arXiv subjects

Victor S. L'vov

Publications and source records attributed to Victor S. L'vov.

At least 73 records · Page 4Linked to original sources

arXiv2004

Simple analytical model for entire turbulent boundary layer over flat plane: from viscous and mixing layers to turbulent logarithmic region

We discuss a simple analytical model of the turbulent boundary layer (TBL) over flat plane. The model offers an analytical description of the profiles of mean velocity and turbulent activity in the entire boundary region, from the viscous sub-layer, through the buffer layer further into the log-law turbulent region. In contrast to various existing interpolation formulas the model allows one to generalize the description of simple TBL of a Newtonian fluid for more complicated flows of turbulent suspensions laden with heavy particles, bubbles, long-chain polymers, to include the gravity acceleration, etc.

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arXiv2004

Saturation of Turbulent Drag Reduction in Dilute Polymer Solutions

Drag reduction by polymers in turbulent wall-bounded flows exhibits universal and non-universal aspects. The universal maximal mean velocity profile was explained in a recent theory. The saturation of this profile and the crossover back to the Newtonian plug are non-universal, depending on Reynolds number Re, concentration of polymer $c_p$ and the degree of polymerization $N_p$. We explain the mechanism of saturation stemming from the finiteness of extensibility of the polymers, predict its dependence on $c_p$ and $N$ in the limit of small $c_p$ and large Re, and present the excellent comparison of our predictions to experiments on drag reduction by DNA.

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arXiv2004

Drag Reduction by Polymers in Wall Bounded Turbulence

We address the mechanism of drag reduction by polymers in turbulent wall bounded flows. On the basis of the equations of fluid mechanics we present a quantitative derivation of the "maximum drag reduction (MDR) asymptote" which is the maximum drag reduction attained by polymers. Based on Newtonian information only we prove the existence of drag reduction, and with one experimental parameter we reach a quantitative agreement with the experimental measurements.

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arXiv2004

Drag Reduction by a Linear Viscosity Profile

Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching of the polymers must increase the viscosity, so why is the drag reduced? A recent theory proposed that drag reduction in agreement with experiments is consistent with the effective viscosity growing linearly with the distance from the wall. With this self consistent solution the reduction in the Reynolds stress overwhelms the increase in viscous drag. In this Letter we show, using Direct Numerical Simulations, that a linear viscosity profile indeed reduces the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.

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arXiv2003

Multi-Zone Shell Model for Turbulent Wall Bounded Flows

We suggested a \emph{Multi-Zone Shell} (MZS) model for wall-bounded flows accounting for the space inhomogeneity in a "piecewise approximation", in which cross-section area of the flow, $S$, is subdivided into "$j$-zones". The area of the first zone, responsible for the core of the flow, $S_1\simeq S/2$, and areas of the next $j$-zones, $S_j$, decrease towards the wall like $S_j\propto 2^{-j}$. In each $j$-zone the statistics of turbulence is assumed to be space homogeneous and is described by the set of "shell velocities" $u_{nj}(t)$ for turbulent fluctuations of the scale $\propto 2^{-n}$. The MZS-model includes a new set of complex variables, $V_j(t)$, $j=1,2,... \infty$, describing the amplitudes of the near wall coherent structures of the scale $s_j\sim 2^{-j}$ and responsible for the mean velocity profile. Suggested MZS-equations of motion for $u_{nj}(t)$ and $V_j(t)$ preserve the actual conservations laws (energy, mechanical and angular momenta), respect the existing symmetries (including Galilean and scale invariance) and account for the type of the non-linearity in the Navier-Stokes equation, dimensional reasoning, etc. The MZS-model qualitatively describes important characteristics of the wall bounded turbulence, e.g., evolution of the mean velocity profile with increasing Reynolds number, $\RE$, from the laminar profile towards the universal logarithmic profile near the flat-plane boundary layer as $\RE\to \infty$.

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arXiv2003

Strong Universality in Forced and Decaying Turbulence

The weak version of universality in turbulence refers to the independence of the scaling exponents of the $n$th order strcuture functions from the statistics of the forcing. The strong version includes universality of the coefficients of the structure functions in the isotropic sector, once normalized by the mean energy flux. We demonstrate that shell models of turbulence exhibit strong universality for both forced and decaying turbulence. The exponents {\em and} the normalized coefficients are time independent in decaying turbulence, forcing independent in forced turbulence, and equal for decaying and forced turbulence. We conjecture that this is also the case for Navier-Stokes turbulence.

