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1
Shrinivas, Ajay B., and Gary R. Hunt. "Confined turbulent entrainment across density interfaces." Journal of Fluid Mechanics 779 (August 14, 2015): 116–43. http://dx.doi.org/10.1017/jfm.2015.366.
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In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.
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Burridge, H. C., and G. R. Hunt. "Entrainment by turbulent fountains." Journal of Fluid Mechanics 790 (February 4, 2016): 407–18. http://dx.doi.org/10.1017/jfm.2016.16.
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Experimental measurements of entrainment by turbulent fountains from circular sources in quiescent uniform environments are presented. Our results span almost four orders of magnitude in the source Froude number ($0.004\leqslant \mathit{Fr}_{0}\leqslant 25$) and thereby encompass the entrainment across all classes of fountain behaviour identified to date. We identify scalings for the entrained volume flux $Q_{E}$, in terms of $\mathit{Fr}_{0}$ and the source volume flux $Q_{0}$, within a number of distinct Froude-number bands corresponding to each class of fountain. Additionally we identify a distinct class of new behaviour, as yet unreported, for $\mathit{Fr}_{0}\lesssim 0.1$.
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Ura, Masaru, and Nobuhiro Matsunaga. "ENTRAINMENT DUE TO MEAN FLOW IN TWO-LAYERED FLUID." Coastal Engineering Proceedings 1, no. 21 (January 29, 1988): 189. http://dx.doi.org/10.9753/icce.v21.189.
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The entrainment phenomena have been investigated across an interface between two-layered stratified flow induced by wind shear stress. The velocities of mean flow, turbulence and entrainment have been measured under three different conditions of water surface by using a wind-wave tank. When the entrainment velocity ue is expressed on the basis of the turbulent quantities at the interface, the turbulent entrainment coefficient E ( = ue/u) is given by E = A-(egl/u2)-3I1 ( A = 0.7). Here Eg, u and 1 are the effective buoyancy, the turbulence intensity and the integral lengthscale of turbulence at the interface, respectively. This result coincides with the relationship of entrainment due to oscillating grid turbulence, in which the mean flow does not exist. When, for the practical purpose, the estimation of ue is made by using the mean velocity Um and the depth h of mixed layer, Em ( - Ue/Um ) = Am•(egh/Um 2)"3/2 is derived from the transformation of E = A-(egl/u2)-3/2. There holds Am = A-Tf between Am and Tf, Tf being a turbulence factor given by (u/Um)4•(1/h)-3/2. It has been found that this relationship is also valid in various types of two-layered stratified flows as well as the wind-induced two-layered flows.
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Abade, Gustavo C., Wojciech W. Grabowski, and Hanna Pawlowska. "Broadening of Cloud Droplet Spectra through Eddy Hopping: Turbulent Entraining Parcel Simulations." Journal of the Atmospheric Sciences 75, no. 10 (October 2018): 3365–79. http://dx.doi.org/10.1175/jas-d-18-0078.1.
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This paper discusses the effects of cloud turbulence, turbulent entrainment, and entrained cloud condensation nuclei (CCN) activation on the evolution of the cloud droplet size spectrum. We simulate an ensemble of idealized turbulent cloud parcels that are subject to entrainment events modeled as a random process. Entrainment events, subsequent turbulent mixing inside the parcel, supersaturation fluctuations, and the resulting stochastic droplet activation and growth by condensation are simulated using a Monte Carlo scheme. Quantities characterizing the turbulence intensity, entrainment rate, CCN concentration, and the mean fraction of environmental air entrained in an event are all specified as independent external parameters. Cloud microphysics is described by applying Lagrangian particles, the so-called superdroplets. These are either unactivated CCN or cloud droplets that grow from activated CCN. The model accounts for the addition of environmental CCN into the cloud by entraining eddies at the cloud edge. Turbulent mixing of the entrained dry air with cloudy air is described using the classical linear relaxation to the mean model. We show that turbulence plays an important role in aiding entrained CCN to activate, and thus broadening the droplet size distribution. These findings are consistent with previous large-eddy simulations (LESs) that consider the impact of variable droplet growth histories on the droplet size spectra in small cumuli. The scheme developed in this work is ready to be used as a stochastic subgrid-scale scheme in LESs of natural clouds.
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Lecoanet, Daniel, and Nadir Jeevanjee. "Entrainment in Resolved, Dry Thermals." Journal of the Atmospheric Sciences 76, no. 12 (November 20, 2019): 3785–801. http://dx.doi.org/10.1175/jas-d-18-0320.1.
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Abstract Entrainment in cumulus convection remains ill understood and difficult to quantify. For instance, entrainment is widely believed to be a fundamentally turbulent process, even though Turner pointed out in 1957 that dry thermals entrain primarily because of buoyancy (via a dynamical constraint requiring an increase in radius r). Furthermore, entrainment has been postulated to obey a 1/r scaling, but this scaling has not been firmly established. Here, we study the classic case of dry thermals in a neutrally stratified environment using fully resolved direct numerical simulation. We combine this with a thermal tracking algorithm that defines a control volume for the thermal at each time, allowing us to directly measure entrainment. We vary the Reynolds number (Re) of our thermals between laminar (Re ≈ 600) and turbulent (Re ≈ 6000) regimes, finding only a 20% variation in entrainment rate ε, supporting the claim that turbulence is not necessary for entrainment. We also directly verify the postulated ε ~ 1/r scaling law.
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Mistry, Dhiren, Jimmy Philip, James R. Dawson, and Ivan Marusic. "Entrainment at multi-scales across the turbulent/non-turbulent interface in an axisymmetric jet." Journal of Fluid Mechanics 802 (August 10, 2016): 690–725. http://dx.doi.org/10.1017/jfm.2016.474.
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We consider the scaling of the mass flux and entrainment velocity across the turbulent/non-turbulent interface (TNTI) in the far field of an axisymmetric jet at high Reynolds number. Time-resolved, simultaneous multi-scale particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) are used to identify and track the TNTI, and directly measure the local entrainment velocity along it. Application of box-counting and spatial-filtering methods, with filter sizes $\unicode[STIX]{x1D6E5}$ spanning over two decades in length, show that the mean length of the TNTI exhibits a power-law behaviour with a fractal dimension $D\approx 0.31{-}0.33$. More importantly, we invoke a multi-scale methodology to confirm that the mean mass flux, which is equal to the product of the entrainment velocity and the surface area, remains constant across the range of filter sizes. The results, within experimental uncertainty, also show that the entrainment velocity along the TNTI exhibits a power-law behaviour with $\unicode[STIX]{x1D6E5}$, such that the entrainment velocity increases with increasing $\unicode[STIX]{x1D6E5}$. In fact, the mean entrainment velocity scales at a rate that balances the scaling of the TNTI length such that the mass flux remains independent of the coarse-grain filter size, as first suggested by Meneveau & Sreenivasan (Phys. Rev. A, vol. 41, no. 4, 1990, pp. 2246–2248). Hence, at the smallest scales the entrainment velocity is small but is balanced by the presence of a very large surface area, whilst at the largest scales the entrainment velocity is large but is balanced by a smaller (smoother) surface area.
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Hallworth, Mark A., Jeremy C. Phillips, Herbert E. Huppert, and R. Stephen J. Sparks. "Entrainment in turbulent gravity currents." Nature 362, no. 6423 (April 1993): 829–31. http://dx.doi.org/10.1038/362829a0.
