Variation of drag coefficients in an interacting drop stream

Variation of the drag coefficient of closely spaced drops in a stream injected into a turbulent flow is studied experimentally. Three different regions are identified. In the first region, close to the injector, drops flow in the wake of each other. The drag coefficient in this region is much smaller (by a factor of 4 to 5) than the standard drag coefficient, and its magnitude is dependent on the drop initial spacing. Shortly downstream of the injection point, the transition region starts, where the drag coefficient increases rapidly approaching the drag coefficient of a single isolated drop. And further down-stream when drops are dispersed significantly, the drag coefficient will behave the same way as a single isolated drop.     

Viscous flow of a suspension of liquid drops in a cylindrical tube

Viscous flow in a circular cylindrical tube containing an infinite line of viscous liquid drops equally spaced along the tube axis is considered under the assumption that a surface tension, sufficiently large, holds the drops in a nearly spherical shape. Three cases are considered: (1) axial translation of the drops, (2) flow of the external fluid past a line of stationary drops, and (3) flow of external fluid and liquid drops under an imposed pressure gradient. Both fluids are taken to be Newtonian and incompressible, and the linearized equations of creeping flow are used.The results show that both drag and pressure drop per sphere increase as the spacing increases at fixed radius and also increase as the radius of the drop increases. The presence of the internal motion reduces the drag and pressure gradients in all cases compared to rigid spheres, particularly for drops approaching the size of the tube.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

Investigation on effect of drag models on flow behavior of power-law fluid–solid two-phase flow in fluidized bed

In this study, a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed. Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop. One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore, and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations. Predictions are compared with experimental data measured by Lali et al., where the computed porosities from model B are closer to the measured data than other models. Furthermore, the predicted pressure drop rises as liquid velocity increases, while it decreases with the increase of particle size. Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient, and particle concentration, granular temperature, granular pressure, and granular viscosity go down accordingly.     

Particle and drop dynamics in the flow behind a shock wave

The results of an investigation of the dynamics of hard particles and liquid drops in the flow behind a transmitted shock wave are presented. From the equation of motion of a particle in the shock wave, relations for the displacement, velocity and acceleration as functions of time and certain velocity-relaxation parameters taking into account the properties of the gas and the aerodynamic drag of the particles are obtained for unsteady flow around the particles at an acceleration of 10³–10⁴ m/s². It is shown that the velocity-relaxation parameters are universal. Approaches to finding the aerodynamic drag of freely-accelerating bodies from the dynamics of their acceleration after being suddenly exposed to the flow are considered. It is established that under these conditions the drop dynamics observed can be well described in terms of the same velocity-relaxation parameters with account for linear growth of the transverse drop size. All the kinematic functions obtained are confirmed experimentally.     

The flow past a sphere in a cylindrical tube: effects of intertia,shear-thinning and elasticity

Numerical solutions are presented for the flow past a sphere placed at the centreline of a cylindrical tube for Reynolds numbers ranging from 0 to 150, using a boundary element method. Fluids are modelled by a variety of constitutive equations including the Newtonian, the Carreau and the Phan-Thien-Tanner models. The influence of inertia, shear-thinning and fluid elasticity on the flow field, drag and the pressure drop force-drag ratio is examined. Some results are compared with available experimental data.     

Heat and mass transfer in the vicinity of an evaporating droplet

The problem of heat and mass transfer around an evaporating water drop located in a superheated steam flow is studied using singular perturbation methods.The perturbation scheme involves three regions and some results concerning the Nusselt number and the drag coefficient of the drop are presented.     

On the force acting on a heated spherical drop moving in a gaseous medium

The flow around a heated spherical drop in a viscous non-isothermal gaseous medium with uniformly distributed constant-power heat sources (sinks) acting inside is theoretically described in the Stokes approximation. It is assumed that the mean temperature of the drop surface can differ substantially from the temperature of the ambient gaseous medium. An analytical expression for the drag force and drift velocity in the gravity field is derived by solving hydrodynamic equations with allowance for the temperature dependence of viscosity, thermal conductivity, and density of the gaseous medium.     

