Nondimensional Dissipation Rate Research Articles

Turbulent characteristics and energy transfer in the far field of active-grid turbulence

Turbulent characteristics in the far field of active-grid turbulence have been investigated through wind tunnel experiments using hot-wire anemometry. Two forcing protocols are employed following previous studies: one is the double-random mode and the other is the open mode with the grid remaining static with minimum blockage. The integral length scale L for the double-random modes slightly decreases with streamwise distance in the far field as observed in the near field of the active-grid turbulence. The nondimensional dissipation rate Cε for the double-random modes is around 0.5. This asymptotic value is different from those reported in previous active-grid turbulence experiments and could be nonuniversal. The equilibrium scaling L/λ=CεReλ/15 (λ is the Taylor microscale and Reλ is the turbulent Reynolds number) with a constant Cε is established in the far field of the double-random modes regardless of active-grid motions. The sum of production and destruction terms in the enstrophy budget equation for homogeneous and isotropic turbulence S+2G/Reλ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient) is proportional to Reλ−1 and close to zero in the present active-grid turbulence, suggesting that the equilibrium scaling is possibly related to the balance between the production and destruction of the enstrophy.

Energy dissipation and enstrophy production/destruction at very low Reynolds numbers in the final stage of the transition period of decay in grid turbulence

Decay characteristics of turbulent kinetic energy and enstrophy in grid turbulence have been investigated in the far downstream region (x/M∼103: x is the downstream distance from the grid, M is the mesh size of the grid) through wind tunnel experiments using hot-wire anemometry, with the lowest turbulent Reynolds number Reλ≈5. The non-dimensional dissipation rate Cε increases rapidly toward the final stage of the transition period of decay and the profile agrees well with previous direct numerical simulation [W. D. McComb et al., “Taylor's (1935) dissipation surrogate reinterpreted,” Phys. Fluids 22, 061704 (2010)] and theoretical estimation [D. Lohse, “Crossover from high to low Reynolds number turbulence,” Phys. Rev. Lett. 73, 3223 (1994)] at very low Reλ in decaying and stationary isotropic turbulence. The present result of Cε is an update on the experimental data in grid turbulence toward a very low Reλ, where measurements have been absent. The energy spectrum in the dissipation range at very low Reλ deviates from a universal form observed at high Reynolds numbers. The decay rate of enstrophy is proportional to S+2G/Reλ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient). It is shown that G and S+2G/Reλ increase rapidly with decreasing Reλ at very low Reλ, indicating that the effect of enstrophy destruction is dominant in the final stage of the transition period of decay. The profiles of S+2G/Reλ against Reλ is well fitted by a power-law function even in the final stage of the transition period of decay.

Non-dimensional energy dissipation rate near the turbulent/non-turbulent interfacial layer in free shear flows and shear free turbulence

The non-dimensional dissipation rate $C_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}L/u^{\prime 3}$, where $\unicode[STIX]{x1D700}$, $L$ and $u^{\prime }$ are the viscous energy dissipation rate, integral length scale of turbulence and root-mean-square of the velocity fluctuations, respectively, is computed and analysed within the turbulent/non-turbulent interfacial (TNTI) layer using direct numerical simulations of a planar jet, mixing layer and shear free turbulence. The TNTI layer that separates the turbulent and non-turbulent regions exists at the edge of free shear turbulent flows and turbulent boundary layers, and comprises both the viscous superlayer and turbulent sublayer regions. The computation of $C_{\unicode[STIX]{x1D700}}$ is made possible by the introduction of an original procedure, based on local volume averages within spheres of radius $r$, combined with conditional sampling as a function of the location with respect to the TNTI layer. The new procedure allows for a detailed investigation of the scale dependence of several turbulent quantities near the TNTI layer. An important achievement of this procedure consists in permitting the computation of the turbulent integral scale within the TNTI layer, which is shown to be approximately constant. Both the non-dimensional dissipation rate and turbulent Reynolds number $Re_{\unicode[STIX]{x1D706}}$ vary in space within the TNTI layer, where two relations are observed: $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$ and $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$. Specifically, whereas the viscous superlayer and part of the turbulent sublayer display $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-2}$, the remaining of the turbulent sublayer exhibits $C_{\unicode[STIX]{x1D700}}\sim Re_{\unicode[STIX]{x1D706}}^{-1}$, which is consistent with non-equilibrium turbulence (Vassilicos, Annu. Rev. Fluid Mech. vol. 47, 2015, pp. 95–114).

