Reynolds Number Dependence Research Articles

Comment on “Evolution of wall shear stress with Reynolds number in fully developed turbulent channel flow experiments”

The recent study by Gubian et al. [Phys. Rev. Fluids 4, 074606 (2019)], based on a new wall-shear-stress sensor in a low-Reynolds-number Re turbulent channel flow, came to the surprising conclusion that the magnitude of the fluctuating wall-shear stress ${\ensuremath{\tau}}_{w,rms}^{+}$ reaches an asymptotic value of 0.44 beyond the friction Reynolds number ${\text{Re}}_{\ensuremath{\tau}}\ensuremath{\approx}600$. This statement is at odds with results from well-established direct numerical simulation (DNS) results that exceed the authors' highest Reynolds number by up to a factor of 5 while exhibiting a clear Reynolds-number dependence. Furthermore, they claim that ``prior estimates of these quantities did not resolve the full range of wall-shear-stress fluctuations, which extended beyond 10 standard deviations above the mean.'' This contradicts high-quality DNS results and calls for a more in-depth explanation, which is given in the present Comment. We shows that the measurements by Gubian et al. suffer from spatial-resolution issues among others, which when accounted for invalidate the statements made of an asymptotic state at ${\text{Re}}_{\ensuremath{\tau}}\ensuremath{\approx}600$ and resurrects the Reynolds-number dependence of ${\ensuremath{\tau}}_{w,rms}^{+}$ for which DNS evidence exists exceeding ${\text{Re}}_{\ensuremath{\tau}}\ensuremath{\approx}600$ by an order of magnitude.

Reply to “Comment on ‘Evolution of wall shear stress with Reynolds number in fully developed turbulent channel flow experiments' ”

\"Orl\"u and Schlatter [R. \"Orl\"u and P. Schlatter, preceding Comment, Phys. Rev. Fluids 5, 127601 (2020)] claim that the evolution of the turbulent intensity of the wall-shear stress (${\ensuremath{\tau}}_{{w}_{RMS}}/\ensuremath{\langle}{\ensuremath{\tau}}_{w}\ensuremath{\rangle}$) to a constant value at sufficiently large Reynolds number in our previous work [P.-A. Gubian et al., Phys. Rev. Fluids 4, 074606 (2019)] is the result of ``strong effects of insufficient spatial resolution'' of the sensor used therein, which, when corrected for, restores a continual Reynolds number dependence of ${\ensuremath{\tau}}_{{w}_{RMS}}/\ensuremath{\langle}{\ensuremath{\tau}}_{w}\ensuremath{\rangle}$. They also argue that the temporal resolution of the sensor used in our work had not been characterized. Herein, it is demonstrated that there are multiple other studies that have shown (by way of direct numerical simulation) that ${\ensuremath{\tau}}_{{w}_{RMS}}/\ensuremath{\langle}{\ensuremath{\tau}}_{w}\ensuremath{\rangle}$ becomes constant and independent of Reynolds number. Moreover, as \"Orl\"u and Schlatter themselves note, their proposed correction substantially overcorrects the data, such that the corrected data are ``too high and strong,'' which is presumably because they are applying a correction designed for velocity measurements to a sensor that (directly) measures wall-shear stress and is governed by different physical principles. Finally, the frequency response of the sensor used in our previous work (a flush-mounted hot-wire sensor in which the hot-wire is installed over a small rectangular cavity in the base of the sensor) has indeed been characterized and documented, by Sturzebecher et al. [D. Sturzebecher et al., Exp. Fluids 31, 294 (2001)]. Therein, the sensor was shown to have a cutoff frequency that exceeds 30 kHz, whereas the highest frequencies of the flow in our previous work did not exceed 10 kHz.

