We propose a systematic derivation method of the Korteweg-de Vries-Burgers (KdVB) equation and nonlinear Schrödinger (NLS) equation for nonlinear waves in bubbly liquids on the basis of appropriate choices of scaling relations of physical parameters. The basic equations are composed of a set of conservation equations for mass and momentum and the equation of bubble dynamics in a two-fluid model. The scaling of parameters is related to the wavelength, frequency, propagation speed, and amplitude of waves concerned. With the help of the method of multiple scales, appropriate choices of the parameter scaling allow us to derive various nonlinear wave equations systematically from a set of basic equations. The result shows that the one-dimensional nonlinear propagation of a long wave with a low frequency is described by the KdVB equation, and that of an envelope of a carrier wave with a high frequency by the NLS equation. Thus, we establish a unified theory of derivation of nonlinear wave equations in bubbly liquids.
This paper presents results of an experimental study of the organized oscillation of incompressible turbulent flow over a rectangular cavity. Experiments were carried out to clarify the oscillation frequencies and the strength of shear layer-cavity trailing edge interaction. LDV and PIV measurements were made in a closed re-circulating water tunnel for rectangular cavities with L/D (L; cavity length, D; cavity depth) =2 and L/D=4. Measurements were also made for a backward facing step flow. Reynolds number based on momentum thickness was 8,300. LDV-measured velocity spectra reveal that self-sustained oscillations due to shear layer-trailing edge interaction are absent or very weak. At the upstream part of the cavity, dominant frequencies of the velocity fluctuations show little variation with L/D, and the frequencies are close to those of the shear layer instability. As the measuring point moves to the downstream, oscillation frequencies decrease and the peaks of the spectra become unclear. PIV-measured instantaneous flow fields show the existence of organized vortical structures in the cavity shear layer. From PIV images, frequencies of cavity oscillation are visually estimated and the results are compared to the LDV-measured velocity spectra.
Vertical sloshing is the liquid surface motion in a container forced to oscillate in the vertical direction. The present paper concerns the vertical sloshing in various equilateral-polygonal-section containers such as octagonal-, heptagonal-, hexagonal-, pentagonal-, square- and triangular-section containers together with a circular-section container, in order to generalise their sloshing modes. As a result, the authors classify the sloshing modes on the basis of the conventional circular-section-container sloshing modes. It is revealed that this modal classification has some advantages over that based on the conventional square-section-container sloshing modes. Furthermore, the stability diagrams for all the equilateral-polygonal-section containers are investigated by both experiments and computations. The present computation is based on a discrete singularity method. The proposed modal classification is useful to predict the eigen frequencies. Specifically speaking, it is found that the equivalent diameter de1 based on the hydraulic mean depth is the most adequate as a characteristic length scale to classify all the sloshing modes. The authors show a unified formula to predict the eigen frequencies, using de1 together with the proposed modal classification.
Flows of complex fluids in a single-screw extruder were numerically simulated using a coupling method of continuum-mechanics-based computations for macroscopic flows and stochastic simulations for fluid mesoscale structures and the advantage of micro-macro simulations for complex fluids was investigated. In the present study, flows of polymer melts and dilute suspensions of disclike particles were considered and a simple model for the extrusion flow in which the flow is approximated by a shallow channel flow, was adopted. In the simulation for polymer melts, the Curtiss-Bird model was applied to represent the dynamics of polymer networks. The orientation behavior of polymers was analyzed and its dependence on the Weissenberg number was investigated. In addition, a low-cost method for plotting the distribution of orientation angle of polymers was proposed. In this method, additional stochastic simulation was performed at a sampling point in the flow field. The dependence of orientation angle on the gap wise position in an extruder was captured in more detail by this method. In the simulation of suspensions of disclike particles, a disclike particle was modeled by an oblate spheroid, and the motion of particles was computed using the stochastic differential equation for orientation vectors of oblate spheroids. The orientation behavior of particles was analyzed to find that the present approach is effective also for the suspension flows.
