Mean Dissipation Rate Research Articles

On local isotropy and scale dependence of pair dispersion in turbulent canopy flows

Canopy flows in the atmospheric surface layer play important economic and ecological roles, governing the dispersion of passive scalars in the environment. The interaction of high-velocity fluid and large-scale surface-mounted obstacles in canopy flows produces drag and causes intense, inhomogeneous and anisotropic turbulence. In this work, we focus on the turbulent dispersion of passive scalars by studying the ‘pair dispersion’ – a statistical measure of relative motion between particles. We analyse the results of a three-dimensional particle tracking velocimetry experiment in a wind-tunnel canopy flow, focusing on small scales. We confirm the existence of local isotropy of pair dispersion at scales smaller than a characteristic shear length scale $L_\varGamma =(\epsilon /\varGamma ^3)^{1/2}$ , where $\epsilon$ and $\varGamma$ are the mean dissipation rate and shear rate, respectively. Furthermore, we show that pair dispersion in this locally isotropic regime is a scale-dependent super-diffusive process, similar to what occurs in homogeneous isotropic turbulent flows. In addition, we measure the pair relative velocity correlation function, showing that its de-correlation occurs in the locally isotropic regime, and discuss the implications of this observation for modelling pair dispersion. Thus, our study extends the fundamental understanding of turbulent pair dispersion to the anisotropic inhomogeneous turbulent canopy flow, bringing valuable information for modelling scalar dispersion in the atmospheric surface layer.

Proof that all dissipation rates are only functions of time for transported joint-normal distributions

It has previously been proven that the conditional dissipation rate to transport a Gaussian distribution is equal to the mean dissipation rate throughout the variables' space and that only a Gaussian distribution can have a conditional dissipation rate that is only a function of time. This article extends both proofs to a joint-normal distribution for any number of dimensions.

Local energy transfer and dissipation in incompressible Hall magnetohydrodynamic turbulence: The coarse-graining approach.

We derive the coarse-graining (CG) equationsof incompressible Hall magnetohydrodynamic (HMHD) turbulence to investigate the local (in space) energy transfer rate as a function of the filtering scale ℓ. First, the CG equationsare space averaged to obtain the analytical expression of the mean cascade rate. Its application to three-dimensional simulations of (weakly compressible) HMHD shows a cascade rate consistent with the value of the mean dissipation rate in the simulations and with the classical estimates based on the "third-order" law. Furthermore, we developed an anisotropic version of CG that allows us to study the magnitude of the cascade rate along different directions with respect to the mean magnetic field. Its implementation on the numerical data with moderate background magnetic field shows a weaker cascade along the magnetic field than in the perpendicular plane, while an isotropic cascade is recovered in the absence of a background field. The strength of the CG approach is further revealed when considering the local-in-space energy transfer, which is shown theoretically and numerically to match at a given position x, when locally averaged over a neighboring region, the (quasi-)local dissipation. Prospects of exploiting this model to investigate local dissipation in spacecraft data are discussed.

Effects of low clay concentrations on nearly isotropic turbulence

Aqueous flows containing clay in suspension are ubiquitous in natural environments and play a pivotal role in many processes. Using a novel approach, we experimentally quantified the effect of low clay concentrations in water-clay flows under nearly isotropic turbulence. The results demonstrate the significant impact of clay on the multiscale dynamics of turbulence even at low clay concentrations, including the turbulent kinetic energy, the mean dissipation rate, the Taylor microscale, and the inertial subrange of the velocity spectrum.

Interplay of periodic dynamics and noise: Insights from a simple adaptive system.

We study the dynamics of a simple adaptive system in the presence of noise and periodic damping. The system is composed by two paths connecting a source and a sink, and the dynamics is governed by equations that usually describe food search of the paradigmatic Physarum polycephalum. In this work we assume that the two paths undergo damping whose relative strength is periodically modulated in time, and we analyze the dynamics in the presence of stochastic forces simulating Gaussian noise. We identify different responses depending on the modulation frequency and on the noise amplitude. At frequencies smaller than the mean dissipation rate, the system tends to switch to the path which minimizes dissipation. Synchronous switching occurs at an optimal noise amplitude which depends on the modulation frequency. This behavior disappears at larger frequencies, where the dynamics can be described by the time-averaged equations. Here we find metastable patterns that exhibit the features of noise-induced resonances.