Aerodynamic noise of high-speed train pantographs: Comparisons between field measurements and an updated component-based prediction model

Aerodynamic noise from pantographs becomes an important source of noise from trains at high speeds. Previous studies have mostly been based on numerical predictions using computational aeroacoustic methods, which require large computing resources, or measurements conducted in a wind tunnel which cannot take all the real conditions into account. A component-based model relying on empirical constants obtained from the literature has been shown to predict aerodynamic noise from pantographs that agrees well with wind tunnel measurements. This model is extended in this paper by making use of simulation results on individual cylinders to refine the model constants and the Reynolds number dependence. In addition, allowance for the effect of incoming turbulence and cylinder aspect ratio is also extended. The updated model shows improved agreement with wind tunnel measurements, particularly at low frequencies. This model is then used to predict pantograph noise in more realistic conditions during train pass-by. The incoming flow conditions in terms of the incident flow speed, the turbulence intensity and the turbulence length scale are estimated from the literature considering the development of the boundary layer along the train roof. The sensitivity of the model to these assumptions is assessed using Monte Carlo simulations. The predicted results are compared with field measurements obtained using microphone array techniques for pantograph on different operational trains. Good agreement is obtained between the predictions and the measurements in terms of the far-field noise spectra and the dependence of noise level on speed. Differences are noted between measured levels for different orientations of the pantograph which according to the model are mainly related to the distance of the pantograph from the front of the train.

Reynolds number dependence of turbulence induced by the Richtmyer–Meshkov instability using direct numerical simulations

Abstract

Reynolds number scaling of burning rates in spherical turbulent premixed flames

Abstract

Experimental investigation of the angle of attack dependence of the flow past a cactus-shaped cylinder with four ribs

The aerodynamic properties of a two-dimensional cactus-shaped cylinder with four ribs are studied experimentally. The cross section of the cylinder corresponds to a modified square cylinder with rounded corners and concave sides inspired by tall succulents with four ribs. The mean aerodynamic coefficients, fluctuating lift coefficient, Strouhal number, and mean surface pressure distribution are measured as a function of the angular orientation for Reynolds numbers ranging from 50,000 to 150,000. Hot-wire measurements are conducted at two locations in the wake for several angles of attack to provide insight into the vortex shedding frequencies. The results show that the studied shape exhibits a strong angle of attack dependence while no Reynolds number dependence is found within the tested range. As for the square cylinder, a critical angle of attack of approximately 13° is observed at which drag and fluctuating lift forces are minimised while mean lift and Strouhal number reach their highest values. Reattachment of the shear layer at and above the critical angle of attack is confirmed using PIV. Overall, succulents with four ribs retain some of the aerodynamic benefits that have been observed for cactus-shaped cylinders with many ribs, albeit over a limited range of angles of attack.

Transitional and turbulent flows in rectangular ducts: budgets and projection in principal mean strain axes

We carry out Direct Numerical Simulation (DNS) of flows in closed rectangular ducts with several aspect ratios. The Navier–Stokes equations are discretised through a second-order finite difference scheme, with non-uniform grids in two directions. The duct cross-sectional area is maintained constant as well as the flow rate, which allows to investigate which is the appropriate length scale in the Reynolds number for a good scaling in the laminar and in the fully turbulent regimes. We find that the Reynolds number based on the half length of the short side leads to a critical Reynolds number which is independent on the aspect ratio (), for ducts with . The mean and rms wall-normal velocity profiles are found to scale with the local value of the friction velocity. At high friction Reynolds numbers, the Reynolds number dependence is similar to that in turbulent plane channels, hence flows in rectangular ducts allow to investigate the Reynolds number dependency through a reduced number of simulations. At low Re, the profiles of the statistics differ from those in the two-dimensional channel due to the interaction of flow structures of different size. The projection of the velocity vector and of the Reynolds stress tensor along the eigenvectors of the strain-rate tensor yields reduced Reynolds stress anisotropy and simple turbulence kinetic energy budgets. We further show that the isotropic rate of dissipation is more difficult to model than the full dissipation rate, whose distribution does not largely differ from that of turbulence kinetic energy production. We expect that this information may be exploited for the development of advanced RANS models for complex flows.

