Taylor Microscale Reynolds Number Research Articles

Some results on the Reynolds number scaling of pressure statistics in isotropic turbulence

Using data from direct numerical simulations in the Reynolds number range 8 ≤ R λ ≤ 1000 , where R λ is the Taylor microscale Reynolds number, we assess the Reynolds number scaling of the microscale and the integral length scale of pressure fluctuations in homogeneous and isotropic turbulence. The root-mean-square (rms) pressure (in kinematic units) is about 0.91 ρ u ′ 2 , where u ′ is the rms velocity in any one direction. The ratio of the pressure microscale to the (longitudinal) velocity Taylor microscale is a constant of about 0.74 for very low Reynolds numbers but increases approximately as 0.17 R λ 1 / 3 at high Reynolds numbers. We discuss these results in the context of the existing theory and provide plausible explanations, based on intermittency, for their observed trends.

Is Stokes number an appropriate indicator for turbulence modulation by particles of Taylor-length-scale size?

It has been established both numerically and experimentally [see, e.g., A. Ferrante and S. E. Elghobashi, “On the physical mechanisms of two-way coupling in particle-laden isotropic turbulence,” Phys. Fluids 15, 315 (2003); T. S. Yang and S. S. Shy, “Two-way interaction between solid particles and homogeneous air turbulence: Particle settling rate and turbulence modification measurements,” J. Fluid Mech. 526, 171 (2005)] that the Stokes number, τp/τk, can be used as an indicator to determine the extent to which small particles (dp<η) modify the turbulence structure for fixed values of their volume fraction and mass fraction, where dp, τp, η, and τk are the particle diameter, the response time, the Kolmogorov length, and time scales, respectively. The objective of the present study is to investigate whether the Stokes number can also be used as an indicator for the modulation of turbulence by particles of the Taylor-length-scale size, i.e., dp∼λ⪢η. We employ direct numerical simulation with an immersed boundary method to fully resolve the flow around 4800 freely moving particles of Taylor-length-scale size (dp∼λ) in decaying isotropic turbulence with initial Taylor-microscale Reynolds number, Reλ=110, using a computational mesh of 5123 grid points. Our results show that particles whose diameter dp∼λ⪢η and have identical τp (but different dp and/or ρp) can have different effects on isotropic turbulence. Thus, τp/τk should not be used as an indicator of turbulence modulation by these “large” particles.

On the role of vortical structures for turbulent mixing using direct numerical simulation and wavelet-based coherent vorticity extraction

The influence of vortical structures on the transport and mixing of passive scalars is investigated. Initial conditions are taken from a direct numerical simulation database of forced homogeneous isotropic turbulence, with passive scalar fluctuations, driven by a uniform mean gradient, are performed for Taylor microscale Reynolds numbers (R λ) of 140 and 240, and Schmidt numbers 1/8 and 1. For each R λ, after reaching a fully developed turbulent regime, which is statistically steady, the Coherent Vorticity Extraction is applied to the flow. It is shown that the coherent part is able to preserve the vortical structures with only less than 4% of wavelet coefficients while retaining 99.9% of energy. In contrast, the incoherent part is structureless and contains negligible energy. By taking the total, coherent and incoherent velocity fields in turn as initial conditions, new simulations were performed without forcing while the uniform mean scalar gradient is maintained. It is found that the results from simulations with total and coherent velocity fields as initial conditions are very similar. In contrast, the time integration of the incoherent flow exhibits its primarily dissipative nature. The evolutions of passive scalars at Schmidt numbers 1/8 and 1 advected by the total, coherent or incoherent velocity suggest that the vortical structures retained in the coherent part play a dominant role in turbulent transport and mixing. Indeed, the total and coherent flows give almost the same time evolution of the scalar variance, scalar flux and mean scalar dissipation, while the incoherent flow only gives rise to weak scalar diffusion.

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

Time evolution of a fractal distribution: Particle concentrations in free-surface turbulence

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal structure, a rare manifestation of a strange attractor observable in real space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions D q ( t ) ( q = 1 to 6) of the floating particles starting with the uniform distribution D q ( 0 ) = 2 for Taylor Microscale Reynolds number R e λ ≃ 160 . Focus is on the time evolution of the correlation dimension D 2 ( t ) as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.

