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We study the free decay of incompressible homogeneous nonaxisymmetric turbulence with and without helicity, by using direct numerical simulations (DNSs). Its initial energy spectrum E(k), say at t = t0, is given by Ck2 + o(k2) in a sufficiently low k-range, where k is the wavenumber, and C is a positive constant independent of k. We focus our attention to turbulence whose anisotropy is generated by the action of simple mean shear, and consider the decay after the sudden release of the mean shear. The DNSs suggest that at a stage where the turbulence evolves in a self-similar way, the large-scale anisotropy characterized by statistical quantities including the relative velocity correlations is persistent, irrespective of the presence or absence of the initial helicity. The DNSs also suggest that E(k) keeps the form Ck2 + o(k2) at low k for t ≥ t0, and C is time-independent. These DNS results are in accordance with those by a theory about a generalization of a turbulence class of what was argued by Saffman [J. Fluid Mech. 27, 581 (1967)].

In high-Reynolds-number turbulence the spatial distribution of velocity fluctuation at small scales is strongly non-uniform. In accordance with the non-uniformity, the distributions of the inertial and viscous forces are also non-uniform. According to direct numerical simulation (DNS) of forced turbulence of an incompressible fluid obeying the Navier–Stokes equation in a periodic box at the Taylor microscale Reynolds number $R_\lambda \approx 1100$ , the average $\langle R_{loc}\rangle$ over the space of the ‘local Reynolds number’ $R_ {loc}$ , which is defined as the ratio of inertial to viscous forces at each point in the flow, is much smaller than the conventional ‘Reynolds number’ given by $Re \equiv UL/\nu$ , where $U$ and $L$ are the characteristic velocity and length of the energy-containing eddies, and $\nu$ is the kinematic viscosity. While both conditional averages of the inertial and viscous forces for a given squared vorticity $\omega ^{2}$ increase with $\omega ^{2}$ at large $\omega ^{2}$ , the conditional average of $R_ {loc}$ is almost independent of $\omega ^{2}$ . A comparison of the DNS field with a random structureless velocity field suggests that the increase in the conditional average of $R_ {loc}$ with $\omega ^{2}$ at large $\omega ^{2}$ is suppressed by the Navier–Stokes dynamics. Something similar is also true for the conditional averages for a given local energy dissipation rate per unit mass. Certain features of intermittency effects such as that on the $Re$ dependence of $\langle R_{loc}\rangle$ are explained by a multi-fractal model by Dubrulle (J. Fluid Mech., vol. 867, 2019, P1).

High-resolution direct numerical simulations of incompressible turbulence with Taylor-scale Reynolds number up to 2250 show that there exists a scaling-range (approximately $100lr/\ensuremath{\eta}l400$), in which the two-point second order velocity structure functions fit well to a simple power law scaling with respect to the distance $r$ between the two points, where $\ensuremath{\eta}$ is the Kolmogorov length scale. However, the prefactor of the power law depends on the Reynolds number.

We study the mixing of a passive scalar in homogeneous turbulent flow with and without anisotropic external force. It is assumed that no scalar source exists and the scalar spectrum at an initial time instant t0 is given by the form Ck^2+o(k^2) at the wave number k→0, where C is independent of k. We have performed direct numerical simulations (DNSs) of the mixing, in which the initial integral length scales of the scalar field are comparable to those of the velocity field and the Schmidt number is unity. The DNSs show that even though the large-scale anisotropy of the velocity field grows with time owing to the external force, its scalar field counterpart remains almost unchanged, i.e., frozen with respect to time, in a state where the scalar field evolves in a self-similar manner. The degree of the anisotropy of each of the velocity and scalar fields is measured by the ratios of the integral length scales in different Cartesian directions. The DNSs also suggest that the scalar spectrum keeps the form Ck^2+o(k^2) at small k for time t(≥t0), and that C is time independent.

The idea of linear response theory well known in the statistical mechanics for thermal equilibrium systems is applied to one-point statistics in the inertial sublayer of wall-bounded turbulence (WBT). A close analogy between the energy transfer from large to small scales in isotropic turbulence and the momentum transfer in the wall normal direction in WBT plays a key role in the application. The application gives estimates of the influence of the finite Reynolds number on the statistics. The estimates are consistent with data by high-resolution direct numerical simulations of turbulent channel flow.

Ferrocene-terminated self-assembled monolayers (SAMs) have been widely studied in the past quarter century to reveal the electrochemical properties of chemically modified electrodes. It has been well-known that the formal potential of the system strongly depends on the local environment of the ferrocene moiety. Although electronic states of ferrocene-terminated SAMs should directly affect the electrochemical properties, knowledge concerning electronic structures with respect to different local environment is very limited. In this study, we performed density functional theory calculations of ferrocene-terminated SAMs with different coadsorbed species to reveal the relationship between the electronic structures and the local environment of ferrocene moieties. Depending on the local electrostatic potential, density of states derived from the highest-occupied molecular orbital (HOMO) and its vicinities (HOMO–1 and HOMO–2) of the ferrocene moiety were found to largely shift with respect to the electrode Fermi ...