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arXiv2003

Effect of particle inertia on the turbulence in a suspension

We propose a one-fluid analytical model for a turbulently flowing dilute suspension, based on modified Navier-Stokes equation with a $k$-dependent effective density of suspension, $ρ_ {eff}(k)$, and an additional damping term $\propto γ_ p(k)$, representing the fluid-particle friction (described by Stokes law). The statistical description of turbulence within the model is simplified by a modification of the usual closure procedure based on the Richardson-Kolmogorov picture of turbulence with a differential approximation for the energy transfer term. The resulting ordinary differential equation for the energy budget is solved analytically for various important limiting cases and numerically in the general case. In the inertial interval of scales we describe analytically two competing effects: the energy suppression due to the fluid particle friction and the energy enhancement during the cascade process due to decrease of the effective density of the small scale motions. An additional suppression or enhancement of the energy density may occur in the viscous subrange, caused by the variation of the extent of the inertial interval due to the combined effect of the fluid-particle friction and the decrease of the kinematic viscosity of the suspensions. The analytical description of the complicated interplay of these effects supported by numerical calculations is presented. Our findings allow one to rationalize the qualitative picture of the isotropic homogeneous turbulence of dilute suspensions as observed in direct numerical simulations.

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arXiv2002

Drag Reduction by Polymers in Turbulent Channel Flows: Energy Redistribution Between Invariant Empirical Modes

We address the phenomenon of drag reduction by dilute polymeric additive to turbulent flows, using Direct Numerical Simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows respectively. The modes are obtained empirically using the Karhunen-Loeve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes as proposed in some previous theories.

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arXiv2002

Scaling Exponents in Anisotropic Hydrodynamic Turbulence

In anisotropic turbulence the correlation functions are decomposed in the irreducible representations of the SO(3) symmetry group (with different "angular momenta" $\ell$). For different values of $\ell$ the second order correlation function is characterized by different scaling exponents $ζ_2(\ell)$. In this paper we compute these scaling exponents in a Direct Interaction Approximation (DIA). By linearizing the DIA equations in small anisotropy we set up a linear operator and find its zero-modes in the inertial interval of scales. Thus the scaling exponents in each $\ell$-sector follow from solvability condition, and are not determined by dimensional analysis. The main result of our calculation is that the scaling exponents $ζ_2(\ell)$ form a strictly increasing spectrum at least until $\ell=6$, guaranteeing that the effects of anisotropy decay as power laws when the scale of observation diminishes. The results of our calculations are compared to available experiments and simulations.

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arXiv2002

Stabilization of Hydrodynamic Flows by Small Viscosity Variations

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers we study the effects of a weakly space-dependent viscosity on the stability of hydrodynamic flows. In a recent Letter [Phys. Rev. Lett. {\bf 87}, 174501, (2001)] we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the "critical layer"). We showed that a layer of weakly space-dependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observation in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the time-dependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a time-dependent solution and is three-dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is {\em not} due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

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arXiv2002

The clustering instability of inertial particles spatial distribution in turbulent flows

A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is a size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associated with the growth of the second and higher moments. A new concept of compressibility of the turbulent diffusion tensor caused by a finite correlation time of an incompressible velocity field is introduced. In this model of the velocity field, the field of Lagrangian trajectories is not divergence-free. A mechanism of saturation of the clustering instability associated with the particles collisions in the clusters is suggested. Applications of the analyzed effects to the dynamics of droplets in the turbulent atmosphere are discussed. An estimated nonlinear level of the saturation of the droplets number density in clouds exceeds by the orders of magnitude their mean number density. The critical size of cloud droplets required for clusters formation is more than $20 μ$m.

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arXiv2002

One-fluid description of turbulently flowing suspension

We suggested a one-fluid model of a turbulent dilute suspension which accounts for the ``two-way'' fluid-particle interactions by $k$-dependent effective density of suspension and additional damping term in the Navier-Stokes equation. We presented analytical description of the particle modification of turbulence including scale invariant suppression of a small $k$ part of turbulent spectrum (independent of the particle response time) and possible enhancemenent of large $k$ region [up to the factor $(1+ϕ)^{2/3}$]. Our results are in agreement with qualitative picture of isotropic homogeneous turbulence of dilute suspensions previously observed in laboratory and numerical experiments.

