Bathtub Vortex Research Articles

Analytical models for critical heights of vortexing in flat and spherical bottom tanks with siphon and bottom drains

Critical height or critical submergence is liquid level at which air-core vortex extends from the free surface into drain hole when a liquid is drained from a container/tank. Extensive analytical and experimental studies have been reported on critical height of bath tub vortex, for liquid draining downward from flat bottom propellant tanks. Rockets making use of liquid propellants mostly employ spherical bottom propellant tanks as well as siphon or upward drain flow. Keeping in view of such practical applications, analytical models are developed for critical height, considering the effects of siphon drain and the shape of tank bottom. Additional design parameters influencing the behavior for each case are identified. Appropriate governing equations and a solution methodology are developed pertinent to the system considered, to predict the critical height for siphon drain and spherical bottom tank independently as well as for both combined. The results indicate that the critical height for spherical bottom tank is higher than for flat bottom tank, due to higher local flow velocity. Siphon geometry can be designed for critical height much less than normal drain from flat bottom tank. These observations are in accordance with the results published in the literature. This paper reports the analytical models and solution methodology to predict the critical height for vortexing for normal draining from spherical tank bottom, siphon draining from both flat and spherical bottom tanks.

Bathtub vortex effect on Torricelli's law

When emptying a water filled container through a bottom hole, a ``bathtub vortex'' may appear and deform the free surface. Torricelli's law, which predicts the flow rate as a function of the water height, has been known for 400 years, but deviations are observed in the presence of such a vortex. This study focuses on the impact of the bathtub vortex on the emptying velocity. Through an experiment on the unsteady draining flow in a rotating tank, we show that Torricelli's law is modified as a function of the surface deformation, and that the draining time is mainly determined by a nondimensional parameter corresponding to the ratio of the outlet size to the Ekman boundary layer thickness.

Analysis of a stable bathtub vortex in a rotating container

Rotating flows with free-surface vortices can be found in many engineering applications, such as pump and turbine intakes, vessels, and nuclear reactors. The need to address rather different flow regions existing in such flows, such as Ekman and Stewartson layers and the line vortex zone, in a coupled manner, makes modeling of free-surface rotating flows very challenging. In this work, the flow field of a free-surface vortex, created in a rotating cylinder with a drain hole in its bottom, is investigated numerically and analytically. Above the drain hole of the cylinder, a free-surface vortex, accompanied by axial velocity, is created. This axial velocity profile is governed by the Ekman boundary layer far from the axis and by the drainage in its proximity. The experiments of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)] on the so-called bathtub vortex are numerically modeled with the finite volume method. The simulations are validated with the available measurements from the experiments. Using the simulation results, self-similar and non-self-similar models, describing the velocity fields in the Ekman boundary layer, are compared and tested. It is shown that the self-similar model is more accurate than the non-self-similar model. It is also demonstrated that the analytical model of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)], when modified as suggested in the present study, is capable of predicting the free-surface profile for low rotation rates. However, for high rotation rates, only the numerical simulation can predict the relation between the flow field within the liquid and the free-surface profile.

A numerical investigation on a capsule-intake of the electrical submersible pump in skid

The Electrical Submersible Pumping (ESP) system is one of the most commonly used artificial lift methods in the petroleum industry and one recent breakthrough in this system is enclosed in a capsule and positioned on the seabed in a skid. As it is a recent technology, with only a small amount of equipment currently in operation, there is a clear need for a greater understanding of the flow within this geometry with the objective of perfecting the design of this promising submarine boosting system. This paper presents a numerical investigation of single-phase flow within the scaled capsule-intake of an ESP system in the Skid considering a model with geometric and dynamic similarities in relation to a real equipment in operation in the Espadarte field, located in the Campos Basin, Brazil. The tridimensional and transient simulation for a case for one mass flow rate and inclination angles are investigated. While neglecting the effects of the diffuser and impeller in the system, the flow field features, axial and radial velocity profiles in the intake region were computed. The numerical results show that the flow in the intake region is dominated by the secondary flow, leading to the formation of bathtub vortex. It is expected that the vortices influence the flow pattern in the intake region, breaking the larger bubbles into smaller bubbles, making the transition from the slug flow pattern to the dispersed bubbles or bubbly pattern in which it would be more difficult to be dragged into the intake.