A numerical prediction method of cavitation erosion is proposed. In this method, the analysis of bubbles in cavitating flows is performed and the intensity of cavitation erosion is evaluated by the impact pressure induced by spherical bubble collapse. In the present study, two-dimensional cavitating flow around the Clark Y 11.7 % hydrofoil is used to examine the proposed numerical prediction method. The proposed numerical method predicts that the intensities of cavitation erosion in noncavitating, attached cavitating and pseudo-supercavitating flows are far weaker than the intensity of cavitation erosion in a transient cavitating flow, and the intensity in the vicinity of the sheet cavity termination is high. These results correspond well to experimental results, and it is confirmed that systematic erosion characteristics are generally captured by this method. Furthermore, the velocity dependence of cavitation erosion is examined, and it is found that the exponent n in the relation between the intensity I and main flow velocity Uin (I ∝ Uinn) becomes large when the bubble radius is large and ranges between 4.3 and 7.0 in the present study. According to the bubble dynamics, the ambient pressure and the rate of increases in pressure increase as the main flow velocity, and the maximum internal pressure increase. Therefore, it is thought that smaller bubbles cause cavitation erosion when the main flow velocity is large.
The characteristics of a four-roll coating system were numerically investigated and compared with experimental data to validate the theoretical models used in this study. In the theoretical models, a film splitting model using a power-law-type equation, a roll-gap model based on elastohydrodynamics, and a flow model from a rotating-cylinder system were applied. The parametric computations for each operational condition revealed the steady and dynamic behaviors of a coating film and liquid films on the coating rolls. The results of the frequency response to the speed disturbances of the coating rolls indicated that the sensitivity of the lowest coating roll to the disturbance was half that of the others; this implies that the requirement for the accuracy of a driving system of the coating roll is not as severe as compared with others. The experimental data and the numerical results at steady state agreed well. Therefore, the theoretical models used in this research were found to be appropriate.
The aerodynamic characteristics of airfoils have been researched in higher Reynolds-number ranges more than 106, in a historic context closely related with the developments of airplanes and fluid machineries in the last century. However, our knowledge is not enough at low and middle Reynolds-number ranges. So, in the present study, we investigate such basic airfoils as a NACA0015, a flat plate and the flat plates with modified fore-face and after-face geometries at Reynolds number Re < 1.0×105, using two- and three-dimensional computations together with wind-tunnel and water-tank experiments. As a result, we have revealed the effect of the Reynolds number Re upon the minimum drag coefficient CDmin. Besides, we have shown the effects of attack angle α upon various aerodynamic characteristics such as the lift coefficient CL, the drag coefficient CD and the lift-to-drag ratio CL/CD at Re = 1.0×102, discussing those effects on the basis of both near-flow-field information and surface-pressure profiles. Such results suggest the importance of sharp leading edges, which implies the possibility of an inversed NACA0015. Furthermore, concerning the flat-plate airfoil, we investigate the influences of fore-face and after-face geometries upon such effects.
This present study investigates liquid flow patterns on the inner surface of a rotary bell cup atomizer and the effect of these flow patterns on the breakup patterns at the edge of the rotary bell cup. It also estimates the atomization characteristics based on the diameter of liquid ligaments issuing from the cup edge. The cup had an outer diameter of 70 mm and the following experimental conditions were used: rotational speed N = 1,000 to 50,000 rpm and liquid flow rate Q = 50 to 300 mL/min. Classification of the breakup pattern at the cup edge revealed that, irrespective of the flow rate, several liquid ligaments were produced at low rotational speeds and fine ligaments, which cause Rayleigh breakup, were generated at high rotational speeds. Classification of the liquid flow pattern on the cup surface revealed that the liquid film on the surface was a smooth film at low rotational speeds and a radial streak film or a partial dry film at high rotational speeds. Good atomization characteristics are expected since the radial streak film and the partial dry film cause fine ligament breakup at the cup edge. It was also demonstrated that the mean droplet size for Rayleigh breakup, which generates uniform droplets, can be effectively estimated by calculations based on the diameter of the ligaments that form on the cup edge.