Observation of internal tides, nonlinear internal waves and mixing in the Lombok Strait, Indonesia

Lombok Strait is one of the main exit passages of the Indonesian throughflow (ITF), connecting the interior Indonesian seas with the Indian Ocean. Satellite images have shown that this strait is also the most remarkable zone of the Indonesian seas for the generation of internal tides, which frequently evolve into internal solitary waves (ISWs). Here we present the first comprehensive high-frequency observations of the internal tide in the strait. The observations were carried out at a fixed position around 8 km from the western coast of Lombok Island at a depth of ~300 m. We confirmed the appearance of the non-linear internal tide with an amplitude of ~90 m at a fairly close distance (~30 km) from its generation site in the Nusa Penida Sills (NPS), with a dominant M2 tidal frequency. High-frequency internal waves associated with vertical velocities reaching 20 cm s-1 were observed within the internal tidal trough. We estimated the turbulent kinetic energy dissipation rate using Thorpe and Ellison scales and found a mean dissipation rate of the order of ~ [10-6-10-7] m2 s-3, with enhanced values in the thermocline layer, possibly driven by shear instabilities. We underlined that the use of the Ellison scale is a promising alternative for indirectly estimating the dissipation rate from high-resolution time-series data sets.

Gravity Wave Breaking and Vortex Ring Formation Observed by PMC Turbo

AbstractPolar mesospheric cloud (PMC) imaging and lidar profiling performed aboard the 5.9‐day PMC Turbo balloon flight from Sweden to northern Canada in July 2018 revealed a wide variety of gravity wave (GW) and instability events occurring nearly continuously at approximately 82 km. We describe one event exhibiting GW breaking and associated vortex rings driven by apparent convective instability. Using PMC Turbo imaging with spatial and temporal resolution of 20 m and 2 s, respectively, we quantify the GW horizontal wavelength, propagation direction, and apparent phase speed. We identify vortex rings with diameters of 2–5 km and horizontal spacing comparable to their size. Lidar data show GW vertical displacements of ±0.3 km. From the data, we find a GW intrinsic frequency and vertical wavelength of 0.009 ± 0.003 rad s−1 and 9 ± 4 km, respectively. We show that these values are consistent with the predictions of numerical simulations of idealized GW breaking. We estimate the momentum deposition rate per unit mass during this event to be 0.04 ± 0.02 m s−2 and show that this value is consistent with the observed GW. Comparison to simulation gives a mean energy dissipation rate for this event of 0.05–0.4 W kg−1, which is consistent with other reported in situ measurements at the Arctic summer mesopause.

Extreme dissipation and intermittency in turbulence at very high Reynolds numbers.

Extreme dissipation events in turbulent flows are rare, but they can be orders of magnitude stronger than the mean dissipation rate. Despite its importance in many small-scale physical processes, there is presently no accurate theory or model for predicting the extrema as a function of the Reynolds number. Here, we introduce a new model for the dissipation probability density function (PDF) based on the concept of significant shear layers, which are thin regions of elevated local mean dissipation. At very high Reynolds numbers, these significant shear layers develop layered substructures. The flow domain is divided into the different layer regions and a background region, each with their own PDF of dissipation. The volume-weighted regional PDFs are combined to obtain the overall PDF, which is subsequently used to determine the dissipation variance and maximum. The model yields Reynolds number scalings for the dissipation maximum and variance, which are in agreement with the available data. Moreover, the power law scaling exponent is found to increase gradually with the Reynolds numbers, which is also consistent with the data. The increasing exponent is shown to have profound implications for turbulence at atmospheric and astrophysical Reynolds numbers. The present results strongly suggest that intermittent significant shear layer structures are key to understanding and quantifying the dissipation extremes, and, more generally, extreme velocity gradients.

Swimming of a uniform deformable sphere in a viscous incompressible fluid with inertia

The swimming of a deformable uniform sphere is studied in second order perturbation theory in the amplitude of the stroke. The effect of the first order reaction force on the first order center of mass velocity is calculated in linear response theory by use of Newton’s equation of motion. The response is characterized by a dipolar admittance, which is shown to be proportional to the translational admittance. As a consequence the mean swimming velocity, calculated in second order perturbation theory, depends on the added mass of the sphere. The mean swimming velocity and the mean rate of dissipation are calculated for several selected strokes.

