Stabilized finite element method for incompressible flows with high Reynolds number

In the following paper, we discuss the exhaustive use and implementation of stabilization finite element methods for the resolution of the 3D time-dependent incompressible Navier–Stokes equations. The proposed method starts by the use of a finite element variational multiscale (VMS) method, which consists in here of a decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales. This choice of decomposition is shown to be favorable for simulating flows at high Reynolds number. We explore the behaviour and accuracy of the proposed approximation on three test cases. First, the lid-driven square cavity at Reynolds number up to 50,000 is compared with the highly resolved numerical simulations and second, the lid-driven cubic cavity up to Re = 12,000 is compared with the experimental data. Finally, we study the flow over a 2D backward-facing step at Re = 42,000. Results show that the present implementation is able to exhibit good stability and accuracy properties for high Reynolds number flows with unstructured meshes.     

Broadband trailing edge noise from a sharp-edged strut

This paper presents experimental data concerning the flow and noise generated by a sharp-edged flat plate at low-to-moderate Reynolds number (Reynolds number based on chord of 2.0 × 10(5) to 5.0 × 10(5)). The data are used to evaluate a variety of semi-empirical trailing edge noise prediction methods. All were found to under-predict noise at lower frequencies. Examination of the velocity spectra in the near wake reveals that there are energetic velocity fluctuations at low frequency about the trailing edge. A semi-empirical model of the surface pressure spectrum is derived for predicting the trailing edge noise at low-to-moderate Reynolds number.     

圆柱绕流问题的初期流动数值模拟

本文继Zheng-huan Teng[1]中介绍的解Navier-stokes方程的椭圆涡团法,研究了一种新的变形涡团法,用以模拟不可压粘性流体绕圆柱的不定常流动。圆柱在静止流体中突然起动并做匀速直线运动。对整个流动区域构造完全Navier-Stokes方程的解并不容易,近十年出现很多数值研究,本文对算法有所推进。把本文的方法称作变形涡团法是因为圆柱边界附近的流体中用椭圆涡团,远离边界时用圆形涡团。计算圆柱绕流比平板绕流在满足附着条件上更为困难,本文分析了怎样在圆柱边界上给出适当的附着条件的数值方法。在算例中雷诺数分别取200、550、3000,得到了不定常边界层分离,二次涡等复杂的物理现象,这些数值结果与近年实验结果[2]是一致的。     

Existence of dual solutions and three-dimensional instability in helical pipe flow

Three-dimensional (3D) direct numerical simulations (DNS) of the viscous incompressible fluid flow through a helical pipe with circular cross section were performed. The flow is governed by three parameters: the Dean number (or the Reynolds number), curvature, and torsion. First, we obtained steady solutions by steady 3D calculations, where dual solutions were found, one was uniform in the pipe axial direction and the other varied very slowly, if torsion exceeded a critical value. Then, the instability of the steady solutions obtained was studied by unsteady 3D calculations. We obtained critical Reynolds numbers of steady to unsteady transition by observing the behaviors of the unsteady solutions. The present results of the critical Reynolds number nearly agreed with those by the 2D linear stability analysis (Yamamoto et al. [9]) except for the lowest critical Reynolds number region, where the present study gave the critical Reynolds number much less than that obtained by the 2D linear stability analysis. We found the vortical structures in the form of a standing wave slightly above the marginal instability state, which is a trigger of explosive 3D instability.     

Numerical Stability Analysis of FDLBM

We analyze the numerical stability of Finite Difference Lattice Boltzmann Method (FDLBM) by means of von Neumann stability analysis. The stability boundary of the FDLBM depends on the BGK relaxation time, the CFL number, the mean flow velocity, and the wavenumber. As the BGK relaxation time is increased at constant CFL number, the stability of the central difference LB scheme may not be ensured. The limits of maximum stable velocity are obtained around 0.39, 0.43, and 0.43 for the central difference, for the explicit upwind difference, and for the semi-implicit upwind difference schemes, respectively. We derive artificial viscosities for every difference scheme and investigate their influence on numerical stability. The requirements for artificial viscosity is consistent with the conditions derived from von Neumann stability analysis. This analysis elucidates that the upwind difference schemes are suitable for simulation of high Reynolds number flows.     

