VELOCITY-ACCELERATION STRUCTURE FUNCTION IN TWO-DIMENSIONAL DECAYING TURBULENCE$^{\bf 1)}$

湍流场中二阶速度加速度结构函数 (velocity-acceleration structure function, VASF) 被认为与尺度间能量或者拟涡能的传递相关,其正负表明传递的方向. 三维湍流中,能量从大尺度向 小尺度传递,VASF 为负. 在二维湍流中,能量反向传递到大尺度,拟涡能正向传递到小尺度,因此理论上 VASF 无论在反向能量级串区还是在正向拟 涡能级串区均为正. 然而,相对于三维湍流中 VASF 的充分研究,二维湍流中 VASF 的正负性迄今尚无实验或数值模拟数据验证. 本文通过三维二维湍流中普适的公式推导,指出在空间非均匀湍流场中,VASF 除了尺度间传递,还受到非均匀项的影响. 一种常见的空间非均匀湍流场是在实验研究中常用的风洞或水洞中,湍流发生装置 (如栅格) 后的湍流. 该流场中,湍流强度随下游位置增大而逐渐衰减,这种衰减则带来空间上的非均匀性. 本文在基于竖直流动皂膜的二维衰减湍流场中,利用拉格朗日粒子追踪法测得在拟涡能级串区的 VASF,并分析各部分的影响. 结果表明,虽然尺度间传递项为正值,但由于衰减带来的非均匀项为负值,使得 VASF 的值为负,使之失去了表征拟涡能传递方向的意义. 因此,在类似风洞、水洞、水槽等衰减流场中对 VASF 的讨论不应忽略非均匀项. 最后对与 VASF 密切相关的弥散过程进行分析,发现后期弥散过程变慢是由于负的 VASF 导致.     

二维衰减湍流的速度加速度结构函数

湍流场中二阶速度加速度结构函数 (velocity-acceleration structure function, VASF) 被认为与尺度间能量或者拟涡能的传递相关,其正负表明传递的方向. 三维湍流中,能量从大尺度向 小尺度传递,VASF 为负. 在二维湍流中,能量反向传递到大尺度,拟涡能正向传递到小尺度,因此理论上 VASF 无论在反向能量级串区还是在正向拟 涡能级串区均为正. 然而,相对于三维湍流中 VASF 的充分研究,二维湍流中 VASF 的正负性迄今尚无实验或数值模拟数据验证. 本文通过三维二维湍流中普适的公式推导,指出在空间非均匀湍流场中,VASF 除了尺度间传递,还受到非均匀项的影响. 一种常见的空间非均匀湍流场是在实验研究中常用的风洞或水洞中,湍流发生装置 (如栅格) 后的湍流. 该流场中,湍流强度随下游位置增大而逐渐衰减,这种衰减则带来空间上的非均匀性. 本文在基于竖直流动皂膜的二维衰减湍流场中,利用拉格朗日粒子追踪法测得在拟涡能级串区的 VASF,并分析各部分的影响. 结果表明,虽然尺度间传递项为正值,但由于衰减带来的非均匀项为负值,使得 VASF 的值为负,使之失去了表征拟涡能传递方向的意义. 因此,在类似风洞、水洞、水槽等衰减流场中对 VASF 的讨论不应忽略非均匀项. 最后对与 VASF 密切相关的弥散过程进行分析,发现后期弥散过程变慢是由于负的 VASF 导致.      

涡度拟能传输惯性区二维湍流标量场方差谱

本文应用非平衡统计力学封闭方法,给出涡度拟能传输惯性区二维湍流标量场方差谱的完整表达式,数值计算该表达式中的比例系数B。由于传输过程的非局部性,B依赖于表征波数变化范围的局部化因子,不再是普适常数。     

MHD Turbulence at the Laboratory Scale: Established Ideas and New Challenges

The properties of MHD turbulence in the electrically conducting fluids available in the laboratory (where the magnetic Reynolds number is significantly smaller than unity) may be summarised as follows:(1) The Alfven waves, even under their degenerated form at this scale, are responsible for a tendency to two-dimensionality. Eddies tend to become aligned with the applied magnetic field and inertia tends to restore isotropy. The competition between these mechanisms results in a spectral law t^-2k^-3.(2) When insulating walls, perpendicular to the magnetic field, are present and close enough to each other, two-dimensionality can be established with a good approximation within the large scales, and the predominant mechanism is the inverse energy cascade.(3) These columnar eddies are nevertheless submitted to a dissipation within the Hartmann boundary layers present at their ends, whose time scale is independent of the wave number. When this damping effect is negligible, ordinary 2D turbulence is observed with k^-5/3 spectra. On the contrary when this (ohmic and viscous) damping is significant this 2D turbulence exhibits k^-3 spectra.Besides these homogeneous (except within the Hartmann layers) conditions, for instance in shear flows such as mixing layers, almost nothing is known except that two-dimensionality may be well established. The first results of a recent experimental investigation (still in development) are presented. Some challenging questions are raised, such as the interpretation of a surprising difference between the transport of momentum and the transport of a scalar quantity (heat) across that layer. A video was shown during the oral presentation of this paper, illustrating the energy transfer toward the large scales and the weakness of the dissipation suffered by this 2D velocity field.     