Drag difference between steady and shocked gas flows passing through a porous body

Experimental and numerical investigations of gas flows through porous materials have been carried out. We have investigated steady and unsteady processes occurring when the gas flow interacts with porous materials. Densities and porosities of the four open-cell-type polyurethane foams which were investigated are kg/m and , with the foams having different structures. Experiments were conducted to determine the steady drag coefficient of the porous material at low Reynolds numbers, evaluated from the pressure drop. The Forchheimer equation was applied to determine the drag. Values of permeability coefficients () in the Forchheimer equation were estimated by comparing computed and experimental results. Results show that the drag coefficient is largely affected by the internal structure of the foam, and has almost no effect on the stress history, while the value of dominates the stress history variation. Differences of 1000 times exist between the steady flow and unsteady shock tube flow values. Received 15 May 1998/ Accepted 15 March 1999     

Friction and heat transfer in drag-reducing surfactant solution flow through curved pipes and elbows

A study of drag-reducing flow in curved pipes was conducted. In contrast to earlier studies we show that if we use a modified definition of drag reduction that includes only the turbulence effects, we observe indeed the same level of drag reduction in both coiled and straight pipes. More complex results showing reduced drag reduction compared to curved pipes were achieved with elbows. Two elbows of different size and type were tested in turbulent flow of both water and drag-reducing surfactant solution. A more elaborate analysis was conducted for a half-inch threaded elbow with a ratio of curvature radius to diameter of 1.2. The pressure drop and heat transfer were measured in a section downstream from the elbow over a distance of x/D = 130 in order to investigate the hydrodynamic and thermal developments of the flow. The pressure drop coefficient of the elbow was calculated for water and a surfactant solution, based on the total increase in pressure drop in the system due to the presence of the elbow. For a larger welded elbow of 6″ diameter some drag reduction was measured for the surfactant solution.     

An improved correlation of the pressure drop in stenotic vessels using Lorentz’s reciprocal theorem

A mathematical model of the human cardiovascular system in conjunction with an accurate lumped model for a stenosis can provide better insights into the pressure wave propagation at pathological conditions. In this study, a theoretical relation between pressure drop and flow rate based on Lorentz’s reciprocal theorem is derived, which offers an identity to describe the relevance of the geometry and the convective momentum transport to the drag force. A voxelbased simulator V-FLOW VOF3 D, where the vessel geometry is expressed by using volume of fluid(VOF) functions, is employed to find the flow distribution in an idealized stenosis vessel and the identity was validated numerically. It is revealed from the correlation that the pressure drop of NS flow in a stenosis vessel can be decomposed into a linear term caused by Stokes flow with the same boundary conditions, and two nonlinear terms. Furthermore, the linear term for the pressure drop of Stokes flow can be summarized as a correlation by using a modified equation of lubrication theory, which gives favorable results compared to the numerical ones. The contribution of the nonlinear terms to the pressure drop was analyzed numerically, and it is found that geometric shape and momentum transport are the primary factors for the enhancement of drag force. This work paves a way to simulate the blood flow and pressure propagation under different stenosis conditions by using 1D mathematical model.     