Superradiance in dispersive black hole analogues

Wave equations containing spatial derivatives which are higher than second order arise naturally in the context of condensed matter systems. The solutions of such equations contain more than two modes and consequently, the range of possible interactions between the different modes is significantly enhanced compared to the two mode case. We develop a framework for analysing the different mode interactions based on the classical turning points of the dispersion relation. We then apply this framework to the scattering of deep water gravity waves with a draining bathtub vortex, a system which constitutes the analogue of a rotating black hole in the non-dispersive limit. In particular, we show that the different scattering outcomes are controlled by the light-ring frequencies, a concept routinely applied in black hole physics, and two new frequencies which are related to the strength of dispersion. We find that the frequency range in which the reflected wave is superradiantly amplified appears as a simple modification to the non-dispersive case. However, the condition to observe this amplification is complicated by the fact that a superradiant mode can be reflected back into the system by scattering with one of the additional modes. We provide estimates for the reflection coefficients in the full dispersive regime.

Higher-derivative analogue Aharonov–Bohm effect, absorption and superresonance

In this paper we consider the acoustic metric obtained from an Abelian Higgs model extended with higher derivative terms in the bosonic sector which describes an acoustic rotating black hole and analyze the phenomena of superresonance, analogue Aharonov–Bohm effect and absorption. We investigate these effects by computing the scalar wave scattering by a draining bathtub vortex and discuss the physical interpretation of higher derivative terms. Furthermore, we show that the absorption does not vanish as the draining parameter tends to zero.

Polariton fluids for analogue gravity physics.

Analogue gravity enables the study of fields on curved space-times in the laboratory. There are numerous experimental platforms in which amplification at the event horizon or the ergoregion has been observed. Here, we demonstrate how optically generating a defect in a polariton microcavity enables the creation of one- and two-dimensional, transsonic fluid flows. We show that this highly tuneable method permits the creation of horizons. Furthermore, we present a rotating geometry akin to the water-wave bathtub vortex. These experiments usher in the possibility of observing stimulated as well as spontaneous amplification by the Hawking, Penrose and Zeld'ovich effects in fluids of light. This article is part of a discussion meeting issue 'The next generation of analogue gravity experiments'.

Modeling of Gas Entrainment Flow Rate by Bathtub Vortex

In the design study on sodium-cooled fast reactors, .it is important to investigate the gas entrainment (GE) phenomena in detail. In fact, a lot of analytical, experimental and numerical studies have been conducted to clarify the onset condition of GE and some GE onset models have been proposed. However, few studies on the modeling of entrained gas flow rate has been conducted due to the difficulty on modeling the gas bubble entrainment at a free surface, which is accompanied by complicated free surface deformation. In this paper, the authors propose a mechanistic model to predict the entrained gas flow rate by a free surface vortex. The model contains the theoretical equation of transient gas core elongation and the empirical equation of critical gas core length for gas bubble detachment. The mechanistic model is applied to predict the entrained gas flow rate in a simple GE experiment. As a result, the predicted results show qualitatively good agreement with the experimental results of the entrained gas flow rate. Therefore, it is confirmed that the proposed mechanistic model can predict the entrained gas flow rate by a free surface vortex.

Left Ventricular Hemodynamics with an Implanted Assist Device: An In Vitro Fluid Dynamics Study.