Investigating asymptotic suction boundary layers using a one-dimensional stochastic turbulence model

The turbulent asymptotic suction boundary layer is studied using a one-dimensional turbulence (ODT) model. ODT is a fully resolved, unsteady stochastic simulation technique. While flow properties reside on a one-dimensional domain, turbulent advection is represented using mapping events whose occurrences are governed by a random process. Due to its reduced spatial dimensionality, ODT achieves major cost reductions compared to three-dimensional (3D) simulations. A comparison to recent direct numerical simulation (DNS) data at moderate Reynolds number (Re = u_∞ / v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively) suggests that the ODT model is capable of reproducing several velocity statistics, i.e. mean velocity and turbulent kinetic energy budgets, while peak turbulent stresses are under-estimated by ODT. Variation of the Reynolds number in the range Re ∈ [333,400,500,1000] shows that ODT can reproduce various trends observed as a result of increased suction in turbulent asymptotic suction boundary layers, i.e. the reduction of Reynolds stresses and enhanced skin friction. While up to Re = 500 our results can be directly compared to recent LES data, the simulation at Re = 1000 is currently not feasible through full 3D simulations, hence ODT may assist the design of future DNS or LES simulations at larger Reynolds numbers.     

Magnetic-field generation in Kolmogorov turbulence

We analyze the initial, kinematic stage of magnetic field evolution in an isotropic and homogeneous turbulent conducting fluid with a rough velocity field, v(l) approximately l(alpha), alpha<1. This regime is relevant to the problem of magnetic field generation in fluids with small magnetic Prandtl number, i.e., with Ohmic resistivity much larger than viscosity. We propose that the smaller the roughness exponent alpha, the larger the magnetic Reynolds number that is needed to excite magnetic fluctuations. This implies that numerical or experimental investigations of magnetohydrodynamic turbulence with small Prandtl numbers need to achieve extremely high resolution in order to describe magnetic phenomena adequately.     

Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow

A simple theoretical analysis and direct numerical simulations on 512(3) grid points suggest that the velocity correlation spectrum tensor in the inertial subrange of homogeneous turbulent shear flow at high Reynolds number is given by a simple form that is an anisotropic function of the wave vector. The tensor is determined by the rate of the strain tensor of the mean flow, the rate of energy dissipation per unit mass, the wave vector, and two nondimensional constants.     

Origination and evolution of turbulent flows in a rotating spherical layer

The object of consideration is the turbulent flows of a viscous incompressible liquid that arises in a wide spherical layer with counter-rotating boundaries (the thickness of the layer equals the radius of the inner sphere). Regimes established when the outer sphere rotates with a constant velocity and the inner one rotates with an increasing velocity are studied in physical and numerical experiments. The averaged meridional circulation and the pulsation profiles of all velocity components are derived by direct calculation. It is found that both observed and simulated turbulent regimes are characterized by the continuous spectrum of velocity pulsation near their formation boundary. In going from the laminar to chaotic regime, the correlation dimension increases stepwise and then slightly varies with increasing Reynolds number in a nonlinear manner.     

Is multiscaling an artifact in the stochastically forced Burgers equation?

We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum approximately 1/k, where k is the wave number. From very high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be falsified by standard tests. We find a new artifact in which logarithmic corrections can appear disguised as anomalous scaling and conclude that bifractal scaling is likely.     

On the variable order dynamics of the nonlinear wake caused by a sedimenting particle

In this work we develop a variable order (VO) differential equation of motion for a spherical particle sedimenting in a quiescent viscous liquid. In particular, we examine the various force terms in the equation of motion and propose a new form for the history drag acting on the particle. We show that the variable order formulation allows for an effective way to express the dynamic transition of the dominant forces over the entire time of the motion of the particle from rest to terminal velocity. The use of VO operators also allows us to examine the evolving dynamics of the wake during sedimentation. Using numerical data from a finite element simulation of a sedimenting particle, we first solve for the order of the derivative that returns the correct decay of the history force. We then propose a relatively simple expression for the history force that is a function of the Reynolds number and particle-to-fluid density ratio. The new history drag expression correlates very well (R²>0.99) with the numerical data for terminal Reynolds numbers ranging from 2.5 to 20, and for particle-to-fluid density ratios of interest in practice (1<β<10).     