Hydraulics and Mixing of the Deep Overflow in the Lifamatola Passage of the Indonesian Seas

AbstractHydrographic measurements recently acquired along the thalweg of the Lifamatola Passage combined with historical moored velocity measurements immediately downstream of the sill are used to study the hydraulics, transport, mixing, and entrainment in the dense overflow. The observations suggest that the mean overflow is nearly critical at the mooring site, suggesting that a weir formula may be appropriate for estimating the overflow transport. Our assessment suggests that the weir formulas corresponding to a rectangular, triangular, or parabolic cross section all result in transports very close to the observation, suggesting their potential usage in long-term monitoring of the overflow transport or parameterizing the transport in numerical models. Analyses also suggest that deep signals within the overflow layer are blocked by the shear flow from propagating upstream, whereas the shallow wave modes of the full-depth continuously stratified flow are able to propagate upstream from the Banda Sea into the Maluku Sea. Strong mixing is found immediately downstream of the sill crest, with Thorpe-scale-based estimates of the mean dissipation rate within the overflow up to 1.1 × 10−7 W kg−1 and the region-averaged diapycnal diffusivity within the downstream overflow in the range of 2.3 × 10−3 to 10.1 × 10−3 m2 s−1. Mixing in the Lifamatola Passage results in 0.6–1.2-Sv (1 Sv ≡ 106 m3 s−1) entrainment transport added to the overflow, enhancing the deep-water renewal in the Banda Sea. A bulk diffusivity coefficient estimated in the deep Banda Sea yields 1.6 × 10−3 ± 5 × 10−4 m2 s−1, with an associated downward turbulent heat flux of 9 W m−2.

Computational fluid dynamics analysis of mixing and gas-liquid mass transfer in wave bag bioreactor.

Single use bioreactors provide an attractive alternative to traditional deep-tank stainless steel bioreactors in process development and more recently manufacturing process. Wave bag bioreactors, in particular, have shown potential applications for cultivation of shear sensitive human and animal cells. However, the lack of knowledge about the complex fluid flow environment prevailing in wave bag bioreactors has so far hampered the development of a scientific rationale for their scale up. In this study, we use computational fluid dynamics (CFD) to investigate the details of the flow field in a 20-L wave bag bioreactor as a function of rocking angle and rocking speed. The results are presented in terms of local and mean velocities, mixing, and energy dissipation rates, which are used to create a process engineering framework for the scale-up of wave bag bioreactors. Proof-of-concept analysis of mixing and fluid flow in the 20-L wave bag bioreactor demonstrates the applicability of the CFD methodology and the temporal and spatial energy dissipation rates integrated and averaged over the liquid volume in the bag provide the means to correlate experimental volumetric oxygen transfer rates (kL a) data with power per unit volume. This correlation could be used as a rule of thumb for scaling up and down the wave bag bioreactors.

Estimation of mean turbulent kinetic energy and temperature variance dissipation rates using a spectral chart method

A method aimed at estimating εk and εθ, respectively, the mean dissipation rates of turbulent kinetic energy k and half the temperature variance θ2/2, is developed for slightly heated turbulent flows of air. It is limited to a Prandtl number near unity and applicable to flows where temperature can be treated as a passive scalar. A significant advantage of the method is that εk and εθ can both be estimated from the measurement of a temperature frequency spectrum, Gθθ(f). The method relies on the collapse in the dissipative range of one-dimensional temperature spectra, ϕθ(k1η), when normalized with εθ, εk, and ν. This collapse ensues from a similarity analysis of scale-by-scale budgets of the second-order structure function for the temperature. A generic spectrum ϕθG(k1η), defined in the wavenumber range 0.07 ≤ k1η ≤ 0.7, is used to construct a spectral chart. The method has been tested in several flows and found to be reliable. In particular, it is tested on the axis of a slightly heated round jet, where εk and εθ can be estimated accurately via the budgets of k and θ2/2, and the agreement between these estimates and the spectral chart results is almost perfect.

