Second-order relative exponent of isotropic turbulence

Theoretical results on the scaling properties of turbulent velocity fields are reported in this letter. Based on the Kolmogorov equation and typical models of the second-order statistical moments (energy spectrum and the second-order structure function), we have studied the relative scaling using the ESS method. It is found that the relative EES scaling exponent S₂ is greater than the real or theoretical inertial range scaling exponent ξ₂, which is attributed to an evident bump in the ESS range.     

Finite-dimensional description of convective Reaction-Diffusion equations

We are concerned with the asymptotic dynamics of a certain type of semilinear parabolic equation, namely,u _t=u _xx+(f(u))_x+g(u)+h(x) on the interval [0,L]. Under the general condition we prove that this equation admits a dissipative dynamical system and it possesses the global attractor. But for largeL > 0, we do not know whether or not an inertial manifold exists. Here we introduce a nonlinear change of variables so that we transform the above equation to a reaction diffusion system which possesses exactly the same asymptotic dynamics. We then prove the existence of an inertial manifold for the transformed equation; thereby we find the ordinary differential equation which describes completely the long-time dynamics of the orginal equation.     

Small scale intermittency in a rough wall turbulent boundary layer

Relatively good quality isotropy is observed in the central region of a turbulent boundary layer developing over a mesh screen rough wall. Spectra of velocity and, more especially, vorticity fluctuations satisfy isotropy over a significant wavenumber range. Inertial range scaling exponents ζ _u2 (p) and ζ _u3 (p) of moments of order p(?8) of increments of the transverse velocity fluctuations u ₂ and u ₃ are significantly smaller than the exponents ζ _u1 (p) of increments of the longitudinal velocity fluctuation u ₁. Exponents inferred from the locally averaged values of squared transverse vorticity fluctuations are only slightly smaller than ζ _u1 (p). The difference between ζ _u1 (p) and ζ _u2 (p) [or ζ _u3 (p)] more likely reflects the departure from isotropy of inertial range scales. There is evidence to suggest that the difference decreases with an increase in the Reynolds number and/or a decrease in the magnitude of the mean shear.     

On the particle inertia-free collision with a partially contaminated spherical bubble

The collision between a contaminated spherical bubble and fine particles in suspension is considered for r_p/r_b ? 1 (r_p being the radius of the particles in suspension and r_b the radius of the bubble). The collision probability or efficiency is defined as the number of particles colliding the bubble surface to the number of particles initially present in the volume swept out by the bubble. In this note we show that the collision probability can be expressed as P_c(r_p/r_b,Re) = g(r_p/r_b)f(Re) for both mobile and immobile interfaces. For partially contaminated bubbles a linear or quadratic dependency in r_p/r_b is found depending on the level of contamination and the value of r_p/r_b. These behaviors are given by the flux of particles near the surface which is controlled by the tangential velocity for mobile interfaces and by the velocity gradient for immobile interfaces. The threshold value (r_p/r_b)_th between the r_p/r_b and (r_p/r_b)² evolution is shown to vary as sin^n(Re)(θ_clean/n(Re))sin(3θ_clean/4), θ_clean being the angle describing the front clean part of the bubble and n(Re) varying from n = 2 to n = 1 from small to large Reynolds number.     

Flow of K-BKZ fluids between parallel plates rotating about distinct axes: shear thinning and inertial effects

It has been shown that for any simple fluid, a flow field of the form u = –Ω[y - g(z)], v = Ω[x – f(z], w = 0, which is appropriate for modeling the flow in a orthogonal rheometer, is dynamically possible. The functions f(z) and g(z) depend on the choice of constitutive equation. In the present paper, these are calculated for a class of K-BKZ fluids which exhibit shear thinning. The results are then used to study the interaction of shear thinning and inertial effects on the flow field in an orthogonal rheometer.     