Aerodynamics of a two-dimensional bluff body with the cross-section of a train

Experiments on the aerodynamics of a two-dimensional bluff body simplified from a China high-speed train in crosswinds were carried out in a wind tunnel. Effects of wind angle of attack α varying in [−20°, 20°] were investigated at a moderate Reynolds number Re = 9.35 × 104 (based on the height of the model). Four typical behaviors of aerodynamics were identified. These behaviors are attributed to the flow structure around the upper and lower halves of the model changing from full to intermittent reattachment, and to full separation with a variation in α. An alternate transition phenomenon, characterized by an alteration between large- and small-amplitude aerodynamic fluctuations, was detected. The frequency of this alteration is about 1/10 of the predominant vortex shedding. In the intervals of the large-amplitude behavior, aerodynamic forces fluctuate periodically with a strong span-wise coherence, which are caused by the anti-symmetric vortex shedding along the stream-wise direction. On the contrary, the aerodynamic forces fluctuating at small amplitudes correspond to a weak span-wise coherence, which are ascribed to the symmetric vortex shedding from the upper and lower halves of the model. Generally, the mean amplitude of the large-amplitude mode is 3 times larger than that of the small one. Finally, the effects of Reynolds number were examined within Re = [9.35 × 104, 2.49 × 105]. Strong Reynolds number dependence was observed on the model with two rounded upper corners.

Aerodynamic Effects of Uniform Blowing and Suction on a NACA4412 Airfoil

We carried out high-fidelity large-eddy simulations to investigate the effects of uniform blowing and uniform suction on the aerodynamic efficiency of a NACA4412 airfoil at the moderate Reynolds number based on chord length and incoming velocity of Re_c=200{,}000. We found that uniform blowing applied at the suction side reduces the aerodynamics efficiency, while uniform suction increases it. This result is due to the combined impact of blowing and suction on skin friction, pressure drag and lift. When applied to the pressure side, uniform blowing improves aerodynamic efficiency. The Reynolds-number dependence of the relative contributions of pressure and friction to the total drag for the reference case is analysed via Reynolds-averaged Navier–Stokes simulations up to Re_c=10{,}000{,}000. The results suggest that our conclusions on the control effect can tentatively be extended to a broader range of Reynolds numbers.

Scaling mean velocity in two-dimensional turbulent wall jets

Studies in the literature on plane turbulent wall jets on flat surfaces, have invariably considered either the nozzle initial conditions or the asymptotic conditions far downstream, as scaling parameters for the streamwise variations of length and velocity scales. These choices, however, do not square with the notion of self similarity which is essentially a "local" concept. We first demonstrate that the streamwise variations of velocity and length scales in wall jets show remarkable scaling with local parameters i.e. there appear to be no imposed length and velocity scales. Next, it is shown that the mean velocity profile data suggest existence of two distinct layers - the wall (inner) layer and the full-free jet (outer) layer. Each of these layers scales on the appropriate length and velocity scales and this scaling is observed to be universal i.e. independent of the local friction Reynolds number. Analysis shows that the overlap of these universal scalings leads to a Reynolds-number-dependent power-law velocity variation in the overlap layer. It is observed that the mean-velocity overlap layer corresponds well to the momentum-balance mesolayer and there appears to be no evidence for an inertial overlap; only the meso-overlap is observed. Introduction of an intermediate variable absorbs the Reynolds-number dependence of the length scale in the overlap layer and this leads to a universal power-law overlap profile for mean velocity in terms of the intermediate variable.

Large and small scales in a turbulent orifice round jet: Reynolds number effects and departures from isotropy

An experimental investigation of the near field of a turbulent orifice jet is performed using high resolution Particle Image Velocimetry, aiming to highlight effects on the flow field due to changes in Reynolds number. The attention is focused onto departures from isotropy for large and small scales, by considering statistics of mean square velocity and velocity derivatives and specifically the non-dimensional ratios of such quantities. The results compare well with available literature data and pointed out that the effects of Reynolds number on large scales are usually small and limited to a region ranging less than seven-ten diameters from the jet outlet. For small scales, such Reynolds number dependence is extended up to ten-fifteen diameters. Farther from the jet exit, Reynolds number dependence almost disappears and all data approach similar asymptotic behaviors. On the other hand, velocity and some velocity derivative statistics clearly show that neither large nor small scale statistics strictly follow the isotropy condition; nonetheless, differences from that condition are limited to a factor which is almost constant in the whole measured field. In order to provide a link between such large and small scale departures from isotropy, a relation among mean square velocity ratios and mean square derivative ratios is proposed and proved to be well verified in the measured region and interval of Reynolds numbers. This relation allows deriving small scale derivative ratios, which are difficult to measure experimentally or to obtain numerically, due to high resolution requirements, from large scale velocity ratios, which are achieved much easier.