Drag reduction using surfactants in a rotating cylinder geometry

Turbulent Couette flow between two circular cylinders has been used for drag reduction experiments using surfactants. In the experiments presented here, only the outer cylinder rotates, the inner cylinder remains at rest and accurate measurements of the torque at the inner cylinder are measured. Water is used as a reference fluid. A drag reducing surfactant called Arquad S-50 (Akzo Nobel Surface Chemistry LLC, Chicago, Ill., USA) (5 mM)+NaSal (12.5 mM) was used as the drag reduction agent. This surfactant can reduce the drag up to 70% (a Reynolds number of about 70,000–150,000) as measured by pressure drop in a pipe flow. Experiments in Couette flow also show drag reduction in the turbulent range. Two arrangements were used, (1) one small trip-wire on the inner cylinder, and (2) four larger trip-wires on the outer cylinder. These trips reduce the critical Reynolds number for transition from laminar to turbulent flow. In case (1), we obtained 18% drag reduction at 5,000<Re<15,000 and in case (2), we obtained an average reduction of about 20% at 2,000<Re<10,000, increasing up to 30% at Re=15,000. The paper also discusses two important problems. First, the shear rate is not constant in the radial gap in circular Couette flow. For non-Newtonian fluids, where the molecular viscosity is a function of the shear rate, this effect must be considered. Second, which viscosity should be used in the Reynolds number? For pipe flow measurements, most authors use the viscosity of the solvent (generally water and Newtonian). For measurements in the Couette flow, we use a different approach, which is described in this paper. We conclude that Couette flow is a useful method for drag reduction investigations. Its advantage is the much smaller geometry in comparison to those of conventional test facilities such as wind tunnels, water, or oil channels or in tubes.     

Metal Foam Heat Exchangers for Heat Transfer Augmentation from a Cylinder in Cross-Flow

A numerical study has been conducted to examine the heat transfer from a metal foam-wrapped solid cylinder in cross-flow. Effects of the key parameters including the free stream velocity and characteristics of metal foam such as porosity, permeability, and form drag coefficient on heat and fluid flow are examined. Being a determining factor in pressure drop and heat transfer increment, the porous layer thickness is changed systematically to observe that there is an optimum layer thickness beyond which the heat transfer does not improve while the pressure drop continues to increase. This has been verified by the application of Bejan’s Intersection of Asymptotes method. Results have been compared to those of a finned-tube heat exchanger to observe much higher heat transfer rate with reasonable excess pressure drop leading to a higher area goodness factor for metal foam-wrapped cylinder.     

Effect of interface deformability on thermocapillary motion of a drop in a tube

The effect of an externally imposed axial temperature gradient on the mobility and deformation of a drop in an otherwise stagnant liquid within an insulated cylindrical tube is investigated. In the absence of bulk transport of momentum and energy, the boundary integral technique is used to obtain the flow and temperature fields inside and outside the deformable drop. The steady drop shapes and the corresponding migration velocities are examined over a wide range of the dimensionless parameters. The steady drop shape is nearly spherical for dimensionless drop sizes <0.5, but becomes slightly elongated in the axial direction for drop sizes comparable to tube diameter. The adverse effect of drop deformation on the effective temperature gradient driving the motion is slightly more pronounced than its favorable effect of reducing drag, thereby leading to a slight reduction in drop mobility with increasing drop deformation. Increasing the viscosity ratio reduces drop deformation and leads to a slight enhancement in the relative mobility (with respect to free thermocapillary motion) of confined drops. When the drop fluid has a lower thermal conductivity than the exterior phase, the presence of the thermally-insulating wall increases the thermal driving force for drop motion (compared to that for the same drop in unbounded domain) by causing more pronounced bending of the isotherms toward the drop. However, the favorable thermal effect of the confining wall is overwhelmed by its retarding hydrodynamic effect, causing the confined drop to always move slower than its unbounded counterpart regardless of the value of the thermal conductivity ratio.     

Two-phase two-component interface drag coefficients in separated phase flows

This work examines the behavior of the interface friction factor or drag coefficient as a means for extending the modeling of separated two-phase flows through the separate consideration of each phase. The model development of this work builds primarily upon the work of Carofano & McManus (1969), Wallis (1970) and Smith (1968). A one-dimensional flow model was developed for the case of vertical upward annular fiow of an air-water mixture with droplet entrainment. The model was developed for the investigation of accelerating flows in a nozzle but is utilized in this study for the investigation of momentum transport occurring in non-accelerating flows. This study presents experimental data showing the behaviour of the flow pressure drop occurring at various flow qualities and gas velocities. Also presented are empirical results for values of the air-water interface drag coefficient as a function of flow quality and gas core Reynolds number. The drag coefficient variation is compared to a previous correlation developed by Wallis 1969).     

