A recent discovery about the inertial range of homogeneous and isotropic turbulence is the saturation of the scaling exponents ζn for large n, defined via structure functions of order n as Sn(r)=〈(δru)n〉=A(n)rζn. We focus on longitudinal structure functions for δru between two positions that are r apart in the same direction as u. In a previous work [], two of the present authors developed a theory for ζn, which agrees with measurements for all n for which reliable data are available, and shows saturation for large n. Here, we derive expressions for the probability density functions of δru for four different states of turbulence, including the asymptotic fourth state defined by the saturation of exponents for large n. This saturation means that the scale separation is violated in favor of strongly coupled quasiordered flow structures, which likely take the form of long and thin (worm-like) structures of length L and thickness l=O(L/Re). Published by the American Physical Society 2024

We provide a theoretical estimation of the relaxation time for the kinetic theoretical representation of homogeneous, isotropic, and stationary turbulent flow. The basic approach depends on the construction of a balance of fluctuation and dissipation between the relaxation process and the self-consistently generated fluctuating force. The kinetic theory representation generalizes the modeling of turbulence so that turbulent viscosity as well as all higher order transport coefficients are determined by the relaxation time. The resulting value of the relaxation time gives an estimation of the turbulent viscosity with the coefficient value comparable with those from most of the representative turbulence models, which have been obtained empirically or semi-empirically.

We accomplish two major tasks. First, we show that the turbulent motion at large scales obeys Gaussian statistics in the interval 0 < Rlambda < 8.8, where Rlambda is the microscale Reynolds number, and that the Gaussian flow breaks down to yield place to anomalous scaling at the universal Reynolds number bounding the inequality above. In the inertial range of turbulence that emerges following the breakdown, the effective Reynolds number based on the turbulent viscosity, Rlambda* assumes this same constant value of about 9. This scenario works also for the emergence of turbulence from an initially non-turbulent state. Second, we derive expressions for the anomalous scaling exponents of structure functions and moments of spatial derivatives, by analyzing the Navier-Stokes equations in the form developed by Hopf. We present a novel procedure to close the Hopf equation, resulting in expressions for zetan in the entire range of allowable moment-order, n, and demonstrate that accounting for the temporal dynamics changes the scaling from normal to anomalous. For large n, the theory predicts the saturation of zetan with n, leading to two inferences: (a) the smallest length scale etan = LRe-1 << LRe-3/4, where Re is the large-scale Reynolds number, and (b) velocity excursions across even the smallest length scales can sometimes be as large as the large scale velocity itself. Theoretical predictions for each of these aspects are shown to be in quantitative agreement with available experimental and numerical data.

Asymptotically large Reynolds number hydrodynamic turbulence is characterized by multi-scaling of moments of velocity increments and spatial derivatives. With decreasing Reynolds number toward $R_{\lambda}=R^{tr}_{\lambda}\approx 9.0$, the anomalous scaling disappears in favor of the "normal" one and close-to-Gaussian probability densities [Yakhot \& Donzis, {\bf 119}, 044501 (2017)]. The nature of this transition and its universality are subjects of this work. Here we consider Benard convection ( Prandtl number $Pr=1$) between infinite horizontal plates. It is shown that in this system the "competition" between Bolgiano and Kolmogorov processes, results in small-scale velocity fluctuations driven by effective "large-scale" Gaussian random temperature field. Therefore, the intermittent dynamics of velocity derivatives are similar or even identical to that in homogeneous and isotropic turbulence generated by the large-scale random forcing. It is shown that low-Rayleigh number instabilities make the problem much more involved and may lead to transition from Gaussian to exponential PDF of the temperature field. The developed {\it mean-field theory} yielded dimensionless heat flux $Nu\propto Ra^{\beta}$ with $\beta\approx 15/56\approx 0.27$, close to the outcome of Chicago experiment. These results point to an unusual small-scale universality of turbulent flows. It is also shown that at $R_{\lambda}\leq 9.0$, a flow "remembers" its laminar background and, therefore, cannot be universal.

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by Yakhot and Donzis [Phys. Rev. Lett. 119, 044501 (2017)] for homogeneous and isotropic turbulence. In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, is much more complex. In this paper, we discuss the Reynolds-number dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-Benard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by Pandey et al. [Nat. Commun. 9, 2118 (2018)]. Different Reynolds numbers l less than or similar to Re-l less than or similar to 1000 which are based on the thickness of the bulk region l and the corresponding rootmean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number Ra = 10(5). A few specific features of the data agree with the theory. The normalized moments of the kinetic energy dissipation rate epsilon(n) show a nonmonotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.

