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We give a hydrodynamical explanation for the chaotic behaviour of a dripping faucet using the results of the stability analysis of a static pendant drop and a proper orthogonal decomposition (POD) of the complete dynamics. We find that the only relevant modes are the two classical normal forms associated with a saddle–node–Andronov bifurcation and a Shilnikov homoclinic bifurcation. This allows us to construct a hierarchy of reduced-order models including maps and ordinary differential equations which are able to qualitatively explain prior experiments and numerical simulations of the governing partial differential equations and provide an explanation for the complexity in dripping. We also provide a new mechanical analogue for the dripping faucet and a simple rationale for the transition from dripping to jetting modes in the flow from a faucet.
High-resolution large-eddy simulation is used to investigate the mean and turbulence properties of a separated flow in a channel constricted by periodically distributed hill-shaped protrusions on one wall that obstruct the channel by 33% of its height and are arranged 9 hill heights apart. The geometry is a modification of an experimental configuration, the adaptation providing an extended region of post-reattachment recovery and allowing high-quality simulations to be performed at acceptable computing costs. The Reynolds number, based on the hill height and the bulk velocity above the crest is 10595. The simulated domain is streamwise as well as spanwise periodic, extending from one hill crest to the next in the streamwise direction and over 4.5 hill heights in the spanwise direction. This arrangement minimizes uncertainties associated with boundary conditions and makes the flow an especially attractive generic test case for validating turbulence closures for statistically two-dimensional separation. The emphasis of the study is on elucidating the turbulence mechanisms associated with separation, recirculation reattachment, acceleration and wall proximity. Hence, careful attention has been paid to resolution, and a body-fitted, low-aspect-ratio, nearly orthogonal numerical grid of close to 5 million nodes has been used. Unusually, the results of two entirely independent simulations with different codes for identical flow and numerical conditions are compared and shown to agree closely. Results are included for mean velocity, Reynolds stresses, anisotropy measures, spectra and budgets for the Reynolds stresses. Moreover, an analysis of structural characteristics is undertaken on the basis of instantaneous realizations, and links to features observed in the statistical results are identified and interpreted. Among a number of interesting features, a distinct ‘splatting’ of eddies on the windward hill side following reattachment is observed, which generates strong spanwise fluctuations that are reflected, statistically, by the spanwise normal stress near the wall exceeding that of the streamwise stress by a substantial margin, despite the absence of spanwise strain.
This paper examines the stability of swirling flows in a non-homogeneous fluid. Density gradients are shown to produce two distinct kinds of instability. The first is the centrifugal instability (CTI) which mainly affects axisymmetric, short-axial-wavelength eigenmodes. The second is the Rayleigh–Taylor instability (RTI) which mainly affects non-axisymmetric, two-dimensional eigenmodes. These instabilities are described for a family of model flows for which the velocity law $V(r)$ corresponds to a Gaussian vortex with radius 1, and the density law $R(r)$ corresponds to a Gaussian distribution characterized by a density contrast $C$ and a characteristic radius $b$. A full map in the ($C, b$)-plane is given for the amplification rate and the structure of the most amplified eigenmode. For small density contrasts ($C\,{<}\,0.5$), the CTI occurs only for $b \,{>}\, 1$ and the RTI for $b \,{\lesssim}\, 0.8$. On the other hand, for high density contrasts ($C \,{>}\, 0.5$), a competition between the two kinds of instabilities is observed. From a fundamental point of view, the nature of the instability depends on the local values of $ G^2\,{=}\,{-}r^{-1}V^2R^{-1}{\rm d}R/{\rm d}r$ and the Rayleigh discriminant $ \Phi\,{=}\,r^{-3}{\rm d}(r^2V^2)/{\rm d}r $. CTI occurs whenever $ G^2\,{>}\,\Phi $ somewhere in the flow. For RTI, a necessary condition is that $G^2\,{>}\,0$ somewhere in the flow. By an asymptotic analysis, we show that this condition is also sufficient in the limit $b\,{\rightarrow}\, 0$, $C\,{\rightarrow}\, 0$. This asymptotic analysis also confirms that shear has a stabilizing effect on RTI and that this instability is strictly analogous to the standard RTI obtained in the case where light fluid is situated below heavier fluid in the presence of gravity.