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Presley, John D., and James W. Telford. "Turbulent entrainment at an inversion." Pure and Applied Geophysics PAGEOPH 127, no. 1 (1988): 117–41. http://dx.doi.org/10.1007/bf00878694.
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Craske, John, Pietro Salizzoni, and Maarten van Reeuwijk. "The turbulent Prandtl number in a pure plume is 3/5." Journal of Fluid Mechanics 822 (June 8, 2017): 774–90. http://dx.doi.org/10.1017/jfm.2017.259.
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We derive a new expression for the entrainment coefficient in a turbulent plume using an equation for the squared mean buoyancy. Consistency of the resulting expression with previous relations for the entrainment coefficient implies that the turbulent Prandtl number in a pure plume is equal to 3/5 when the mean profiles of velocity and buoyancy have a Gaussian form of equal width. Entrainment can be understood in terms of the volume flux, the production of turbulence kinetic energy or the production of scalar variance for either active or passive variables. The equivalence of these points of view indicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbers depend on the Richardson number of the flow, the ambient stratification and the relative widths of the velocity and scalar profiles. The general framework is valid for self-similar plumes, which are characterised by a power-law scaling. For jets and pure plumes it is shown that the derived relations are in reasonably good agreement with results from direct numerical simulations and experiments.
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de Lozar, Alberto, and Juan Pedro Mellado. "Direct Numerical Simulations of a Smoke Cloud–Top Mixing Layer as a Model for Stratocumuli." Journal of the Atmospheric Sciences 70, no. 8 (August 1, 2013): 2356–75. http://dx.doi.org/10.1175/jas-d-12-0333.1.
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Abstract A radiatively driven cloud-top mixing layer is investigated using direct numerical simulations. This configuration mimics the mixing process across the inversion that bounds the stratocumulus-topped boundary layer. The main focus of this paper is on small-scale turbulence. The finest resolution (7.4 cm) is about two orders of magnitude finer than that in cloud large-eddy simulations (LES). A one-dimensional horizontally averaged model is employed for the radiation. The results show that the definition of the inversion point with the mean buoyancy of 〈b〉(zi) = 0 leads to convective turbulent scalings in the cloud bulk consistent with the Deardorff theory. Three mechanisms contribute to the entrainment by cooling the inversion layer: a molecular flux, a turbulent flux, and the direct radiative cooling by the smoke inside the inversion layer. In the simulations the molecular flux is negligible, but the direct cooling reaches values comparable to the turbulent flux as the inversion layer thickens. The results suggest that the direct cooling might be overestimated in less-resolved models like LES, resulting in an excessive entrainment. The scaled turbulent flux is independent of the stratification for the range of Richardson numbers studied here. As suggested by earlier studies, the turbulent entrainment only occurs at the small scales and eddies larger than approximately four optical lengths (60 m in a typical stratocumulus cloud) perform little or no entrainment. Based on those results, a parameterization is proposed that accounts for a large part (50%–100%) of the entrainment velocities measured in the Second Dynamics and Chemistry of the Marine Stratocumulus (DYCOMS II) campaign.
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Briggs, D. A., J. H. Ferziger, J. R. Koseff, and S. G. Monismith. "Entrainment in a shear-free turbulent mixing layer." Journal of Fluid Mechanics 310 (March 10, 1996): 215–41. http://dx.doi.org/10.1017/s0022112096001784.
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Results from a direct numerical simulation of a shear-free turbulent mixing layer are presented. The mixing mechanisms associated with the turbulence are isolated. In the first set of simulations, the turbulent mixing layer decays as energy is exchanged between the layers. Energy spectra with E(k) ∼ k2 and E(k) ∼ k4 dependence at low wavenumber are used to initialize the flow to investigate the effect of initial conditions. The intermittency of the mixing layer is quantified by the skewness and kurtosis of the velocity fields: results compare well with the shearless mixing layer experiments of Veeravalli & Warhaft (1989). Eddies of size of the integral scale (k3/2/∈) penetrate the mixing layer intermittently, transporting energy and causing the layer to grow. The turbulence in the mixing layer can be characterized by eddies with relatively large vertical kinetic energy and vertical length scale. In the second set of simulations, a forced mixing layer is created by continuously supplying energy in a local region to maintain a stationary kinetic energy profile. Assuming the spatial decay of r.m.s. velocity is of the form u &∞ yn, predictions of common two-equation turbulence models yield values of n ranging from -1.25 to -2.5. An exponent of -1.35 is calculated from the forced mixing layer simulation. In comparison, oscillating grid experiments yield decay exponents between n = -1 (Hannoun et al. 1989) and n = -1.5 (Nokes 1988). Reynolds numbers of 40 and 58, based on Taylor microscale, are obtained in the decaying and forced simulations, respectively. Components of the turbulence models proposed by Mellor & Yamada (1986) and Hanjalić & Launder (1972) are analysed. Although the isotropic models underpredict the turbulence transport, more complicated anisotropic models do not represent a significant improvement. Models for the pressure-strain tensor, based on the anisotropy tensor, performed adequately.
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van Reeuwijk, Maarten, and Markus Holzner. "The turbulence boundary of a temporal jet." Journal of Fluid Mechanics 739 (December 18, 2013): 254–75. http://dx.doi.org/10.1017/jfm.2013.613.
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AbstractWe examine the structure of the turbulence boundary of a temporal plane jet at$\mathit{Re}= 5000$using statistics conditioned on the enstrophy. The data is obtained by direct numerical simulation and threshold values span 24 orders of magnitude, ranging from essentially irrotational fluid outside the jet to fully turbulent fluid in the jet core. We use two independent estimators for the local entrainment velocity${v}_{n} $based on the enstrophy budget. The data show clear evidence for the existence of a viscous superlayer (VSL) that envelopes the turbulence. The VSL is a nearly one-dimensional layer with low surface curvature. We find that both its area and viscous transport velocity adjust to the imposed rate of entrainment so that the integral entrainment flux is independent of threshold, although low-Reynolds-number effects play a role for the case under consideration. This threshold independence is consistent with the inviscid nature of the integral rate of entrainment. A theoretical model of the VSL is developed that is in reasonably good agreement with the data and predicts that the contribution of viscous transport and dissipation to interface propagation have magnitude$2{v}_{n} $and$- {v}_{n} $, respectively. We further identify a turbulent core region (TC) and a buffer region (BR) connecting the VSL and the TC. The BR grows in time and inviscid enstrophy production is important in this region. The BR shows many similarities with the turbulent–non-turbulent interface (TNTI), although the TNTI seems to extend into the TC. The average distance between the TC and the VSL, i.e. the BR thickness is about 10 Kolmogorov length scales or half a Taylor length scale, indicating that intense turbulent flow regions and viscosity-dominated regions are in close proximity.
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GERASHCHENKO, S., G. GOOD, and Z. WARHAFT. "Entrainment and mixing of water droplets across a shearless turbulent interface with and without gravitational effects." Journal of Fluid Mechanics 668 (January 26, 2011): 293–303. http://dx.doi.org/10.1017/s002211201000577x.
