Inertial Range Scaling Research Articles

Scaling relations for a randomly advected passive scalar field.

A recent ansatz for dissipation terms gave anomalous inertial-range scaling exponents ({proportional_to}{ital n}{sup 1/2},{ital n}{r_arrow}{infinity}) for the {ital n}th-order structure functions of a passive scalar field advected by a random velocity field. Analysis of a series expansion for the conditional mean of a dissipation term suggests that the ansatz gives the only possible anomalous scaling. Anomaly of inertial-range scaling is supported by realizability inequalities on the dissipation field. Predictions for conditional means and structure functions are compared with simulations.

Comment on "Extended self-similarity in turbulent flows"

A recent paper [R. Benzi et al., Phys. Rev. E 48, R29 (1993)] concluded that the scaling of turbulent structure functions extends deeper into the viscous subrange than what has been assumed so far. We comment on this notion of ``extended selfsimilarity.'' We verify the (approximate) universal character of the viscous subrange scaling of structure functions but show that it is different from their inertial range scaling. The effect is small (although significant) in laboratory flows and it has profound consequences for the multifractal model.

Direct numerical simulation of two-particle relative diffusion in isotropic turbulence

The relative diffusion of fluid particle pairs in statistically stationary isotropic turbulence is studied by direct numerical simulation, at a Taylor-scale Reynolds number of about 90. The growth of two-particle separation exhibits asymptotic stages at small and large diffusion times. Through the two-particle separation, particle–pair velocity correlations are closely related to the Eulerian spatial structure of the turbulence. At large times, the square of the separation distance has a chi-square probability distribution. At the moderate Reynolds number of the simulations, for this asymptotic distribution to be reached before the particles begin to move independently of each other, the initial separation must be small compared to the Kolmogorov scale. In an inertial frame moving with the initial particle velocities, the velocity increments of two fluid particles become uncorrelated only if their initial velocities are uncorrelated, which requires their initial separation be large compared to the integral length scale. For sufficiently large initial separations, the relative velocity increments and mean-square dispersion in this moving frame display a resemblance to inertial range scaling, but with a proportionality constant that is much smaller than classical estimates. At large times, the degree of preferential alignment between the separation and relative velocity vectors is weak, but the product of the separation distance and the velocity component projected along the separation vector is sustained on average.

The renormalization group method in statistical hydrodynamics

This paper gives a first principles formulation of a renormalization group (RG) method appropriate to study of turbulence in incompressible fluids governed by Navier–Stokes equations. The present method is a momentum-shell RG of Kadanoff–Wilson type based upon the Martin–Siggia–Rose (MSR) field-theory formulation of stochastic dynamics. A simple set of diagrammatic rules are developed which are exact within perturbation theory (unlike the well-known Ma–Mazenko prescriptions). It is also shown that the claim of Yakhot and Orszag (1986) is false that higher-order terms are irrelevant in the ε expansion RG for randomly forced Navier–Stokes (RFNS) with power-law force spectrum F̂(k)=D0k−d+(4−ε). In fact, as a consequence of Galilei covariance, there are an infinite number of higher-order nonlinear terms marginal by power counting in the RG analysis of the power-law RFNS, even when ε≪4. The difficulty does not occur in the Forster–Nelson–Stephen (FNS) RG analysis of thermal fluctuations in an equilibrium NS fluid, which justifies a linear regression law for d≳2. On the other hand, the problem occurs also at the nontrivial fixed point in the FNS Model A, or its Burgers analog, when d<2. The marginal terms can still be present at the strong-coupling fixed point in true NS turbulence. If so, infinitely many fixed points may exist in turbulence and be associated to a somewhat surprising phenomenon: nonuniversality of the inertial-range scaling laws depending upon the dissipation-range dynamics.

Anomalous scaling of velocity structure functions in turbulence: a new approach

In this letter we report experimental evidence about the possibility to derive the n-order structure function (in fully developed turbulence) in terms of locally averaged dissipation rate er, and the third order structure function. The results presented here are a generalization of the 1962 Oboukhov hypothesis about the scaling in the inertial range. The results are in accordance with the Extended Self Similarity (ESS), a concept recently introduced in the literature. In fact we reproduce the n-order structure function not only in the region where ESS is valid but in a much broader range.