The turbulence vorticity as a window to the physics of friction-drag reduction by oscillatory wall motion

DNS data for channel flow, subjected to spanwise (in-plane) wall oscillations at a friction Reynolds number of 1025, are used to examine the turbulence interactions that cause the observed substantial reduction in drag provoked by the forcing. Following a review of pertinent interactions between the forcing-induced unsteady Stokes strain and the Reynolds stresses, identified in previous work by the present authors, attention is focused on the equations governing the components of the enstrophy, with particular emphasis placed on the wall-normal and the spanwise components. The specific objective is to study the mechanisms by which the Stokes strain modifies the enstrophy field, and thus the turbulent stresses. As such, the present analysis sheds fresh light on the drag-reduction processes, illuminating the interactions from a different perspective than that analysed in previous work. The investigation focuses on the periodic rise and fall in the drag and phase-averaged properties during the actuation cycle at sub-optimal actuation conditions, in which case the drag oscillates by around ±2% around the time-averaged 20% drag-reduction margin. The results bring out the important role played by specific strain-related production terms in the enstrophy-component equations, and also identifies vortex tilting/stretching in regions of high skewness as being responsible for the observed strong increase in the spanwise enstrophy components during the drag-reduction phase. Simultaneously, the wall-normal enstrophy component, closely associated with near-wall streak intensity, diminishes, mainly as a result of a reduction in a production term that involves the correlation between wall-normal vorticity fluctuations and the spanwise derivative of wall-normal-velocity fluctuations, which pre-multiplies the streamwise shear strain.     

Transient response of enstrophy transport to opposition control in turbulent channel flow

The transient response of the turbulent enstrophy transport to opposition control in the turbulent channel flow is studied with the aid of direct numerical simulation. It is found that the streamwise enstrophy and the spanwise enstrophy are suppressed by the attenuation of the stretching terms at first, while the vertical enstrophy is reduced by inhibiting the tilt of the mean shear. In the initial period of the control, the streamwise enstrophy evolves much slower than the other two components. The vertical vorticity component exhibits a rapid monotonic decrease and also plays an important role in the attenuation of the other two components.     

Minimax principle on energy dissipation of incompressible shear flow

The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin＇s variational principle characterizes the upper bounds （maxi- mum） of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical point of view was adopted to recast Doering-Constantin＇s formu- lation into a minimax principle for the energy dissipation of an incompressible shear flow. Then, the Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin＇s variational principle and Howard- Busse＇s statistical turbulence theory.     

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.     

Geometrical Properties of the Resolved-Scale Velocity and Temperature Fields Predicted using Large-Eddy Simulation

In this paper, local geometrical properties of the velocity and temperature fields of combined forced and natural convection in a vertical slot are studied using large-eddy simulation based on both numerical and analytical approaches. Previous studies on turbulence geometrical statistics appearing in the literature have primarily focused on either isothermal or isotropic turbulent flows; whereas in this work, we extend the scope of research to investigation of a wall-bounded thermal flow. In particular, we focus on studying the resolved helicity, enstrophy generation, local vortex stretching, and a variety of characteristic geometrical alignment patterns between the resolved velocity, vorticity, temperature gradient, subgrid-scale heat flux and the eigenvectors of the resolved strain rate tensor. In order to quantify the effect of buoyancy on the geometrical properties of the thermal flow field, a systematic comparative analysis has been conducted based on three different flow regimes (i.e., viscous sublayer, buffer layer and logarithmic layer) in both the hot and cold wall regions. The near-wall restriction on the geometrical property of the thermal flow field has been analyzed and some interesting wall-limiting geometrical alignment patterns in the form of Dirac delta functions are also reported.     