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We describe experiments of the entrainment and mixing of water (sub-Kolmogorov scale) droplets across a turbulent–non-turbulent interface (TNI) as well a turbulent–turbulent interface (TTI) in shearless grid turbulence, over a time scale in which evaporation is insignificant. The flow is produced by means of a splitter plate with an active grid and water sprays on one side and screens or an active grid on the other side. The Taylor microscale Reλ on the turbulent side is 275 and the average dissipation scale Stokes number, Stη ≈ 0.2, and based on the integral scale, Stl ≈ 0.003. By changing the orientation of the grid system, gravitational effects may be excluded or included. We show that in the absence of gravity, for the Stokes number range studied (0.06 ≤ Stη ≤ 1.33), the droplet distribution does not change across the interface. With gravity, the larger drops are selectively mixed and this is more pronounced for the TNI than for the TTI. The particle concentration distribution is an error function for the TTI but departs significantly for the TNI due to the intermittency in the flow. In terms of particle concentration, the entrainment is most efficient for the TTI with gravity. The results are related to droplet entrainment in clouds.
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Khorsandi, B., S. Gaskin, and L. Mydlarski. "Effect of background turbulence on an axisymmetric turbulent jet." Journal of Fluid Mechanics 736 (November 4, 2013): 250–86. http://dx.doi.org/10.1017/jfm.2013.465.
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AbstractThe effect of different levels of background turbulence on the dynamics and mixing of an axisymmetric turbulent jet at different Reynolds numbers has been investigated. Approximately homogeneous and isotropic background turbulence was generated by a random jet array and had a negligible mean flow (${\langle {U}_{\alpha } \rangle }/ {u}_{\alpha \mathit{rms}} \ll 1$). Velocity measurements of a jet issuing into two different levels of background turbulence were conducted for three different jet Reynolds numbers. The results showed that the mean axial velocities decay faster with increasing level of background turbulence (compared with a jet in quiescent surroundings), while the mean radial velocities increase, especially close to the edges of the jet. Furthermore, the axial root-mean-square velocities of the jet increased in the presence of background turbulence, as did the jet’s width. However, the mass flow rate of the jet decreased, from which it can be inferred that the entrainment into the jet is reduced in a turbulent background. The effect of background turbulence on the entrainment mechanisms is discussed.
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Chauhan, Kapil, Jimmy Philip, Charitha M. de Silva, Nicholas Hutchins, and Ivan Marusic. "The turbulent/non-turbulent interface and entrainment in a boundary layer." Journal of Fluid Mechanics 742 (February 21, 2014): 119–51. http://dx.doi.org/10.1017/jfm.2013.641.
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AbstractThe turbulent/non-turbulent interface in a zero-pressure-gradient turbulent boundary layer at high Reynolds number ($\mathit{Re}_\tau =14\, 500$) is examined using particle image velocimetry. An experimental set-up is utilized that employs multiple high-resolution cameras to capture a large field of view that extends $2\delta \times 1.1\delta $ in the streamwise/wall-normal plane with an unprecedented dynamic range. The interface is detected using a criteria of local turbulent kinetic energy and proves to be an effective method for boundary layers. The presence of a turbulent/non-turbulent superlayer is corroborated by the presence of a jump for the conditionally averaged streamwise velocity across the interface. The steep change in velocity is accompanied by a discontinuity in vorticity and a sharp rise in the Reynolds shear stress. The conditional statistics at the interface are in quantitative agreement with the superlayer equations outlined by Reynolds (J. Fluid Mech., vol. 54, 1972, pp. 481–488). Further analysis introduces the mass flux as a physically relevant parameter that provides a direct quantitative insight into the entrainment. Consistency of this approach is first established via the equality of mean entrainment calculations obtained using three different methods, namely, conditional, instantaneous and mean equations of motion. By means of ‘mass-flux spectra’ it is shown that the boundary-layer entrainment is characterized by two distinctive length scales which appear to be associated with a two-stage entrainment process and have a substantial scale separation.
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ITO, Kei, Hiroyuki OHSHIMA, and Yasutomo IMAI. "ICONE19-44127 STUDY ON TURBULENT MODELING IN GAS ENTRAINMENT EVALUATION METHOD." Proceedings of the International Conference on Nuclear Engineering (ICONE) 2011.19 (2011): _ICONE1944. http://dx.doi.org/10.1299/jsmeicone.2011.19._icone1944_40.
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McHUGH, SEAN T., and SILVANA S. S. CARDOSO. "Turbulent entrainment into inert and reacting multiphase plumes." Journal of Fluid Mechanics 682 (July 15, 2011): 577–89. http://dx.doi.org/10.1017/jfm.2011.246.
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Theoretical predictions and experimental results for turbulent entrainment in inert and reacting, multiphase plumes are presented. It is shown that in an inert, pure plume, the entrainment coefficient is approximately constant with downstream distance. In a reacting plume, in which buoyancy is depleted by chemical reaction, the entrainment coefficient decreases strongly with distance from the source owing mainly to a decrease in the Richardson number. The effect on entrainment of the drift in the velocity and buoyancy distributions in the radial direction, i.e. the similarity drift introduced by Kaminski, Tait & Carazzo (J. Fluid Mech., vol. 526, 2005, pp. 361–76), is found to increase with downstream distance and with the reaction rate but, on laboratory-scale experiments, remains small compared to the contribution to entrainment from the turbulent stresses and buoyancy.
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Shrinivas, Ajay B., and Gary R. Hunt. "Unconfined turbulent entrainment across density interfaces." Journal of Fluid Mechanics 757 (September 23, 2014): 573–98. http://dx.doi.org/10.1017/jfm.2014.474.
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AbstractWe present theoretical models describing the quasi-steady downward transport of buoyant fluid across a gravitationally stable density interface separating two unbounded quiescent fluid masses. The primary transport mechanism is turbulent entrainment resulting from the localised impingement of a vertically forced high-Reynolds-number axisymmetric jet with steady source conditions. The entrainment across the interface is examined in the large-time asymptotic state, wherein the interfacial gravity current, formed by the fluid entrained from the upper layer and the jet, becomes infinitesimally thin and a two-layer stratification persists. Characterising flows with small interfacial Froude numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}({{\mathrm{Fr}}}_i)$ as an axisymmetric semi-ellipsoidal impingement dome, we combine conservation equations with a mechanistic model of entrainment and reveal that, in this regime, the dimensionless entrainment flux $E_i$ across the interface follows the power law $E_i = 0.24{{\mathrm{Fr}}}_i^2$. For large-${{\mathrm{Fr}}}_i$ impingements, modelled as a fully penetrating turbulent fountain, we show that $E_i$ no longer scales with ${{\mathrm{Fr}}}_i^2$, but linearly on ${{\mathrm{Fr}}}_i$, following $E_i = 0.42{{\mathrm{Fr}}}_i$. We establish the intermediate range of ${{\mathrm{Fr}}}_i$ over which there is a transition between these quadratic and linear power laws, thus enabling us to classify the dynamics of entrainment across the interface into three distinct regimes. Finally, the close agreement of our solutions with existing experimental results is illustrated.
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Cao, Zhixian. "Turbulent Bursting-Based Sediment Entrainment Function." Journal of Hydraulic Engineering 123, no. 3 (March 1997): 233–36. http://dx.doi.org/10.1061/(asce)0733-9429(1997)123:3(233).
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Li, Guang-Xing, Keping Qiu, Friedrich Wyrowski, and Karl Menten. "Turbulent entrainment origin of protostellar outflows." Astronomy & Astrophysics 559 (October 30, 2013): A23. http://dx.doi.org/10.1051/0004-6361/201220581.
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Raga, A. C., and J. Cantó. "Turbulent Entrainment in Mira's Cometary Tail." Astrophysical Journal 685, no. 2 (August 28, 2008): L141—L144. http://dx.doi.org/10.1086/592500.