We consider freely decaying homogeneous anisotropic turbulence whose energy spectrum E(k) at k→0 is given by E(k)=Ck2+o(k2) at an initial instant, where k is the wave number and C is a k-independent positive number. An argument is given to show that there are an infinite number of invariants characterizing the large-scale structure of the turbulence. This is a generalization of Saffman's argument, which shows the existence of a finite number of invariants [P. G. Saffman, J. Fluid Mech. 27, 581 (1967)]. By applying a similar argument to homogeneous anisotropic passive scalar turbulence without any scalar source, we show that there are an infinite number of invariants characterizing the large-scale structure of passive scalar fields. Theoretical analysis based on the invariance and a self-similarity assumption for the large-scale evolution shows that the anisotropy of the velocity and passive scalar fields is persistent at large scales. The decay laws of the velocity and passive scalar fields are derived by a simple dimensional analysis.

The electrochemical properties of chemically modified electrodes have long been a significant focus of research. Although the electronic states are directly related to the electrochemical properties, there have been only limited systematic efforts to reveal the electronic structures of adsorbed redox molecules with respect to the local environment of the redox center. In this study, density functional theory (DFT) calculations were performed for ferrocene-terminated self-assembled monolayers with different electron-donating abilities, which can be regarded as the simplest class of chemically modified electrodes. We revealed that the local electrostatic potentials, which are changed by the electron donating/withdrawing functional groups at the ferrocene moiety and the dipole field of coadsorbed inert molecules, practically determine the density of states derived from the highest occupied molecular orbital (HOMO) and its vicinities (HOMO−1 and HOMO−2) with respect to the electrode Fermi level. Therefore, to design new, sophisticated electrodes with chemical modification, one should consider not only the electronic properties of the constituent molecules, but also the local electrostatic potentials formed by these molecules and coadsorbed inert molecules.

In theories of turbulence, it is assumed that in an appropriate small-scale range, called inertial subrange (ISR), of turbulence at sufficiently high Reynolds number (Re) and far from fluid boundaries, the statistics are isotropic and have a certain kind of universality that is insensitive to Re and the details of large-scale flow conditions. However, in any real turbulent flow, Re must be finite however large it may be, and the statistics are in general anisotropic, in a strict sense, due to the influence of anisotropic external force or factor, say disturbance X, such as the mean shear, density stratification, etc.. Attempts have been made to estimate the influence of X on the statistics of the ISR of turbulent flows in unbounded domain at high Reynolds number, by using an idea similar to that of Linear Response Theory (LRT) that has been used in the study of systems at or near thermal equilibrium state. The use of the idea gives estimates of the influence of X on one-time averages such as the correlation spectrum in the ISR, where u_i(k) is the Fourier transform of the fluctuating velocity component in the i-th Cartesian direction and k is the wave vector. Recently, the idea of LRT has been applied to statistics in a layer called inertial sublayer (ISL) of turbulent channel flow, which is one of the most canonical wall bounded flows. The application is based on a similarity between the energy cascade in the ISR and the momentum transfer in the ISL. The application gives estimates of the influence of the finiteness of viscosity (or Re) and the distance from the wall on the one-point one-time averages , and in the ISL, where u, v and w are respectively the fluctuating velocities in the stream-wise, span-wise and wall-normal directions. The idea of LRT is applicable, at least formally, also to turbulent diffusion. It gives estimates of the influence of the finite Re on the single- and pair- particle dispersion in high-Re turbulence for an appropriate time interval.

The position- and direction-dependence of length scales in turbulent channel flow was studied using the data from a series of direct numerical simulations (DNSs) of turbulent channel flow with the ...

