Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System

The adverse pressure gradient induced by a surface-mounted obstacle in a turbulent boundary layer causes the approaching flow to separate and form a dynamically rich horseshoe vortex system (HSV) in the junction of the obstacle with the wall. The Reynolds number of the flow (Re) is one of the important parameters that control the rich coherent dynamics of the vortex, which are known to give rise to low-frequency, bimodal fluctuations of the velocity field (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). We carry out detached eddy simulations (DES) of the flow past a circular cylinder mounted on a rectangular channel for Re = 2.0 × 10⁴ and 3.9 × 10⁴ (Dargahi, Exp Fluids 8:1–12, 1989) in order to systematically investigate the effect of the Reynolds number on the HSV dynamics. The computed results are compared with each other and with previous experimental and computational results for a related junction flow at a much higher Reynolds number (Re = 1.15 × 10⁵) (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). The computed results reveal significant variations with Re in terms of the mean-flow quantities, turbulence statistics, and the coherent dynamics of the turbulent HSV. For Re = 2.0 × 10⁴ the HSV system consists of a large number of necklace-type vortices that are shed periodically at higher frequencies than those observed in the Re = 3.9 × 10⁴ case. For this latter case the number of large-scale vortical structures that comprise the instantaneous HSV system is reduced significantly and the flow dynamics becomes quasi-periodic. For both cases, we show that the instantaneous flowfields are dominated by eruptions of wall-generated vorticity associated with the growth of hairpin vortices that wrap around and disorganize the primary HSV system. The intensity and frequency of these eruptions, however, appears to diminish rapidly with decreasing Re. In the high Re case the HSV system consists of a single, highly energetic, large-scale necklace vortex that is aperiodically disorganized by the growth of the hairpin mode. Regardless of the Re, we find pockets in the junction region within which the histograms of velocity fluctuations are bimodal as has also been observed in several previous experimental studies.     

Very high Reynolds number boundary layers over 3D sparse roughness and obstacles: the mean flow

Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers (61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H ⁺ < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness, the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the turbulent component increases as the roughness effects extend across the boundary layer.     

Measurements and scaling of wall shear stress fluctuations

Measurements of velocity and wall shear stress fluctuations were made in an external turbulent boundary layer developed over a towed surface-piercing flat plate. An array of eight flush-mounted wall shear stress sensors was used to compute the space-time correlation function. A methodology for in situ calibration of the sensors for ship hydrodynamic applications is presented. The intensity of the wall shear stress fluctuations, τ _rms/τ _avg was measured as 0.25 and 0.36 for R _θ =3,150 and 2,160 respectively. The probability density is shown to exhibit positive skewness, and lack of flow reversals at the wall. Correlations between velocity and wall shear stress fluctuations are shown to collapse with outer boundary layer length and velocity scales, verifying the existence of large-scale coherent structures which convect and decay along the wall at an angle of inclination varying from 10 to 13° over the range of Reynolds numbers investigated. The wall shear stress convection velocity determined from narrow band correlation measurements is shown to scale with outer variables. The space-time correlation of the wall shear is shown to exhibit a well-defined convective ridge, and to decay 80% over approximately 3δ for R _θ =3,150. Published online: 7 November 2002     

Influence of unsteady wake on a turbulent separation bubble

 An experimental study was made of turbulent separated and reattaching flow over a blunt body, where unsteady wake was generated by a spoked-wheel type of wake generator with cylindrical rods. The influence of unsteady wake was scrutinized by altering the rotation direction (clockwise and counter-clockwise) and the normalized passing frequency (0 ≤ St _H  ≤ 0.20). The Reynolds number based on the cylindrical rod was Re _d =375. A phase-averaging technique was employed to characterize the unsteady wake. The effect of different rotation directions, which gave a significant reduction of x _R , was examined in detail. The wall pressure fluctuations on the blunt body were analyzed in terms of the spectrum and the coherence. Received: 15 January 2001 / Accepted: 17 July 2001     

Statistical properties of wall shear stress fluctuations in turbulent channel flows

Instantaneous velocity and wall shear stress measurements are conducted in a turbulent channel flow in the Kármán number range of Re_τ = 74–400. A one-dimensional LDA system is used to measure the streamwise velocity fluctuations, and an electrochemical technique is utilized to measure the instantaneous wall shear stress. For the latter, frequency response and nonuniform correction methods are used to provide an accurate, well-resolved wall statistics database. The Reynolds number dependency of the statistical wall quantities is carefully investigated. The corrected relative wall shear stress fluctuations fit well with the best DNS data available and meet the need for clarification of the small discrepancy observed in the literature between the experimental and numerical results of such quantities. Higher-order statistics of the wall shear stress, spectra, and the turbulence kinetic energy budget at the wall are also investigated. The present paper shows that the electrochemical technique is a powerful experimental method for hydrodynamic studies involving highly unsteady flows. The study brings with it important consequences, especially in the context of the current debate regarding the appropriate scaling as well as the validation of new predictive models of near-wall turbulence.     