On one-eighth law in unforced two-dimensional turbulence

We critically revisit the various attempts to prove one-eighth law in two-dimensional (2D) turbulence and reconcile them. Herein, the one-eighth law has been proved for unforced 2D incompressible high Reynolds number turbulence. An exact expression of the time derivative of two-point second order velocity correlation function is also derived for the enstrophy cascade dominated regime.     

Lagrangian view of time irreversibility of fluid turbulence

A turbulent flow is maintained by an external supply of kinetic energy, which is eventually dissipated into heat at steep velocity gradients. The scale at which energy is supplied greatly differs from the scale at which energy is dissipated, the more so as the turbulent intensity(the Reynolds number) is larger. The resulting energy flux over the range of scales, intermediate between energy injection and dissipation, acts as a source of time irreversibility. As it is now possible to follow accurately fluid particles in a turbulent flow field, both from laboratory experiments and from numerical simulations, a natural question arises: how do we detect time irreversibility from these Lagrangian data? Here we discuss recent results concerning this problem. For Lagrangian statistics involving more than one fluid particle, the distance between fluid particles introduces an intrinsic length scale into the problem. The evolution of quantities dependent on the relative motion between these fluid particles, including the kinetic energy in the relative motion, or the configuration of an initially isotropic structure can be related to the equal-time correlation functions of the velocity field, and is therefore sensitive to the energy flux through scales, hence to the irreversibility of the flow. In contrast, for singleparticle Lagrangian statistics, the most often studied velocity structure functions cannot distinguish the "arrow of time". Recent observations from experimental and numerical simulation data, however, show that the change of kinetic energy following the particle motion, is sensitive to time-reversal. We end the survey with a brief discussion of the implication of this line of work.     

Peak position of dissipation spectrum in turbulent boundary layers

The peak locations of energy dissipation spectrum (y(+)(kp)) are investigated experimentally in the low Reynolds number (R(theta)<5000) zero-pressure-gradient turbulent boundary layers. These peaks are scaled by the relation y(+)(kp)=2(u(*)delta/nu)(1/2)=2R(1/2)(*), where u(*) is friction velocity and delta is the boundary layer thickness. They are located close to the peak positions of the Reynolds shear stress (y(+)(p)), that is, y(+)(kp) approximately equal to y(+)(p). This result predicts that the small-scale structures residing in high-energy dissipation regions concentrate around y(+)(p).     

A theoretical model of turbulent fiber suspension and its application to the channel flow

A theoretical model of turbulent fiber suspension is developed by deriving the equations of Reynolds averaged Navier-Stokes,turbulence kinetic energy and turbulence dissipation rate with the additional term of fibers.In order to close the above equations,the equation of probability distribution function for mean fiber orientation is also derived.The theoretical model is applied to the turbulent channel flow and the corresponding equations are solved numerically.The numerical results are verified by comparisons with the experimental ones.The effects of Reynolds number,fiber concentration and fiber aspect-ratio on the velocity profile,turbulent kinetic energy and turbulent dissipation rate are analyzed.Based on the numerical data,the expression for the velocity profile in the turbulent fiber suspension channel flow,which includes the effect of Reynolds number,fiber concentration and aspect-ratio,is proposed.     

Continuous scanning, laser imaging velocimetry

Careful exploitation of the anisotropy native to late time stratified and rotating flows permits the use of a laser scanning measurement technique to simultaneously resolve the 2D velocity field in O(100) slices. The technique relies on getting the Reynolds number from the length scale while keeping the velocity small, this provides a characteristic time scale that is sufficiently large to permit full 3D scanning through the measurement volume in a relatively short time. As the vertical velocity component of these late time stratified flows is effectively zero, all components of the deformation tensor are resolved. 3D, time resolved measurements of the vorticity and enstrophy fields associated with stratified rotating flows such as vortex dipoles, monopoles and wakes are presented.     