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arXiv2002

Quasi-Gaussian Statistics of Hydrodynamic Turbulence in 3/4+εdimensions

The statistics of 2-dimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussian like in equilibrium. The skewness $\C S \equiv S_3(R)/S^{3/2}_2(R)$ was measured as $\C S_{\text{exp}}\approx 0.03$. This contradiction is lifted by understanding that 2-dimensional turbulence is not far from a situation with equi-partition of enstrophy, which exist as true thermodynamic equilibrium with K41 exponents in space dimension of $d=4/3$. We evaluate theoretically the skewness $\C S(d)$ in dimensions ${4/3}\le d\le 2$, show that $\C S(d)=0$ at $d=4/3$, and that it remains as small as $\C S_{\text{exp}}$ in 2-dimensions.

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arXiv2002

Inverse Cascade Regime in Shell Models of 2-Dimensional Turbulence

We consider shell models that display an inverse energy cascade similar to 2-dimensional turbulence (together with a direct cascade of an enstrophy-like invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a "quasi-equilibrium" situation with equipartition of the energy among the shells. We show analytically that the quasi-equilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with K41 scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.

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arXiv2001

Retardation of the onset of yurbulence by minor viscosity contrasts

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers we study the effects of minor viscosity contrasts on the stability of hydrodynamic flows. The key player is a localized region where the energy of fluctuations is produced by interactions with the mean flow (the "critical layer"). We show that a layer of weakly space-dependent viscosity placed near the critical layer can have very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. The effect is not due to a modified dissipation (as is assumed in theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

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arXiv2001

Quasi-solitons and asymptotic multiscaling in shell models of turbulence

A variation principle is suggested to find self-similar solitary solutions (``solitons'') of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(0.001). It is found how the soliton shape, propagation time $t_n$ and the dynamical exponent $z_0$ (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of $z$ the author discovered ``quasi-solitons'' which approximate with high accuracy corresponding self-similar equations for an interval of times from $-\infty$ to some time in the vicinity of the peak maximum or even after it. The conjecture is that the trajectories in the vicinity of the quasi-solitons (with continuous spectra of $z$) provide an essential contribution to the multiscaling statistics of high-order correlation functions, referred as an ``asymptotic multiscaling''. This contribution may be even more important than that of the trajectories in the vicinity of the exact soliton with a fixed value $z_0$. Moreover there are no solitons in some region of the parameters where quasi-solitons provide dominant contribution to the asymptotic multiscaling

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arXiv2000

Outliers, Extreme Events and Multiscaling

Extreme events have an important role which is sometime catastrophic in a variety of natural phenomena including climate, earthquakes and turbulence, as well as in man-made environments like financial markets. Statistical analysis and predictions in such systems are complicated by the fact that on the one hand extreme events may appear as "outliers" whose statistical properties do not seem to conform with the bulk of the data, and on the other hands they dominate the (fat) tails of probability distributions and the scaling of high moments, leading to "abnormal" or "multi"-scaling. We employ a shell model of turbulence to show that it is very useful to examine in detail the dynamics of onset and demise of extreme events. Doing so may reveal dynamical scaling properties of the extreme events that are characteristic to them, and not shared by the bulk of the fluctuations. As the extreme events dominate the tails of the distribution functions, knowledge of their dynamical scaling properties can be turned into a prediction of the functional form of the tails. We show that from the analysis of relatively short time horizons (in which the extreme events appear as outliers) we can predict the tails of the probability distribution functions, in agreement with data collected in very much longer time horizons. The conclusion is that events that may appear unpredictable on relatively short time horizons are actually a consistent part of a multiscaling statistics on longer time horizons.

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arXiv2000

Analytic Calculation of the Anomalous Exponents in Turbulence: Using the Fusion Rules to Flush Out a Small Parameter

The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorov's statistics of 1941 acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents $ζ_n$ of the $n$th order structure functions can be computed analytically. The crux of the method consists of renormalizing a 4-point interaction amplitude on the basis of the fusion rules. The novelty is that this amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of $ζ_2$, $δ_2= ζ_2-2/3\approx 0.03$. Higher order interaction amplitudes are shown to be even smaller. The corrections to K41 to $O(δ_2)$ result from standard logarithmically divergent ladder-diagrams in which the 4-point interaction acts as a "rung". The theory allows a calculation of the anomalous exponents $ζ_n$ in powers of the small parameter $δ_2$. The $n$-dependence of the scaling exponents $ζ_n$ stems from pure combinatorics of the ladder diagrams. We calculate the exponents $ζ_n$ up to $O(δ_2^3)$. Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter $δ_2$. The actual evaluation of the small parameter $δ_2$ from first principles requires additional developments that are outside the scope of this paper.

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