Fluid flows interacting with deformable wavy channel are simulated by a newly developed full-Eulerian simulation method. A single set of the governing equations for fluid and solid is employed, and a volume-of-fluid (VOF) function is used for describing the multi-component geometry. The stress field is defined by volume-averaging the stresses of the individual components through VOF. The temporal change in the solid deformation is described on the Eulerian frame by updating a left Cauchy-Green deformation tensor. The SMAC method is employed, and a second-order Adams-Bashforth and Crank-Nicolson methods are applied for time-updating the momentum equation. The spatial derivatives are discretized by the second-order central difference, except for the advections of the VOF and the left Cauchy-Green tensor (fifth-order WENO scheme). The full-Eulerian fluid-structure coupling method is applied to a pressure-driven elastic wavy channel of sinusoidal wall geometry to study the interaction between a fluid and elastic object. The elastic walls oscillate as it interacts with the fluid, and the transient phenomenon to steady state is simulated. With a neo-Hookean viscoelastic model as a constitutive law, the obtained numerical results of the elastic wall deformations (for different moduli of elasticity) show good agreements with the theoretical prediction employing the lubrication and steady Stokes approximations. Also, under some pulsating pressure conditions, a nonlinear behavior of the flow rate is studied by varying the amplitude of the pressure difference.
This study focuses on water hammer pumps that can effectively use the water hammer phenomenon and allow fluid transport without drive sources, such as electric motors. An understanding of water hammer pumps' operating conditions and an evaluation of their basic hydrodynamic characteristics are significant for determining whether they can be widely used as an energy-saving device in the future. However, these conventional studies have not described the pump performance in terms of pump head and flow rate, common measures indicating the performance of pumps, and are not useful in fully evaluating the pump characteristics. As a first stage for the understanding of water hammer pump performance in comparison to the characteristics of typical turbo pumps, this study focuses on understanding the basic hydrodynamic characteristics of water hammer pumps and experimentally examines how the hydrodynamic characteristics are affected by the inner diameter of the drive and lifting pipes, the form and capacity of the air chamber, and the angle of the drive pipe, which are believed to be representative geometric form factors.
Continuous flexible thin material, such as paper, textiles, and plastic films, are referred to as webs. At present, web machines are used in various fields. However, a number of problems regarding web handling industries remain to be solved. Generally, webs are transported through rollers. When a web is transported at high speed, the surrounding air enters the gap between the web and the roller and forms an air film. The air film prevents scratches and stains because it eliminates the contact between the web and the roller. However, the air film generates slip due to the reduced traction, which is a serious problem for web production. Therefore, control of the air film thickness has become important. In the present study, measurement of the air film thickness is conducted using a PET film as the web. An experiment is carried out to examine the effect of changing the relative velocity between the web and roller. In addition, flow visualization is conducted in the inlet and outlet areas of the web wrap region using particle image velocimetry (PIV). As a result, the effect of the air flow between the web and the roller on the air film formation and the characteristics of the flow near the inlet and outlet areas are clarified.
Direct numerical simulations (DNS) of turbulent flow through an asymmetric plane diffuser, consisting of a flat wall and an inclined wall, were performed using a high-order finite difference method, to reveal the relationship between turbulence and flow separations under negative pressure gradients. A typical feature of this flow field is the non-steady three-dimensional separation region formed near the inclined wall. At the opening of the diffuser, low speed streaks in wall turbulence cause the initial non-steady separation layer, which includes small-scale reverse flow regions. Because turbulent eddies growing from this initial separation suppress the separated flow, a small reattachment section is formed downstream. When turbulent eddies start to decay and the negative pressure gradient begins to dominate, the flow separates again, and then a large-scale separation region is formed. In summary, the intensity of turbulent eddies affects the formation of the separation and reattachment regions near the inclined wall.
