Emil Hopfinger

High-order velocity structure functions in turbulent shear flows

Measurements are presented of the velocity structure function on the axis of a turbulent jet at Reynolds numbers R λ ≤ 852 and in a turbulent duct flow at R λ = 515. Moments of the structure function up to the eighteenth order were calculated, primarily with a view to establish accurately the dependence on the order of the inertial range power-law exponent and to draw conclusions about the distribution of energy transfer in the inertial range. Adequate definition of the probability density of the structure function was achieved only for moments of order n ≤ 10. It is shown, however, that, although the values of moments of n > 10 diverges from their true values, the dependence of the moment of the structure function on the separation r is still given to a fair accuracy for moments up to n ≈ 18. The results demonstrate that the inertial-range power-law exponent is closely approximated by a quadratic dependence on the power which for lower-order moments ( n [lsim ] 12) would be consistent with a lognormal distribution. Higher-order moments diverge, however, from a lognormal distribution, which gives weight to Mandelbrots (1971) conjecture that ‘Kolmogorovs third hypothesis’ is untenable in the strict sense. The intermittency parameter μ, appearing in the power-law exponent, has been determined from sixth-order moments 〈(δμ) 6 〉 ∼ r 2−μ to be μ = 0.2 ± 0.05. This value coincides with that determined from non-centred dissipation correlations measured in identical conditions.

Break-up and atomization of a round water jet by a high-speed annular air jet

The near- and far-field break-up and atomization of a water jet by a high-speed annular air jet are examined by means of high-speed flow visualizations and phase Doppler particle sizing techniques. Visualization of the jets near field and measurements of the frequencies associated with the gas–liquid interfacial instabilities are used to study the underlying physical mechanisms involved in the primary break-up of the water jet. This process is shown to consist of the stripping of water sheets, or ligaments, which subsequently break into smaller lumps or drops. An entrainment model of the near-field stripping of the liquid is proposed, and shown to describe the measured liquid shedding frequencies. This simplified model explains qualitatively the dependence of the shedding frequency on the air/water momentum ratio in both initially laminar and turbulent water jets. The role of the secondary liquid break-up in the far-field atomization of the water jet is also investigated, and an attempt is made to apply the classical concepts of local isotropy to explain qualitatively the measurement of the far-field droplet size distribution and its dependence on the water to air mass and momentum ratios. Models accounting for the effect of the local turbulent dissipation rate in the gas on both the break-up and coalescence of the droplets are developed and compared with the measurements of the variation of the droplet size along the jets centreline. The total flux of kinetic energy supplied by the gas per unit total mass of the spray jet was found to be the primary parameter determining the secondary break-up and coalescence of the droplets in the far field.

Flow regimes of large-velocity-ratio coaxial jets

An investigation of the near-eld flow structure of coaxial jets with large outer to inner velocity ratio ru has been conducted. Since in all cases ru > 1, the outer jet dominates the near-eld flow structure. Two flow regimes are identied depending on whether ru is larger or smaller than a critical value ruc. When ru r uc, the inner potential cone is truncated and is followed by an unsteady recirculation bubble with low-frequency oscillation. The transition from one regime to another is explained by a simple model whose ingredients are the turbulent entrainment rate, governed by the outer-jet mixing layers and mass conservation. This model satisfactorily predicts the dependence of the inner potential cone length on ru and the critical velocity ratio ruc. The recirculation bubble has a wake-type instability. It oscillates at a low frequency and a large amplitude compared to the Kelvin{Helmholtz mode. Angular cross-correlations in the plane parallel to the jet outlet show moreover that this oscillation displays an azimuthal precession such that the rotation time of the phase of the oscillation equals the oscillation period. These salient features are discussed in the framework of the nonlinear delayed saturation (NLDS) model. Coaxial jets are a simple way by which two fluid streams can be mixed and this conguration is used for instance in combustion chambers of rocket engines. Often, one of the jets (the inner one) is in a liquid state and has to be atomized by a high-speed annular gas jet. This process, known as airblast atomization, has received considerable attention (Lefebvre 1989) during the past few decades. Most of the time the experiments have been aimed at characterizing the spray and have not allowed an analysis of the near-eld flow structure and the instabilities in any detail. Leaving aside surface tension eects, the important parameters in this problem are the momentum flux ratio between the two streams M = 2U 2=1U 2 and the ratio of the outer to the inner nozzle diameters = D2=D1. When the fluid densities are the same, the momentum flux ratio reduces to the velocity ratio of the outer to inner jet ru = U2=U1. The near-eld flow structure of coaxial jets in homogeneous fluids is, therefore, expected to be relevant to the understanding of liquid jet atomization. In the coaxial water jets studied here, quantitative flow visualizations can be used which are particularly helpful in the understanding of the interaction of dierent mixing layers present in the near eld. This is well demonstrated by the laser-induced-fluorescence visualizations of coaxial water jets with 0:59 6 ru 6 4:16, performed by Dahm,