Kolmogorov scaling of turbulent flow in the vicinity of the wall

How to scale even the simplest of turbulent flows continues to be a cause for considerable controversy. In the present research, a data base compiling results from channel flow direct numerical simulations and turbulent boundary layer experiments is employed to investigate the properties of shear and normal Reynolds stresses very close to the wall. Two types of scaling based on Kolmogorov length and velocity scales are analyzed. It is shown that it is highly likely that large length scales of the order of the channel half-width or the boundary layer thickness play an important role even in the innermost regions of wall-bounded turbulent flows, which hints at the persistence of Reynolds number effects in even high Reynolds number flows.     

Statistical-mechanical predictions and Navier-Stokes dynamics of two-dimensional flows on a bounded domain

In this paper the applicability of a statistical-mechanical theory to freely decaying two-dimensional (2D) turbulence on a bounded domain is investigated. We consider an ensemble of direct numerical simulations in a square box with stress-free boundaries, with a Reynolds number that is of the same order as in experiments on 2D decaying Navier-Stokes turbulence. The results of these simulations are compared with the corresponding statistical equilibria, calculated from different stages of the evolution. It is shown that the statistical equilibria calculated from early times of the Navier-Stokes evolution do not correspond to the dynamical quasistationary states. At best, the global topological structure is correctly predicted from a relatively late time in the Navier-Stokes evolution, when the quasistationary state has almost been reached. This failure of the (basically inviscid) statistical-mechanical theory is related to viscous dissipation and net leakage of vorticity in the Navier-Stokes dynamics at moderate values of the Reynolds number.     

Lattice Boltzmann study of three-dimensional immiscible Rayleigh–Taylor instability in turbulent mixing stage

In this paper, we numerically studied the late-time evolutional mechanism of three-dimensional (3D) single-mode immiscible Rayleigh–Taylor instability (RTI) by using an improved lattice Boltzmann multiphase method implemented on graphics processing units. The influences of extensive dimensionless Reynolds numbers and Atwood numbers on phase interfacial dynamics, spike and bubble growth were investigated in details. The longtime numerical experiments indicate that the development of 3D singlemode RTI with a high Reynolds number can be summarized into four different stages: linear growth stage, saturated velocity growth stage, reacceleration stage and turbulent mixing stage. A series of complex interfacial structures with large topological changes can be observed at the turbulent mixing stage, which always preserve the symmetries with respect to the middle axis for a low Atwood number, and the lines of symmetry within spike and bubble are broken as the Atwood number is increased. Five statistical methods for computing the spike and bubble growth rates were then analyzed to reveal the growth law of 3D single-mode RTI in turbulent mixing stage. It is found that the spike late-time growth rate shows an overall increase with the Atwood number, while the bubble growth rate experiences a slight decrease with the Atwood number at first and then basically maintains a steady value of around 0.1. When the Reynolds number decreases, the later stages cannot be reached gradually and the evolution of phase interface presents a laminar flow state.     

非均衡湍流研究进展

Richardson-Kolmogorov能量级串理论是湍流研究中最重要的基础理论,其中一个推论是能量的传输和耗散应当是均衡的,对应于耗散系数${C_\mathit{\boldsymbol{\epsilon}}}$为常数.然而近些年的实验及数值模拟都发现了不符合Richardson-Kolmogorov能量级串理论的非均衡耗散律,即使在此区域内Reynolds数足够高,能谱满足Kolmogorov的-5/3标度律,${C_\mathit{\boldsymbol{\epsilon}}}$也不为常数,而满足${C_\mathit{\boldsymbol{\epsilon}}}$~Re_Im/Re_Ln,其中m≈1≈n,Re_I为入口Reynolds数,Re_L为以积分尺度为特征长度的当地Reynolds数.近三年来又发现流向速度梯度扭率S_k和Lagrange速度梯度自相关的时间演化Φ'_ijij也可以用来度量非均衡湍流现象,为非均衡湍流的研究开辟了新路.