The diameter of small bubbles separated from an air jet injected into a turbulent boundary layer from a nozzle on the wall is evaluated. There exist three patterns of bubble separation that are dependent on the jet velocity, i.e., single bubble, coalesced bubbles, and a continuous air jet. The mechanism of bubble separation from the jet is theoretically considered for the single-bubble region. First, the bubble volume is calculated when the force separating the bubbles, such as the drag or lift from the free stream, overcomes the surface tension acting on the wall. Then the total volume of the separating bubble is obtained by adding the volume supplied from the nozzle until the final separation. The diameter of a single bubble was measured for various experimental conditions. The results are well approximated by the results obtained from the theoretical considerations described in the present study.
A flow structure around an intermediate standing baffle in a rectangular open channel has been investigated experimentally. The instantaneous vertical velocity components were successfully measured using an Ultrasonic Velocity Profiler (UVP). Various spatial distributions such as profiles of the vertical time-averaged velocity and relative turbulent intensities at various vertical measuring lines around the baffle indicate how the flow structure changes from up- to downstream of the baffle. At the baffle's upstream they indicate the flow structure of the uprising flow. But behind the baffle indications of vortex shedding and flow separation such as the prominent peak values in the relative turbulent intensity profiles is observed. Also, spatio-temporal distributions of the vertical velocity at up- and downstream sections confirm the existence of periodic change of flow direction near the edge of the baffle at its downstream which can be attributed to the vortex shedding from the baffle edge. In addition, space-dependent power spectra indicate the existence of some peak structures near the baffle edge height at its downstream. For these sections existence of peak values in the space distribution of two frequency modes could be confirmed corresponding to the vortex shedding due to the existence of the baffle. Furthermore, by using multi-line method and multiplexer the peak of the absolute value of the normalized two-point cross-correlation coefficients between vertical velocity fluctuations could be obtained to evaluate the effect of the baffle on the degree of correlation between vertical velocity fluctuations at upstream points with that of downstream ones. It has been found quantitatively that a baffle acts as a barrier wall and causes the degree of correlations to be decreased significantly from the vicinity of its edge height to the channel bed. Also existence of a local peak region in between the baffle edge height and free surface was found in color maps of degree of correlation. It was found that the degree of correlation decreases gradually from the peak region to the proximity of the baffle edge height. Also the decrease in the degree of correlation from that peak to the free surface was captured which can be attributed to the effect of free surface. Thus, flow around the baffle can be characterized into two regions with very different characteristics.
In this study, effects of the tip cavity with various depths, widths, and locations on the leakage flow and performance of an axial turbine cascade have been investigated numerically. The blade was a linear model of the tip section of the GE-E3 high-pressure turbine first-stage rotor blade. The Delayed Detached Eddy Simulation (DDES) model was used in the simulations. The computational results showed that the leakage mass flow rate and mass-averaged total pressure loss decreased as the depth and width of the tip cavity increased. And, it was shown that the cavity near the pressure side is more effective than that near the suction side. These effects depend on a vortex generated behind the pressure side in the cavity, which is changed with the depth, width, and location. The vortex entrains the leakage flow through the clearance toward the bottom of the cavity and in the chord wise direction, which reduces the flow leaking out.
Direct numerical simulation (DNS) of a feedback-controlled turbulent channel flow at Reτ = 640 is carried out. As an idealized feedback control, we selectively damp either the small scale wall-normal velocity fluctuations (defined as those with the spanwise wavelength smaller than 300 wall units) or the large scale fluctuations (the spanwise wavelength larger than 300 wall units). The present DNS reveals that the control of small scale fluctuations leads to more drag reduction than that of large scale fluctuations. When the small scale fluctuation is damped, the friction drag is reduced by the amount corresponding to the absence of small scale fluctuation. In contrast, when the large scale fluctuation is damped, the friction drag reduction is much less than that expected from the absence of large scale fluctuation. In the latter case, the contribution from the small scale fluctuation to the friction drag is found to be drastically increased due to the reduction of pressure fluctuation and destruction of Reynolds shear stress.
