Transport equations for the normalized nth-order moments of velocity derivatives in grid turbulence

Transport equations for the normalized moments of the longitudinal velocity derivative ${F_{n + 1}}$ (here, $n$ is $1, 2, 3\ldots$ ) are derived from the Navier–Stokes (N–S) equations for shearless grid turbulence. The effect of the (large-scale) streamwise advection of ${F_{n + 1}}$ by the mean velocity on the normalized moments of the velocity derivatives can be expressed as $C_1 {F_{n + 1}}/Re_\lambda$ , where $C_1$ is a constant and $Re_\lambda$ is the Taylor microscale Reynolds number. Transport equations for the normalized odd moments of the transverse velocity derivatives ${F_{y,n + 1}}$ (here, $n$ is 2, 4, 6), which should be zero if local isotropy is satisfied, are also derived and discussed in sheared and shearless grid turbulence. The effect of the (large-scale) streamwise advection term on the normalized moments of the velocity derivatives can also be expressed in the form $C_2 {F_{y,n + 1}}/Re_\lambda$ , where $C_2$ is a constant. Finally, the contribution of the mean shear in the transport equation for ${F_{n + 1}}$ can be modelled as $15 B/Re_\lambda$ , where $B$ ( $=S^*{S_{s,n + 1}}$ ) is the product of the non-dimensional shear parameter $S^*$ and the normalized mixed longitudinal-transverse velocity derivatives ${{S_{s,n + 1}}}$ ; if local isotropy is satisfied, $S_{s,n + 1}$ should be zero. These results indicate that if ${F_{n + 1}}$ , ${F_{y,n + 1}}$ and $B$ do not increase as rapidly as $Re_\lambda$ , then the effect of the large-scale structures on small-scale turbulence will disappear when $Re_\lambda$ becomes sufficiently large.

RESEARCH OF EXTERNAL MASS TRANSFER PROCESSES FOR ADSORPTION FROM SOLUTIONS IN A APPARATUS WITH STIRRING

A study has been mode of transport-controlled mass transfer-controlled to particles suspended in a stirred vessel. The motion of particle in a fluid was examined and a method of predicting relative velocities in terms of Kolmogoroff’s theory of local isotropic turbulence for mass transfer was outlined. To provide a more concrete visualization of complex wave form of turbulence, the concepts of eddies, of eddy velocity, scale (or wave number) and energy spectrum, have proved convenient. Large scale motions of scale contain almost all of the energy and they are directly responsible for energy diffusion throughout the stirring vessel by kinetic and pressure energies. However, almost no energy is dissipated by the large-scale energy-containing eddies. A scale of motion less than is responsible for convective energy transfer to even smaller eddy sires. At still smaller eddy scales, close to a characteristic microscale, both viscous energy dissipation and convection are the rule. The last range of eddies has been termed the universal equilibrium range. It has been further divided into a low eddy size region, the viscous dissipation subrange, and a larger eddy size region, the inertial convection subrange. Measurements of energy spectrum in mixing vessel are shown that there is a range, where the so called -(5/3) power law is effective. Accordingly, the theory of local isotropy of Kolmogoroff can be applied because existence of the internal subrange. As the integrated value of local energy dissipation rate agrees with the power per unit mass of liquid from the impeller, almost all energy from the impeller is viscous dissipated in eddies of microscale. The correlation for mass transfer to particles suspended in a stirred vessel is recommended. The results of experimental study are approximately 12 % above the predicted values.

Anomalous Scaling in the Kinematic Magnetohydrodynamic Turbulence

The problem of the anomalous scaling in the kinematic magnetohydrodynamic turbulence is investigated using the field theoretic renormalization group method and the operator product expansion technique. The anomalous dimensions of all leading composite operators, which drive the anomalous scaling of the correlation functions of a weak passive magnetic field, are determined up to the second order of the perturbation theory (i.e., in the two-loop approximation in the field theoretic terminology) in the presence of a large scale anisotropy for physically the most interesting three-dimensional case. It is shown that the leading role in the anomalous scaling properties of the model is played by the anomalous dimensions of the composite operators near the isotropic shell, in accordance with the Kolmogorov’s local isotropy restoration hypothesis. The importance of the two-loop corrections to the anomalous dimensions of the leading composite operators is demonstrated.

Patching over Berkovich curves and quadratic forms

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local–global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the $u$-invariant. The patching method we adapt was introduced by Harbater and Hartmann [Patching over fields, Israel J. Math. 176 (2010), 61–107] and further developed by these two authors and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263]. The results presented in this paper generalize those of Harbater, Hartmann, and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263] on the local–global principle and quadratic forms.

