Modulation of near-wall turbulence in the transitionally rough regime

Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the roughness size increases, the background turbulence begins to be modified, notably by the increase of energy for short, wide wavelengths, which is consistent with the appearance of a shear-flow instability of the mean flow. A laminar model is presented to estimate the roughness-coherent contribution, as well as the displacement height and the velocity at the roughness crests. Based on the effects observed in the background turbulence, the roughness function is decomposed into different terms to analyse different contributions to the change in drag, laying the foundations for a predictive model

The effect of slip and surface texture on turbulence over superhydrophobic surfaces

Superhydrophobic surfaces are able to entrap gas pockets in between surface roughness elements when submerged in water. These entrapped gas pockets give these surfaces the potential to reduce drag due to the overlying flow being able to locally slip over the gas pockets, resulting in a mean slip at the surface. In this work we assess the separate effects that surface slip and surface texture have on turbulence over superhydrophobic surfaces. We show that the direct effect of surface slip does not modify the dynamics of the overlying turbulence, which remains canonical or smooth-wall like. The surface drag is governed by the difference between two virtual origins, the virtual origin of the mean flow and the virtual origin experienced by the overlying turbulence, in an extension of the theory from Luchini, Manzo &amp; Pozzi (J. Fluid Mech., vol. 228, 1991, pp. 87–109) for riblets. Streamwise slip deepens the virtual origin of the mean flow, while spanwise slip deepens the virtual origin perceived by the overlying turbulence. Drag reduction is then proportional to the difference between the two virtual origins. We decompose the near-wall flow into background-turbulence and texture-coherent components, and show that the background-turbulence component experiences the surface as homogeneous slip lengths. The validity of the slip-length model can then be extended to larger texture size $L^{+}$ than thought in previous studies. For $L^{+}\gtrsim 25$, however, we observe that a nonlinear interaction with the texture-coherent flow develops that alters the dynamics of the background turbulence, exhibiting a modified distribution of turbulent energy across length scales. This has the effect of reducing the velocity increment \unicode[STIX]{x0394}U^{+} compared to that predicted using homogeneous slip lengths and sets the upper limit of applicability of slip-length models.</jats:p

Manifestation of anisotropy persistence in the hierarchies of MHD scaling exponents

The first example of a turbulent system where the failure of the hypothesis of small-scale isotropy restoration is detectable both in the `flattening' of the inertial-range scaling exponent hierarchy, and in the behavior of odd-order dimensionless ratios, e.g., skewness and hyperskewness, is presented. Specifically, within the kinematic approximation in magnetohydrodynamical turbulence, we show that for compressible flows, the isotropic contribution to the scaling of magnetic correlation functions and the first anisotropic ones may become practically indistinguishable. Moreover, skewness factor now diverges as the P\'eclet number goes to infinity, a further indication of small-scale anisotropy.Comment: 4 pages Latex, 1 figur

Neural Decision Boundaries for Maximal Information Transmission

We consider here how to separate multidimensional signals into two categories, such that the binary decision transmits the maximum possible information transmitted about those signals. Our motivation comes from the nervous system, where neurons process multidimensional signals into a binary sequence of responses (spikes). In a small noise limit, we derive a general equation for the decision boundary that locally relates its curvature to the probability distribution of inputs. We show that for Gaussian inputs the optimal boundaries are planar, but for non-Gaussian inputs the curvature is nonzero. As an example, we consider exponentially distributed inputs, which are known to approximate a variety of signals from natural environment.Comment: 5 pages, 3 figure

Exact Resummations in the Theory of Hydrodynamic Turbulence: III. Scenarios for Anomalous Scaling and Intermittency