Decaying turbulence in soap films: energy and enstrophy evolution

This experimental study of quasi-two-dimensional grid turbulence in gravity-driven soap-film flow focuses on the differences between the behavior of the flow and the theoretical picture of two-dimensional turbulence. A previously unattainable quality of velocity field acquisition facilitates simultaneous measurement of velocity field features in the scale range spanning over two orders of magnitude. The highly-resolved flow field data are analyzed statistically in terms of velocity structure functions, as well as energy and enstrophy averages at different downstream positions. We find the rate of decay of these averages to be quantifiably greater than the predictions of the two-dimensional turbulence theory. This increased decay is likely to be the manifestation of the extra dissipation mechanism present in soap-film flows and prominent on the larger scales—air drag. The structure function analysis confirms the notion. This research was supported by Los Alamos National Laboratory, task order BG109.     

Isoenstrophy points and surfaces in turbulent flow and mixing

Vorticity ω magnitude is measured by the enstrophy field ω². Equations describing the motion of surfaces of constant enstrophy, and lines and points of extreme enstrophy, are derived. The purpose is to develop better tools for studies of small scale processes of turbulence and turbulent mixing.     

FST方法分析二维Rayleigh-Bénard湍流热对流系统的能量输运

本文应用空间滤波方法：FST(Filter-space technique)方法,研究二维Rayleigh-Bénard(RB)湍流热对流系统中湍动能、热能和拟涡能的能量输运．研究中Rayleigh数(Ra)选取为1x10^8、1x10^9和1x10^10,Prandtl数(Pr)固定为4.38．我们展示了的结果表明,在二维RB系统中,三个Ra数下全场的平均湍动能和平均拟涡能在不同滤波尺度下的能量输运与Kraichnan在1967年预测的二维湍流中的级串理论有所偏差,而中心区域的能量都是向小尺度输运的．结果还揭示了瞬时能量输运的一些局部特性,包括它们在小尺度上不对称的分布．     

Two-Dimensional Turbulence with Rigid Circular Walls

We report numerical computations of decaying two-dimensional Navier--Stokes turbulence inside a circular rigid boundary. We summarize previously reported calculations involving no-slip boundary conditions and present results with higher spatial resolution than achieved before (with, however, no qualitative changes in the observed behavior). We then report new results with stress-free boundary conditions (for a viscous fluid, but bounded by a perfectly slippery wall). The method used is spectral, involving expansions of the fields in orthonormal sets of functions which obey two boundary conditions (circular analogues of the Chandrasekhar–Reid functions). The computation takes place entirely in the spectral space. Large-scale Reynolds numbers are typically less than a thousand. Interest focuses on the role played by angular momentum, in determining the decay of the turbulence with no-slip boundary conditions, and the role of possible other ideal invariants in the stress-free case. Received 30 September 1996 and accepted 5 February 1997     

Direct simulation of two‐dimensional turbulent flow over a surface‐mounted obstacle

Two‐dimensional turbulent flow over a surface‐mounted obstacle is studied as a numerical experiment that takes place in a wind tunnel. The transient Navier–Stokes equations are solved directly with Galerkin finite elements. The Reynolds number defined with respect to the height of the wind tunnel is 12 518. Instantaneous streamline patterns are shown that give a complete picture of the flow phenomena. Energy and enstrophy spectra yield the dual cascade of two‐dimensional turbulence and the ?1 power law decay of enstrophy. Mean values of velocities and root mean square fluctuations are compared with the available experimental results. Other statistical characteristics of turbulence such as Eulerian autocorrelation coefficients, longitudinal and lateral coefficients are also computed. Finally, oscillation diagrams of computed velocity fluctuations yield the chaotic behaviour of turbulence. Copyright © 2007 John Wiley & Sons, Ltd.     

Ideal Turbulence

Ideal turbulence is a mathematical phenomenon which occurs in certain infinite-dimensional deterministic dynamical systems and implies that the attractor of a system lies off the phase space and among the attractor points there are fractal or even random functions. A mathematically rigorous definition of ideal turbulence is based on standard notions of dynamical systems theory and chaos theory. Ideal turbulence is observed in various idealized models of real distributed systems of electrodynamics, acoustics, radiophysics, etc. In systems without internal resistance, cascade processes are capable to birth structures of arbitrarily small scale and even to cause stochastization of the systems. Just these phenomena are inherent in ideal turbulence and they help to understand the mathematical scenarios for many features of real turbulence.     