Initial breakup of a small-diameter liquid jet by a high-speed gas stream

The breakup and atomization of a liquid jet by a high-speed gas stream is a multi-parameter problem. A generic case, of interest for practical applications, is that where the gas stream is coaxial with the liquid jet and the gas to liquid nozzle area ratio is of order one. Here we consider the situation where a small diameter liquid jet is exposed to a high-speed, large diameter gas jet (gas to liquid nozzle area ratio of order 100); the liquid to gas mass flux ratio is thus small. It is shown that all of the breakup takes place in the near-field and atomization is completed within the potential cone of the gas jet. The resulting drop size does not depend on whether the liquid is injected parallel or perpendicular to the gas stream. It depends primarily on gas velocity and to a weaker extent on the liquid mass flux. It is argued that the mechanism of atomization is similar to that of a liquid drop, suddenly exposed to a high-speed gas stream. The results are analyzed in the context of stripping models of the Kelvin-Helmholtz and Rayleigh-Taylor type(Joseph, D.D. et al., Int. J. Multiphase Flow, 25, 1999, pp. 1263-1303) and illustrated by high-speed video images.The situation of a small-diameter liquid jet exposed to a large-diameter high-speed gas jet (gas-to-liquid nozzle area ratio of order 100 to 1000) is investigated experimentally. Flow visualization and particle-sizing techniques are employed to examine the initial jet breakup process and primary liquid atomization. Observations of the initial breakup of the liquid jet in the near-nozzle region, combined with droplet-size mea-surements, are used in an effort to elucidate the dominant mechanism of primary breakup of the liquid. It is shown that for large aerodynamic Weber numbers, the bulk of the liquid atomization is completed within a few gas-jet diameters of the nozzle exit, inside of the potential cone of the gas flow. Breakup is therefore completed within the zone of constant ambient gas velocity. It is argued that the mechanism of initial jet breakup is similar to that of a liquid drop suddenly exposed to a high-speed gas stream. A phenomenological breakup model is proposed for the initial droplet size, based upon the accelerative, secondary destabilization (via Rayleigh–Taylor instability) of the liquid wave crests resulting from the primary Kelvin–Helmholtz instability of the liquid jet surface. Primary mean droplet sizes are shown to scale well on the most unstable Rayleigh–Taylor wavelength, and the dependence of the droplet diameter on both the atomizing gas velocity and the liquid surface tension are successfully captured by the proposed breakup model.

Experiments on vortex breakdown in a confined flow generated by a rotating disc

The steady-state flow generated by a rotating bottom in a closed cylindrical container and the resulting vortex breakdown bubbles have been studied experimentally. By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, we examined the way in which the formation and structure of the breakdown bubbles depend on the surrounding flow. Details of the flow were visualized by means of the electrolytic precipitation technique, whereas a particle tracking technique was used to characterize the whole flow field. We found that the breakdown bubbles inside the container flow are in many ways similar to those in vortex tubes. First, the bubbles are open with in- and outflow and second, their structure is, like in the case of vortex breakdown in pipe flows, highly axisymmetric on the upstream side of the bubble and asymmetric on their rear side. However, and surprisingly, we observed bubbles which are open and stationary at the same time. This shows that open breakdown bubbles are not necessarily the result of periodic oscillations of the recirculation zone. The asymmetry of the flow structure is found to be related to the existence of asymmetric flow separations on the container wall. If the angular velocity of the rotating bottom is increased the evolution of the breakdown bubbles is different in both configurations: in the rigid cover case the breakdown bubbles disappear but persist in the free surface case.

