Mean Energy Dissipation Rate Research Articles

Horizontal velocity structure functions in the upper troposphere and lower stratosphere: 1. Observations

We compute horizontal velocity structure functions using quasiglobal data accumulated by specially equipped commercial aircraft on 7630 flights from August 1994 to December 1997. Using the ozone concentration measurements, we classify the results as tropospheric or stratospheric. We further divide the results into four absolute latitude bands. For separation distance r between 10 and 100 km, the lower stratospheric diagonal third‐order structure functions are proportional to negative r. This implies a downscale energy cascade, and we estimate the mean energy dissipation rate to be 〈ε〉 ≈ 6 × 10−5 m2 s−3. For r between 300 and 1500 km, a positive r3 dependence was visible for the polar stratospheric data. This may be the result of a two‐dimensional (2D) turbulence downscale enstrophy cascade, and we estimate the average enstrophy flux to be Πω ≈ 2 × 10−15 s−3 and the energy spectral constant to be κ ≈ 2. The negative sign of these third‐order functions at mesoscales in both the upper troposphere and lower stratosphere provide no support for an inverse energy cascade 2D turbulence. At scales above ∼100 km, the second‐order structure functions increase with latitude in the troposphere and decrease with latitude in the stratosphere. The off‐diagonal third‐order functions in the stratosphere show a remarkably clean negative r2 dependency from 10 to 1000 km in scale.

Turbulent energy scale budget equations in a fully developed channel flow

Kolmogorov's equation, which relates second- and third-order moments of the velocity increment, provides a simple method for estimating the mean energy dissipation rate 〈ε〉 for homogeneous and isotropic turbulence. However, this equation is usually not verified in small to moderate Reynolds number flows. This is due partly to the lack of isotropy in either sheared or non-sheared flows, and, more importantly, to the influence, which is flow specific, of the inhomogeneous and anisotropic large scales. These shortcomings are examined in the context of the central region of a turbulent channel flow. In this case, we propose a generalized form of Kolmogorov's equation, which includes some additional terms reflecting the large-scale turbulent diffusion acting from the walls through to the centreline of the channel. For moderate Reynolds numbers, the mean turbulent energy transferred at a scale r also contains a large-scale contribution, reflecting the non-homogeneity of these scales. There is reasonable agreement between the new equation and hot-wire measurements in the central region of a fully developed channel flow.

Effect of initial conditions on the mean energy dissipation rate and the scaling exponent

Implications of the expectation that the mean energy dissipation rate <epsilon> should become independent of viscosity at sufficiently large values of R(lambda), the Taylor microscale Reynolds number, are examined within the framework of small-scale intermittency and an adequate description of the second-order velocity structure function over the dissipative and inertial ranges. For nominally the same flow, a two-dimensional wake, but with different initial conditions, values of C(epsilon) identical with<epsilon>L/u('3), the scaling exponent and the Kolmogorov constant differ at the same R(lambda).

Scaling of mean temperature dissipation rate

Expressions for Cε and Cεθ, the dimensionless mean energy and temperature dissipation rates, can be reasonably reconciled with measured and numerical data at small to moderate Rλ only if the Kolmogorov constant and more especially the Obukhov–Corrsin constant are assumed to depend on the flow and the Reynolds number.

Transport equations for the mean energy and temperature dissipation rates in grid turbulence

Simultaneous measurements of velocity and temperature fluctuations were made in the turbulent flow downstream of a grid-heated screen combination. The magnitudes of n and m, the power-law exponents for the decay of turbulent energy and temperature variances, are nearly equal. This approximate equality is in conformity with the locations of the peaks in the three-dimensional turbulent energy and temperature spectra. The values of n and m are consistent with the isotropic forms of the transport equations for the mean turbulent energy and temperature variance. They are also consistent, allowing for the measurement accuracy, with the isotropic forms of the equations for the mean turbulent energy and temperature dissipation rates.