A lattice Boltzmann method (LBM) is proposed for the simulation of natural convection in anisotropic porous media with the generalized non-Darcy model. The Brinkman-Forchheimer momentum equation is recovered from a kinetic equation for the distribution function, which has a forcing term to introduce anisotropy of permeability. The LBM can analytically calculate fluid velocity using the inverse matrix for the Brinkman equation. The simulation results for a Poiseuille flow of anisotropic porous media demonstrate that the proposed LBM for the Brinkman equation has second-order accuracy in space for any permeability ratio. Since the forcing term of the generalized equation contains a nonlinear (Forchheimer) term containing the quadratic form of the velocity and the permeability tensor, the LBM must use a free-derivative optimization method. In the numerical simulation of natural convection, the streamlines and isotherms obtained by the LBM agree well with those of the finite difference method (FDM), as well as with those of previous investigations of various fundamental parameters, such as the Rayleigh number, the inclination of the principal permeability direction, and the permeability ratio. The numerical results of the LBM show good agreement with the reference solutions for the value of the stream function at the center of the primary vortex and the average Nusselt number.
Earlier wind tunnel experiments supported by visualization in water flow carried out by the present authors showed that two types of longitudinal vortices shed periodically from a cruciform two identical circular cylinder system in uniform flow. These longitudinal vortices induce resonance-like oscillations over a wide velocity range accompanied by the synchronization phenomenon. The specific aim of this work is to investigate the universality of the longitudinal vortex shedding by carrying out experiments in both water and air flows. In addition to the velocity, the alternating lift force loading on the upstream cylinder in water flow is measured by using load-cells. The two longitudinal vortices are confirmed to shed periodically for small size systems in water flow in spite of large difference in the experimental conditions. Relationships of the Strouhal number and the lift force coefficient for the vortex shedding against Reynolds number agree well with the previous data except for the peculiar vanishing of the necklace vortex when Re is larger than 22000. The cause of this phenomenon is left for further investigation.
This paper deals with the measurement and numerical simulation of the characteristics of ion drag pumps. The pressure-flow rate characteristics and the flow fields of ion drag pumps with different configurations are measured and are numerically simulated. Measured and simulated results are compared with each other. The value of the ionic mobility and the relation between the electric field strength and the injected charge density are determined so that the numerical simulation results agree well with measured results for a certain standard configuration of ion drag pump. It is shown that the pump characteristics can be simulated with a relatively good accuracy by the numerical simulation and that an appropriate selection of the value of the ionic mobility is essential to the success in numerical simulation. It is also shown that the accuracy of the numerical simulations for the pumps with configurations different from the standard one is lowered when the relation between the electric field strength and the injected charge density for the standard pump is used.
Measurement-integrated (MI) simulation is a numerical flow analysis method with a feedback mechanism from measurement of a real flow. It correctly reproduces a real flow under inherent ambiguity in a mathematical model or a computational condition. In this paper we theoretically investigated the destabilization phenomenon of MI simulation, in which analysis error suddenly increases at some critical feedback gain. This phenomenon has been considered as instability of a closed-loop feedback system, but present study treated it as that of a numerical scheme. First, the mechanism of the destabilization phenomenon was investigated based on the sufficient condition of the convergence of iterative calculation of existing MI simulation. It was found that the feedback signal in the source term destabilized the iterative calculation. Then, a new MI simulation scheme was derived by evaluating the feedback signal in the linear term to remove the cause of the destabilization. The validity of the present theoretical analysis was verified for examples treated in former studies of MI simulations: blood flow in an aneurismal aorta with ultrasonic measurement, blood flow in a cerebral aneurism with magnetic resonance measurement, Karman vortex street behind a square cylinder with PIV measurement, and fully developed turbulent flow in a square duct with ideal measurement. Occurrences of destabilization phenomenon in all the examples were well explained by the condition of this study, especially for cases of relatively small time steps and large feedback gains. Furthermore, the new MI simulation scheme realized the analysis without the destabilization phenomenon. The present theoretical result confirming that the destabilization phenomenon is not the instability of the feedback system but that of a numerical scheme is generally applicable to MI simulations using the velocity error for the feedback signal.