Extending the concept of machine acceleration used to model the bubble-particle detachment in flotation. Part 2 – Machine acceleration of solid particles in water

The machine acceleration introduced by H. J. Schulze has been a cornerstone of the bubble-particle (BP) detachment theory. It connects the mean dissipation rate of turbulence energy of water in a flotation machine with the BP detachment. In a recent paper, we scrutinized the theory and recommended its extension to cover both the longitudinal (centrifugation) and transverse (shear) effects of water turbulence. Since there is a significant difference in the densities of water, solid particles, and air bubbles, it is unknown if the available determination of the machine acceleration using the “water particles” is a good approximation for the bubble-particle aggregates. This paper aims to provide an answer to this fundamental question. For simplicity, this paper focuses on the acceleration of solid particles in water. Specifically, we use the Basset-Boussinesq-Oseen equation to quantify the particle acceleration. The correlation method in combination with the isotropic turbulence theory is applied to predict the particle acceleration components for the full range of turbulence length scale. Comparing with Schulze’s model, our recent model shows that the particle inertia should be considered in calculating the machine acceleration, especially for large and/or heavy particles and high energy dissipation rate. Our numerical results show that the machine acceleration decreases with increasing particle size and density and increases with increasing dissipation rate of turbulent kinetic energy and turbulence intensity.

Shallow water wave turbulence

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

An empirical expression for on the axis of a slightly heated turbulent round jet

Self-preservation analyses of the equations for the mean temperature and the second-order temperature structure function on the axis of a slightly heated turbulent round jet are exploited in an attempt to develop an analytical expression for$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$, the mean dissipation rate of$\overline{\unicode[STIX]{x1D703}^{2}}/2$, where$\overline{\unicode[STIX]{x1D703}^{2}}$is the temperature variance. The analytical approach follows that of Thiessetet al.(J. Fluid Mech., vol. 748, 2014, R2) who developed an expression for$\unicode[STIX]{x1D716}_{k}$, the mean turbulent kinetic energy dissipation rate, using the transport equation for$\overline{(\unicode[STIX]{x1D6FF}u)^{2}}$, the second-order velocity structure function. Experimental data show that complete self-preservation for all scales of motion is very well satisfied along the jet axis for streamwise distances larger than approximately 30 times the nozzle diameter. This validation of the analytical results is of particular interest as it provides justification and confidence in the analytical derivation of power laws representing the streamwise evolution of different physical quantities along the axis, such as:$\unicode[STIX]{x1D702}$,$\unicode[STIX]{x1D706}$,$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$,$R_{U}$,$R_{\unicode[STIX]{x1D6E9}}$(all representing characteristic length scales), the mean temperature excess$\unicode[STIX]{x1D6E9}_{0}$, the mixed velocity–temperature moments$\overline{u\unicode[STIX]{x1D703}^{2}}$,$\overline{v\unicode[STIX]{x1D703}^{2}}$and$\overline{\unicode[STIX]{x1D703}^{2}}$and$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Simple models are proposed for$\overline{u\unicode[STIX]{x1D703}^{2}}$and$\overline{v\unicode[STIX]{x1D703}^{2}}$in order to derive an analytical expression for$A_{\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}}$, the prefactor of the power law describing the streamwise evolution of$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}$. Further, expressions are also derived for the turbulent Péclet number and the thermal-to-mechanical time scale ratio. These expressions involve global parameters that are most likely to be influenced by the initial and/or boundary conditions and are therefore expected to be flow dependent.

Influence of Impeller Geometry on Hydromechanical Stress in Stirred Liquid/Liquid Dispersions

Hydromechanical stress is a crucial parameter for a broad range of multiphase processes in the field of (bio)chemical engineering. The effect of impeller type and geometry on hydromechanical stress in stirred tanks is important. The present study aims at characterizing conventional and new impeller types in terms of particle stress. A two-phase liquid/liquid noncoalescing dispersion system is employed, and the drop breakage is monitored in-line in a stirred tank. The published effects of agitation on drop deformation are confirmed and expanded significantly for five modified new impeller types. Radial impellers are advantageous for applications where low shear conditions are desired. A modified propeller with a peripheral ring and the developed wave-ribbon impellers present remarkable results by producing significantly low and high hydromechanical stress, respectively. The results obtained are correlated in terms of mean and maximum energy dissipation rate, as well as circulation frequency in the impeller...

Global energy fluxes in turbulent channels with flow control

This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skin-friction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation). Direct numerical simulations (DNS) of different control strategies are performed at constant power input (CPI), so that at statistical equilibrium, each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow. A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wall-normal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the control-dependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their inter-relations and connection to flow control, so that a clear target for flow control can be identified.