Reynolds Number Dependence of Velocity Structure Functions in a Turbulent Pipe Flow

Low-order moments of the increments δu andδv where u and v are the axial and radial velocity fluctuations respectively, have been obtained using single and X-hot wires mainly on the axis of a fully developed pipe flow for different values of the Taylor microscale Reynolds numberR λ. The mean energy dissipation rate〉e〈 was inferred from the uspectrum after the latter was corrected for the spatial resolution of the hot-wire probes. The corrected Kolmogorov-normalized second-order structure functions show a continuous evolution withR λ. In particular, the scaling exponentζ v , corresponding to the v structure function, continues to increase with R λ in contrast to the nearly unchanged value of ζ u . The Kolmogorov constant for δu shows a smaller rate of increase with R λ than that forδv. The level of agreement with local isotropy is examined in the context of the competing influences ofR λ and the mean shear. There is close but not perfect agreement between the present results on the pipe axis and those on the centreline of a fully developed channel flow.

Surface roughness effects in turbulent boundary layers

The effects of surface roughness on a turbulent boundary layer are investigated by comparing measurements over two rough walls with measurements from a smooth wall boundary layer. The two rough surfaces have very different surface geometries although designed to produce the same roughness function, i.e. to have nominally the same effect on the mean velocity profile. Different turbulent transport characteristics are observed for the rough surfaces. Substantial effects on the stresses occur throughout the layer showing that the roughness effects are not confined to the wall region. The turbulent energy production and the turbulent diffusion are significantly different between the two rough surfaces, the diffusion having opposite sign in the region γ/δ < 0.5. Although velocity spectra exhibit differences between the three surfaces, the mean energy dissipation rate does not appear to be significantly affected by the roughness.

Three-component vorticity measurements in a turbulent grid flow

All components of the fluctuating vorticity vector have been measured in decaying grid turbulence using a vorticity probe of relatively simple geometry (four X-probes, i.e. a total of eight hot wires). The data indicate that local isotropy is more closely satisfied than global isotropy, the r.m.s. vorticities being more nearly equal than the r.m.s. velocities. Two checks indicate that the performance of the probe is satisfactory. Firstly, the fully measured mean energy dissipation rate 〈ε〉 is in good agreement with the value inferred from the rate of decay of the mean turbulent energy 〈q2〉 in the quasi-homogeneous region; the isotropic mean energy dissipation rate 〈εiso〉 agrees closely with this value even though individual elements of 〈ε〉 indicate departures from isotropy. Secondly, the measured decay rate of the mean-square vorticity 〈ω2〉 is consistent with that of 〈q2〉 and in reasonable agreement with the isotropic form of the transport equation for 〈ω2〉. Although 〈ε〉≃〈εiso〉, there are discernible differences between the statistics of ε and εiso; in particular, εiso is poorly correlated with either ε or ω2. The behaviour of velocity increments has been examined over a narrow range of separations for which the third-order longitudinal velocity structure function is approximately linear. In this range, transverse velocity increments show larger departures than longitudinal increments from predictions of Kolmogorov (1941). The data indicate that this discrepancy is only partly associated with differences between statistics of locally averaged ε and ω2, the latter remaining more intermittent than the former across this range. It is more likely caused by a departure from isotropy due to the small value of Rλ, the Taylor microscale Reynolds number, in this experiment.

The structure and dynamics of the Bottom Boundary Layer in shallow sea areas without tidal influence: an experimental approach