Characteristics of the turbulent energy dissipation rate in a cylinder wake

This work aims to improve our understanding of the turbulent energy dissipation rate in the wake of a circular cylinder. Ten of the twelve velocity derivative terms which make up the energy dissipation rate are simultaneously obtained with a probe composed of four X-wires. Measurements are made in the plane of mean shear at$x/d=10$, 20 and 40, where$x$is the streamwise distance from the cylinder axis and$d$is the cylinder diameter, at a Reynolds number of$2.5\times 10^{3}$based on$d$and free-stream velocity. Both statistical and topological features of the velocity derivatives as well as the energy dissipation rate, approximated by a surrogate based on the assumption of homogeneity in the transverse plane, are examined. The spectra of the velocity derivatives indicate that local axisymmetry is first satisfied at higher wavenumbers while the departure at lower wavenumbers is caused by the Kármán vortex street. The spectral method proposed by Djenidi & Antonia (Exp. Fluids, vol. 53, 2012, pp. 1005–1013) based on the universality of the dissipation range of the longitudinal velocity spectrum normalized by the Kolmogorov scales also applies in the present flow despite the strong perturbation from the Kármán vortex street and violation of local isotropy at small$x/d$. The appropriateness of the spectral chart method is consistent with Antoniaet al.’s (Phys. Fluids, vol. 26, 2014, 45105) observation that the two major assumptions in Kolmogorov’s first similarity hypothesis, i.e. very large Taylor microscale Reynolds number and local isotropy, can be significantly relaxed. The data also indicate that vorticity spectra are more sensitive, when testing the first similarity hypothesis, than velocity spectra. They also reveal that the velocity derivatives$\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y$and$\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x$play an important role in the interaction between large and small scales in the present flow. The phase-averaged data indicate that the energy dissipation is concentrated mostly within the coherent spanwise vortex rollers, in contrast with the model of Hussain (J. Fluid Mech., vol. 173, 1986, pp. 303–356) and Hussain & Hayakawa (J. Fluid Mech., vol. 180, 1987, p. 193), who conjectured that it resides mainly in regions of strong turbulent mixing.

Generalised higher-order Kolmogorov scales

Kolmogorov introduced dissipative scales based on the mean dissipation $\langle {\it\varepsilon}\rangle$ and the viscosity ${\it\nu}$, namely the Kolmogorov length ${\it\eta}=({\it\nu}^{3}/\langle {\it\varepsilon}\rangle )^{1/4}$ and the velocity $u_{{\it\eta}}=({\it\nu}\langle {\it\varepsilon}\rangle )^{1/4}$. However, the existence of smaller scales has been discussed in the literature based on phenomenological intermittency models. Here, we introduce exact dissipative scales for the even-order longitudinal structure functions. The derivation is based on exact relations between even-order moments of the longitudinal velocity gradient $(\partial u_{1}/\partial x_{1})^{2m}$ and the dissipation $\langle {\it\varepsilon}^{m}\rangle$. We then find a new length scale ${\it\eta}_{C,m}=({\it\nu}^{3}/\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$ and $u_{C,m}=({\it\nu}\langle {\it\varepsilon}^{m/2}\rangle ^{2/m})^{1/4}$, i.e. the dissipative scales depend rather on the moments of the dissipation $\langle {\it\varepsilon}^{m/2}\rangle$ and thus the full probability density function (p.d.f.) $P({\it\varepsilon})$ instead of powers of the mean $\langle {\it\varepsilon}\rangle ^{m/2}$. The results presented here are exact for longitudinal even-ordered structure functions under the assumptions of (local) isotropy, (local) homogeneity and incompressibility, and we find them to hold empirically also for the mixed and transverse as well as odd orders. We use direct numerical simulations (DNS) with Reynolds numbers from $Re_{{\it\lambda}}=88$ up to $Re_{{\it\lambda}}=754$ to compare the different scalings. We find that indeed $P({\it\varepsilon})$ or, more precisely, the scaling of $\langle {\it\varepsilon}^{m/2}\rangle /\langle {\it\varepsilon}\rangle ^{m/2}$ as a function of the Reynolds number is a key parameter, as it determines the ratio ${\it\eta}_{C,m}/{\it\eta}$ as well as the scaling of the moments of the velocity gradient p.d.f. As ${\it\eta}_{C,m}$ is smaller than ${\it\eta}$, this leads to a modification of the estimate of grid points required for DNS.