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van Reeuwijk, Maarten, Markus Holzner, and C. P. Caulfield. "Mixing and entrainment are suppressed in inclined gravity currents." Journal of Fluid Mechanics 873 (June 28, 2019): 786–815. http://dx.doi.org/10.1017/jfm.2019.430.
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We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $\mathit{Ri}_{f}$, gradient Richardson number $\mathit{Ri}_{g}$ and turbulent flux coefficient $\unicode[STIX]{x1D6E4}$ are constant across a large portion of the depth. Changing the slope angle $\unicode[STIX]{x1D6FC}$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $\mathit{Ri}_{max}\approx 0.15$ as $\unicode[STIX]{x1D6FC}\rightarrow 0$ at which the entrainment coefficient $E\rightarrow 0$. The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as $\unicode[STIX]{x1D6FC}\rightarrow 0$, due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small $\unicode[STIX]{x1D6FC}$ which is able to predict accurately $\mathit{Ri}_{f}$, $\mathit{Ri}_{g}$ and $\unicode[STIX]{x1D6E4}$ as a function of $\unicode[STIX]{x1D6FC}$ and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$, which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.
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Krug, Dominik, Markus Holzner, Beat Lüthi, Marc Wolf, Wolfgang Kinzelbach, and Arkady Tsinober. "The turbulent/non-turbulent interface in an inclined dense gravity current." Journal of Fluid Mechanics 765 (January 20, 2015): 303–24. http://dx.doi.org/10.1017/jfm.2014.738.
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AbstractWe present an experimental investigation of entrainment and the dynamics near the turbulent/non-turbulent interface in a dense gravity current. The main goal of the study is to investigate changes in the interfacial physics due to the presence of stratification and to examine their impact on the entrainment rate. To this end, three-dimensional data sets of the density and the velocity fields are obtained through a combined scanning particle tracking velocimetry/laser-induced fluorescence approach for two different stratification levels with inflow Richardson numbers of $\mathit{Ri}_{0}=0.23$ and $\mathit{Ri}_{0}=0.46$, respectively, at a Reynolds number around $\mathit{Re}_{0}=3700$. An analysis conditioned on the instantaneous position of the turbulent/non-turbulent interface as defined by a threshold on enstrophy reveals an interfacial region that is in many aspects independent of the initial level of stratification. This is reflected most prominently in matching peaks of the gradient Richardson number $\mathit{Ri}_{g}\approx 0.1$ located approximately $10{\it\eta}$ from the position of the interface inside the turbulent region, where ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ is the Kolmogorov scale, and ${\it\nu}$ and ${\it\epsilon}$ denote the kinematic viscosity and the rate of turbulent dissipation, respectively. A possible explanation for this finding is offered in terms of a cyclic evolution in the interaction of stratification and shear involving the buildup of density and velocity gradients through inviscid amplification and their subsequent depletion through molecular effects and pressure. In accordance with the close agreement of the interfacial properties for the two cases, no significant differences were found for the local entrainment velocity, $v_{n}$ (defined as the propagation velocity of an enstrophy isosurface relative to the fluid), at different initial stratification levels. Moreover, we find that the baroclinic torque does not contribute significantly to the local entrainment velocity. Comparing results for the surface area of the convoluted interface to estimates from fractal scaling theory, we identify differences in the interface geometry as the major factor in the reduction of the entrainment rate due to density stratification.
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Aspden, A. J., N. Nikiforakis, J. B. Bell, and Stuart B. Dalziel. "Turbulent jets with off-source heating." Journal of Fluid Mechanics 824 (July 11, 2017): 766–84. http://dx.doi.org/10.1017/jfm.2017.272.
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Motivated by anomalous entrainment behaviour in cumulus clouds, Bhat et al. (Exp. Fluids, vol. 7, 1989, pp. 99–102) pioneered a laboratory experiment to study turbulent jets subjected to a volumetric heating away from the momentum source. The study concluded that the use of a constant entrainment coefficient was insufficient for the flow, and that the results did not confirm the analysis of Hunt (Recent Research Advances in the Fluid Mechanics of Turbulent Jets and Plumes, 1994, pp. 309–334, Kluwer Academic), which suggested that an increase in relative turbulent transport of streamwise momentum could lead to a decrease in entrainment. The present paper re-evaluates theoretical aspects of both studies, and includes a decomposition of the factors contributing to entrainment. The reworked analysis is then used to examine three-dimensional numerical simulations of turbulent jets with off-source heating. The data are consistent with previous work, but give deeper insight not easily obtainable through experiment. Specifically, direct measurement of flux integrals shows that previous inference from experimental measurements of centreline velocity and profile widths under the assumption of self-similarity can lead to underestimation of the mass flux by over 50 % in some cases. Radial profiles of temperature, radial velocity and turbulent correlations show significant departures from self-similarity. The flux measurements show that there is actually an increase in the entrainment coefficient with heating, and that it is locally enhanced by positive forcing and decreased by an increase in turbulent transport of streamwise momentum, thereby confirming the essence of the original proposal of Hunt.
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McWilliams, James C., Edward Huckle, Junhong Liang, and Peter P. Sullivan. "Langmuir Turbulence in Swell." Journal of Physical Oceanography 44, no. 3 (March 1, 2014): 870–90. http://dx.doi.org/10.1175/jpo-d-13-0122.1.
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Abstract The problem is posed and solved for the oceanic surface boundary layer in the presence of wind stress, stable density stratification, equilibrium wind-waves, and remotely generated swell-waves. The addition of swell causes an amplification of the Lagrangian-mean current and rotation toward the swell-wave direction, a fattening of the Ekman velocity spiral and associated vertical Reynolds stress profile, an amplification of the inertial current response, an enhancement of turbulent variance and buoyancy entrainment rate from the pycnocline, and—for very large swell—an upscaling of the coherent Langmuir circulation patterns. Implications are discussed for the parameterization of Langmuir turbulence influences on the mean current profile and the material entrainment rate in oceanic circulation models. In particular, even though the turbulent kinetic energy monotonically increases with wave amplitude inversely expressed by the turbulent Langmuir number La, the Lagrangian shear eddy viscosity profile κL(z) is a nonmonotonic function of La, first increasing with increasing wave amplitude up to approximately the wind-wave equilibrium level, then decreasing with additional swell-wave amplitude. In contrast, the pycnocline entrainment rate is a monotonic function ~La−2.
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MUSCULUS, MARK P. B. "Entrainment waves in decelerating transient turbulent jets." Journal of Fluid Mechanics 638 (October 1, 2009): 117–40. http://dx.doi.org/10.1017/s0022112009990826.
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A simplified one-dimensional partial differential equation for the integral axial momentum flux during the deceleration phase of single-pulsed transient incompressible jets is derived and solved analytically. The wave speed of the derived first-order nonlinear wave equation shows that the momentum flux transient from the deceleration phase propagates downstream at twice the initial jet penetration rate. Transient-jet velocity data from the existing literature is shown to be consistent with this derivation, and an algebraic analytical solution matches the measured timing and decay of axial velocity after the deceleration transient. The solution also shows that a wave of increased entrainment accompanies the deceleration transient as it travels downstream through the jet. In the long-time limit, the peak entrainment rate at the leading edge of this ‘entrainment wave’ approaches an asymptotic value of three times that of the initial steady jet. The rate of approach to the asymptotic behaviour is controlled by the deceleration rate, which suggests that rate-shaping may be tailored to achieve a desired mixing state at a given time after the end of a single-pulsed jet. In the wake of the entrainment wave, the absolute entrainment rate eventually decays to zero. The local injected fluid concentration also decays, however, so that entrainment rate relative to the local concentration of injected fluid remains higher than in the initial steady jet. An analysis of diesel engine fuel-jets is provided as one example of a transient-jet application in which the considerable increase in the mixing rate after the deceleration phase has important implications.