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws $S_n(R)\sim R^{\zeta_n}$, and the correlation of energy dissipation $K_{\epsilon\epsilon}(R) \sim R^{-\mu}$. The goal is to understand the exponents $\zeta_n$ and $\mu$ from first principles. In paper II of this series it was shown that the existence of an ultraviolet scale (the dissipation scale $\eta$) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted $\Delta$. The exact resummation of ladder diagrams resulted in the calculation of $\Delta$ which satisfies the scaling relation $\Delta=2-\zeta_2$. In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e. $\zeta_n$ not being linear in $n$) with the renormalization scale being the infrared outer scale of turbulence $L$. It is shown that deviations from K41 scaling of $S_n(R)$ ($\zeta_n\neq n/3$) must appear if the correlation of dissipation is mixing (i.e. $\mu>0$). We derive an exact scaling relation $\mu = 2\zeta_2-\zeta_4$. We present analytic expressions for $\zeta_n$ for all $n$ and discuss their relation to experimental data. One surprising prediction is that the time decay constant $\tau_n(R)\propto R^{z_n}$ of $S_n(R)$ scales independently of $n$: the dynamic scaling exponent $z_n$ is the same for all $n$-order quantities, $z_n=\zeta_2$.Comment: PRE submitted, 22 pages + 11 figures, REVTeX. The Eps files of figures will be FTPed by request to [email protected]

Intrinsic gain modulation and adaptive neural coding

In many cases, the computation of a neural system can be reduced to a receptive field, or a set of linear filters, and a thresholding function, or gain curve, which determines the firing probability; this is known as a linear/nonlinear model. In some forms of sensory adaptation, these linear filters and gain curve adjust very rapidly to changes in the variance of a randomly varying driving input. An apparently similar but previously unrelated issue is the observation of gain control by background noise in cortical neurons: the slope of the firing rate vs current (f-I) curve changes with the variance of background random input. Here, we show a direct correspondence between these two observations by relating variance-dependent changes in the gain of f-I curves to characteristics of the changing empirical linear/nonlinear model obtained by sampling. In the case that the underlying system is fixed, we derive relationships relating the change of the gain with respect to both mean and variance with the receptive fields derived from reverse correlation on a white noise stimulus. Using two conductance-based model neurons that display distinct gain modulation properties through a simple change in parameters, we show that coding properties of both these models quantitatively satisfy the predicted relationships. Our results describe how both variance-dependent gain modulation and adaptive neural computation result from intrinsic nonlinearity.Comment: 24 pages, 4 figures, 1 supporting informatio

Towards a Nonperturbative Theory of Hydrodynamic Turbulence:Fusion Rules, Exact Bridge Relations and Anomalous Viscous Scaling Functions

In this paper we derive here, on the basis of the NS eqs. a set of fusion rules for correlations of velocity differences when all the separation are in the inertial interval. Using this we consider the standard hierarchy of equations relating the $n$-th order correlations (originating from the viscous term in the NS eq.) to $n+1$'th order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain is decoupled in the sense that the correlations of $n+1$'th order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of eqs. when some separations go to zero we derive a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every n'th order correlation function of velocity differences {\cal F}_n(\B.R_1,\B.R_2,\dots) exhibits its own cross-over length $\eta_{n}$ to dissipative behavior as a function of, say, $R_1$. This length depends on $n$ {and on the remaining separations} $R_2,R_3,\dots$. When all these separations are of the same order $R$ this length scales like $\eta_n(R)\sim \eta (R/L)^{x_n}$ with $x_n=(\zeta_n-\zeta_{n+1}+\zeta_3-\zeta_2)/(2-\zeta_2)$, with $\zeta_n$ being the scaling exponent of the $n$'th order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents $\zeta_n$ of the $n$'th order structure functions. One of these relations is the well known ``bridge relation" for the scaling exponent of dissipation fluctuations $\mu=2-\zeta_6$.Comment: PRE, Submitted. REVTeX, 18 pages, 7 figures (not included) PS Source of the paper with figures avalable at http://lvov.weizmann.ac.il/onlinelist.htm

Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence

Statistical model of strongly anisotropic fully developed turbulence of the weakly compressible fluid is considered by means of the field theoretic renormalization group. The corrections due to compressibility to the infrared form of the kinetic energy spectrum have been calculated in the leading order in Mach number expansion. Furthermore, in this approximation the validity of the Kolmogorov hypothesis on the independence of dissipation length of velocity correlation functions in the inertial range has been proved.Comment: REVTEX file with EPS figure

Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ε3\varepsilon^{3}

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function \propto\delta(t-t')/k^{d+\eps}. Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the $\varepsilon$ expansion to order $\varepsilon^{3}$ (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the $\varepsilon$ expansion are discussed; the improved ``inverse'' $\varepsilon$ expansion is proposed and the comparison with the existing nonperturbative results is given.Comment: 34 pages, 5 figures, REVTe