Direct transition from low Reynolds number "weak" Gaussian turbulence to fully developed "strong" turbulence at a critical Reynolds number R^tr_λ≈ 8.91 has recently been theoretically predicted and tested in high resolution numerical simulations of V. Yakhot & D. A. Donzis, Phys. Rev. Lett. 119, 044501 (2017) & PhysicaD, 384-385, 12 (2018) on an example of a flow excited by a Gaussian random force. The matching between the low-Reynolds number Gaussian asymptotic (Re<>Re^tr), led to closed approximate equation for exponents of moments of derivatives in a good agreement with experimental data. In this paper we study transition to turbulence in Benard (RB) convection where, depending on the Rayleigh number, turbulence is produced by both weak instabilities of the bulk flow and, the plume-generating instabilities of the wall boundary layers. The developed theory explains non-monotonic behavior of the low-Reynolds - number moments of velocity derivatives M_2n (Re)=(∂_xv_x)^2n/[(∂_xv_x)^2]^nobserved in direct numerical simulations of Schumacher et.al (Phys.Rev.E, 98,033120 (2018)). In the high-Reynolds number limit, the moments are given byM_2n∝ Re^ρ_2nwith the exponentsρ_2nslightly different from those in a Gaussian-stirring case of Refs. [3]-[4]. This may be related to universality classes defined by production mechanisms.

To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and dissipation rate, <((v(x + r) - v(x))(n))over bar> proportional to r(zeta n) and (epsilon(n)) over bar proportional to Re-dn, respectively. In high Reynolds number flows, the moments of different orders cannot be simply related to each other which is the signature of anomalous scaling, one of the most puzzling features of turbulent flows. High-order moments are related to extreme, rare events and our ability to quantitatively describe them is crucially important for meteorology, heat, mass transfer and other applications. In this work we present a solution to this problem in the particular case of the Navier-Stokes equations driven by a random force. A novel aspect of this work is that, unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers Re-tr where the first emergence of anomalous scaling is observed out of a low-Re Gaussian background. The obtained closed expressions for anomalous scaling exponents and cl, which depend on the transition Reynolds number, agree well with experimental and numerical data in the literature and, when n >> 1, d(n) approximate to 0.19n ln(n). The theory yields the energy spectrum E(k) proportional to k(-zeta 2-1) I with zeta(2) approximate to 0.699, different from the outcome of Kolmogorov's theory. It is also argued that fluctuations of dissipation rate and those of the transition point itself are responsible for both, deviation from Gaussian statistics and multiscaling of velocity field. (C) 2018 Elsevier B.V. All rights reserved.

To characterize fluctuations in a turbulent flow, one usually studies different moments of velocity increments and/or dissipation rate, $\overline{(v(x+r)-v(x))^{n}}\propto r^{\zeta_{n}}$ and $\overline{{\cal E}^{n}}\propto Re^{d_{n}}$, respectively. In high Reynolds number flows, the moments of different orders with $n\neq m$ cannot be simply related to each other which is the signature of anomalous scaling. In this work we present a solution to this problem in the particular case of the Navier-Stokes equations driven by a random force. A novel aspect of this work is that unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers $Re^{tr}$ of the first emergence of anomalous scaling out of low-$Re$ Gaussian background. The obtained closed expressions for anomalous scaling exponents $\zeta_{n}$ and $d_{n}$ agree well with available in literature experimental and numerical data and, when $n\gg 1$, $d_{n}\approx 0.19n \ln(n)$. The theory yields the energy spectrum $E(k)\propto k^{-\zeta_{2}-1}$ with $\zeta_{2}\approx 0.699$, different from the outcome of Kolmogorov's theory. It is also shown that fluctuations of dissipation rate are responsible for both: deviation from Gaussian statistics and multiscaling of velocity field.

We explore the scaling behavior of an unsteady flow that is generated by an oscillating body of finite size in a gas. If the gas is gradually rarefied, the Navier-Stokes equations begin to fail and a kinetic description of the flow becomes more appropriate. The failure of the Navier-Stokes equations can be thought to take place via two different physical mechanisms: either the continuum hypothesis breaks down as a result of a finite size effect or local equilibrium is violated due to the high rate of strain. By independently tuning the relevant linear dimension and the frequency of the oscillating body, we can experimentally observe these two different physical mechanisms. All the experimental data, however, can be collapsed using a single dimensionless scaling parameter that combines the relevant linear dimension and the frequency of the body. This proposed Knudsen number for an unsteady flow is rooted in a fundamental symmetry principle, namely, Galilean invariance.