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Noriega-Crespo, Alberto. "Turbulent Mixing Layers in Herbig-Haro and Embedded H2 Jets." Symposium - International Astronomical Union 182 (1997): 103–10. http://dx.doi.org/10.1017/s0074180900061581.
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Outflows arising from young stellar objects interact with their surrounding medium through different mechanisms, such as shocks, turbulence and /or entrainment. These two last mechanisms have recently begun to be fully developed in an effort to understand the coupling of the stellar jet gas (mostly atomic) with the molecular environment. Observationally a number of objects are being studied in the near infrared and millimeter wavelengths to map the warm (H2) and cold (CO) molecular gas, respectively. In this paper we discuss some of the properties in the near infrared of various embedded jets in the context of turbulent entrainment.
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BLOOMFIELD, LYNN J., and ROSS C. KERR. "A theoretical model of a turbulent fountain." Journal of Fluid Mechanics 424 (November 16, 2000): 197–216. http://dx.doi.org/10.1017/s0022112000001907.
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A theoretical model of axisymmetric turbulent fountains in both homogeneous and stratified fluids is developed. The model quantifies the entrainment of ambient fluid into the initial fountain upflow, and the entrainment of fluid from both the upflow and environment into the subsequently formed downflow. Four different variations of the model are considered, comprising the two most reasonable formulations of the body forces acting on the ‘double’ structure and two formulations of the rate of entrainment between the flows. The four model variations are tested by comparing the predictions from each of them with experimental measurements of fountains in homogeneous and stratified fluids.
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Falcone, Anthony M., and Joseph C. Cataldo. "Entrainment Velocity in an Axisymmetric Turbulent Jet." Journal of Fluids Engineering 125, no. 4 (July 1, 2003): 620–27. http://dx.doi.org/10.1115/1.1595674.
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Mean radial and turbulent radial velocity profiles were measured in a circular jet at up to 40 jet diameters downstream of the jet exit using an LDA. The mean radial velocity in the ambient reservoir (the entrainment velocity) is found to be inversely proportional to the radial distance from the jet centerline. The coefficient of proportionality, c, increases in the zone of flow establishment and reaches a constant after the transition zone. It is suggested that the traditional definition of entrainment velocity, which maintains direct proportionality to the local jet velocity by the entrainment coefficient, should be augmented by this inverse function.
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Breda, M., and O. R. H. Buxton. "Behaviour of small-scale turbulence in the turbulent/non-turbulent interface region of developing turbulent jets." Journal of Fluid Mechanics 879 (September 20, 2019): 187–216. http://dx.doi.org/10.1017/jfm.2019.676.
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Tomographic particle image velocimetry experiments were conducted in the near and intermediate fields of two different types of jet, one fitted with a circular orifice and another fitted with a repeating-fractal-pattern orifice. Breda & Buxton (J. Vis., vol. 21 (4), 2018, pp. 525–532; Phys. Fluids, vol. 30, 2018, 035109) showed that this fractal geometry suppressed the large-scale coherent structures present in the near field and affected the rate of entrainment of background fluid into, and subsequent development of, the fractal jet, relative to the round jet. In light of these findings we now examine the modification of the turbulent/non-turbulent interface (TNTI) and spatial evolution of the small-scale behaviour of these different jets, which are both important factors behind determining the entrainment rate. This evolution is examined in both the streamwise direction and within the TNTI itself where the fluid adapts from a non-turbulent state, initially through the direct action of viscosity and then through nonlinear inertial processes, to the state of the turbulence within the bulk of the flow over a short distance. We show that the suppression of the coherent structures in the fractal jet leads to a less contorted interface, with large-scale excursions of the inner TNTI (that between the jet’s azimuthal shear layer and the potential core) being suppressed. Further downstream, the behaviour of the TNTI is shown to be comparable for both jets. The velocity gradients develop into a canonical state with streamwise distance, manifested as the development of the classical tear-drop shaped contours of the statistical distribution of the velocity-gradient-tensor invariants $\mathit{Q}$ and $\mathit{R}$. The velocity gradients also develop spatially through the TNTI from the irrotational boundary to the bulk flow; in particular, there is a strong small-scale anisotropy in this region. This strong inhomogeneity of the velocity gradients in the TNTI region has strong consequences for the scaling of the thickness of the TNTI in these spatially developing flows since both the Taylor and Kolmogorov length scales are directly computed from the velocity gradients.
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Carlotti, P., and G. R. Hunt. "An entrainment model for lazy turbulent plumes." Journal of Fluid Mechanics 811 (December 15, 2016): 682–700. http://dx.doi.org/10.1017/jfm.2016.714.
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An entrainment model for lazy turbulent plumes is proposed, the resulting solutions of the plume conservation equations are developed and the implications for plume behaviour are considered and compared with the available experimental data. Indeed, the applicability of the classic solutions of the conservation equations subject to source conditions that produce lazy plumes, i.e. those with suitably high source Richardson number, contains an essential weakness: the underlying assumption of a constant entrainment coefficient. While entrainment models prescribing the dependence of the entrainment coefficient on the local Richardson number have been proposed for forced plumes, corresponding formulations for lazy plumes have not until now been considered. In the context of saline plumes, the model is applied directly. For hot gaseous plumes, we use a modified definition of buoyancy flux to recover a constant buoyancy flux in a non-stratified environment, despite the specific heat varying with the temperature. After a brief review of existing forced-plume formulations of entrainment, a power-law variation is adopted for the lazy plume. The plume equations are solved for the parameter $0\leqslant \unicode[STIX]{x1D714}<1$, where $\unicode[STIX]{x1D714}$ denotes the exponent of the power law. The cases of pure plumes and lazy plumes are then analysed in more detail; to the best of our knowledge this represents the first modelling of variable entrainment for lazy plumes. Specifically, it is shown that classic plume theory is recovered for $\unicode[STIX]{x1D714}=0$, while for $\unicode[STIX]{x1D714}=1/5$ the plume equations may be solved using usual functions (notably polynomials) only. The results of the models for these cases are very similar, which advocates the idea of selecting systematically $\unicode[STIX]{x1D714}=1/5$, instead of $\unicode[STIX]{x1D714}=0$, for cases where the effect of variation of entrainment is weak, since the new model leads to simple calculations. In the case of very lazy plumes, it is shown that, provided that a relevant value of $\unicode[STIX]{x1D714}$ is chosen, the new model reproduces the available experimental results well.
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Robinson, F. J., and S. C. Sherwood. "Modeling the Impact of Convective Entrainment on the Tropical Tropopause." Journal of the Atmospheric Sciences 63, no. 3 (March 1, 2006): 1013–27. http://dx.doi.org/10.1175/jas3673.1.