Comparison of turbulent channel and pipe flows with varying Reynolds number

Single normal hot-wire measurements of the streamwise component of velocity were taken in fully developed turbulent channel and pipe flows for matched friction Reynolds numbers ranging from 1,000 ≤ Re _τ ≤ 3,000. A total of 27 velocity profile measurements were taken with a systematic variation in the inner-scaled hot-wire sensor length l ⁺ and the hot-wire length-to-diameter ratio (l/d). It was observed that for constant l ⁺ = 22 and l/d >~200l/d \gtrsim 200, the near-wall peak in turbulence intensity rises with Reynolds number in both channels and pipes. This is in contrast to Hultmark et al. in J Fluid Mech 649:103–113, (2010), who report no growth in the near-wall peak turbulence intensity for pipe flow with l ⁺ = 20. Further, it was found that channel and pipe flows have very similar streamwise velocity statistics and energy spectra over this range of Reynolds numbers, with the only difference observed in the outer region of the mean velocity profile. Measurements where l ⁺ and l/d were systematically varied reveal that l ⁺ effects are akin to spatial filtering and that increasing sensor size will lead to attenuation of an increasingly large range of small scales. In contrast, when l/d was insufficient, the measured energy is attenuated over a very broad range of scales. These findings are in agreement with similar studies in boundary layer flows and highlight the need to carefully consider sensor and anemometry parameters when comparing flows across different geometries and when drawing conclusions regarding the Reynolds number dependency of measured turbulence statistics. With an emphasis on accuracy, measurement resolution and wall proximity, these measurements are taken at comparable Reynolds numbers to currently available DNS data sets of turbulent channel/pipe flows and are intended to serve as a database for comparison between physical and numerical experiments.     

Synchronized analysis of an unsteady laminar flow downstream of a circular cylinder centred between two parallel walls using PIV and mass transfer probes

Experiments are carried out in the wake of a cylinder of d _c  = 10 mm diameter placed symmetrically between two parallel walls with a blockage ratio r = 1/3 and a Reynolds number varying between 75 ≤ Re ≤ 277. Particle image velocimetry is exerted to obtain the instantaneous velocity components in the cylinder wake. A snapshot proper orthogonal decomposition (POD) is also applied to these PIV results in order to extract the dominant modes through the implementation of an inhomogeneous filtering of these different snapshots, apart from an interpolation to estimate the wall shear rate at the lower wall downstream the cylinder. Mass transfer circular probes are placed at the lower wall downstream this obstacle so as to further determine the time evolution of the wall shear rate, by bringing the inverse method to bear on the convective-diffusion equation. Comparisons between the two synchronized techniques demonstrate that electrochemical method can give more accurate information about the coherent structures present in the flow and about the interaction of the von Kármán vortices with the walls of the tunnel as well. The comparison between the two measurement techniques in the flow regions concerns the spatiotemporal evolutions of the wall shear rate obtained from PIV measurements and the wall shear rate using mass transfer probes. Discrepancy between the PIV measurements and the electrochemical ones near the wall, where the secondary vortices P ₁′ are generated at wall, are caused by a PIV bias and a limitations of the singular mass transfer probes.     

Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity

The swirling flow between a rotating lid and a stationary cylinder is studied experimentally. The flow is governed by two parameters: the ratio of container height to disk radius, h, and the Reynolds number, Re, based on the disk angular velocity, cylinder radius and kinematic viscosity of the working liquid. For the first time, the onset of three-dimensional flow behavior is measured by combining the high spatial resolution of particle image velocimetry and the temporal accuracy of laser Doppler anemometry. A detailed mapping of the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow is investigated for 1 ≥ h ≥ 3.5. The flow is characterized by the development of azimuthal modes of different wave numbers. A range of different modes is detected and critical Reynolds numbers and associated frequencies are identified. The results are compared to the numerical stability analysis of Gelfgat et al. (J Fluid Mech 438:363–377, 2001). In most cases, the measured onset of three-dimensionality is in good agreement with the numerical results and disagreements can be explained by bifurcations not accounted for by the numerical stability analysis.     

Experiment on smooth,circular cylinders in cross-flow in the critical Reynolds number regime

Experiments were conducted for 2D circular cylinders at Reynolds numbers in the range of 1.73 × 10⁵–5.86 × 10⁵. In the experiment, two circular cylinder models made of acrylic and stainless steel, respectively, were employed, which have similar dimensions but different surface roughness. Particular attention was paid to the unsteady flow behaviors inferred by the signals obtained from the pressure taps on the cylinder models and by a hot-wire probe in the near-wake region. At Reynolds numbers pertaining to the initial transition from the subcritical to the critical regimes, pronounced pressure fluctuations were measured on the surfaces of both cylinder models, which were attributed to the excursion of unsteady flow separation over a large circumferential region. At the Reynolds numbers almost reaching the one-bubble state, it was noted that the development of separation bubble might switch from one side to the other with time. Wavelet analysis of the pressure signals measured simultaneously at θ = ±90° further revealed that when no separation bubble was developed, the instantaneous vortex-shedding frequencies could be clearly resolved, about 0.2, in terms of the Strouhal number. The results of oil-film flow visualization on the stainless steel cylinder of the one-bubble and two-bubble states showed that the flow reattachment region downstream of a separation bubble appeared not uniform along the span of the model. Thus, the three dimensionality was quite evident.     