Temperature correlations with vorticity and velocity in a turbulent cylinder wake

This work aims to understand the difference in the correlations between the fluctuating temperature and the vorticity from that between the fluctuating temperature and the velocity in a turbulent cylinder near wake. Measurements are made at x/d = 10, 20 and 40, where x is the streamwise distance from the cylinder axis and d is the cylinder diameter, with a Reynolds number of 2.5×10³ based on d and the free-stream velocity. The three components of the fluctuating velocity vector u_i(i = 1, 2 and 3), vorticity vector ω_i (i = 1, 2 and 3), and temperature θ in the plane of the mean shear are measured simultaneously with a multi-wire probe consisting of four X-hotwires and four cold wires. It is found that at x/d = 10, both correlations between u_iand θ and between ω_i and θ predominantly take place at St = 0.21, due to the concentric distribution of the Kármán vortices and the heat. With increasing x/d, the correlation between ω_i (i = 1, 2 and 3) and θ drops rapidly, as a result of the weakened Kármán vortices; in contrast, the correlation between u₁ and θ increases appreciably, largely due to an enhanced correlation between u₁ and θ at low frequencies or scales of motions larger than the Kármán vortex. The slowly decreasing (along x) two-point autocorrelations of u₁ and θ suggest that the very-large-scale motions (VLSMs) found in wall flows occur also in the turbulent wake and are responsible for the high correlation between u₁ and θ at low frequencies.     

Finite-dimensional asymptotic behavior of some semilinear damped hyperbolic problems

We prove that the solution semigroup $$S_t \left[ {u_0 ,v_0 } \right] = \left[ {u(t),u_t (t)} \right]$$ generated by the evolutionary problem $$\left\{ P \right\}\left\{ \begin{gathered} u_{tt} + g(u_t ) + Lu + f(u) = 0, t \geqslant 0 \hfill \\ u(0) = u_0 , u_t (0) = \upsilon _0 \hfill \\ \end{gathered} \right.$$ possesses a global attractorA in the energy spaceE _o=V×L ²(Ω). Moreover,A is contained in a finite-dimensional inertial setA attracting bounded subsets ofE ₁=D(L)×V exponentially with growing time.     

On the dispersion of a solute in oscillating flow through a channel

The paper presents an exact analysis of the dispersion of a solute in an incompressible viscous fluid flowing slowly in a parallel plate channel under the influence of a periodic pressure gradient. Using a generalised dispersion model which is valid for all times after the solute injection, the diffusion coefficientsK _i (τ)(i=1,2,3,…) are determined as functions of timeτ when the initial distribution of the solute is in the form of a slug of finite extent. The second coefficientK ₂(τ) gives a measure of the longitudinal dispersion of the solute due to the combined influence of molecular diffusion and nonuniform velocity across the channel cross-section. The analysis leads to the novel result thatK ₂(τ) consists of a steady partS and a fluctuating partD ₂(τ) due to the pulsatility of the flow. It is shown thatS increases with increase inλ (the amplitude of pressure pulsation) for small values ofω (the frequency of the pulsation). But for largeω, S decreases with increase inλ. It is also found that for fixedλ, there is very little fluctuation inD ₂(τ) forω=1, butD ₂(τ) shows fluctuation with large amplitude whenω slightly exceeds unity. The amplitude ofD ₂(τ) then decreases with further increase inω. Thus the variation of bothS andD ₂(τ) withω is non-monotonic. Finally,? _m , the average concentration of the solute over the channel cross-section is determined for various values ofλ andω.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

High accuracy wave simulation – Revised derivation,numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

The nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [t_n, t_n+1] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax–Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models – the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.     