Comparison of the lattice Boltzmann and pseudo-spectral methods for decaying turbulence: Low-order statistics

We conduct a detailed comparison of the lattice Boltzmann equation (LBE) and the pseudo-spectral (PS) methods for direct numerical simulations (DNS) of the decaying homogeneous isotropic turbulence in a three-dimensional periodic cube. We use a mesh size of N 3 = 128 3 and the Taylor micro-scale Reynolds number 24.35 ⩽ Re λ ⩽ 72.37 , and carry out all simulations to t ≈ 30 τ 0 , where τ 0 is the turbulence turnover time. In the PS method, the second-order Adam–Bashforth scheme is used to numerically integrate the nonlinear term while the viscous term is treated exactly. We compare the following quantities computed by the LBE and PS methods: instantaneous velocity u and vorticity ω fields, and statistical quantities such as, the total energy K ( t ) and the energy spectrum E ( k , t ) , the dissipation rate ε ( t ) , the root-mean-squared (rms) pressure fluctuation δ p ( t ) and the pressure spectrum P ( k , t ) , and the skewness and flatness of the velocity derivative. Our results show that the LBE method performs very well when compared to the PS method in terms of accuracy and efficiency: the instantaneous flow fields, u and ω , and all the statistical quantities — except the rms pressure fluctuation δ p ( t ) and the pressure spectrum P ( k , t ) — computed from the LBE and PS methods agree well with each other, provided that the initial flow field is adequately resolved by both methods. We note that δ p ( t ) and P ( k , t ) computed from the two methods agree with each other in a period of time much shorter than that for other quantities, indicating that the pressure field p computed by using the LBE method is less accurate than other quantities. The skewness and flatness computed from the LBE method contain high-frequency oscillations due to acoustic waves in the system, which are absent in PS methods. Our results indicate that the resolution requirement for the LBE method is δ x / η 0 ⩽ 1.0 , approximately twice of the requirement for PS methods, where δ x and η 0 are the grid spacing and the initial Kolmogorov length, respectively. Overall, the LBE method is shown to be a reliable and accurate method for the DNS of decaying turbulence.

Gas-kinetic schemes for direct numerical simulations of compressible homogeneous turbulence

We apply the gas-kinetic scheme (GKS) for the direct numerical simulations (DNSs) of compressible decaying homogeneous isotropic turbulence (DHIT). We intend to study the accuracy, stability, and efficiency of the gas-kinetic scheme for DNS of compressible homogeneous turbulence depending on both flow conditions and numerics. In particular, we study the GKS with multidimensional, quasi-one-dimensional, dimensional-splitting, and smooth-flow approximations. We simulate the compressible DHIT with the Taylor microscale Reynolds number Re(lambda)=72.0 and the turbulence Mach number Ma(t) between 0.1 and 0.6. We compute the low-order statistical quantities including the total kinetic energy K(t), the dissipation rate epsilon(t), the skewness S(u)(t), and the flatness F(u)(t) of the velocity field u(x,t). We assess the effects on the turbulence statistics due to the approximations made in the treatment of fluxes, the flux limiter, the accuracy of the interpolation, and the bulk viscosity. Our results show that the GKS is adequate for DNS of compressible homogeneous turbulence as far as the low-order turbulence statistics are concerned.

Coherent vorticity extraction in 3D homogeneous isotropic turbulence: influence of the Reynolds number and geometrical statistics

The coherent vorticity extraction method (CVE) is based on the nonlinear filtering of the vorticity field projected onto an orthonormal wavelet basis made of compactly supported functions. CVE decomposes each turbulent flow realization into two orthogonal components: a coherent and an incoherent random flow. They both contribute to all scales in the inertial range, but exhibit different statistical behavior. We apply CVE to 256 3 subcubes extracted from 3D homogeneous isotropic turbulent flows at different Taylor microscale Reynolds numbers (Rλ = 140, 240 and 680), computed by a direct numerical simulation (DNS) at different resolutions (N = 256 3 , 512 3 and 2048 3 ), respectively. We compare the total, coherent and incoherent vorticity fields obtained by using CVE and show that few wavelets coefficients are sufficient to represent the coherent vortices of the flows. Geometrical statistics in term of helicity are also analyzed and the λ2 criterion is applied to the filtered flow fields.