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Abstract Simulations with the Weather Research and Forecasting (WRF) cloud-resolving model of deep moist convective events reveal net cooling near the tropopause (∼15–18 km above ground), caused by a combination of large-scale ascent and small-scale cooling by the irreversible mixing of turbulent eddies overshooting their level of neutral buoyancy. The turbulent cooling occurred at all CAPE values investigated (local peak values ranging from 1900 to 3500 J kg−1) and was robust to grid resolution, subgrid-scale turbulence parameterization, horizontal domain size, model dimension, and treatment of ice microphysics. The ratio of the maximum downward heat flux in the tropopause to the maximum tropospheric upward heat flux was close to 0.1. This value was independent of CAPE but was affected by changes in microphysics or subgrid-scale turbulence parameterization. The convective cooling peaked roughly 1 km above the cold point in the background input sounding and the mean cloud- and (turbulent kinetic energy) TKE-top heights, which were all near 16.5 km above ground. It was associated with turbulent entrainment of stratospheric air from as high as 18.25 km into the troposphere. Typical cooling in the experiments was of order 1 K during convective events that produced order 10 mm of precipitation, which implied a significant contribution to the tropopause energy budget. Given the sharp concentration gradients and long residence times near the cold point, even such a small entrainment rate is likely consequential for the transport and ambient distribution of trace gases such as water vapor and ozone, and probably helps to explain the gradual increase of ozone typically observed below the tropical tropopause.
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Nokes, R. I. "On the entrainment rate across a density interface." Journal of Fluid Mechanics 188 (March 1988): 185–204. http://dx.doi.org/10.1017/s0022112088000692.
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Mixed-layer deepening due to grid-generated turbulence is studied experimentally with the aim of explaining the contradictory results of previous studies. Entrainment rates are calculated at fixed distances from the grid in order to avoid the necessity of using an empirical expression for the decay of the turbulent velocity scale. It is shown that an incorrect form of this decay law can cause large errors in the predicted Richardson number dependence of the entrainment rate. For this study this dependence can be expressed as a power law of the form E = KRi−1,2. The spread of the results imply that an error of at least ± 10% is realistic in the determination of the exponent.The turbulent velocity decay law is also deduced from the data, and it is found that the decay cannot be represented by a simple power law. Indeed two distinct flow regions, with differing decay rates, are present.
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Neamtu-Halic, Marius M., Dominik Krug, George Haller, and Markus Holzner. "Lagrangian coherent structures and entrainment near the turbulent/non-turbulent interface of a gravity current." Journal of Fluid Mechanics 877 (August 27, 2019): 824–43. http://dx.doi.org/10.1017/jfm.2019.635.
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In this paper, we employ the theory of Lagrangian coherent structures for three-dimensional vortex eduction and investigate the effect of large-scale vortical structures on the turbulent/non-turbulent interface (TNTI) and entrainment of a gravity current. The gravity current is realized experimentally and different levels of stratification are examined. For flow measurements, we use a multivolume three-dimensional particle tracking velocimetry technique. To identify vortical Lagrangian coherent structures (VLCSs), a fully automated three-dimensional extraction algorithm for multiple flow structures based on the so-called Lagrangian-averaged vorticity deviation method is implemented. The size, the orientation and the shape of the VLCSs are analysed and the results show that these characteristics depend only weakly on the strength of the stratification. Through conditional analysis, we provide evidence that VLCSs modulate the average TNTI height, consequently affecting the entrainment process. Furthermore, VLCSs influence the local entrainment velocity and organize the flow field on both the turbulent and non-turbulent sides of the gravity current boundary.
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Fodor, Katherine, and Juan Pedro Mellado. "New Insights into Wind Shear Effects on Entrainment in Convective Boundary Layers Using Conditional Analysis." Journal of the Atmospheric Sciences 77, no. 9 (September 1, 2020): 3227–48. http://dx.doi.org/10.1175/jas-d-19-0345.1.
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Abstract Conventional analysis has shown that strong wind shear enhances the entrainment buoyancy flux in the convective boundary layer. By conditioning the entrainment zone into regions of turbulent (i.e., strongly vortical) and nonturbulent (i.e., weakly vortical) flow, some unexpected aspects of this process are revealed. It is found that turbulent regions contribute the most to the entrainment buoyancy flux, but that as wind shear increases, the magnitude of the buoyancy flux in turbulent regions remains approximately constant, or even decreases, despite substantially stronger buoyancy fluctuations. The reason is that the correlation between buoyancy and vertical velocity fluctuations decreases with increasing wind shear, to the extent that it compensates the stronger buoyancy fluctuations. In free convection, this correlation is high because the vertical velocity is mainly determined by the buoyancy force acting in the same direction. Under strong shear conditions, buoyancy is no longer the only external source of vertical velocity fluctuations and their correlation consequently decreases. Hence, shear enhancement of the buoyancy flux in the entrainment zone is primarily due to an increase of the turbulent area fraction, rather than a change of flux inside the turbulent regions.
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Mistry, Dhiren, Jimmy Philip, and James R. Dawson. "Kinematics of local entrainment and detrainment in a turbulent jet." Journal of Fluid Mechanics 871 (May 30, 2019): 896–924. http://dx.doi.org/10.1017/jfm.2019.327.
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In this paper we investigate the continuous, local exchange of fluid elements as they are entrained and detrained across the turbulent/non-turbulent interface (TNTI) in a high Reynolds number axisymmetric jet. To elucidate characteristic kinematic features of local entrainment and detrainment processes, simultaneous high-speed particle image velocimetry and planar laser-induced fluorescence measurements were undertaken. Using an interface-tracking technique, we evaluate and analyse the conditional dependence of local entrainment velocity in a frame of reference moving with the TNTI in terms of the interface geometry and the local flow field. We find that the local entrainment velocity is intermittent with a characteristic length scale of the order of the Taylor micro-scale and that the contribution to the net entrainment rate arises from the imbalance between local entrainment and detrainment rates that occurs with a ratio of two parts of entrainment to one part detrainment. On average, an increase in local entrainment is correlated with excursions of the TNTI towards jet centreline into regions of higher streamwise momentum, convex surface curvature facing the turbulent side of the jet and along the leading edges of the interface. In contrast, detrainment is correlated with excursions of the TNTI away from the jet centreline into regions of lower streamwise momentum, concave surface curvature and along the trailing edge. We find that strong entrainment is characterised by a local counterflow velocity field in the frame of reference moving with the TNTI which enhances the transport of rotational and irrotational fluid elements. On the other hand, detrainment is characterised by locally uniform flow fields with the local fluid velocity on either side of the TNTI advecting in the same direction. These local flow patterns and the strength of entrainment or detrainment rates are also observed to be strongly influenced by the presence and relative strength of vortical structures which are of the order of the Taylor micro-scale that populate the turbulent region along the jet boundary.
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Jahanbakhshi, Reza, and Cyrus K. Madnia. "Entrainment in a compressible turbulent shear layer." Journal of Fluid Mechanics 797 (May 24, 2016): 564–603. http://dx.doi.org/10.1017/jfm.2016.296.