The Role of Reynolds Number Effect and Tip Leakage in Compressor Geometry Scaling at Low Turbulent Reynolds Numbers

Abstract In compressor design, a convenient way to save time is to scale an existing geometry to required specifications, rather than developing a new design. The approach works well when scaling compressors of similar size at high Reynolds numbers but becomes more complex when applied to small-scale machines. Besides the well-understood increase in surface friction due to increased relative surface roughness, two other main problems specific to small-scale turbomachinery can be specified: (1) the Reynolds number effect, describing the non-linear dependency of surface friction on Reynolds number and (2) increased relative tip clearance resulting from manufacturing limitations. This paper investigates the role of both effects in a geometric scaling process, as used by a designer. The work is based on numerical models derived from an experimentally validated geometry. First, the effects of geometric scaling on compressor performance are assessed analytically. Second, prediction capabilities of reduced-order models from the public domain are assessed. In addition to design point assessment, often found in other publications, the models are tested at off-design. Third, the impact of tip leakage on compressor performance and its Reynolds number dependency is assessed. Here, geometries of different scale and with different tip clearances are investigated numerically. Fourth, a detailed investigation regarding tip leakage driving mechanisms is carried out and design recommendations to improve small-scale compressor performance are provided.

Reynolds Number Dependence of Zero Pressure Gradient Turbulent Boundary Layers Including Third-Order Moments and Spatial Correlations

Abstract Measurements were made in zero pressure gradient turbulent boundary layers on a smooth wall, at momentum thickness Reynolds numbers, ranging from 800 to 6340. The experiments were conducted in a recirculating water tunnel. Two-component velocity profiles were acquired using laser Doppler velocimetry at five streamwise stations and three different freestream velocities. Velocity field measurements were acquired using particle image velocimetry in streamwise-wall normal and streamwise–spanwise planes. Profiles of mean velocity and turbulence statistics including the Reynolds normal and shear stresses, and triple products of the velocity fluctuations are presented in both inner and outer coordinates. Variations in the profiles at representative distances from the wall are presented and quantified as functions of Reynolds number. The triple products are explained in terms of transport of Reynolds stresses though motions associated with quadrant analysis, and variation with Reynolds number is consistent with that of Reynolds stresses. The structure of turbulence was considered through two-point correlations of the fluctuations in velocity fields. In general, the shape and inclination angles of the structures did not change with Reynolds number, but some streamwise and spanwise growth was observed as Reynolds number increased.

Turbulent flows in square ducts: physical insight and suggestions for turbulence modellers

ABSTRACTWe carry out Direct Numerical Simulation (DNS) of flows in a straight duct with square cross-section. At high-friction Reynolds numbers, the Reynolds number dependence is similar to turbulent plane channels, hence flows in a square duct allows to investigate the Reynolds number dependency through a reduced number of simulations. Secondary motion play a minor role in this regime. At low Re, the profiles of the statistics differ from those in the two-dimensional channel due to the interaction of flow structures with different size. Large flow scales due to the mean motion are responsible for creating turbulence anisotropy in wall-bounded flows, causing serious difficulties for RANS turbulence closures. Projection of velocity vectors and of the Reynolds stress tensor along the eigenvectors of the mean strain-rate tensor yields reduced Reynolds stress anisotropy, and simple turbulence kinetic energy budgets. We show that the isotropic rate of dissipation is more difficult to model than the full dissipation rate, whose distribution does not largely differ from that of turbulence kinetic energy production. We expect that this information may be exploited for the development of advanced RANS models for complex flows.