Effect of the Scalar Injection Mechanism on Passive Scalar Structure Functions in a Turbulent Flow

Higher-order passive scalar (temperature) structure functions are measured in the turbulent wake of a circular cylinder at a Taylor-microscale Reynolds number (Rlambda) of 370. The scalar is injected by two different means: (i) heating of the cylinder and (ii) use of a mandoline. Even though the second-order statistics (e.g., power spectra, second-order structure functions) of the scalar field are experimentally indistinguishable in the inertial and dissipative ranges, we observe notable differences in the inertial-range scaling exponents (xin) of the scalar structure functions at higher orders. The implication is therefore that the variations in previous estimates of xin may be attributable to differences in the scalar field initial conditions (and may not be deemed characteristic of a universal nature of the small-scale statistics of turbulent passive scalars).

Flume experiments on intermittency and zero-crossing properties of canopy turbulence

How the presence of a canopy alters the clustering and the fine scale intermittency exponents and any possible connections between them remains a vexing research problem in canopy turbulence. To begin progress on this problem, detailed flume experiments in which the longitudinal and vertical velocity time series were acquired using laser Doppler anemometry within and above a uniform canopy composed of densely arrayed rods. The time series analysis made use of the telegraphic approximation (TA) and phase-randomization (PR) methods. The TA preserved the so-called zero-crossing properties in the original turbulent velocity time series but eliminated amplitude variations, while the PR generated surrogate data that preserved the spectral scaling laws in the velocity series but randomized the acceleration statistics. Based on these experiments, it was shown that the variations in the dissipation intermittency exponents were well described by the Taylor microscale Reynolds number (Reλ) within and above the canopy. In terms of clustering, quantified here using the variance in zero-crossing density across scales, two scaling regimes emerged. For spatial scales much larger than the canopy height hc, representing the canonical scale of the vortices dominating the flow, no significant clustering was detected. For spatial scales much smaller than hc, significant clustering was discernable and follows an extensive scaling law inside the canopy. Moreover, the canopy signatures on the clustering scaling laws were weak. When repeating these clustering measures on the PR data, the results were indistinguishable from the original series. Hence, clustering exponents derived from variances in zero-crossing density across scales primarily depended on the velocity correlation function and not on the distributional properties of the acceleration. In terms of the connection between dissipation intermittency and clustering exponents, there was no significant relationship. While the former varied significantly with Reλ, the latter showed only minor variations within and above the canopy sublayer.

Dynamics Behaviors and Scaling in Intermittent Turbulence of a Shell Model

In this paper, the dynamics behaviors on f0-δ parameter surface is investigated for Gledzer–Ohkitani–Yamada model. We indicate the type of intermittency chaos transitions is saddle node bifurcation. We plot phase diagram on f0-δ parameter surface, which is divided into periodic, quasi-periodic, and intermittent chaos areas. By means of varying Taylor-microscale Reynolds number, we calculate the extended self-similarity of velocity structure function.

Effect of freestream isotropic turbulence on heat transfer from a sphere

We consider direct numerical simulation (DNS) based on pseudospectral methods to study the heat transfer around a stationary sphere held at a constant temperature and subject to an ambient turbulent velocity and temperature condition. The sphere Reynolds number is in the range of 63–400, and the sphere diameter (d) varies from one to eight times the Kolmogorov scale (η). The ambient turbulent field is isotropic, and the Taylor microscale Reynolds number Rλ varies from 38 to 240. Results from two sets of DNS are presented. In the first set, the ambient velocity field is turbulent, but the ambient temperature is held constant. In the second set of simulations, both the ambient velocity and the temperature fields are turbulent. These two sets of simulations allow us to isolate the role of freestream velocity fluctuations and temperature fluctuations in modifying the mean and time-dependent heat transfer from the sphere. The mean Nusselt number is observed to be independent of Rλ. It is shown that the freestream turbulence does not have any significant effect on the mean Nusselt number, and the available correlations for a steady and uniform ambient can predict the mean Nusselt number under the turbulent ambient condition. The instantaneous Nusselt number, however, can differ significantly from the mean, and can be negative in case of large temperature fluctuation in the far field. The instantaneous Nusselt number obtained from the DNS is analyzed and compared with the analytical expression for the unsteady thermal response of a sphere. It is shown that the thermal added-mass effect is small for d/η≈1 but introduces spurious oscillation at higher d. The thermal history effect is shown to be insignificant for all d/η. Properties of the thermal wake in the presence of the turbulent velocity and temperature fields are studied. The mean thermal wake is observed to be shorter in streamwise direction and wider in crossflow direction in a turbulent ambient than that in a steady and uniform ambient. The mean thermal wake is shown to behave like a self-preserving laminar wake. It is also shown that the fluctuation in the ambient temperature is damped out in the sphere wake, and the large variation observed in the instantaneous Nusselt number arises from the variation in the local heat transfer near the front stagnation area of the sphere rather than the wake.