Left ventricle assist devices (VADs) aid the heart pumping blood into the systemic circulation and grant the required cardiac output (CO) when the heart itself cannot provide it. However, it is unclear how effective these devices are at restoring not only physiological CO values but also normal intraventricular hemodynamics. In this work, the modified hemodynamics due to a VAD implantation is studied in vitro using an elastic ventricle made of silicone, which is incorporated into a pulse-duplicator setup prescribing a realistic pulsatile flow. Thereafter, a continuous axial pump is connected at the ventricle apex to mimic a VAD and its effect on the ventricular hemodynamics is investigated as a function of the pump flow suction. Using particle image velocimetry (PIV), we observe that the continuous pump flow effectively provides unloading on the ventricle and yields an increased CO. Conversely, the continuous blood suction from the ventricle apex deeply alters the hemodynamics and, in addition, the VAD obstruction in the ventricle behaves as a bluff body that affects the vorticity distribution in the LV thus creating a stagnant region at the ventricle apex. This phenomenon is rationalized by measuring in a modified set-up the benefits on the hemodynamics of a flush-mounted device. Additionally, the suction operated by the VAD reduces the ventricular pressure and yields an increase in the swirling motion around the ventricle axis, in a similar fashion as the bath-tub vortex effect, thus further modifying the intraventricular hemodynamics with respect to healthy conditions.

Scattering of surface shallow water waves on a draining bathtub vortex

In the linear approximation we study long wave scattering on an axially symmetric flow in a shallow water basin with a drain in the center. This classical problem can be considered as a model of wave scattering on a rotating black hole. For the low frequencies analytic solutions are derived to describe both pure potential perturbations (surface gravity waves), and perturbations with nonzero potential vorticity. For moderate frequencies such solutions are obtained numerically and illustrated graphically. It is shown that there are two processes governing the dynamics of perturbations, namely, the scattering of incident gravity waves by central vortex, and the emission of gravity waves stimulated by the potential vorticity. Some aspects of their joint action are discussed.

Dependence of instability to induce a bathtub vortex in a rectangular vessel on the aspect ratio of the horizontal cross section

Dependence of the linear instability to induce a bathtub vortex in a rectangular vessel on the horizontal aspect ratio of the vessel is investigated in the present paper. Water comes in through inlet sections on the top of two facing side walls, and is drained out through a pipe attached to the drain hole at the center on the bottom of the vessel. Water surface is assumed to be always flat without any dip even after instability yielded a large-scale vortex above the drain hole. The flow is steady and symmetric with respect to both vertical center planes along the streamwise direction and perpendicular to it at small Reynolds numbers, but becomes unstable to infinitesimal disturbances inevitably included in the flow at Reynolds numbers larger than a critical value. The steady symmetric flow is numerically obtained, and its stability is analyzed by numerically solving the linear disturbance equation. The instability is confirmed to lead to induction of a so-called bathtub vortex. It is found that there exists a threshold aspect ratio of the rectangular cross section for the instability, and the threshold value is determined.

Black Hole Quasibound States from a Draining Bathtub Vortex Flow.

Quasinormal modes are a set of damped resonances that describe how an excited open system is driven back to equilibrium. In gravitational physics these modes characterize the ringdown of a perturbed black hole, e.g., following a binary black hole merger. A careful analysis of the ringdown spectrum reveals the properties of the black hole, such as its angular momentum and mass. In more complex gravitational systems, the spectrum might depend on more parameters and hence allows us to search for new physics. We present a hydrodynamic analog of a rotating black hole that illustrates how the presence of extra structure affects the quasinormal mode spectrum. The analogy is obtained by considering wave scattering on a draining bathtub vortex flow. We show that due to vorticity of the background flow, the resulting field theory corresponds to a scalar field on an effective curved spacetime which acquires a local mass in the vortex core. The obtained quasinormal mode spectrum exhibits long-lived trapped modes, commonly known as quasibound states. Our findings can be tested in future experiments building upon recent successful implementations of analog rotating black holes.

Observation of the spiral flow and vortex induced by a suction pump in superfluid 4He

A suction flow generates a whirlpool, namely a bathtub vortex, in a classical fluid; in contrast, rotating containers, which are usually used for studies of superfluid helium, can produce only simple solid rotation. In the present work, the superfluid flow and concentrated quantized vortices induced by a cryogenic motor immersed in superfluid 4He were investigated. Using a motor with six blades in a cylinder caused the free surface of the superfluid 4He to take on a parabolic shape, indicating that the motor produces a rotating superfluid flow. To drive a suction flow in superfluid helium, the motor was mounted in a cylindrical container with a small hole at the center of the top and a slit at the side, acting as a superfluid pump. This pump was successfully used to generate a spiral flow and a vortex with a funnel-shaped core in superfluid 4He, suggesting that the resulting suction flow transports and centralizes quantized vortices to the suction hole, increasing the vortex circulation and sucking the free surface of the superfluid down.

