arXiv subjects

W. D. McComb

Publications and source records attributed to W. D. McComb.

14 recordsLinked to original sources

arXiv2021

Does intermittency affect the inertial transfer rate in stationary isotropic turbulence?

Direct numerical simulations of the forced Navier-Stokes equations were performed, in which each shell-averaged quantity evolved from a value appropriate to an initial Gaussian state, to fluctuate about a mean value. Once the transient had passed, mean values (and their associated statistics) were obtained by sampling the evolved time-series at intervals of the order of an eddy-turnover time. This was repeated for a range of Taylor-Reynolds numbers from 10.6 to 335.2. With increasing Reynolds number, our results for energy spectra, transfer spectra and inertial flux supported the Kolmogorov-Obukhov picture of turbulent energy transfer. In particular, we observed the onset of scale-invariance of the inertial flux, accompanied by the onset of the -5/3 power law in the energy spectrum for the corresponding inertial range of wavenumbers. Detailed comparisons showed that our results were in agreement with those found in many other investigations. Flow visualization methods were used to study the internal intermittency. This phenomenon is seen in single realisations but was found to average out with increasing number of realisations under ensemble-averaging. Following a critical review of the literature relating to the controversy about intermittency effects versus finite-Reynolds number corrections, it was concluded that, for the case of stationary isotropic turbulence, internal intermittency cannot affect the Kolmogorov-Obukhov picture, as this is constructed entirely in terms of ensemble-averaged mean quantities.

physics.flu-dyn↗

arXiv2021

The infinite Reynolds number limit and the quasi dissipative anomaly

From a critical review of Onsager's pioneering paper which was published in 1949 we conclude that his analysis was at odds with other workers in the field because instead of taking the infinite Reynolds number limit he simply set the viscosity equal to zero which is not the same thing.

physics.flu-dyn↗

arXiv2020

A modified Lin equation for the energy balance in isotropic turbulence

A new form of the well known Lin equation for the energy balance in isotropic turbulence is given which takes account of the filtered-partitioned transfer spectra which were used to resolve the scale paradox. It is argued that this new formulation provides a new starting point for both theoretical and numerical methods which may lead to a better understanding of the underlying energy processes in turbulence.

physics.flu-dyn↗

arXiv2018

The effect of Kolmogorov (1962) scaling on the universality of turbulence energy spectra

It has long been established that turbulence energy spectra scale on the Kolmogorov (1941) variables over a wide range of Reynolds numbers and in vastly different physical systems, depending only on the dissipation rate, the kinematic viscosity and the wavenumber. On the other hand, the analogous study of structure functions in real space is strongly influenced by the Kolmogorov (1962) refined theory, which introduced a dependence on a large length scale Lext, characteristic of the system size. If such a dependence exists it is surprising that it does not show up in the study of wavenumber spectra, where the different physical systems suggest that Lext can vary by up to five orders of magnitude. Here we use an order of magnitude calculation to suggest that scaling according to Kolmogorov (1962) would destroy the observed asymptotic universality of energy spectra at large wavenumbers.

physics.flu-dyn↗

arXiv2018

Onset criteria for freely decaying isotropic turbulence

From DNS of turbulence decaying from specified initial conditions for the range of initial Taylor-Reynolds numbers 2.58 < Rλ(0) < 358.6, it was found that the shape of the iconic curve of dimensionless dissipation versus Reynolds number depended strongly on the choice of measurement time. For our preferred time, a composite based on peak values in the dissipation and inertial transfer curves, the result was virtually identical to the forced, stationary case. In the course of studying onset criteria, we found that the exponent for the power-law decay of the energy decreased with increasing Reynolds number and lay in the range 1.35 < n < 2.60. An additional run was performed, using the data from a stationary, forced simulation with Rλ = 335 for the initial condition. The results of this suggested that the time taken for energy to pass through the cascade was about one half of an initial eddy turnover time.

physics.flu-dyn↗

arXiv2017

The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence

An expression for the dimensionless dissipation rate was derived from the Karman-Howarth equation by asymptotic expansion of the second- and third- order structure functions in powers of the inverse Reynolds number. The implications of the time-derivative term for the assumption of local stationarity (or local equilibrium) which underpins the derivation of the Kolmogorov `4/5' law for the third-order structure function were studied. It was concluded that neglect of the time-derivative cannot be justified by reason of restriction to certain scales (the inertial range) nor to large Reynolds numbers. In principle, therefore, the hypothesis cannot be correct, although it may be a good approximation. It follows, at least in principle, that the quantitative aspects of the hypothesis of local stationarity could be tested by a comparison of the asymptotic dimensionless dissipation rate for free decay with that for the stationary case. But in practice this is complicated by the absence of an agreed evolution time for making the measurements during the decay. However, we can assess the quantitative error involved in using the hypothesis by comparing the exact asymptotic value of the dimensionless dissipation in free decay calculated on the assumption of local stationarity to the experimentally determined value (e.g. by means of direct numerical simulation), as this relationship holds for all measuring times. Should the assumption of local stationarity lead to significant error, then the `4/5' law needs to be corrected. Despite this, scale invariance in wavenumber space appears to hold in the formal limit of infinite Reynolds numbers, which implies that the `-5/3' energy spectrum does not require correction in this limit.