Examination of Kolmogorov's 4/5 Law by High-Resolution Direct Numerical Simulation Data of Turbulence

Kolmogorov's 4/5 law is examined quantitatively by using the data of direct numerical simulations of incompressible turbulence in a periodic box with the number of grid points and the Taylor micro-scale Reynolds number 2048 3 and R λ ≈732, respectively. The examination is based on a generalization of the Kármán–Howarth–Kolmogorov (KHK) equation for isotropic turbulence to anitropic turbulence. An emphasis is put on the influence of the finite Reynolds number, length-scale and the weak but finite anisotropy. The analysis of the DNS data gives a quantitative idea on the role of the viscous and forcing terms in the KHK equation. It also suggests that the anisotropy has substantial influence on the 4/5 law for a wide scale-range.

A Lagrangian investigation of the small-scale features of turbulent entrainment through particle tracking and direct numerical simulation

We report an analysis of small-scale enstrophy ω2and rate of strains2dynamics in the proximity of the turbulent/non-turbulent interface in a flow without strong mean shear. The techniques used are three-dimensional particle tracking (3D-PTV), allowing the field of velocity derivatives to be measured and followed in a Lagrangian manner, and direct numerical simulations (DNS). In both experiment and simulation the Taylor-microscale Reynolds number isReλ= 50. The results are based on the Lagrangian viewpoint with the main focus on flow particle tracers crossing the turbulent/non-turbulent interface. This approach allowed a direct investigation of the key physical processes underlying the entrainment phenomenon and revealed the role of small-scale non-local, inviscid and viscous processes. We found that the entrainment mechanism is initiated by self-amplification ofs2through the combined effect of strain production and pressure--strain interaction. This process is followed by a sharp change of ω2induced mostly by production due to viscous effects. The influence of inviscid production is initially small but gradually increasing, whereas viscous production changes abruptly towards the destruction of ω2. Finally, shortly after the crossing of the turbulent/non-turbulent interface, production and dissipation of both enstrophy and strain reach a balance. The characteristic time scale of the described processes is the Kolmogorov time scale, τη. Locally, the characteristic velocity of the fluid relative to the turbulent/non-turbulent interface is the Kolmogorov velocity,uη.

Wavelet-based statistics of energy transfer in high-resolution direct numerical simulation of three-dimensional homogeneous isotropic turbulence

Small-scale statistics of wavelet-based quantities of turbulence are examined, with particular attention to t α,β[ ι ], which is the wavelet-based transfer of the wavelet-based energy e α[ ι ] at location ι and scale r α to all scales smaller than r β (⩽r α). A divergence-free orthonormal wavelet decomposition method is applied to direct numerical simulation data of three-dimensional incompressible homogeneous isotropic turbulence with grid points and Taylor microscale Reynolds numbers up to 20483 and 732, respectively. We propose a new statistical measure for the degree of scale locality of wavelet-based energy flux to scales smaller than r β constructed from t α,β[ ι ]. At R λ = 732, the measure retains a constant value in the inertial subrange, a value about two-thirds as large as that in the dissipation range, though the flux is local in scale throughout the range, from the inertial subrange to the dissipation range. We confirm that the influence of energy transfer from the large-scale flow on the flux becomes smaller with decreasing scales and negligible in the dissipation range. Scale-dependent spatial correlation between t α,β[ ι ] and e α[ ι ] is discussed. We find that, when r β is in the inertial range, the conditional correlation between forward transfer, which is t α,β[ ι ] at the positions satisfying t α,β[ ι ]>0, and e α[ ι ] becomes stronger as scale r α increases from r β, while this behavior is reversed when r β is in the dissipation range.