THE MULTIFRACTAL FORMALISM REVISITED WITH WAVELETS

The multifractal formalism originally introduced for singular measures is revisited using the wavelet transform. This new approach is based on the definition of partition functions from the wavelet transform modulus maxima. We demonstrate that the f(α) singularity spectrum can be readily determined from the scaling behavior of these partition functions. We show that this method provides a natural generalization of the classical box-counting techniques to fractal functions (the wavelets actually play the role of “generalized boxes”). We report on a systematic comparison between this alternative method and the structure function approach which is commonly used in the context of fully developed turbulence. We comment on the intrinsic limitations of the structure functions which possess fundamental drawbacks and do not provide a full characterization of the singularities of a signal in many cases. We show that our method based on the wavelet transform modulus maxima decomposition works in most situations and is likely to be the ground of a unified multifractal description of singular distributions. Our theoretical considerations are both illustrated on pedagogical examples, e.g., generalized devil staircases and fractional Brownian motions, and supported by numerical simulations. Recent applications of the wavelet transform modulus maxima method to experimental turbulent velocity signals at inertial range scales are compared to previous measurements based on the structure function approach. A similar analysis is carried out for the locally averaged dissipation and the validity of the Kolmogorov’s refined similarity hypothesis is discussed. To conclude, we elaborate on a wavelet based technique which goes further than a simple statistical characterization of the scaling properties of fractal objects and provides a very promising tool for solving the inverse fractal problem, i.e., for uncovering their construction rule in terms of a discrete dynamical system.

Vortical scales for two- and three-dimensional turbulence.

Correlation analysis of two-dimensional (2D) turbulence is performed on the basis of the Navier-Stokes equations. It is found that within the inertial range of scales (L ${\mathrm{Re}}^{\mathrm{\ensuremath{-}}1/2}$\ensuremath{\ll}r\ensuremath{\ll}L; L is the external scale, Re is the Reynolds number) there is a physically distinguished scale ${\mathit{l}}_{\mathit{c}}$=L ${\mathrm{Re}}^{\mathrm{\ensuremath{-}}1/4}$---a natural scale for coherent vortex patches. A hierarchy of vortical scales for 2D and 3D turbulence is found from a covariance analysis of more detailed spatial structure of the vorticity field.

Scaling functions and scaling exponents in turbulence.

We extend the recent work of Sirovich, Smith, and Yakhot (unpublished) and obtain for structure functions of arbitrary order an expression that is uniformly valid for the dissipation as well as the inertial range of scales. We compare the expression with experimental data obtained in a moderate-Reynolds-number turbulent boundary layer and find good agreement. This enables a more definitive determination of the scaling exponents and intermittency corrections than has been possible in the past. The results are substantiated by several consistency checks

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameterɛ, which measures the strength of the infrared divergence: forɛ 2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrodinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.

Statistical balance of vorticity and a new scale for vortical structures in turbulence.

The balance of one-point and two-point statistical characterics of vorticity, is considered on the basis of the Navier-Stokes equations. It is shown that within the inertial range of scales (L ${\mathrm{Re}}^{\mathrm{\ensuremath{-}}3/4}$\ensuremath{\ll}r\ensuremath{\ll}L, L external scale, Re Reynolds number) there is a physically distinguished scale ${\mathit{l}}_{\mathit{s}}$\ensuremath{\sim}L ${\mathrm{Re}}^{\mathrm{\ensuremath{-}}3/10}$. The balance of vortical correlations with scales r\ensuremath{\ge}${\mathit{l}}_{\mathit{s}}$ is directly affected by the large-scale motion. ${\mathit{l}}_{\mathit{s}}$ is a natural length scale for the ``vortex strings,'' observed experimentally and numerically in three-dimensional turbulent flows. The twist of vortex lines in the internal structure of vortex strings is also briefly discussed.

Linear Eddy Modeling of Entrainment and Mixing in Stratus Clouds

Mixing of entrained air in stratus clouds is an important but poorly understood process. It is a crucial ingredient of cloud-top entrainment instability (CEI). CEI has been proposed as a breakup mechanism for stratus clouds. A recently developed model called the linear eddy model was used to simulate mixing of air entrained into stratus clouds. The linear eddy approach involves stochastic simulation on a one-dimensional domain with sufficient resolution to include all physically relevant length scales. In each realization, molecular diffusion is implemented explicitly, while a sequence of statistically independent “rearrangement events” represents the effect of turbulent eddies. Inertial range scaling is incorporated. The linear eddy model was used to simulate the mixing of one or more wisps of entrained air with a specified volume of cloud-topped boundary layer (CTBL) air. The volume was idealized to be a horizontal slab of fluid that travels from the top of the CTBL down to the surface in the descending branch of a large convective eddy. The probability density function of the mixing fraction of entrained air was determined from linear eddy model simulations as a function of time for a mean mixing fraction of 0.05 and three wisp sizes. The effect of the mixing on the mean buoyancy of the downdraft could then be calculated given a specification of the buoyancy as a function of mixing fraction. In the simulations, the entrained air did not completely mix with cloudy air just below the CTBL top, nor was uniform saturation maintained. Furthermore, when buoyancy functions typical of observed CTBLs were used, the mean downdraft buoyancy due to entrainment and mixing integrated over the cloud layer remained positive. This suggests that CEI is unlikely in stratocumulus. An additional conclusion is that using reduced spatial resolutions typical of published large-eddy simulations (LES) of CTBLs in mixing simulations significantly underestimates the buoyancy in the cloud layer near cloud top. This may explain why low-resolution LFS simulations have exhibited CEI under conditions for which CEI is not observed in the atmosphere.