physics.flu-dyn↗

arXiv2015

Self-organization and transition to turbulence in isotropic fluid motion driven by negative damping at low wavenumbers

We observe a symmetry-breaking transition from a turbulent to a self-organized state in direct numerical simulation of the Navier-Stokes equation at very low Reynolds number. In this self-organized state the kinetic energy is contained only in modes at the lowest resolved wavenumber, the skewness vanishes, and visualization of the flows shows a lack of small-scale structure, with the vorticity and velocity vectors becoming aligned (a Beltrami flow).

physics.flu-dyn↗

arXiv2015

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 Kármán-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 $R_L=5875$ ($R_λ=435$), 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 $C_{\varepsilon,\infty} = 0.468 \pm 0.006$.

physics.flu-dyn↗

arXiv2014

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 $ζ_n$ 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 $ζ_2$ decreases as the Taylor-Reynolds number increases, with $ζ_2 \to 0.679 \pm 0.013$ as $R_λ \to \infty$. 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.

physics.flu-dyn↗

arXiv2014

The infrared properties of the energy spectrum in freely decaying isotropic turbulence

The low wavenumber expansion of the energy spectrum takes the well known form: $ E(k,t) = E_2(t) k^2 + E_4(t) k^4 + ... $, where the coefficients are weighted integrals against the correlation function $C(r,t)$. We show that expressing $E(k,t)$ in terms of the longitudinal correlation function $f(r,t)$ immediately yields $E_2(t)=0$ by cancellation. We verify that the same result is obtained using the correlation function $C(r,t)$, provided only that $f(r,t)$ falls off faster than $r^{-3}$ at large values of $r$. As power-law forms are widely studied for the purpose of establishing bounds, we consider the family of model correlations $f(r,t)=α_n(t)r^{-n}$, for positive integer $n$, at large values of the separation $r$. We find that for the special case $n=3$, the relationship connecting $f(r,t)$ and $C(r,t)$ becomes indeterminate, and (exceptionally) $E_2 \neq 0$, but that this solution is unphysical in that the viscous term in the Kármán-Howarth equation vanishes. Lastly, we show that $E_4(t)$ is independent of time, without needing to assume the exponential decrease of correlation functions at large distances.

physics.flu-dyn↗

arXiv2012

Eulerian Field-Theoretic Closure Formalisms for Fluid Turbulence

The formalisms of Wyld [2] and Martin, Siggia, and Rose (MSR) [3] address the closure problem of a statistical treatment of homogeneous isotropic turbulence (HIT) based on techniques primarily developed for quantum field theory. In the Wyld formalism, there is a well-known double-counting problem, for which an {\it ad hoc} solution was suggested by Lee [44]. We show how to implement this correction in a more natural way from the basic equations of the formalism. This leads to what we call the "Improved Wyld-Lee Renormalized Perturbation Theory". MSR had noted that their formalism had more vertex functions than Wyld's formalism and based on this felt Wyld's formalism was incorrect. However a careful comparison of both formalisms here shows that the Wyld formalism follows a different procedure to that of the MSR formalism and so the treatment of vertex corrections appears in different ways in the two formalisms. Taking that into account, along with clarifications made to both formalisms, we find that they are equivalent and we demonstrate this up to fourth order.

cond-mat.stat-mech↗

arXiv2006

Non-local modulation of the energy cascade in broad-band forced turbulence

Classically, large-scale forced turbulence is characterized by a transfer of energy from large to small scales via nonlinear interactions. We have investigated the changes in this energy transfer process in broad-band forced turbulence where an additional perturbation of flow at smaller scales is introduced. The modulation of the energy dynamics via the introduction of forcing at smaller scales occurs not only in the forced region but also in a broad range of length-scales outside the forced bands due to non-local triad interactions. Broad-band forcing changes the energy distribution and energy transfer function in a characteristic manner leading to a significant modulation of the turbulence. We studied the changes in this transfer of energy when changing the strength and location of the small-scale forcing support. The energy content in the larger scales was observed to decrease, while the energy transport power for scales in between the large and small scale forcing regions was enhanced. This was investigated further in terms of the detailed transfer function between the triad contributions and observing the long-time statistics of the flow. The energy is transferred toward smaller scales not only by wavenumbers of similar size as in the case of large-scale forced turbulence, but by a much wider extent of scales that can be externally controlled.

physics.flu-dyn↗

arXiv2005

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.

nlin.CD↗

arXiv2000

A renormalized expression for the turbulent energy dissipation rate

Conditional elimination of degrees of freedom is shown to lead to an exact expression for the rate of turbulent energy dissipation in terms of a renormalized viscosity and a correction. The correction is neglected on the basis of a previous hypothesis [W.D. McComb and C. Johnston, J.Phys.A v33 L15 (2000)] that there is a range of parameters for which a quasi-stochastic estimate is a good approximation to the exact conditional average. This hypothesis was tested by a perturbative calculation to second order in the local Reynolds number, and the Kolmogorov prefactor (taken as a measure of the renormalized dissipation rate) was found to reach a fixed point which was insensitive to initial values of the kinematic viscosity and to values of the spatial rescaling factor h in the range 0.4 <= h <= 0.8.

cond-mat↗