A global stability analysis of Kelvin's vortex ring to three-dimensional disturbances of infinitesimal amplitude is made. The basic state is a steady asymptotic solution of the Euler equations, in powers of the ratio $\epsilon$ of the core radius to the ring radius, for an axisymmetric vortex ring with vorticity proportional to the distance from the symmetric axis. The effect of ring curvature appears at first order, in the form of a dipole field, and a local straining field, which is a quadrupole field, follows at second order. The eigenvalue problem of the Euler equations, retaining the terms to first order, is solved in closed form, in terms of the Bessel and the modified Bessel functions. We show that the dipole field causes a parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by 1. The most unstable mode occurs in the short-wavelength limit, under the constraint that the radial and the azimuthal wavenumbers are of the same magnitude, and the limiting value of maximum growth rate coincides with the value 165/256$\epsilon$ obtained by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, p. 3151) by means of the geometric optics method. The instability mechanism is traced to stretching of disturbance vorticity in the toroidal direction. In the absence of viscosity, the dipole effect outweighs the straining field effect of $O(\epsilon^2)$ known as the Moore–Saffman–Tsai–Widnall instability. The viscosity acts to damp the former preferentially and these effects compete with each other.
In this study spanwise correlation measurements and smoke flow visualization were performed on vortex shedding behind a normal plate. For Reynolds numbers in a range between 1800 and 27000, the hot-wire signals measured were analysed by a wavelet transformation, from which the instantaneous properties of vortex shedding were obtained and examined. Results show that the phase difference of vortex shedding detected at two spanwise locations, separated by twice the characteristic length, can be as high as 35$^\circ$. A correlation analysis further shows that large spanwise phase differences occur when small fluctuating amplitudes in the vortex shedding signals are measured. Smoke-wire visualization performed at Reynolds number 1800 indicates that the formation of shedding vortex can be divided into two distinct situations, namely, one featuring a long formation region, called Mode L; and the other featuring a short formation region, called Mode S. In Mode S, the three-dimensionality of vortex formation appears to be very pronounced, and the secondary vortices are clearly present in the separated shear layer. The events of Mode S occupy less than 5% of the total time measured, and are called the burst events in this study.
The two-dimensional boundary-layer flow over a cooled/heated flat plate is investigated. A cooled plate (with a free-stream flow and wall temperature distribution which admit similarity solutions) is shown to support non-modal disturbances, which grow algebraically with distance downstream from the leading edge of the plate. In a number of flow regimes, these modes have diminishingly small wavelength, which may be studied in detail using asymptotic analysis.
Corresponding non-self-similar solutions are also investigated. It is found that there are important regimes in which if the temperature of the plate varies (in such a way as to break self-similarity), then standard numerical schemes exhibit a breakdown at a finite distance downstream. This breakdown is analysed, and shown to be related to very short-scale disturbance modes, which manifest themselves in the spontaneous formation of an essential singularity at a finite downstream location. We show how these difficulties can be overcome by treating the problem in a quasi-elliptic manner, in particular by prescribing suitable downstream (in addition to upstream) boundary conditions.
This paper describes experiments on small solid particle settling behaviour in stationary homogeneous isotropic air turbulence. We present here a new methodology using a recently developed cruciform apparatus: a large horizontal cylindrical vessel equipped with a pair of counter-rotating fans and perforated plates at each end is used to generate stationary near-isotropic turbulence in the core region between the two perforated plates and a long vertical vessel is used to supply heavy descending particles from its top. This novel experimental design, without the unwanted influences from the injection of particles, the mean flow, and the decay of turbulence, allows direct imaging and velocity measurements of the two-way interaction between heavy particles and homogeneous isotropic turbulence. Consequently, the spatiotemporal responses of both fluid turbulence and particle settling can be determined by high-speed digital particle image velocimetry and accelerometry, together with the wavelet transform analysis for the first time. Hence, experimental information on and thereby understanding of the particle settling rate, preferential accumulation, and turbulence modification due to the presence of the particles is obtained.