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Direct numerical simulations (DNS) of temporally evolving shear layers have been performed to study the entrainment of irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). Four cases with convective Mach number from 0.2 to 1.8 are used. Entrainment is studied via two mechanisms; nibbling, considered as vorticity diffusion across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. The mass flow rate due to nibbling is calculated as the product of the entrained mass flux with the surface area of the TNTI. It is found that increasing the convective Mach number results in a decrease of the average entrained mass flux and the surface area of the TNTI. For the incompressible shear layer the local entrained mass flux is shown to be highly correlated with the viscous terms. However, as the convective Mach number increases, the mass fluxes due to the baroclinic and the dilatation terms play a more important role in the local entrainment process. It is observed that the entrained mass flux is dependent on the local dilatation and geometrical shape of the TNTI. For a compressible shear layer, most of the entrainment of the irrotational flow into the turbulent region due to nibbling is associated with the compressed regions on the TNTI. As the convective Mach number increases, the percentage of the compressed regions on the TNTI decreases, resulting in a reduction of the average entrained mass flux. It is also shown that the local shape of the interface, looking from the turbulent region, is dominated by concave shaped surfaces with radii of curvature of the order of the Taylor length scale. The average entrained mass flux is found to be larger on highly curved concave shaped surfaces regardless of the level of dilatation. The mass fluxes due to vortex stretching, baroclinic torque and the shear stress/density gradient terms are weak functions of the local curvatures on the TNTI, whereas the mass fluxes due to dilatation and viscous diffusion plus the viscous dissipation terms have a stronger dependency on the local curvatures. As the convective Mach number increases, the probability of finding highly curved concave shaped surfaces on the TNTI decreases, whereas the probability of finding flatter concave and convex shaped surfaces increases. This results in a decrease of the average entrained mass flux across the TNTI. Similar to the previous works on jets, the results show that the contribution of the engulfment to the total entrainment is small for both incompressible and compressible mixing layers. It is also shown that increasing the convective Mach number decreases the velocities associated with the entrainment, i.e. induced velocity, boundary entrainment velocity and local entrainment velocity.
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Dupont, P., S. Piponniau, and J. P. Dussauge. "Compressible mixing layer in shock-induced separation." Journal of Fluid Mechanics 863 (January 28, 2019): 620–43. http://dx.doi.org/10.1017/jfm.2018.987.
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Unsteadiness in separated shock–boundary layer interactions have been previously analysed in order to propose a scenario of entrainment–discharge as the origin of unsteadiness. It was assumed that the fluid in the separated zone is entrained by the free shear layer formed at its edge, and that this layer follows the properties of the canonical mixing layer. This last point is addressed by reanalysing the velocity measurements in an oblique shock reflection at a nominal Mach number of 2.3 and for two cases of flow deviation ($8^{\circ }$ and $9.5^{\circ }$). The rate of spatial growth of this layer is evaluated from the spatial growth of the turbulent stress profiles. Moreover, the entrainment velocity at the edge of the layer is determined from the mean velocity profiles. It is shown that the values of turbulent shear stress, spreading rate and entrainment velocity are consistent, and that they follow the classical laws for turbulent transport in compressible shear layers. Moreover, the measurements suggest that the vertical normal stress is sensitive to compressibility, so that the anisotropy of turbulence is affected by high Mach numbers. Dimensional considerations proposed by Brown & Roshko (J. Fluid Mech., vol. 64, 1974, 775–781) are reformulated to explain this observed trend.
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Jahanbakhshi, Reza, and Cyrus K. Madnia. "The effect of heat release on the entrainment in a turbulent mixing layer." Journal of Fluid Mechanics 844 (April 3, 2018): 92–126. http://dx.doi.org/10.1017/jfm.2018.122.
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Direct numerical simulations of a temporally evolving compressible reacting mixing layer have been performed to study the entrainment of the irrotational flow into the turbulent region across the turbulent/non-turbulent interface (TNTI). In order to study the effects of heat release and interaction of the flame with the TNTI on turbulence several cases with different heat release levels, $Q$, and stoichiometric mixture fractions are chosen for the simulations with the highest opted value for $Q$ corresponding to hydrogen combustion in air. The combustion is mimicked by a one-step irreversible global reaction, and infinitely fast chemistry approximation is used to compute the species mass fractions. Entrainment is studied via two mechanisms: nibbling, considered as the vorticity transport across the TNTI, and engulfment, the drawing of the pockets of the outside irrotational fluid into the turbulent region. As the level of heat release increases, the total entrained mass flow rate into the mixing layer decreases. In a reacting mixing layer by increasing the heat release rate, the mass flow rate due to nibbling is shown to decrease mostly due to a reduction of the local entrainment velocity, while the surface area of the TNTI does not change significantly. It is also observed that nibbling is a viscous dominated mechanism in non-reacting flows, whereas it is mostly carried out by inviscid terms in reacting flows with high level of heat release. The contribution of the engulfment to entrainment is small for the non-reacting mixing layers, while mass flow rate due to engulfment can constitute close to 40 % of the total entrainment in reacting cases. This increase is primarily related to a decrease of entrained mass flow rate due to nibbling, while the entrained mass flow rate due to engulfment does not change significantly in reacting cases. It is shown that the total entrained mass flow rate in reacting and non-reacting compressible mixing layers can be estimated from an expression containing the convective Mach number and the density change due to heat release.
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Glezer, Ari, and Donald Coles. "An experimental study of a turbulent vortex ring." Journal of Fluid Mechanics 211 (February 1990): 243–83. http://dx.doi.org/10.1017/s0022112090001562.
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A turbulent vortex ring having a relatively thin core is formed in water by a momentary jet discharge from an orifice in a submerged plate. The necessary impulse is provided by a pressurized reservoir and is controlled by a fast programmable solenoid valve. The main aim of the research is to verify the similarity properties of the mean flow, as defined by ensemble averaging, and to find the distribution of mean vorticity, turbulent energy, and other quantities in the appropriate non-steady similarity coordinates. The velocity field of the vortex is measured for numerous realizations with the aid of a two-channel tracking laser-Doppler velocimeter. The problem of dispersion in the trajectories of the individual rings is overcome by development of a signature-recognition technique in two variables. It is found that the turbulence intensity is largest near the vortex core and that at least the radial component is not negligible in the near wake. The slow growth of the ring structure is controlled by a slight excess of entrainment over de-entrainment. An important inference is that the growth process and the process of turbulence production probably involve secondary vortices wrapped around the core in azimuthal planes.
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KAMINSKI, EDOUARD, STEPHEN TAIT, and GUILLAUME CARAZZO. "Turbulent entrainment in jets with arbitrary buoyancy." Journal of Fluid Mechanics 526 (March 10, 2005): 361–76. http://dx.doi.org/10.1017/s0022112004003209.
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Mellado, Juan Pedro. "Turbulent Entrainment in the Atmospheric Boundary Layer." PAMM 14, no. 1 (December 2014): 651–52. http://dx.doi.org/10.1002/pamm.201410309.
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Jacobson, M. R., and F. Y. Testik. "Turbulent entrainment into fluid mud gravity currents." Environmental Fluid Mechanics 14, no. 2 (February 27, 2014): 541–63. http://dx.doi.org/10.1007/s10652-014-9344-5.
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Breidenthal, Robert E. "The effect of acceleration on turbulent entrainment." Physica Scripta T132 (December 2008): 014001. http://dx.doi.org/10.1088/0031-8949/2008/t132/014001.
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Zhou, Y., and J. C. Vassilicos. "Related self-similar statistics of the turbulent/non-turbulent interface and the turbulence dissipation." Journal of Fluid Mechanics 821 (May 25, 2017): 440–57. http://dx.doi.org/10.1017/jfm.2017.262.
