravity calculations. The liquid phases can be separated, partially mixed, or fully mixed. The bottom aqueous film velocity was calculated using the law of the wall formulation, and was used to calculate the flowing aqueous phase fraction.The model predictions of different flow characteristics for two and/or three-phase flows were compared with available experimental data. The pressure gradient, wave amplitude, and aqueous phase fraction predictions were in good agreement with the experimental data. However, the liquid holdup predictions were slightly under-predicted by the model. Overall, an acceptable agreement was observed for all cases.Most of the common multiphase stratified flow models are developed with the assumption of steady-state conditions and with constant interfacial friction factor value. This study proposes a novel method to model stratified flow. The predictions are in acceptable agreement with experimental data conducted under stratified wavy flow pattern conditions.     

Do huge waves exist in horizontal gas-liquid pipe flow?

Huge waves are periodic interfacial structures which are observed in vertical co-current gas-liquid two-phase flow under churn and the transition between churn and annular flows. Published data examining vertical gas-liquid flow indicate that a huge wave has either a continuous gas core surrounded by a large-scale interfacial wave or a core with a highly-agitated mixture of gas and liquid.Employing a Wire-Mesh Sensor (WMS), the spatio/temporal investigation of high flow rate horizontal air-water flow divulged some recurrent liquid structures (one may call pseudo-slugs) analogous to huge waves of (vertical) churn flow. In both cases, the blow-through (penetration of gas into the liquid structure) was the most manifest feature.Different qualitative and quantitative methods were employed to compare the behavior of pseudo-slug to churn flow. The quantitative measures included Probability Density Function analysis (PDF), distribution coefficient in drift flux model, structural velocity, core average velocity, interfacial friction factor, and slippage number. Both flow regimes demonstrated similar behavior.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

A new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles. To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved. Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley & Sons, Ltd.     

The spatial inhomogeneity of turbulence above a fully rough, packed bed in open channel flow

Single point turbulence statistics measured directly above and in close proximity to the wall in a fully developed, fully rough, turbulent open channel flow are reported. In order to investigate the spatial inhomogeneity of the turbulence, the measurements were obtained over a matrix of measurement points in a plane parallel to the roughness-bed surface. The measurements were obtained with a three-component laser Doppler velocimeter (3D-LDV) system. The turbulence statistics associated with the vertical velocity component, including conditioned mean vertical velocities, rms distributions, and mean vertical momentum fluxes are emphasized. For the Reynolds and Froude numbers associated with this investigation, and with the specific roughness geometry employed in this study (a packed bed of uniform-diameter spheres), it is found that the distribution of the local mean vertical velocity, <w>, has non-zero contributions over the roughness pattern and that this contributes to a mean net vertical momentum flux into the roughness bed. However, the net vertical momentum flux due to turbulent fluctuations is positive out of the bed, consistent with smooth-wall behavior. These results are relevant to the study of sediment entrainment and suspension/deposition as well as the exchange and transport of chemical species between the channel core flow and the fluid within the roughness bed. Received: 21 July 1998/Accepted: 20 November 1999     

Flow around submerged structures subjected to shallow submergence over plane bed

The results of an experimental investigation on the flow field around submerged structures on horizontal plane beds, measured by an acoustic Doppler velocimeter (ADV), are presented. Experiments were conducted for various conditions of submergence, having submergence factors ranging from 1.0 to 2.0 and average flow velocity ranging from 0.25 to 0.51 m/s. The Froude number and the Reynolds number of the approaching flow for different runs are in the range of 0.18–0.42 and 50 000–76 500, respectively. The vertical distributions of time-averaged three dimensional velocity components and turbulence intensity components at different radial distances from the submerged structures are plotted. Deceleration and acceleration of the approaching flow around the submerged body are evident from the vertical distributions of the horizontal velocity component, whereas the lifting and diving nature of the flow are indicated by the vertical velocity component distributions. The vertical distributions of the horizontal velocity component indicate reduction of 30% of the non-dimensional time-averaged horizontal velocity component magnitude for the cylinder of diameter 11.5 cm in comparison to the cylinder of diameter 10 cm. Also, there is an increase of 10–25% in the horizontal velocity component at different radial sections. The flow is three dimensional in the downstream of the submerged structure. The velocity and the turbulent intensity components are also well predicted by FLUENT. The flow characteristics in the wake and the induced bed shear stress are also analyzed with FLUENT.The profiles of non-dimensional shear velocity deviate from the log law in the wake and the far downstream directions. The scour prone regions may be identified from the profiles of the induced bed shear stress around the submerged structure.     

Order and disorder in particle–liquid flows in a pipe

The spherical expanded polystyrene particle–oil two-phase flow in a vertical pipe was used to simulate the dispersed phase distribution in laminar bubbly flows. A three-dimensional particle image tracking technique was used to track the particles in the flow to study the ordered structure of dispersed phase distribution and its transition to disorder. The ordered structures behaved as particle strings aligned in the flow direction as induced by the flow shear. The structures were quite durable in high liquid velocity flows and dispersed gradually as the liquid velocity decreased. In lower velocity flows, the particles tended to form clusters in the horizontal direction, as predicted by potential theory for spherical bubbles rising in a quiescent inviscid liquid and as observed in experiments on non-shear bubbly water flows.     

Modeling of dynamics, heat transfer, and combustion in two-phase turbulent flows: 1. Isothermal flows

This paper presents a review of authors' collective works in the field of two-phase flow modeling done in the past few decades. The paper is aimed at the construction of mathematical models for simulation of particle-laden turbulent flows. A kinetic equation was obtained for the probability density function (PDF) of the particle velocity distribution in turbulent flows. The proposed kinetic equation describes both the interaction of particles with turbulent eddies of the carrier phase and particle-particle collisions. This PDF equation is used for the derivation of different schemes describing turbulent momentum transfer in the dispersed particle phase. The turbulent characteristics of the gaseous phase are calculated on the basis of the k - turbulence model with a modulation effect of particles on the turbulence.

The constructed models have been applied to the calculation of various two-phase gas-particle turbulent flows in jets and channels as well as particle deposition in tubes and separators. For validating the theoretical and numerical results, a wide range of comparisons with experimental data from Russian and foreign sources has been done.     

Effects of a transverse flows with variable viscosity in micropolar fluids

A regular perturbation analysis is presented for three laminar natural convection flows in micropolar fluids in liquids with temperature dependent viscosity: a freely-rising plane plume, the flow above a horizontal line source on an adiabatic surface (a plane wall plume) and the flow adjacent to a vertical uniform flux surface. While these flows have well-known power-low similarity solutions when the fluid viscosity is taken to be constant, they are non-similar when the viscosity is considered to a function of temperature. A single similar flow, that adjacent to a vertical isothermal surface, is also analysed for comparison in order to estimate the extent of validity of perturbation analysis. The formulation used here provides a unified treatment of variable viscosity effects on those four flows. Computed first-order perturbation quantities are presented for all four flows. Numerical results for velocity, angular velocity and thermal functions has been shown graphically or tabulated for different values of micropolar parameters. Received on 20 October 1997