The structure of the near wake of a sphere moving horizontally in a stratified fluid

We present experimental results for the wake structure of spheres moving in homogeneous and stratified fluid. In homogeneous fluid, the results of Kim & Durbin (1988) are confirmed and it is shown that the two characteristic frequencies of the wake correspond to two instability modes, the Kelvin-Helmholtz instability and a spiral instability. For the stratified wake four general regimes have been identified, depending principally on the Froude number F. For F > 4.5 the near wake is similar to the homogeneous case, and for F < 0.8 it corresponds to a triple-layer flow with two lee waves, of amplitude linear in F, surrounding a layer dominated by quasi-twodimensional motion. Froude numbers close to one (F~]0.8,1.5[ ) give rise to a saturated lee wave of amplitude equal to half the sphere radius, which suppresses the separation region or splits it into two. Between F = 1.5 and 4.5 a more complex regime exists where the wake recovers progressively its behaviour in homogeneous fluid: the axisymmetry of the recirculating zone, the Kelvin-Helmholtz instability and, finally, the spiral instability.

Observations of vortex breakdown in an open cylindrical container with a rotating bottom

A laser induced fluorescent dye technique was used to visualize the steady-state flow driven by a rotating bottom in an open, cylindrical container. The flow behaviour and the vortex breakdown conditions were studied as a function of the container aspect ratio H/R and the Reynolds number Re = ΩR2/v. Like in the closed container configuration, previously studied by Vogel (1968) and Escudier (1984), vortex breakdown occurs in a certain parameter range (H/R, Re). However, in the free surface configuration vortex breakdown conditions as well as the forms of the breakdown bubbles differ notably from what is observed in the closed container configuration. In particular, it is found that as Re is increased, the breakdown bubbles get attached to the free surface and grow in diameter.

Internal waves produced by the turbulent wake of a sphere moving horizontally in a stratified fluid

The internal gravity wave field generated by a sphere towed in a stratified fluid was studied in the Froude number range 1.5 < F < 12.7, where Fis defined with the radius of the sphere. The Reynolds number was sufficiently large for the wake to be turbulent (Re~[380,30000]). A fluorescent dye technique was used to differentiate waves generated by the sphere, called lee waves, from the internal waves, called random waves, emitted by the turbulent wake. We demonstrate that the lee waves are well predicted by linear theory and that the random waves due to the turbulence are related to the coherent structures of the wake. The Strouhal number of these structures depends on F when F 5 4.5. Locally, these waves behave like transient internal waves emitted by impulsively moving bodies.

Charge Transport by Self‐Generated Turbulence in Insulating Liquids Submitted to Unipolar Injection

During unipolar (or ambipolar) injection of charge carriers into an insulating liquid, a liquid motion is generally observed. This motion, generated by the charge carriers, is thought to be responsible for the high carrier mobility which is about 2×10−3 cm3 V−1 sec−1 in nitrobenzene, i.e., an order of magnitude higher than the true mobility value (1.9×10−4 cm2 V−1 sec−1). Recent schlieren photographs strongly support the assumption that when the applied voltage is high enough, for nitrobenzene of the order 103‐104 V, the “self‐generated” motion is turbulent shortly after the voltage is applied. This turbulence invades the undisturbed fluid at practically constant velocity. Neglecting the true carrier mobility and making certain assumptions concerning the turbulent structure, the charge transport and consequently the entrainment can be described to a good approximation by a Lagrangian diffusion process and the charge distribution is governed by a diffusion equation. The latter remains valid during the stea...

Internal waves generated by a moving sphere and its wake in a stratified fluid

The internal gravity waves and the turbulent wake of a sphere moving through stratified fluid were studied by the fluorescent dye technique. The Reynolds number Re=U·2a/v was kept nearly constant at about 3 · 103 and the Froude number F;U/a N ranged from 0.5 to 12.5. It is observed that waves generated by the body are dominant only when F<4 and are replaced by waves generated by the large scale coherent structures of the wake when F>4.