Sequential transitions of bathtub vortex flow

A bathtub vortex has been found to be induced autonomously by instability of the steady symmetric flow if the bathtub has a rectangular cross section. A new study finds that the vortex axis deviates from the center of the drain hole by further instability and even further instability causes temporal oscillation of the vortex.

Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy

The scale, inflexion point and maximum point are important scaling parameters for studying growth phenomena with a size following the lognormal function. The width of the size function and its entropy depend on the scale parameter (or the standard deviation) and measure the relative importance of production and dissipation involved in the growth process. The Shannon entropy increases monotonically with the scale parameter, but the slope has a minimum at p 6/6. This value has been used previously to study spreading of spray and epidemical cases. In this paper, this approach of minimizing this entropy slope is discussed in a broader sense and applied to obtain the relationship between the inflexion point and maximum point. It is shown that this relationship is determined by the base of natural logarithm e ' 2.718 and exhibits some geometrical similarity to the minimal surface energy principle. The known data from a number of problems, including the swirling rate of the bathtub vortex, more data of droplet splashing, population growth, distribution of strokes in Chinese language characters and velocity profile of a turbulent jet, are used to assess to what extent the approach of minimizing the entropy slope can be regarded as useful.

Influence of fish swimming on the flow pattern of circular tanks

The effect of swimming fish on the average velocity and velocity profile of a circular tank was studied. Working with different inlet diameters and flow rates, nine different impulse forces (configurations) were evaluated. Each configuration was tested with and without fish, and the effects of two different fish sizes were compared.The velocity profiles in experiments with fish presented a considerable reduction in velocity in the centre of the tank near the outlet, which was a consequence of the increase in the kinematic eddy viscosity due to the turbulence introduced by fish swimming. A flattening of the angular velocity profile was observed in the central area of the tank, which had a radius of about 0.3m (18% of the total volume of the tank).A previous model proposed by Oca and Masaló (2013) was modified in order to better describe the distribution of velocities in the central volume of a tank with swimming fish. The proposed modification was based on Burgers⿿ proposal for a bathtub vortex, which implies the determination of the parameter (1⿿e⿿αr2), where r is the radius and the α values were experimentally obtained for each tank configuration, in which they increased with the impulse force.The average velocities in the tank were proportional to the square root of the impulse force in experiments with and without fish. Experiments with fish presented lower average velocities, which imply higher tank resistance coefficients. At similar stocking densities (14.6kg/m3), the increase in the tank resistance coefficients obtained with small fish sizes (154g) were slightly higher than those obtained with bigger fish sizes (330g).

Free sink vortex Ekman suction-extraction evolution mechanism

The suction-extraction phenomenon occurs in the formation process of free sink vortex (bathtub vortex), and it is a complex gas-liquid coupling matter. The Ekman layer coupling and its evolution mechanism involved in the above matter possess important scientific value and practical engineering significance. To address the above issue, a modeling and analytical method for the Ekman suction-extraction evolution mechanism of free sink vortex is proposed.#br#Based on the multiphase volume of fluid model and turbulent kinetic energy-dissipation (k-ε) model, a gas-liquid two-phase fluid dynamic model for free sink vortex Ekman suction-extraction is set up. Considering the rotating and shearing characteristics of sink vortex, a two-phase surface is reconstructed by piecewise linear interface construction method. Based on the above models, the internal relations between initial rotation velocity component, drainage capacity and Ekman suction-extraction are investigated, and the corresponding flow field profile regularities are revealed.#br#According to the results of a series of numerical instances, some regularities are obtained as follows. 1) If the initial velocity disturbance is variable, the distances of the suction and extraction holes from the container bottom both keep constant. In the suction stage, the suction hole is located at a fixed position above the container bottom surface, and the extraction hole is in the plane of the bottom surface. 2) If the initial disturbance is enhanced, the rotation velocity of the suction stage increases, and the suction and extraction heights and Ekman layer thickness become larger, while Ekman suction-extraction intensities of suction, extraction and penetration stages turn weaker. 3) If the initial disturbance is invariable, the heights of Ekman suction and extraction remain unchanged, and are independent of drainage capacity. 4) The small-scale vortexes separated from the large-scale ones in the bottom corner of container take on a phenomenon of flow around by a right-angle, which is caused by the viscosity of Ekman boundary layer and the potential flow of the sink hole. 5) Considering the stream line profiles of suction and extraction stages, the dispersion of stream lines keeps constant with the time going by, and the stream lines near the central region of vortex tend to be converged by increasing the gas-liquid coupling.#br#In general, the results can offer useful reference to the research work of free sink vortex formation mechanism, and provide technical supports for vortex suppression control of the areas of metallurgy pouring, chemistry separation and hydraulic drainage. The subsequent researches of the fractal based sink vortex evolution mechanism and lattice Boltzmann based phase surface tracing will be carried out.