This report describes extensive investigations of the near bottom layer of the Western Baltic (Mecklenburg Bight, Darss Sill and Arkona Basin) which were conducted over a 5 year period to determine the typical structure, vertical thickness, vertical turbulence structure, and spatial and temporal variability of this water mass with regard to the area's particular hydrographic conditions. Series of vertical profiles were obtained using the microstructure profiler MSS86, which is capable of measuring high resolution profiles of temperature, conductivity, current shear, light attenuation and pressure down to the seafloor. The near bottom current structure was simultaneously measured with conventional current metres at fixed depths. A typical vertical density structure of the near bottom layer was found. At all investigation sites the Bottom Boundary Layer was separated from the overlying water mass by a well pronounced thermohaline pycnocline. A homogeneous water layer was situated above the bottom with a mean thickness of 2.2 m and typical variation between 0.5 and 3.5 m. The thickness of both the homogeneous layer and of the near bottom layer vary considerably. It is suggested that horizontal advection is responsible for these fluctuations in thickness. The variation in thickness of the Homogeneous Layer is independent of the local mean current velocity, wind speed and energy dissipation rate. Over periods of about 2 days the thickness of the Homogeneous Layer is determined by the average wind speed. The Bottom Boundary Layer shows its own characteristic dynamic, which is largely decoupled from that of the remaining water body. A logarithmic layer was generally not resolved by the current measurements. From dissipation rate measurements, the wall layer was determined to be 0.9 m thick. There was no significant correlation between the dissipation rate and the local wind speed, or between the dissipation rate and local mean current u 100. This means that any simple parameterisation relating u 100 or friction velocity to the locally produced turbulence and consequently to the resuspension of sediment is probably not applicable to shallow sea areas with properties like the Western Baltic. The investigation of sediment concentration in the BBL illustrates the importance of local effects combined with advection. The sediment stratified layer covers only the bottom most 50 cm.

Second- and third-order longitudinal velocity structure functions in a fully developed turbulent channel flow

Second- and third-order moments of temporal increments of the longitudinal velocity fluctuation have been measured using hot wires in a fully developed turbulent channel flow and analyzed using values of the mean energy dissipation rate inferred from direct numerical simulation data for the same flow and Reynolds numbers. The results indicate that dissipative range scales approach isotropy relatively rapidly when the mean shear is negligible. For larger inertial range type scales, the approach is slower. The rate at which isotropy is approached seems slower for third-order than second-order structure functions. This rate depends more on the mean shear than on the turbulent Reynolds number. Two-point laser Doppler anemometry measurements provide reasonable support for the use of Taylor’s hypothesis, at least outside the buffer region.

Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring

The mean rates of energy injection and energy dissipation in steady regimes of turbulence are measured in two types of flow confined in closed cells. The first flow is generated by counterrotating stirrers and the second is a Couette-Taylor flow. In these two experiments the solid surfaces that set the fluid into motion are at first smooth, so that everywhere the velocity of the stirrers is locally parallel to its surface. In all such cases the mean rate of energy dissipation does not satisfy the scaling expected from Kolmogorov theory. When blades perpendicular to the motion are added to the stirring surfaces the Kolmogorov scaling is observed in all the large range of Reynolds numbers (10,Re,10) investigated. However, with either smooth or rough stirring the measurements of the pressure fluctuations exhibit no Reynolds number dependence. This demonstrates that, though the smooth stirrers are less efficient in setting the fluid into motion, their efficiency is independent of the Reynolds number so that the Kolmogorov scaling characterizes, in all cases, the dissipation in the bulk of the fluid. The difference in the global behaviors corresponds to a different balance between the role of the different regions of the flow. With smooth stirrers the dissipation in the bulk is weaker than the Reynoldsnumber-dependent dissipation in the boundary layers. With rough ~or inertial! stirrers the dissipation in the bulk dominates, hence the Kolmogorovian global behavior. @S1063-651X~97!11606-3#

The effect of small-scale fluid motion on the green alga Scenedesmus quadricauda

The effect of shear flow on the green alga Scenedesmus quadricauda grown in Bristol's medium wastested. The shear flow was generated using a Couettetype rotating cylinder apparatus. Growth of Scenedesmus quadricauda, measured in terms ofchlorophyll a concentration, was inhibited underdifferent fluid motions. Inhibition was mostpronounced at high Reynolds number (Re) and thecorresponding mean rate of energy dissipation(e). Algal growth was maximum during thestagnant fluid flow experiment. The flocs comprised of dead and living cells of algae formed as a resultof shear flow. Cell morphometry did not changeconsistently under different flow conditions but celldestruction was evident. A two step model isproposed, relating algal growth as a function of Re,and e. The attenuation factor, φF for growth limiting conditions underdifferent fluid motions, was defined as the ratio of the algal growth rate constant to the maximum algalgrowth constant under stagnant fluid flowconditions.

Development and Mixing Characteristics of Foldiny Anchor for Round-Bottomed Flask.