We consider the transition to strong turbulence in an infinite fluid stirred by a Gaussian random force. The transition is defined as a first appearance of anomalous scaling of normalized moments of velocity derivatives (dissipation rates) emerging from the low-Reynolds-number Gaussian background. It is shown that, due to multiscaling, strongly intermittent rare events can be quantitatively described in terms of an infinite number of different "Reynolds numbers" reflecting a multitude of anomalous scaling exponents. The theoretically predicted transition disappears at R_{λ}≤3. The developed theory is in quantitative agreement with the outcome of large-scale numerical simulations.

Turbulence problem is often considered as "the last unsolved problem of classical physics". It is due to strong interaction between velocity and/or velocity gradient fluctuations, a high Reynolds number flow is a fascinating mixture of purely random, close to Gaussian, fields and coherent structures where substantial fraction of kinetic energy is dissipated into heat. To evaluate intensity of fluctuations, one usually studies different moments of velocity increments and/or dissipation rate, characterized by scaling exponents $\zeta_{n}$ and $d_{n}$, respectively. In high Reynolds number flows, the moments of different orders with $n\neq m$ cannot be simply related to each other, which is the signature of anomalous scaling, making this problem "the last unsolvable". No perturbative treatment can lead to quantitative description of this feature. In this work the expressions for the moments of dissipation rate $e_{n}=\overline{{\cal E}^{n}}\propto Re^{d_{n}}$ and those of velocity derivatives $M_{2n}=\overline{(\partial_{x}u_{x})^{2n}}\propto \frac{v_{o}^{2n}}{L^{2n}}Re^{\rho_{2n}}$ are derived for an infinite fluid stirred by a white-in-time Gaussian random force supported in the vicinity of the wave number $k_{f}\approx \frac{2\pi}{L}=O(1)$, where $v_{0}$ and $L$ are characteristic velocity and integral scale, respectively. A novel aspect of this work is that unlike previous efforts which aimed at seeking solutions around the infinite Reynolds number limit, we concentrate on the vicinity of transitional Reynolds numbers $Re^{tr}$ of the first emergence of anomalous scaling out of Low-Re Gaussian background. The obtained closed expressions for anomalous scaling exponents $d_{n}$ and $\rho_{n}$ agree well with available in literature experimental and numerical data and, when $n\gg 1$, $d_{n}\approx 0.3n \ln(n)$.

The expression for the moments of dissipation rate $e_{n}=\overline{{\cal E}^{n}}\propto Re^{d_{n}}$ in strong turbulence is derived for an infinite fluid stirred by a Gaussian random force supported in the vicinity of wave number $k_{f}\approx \frac{2\pi}{L}$ where $L$ is the integral scale. The obtained closed expression for anomalous scaling exponents $d_{n}$ agrees well with available in literature experimental and numerical data in the range $n\leq 5$ and, as $n\rightarrow\infty$, $d_{n}\rightarrow 0.3n \ln(n)$. The anomalous scaling of velocity derivatives is a result of dynamic coupling of dissipation rate and fluctuations of critical Reynolds number.

Renormalization or coarse-graining applied to basic equations governing multi -scale phenomena, leading to effective equations for large-scale properties is often called model-building. Unlike fluids in thermodynamic equilibrium, in case of high-Reynolds number turbulent flows the procedure leads to generation of an infinite number relevant high-order nonlinearities which are hard to deal with. In this paper, based on the recently discovered universality of transition to strongly non-Gaussian (anomalous) statistics of velocity derivatives, we show that in the infrared limit \(k\rightarrow 2\pi /L\), where \(L\) is the integral scale corresponding to the top of inertial range, the lowest-order contributions to the renormalized perturbation expansion give asymptotically exact equations for the large-scale features of the flow. The quality of the derived models is demonstrated on a few examples of complex flows. At the small scales \(\varDelta < L\), an infinite number of \(O(1)\) non-linear terms, generated by the procedure invalidate low-order models widely used for Large-Eddy-Simulations (LES) of turbulent flows.

Single-crystal diamond nanomechanical resonators are being developed for countless applications. A number of these applications require that the resonator be operated in a fluid, that is, a gas or a liquid. Here, we investigate the fluid dynamics of single-crystal diamond nanomechanical resonators in the form of nanocantilevers. First, we measure the pressure-dependent dissipation of diamond nanocantilevers with different linear dimensions and frequencies in three gases, He, N2, and Ar. We observe that a subtle interplay between the length scale and the frequency governs the scaling of the fluidic dissipation. Second, we obtain a comparison of the surface accommodation of different gases on the diamond surface by analyzing the dissipation in the molecular flow regime. Finally, we measure the thermal fluctuations of the nanocantilevers in water and compare the observed dissipation and frequency shifts with theoretical predictions. These findings set the stage for developing diamond nanomechanical resonators operable in fluids.