A local lagrangian analysis of passive particle advection in a gas flow field

A local analysis is performed to study the departure from passive advection by small inertial particles based on a Lagrangian framework. The analysis considers heavy particles immersed in a gaseous flow and is restricted to short times, making it relevant to the PIV technique. A necessary (but not sufficient condition) for passive particle advection of inertial particles is that the quantity Λ_maxτ_p be much smaller than one, where Λ_max is the largest modulus of the eigenvalues corresponding to the velocity gradient tensor. This allows for the inertial and passive time scales to match beyond the initial transient, and consequently for the respective trajectories to remain relatively close. Due to this important role regarding advection behavior, Λ_maxτ_p is offered as a definition of a local Stokes number, St_Λ. Since this quantity is a field quantity, it directly provides indication of when and where passive advection by particles can be expected. A departure equation is obtained in one-dimension, where the influence of initial velocity and gravity are explicitly shown. If the flow is irrotational, the higher dimensional analysis reduces to a set of decoupled one-dimensional equations acting along each respective eigenvector of the velocity gradient tensor. A similar expression is found for the case of a purely temporal flow field.     

The comparison model method: A new arithmetic approach to the discrete inverse problem of groundwater hydrology

Let the steady-state pressure z(·) of a fluid in a one-dimensional domain be governed by the equation d _x (a d _x z) = f subject to Dirichlet boundary conditions. We consider the identification of the transmissivity a (·), given f(·), and measured pressure z(·) by the comparison model method, a direct method which has been known and applied for some time but lacked theoretical background. With reference to a domain in one spatial dimension, we examine both the infinite-(‘continuous’) and finite-(discrete) dimensional cases. In the former, the method is based on the solution p(·) of an auxiliary flow equation, where f(·) and the two-point boundary conditions are unchanged with respect to the original or z(·) equation, whereas a tentative constant value b is assigned to the auxiliary transmissivity. The ratio of the first derivatives of p(·) and z(·) multiplied by b yields a solution ã(·) to the inverse problem. We examine in detail the nonuniqueness of ã(·) as a function of b, some cases where existence implies uniqueness, the role of positivity constraints, and a special feature: self-identifiability. We then translate all available results into the discrete case, where the good unknowns for the inverse problem are the internode coefficients. Several algebraic and numerical examples are presented.     

Near-wake flow characteristics of a circular cylinder close to a wall

Flow characteristics in the near wake of a circular cylinder located close to a fully developed turbulent boundary layer are investigated experimentally using particle image velocimetry (PIV). The Reynolds number based on the cylinder diameter (D) is 1.2×10⁴ and the incident boundary layer thickness (δ) is 0.4D. Detailed velocity and vorticity fields in the wake region (0<x/D<6) are given for various gap heights (S) between the cylinder and the wall, with S/D ranging from 0.1 to 1.0. Both the ensemble-averaged (including the mean velocity vectors and Reynolds stress) and the instantaneous flow fields are strongly dependent on S/D. Results reveal that for S/D⩾0.3, the flow is characterized by the periodic, Kármán-like vortex shedding from the upper and lower sides of the cylinder. The shed vortices and their evolution are revealed by analyzing the instantaneous flow fields using various vortex identification methods, including Galilean decomposition of velocity vectors, calculation of vorticity and swirling strength. For small and intermediate gap ratios (S/D⩽0.6), the wake flow develops a distinct asymmetry about the cylinder centreline; however, some flow quantities, such as the Strouhal number and the convection velocity of the shed vortex, keep roughly constant and virtually independent of S/D.     

Space-time correlations in turbulent Rayleigh–Bnard convection

The recent development of the elliptic model(He,et al. Phy. Rev. E, 2006), which predicts that the space-time correlation function Cu(r, τ) in a turbulent flow has a scaling form Cu(rE, 0) with rEbeing a combined space-time separation involving spatial separation r and time delay τ, has stimulated considerable experimental efforts aimed at testing the model in various turbulent flows. In this paper, we review some recent experimental investigations of the space-time correlation function in turbulent Rayleigh–B′enard convection. The experiments conducted at different representative locations in the convection cell confirmed the predictions of the elliptic model for the velocity field and passive scalar field, such as local temperature and shadowgraph images.The understanding of the functional form of Cu(r, τ) has a wide variety of applications in the analysis of experimental and numerical data and in the study of the statistical properties of small-scale turbulence. A few examples are discussed in the review.     