Analysis of the turbulent flow and trailing vortices induced by new design grooved blade impellers in a baffled tank

U or V shaped grooves have shown reduction in energy and power consumption in blunt bodies immersed in turbulent streams. In this work, the performance of two new design turbines with four blades rotated 45°, one with U and the other with V grooved shape blades, were evaluated in a baffled stirred tank reactor and they were compared with the one of the regular 4PBT. The power consumption, mean velocities, turbulent kinetic energy production (TKE), energy dissipation rate (ε) and trailing vortices induced by new design impellers with grooved walls and 4PBT were evaluated experimentally and modeled by detached eddy simulation (DES) approach, which has not been tested in blade walls with small perturbations. Validation of the results was carried out by PIV (Particle image velocimetry) experimental tests. The evolution of new and distinctive trailing vortices located in the upper and lower faces of the grooved new design impellers is presented. They were measured by angled resolved PIV technique, and visualized in three dimensions by the DES method. The V-grooves shaped impeller generated a larger region of influence of the turbulent kinetic energy along the discharge region than the other impellers. Moreover, presented higher mean dissipation rate levels for the most of the measured angles and induced higher number of trailing vortices in the blades vicinity than the other turbines. It was found that in a wide range of the Reynolds number (Re) (40 × 103 < Re < 125 × 103) the dissipated power measured experimentally from shaft, presented reductions of the order of 6% and 4% for the U and V grooved new models, respectively, compared with that of the regular 4PBT impeller.

One-point and two-point statistics of homogeneous isotropic decaying turbulence with variable viscosity

The decay of homogeneous isotropic turbulence in a variable viscosity fluid with a viscosity ratio up to 15 is analyzed by means of highly resolved direct numerical simulations (DNS) at low Reynolds numbers. The question addressed by the present work is how quantities such as the kinetic energy and the associated dissipation rate, as well as the inter-scale transport mechanism of turbulence are changed by local fluctuations of the viscosity. The comparison is performed with respect to the decaying homogeneous isotropic turbulence with constant viscosity (CV), equal to the mean value of the variable viscosity (VV). From the one-point budget equation of the turbulent kinetic energy, it is shown that the mean dissipation rate is nearly unchanged by variable viscosity effects. This result is explained by a negative correlation between the local viscosity and the local velocity gradients. However, the dissipation is a highly fluctuating quantity with a strong level of intermittency. From a statistical analysis it is shown that turbulent flows with variable viscosity are characterized by an enhanced level of small-scale intermittency with respect to CV flow, which results in the presence of smaller length scales. The effect of variable viscosity on the turbulent cascade is analyzed by a budget equation for the velocity structure function. From DNS it is shown that viscosity gradients contribute to the inter-scale transport mechanism in the form of an inverse transport, where information propagates from the small scales to the large scales.

Universality of local dissipation scales in turbulent boundary layer flows with and without free-stream turbulence

Measurements of the small-scale dissipation statistics of turbulent boundary layer flows with and without free-stream turbulence are reported for Reτ ≈ 1000 (Reθ ≈ 2000). The scaling of the dissipation scale distribution is examined in these two boundary conditions. Results demonstrated that the local large-scale Reynolds number based on the measured longitudinal integral length scale fails to properly normalize the dissipation scale distribution near the wall in these two free-stream conditions due to the imperfect characterization of the upper bound of the inertial cascade by the integral length scale. A surrogate found from turbulent kinetic energy and mean dissipation rate only moderately improved the scaling of the dissipation scales, relative to the measured integral length scale. When a length scale based on the distance from the wall [as suggested by Bailey and Witte, “On the universality of local dissipation scales in turbulent channel flow,” J. Fluid Mech. 786, 234–252 (2015)] was utilized to scale the dissipation scale distribution, in the region near the wall, there was a noticeable improvement in the collapse of the normalized distribution of dissipation scales. In addition, unlike in channel flows, in the outer layer of the turbulent boundary layer, the normalized distributions of the local dissipation scales were observed to be dependent on the wall-normal position. This was found to be attributable to the presence of external intermittency in the outer layer as the presence of free-stream turbulence was found to restore the scaling behavior by replacing the intermittent laminar flow with turbulent flow.