Investigation on bathtub vortex flow field by Particle Image Velocimetry

An experimental investigation on bathtub vortices has been performed by using Particle Image Velocimetry (PIV). Velocity fields associated to the free surface vortex were obtained at three horizontal planes and four Reynolds numbers, i.e. between 2400 and 11,000 (calculated with reference to the exit hole diameter and the mean exit velocity). Due to the unsteady behavior of the flow field, the vortex center positions have been identified and the vortex paths were reconstructed for all experiments. Average velocity fields have been calculated by aligning the vortex centers at each frame, in order to derive radial and tangential velocity profiles, to be compared at different Reynolds numbers and measurement planes. The results show that the radial motion assumes a potential behavior when it is near the exit hole, scaling quite well with the average exit velocity (thus with the corresponding Reynolds number). On the other hand, the tangential component is well approximated by the Rankine’s flow potential solution only near the free surface, the tangential velocity peak increment not being linearly proportional to the outlet velocity. Vorticity fields and circulation profiles have been derived from the measured velocity fields and discussed. Turbulence fluctuations statistical analysis gives also evidence of a clear dependence on Reynolds number and distance from the exit hole.

Control of the intensity of a bathtub vortex by acting on the upstream flow

In a motor vehicle, the cowl box is a volume located at the bottom of the windshield. It collects rainwater and drains it to provide clean dry air to the passenger compartment through the ventilation system. When rainwater accumulates in the box, a bathtub vortex forms above the drain pipe. This vortex sucks up air into the pipe and creates an air core that decreases the water drain rate in the pipe. This increases the water level in the box and can cause a water overflow into the ventilation system. The behavior of this bathtub vortex was experimentally studied using a simplified geometry representative of a real cowl box. The inlet water flow rate was controlled and a capacitive probe measured the water level in the box. The flow was studied using Particle Image Velocimetry to measure the velocity field around the vortex. The flow pattern is described using these data. Due to geometry and inlet conditions, the upstream flow forces the vortex counter-clockwise. It also shifts the vortex axis away from the drain axis. In this configuration, the upstream flow is strongly asymmetric and feeds the vortex using less than half the width of the box. Based on these observations, a device was tested in order to reduce the vortex intensity and consequently the water level. Resulting velocity fields showed a better distribution of the upstream flow. The vortex intensity decreased by up to 55% and the water level by up to 53%. While there was still a vortex with an air core, the water level was therefore significantly reduced. These results are particularly interesting for cowl box design: as this device can guarantee a lower water level, the cowl box depth can be reduced and space can be saved.

Bathtub vortex induced by instability.

The driving mechanism and the swirl direction of the bathtub vortex are investigated by the linear stability analysis of the no-vortex flow as well as numerical simulations. We find that only systems having plane symmetries with respect to vertical planes deserve research for the swirl direction. The bathtub vortex appearing in a vessel with a rectangular cross section having a drain hole at the center of the bottom is proved to be induced by instability when the flow rate exceeds a threshold. The Coriolis force is capable of determining the swirl direction to be cyclonic.