Three-dimensional tomographic time dependent PIV measurements of high Reynolds number (Re) laboratory turbulence are presented which show the existence of long-lived, highly sheared thin layer eddy structures with thickness of the order of the Taylor microscale and internal fluctuations. Highly sheared layer structures are also observed in direct numerical simulations of homogeneous turbulence at higher values of Re (Ishihara et al., Annu Rev Fluid Mech 41:165–180, 2009). But in the latter simulation, where the fluctuations are more intense, the layer thickness is greater. A rapid distortion model describes the structure and spectra for the velocity fluctuations outside and within ‘significant’ layers; their spectra are similar to the Kolmogorov (C R Acad Sci URSS 30:299–303, 1941) and Obukhov (Dokl Akad Nauk SSSR 32:22–24, 1941) statistical model (KO) for the whole flow. As larger-scale eddy motions are blocked by the shear layers, they distort smaller-scale eddies leading to local zones of down-scale and up-scale transfer of energy. Thence the energy spectrum for high wave number k is \(E_X (k)\sim Bk^{-2p}\). The exponent p depends on the forms of the large eddies. The non-linear interactions between the distorted inhomogeneous eddies produce a steady local structure, which implies that 2p = 5/3 and a flux of energy into the thin-layers balancing the intense dissipation, which is much greater than the mean \(\left \). Thence \(B\sim\left ^{2/3}\) as in KO. Within the thin layers the inward flux energises extended vortices whose thickness and spacing are comparable with the viscous microscale. Although peak values of vorticity and velocity of these vortices greatly exceed those based on the KO scaling, the form of the viscous range spectrum is consistent with their model.

Using direct numerical simulation of turbulence in a periodic box driven by homogeneous forcing, with a maximum of 40963 grid points and Taylor micro-scale Reynolds numbers R λ up to 1131, it is shown that there is a transition in the forms of the significant, high vorticity, intermittent structures, from isolated vortices when R λ is less than 102 to complex thin-shear layers when R λ exceeds about 103. Both the distance between the layers and their widths are comparable with the integral length scale. The thickness of each of the layers is of the order of the Taylor micro-scale λ. Across the layers the velocity ‘jumps’ are of the order of the rms velocity u o of the whole flow. Within the significant layers, elongated vortical eddies are generated, with microscale thickness l v ~10η ≪ λ, with associated peak values of vorticity as large as 35ω rms and with velocity jumps as large as 3.4u o , where η is the Kolmogorov micro scale and ω rms the rms vorticity. The dominant vortical eddies in the layers, which are approximately parallel to the vorticity averaged over the layers, are separated by distances of order l v . The close packing leads to high average energy dissipation inside the layer, as large as ten times the mean rate of energy dissipation over the whole flow. The interfaces of the layers act partly as a barrier to the fluctuations outside the layer. However, there is a net energy flux into the small scale eddies within the thin layers from the larger scale motions outside the layer.

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which $E(k\ensuremath{\rightarrow} 0)\ensuremath{\sim} {k}^{2} $, where $E$ is the energy spectrum and $k$ the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor ${\Phi }_{ij} (\mathbi{k})$ at small $k$. These are ${M}_{\parallel } = {\Phi }_{\parallel } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$ and ${M}_{\perp } = 2{\Phi }_{\perp } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$, where the subscripts $\parallel $ and $\perp $ indicate quantities parallel and perpendicular to the axis of symmetry. Since ${M}_{\parallel } \ensuremath{\sim} { u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $ and ${M}_{\perp } \ensuremath{\sim} { u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $, $u$ and $\ell $ being integral scales, self-similarity of the large scales (when it applies) demands ${ u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $ and ${ u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $. This, in turn, requires that ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that ${M}_{\parallel } $ and ${M}_{\perp } $ are indeed invariants and that, in the fully developed state, ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} = \text{constant} $. Somewhat surprisingly, the same simulations also show that ${\ell }_{\parallel } / {\ell }_{\perp } $ is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ and ${\ell }_{\parallel } / {\ell }_{\perp } $ are both constant, the integral scales should evolve as ${ u}_{\perp }^{2} \ensuremath{\sim} { u}_{\parallel }^{2} \ensuremath{\sim} {t}^{\ensuremath{-} 6/ 5} $ and ${\ell }_{\perp } \ensuremath{\sim} {\ell }_{\parallel } \ensuremath{\sim} {t}^{2/ 5} $, irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.

We measured the correlation spectrum ${\hat {Q} }_{p} (\mathbi{k})$ of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number ${R}_{\lambda } $ up to ${\simeq }700$ by a newly developed pressure probe with spatial and temporal resolutions that are sufficient to analyse inertial-subrange statistics. The influence of the mean velocity gradient tensor ${S}_{ij} $ in the mixing layer, which is almost constant near its centreline, is studied using an idea similar to that underlying the linear response theory developed in statistical mechanics for systems at or near thermal equilibrium. If we write the spectrum ${\hat {Q} }_{p} (\mathbi{k})$ as ${\hat {Q} }_{p} (\mathbi{k})= { \hat {Q} }_{p}^{(0)} (\mathbi{k})+ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$, where ${ \hat {Q} }_{p}^{(0)} (\mathbi{k})$ is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small ${S}_{ij} $ the deviation $ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$ due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to ${S}_{ij} $, $k$ and the mean energy dissipation rate. We also measured the pressure–velocity and velocity–velocity correlation spectra. Deviations from isotropy due to shear are shown to be approximately proportional to ${S}_{ij} $ at large ${R}_{\lambda } $.