A folding anchor impeller was developed for a round-bottomed flask with a small neck, which is frequently used for research and development of new materials and technologies in the laboratories of industries and universities. The power consumption and mixing time were measured over a wide range of Reynolds number and compared with those obtained for two traditional types of impellers for relatively high viscous liquids in a round-bottomed flask and those calculated for a helical ribbon impeller set in a cylindrical vessel. It was found that the power correlation developed for paddle impellers in a spherical agitated vessel was also applicable for data obtained for the folding anchor impeller. The power consumption for the folding anchor impeller was much higher than those of two traditional impellers, but almost the same with that for a double helical ribbon impeller. On the other hand, the mixing time for the folding anchor impeller was less than one-fifth those obtained for two traditional impellers, and it was comparable to that calculated for a helical ribbon impeller according to the correlation. Consequently, the folding anchor impeller needs the lowest mean energy dissipation rate to attain the same mixing time among the impellers considered in this work.

Turbulent breakage of filamentous microorganisms in submerged culture in mechanically stirred bioreactors

A discussion is presented suggesting that hyphal breakage in mechanically stirred fermenters is likely to occur by direct tensile stresses acting on the opposite ends of the hypha of filamentous microorganisms. These stresses originate from the dynamic pressure fluctuations of eddies with scales in the inertial convection subrange of the turbulent energy spectrum. The maximum strain energy theory of failure of an elastic material under this mode of stress is used to set up a relationship between the stable length of hyphae and some of the physical parameters affecting it, including the mean energy dissipation rate in the bioreactor. Experimental data are reported for the rate of breakage of hyphae obtained under different operating conditions during the fermentation of Penicillium chrysogenum in a 7 and 150 1 mechanically stirred bioreactor. Data on the stable size of hyphae agree well with the model based on the maximum energy criterion. The experimental data further suggest that hyphal breakage is approximately a first-order kinetic process with a rate constant which appears to be moderately dependent on the product of the mean energy dissipation rate and the reciprocal of the impeller circulation time.

Prediction of mechanical damage to animal cells in turbulence.

In previous work a model was proposed for estimation of disruption of animal cells in turbulent capillary flows using information about the hydrodynamics, and cell mechanical properties determined by micromanipulation. The model assumed that the capillary flow consists of a laminar sublayer and a homogeneous turbulent region, and within the latter eddies of sizes similar to or smaller than the cells interact with those cells, causing local surface deformations. The proposed mechanism of cell damage was that such deformations result in an increase in membrane tension and surface energy, and that a cell disrupts when its bursting membrane tension and bursting surface energy are exceeded. The surface energy of the cells was estimated from the kinetic energy of appropriate sized eddies. To test the model, cells were disrupted in turbulent flows in capillaries at mean energy dissipation rates ranging from 800 to 2 x 10(4) Wkg-1. The model assumed that the specific lysis rate is almost independent of the number of passes, which was verified by the experimental data. The implication was that despite the damage the cell mechanical properties did not change markedly during multiple recirculations through the capillaries. On average the model underestimated the cell disruption by about 15%. Although the model gave reasonably good predictions, it lacks proper explanation of the independence of the specific lysis rate on the number of passes. In this paper it is shown that this problem can be resolved in principle by consideration of the localisation of the energy dissipation in turbulent capillary flows. The necessity of further modelling of cell-turbulence interactions is demonstrated.

Effect of Particle-Impeller Impact Rate on Properties of Product Crystals in Agitated Crystallizers of Different Sizes.

Crystallization experiments of aluminium potassium sulfate were carried out in agitated crystallizers of different sizes equipped with six-bladed Rushton disc turbines. Properties of the product crystals, that is the agglomerate ratio, the size distribution, the Sauter mean diameter and the total number of product crystals, were measured and the particle-impeller impact rate was calculated according to a correlation previously derived by the authors. By comparing these results, the effects of particle-impeller impact rate on the properties of product crystals were investigated. As a result, it was found that the total number of product crystals increases, the agglomerate ratio decreases and the size distribution becomes narrow with an increase in impact rate. The results obtained for agitated crystallizers of different sizes indicated that the same agglomerate ratio, size distribution or weight mean diameter of the product crystals is given at a constant mean energy dissipation rate throughout the vessel. On the other hand, the same dimensionless number of product crystals is obtained at a constant impeller tip speed for geometrically similar systems

