Spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence

The pseudospectral method, in conjunction with a new technique for obtaining scaling exponents $\zeta_n$ from the structure functions $S_n(r)$, is presented as an alternative to the extended self-similarity (ESS) method and the use of generalized structure functions. We propose plotting the ratio $|S_n(r)/S_3(r)|$ against the separation $r$ in accordance with a standard technique for analysing experimental data. This method differs from the ESS technique, which plots $S_n(r)$ against $S_3(r)$, with the assumption $S_3(r) \sim r$. Using our method for the particular case of $S_2(r)$ we obtain the new result that the exponent $\zeta_2$ decreases as the Taylor-Reynolds number increases, with $\zeta_2 \to 0.679 \pm 0.013$ as $R_{\lambda} \to \infty$. This supports the idea of finite-viscosity corrections to the K41 prediction for $S_2$, and is the opposite of the result obtained by ESS. The pseudospectral method also permits the forcing to be taken into account exactly through the calculation of the energy input in real space from the work spectrum of the stirring forces.Comment: 31 pages including appendices, 10 figure

Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation

Procedures for time-ordering the covariance function, as given in a previous paper (K. Kiyani and W.D. McComb Phys. Rev. E 70, 066303 (2004)), are extended and used to show that the response function associated at second order with the Kraichnan-Wyld perturbation series can be determined by a local (in wavenumber) energy balance. These time-ordering procedures also allow the two-time formulation to be reduced to time-independent form by means of exponential approximations and it is verified that the response equation does not have an infra-red divergence at infinite Reynolds number. Lastly, single-time Markovianised closure equations (stated in the previous paper above) are derived and shown to be compatible with the Kolmogorov distribution without the need to introduce an ad hoc constant.Comment: 12 page

Energy transfer and dissipation in forced isotropic turbulence

A model for the Reynolds number dependence of the dimensionless dissipation rate $C_{\varepsilon}$ was derived from the dimensionless K\'{a}rm\'{a}n-Howarth equation, resulting in $C_{\varepsilon}=C_{\varepsilon, \infty} + C/R_L + O(1/R_L^2)$, where $R_L$ is the integral scale Reynolds number. The coefficients $C$ and $C_{\varepsilon,\infty}$ arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to $R_L=5875$ ($R_\lambda=435$), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law $R_L^n$ with exponent value $n = -1.000\pm 0.009$, and that this decay of $C_{\varepsilon}$ was actually due to the increase in the Taylor surrogate $U^3/L$. The model equation was fitted to data from the DNS which resulted in the value $C=18.9\pm 1.3$ and in an asymptotic value for $C_\varepsilon$ in the infinite Reynolds number limit of $C_{\varepsilon,\infty} = 0.468 \pm 0.006$.Comment: 26 pages including references and 6 figures. arXiv admin note: text overlap with arXiv:1307.457

Re-examination of the infra-red properties of randomly stirred hydrodynamics

Dynamic renormalization group (RG) methods were originally used by Forster, Nelson and Stephen (FNS) to study the large-scale behaviour of randomly-stirred, incompressible fluids governed by the Navier-Stokes equations. Similar calculations using a variety of methods have been performed since, but have led to a discrepancy in results. In this paper, we carefully re-examine in $d$-dimensions the approaches used to calculate the renormalized viscosity increment and, by including an additional constraint which is neglected in many procedures, conclude that the original result of FNS is correct. By explicitly using step functions to control the domain of integration, we calculate a non-zero correction caused by boundary terms which cannot be ignored. We then go on to analyze how the noise renormalization, absent in many approaches, contributes an ${\mathcal O}(k^2)$ correction to the force autocorrelation and show conditions for this to be taken as a renormalization of the noise coefficient. Following this, we discuss the applicability of this RG procedure to the calculation of the inertial range properties of fluid turbulence.Comment: 16 pages, 6 figure

Texture, twinning and metastable "tetragonal" phase in ultrathin films of HfO₂ on a Si substrate

Thin HfO<sub>2</sub> films grown on the lightly oxidised surface of (100) Si wafers have been examined using dark-field transmission electron microscopy and selected area electron diffraction in plan view. The polycrystalline film has a grain size of the order of 100 nm and many of the grains show evidence of twinning on (110) and (001) planes. Diffraction studies showed that the film had a strong [110] out-of-plane texture, and that a tiny volume fraction of a metastable (possibly tetragonal) phase was retained. The reasons for the texture, twinning and the retention of the metastable phase are discussed

Gauge symmetry and Slavnov-Taylor identities for randomly stirred fluids

The path integral for randomly forced incompressible fluids is shown to have an underlying Becchi-Rouet-Stora (BRS) symmetry as a consequence of Galilean invariance. This symmetry must be respected to have a consistent generating functional, free from both an overall infinite factor and spurious relations amongst correlation functions. We present a procedure for respecting this BRS symmetry, akin to gauge fixing in quantum field theory. Relations are derived between correlation functions of this gauge fixed, BRS symmetric theory, analogous to the Slavnov-Taylor identities of quantum field theory.Comment: 5 pages, no figures, In Press Physical Review Letters, 200

Crystal Structure of the ZrO Phase at Zirconium/Zirconium Oxide Interfaces

Zirconium-based alloys are used in water-cooled nuclear reactors for both nuclear fuel cladding and structural components. Under this harsh environment, the main factor limiting the service life of zirconium cladding, and hence fuel burn-up efficiency, is water corrosion. This oxidation process has recently been linked to the presence of a sub-oxide phase with well-defined composition but unknown structure at the metal–oxide interface. In this paper, the combination of first-principles materials modeling and high-resolution electron microscopy is used to identify the structure of this sub-oxide phase, bringing us a step closer to developing strategies to mitigate aqueous oxidation in Zr alloys and prolong the operational lifetime of commercial fuel cladding alloys

A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence

A statistical closure of the Navier-Stokes hierarchy which leads to equations for the two-point, two-time covariance of the velocity field for stationary, homogeneous isotropic turbulence is presented. It is a generalisation of the self-consistent field method due to Edwards (1964) for the stationary, single-time velocity covariance. The probability distribution functional P [u, t] is obtained, in the form of a series, from the Liouville equation by means of a perturbation expansion about a Gaussian distribution, which is chosen to give the exact two-point, two-time covariance. The triple moment is calculated in terms of an ensemble-averaged infinitesimal velocity-field propagator, and shown to yield the Edwards result as a special case. The use of a Gaussian zero-order distribution has been found to justify the introduction of a fluctuation-response relation, which is in accord with modern dynamical theories. In a sense this work completes the analogy drawn by Edwards between turbulence and Brownian motion. Originally Edwards had shown that the noise input was determined by the correlation of the velocity field with the externally applied stirring forces but was unable to determine the system response. Now we find that the system response is determined by the correlation of the velocity field with internal quasi-entropic forces. This analysis is valid to all orders of perturbation theory, and allows the recovery of the Local Energy Transfer (LET) theory, which had previously been derived by more heuristical methods. The LET theory is known to be in good agreement with experimental results. It is also unique among two-point statistical closures in displaying an acceptable (i.e. non-Markovian) relationship between the transfer spectrum and the system response, in accordance with experimental results. As a result of the latter property, it is compatible with the Kolmogorov (K41) spectral phenomenology