Controls on Turbulent Mixing in a Strongly Stratified and Sheared Tidal River Plume

AbstractConsiderable effort has been made to parameterize turbulent kinetic energy (TKE) dissipation rate ε and mixing in buoyant plumes and stratified shear flows. Here, a parameterization based on Kunze et al. is examined, which estimates ε as the amount of energy contained in an unstable shear layer (Ri < Ric) that must be dissipated to increase the Richardson number Ri = N2/S2 to a critical value Ric within a turbulent decay time scale. Observations from the tidal Columbia River plume are used to quantitatively assess the relevant parameters controlling ε over a range of tidal and river discharge forcings. Observed ε is found to be characterized by Kunze et al.’s form within a factor of 2, while exhibiting slightly decreased skill near Ri = Ric. Observed dissipation rates are compared to estimates from a constant interfacial drag formulation that neglects the direct effects of stratification. This is found to be appropriate in energetic regimes when the bulk-averaged Richardson number Rib is less than Ric/4. However, when Rib > Ric/4, the effects of stratification must be included. Similarly, ε scaled by the bulk velocity and density differences over the plume displays a clear dependence on Rib, decreasing as Rib approaches Ric. The Kunze et al. ε parameterization is modified to form an expression for the nondimensional dissipation rate that is solely a function of Rib, displaying good agreement with the observations. It is suggested that this formulation is broadly applicable for unstable to marginally unstable stratified shear flows.

Complete self-preservation on the axis of a turbulent round jet

Self-preservation (SP) solutions on the axis of a turbulent round jet are derived for the transport equation of the second-order structure function of the turbulent kinetic energy ($k$), which may be interpreted as a scale-by-scale (s.b.s.) energy budget. The analysis shows that the mean turbulent energy dissipation rate, $\overline{{\it\epsilon}}$, evolves like $x^{-4}$ ($x$ is the streamwise direction). It is important to stress that this derivation does not use the constancy of the non-dimensional dissipation rate parameter $C_{{\it\epsilon}}=\overline{{\it\epsilon}}u^{\prime 3}/L_{u}$ ($L_{u}$ and $u^{\prime }$ are the integral length scale and root mean square of the longitudinal velocity fluctuation respectively). We show, in fact, that the constancy of $C_{{\it\epsilon}}$ is simply a consequence of complete SP (i.e. SP at all scales of motion). The significance of the analysis relates to the fact that the SP requirements for the mean velocity and mean turbulent kinetic energy (i.e. $U\sim x^{-1}$ and $k\sim x^{-2}$ respectively) are derived without invoking the transport equations for $U$ and $k$. Experimental hot-wire data along the axis of a turbulent round jet show that, after a transient downstream distance which increases with Reynolds number, the turbulence statistics comply with complete SP. For example, the measured $\overline{{\it\epsilon}}$ agrees well with the SP prediction, i.e. $\overline{{\it\epsilon}}\sim x^{-4}$, while the Taylor microscale Reynolds number $Re_{{\it\lambda}}$ remains constant. The analytical expression for the prefactor $A_{{\it\epsilon}}$ for $\overline{{\it\epsilon}}\sim (x-x_{o})^{-4}$ (where $x_{o}$ is a virtual origin), first developed by Thiesset et al. (J. Fluid Mech., vol. 748, 2014, R2) and rederived here solely from the SP analysis of the s.b.s. energy budget, is validated and provides a relatively simple and accurate method for estimating $\overline{{\it\epsilon}}$ along the axis of a turbulent round jet.

The inertial subrange in turbulent pipe flow: centreline

The inertial-subrange scaling of the axial velocity component is examined for the centreline of turbulent pipe flow for Reynolds numbers in the range $249\leqslant Re_{{\it\lambda}}\leqslant 986$. Estimates of the dissipation rate are made by both integration of the one-dimensional dissipation spectrum and the third-order moment of the structure function. In neither case does the non-dimensional dissipation rate asymptote to a constant; rather than decreasing, it increases indefinitely with Reynolds number. Complete similarity of the inertial range spectra is not evident: there is little support for the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 32, 1941a, pp. 16–18; Dokl. Akad. Nauk SSSR, vol. 30, 1941b, pp. 301–305) and the effects of Reynolds number are not well represented by Kolmogorov’s ‘extended similarity hypothesis’ (J. Fluid Mech., vol. 13, 1962, pp. 82–85). The second-order moment of the structure function does not show a constant value, even when compensated by the extended similarity hypothesis. When corrected for the effects of finite Reynolds number, the third-order moments of the structure function accurately support the ‘four-fifths law’, but they do not show a clear plateau. In common with recent work in grid turbulence, non-equilibrium effects can be represented by a heuristic scaling that includes a global Reynolds number as well as a local one. It is likely that non-equilibrium effects appear to be particular to the nature of the boundary conditions. Here, the principal effects of the boundary conditions appear through finite turbulent transport at the pipe centreline, which constitutes a source or a sink at each wavenumber.