Experimental study on clustering of large particles in homogeneous turbulent flow

Results of simultaneous measurements of fluid flow and motion of large solid particles in homogeneous turbulent flow (at the Taylor microscale Reynolds number Re λ = 250) are presented. Velocity, acceleration and spatial distribution of particles in a three-dimensional volume along with the surrounding flow velocity and velocity derivatives fields (incl. vorticity and the strain-rate tensor) are obtained by means of two synchronized and cross-calibrated three-dimensional particle tracking velocimetry (3D-PTV) systems. The main focus of the present investigation is on the two-way coupling between the turbulent flow and solid particles which are significantly larger (∼900 μm) than the Kolmogorov length scale (∼200 μm) and heavier than the surrounding liquid (ρ p = 1400 kg m−3>ρ f = 1000 kg m−3). Joint statistics of the local particle concentration and the local strain and vorticity fields indicate that (1) large particles tend to cluster in strain-dominated regions and that (2) preferential concentration (clusters and voids) occur on scales comparable with the Taylor microscale, λ. We infer that the observed clustering of large particles can be related to the same mechanisms applying for clustering of sub-Kolmogorov size particles and that, in analogy, the time scale λ/u r.m.s. can replace the Kolmogorov time scale as a normalization of the particle response time in the definition of the Stokes number.

Particles for tracing turbulent liquid helium

We address the problem of making quantitative measurements of local flow velocities in turbulent liquid helium, using tracer particles. We survey and evaluate presently available particles and previous work to establish the need to develop new particles for the purpose. We present the first practical solution for visualizing fluid motions using a suspension of solid hydrogen particles with diameters of about 2 μm. The hydrogen particles can be used to study flows with Taylor-microscale Reynolds numbers between 85 and 775. The particles can be used equally well with the PIV, LDV, or particle-tracking techniques.

Acceleration Correlations and Pressure Structure Functions in High-Reynolds Number Turbulence

We present measurements of fluid particle accelerations in turbulent water flow between counterrotating disks using three-dimensional Lagrangian particle tracking. By simultaneously following multiple particles with sub-Kolmogorov-time-scale temporal resolution, we measured the spatial correlation of fluid particle acceleration at Taylor microscale Reynolds numbers between 200 and 690. We also obtained indirect, nonintrusive measurements of the Eulerian pressure structure functions by integrating the acceleration correlations. Our measurements are in good agreement with the theoretical predictions of the acceleration correlations and the pressure structure function in isotropic high-Reynolds number turbulence by Obukhov and Yaglom in 1951 [Prikl. Mat. Mekh. 15, 3 (1951)]. The measured pressure structure functions display K41 scaling in the inertial range.

Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint

Coherent vortices are extracted from data obtained by direct numerical simulation (DNS) of three-dimensional homogeneous isotropic turbulence performed for different Taylor microscale Reynolds numbers, ranging from Reλ=167 to 732, in order to study their role with respect to the flow intermittency. The wavelet-based extraction method assumes that coherent vortices are what remains after denoising, without requiring any template of their shape. Hypotheses are only made on the noise that, as the simplest guess, is considered to be additive, Gaussian, and white. The vorticity vector field is projected onto an orthogonal wavelet basis, and the coefficients whose moduli are larger than a given threshold are reconstructed in physical space, the threshold value depending on the enstrophy and the resolution of the field, which are both known a priori. The DNS dataset, computed with a dealiased pseudospectral method at resolutions N=2563,5123,10243, and 20483, is analyzed. It shows that, as the Reynolds number increases, the percentage of wavelet coefficients representing the coherent vortices decreases; i.e., flow intermittency increases. Although the number of degrees of freedom necessary to track the coherent vortices remains small (e.g., 2.6% of N=20483 for Reλ=732), it preserves the nonlinear dynamics of the flow. It is thus conjectured that using the wavelet representation the number of degrees of freedom to compute fully developed turbulent flows could be reduced in comparison to the standard estimation based on Kolmogorov’s theory.