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The scalings of the local entrainment velocity$v_{n}$of the turbulent/non-turbulent interface and of the turbulence dissipation rate are closely related to each other in an axisymmetric and self-similar turbulent wake. The turbulence dissipation scaling implied by the Kolmogorov equilibrium cascade phenomenology is consistent with a Kolmogorov scaling of$v_{n}$whereas the non-equilibrium dissipation scaling reported for various turbulent flows in Vassilicos (Annu. Rev. Fluid Mech., vol. 47, 2015, pp. 95–114), Dairayet al.(J. Fluid Mech., vol. 781, 2015, pp. 166–195), Goto & Vassilicos (Phys. Lett. A, vol. 379 (16), 2015, pp. 1144–1148) and Obligadoet al.(Phys. Rev. Fluids, vol. 1 (4), 2016, 044409) is consistent with a different scaling of $v_{n}$. We present results from a direct numerical simulation of a spatially developing axisymmetric and self-similar turbulent wake which supports this conclusion and the assumptions that it is based on.
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Lopes, Pedro, Rita F. Carvalho, and Jorge Leandro. "Numerical and experimental study of the fundamental flow characteristics of a 3D gully box under drainage." Water Science and Technology 75, no. 9 (February 22, 2017): 2204–15. http://dx.doi.org/10.2166/wst.2017.071.
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Numerical studies regarding the influence of entrapped air on the hydraulic performance of gullies are nonexistent. This is due to the lack of a model that simulates the air-entrainment phenomena and consequently the entrapped air. In this work, we used experimental data to validate an air-entrainment model that uses a Volume-of-Fluid based method to detect the interface and the Shear-stress transport k-ω turbulence model. The air is detected in a sub-grid scale, generated by a source term and transported using a slip velocity formulation. Results are shown in terms of free-surface elevation, velocity profiles, turbulent kinetic energy and discharge coefficients. The air-entrainment model allied to the turbulence model showed a good accuracy in the prediction of the zones of the gully where the air is more concentrated.
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Turner, J. S. "Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows." Journal of Fluid Mechanics 173 (December 1986): 431–71. http://dx.doi.org/10.1017/s0022112086001222.
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The entrainment assumption, relating the inflow velocity to the local mean velocity of a turbulent flow, has been used successfully to describe natural phenomena over a wide range of scales. Its first application was to plumes rising in stably stratified surroundings, and it has been extended to inclined plumes (gravity currents) and related problems by adding the effect of buoyancy forces, which inhibit mixing across a density interface. More recently, the influence of viscosity differences between a turbulent flow and its surroundings has been studied. This paper surveys the background theory and the laboratory experiments that have been used to understand and quantify each of these phenomena, and discusses their applications in the atmosphere, the ocean and various geological contexts.
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Carlier, Johan, and Kodjovi Sodjavi. "Turbulent mixing and entrainment in a stratified horizontal plane shear layer: joint velocity–temperature analysis of experimental data." Journal of Fluid Mechanics 806 (October 10, 2016): 542–79. http://dx.doi.org/10.1017/jfm.2016.592.
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Buoyancy effects on the turbulent mixing and entrainment processes were analysed in the case of a stratified plane shear layer between two horizontal air flows in conditions leading to relatively low values of the flux Richardson number ($|Ri_{f}|_{max}\simeq 0.02$). Velocity and temperature measurements were made with a single$\times$-wire probe thermo-anemometry technique, using multi-overheat sequences to deliver simultaneous velocity–temperature data at high frequency. The spatial resolution was found to be fine enough, in relation to the dissipative scale and the thermal diffusive scale, to avoid false mixing enhancement in the analysis of the physical mechanisms through velocity–temperature coupling in statistical turbulence quantities. Probability density functions (PDFs) and joint probability density functions (JPDFs) were used to distinguish between the different mechanisms involved in turbulent mixing, namely entrainment, engulfing, nibbling and mixing, and point to the contribution of entrainment in the mixing process. When comparing an unstably stratified configuration to its stably stratified equivalent, no significant difference could be seen in the PDF and JPDF quantities, but a conditional analysis based on temperature thresholding enabled a separation between mixed fluid and two distinct sets of events associated with unmixed fluid entrained from the hot and cold sides of the mixing layer into the mixing layer. This separation allowed a direct calculation of the entrainment velocities on both sides of the mixing layer. A significant increase of the total entrainment could be seen in the case of unstably stratified configuration. The entrainment ratios were compared to their prediction by the Dimotakis model and both a rather good relevance of the model and some need for improvement were found from the comparison. It was hypothesised that the improvement should come from better taking into account the distinct contributions of nibbling and engulfing inside the process of entrainment and mixing.
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Kirshbaum, Daniel J. "Numerical Simulations of Orographic Convection across Multiple Gray Zones." Journal of the Atmospheric Sciences 77, no. 10 (October 1, 2020): 3301–20. http://dx.doi.org/10.1175/jas-d-20-0035.1.
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AbstractIdealized simulations are used to determine the sensitivity of moist orographic convection to horizontal grid spacing Δh. In simulated mechanically (MECH) and thermally (THERM) forced convection over an isolated ridge, Δh is varied systematically over both the deep-convection (Δh ~ 10–1 km) and turbulence (Δh ~ 1 km–100 m) gray zones. To aid physical interpretation, a new parcel-based bulk entrainment/detrainment diagnosis for horizontally heterogeneous flows is developed. Within the deep-convection gray zone, the Δh sensitivity is dominated by differences in parameterized versus explicit convection; the former initiates convection too far upstream of the ridge (MECH) and too early in the diurnal heating cycle (THERM). These errors stem in part from a large underprediction of parameterized entrainment and detrainment. Within the turbulence gray zone, sensitivities to Δh arise from the representation of both subcloud- and cloud-layer turbulence. As Δh is decreased, MECH exhibits stronger cloud-layer entrainment to enhance the convective mass flux Mco, while THERM exhibits stronger detrainment to suppress Mco and delay convection initiation. The latter is reinforced by increased subcloud turbulence at smaller Δh, which leads to drying and diffusion of the central updraft responsible for initiating moist convection. Numerical convergence to a robust solution occurs only in THERM, which develops a fully turbulent flow with a resolved inertial subrange (for Δh ≤ 250 m). In MECH, by contrast, turbulent transition occurs within the orographic cloud, the details of which depend on both physical location and Δh.
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Matulka, A., P. López, J. M. Redondo, and A. Tarquis. "On the entrainment coefficient in a forced plume: quantitative effects of source parameters." Nonlinear Processes in Geophysics 21, no. 1 (February 24, 2014): 269–78. http://dx.doi.org/10.5194/npg-21-269-2014.
Full textAbstract:
Abstract. The behavior of a forced plume is mainly controlled by the source buoyancy and momentum fluxes and the efficiency of turbulent mixing between the plume and the ambient fluid (stratified or not). The interaction between the plume and the ambient fluid controls the plume dynamics and is usually represented by the entrainment coefficient αE. Commonly used one-dimensional models incorporating a constant entrainment coefficient are fundamental and very useful for predictions in geophysical flows and industrial situations. Nevertheless, if the basic geometry of the flow changes, or the type of source or the environmental fluid conditions (e.g., level of turbulence, shear, ambient stratification, presence of internal waves), new models allowing for variable entrainment are necessary. The presented paper is an experimental study based on a set of turbulent plume experiments in a calm unstratified ambient fluid under different source conditions (represented by different buoyancy and momentum fluxes). The main result is that the entrainment coefficient is not a constant and clearly varies in time within the same plume independently of the buoyancy and the source position. This paper also analyzes the influence of the source conditions on the mentioned time evolution. The measured entrainment coefficient αE has considerable variability. It ranges between 0.26 and 0.9 for variable Atwood number experiments and between 0.16 and 0.55 for variable source position experiments. As is observed, values are greater than the traditional standard value of Morton et al. (1956) for plumes and jets, which is about 0.13.