Study of forced turbulence and its modulation by finite-size solid particles using the lattice Boltzmann approach

The paper describes application of the mesoscopic lattice Boltzmann (LB) method to the simulations of both single-phase turbulence and particle-laden turbulence which are maintained by a large-scale forcing. The disturbance flows around finite-size solid particles are resolved, providing the opportunity to study the detailed interactions between fluid turbulence and solid particles at the particle–fluid interfaces. Specifically, a nonuniform time-dependent stochastic forcing scheme is implemented within the mesoscopic multiple-relaxation-time LB approach. The statistics of single-phase forced turbulence obtained from the LB approach are found to be in excellent agreement with those from the pseudo-spectral simulations, provided that the grid resolution in the LB simulation is doubled. It is shown that the flow statistics is not sensitive to the velocity scale used for the LB simulation. Preliminary results on forced turbulence laden with non-sedimenting solid particles at a particle-to-fluid density ratio of 5, solid volume fraction of 0.102, and particle diameter to Kolmogorov length ratio of 8.05 are interpreted, using a systematic analysis conducted at three levels: whole-field, phase-partitioned, and profiles as a function of distance from the surface of solid particles. It is found that the particle-laden turbulence is much more dissipative in terms of the non-dimensional dissipation rate, due to both reduction of the effective flow Reynolds number and the viscous boundary layer on the surfaces of solid particles. The thickness of the boundary layer is found to be about 0.4rp. While this boundary layer region accounts for 19.5% of the space within the fluid, it contributes to 57.5% of total viscous dissipation. The vorticity magnitude exhibits a maximum inside the boundary layer and a minimum outside the boundary layer, showing detachment of the vorticity structure from the solid surface. The sharp gradients near the particle surface contribute dominantly to the value of velocity derivative flatness, making the flatness in particle-laden flow much larger than that of single-phase turbulence. In the spectral space, presence of solid particles attenuates energy at large scales including the forcing shells and augments energy at the small scales. The pivot wavenumber is found to be very similar to the value previously found in decaying particle-laden turbulence under the similar parameter setting.

Characteristics of Langmuir Turbulence in the Ocean Mixed Layer

Abstract This study uses large-eddy simulation (LES) to investigate the characteristics of Langmuir turbulence through the turbulent kinetic energy (TKE) budget. Based on an analysis of the TKE budget a velocity scale for Langmuir turbulence is proposed. The velocity scale depends on both the friction velocity and the surface Stokes drift associated with the wave field. The scaling leads to unique profiles of nondimensional dissipation rate and velocity component variances when the Stokes drift of the wave field is sufficiently large compared to the surface friction velocity. The existence of such a scaling shows that Langmuir turbulence can be considered as a turbulence regime in its own right, rather than a modification of shear-driven turbulence. Comparisons are made between the LES results and observations, but the lack of information concerning the wave field means these are mainly restricted to comparing profile shapes. The shapes of the LES profiles are consistent with observed profiles. The dissipation length scale for Langmuir turbulence is found to be similar to the dissipation length scale in the shear-driven boundary layer. Beyond this it is not possible to test the proposed scaling directly using available data. Entrainment at the base of the mixed layer is shown to be significantly enhanced over that due to normal shear turbulence.