Reynolds方程在纹理表面动压润滑计算中的有效性评价

从理论上阐述了纹理表面动压润滑计算中决定Reynolds方程有效性的两个关键因素为油膜厚度与纹理特征长度的比值h/L和缩减的雷诺数re;只有当h/L和re同时趋近于零时Reynolds方程才能够获得准确的结果,并由此在h/L-Re平面上标注了Reynolds方程的适用范围.继而以二维矩形沟槽为实例,采用数值方法计算了h/L和re对Reynolds方程误差的影响规律;分析了Reynolds方程在不同条件下的失效机制;分析了矩形沟槽纹理表面Reynolds方程有效性的评价标准:当缩减的雷诺数re小于0.20,并且h/L小于0.015时能够保证Reynolds方程的误差在10%以下.     

Characteristic lengths and maximum entropy estimation from probe signals in the ellipsoidal bubble regime

The bubble size, surface and volume distributions in two and three phase flows are essential to determine energy and mass transfer processes. The traditional approaches commonly use a conditional probability density function of chord-lengths to calculate the bubble size distribution, when the bubble size, shape and velocity are known. However, the approach used in this paper obtains the above distributions from statistical relations, requiring only the moments inferred from the measurements given by a sampling probe. Using image analysis of bubbles injected in a water tank, and placing an ideal probe on the image, a sample of bubble diameter, shape factor and velocity angle are obtained. The samples of the bubble chord-length are synthetically generated from these variables. Thus, we propose a semi-parametric approach based on the maximum entropy (MaxEnt) distribution estimation subjected to a number of moment constraints avoiding the use of the complex backward transformation. Therefore, the method allows us to obtain the distributions in close form. The probability density functions of the most important length scales (D, D₂₀, D₃₀, D₃₂), obtained applying the semi-parametric approach proposed here in the ellipsoidal bubble regime, are compared with experimental measurements.     

Scalar Dissipation Rate Transport in the Context of Large Eddy Simulations for Turbulent Premixed Flames with Non-Unity Lewis Number

The effects of global Lewis number Le on the statistical behaviour of the unclosed terms in the transport equation of the Favre-filtered scalar dissipation rate (SDR) Ñ _c have been analysed using a Direct Numerical Simulation (DNS) database of freely propagating statistically planer turbulent premixed flames with Le ranging from 0.34 to 1.2. The DNS data has been explicitly filtered to analyse the statistical behaviour of the unclosed terms in the SDR transport equation arising from turbulent transport T ₁, density variation due to heat release T ₂, scalar-turbulence interaction T ₃, reaction rate gradient T ₄, molecular dissipation (?D ₂) and diffusivity gradients f(D) in the context of Large Eddy Simulations (LES). It Le has significant effects on the magnitudes of T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D). Moreover, both qualitative and quantitative behaviours of the unclosed terms T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D) are found to be significantly affected by the LES filter width Δ, which have been explained based on a detailed scaling analysis. Both scaling analysis and DNS data suggest that T ₂, T ₃, T ₄, (?D ₂) and f(D) remain leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport for LES. The scaling estimates of leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport has been utilised to discuss the possibility of extending an existing SDR model for Reynolds Averaged Navier Stokes (RANS) simulation for SDR \(\tilde {{N}}_{c} \) closure in the context of LES of turbulent premixed combustion.     

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.     

Effects of the Reynolds number on a scale-similarity model of Lagrangian velocity correlations in isotropic turbulent flows

A scale-similarity model of a two-point two-time Lagrangian velocity correlation(LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows(HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313(2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values(R_λ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity.Moreover, the dispersion velocity V normalized by the Kolmogorov velocity V_η≡η/τ_η in which η and τ_η are the Kolmogorov space and time scales, respectively, scales with the Reynolds number R_λ as V/V_η∝ R_λ~(1.39) obtained from the numerical data.