Flow and Mixing Fields for Transported Scalar PDF Simulations of a Piloted Jet Diffusion Flame (‘Delft Flame III’)

Numerical simulation results are presented for ‘Delft Flame III’, a piloted jet diffusion flame with strong turbulence–chemistry interaction. While pilot flames emerge from 12 separate holes in the experiments, the simulations are performed on a rectangular grid, under the assumption of axisymmetry. In the first part of the paper, flow and mixing field results are presented with a non-linear first order k–ε model, with the transport equation for ε based on a modeled enstrophy transport equation, for cold and reactive flows. For the latter, the turbulence model is applied in combination with pre-assumed β-PDF modeling for the turbulence–chemistry interaction. The mixture fraction serves as conserved scalar. Two chemistry models are considered: chemical equilibrium and a steady laminar flamelet model. The importance of the turbulence model is highlighted. The influence of the chemistry model is noticeable too. A procedure is described to construct appropriate inlet boundary conditions. Still, the generation of accurate inlet boundary conditions is shown to be far less important, their effect being local, close to the nozzle exit. In the second part of the paper, results are presented with the transported scalar PDF approach as turbulence–chemistry interaction model. A C₁ skeletal scheme serves as chemistry model, while the EMST method is applied as micro-mixing model. For the transported PDF simulations, the model for the pilot flames, as an energy source term in the mean enthalpy transport equation, is important with respect to the accuracy of the flow field predictions. It is explained that the strong influence on the flow and mixing field is through the turbulent shear stress force in the region, close to the nozzle exit.     

Jump conditions for filtered quantities at an under-resolved discontinuous interface. Part 2: A priori tests

In order to study and validate the jump conditions established in part 1, we realize a priori tests thanks to the data of a 3D Direct Numerical Simulation (DNS) of a strongly deformable bubble in a spatially decaying turbulence. The complex interactions between interface and turbulence are fully resolved. An explicit filtering of the DNS has been employed to evaluate the filtered quantities and to check the potential of the models for two-phase flows in the Interface and Subgrid Scales (ISS) modeling case proposed in part 1. The ISS concept is our proposal of a two-phase equivalent for the one-phase Large Eddy Simulation (LES) modeling case with sharp-interfaces. In this concept, bubbles remain bigger than the mesh size. Due to the impossibility to define a filter equivalent to the matched asymptotic expansions, we only test the modeling of the equivalent interface transport (the momentum jump conditions are not tested in this article, but will deserve additional results in a posteriori tests). Because the closure of the transport equation of the under-resolved discontinuous interface requires more modeling assumptions than the closure of the momentum equation, we think that the most relevant test has been done. The a priori tests realized show excellent agreement between the ISS models and the real contributions.     

KOLMOGOROV BEHAVIOR OF NEAR-WALL TURBULENCE AND ITS APPLICATION IN TURBULENCE MODELING

The near-wall behavior of turbulence is re-examined in a way different from that proposed by Hanjalic and Launder¹ and followers^2,3,4,5. It is shown that at a certain distance from the wall, all energetic large eddies will reduce to Kolmogorov eddies (the smallest eddies in turbulence). All the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, are characterised by Kolmogorov microscales. According t o this Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, dissipation rate, etc. at the location where the large eddies become “Kolmogorov” eddies, can be estimated by using both direct numerical simulation (DNS) data and asymptotic analysis of near-wall turbulence. This information will provide useful boundary conditions for the turbulent transport equations. As a n example, the concept is incorporated in the standard κ - εmodel which is then applied t o channel and boundary layer flows. Using appropriate boundary conditions (based on Kolmogorov behaviour of near-wall turbulence), there is no need for any wall-modification to the κ - ε equations (including model constants). Results compare very well with the DNS and experimental data.     

Three-component velocity field measurements of propeller wake using a stereoscopic PIV technique

A stereoscopic PIV (Particle Image Velocimetry) technique was used to measure the three-dimensional flow structure of the turbulent wake behind a marine propeller with five blades. The out-of-plane velocity component was determined using two CCD cameras with an angular displacement configuration. Four hundred instantaneous velocity fields were measured for each of four different blade phases, and ensemble averaged in order to find the spatial evolution of the propeller wake in the region from the trailing edge up to one propeller diameter (D) downstream. The influence of propeller loading conditions on the wake structure was also investigated by measuring the velocity fields at three advance ratios (J=0.59, 0.72 and 0.88). The phase-averaged velocity fields revealed that a viscous wake formed by the boundary layers developed along the blade surfaces. Tip vortices were generated periodically and the slipstream contracted in the near-wake region. The out-of-plane velocity component and strain rate had large values at the locations of the tip and trailing vortices. As the flow moved downstream, the turbulence intensity, the strength of the tip vortices, and the magnitude of the out-of-plane velocity component at trailing vortices all decreased due to effects such as viscous dissipation, turbulence diffusion, and blade-to-blade interaction.