Turbulence Spectra and Eddy Diffusivity over Forests

The main objectives of this observational study are to examine the stability dependence of velocity and air temperature spectra and to employ the spectral quantities to establish relations for eddy diffusivity over forests. The datasets chosen for the analysis were collected above the Browns River forest and the Camp Borden forest over a wide range of stability conditions. Under neutral and unstable conditions the nondimensional dissipation rate of turbulent kinetic energy (TKE) over the forests is lower than that from its Monin-Obukhov similarity (MOS) function for the smooth-wall surface layer. The agreement is somewhat better under stable conditions but a large scatter is evident. When the frequency is made nondimensional by the height of the stand (h) and the longitudinal velocity at this height (uh, the Kaimal spectral model for neutral air describes the observations very well. The eddy diffusivity formulation K = c σ4w/ε provides a promising alternative to the MOS approach, where σw is the standard deviation of the vertical velocity and ε TKE dissipation rate. Current datasets yield a constant of 0.43 for c for sensible heat in neutral and stable air, a value very close to that for the smooth-wall surface layer. It is postulated that c is a conservative parameter for sensible heat in the unstable air, its value probably falling between 0.41 and 0.54. In the absence of ε data, it is possible to estimate K from measurements of the local mean wind u and air stability. As a special case, it is shown that K = 0.27(uh/uh)σw under neutral stability. This relation is then used to establish a profile model for wind speed and scalar concentration in the roughness sublayer. The analysis points out that uh and h are important scaling parameters in attempts to formulate quantitative relations for turbulence over tall vegetation.

Intermittency, the second-order structure function, and the turbulent energy-dissipation rate.

In the context of an interpolation formula for a second-order structure function, Grossmann [Phys. Rev. E 51, 6275 (1995)] considered various implications of the asymptotic behavior of the energy dissipation rate for inertial range intermittency. We reconsider the issue and show that the tendency of the nondimensional dissipation rate to asymptotically approach a constant is consistent with finite intermittency corrections. By extending Lohse's ideas [Phys. Rev. Lett. 73, 3223 (1994)] put forth in a nonintermittent setting, we compute for intermittent turbulence the Reynolds number dependence of the nondimensional dissipation rate and show that the result compares favorably with experimental data.

Fluxes across a thermohaline interface

Measurements of velocity and temperature microstructure and hydrography were made with a towed vehicle moving in and around a single interface in a double-diffusive staircase. The interface was traversed 222 times in a saw-tooth pattern over a track 35 km long. The salinity and potential temperature and density in the mixed layers adjacent to the interface were spatially uniform except for one 8 km long anomaly. The rate of dissipation of kinetic energy was uniformly low in the interface and in the mixed layers, except for one section 600 m long where a Kelvin-Helmholtz instability generated turbulence. For the non-turbulent section of the interface, the mean rate of dissipation was 30.2 × 10 −10 W kg −1 in the mixed layers and 9.5 × 10 −10 W kg −1 in the interface. The non-dimensional dissipation rate, ϵ/vN 2, was almost always less than 16 in the interface and therfore, there was no turblent buoyancy flux according to Rohr et al. (1988, Journal of Fluid Mechanics, 195, 77–111). The average double-diffusive flux of buoyancy by heat was 3.6 × 10 −10 W kg −1. Under certain assumptions the ratio of the flux of buoyancy by heat and salt can be estimated to be 0.53 ± 0.10, in good agreement with laboratory and theoretical estimates for salt fingers. The average Cox number was about 8 in the interface, consistent with the theories of Stern (1975, Ocean circulation physics, Academic Press) and Kunze (1987, Journal of Marine Research, 45 533–556), but displayed an inverse dependence on the vertical temperature gradient which was not predicted. As a result, the flux of buoyancy, as well as the individual contributions by heat and salt, were independent of the local mean vertical temperature gradient and the buoyancy frequency. The length of the turbulent section of the interface was only 1.7% of the total length observed. However, the turbulence was intense—the mean rate of dissipation was 2.5 × 10 −8 W kg −1—and may have sufficiently enhanced the flux of heat to increase the net flux ratio to 0.72, which would be consistent with the large-scale changes in layer properties reported by Schmitt (1987 EOS, Transactions of the American Geophysical Union, 68, 57–70) and the O/(10 km) scale changes observed in this study.

The Kolmogorov constant for carbon dioxide in the atmospheric surface layer over a paddy field

From measurements in the atmospheric surface layer over a paddy field, the Kolmogorov constants for CO2 and longitudinal wind velocity were obtained. In this study, the nondimensional dissipation rate Φnc = (1−16ξv)-1/2 for CO2 variance and Φe = (1−16ξv)-1/4 − ξv for turbulent energy were used, assuming the equality of the local production term and the local dissipation term, and neglecting the divergence flux term in the budget equation. The value of the constant for CO2 was consistent with recent determinations for temperature and humidity. The constant for longitudinal wind velocity showed good agreement with other recent observations.