We found that the particle settling velocity (${V}_{s})$ is much greater than the terminal velocity (${V}_{t})$ in still fluid for which the value of (${V}_{s}\,{-}\, {V}_{t})$ reaches a maximum of 0.13$u^\prime $ when the Stokes number $\hbox{\it St}\,{ =}\,\tau_{p}/\tau_{k}\,{\approx}\,$1 and ${V}_{t}/u^\prime \,{\approx}\,$0.5 at $\hbox{\it Re}_{\lambda }\,{=}\,$120 and $\hbox{\it Re}_{p} \,{<}\,$1, in good agreement with previous numerical results, where $\tau _{p}$ is the particle's relaxation time, $\tau _{k}$ is the Kolmogorov time scale, $u^\prime$ is the energy-weighted r.m.s. turbulent intensity, and $\hbox{\it Re}_{\lambda}$ and $\hbox{\it Re}_{p}$ are the Reynolds numbers based on the Taylor microscale ($\lambda$) and the mean diameter of particles, respectively. Non-uniform particle concentration fields are observed and most significant when $\hbox{\it St}\,{\approx}\,$1.0, at which the particle clusters accumulate preferentially around the outer perimeter of small intense banana-shaped vortical structures. These clusters can turn and stretch banana-shaped vortical structures toward the gravitational direction and thus significantly increase the mean settling rate especially when $\hbox{\it St}\,{ =}\,1$. From spatiotemporal analysis of the flatness factor, it is found that the characteristic length and time scales of these preferential particle clusters are related to the spacing between the adjacent intense vorticity structures of the order $\lambda$ and the time passage of these clustering structures of the order $\tau _{k}$, respectively. By comparing the average frequency spectra between laden (heavy particle) and unladen (neutral particle) turbulent flows over the measurement field at a fixed $\hbox{\it Re}_{\lambda }\,{=}\,$120, turbulence augmentation is found for most frequencies in the gravitational direction, especially for $\hbox{\it St}\,{\ge}\,$1. In the transverse direction, augmentation occurs only at higher frequencies beyond the Taylor microscale for all values of $\hbox{\it St}$ studied varying from 0.36 to 1.9. The increase in the size of energy spectra (turbulence augmentation) due to the presence of heavy particles is greatest at $\tau_{k}^{ - 1 }$ when $\hbox{\it St}\,{\approx}\,$1.0. Furthermore, the slip velocities between fluid turbulence and heavy particles can stimulate the laden turbulent flow to become more intermittent in the dissipation range. Finally, a simple energy balance model for turbulence modification is given to explain these results and areas for further study identified.
Using small-amplitude expansions, we discuss nonlinear effects in the reflection from a sloping wall of a time-harmonic (frequency $\omega$) plane-wave beam of finite cross-section in a uniformly stratified Boussinesq fluid with constant buoyancy frequency $N_{0}$. The linear solution features the incident and a reflected beam, also of frequency $\omega$, that is found on the same (opposite) side to the vertical as the incident beam if the angle of incidence relative to the horizontal is less (greater) than the wall inclination. As each of these beams is an exact nonlinear solution, nonlinear interactions are confined solely in the vicinity of the wall where the two beams meet. At higher orders, this interaction region acts as a source of a mean and higher-harmonic disturbances with frequencies $n\omega$ ($n\,{=}\,2,3,\ldots$); for $n\omega\,{<}\,N_{0}$ the latter radiate in the far field, forming additional reflected beams along $\sin^{-1}(n\omega/N_{0})$ to the horizontal. Depending on the flow geometry, higher-harmonic beams can be found on the opposite side of the vertical from the primary reflected beam. Using the same approach, we also discuss collisions of two beams propagating in different directions. Nonlinear interactions in the vicinity of the collision region induce secondary beams with frequencies equal to the sum and difference of those of the colliding beams. The predictions of the steady-state theory are illustrated by specific examples and compared against unsteady numerical simulations.
The geometrically different planforms of near-wall plume structure in turbulent natural convection, visualized by driving the convection using concentration differences across a membrane, are shown to have a common multifractal spectrum of singularities for Rayleigh numbers in the range $10^{10}$–$10^{11}$ at Schmidt number of 602. The scaling is seen for a length scale range of $2^5$ and is independent of the Rayleigh number, the flux, the strength and nature of the large-scale flow, and the aspect ratio. Similar scaling is observed for the plume structures obtained in the presence of a weak flow across the membrane. This common non-trivial spatial scaling is proposed to be due to the same underlying generating process for the near-wall plume structures.
The variation in the drag coefficient for low-Reynolds-number flow past rings orientated normal to the direction of flow is investigated numerically. An aspect ratio parameter is used for a ring, which describes at its limits a sphere and a circular cylinder. This enables a continuous range of bodies between a sphere and a circular cylinder to be studied.
The computed drag coefficients for the flow past rings at the minimum and maximum aspect ratio limits are compared with the measured and computed drag coefficients reported for the sphere and the circular cylinder. Some interesting features of the behaviour of the drag coefficients with variation of Reynolds number and aspect ratio emerge from the study. These include the decrease in the aspect ratio at which the minimum drag coefficient occurs as the Reynolds number is increased, from $\hbox{\it Ar} \,{\approx}\, 5$ at $\hbox{\it Re} \,{=}\, 1$ to $\hbox{\it Ar} \,{\approx}\, 1$ at $\hbox{\it Re} \,{=}\, 200$. In addition, a substantial decrease in the pressure component of the drag coefficient is observed after the onset of three-dimensional flow while the viscous contribution is similar to that for flow with imposed axisymmetry. Typically, the sudden reduction in drag caused by transition to Mode A shedding is 6%, which is consistent with the behaviour for flow past a circular cylinder. Power-law fits to the drag coefficient for $\hbox{\it Re} \,{\lesssim}\, 100$ are provided, which are accurate within approximately 2%.