Renormalization group and operator product expansion in turbulence: Shell models.

A general role of the renormalization group (RG) in the theory of fully developed turbulence is proposed, with the simple case of the shell models as an illustrative example. A Wilson-type RG is defined, i.e., a transformation in a space of shell-dynamics ``subgrid models'' with fixed uv cutoff, for a class of theories with fixed mean dissipation and strength of quadratic nonlinearity. It is explained that, if a zero-viscosity limit exists, then its ``subgrid'' dynamics below the cutoff is necessarily (near) a fixed point of the RG transformation. Conversely, any RG fixed-point subgrid model is associated to a zero-viscosity limit. By means of an ``asymptotic completeness'' assumption for the fixed point, a high shell-number expansion is established, analogous to the operator product expansion (OPE) of field theory. This expansion predicts characteristic ``multifractal scaling'' for shell variable moments and also relations between inertial and dissipation range scaling exponents. Furthermore, under the plausible assumption of an ``additive OPE,'' a predicted scaling form for two-point moment correlations is established. The results of this paper are nonperturbative but only of a qualitative character, based upon precise assumptions about the fixed-point theory. However, we also discuss the possibility of an implementation of RG by numerical methods (Monte Carlo, decimation, etc.) or perturbation expansion to test the assumptions and to make a quantitative evaluation of the scaling exponents. The relation of RG to naive ``cascade ansatz'' is also discussed.

Author index

In this study, the stability and diffusive characteristics of second-order Roe-MUSCL in conjunction with three Runge–Kutta-based temporal schemes are investigated for implicit large-eddy simulation (ILES) of the Taylor–Green vortex transitional flow. The excessive dissipation inherent to Roe-MUSCL is mitigated using the Thornber et al. [12] and Bidadi–Rani [8] approaches, referred to as the low-Mach-number and current modifications LM [12] and CM, respectively. The temporal schemes investigated are those of Jameson et al. [3], Shu and Osher [15], and Spiteri and Ruuth [16]. The time evolution of statistics such as turbulent kinetic energy (TKE) and enstrophy are shown for various combinations of modified Roe-MUSCL and temporal schemes at 643 and 1283 grid resolutions. Also, the spectra of TKE, and of the numerical viscosity and dissipation-rate inherent to the modified schemes are presented. In general, it is observed that Shu–Osher method is the most stable among the three time schemes. For the 643 grid case, LM [12] resulted in a stable solution when combined with all three time schemes, whereas CM maintained stability only with Shu–Osher. The dissipation rate spectra reveal that the instability arises due to the insufficient rate of energy transfer from the energy-containing to the inertial-range scales. The instability of Jameson et al. and Spiteri–Ruuth schemes (with CM) is manifested as the larger magnitude of the slope of dissipation rate spectrum at low wavenumbers. For the 1283 grid simulations, temporal evolution of TKE showed that only Shu–Osher with LM remained stable. For the cases where both LM and CM are stable, the former exhibited better accuracy in capturing transition and the subsequent decaying turbulent flow. Estimates of effective Reynolds number from current ILES simulations show good agreement with actual Reynolds numbers in prior direct numerical simulations.

Inertial range scaling of laminar shear flow as a model of turbulent transport

Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport.

Density variations in weakly compressible flows

Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible Navier–Stokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the Kolmogorov–Obukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium.

Intermittency and non-gaussian statistics in turbulence

Intermittency effects in turbulence are discussed from a dynamical point of view. A two-fluid model is developed to describe quantitatively the non-gaussian statistics of turbulence at small scales. With a self-similarity argument, the model gives rise to the entire set of inertial range scaling exponents for normalized velocity structure functions. The results are in excellent agreement with experimental and numerical measurements. The model suggests a physical mechanism of intermittency, namely the self-interaction of turbulence structures.