Reynolds number dependence of turbulent cascade in channel flow

Reynolds number dependence of turbulent cascade in channel flow

Study on the Reynolds Number Dependence of Contraction Coefficient of the Flow through Sluice Gates and the Discharge Coefficient Evaluation by the Analysis of Experimental Data

Study on the Reynolds Number Dependence of Contraction Coefficient of the Flow through Sluice Gates and the Discharge Coefficient Evaluation by the Analysis of Experimental Data

Reynolds number dependence of high Schmidt number scalar mixing in liquid axisymmetric jets

Reynolds number dependence of high Schmidt number scalar mixing in liquid axisymmetric jets

Reynolds Number Dependence of the Structure Functions in Homogeneous Turbulence

We compare the predictions of stochastic closure theory (SCT) (Birnir in J Nonlinear Sci 23:657–688, 2013a. https://doi.org/10.1007/s00332-012-9164-z) with experimental measurements of homogeneous turbulence made in the variable density turbulence tunnel (Bodenschatz et al. in Rev Sci Instrum 85(9):093908, 2014) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen. While the general form of SCT contains infinitely many free parameters, the data permit us to reduce the number to seven, only three of which are active over the entire range of Taylor–Reynolds numbers. Of these three, one parameter characterizes the variance of the mean-field noise in SCT and another characterizes the rate in the large deviations of the mean. The third parameter is the decay exponent of the Fourier variables in the Fourier expansion of the noise, which characterizes the smoothness of the turbulent velocity. SCT compares favorably with velocity structure functions measured in the experiment. We considered even-order structure functions ranging in order from two to eight as well as the third-order structure functions at five Taylor–Reynolds numbers (\(R_\lambda \)) between 110 and 1450. The comparisons highlight several advantages of the SCT, which include explicit predictions for the structure functions at any scale and for any Reynolds number. We observed that finite-\(R_\lambda \) corrections, for instance, are important even at the highest Reynolds numbers produced in the experiments. SCT gives us the correct basis function to express all the moments of the velocity differences in turbulence in Fourier space. These turn out to be powers of the sine function indexed by the wavenumbers. Here, the power of the sine function is the same as the order of the moment of the velocity differences (structure functions). The SCT produces the coefficients of the series and so determines the statistical quantities that characterize the small scales in turbulence. It also characterizes the random force acting on the fluid in the stochastic Navier–Stokes equation, as described in this paper.

Neural network models for the anisotropic Reynolds stress tensor in turbulent channel flow

Reynolds-averaged Navier-Stokes (RANS) equations are presently one of the most popular models for simulating turbulence. Performing RANS simulation requires additional modelling for the anisotropic Reynolds stress tensor, but traditional Reynolds stress closure models lead to only partially reliable predictions. Recently, data-driven turbulence models for the Reynolds anisotropy tensor involving novel machine learning techniques have garnered considerable attention and have been rapidly developed. Focusing on modelling the Reynolds stress closure for the specific case of turbulent channel flow, this paper proposes three modifications to a standard neural network to account for the no-slip boundary condition of the anisotropy tensor, the Reynolds number dependence, and spatial non-locality. The modified models are shown to provide increased predicative accuracy compared to the standard neural network when they are trained and tested on channel flow at different Reynolds numbers. The best performance is yielded by the model combining the boundary condition enforcement and Reynolds number injection. This model also outperforms the Tensor Basis Neural Network in Ling et al. [Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J Fluid Mech. 2016;807:155–166] on the turbulent channel flow dataset.

Morphological classification of disintegration behavior of viscoelastic simulant gel propellant in coaxial streams

In this study, the spray structure of simulant gel propellant was analyzed morphologically and qualitatively to understand the basic characteristics of the gelled jet disintegration process in a shear coaxial environment. Categorization of breakup process was conducted based on the water jet breakup theory with generalized Reynolds number of the Herschel–Bulkley model and aerodynamic Weber number. Similar breakup types and disintegration process comparable to Newtonian fluids were depicted. However, the disintegration development was relatively slow compared to Newtonian cases, and the spray structures were quite different especially in fiber-type breakup due to its inherent viscosity change and viscoelastic properties. In the case of gel propellant, a strong Reynolds number dependency was observed under a certain critical value of the generalized Reynolds number, whereas beyond this value, breakup regime can be categorized with only the Weber number as in Newtonian cases. The border of Reynolds number dependency was shown at approximately $$\text{Re}_{\text{gen}} = 200$$ for all simulant gel contents cases considered in this study. The Weber number ranges for the three breakup modes (Rayleigh type, membrane type and fiber type) were found, and a new breakup mode showing a more intricate gel disintegration than the Newtonian one is proposed. .