Effects of the extensional rate on two-dimensional turbulence of semi-dilute polymer solution flows

An experimental study has been performed to investigate the relationship between the extensional viscosity of polymers and the turbulent drag reduction. In order to obtain the flow which is mostly dominated by extensional flow, two-dimensional (2D) grid turbulence made by flowing soap films was used. Extensional rate added to the flow was controlled by changing the conformation of the grid. Polyethylene oxide, as a flexible polymer, and hydroxypropyl cellulose, as a rigid rod-like polymer were added to the flow. Several extensional rates affect polymer behaviors, which induce different effects. Drag reduction effects of polymers under several extensional rates were visualized and analyzed by image processing. Rheological properties of the polymer solutions were also measured by a rheometer. The results indicated that the mechanisms of energy transfer are different in the streamwise and normal directions. The critical concentration to observe drag reduction effects in 2D turbulence was changed by the extensional rate. When extensional rate is higher, the effects were started to observe from lower concentration. These results were confirmed to correspond to the drag reduction effects obtained by classical pressure drop experiments in a pipe flow.     

A Legendre‐spectral element method for flow and heat transfer about an accelerating droplet

The problem of flow and heat transfer associated with a spherical droplet accelerated from rest under gravitational force is studied using a Legendre‐spectral element method in conjunction with a mixed time integration procedure to advance the solution in time. An influence matrix technique that exploits the superposition principle is adapted to resolve the lack of vorticity boundary conditions and to decouple the equations from the interfacial couplings. The computed flow and temperature fields, the drag coefficient, the Nusselt number, and the interfacial velocity and vorticity are presented for a drop moving vertically in a quiescent gas of infinite extent to illustrate the evolution of the flow and temperature fields. Comparison of the predicted drag coefficient and the Nusselt number against previous numerical and experimental results indicate good agreement. Copyright © 2000 John Wiley & Sons, Ltd.     

The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.      

Enhanced drag in pipe turbulent flow by an aqueous electrolyte: An electroviscous effect

Drag enhancement is reported for turbulent pipe flow of aqueous electrolyte solutions. No electroviscous effect was obtained with laminar flow. Nor was any unusual pressure drop observed for laminar or turbulent flow of non-electrolyte aqueous solutions such as sugar. An electroviscous theory was advanced that predicted the drag enhancement for a 1/1 electrolyte solution. The theory depended on consideration of Debye length.     

Active control of flow past an elliptic cylinder using an artificial neural network trained by deep reinforcement learning

The active control of flow past an elliptical cylinder using the deep reinforcement learning (DRL) method is conducted. The axis ratio of the elliptical cylinder $\Gamma$ varies from 1.2 to 2.0, and four angles of attack $\alpha=0^\circ, 15^\circ, 30^\circ$, and $45^\circ$ are taken into consideration for a fixed Reynolds number $Re=100$. The mass flow rates of two synthetic jets imposed on different positions of the cylinder $\theta_1$ and $\theta_2$ are trained to control the flow. The optimal jet placement that achieves the highest drag reduction is determined for each case. For a low axis ratio ellipse, i.e., $\Gamma=1.2$, the controlled results at $\alpha=0^\circ$ are similar to those for a circular cylinder with control jets applied at $\theta_1=90^\circ$ and $\theta_2=270^\circ$. It is found that either applying the jets asymmetrically or increasing the angle of attack can achieve a higher drag reduction rate, which, however, is accompanied by increased fluctuation. The control jets elongate the vortex shedding, and reduce the pressure drop. Meanwhile, the flow topology is modified at a high angle of attack. For an ellipse with a relatively higher axis ratio, i.e., $\Gamma\ge1.6$, the drag reduction is achieved for all the angles of attack studied. The larger the angle of attack is, the higher the drag reduction ratio is. The increased fluctuation in the drag coefficient under control is encountered, regardless of the position of the control jets. The control jets modify the flow topology by inducing an external vortex near the wall, causing the drag reduction. The results suggest that the DRL can learn an active control strategy for the present configuration.