Experiments revealing small impact of turbulence on the energy budget of the mesosphere and lower thermosphere

We have measured a total of 17 in situ profiles of small‐scale density fluctuations (typical resolution: meters) in the lower thermosphere and upper mesosphere, which are used to derive turbulent parameters, such as the turbulent energy dissipation rate ε, the turbulent diffusion coefficient K, and the mean turbulent velocity wturb. The accuracy of the absolute numbers is unprecedented thanks to the very high spatial resolution and a recently improved data analysis procedure. Concentrating on the 12 flights which were performed during winter conditions at high latitudes (69°N), we find mean energy dissipation rates of 1–2 mW kg−1 in the lower mesosphere (&lt;75 km) and 10–20 mW kg−1 in the upper mesosphere and lower thermosphere (&lt;100 km). The corresponding heating rates are approximately 0.1 and 1 K d−l, respectively. These values are at least 1 order of magnitude smaller than most of the previous measurements and are also significantly smaller than typical values assumed in models. Our observations suggest that the heating effect of turbulence is negligible compared to the most prevailing terms of the heat budget. It can be shown by theoretical considerations involving the turbulent energy budget equation that cooling by turbulent heat conduction is also negligible if ε is small.

Estimation of disruption of animal cells by turbulent capillary flow

Disruption of animal cells in turbulent capillary flows has been predicted from a model of cell-hydrodynamic interactions using cell mechanical properties determined by micromanipulation. Eddies of sizes similar to or smaller than the cells are presumed to interact with those cells, causing local surface deformations. The proposed mechanism of cell damage is that such deformations result in an increase in membrane tension and surface energy and that a cell disrupts when its bursting membrane tension and bursting surface energy are exceeded. The surface energy of the cells is estimated from the kinetic energy of appropriately sized eddies. To test the model, cells were disrupted in turbulent flows in capillaries at mean energy dissipation rates up to 2 x 10(4) m(2)/s(3). In all cases the model underestimated the cell disruption by about 15%. Such good agreement implies that the approach of the model to the complicated phenomena of cell turbulence interactions is reasonable.

A comparison of Doppler and spaced antenna radar techniques for the measurement of turbulent energy dissipation rates

There has been concern in the past that spaced antenna radar measurements of energy dissipation rates in the Mesosphere using broad beam arrays may be contaminated by the effects of atmospheric gravity waves. To investigate this possibility, a 1.98 MHz Doppler radar operated by the University of Adelaide was used in two separate configurations; a narrow vertical beam (4.5° half power half width) and 3 smaller spaced antenna systems with broad beams (20° half power half width) were used to simultaneously measure energy dissipation rates in the height region 80 to 100km. The 3 broad beams were also used to determine winds by the spaced antenna method. Although extraction of the energy dissipation rate parameter is a difficult process, temporally and spatially coincident data were obtained enabling some important comparisons to be made. Three years of such data were analysed and the following results noted: (1) the mean energy dissipation rates as measured by the two radars are comparable, (2) the variance of the data as measured by the Doppler radar is larger than that measured by the spaced antenna radar, (3) The spaced antenna radar does not generally over‐estimate energy dissipation rates at moderate and large values, but when turbulence is very weak, it may do so. The degree of over‐estimation is not as serious as has been inferred in the past. However, there is considerable scatter in the short term ratio of measurements using the two methods, which emphasizes the large degree of spatial variability of turbulence strengths in the middle atmosphere.

Log-Stable Distribution and Intermittency of Turbulence

The logarithm of the breakdown coefficient e r /e l , e r being the mean energy dissipation rate averaged over a sphere of radius r is shown, under a similarity assumption, to obey a stable distribution, the characteristic function of which is given by ϕ( z | r / l )=( r / l ) (µ/2 α -2)[i z -( z e $ i π/2 ) α ] , where µ>0 and 0 0, as e→∞. The present results include the log-normal theory for α=2 and coincide with the prediction of µ p due to the β-model in the limit α→0.