Scale-dependent intermittency and coherence in turbulence

The statistics of isotropic homogeneous decaying at moderately large Reynolds number are studied in detail using a Fourier-space band-filtering method on flow fields obtained by direct numerical simulation. Two distinct aspects of the non-Gaussianity of the turbulent field are discussed: The nonzero derivative skewness, related to energy transfer, is shown to be essentially a large-scale property, while intermittency measured by the flatness factor is basically a dissipation range phenomenon. In addition, elongated coherent structures of single velocity components at inertial range scales are shown to align with the velocity component, leading to depletion of the nonlinear interaction. Finally, it is shown that the statistics of the dissipation may be simply related to the statistics of the spatial derivative and small-scale velocity fields.

A generalized Langevin model for turbulent flows

A Langevin model appropriate to constant property turbulent flows is developed from the general equation for the fluid particle velocity increment proposed by Pope in an earlier paper [Phys. Fluids 26, 404 (1983)]. This model can be viewed as an analogy between the turbulent velocity of a fluid particle and the velocity of a particle undergoing Brownian motion. It is consistent with Kolmogorov’s inertial range scaling, satisfies realizability, and is consistent with second-order closure models. The objective of the present work is to determine the form of a second-order tensor appearing in the general model equation as a function of local mean quantities. While the model is not restricted to homogeneous turbulence, the second-order tensor is evaluated by considering the evolution of the Reynolds stresses in homogeneous flows. A functional form for the tensor is chosen that is linear in the normalized anisotropy tensor and in the mean velocity gradients. The resulting coefficients are evaluated by matching the modeled Reynolds stress evolution to experimental data in homogeneous flows. Constraints are applied to ensure consistency with rapid distortion theory and to satisfy a consistency condition in the limit of two-dimensional turbulence. A set of coefficients is presented for which the model yields good agreement with available data in homogeneous flows.

The basis for, and some limitations of, the Langevin equation in atmospheric relative dispersion modelling

The use of the Langevin equation to model turbulent dispersion, particularly in the atmosphere, is examined. The essential feature of the Langevin equation is that fluid particle accelerations are uncorrelated, a good approximation in high Reynolds number three-dimensional turbulence. It thus satisfies all the wellknown inertial range scaling laws. An important consequence of these laws is the equivalence of conditioned one-particle dispersion and relative dispersion. Relative dispersion on global scales is not well represented by the Langevin model, at least in part because motion on these scales is quasi-two-dimensional. On smaller scales, the well-known lack of an upper limit to the scale of turbulent kinetic energy throws doubt on the applicability of the Langevin equation in toto, although a three-dimensional inertial range stage of dispersion may be reasonably well represented. Use of the Langevin equation to directly model relative velocities results in a Gaussian probability density for particle separation. This is unrealistic and leads to an incorrect representation of concentration fluctuations. However, a relative dispersion model based on an appropriate combination of a pair of Langevin equations does realistically model concentration fluctuations.

A simple dynamical model of intermittent fully developed turbulence

We present a phenomenological model of intermittency called the P-model and related to the Novikov-Stewart (1964) model. The key assumption is that in scales N &2-” only a fraction /3n of the total space has an appreciable excitation. The model, the idea of which owes much to Kraichnan (1972, 1974)’ is dynamical in the sense that we work entirely with inertial-range quantities such as velocity amplitudes, eddy turnover times and energy transfer. This gives more physical insight than the traditional approach based on probabilistic models of the dissipation. The P-model leads in an elementary way to the concept of the self-similarity dimension D, a special case of Mandelbrot’s (1974, 1976) ‘fractal dimension’. For threedimensional turbulence, the correction B to the Q exponent of the energy spectrum is equal to +( 3 - D) and is related to the exponent p of the dissipation correlation function by B = Qp (0.17 for the currently accepted value). This is a borderline case of the Mandelbrot inequality B < Qp. It is shown in the appendix that this inequality may be derived from the Navier-Stokes equation under the strong, but plausible, assumption that the inertial-range scaling laws for second- and fourth-order moments have the same viscous cut-off. The predictions of the P-model for the spectrum and for higher-order statistics are in agreement with recent conjectures based on analogies with critical phenomena (Nelkin 1975) but generally diasgree with the 1962 Kolmogorov lognormal model. However, the sixth-order structure function (8v6(Z)) and the dissipation correlation function (e(r) e(r + 1)) are related by