The roles of the Coriolis force and the convection associated with the fluid motion in the formation of columnar eddies in freely decaying homogeneous rotating turbulence at a moderate Rossby number are studied by direct numerical simulation of the Navier–Stokes equations in a periodic box. The simulated field is compared with a series of artificial fields generated by switching off the nonlinear and viscous terms in the Navier–Stokes equation at given instants. The comparison shows that, without the nonlinear convection effect, the Coriolis force cannot sustain the substantial growth in the direction parallel to the rotational axis of the length scale defined on the basis of the two-point correlation of the square of the vorticity, i.e. cannot sustain the formation of the columnar eddies. The length scale characterizes well the intuitive impression from visualization of flow obeying the dynamics with or without the nonlinear effect. It is shown that the lack of substantial growth is insensitive to the scale of the eddies, the box size and viscosity, at least in the case studied here.

We consider the large-scale structure of freely decaying incompressible homogeneous anisotropic helical turbulence, whose energy spectrum E(k) is given by E(k)=Ck^2+o(k^2) at k→0. Here k=|k|,k is the wave vector, and C is a dynamical invariant. The helicity spectrum H(k) is given by H(k)=Chk^3+o(k^3) at k→0, where Ch is in general nonzero in helical turbulence. By generalizing Saffman's argument for nonhelical turbulence [Saffman, J. Fluid Mech. 27, 581 (1967)] to helical turbulence, it is shown that Ch is another dynamical invariant. We present a theoretical analysis based on the time independence of the O(k^0) term of the velocity correlation spectral tensor at k→0 and a self-similarity assumption of flow evolution at large scales including the energy containing range scales. The analysis suggests that if the O(k^0) term is reflection asymmetric at an initial instant, the turbulence does not relax to any reflection symmetric state at the large scales. A simple dimensional analysis yields the decay rates of the helicity and kinetic energy in the fully developed turbulence state. The theoretical results agree with results obtained by direct numerical simulation of incompressible helical turbulence in a periodic box.

Two-dimensional direct numerical simulation (DNS) was applied to auto-ignition of n-heptane/air mixture. The diffusion of intermediate species was studied in terms of the influence on the ignition delay. The chemical reaction process. The simplified reaction mechanism was used in this study. Likewise, the influence of turbulence was investigated by comparing the results of 0 dimensional simulation and two-dimensional direct simulation of non-turbulence condition. In conclusions, the ignition delay was strongly affected by the different diffusion speeds of intermediate species. It was shown that the outflow of OH radical having larger diffusion coefficient decreases the reaction rate at the high temperature point. On the contrary, the reaction rate was enhanced by the influx of OH radical at the low temperature point.

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

We review studies of the statistics of isotropic turbulence in an incompressible fluid at high Reynolds numbers using direct numerical simulation (DNS) from the viewpoint of fundamental physics. The Reynolds number achieved by the largest DNS, with 40963 grid points, is comparable with the largest Reynolds number in laboratory experiments. The high-quality DNS data in the inertial subrange and the dissipative range enable the examination of detailed statistics at small scales, such as the normalized energy-dissipation rate, energy and energy-flux spectra, the intermittency of the velocity gradients and increments, scaling exponents, and flow-field structure. We emphasize basic questions of turbulence, universality in the sense of Kolmogorov's theory, and the dependence of the statistics on the Reynolds number and scale.

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

A review is made of a two-point closure approximation called Lagrangian renormalized approximation (LRA), which is derived by a simple truncation of systematic Lagrangian renormalized expansions and is free from any ad hoc adjusting parameters or quantities. Emphasis is given to three key ideas: (i) representative, (ii) mapping and (iii) renormalized expansion underlying the LRA. The nature of the method of the expansion is explained by simple models, and general properties of the expansion as well as the truncated approximation are discussed in a general framework. A brief survey of the consequences of the LRA applied to three- and two-dimensional turbulences and passive scalar fields convected by turbulence is also presented.

The statistics of energy transfer is studied by using the data of a series of high-resolution direct numerical simulations of incompressible homogeneous turbulence in a periodic box with the Taylor micro-scale Reynolds number R λ and grid points up to approximately 1130 and 4096 3 , respectively. The data show that the energy transfer T across the wave number k is highly intermittent and the skewness S and flatness F of T increase with k approximately as S ∝( k L ) α S , F ∝( k L ) α F in the inertial subrange, where α S ∼2/3, α F ∼1 and L the characteristic length scale of energy containing eddies. The comparison between the statistics of T , the energy dissipation rate e and its average e r over a domain of scale r shows that T is less intermittent than e, while there is a certain similarity between the probability distribution functions of T and e r .

The energy spectrum in the near dissipation range of turbulence is studied by analyzing the data of a series of high-resolution direct numerical simulations of incompressible homogeneous turbulence...