The separation of pairs of particles within turbulent flow fields constructed using the ‘kinematic simulation’ method is explored. A consequence of the way the flow is constructed is that, in contrast to real turbulence, there is no ‘sweeping’ of the smaller eddies by the larger eddies. The implications of this are investigated. A simple phenoenological argument is presented which predicts that the mean-square separation of the particle pairs should grow like $t^6$ in kinematic simulation. Simulations support this result for the case where a large mean velocity is added to the flow to exaggerate the sweeping problem and the inertial subrange is sufficiently long. In the absence of a large mean velocity, the situation is more complex with the simple phenomenological argument failing in the parts of the flow where the velocity is much smaller than the r.m.s. velocity and where there is no sweeping problem. The separation process then follows $t^6$ in the bulk of the flow but follows Richardson's classical $t^3$ law in regions where the velocity is much smaller than the r.m.s. velocity. Because of the way the size of these regions varies in time, the resulting mean-square separation grows like $t^{9/2}$. Both the $t^6$ and $t^{9/2}$ behaviours contrast with the classical Richardson $t^3$ law, which is believed to hold in reality, and raise questions about the applicability of the kinematic simulation approach to the separation of pairs in real turbulent flows.
The combined effects of favourable pressure gradient and streamline curvature were studied experimentally using an approximately homogeneous uniformly sheared turbulence. The shear flow was initially generated in a straight wind tunnel, where the turbulence was allowed to develop a fixed stress anisotropy, and then subsequently directed into a curved wind-tunnel test section. Streamwise pressure gradients were applied by convergence of the curved tunnel walls in the plane of the mean shear. In one set of experiments, convergence was applied in the first half of the curved test section, but not in the second half. In another set of experiments, the convergence was applied in the second half of the curved test section, but not in the first. This arrangement permitted the study of application and removal of streamwise pressure gradient to curved shear flow. Measurements showing the response of the turbulence stresses to the changing mean strain rates are reported and are consistent with previous studies which show that stabilizing curvature diminishes the turbulence energy and stresses. The addition of the streamwise strain rate associated with favourable pressure gradient was observed to have the effect of further diminishing the turbulence activity and its overall anisotropy. However, the important shear component of the anisotropy was increased above what it would be under the influence of curvature alone. The removal of streamwise strain rate caused the turbulence to recover a structure similar to that measured for uniformly curved shear flow; although this adjustment included an increase in the shear component of anisotropy prior to its gradual relaxation.
The principal direction of the Reynolds stress tensor was found to be closely related to the principal direction of the mean strain rate tensor in the present flows. This result was also found to be valid in the outer layer of accelerating curved boundary layers. A relationship between the direction of the principal mean strain rate and the mean flow curvature and streamwise strain rate was formulated to explain how each influences the state of turbulence stress.
A space–time filtering approach is used to divide an unbounded turbulent flow into its radiating and non-radiating components. The result is used to investigate the possibility of identifying the true sources of the sound generated by this flow.
The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation \[\partial_t(-\Delta)^{1/2}\psi+J\big(\psi,(-\Delta)^{1/2}\psi\big)=\mu\Delta\psi+f\] is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1\,{=}\,\langle[(-\Delta)^{1/4}\psi]^2\rangle/2$ and $\Psi_2\,{=}\,\langle[(-\Delta)^{1/2}\psi]^2\rangle/2$ (kinetic energy), where $\langle{\bm \cdot}\rangle$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.
Explosive volcanic jets present an unusual dynamic situation of reversing buoyancy. Their initially negative buoyancy with respect to ambient fluid first opposes the motion, but can change sign to drive a convective plume if a sufficient amount of entrainment occurs. The key unknown is the entrainment behaviour for the initial flow regime in which buoyancy acts against the momentum jet. To describe and constrain this regime, we present an experimental study of entrainment into turbulent jets of negative and reversing buoyancy. Using an original technique based on the influence of the injection radius on the threshold between buoyant convection and partial collapse, we show that entrainment is significantly reduced by negative buoyancy. We develop a new theoretical parameterization of entrainment as a function of the local (negative) Richardson number that (i) predicts the observed reduction of entrainment and (ii) introduces a similarity drift in the velocity and buoyancy profiles as a function of distance from source. This similarity drift allows us to reconcile the different estimates found in the literature for entrainment in plumes.