Wall pressure fluctuations and topology in separated flows over a forward-facing step

Laboratory measurements of wall pressure fluctuations and aerodynamic fields were made in separated flows over a forward facing step (h = 30, 40 and 50 mm with U _e = 15–40 m/s). An array of 16 off-set pressure probes extending in the streamwise and the spanwise directions was especially developed for sensing the wall pressure fluctuations. The flow field was also investigated by wall flow visualizations and PIV to analyze the flow topology in an open section wind tunnel. The results show a different behavior of the flow depending on the aspect ratio l/h and δ/h for high Reynolds numbers. The space time correlations between the wall pressure and the velocity fields were highlighted. The results show that high levels of these correlations are located at the top of the recirculation bubble, mainly in the shear layer and are extended downstream of the re-attachment point. Indeed, the results indicate that the flapping motion at the separation is important in the flow organization at the re-attachment point.     

Degradation of homogeneous polymer solutions in high shear turbulent pipe flow

This study quantifies degradation of polyethylene oxide (PEO) and polyacrylamide (PAM) polymer solutions in large diameter (2.72 cm) turbulent pipe flow at Reynolds numbers to 3 × 10⁵ and shear rates greater than 10⁵ 1/s. The present results support a universal scaling law for polymer chain scission reported by Vanapalli et al. (2006) that predicts the maximum chain drag force to be proportional to Re ^3/2, validating this scaling law at higher Reynolds numbers than prior studies. Use of this scaling gives estimated backbone bond strengths from PEO and PAM of 3.2 and 3.8 nN, respectively. Additionally, with the use of synthetic seawater as a solvent the onset of drag reduction occurred at higher shear rates relative to the pure water solvent solutions, but had little influence on the extent of degradation at higher shear rates. These results are significant for large diameter pipe flow applications that use polymers to reduce drag.     

Turbulence Modelling of Unsteady Turbulent Flows Using the Stress Strain Lag Model

This paper reports the application of a recently developed turbulence modelling scheme known as the C _as model. This model was specifically developed to capture the effects of stress-strain misalignment observed in turbulent flows with mean unsteadiness. Earlier work has reported the approach applied within a linear k-ε modelling framework, and some initial testing of it within the k-ω SST model of Menter (AIAA J 32:1598–1605, 1994). The resulting k-ε-C _as and SST-C _as models have been shown to result in some of the advantages of a full Reynolds Stress transport Model (RSM), whilst retaining the computational efficiency and stability benefits of a eddy viscosity model (EVM). Here, the development of the the high-Reynolds-number version of the C _as model is outlined, with some example applications to steady and unsteady homogeneous shear flows. The SST-C _as form of the model is then applied to further, more challenging cases of 2-D flow around a NACA0012 aerofoil beyond stall and the 3-D flow around a circular cylinder in a square duct, both being flows which exhibit large, unsteady, separated flow regions. The predictions returned by a range of other common turbulence modelling schemes are included for comparison and the SST-C _as scheme is shown to return generally good results, comparable in some respects to those obtainable from far more complex schemes, for only moderate computing resource requirements.     

Numerical simulation of fluid flow and heat transfer characteristics in channel with V corrugated plates

A detailed numerical study is carried out to investigate fluid flow and heat transfer characteristics in a channel with heated V corrugated upper and lower plates. The parameters studied include the Reynolds number (Re = 2,000–5,500), angles of V corrugated plates (θ = 20°, 40°, 60°), and constant heat fluxs (q″ = 580, 830, 1,090 W/m²). Numerical results have been validated using the experimented data reported by Naphon, and a good agreement has been found. The angles of V corrugated plates (θ) and the Reynolds number are demonstrated to significantly affect the fluid flow and the heat transfer rate. Increasing the angles of V corrugated plates can make the heat transfer performance become better. The increasing Reynolds number leads to a more complex fluid flow and heat transfer rate. The numerical calculations with a non-equilibrium wall function have a better accuracy than with a standard wall function for solving high Reynolds numbers or complex flow problems.     

Characterizing developing adverse pressure gradient flows subject to surface roughness

An experimental study was conducted to examine the effects of surface roughness and adverse pressure gradient (APG) on the development of a turbulent boundary layer. Hot-wire anemometry measurements were carried out using single and X-wire probes in all regions of a developing APG flow in an open return wind tunnel test section. The same experimental conditions (i.e., T _∞, U _ref, and C _p) were maintained for smooth, k ⁺ = 0, and rough, k ⁺ = 41–60, surfaces with Reynolds number based on momentum thickness, 3,000 < Re _θ < 40,000. The experiment was carefully designed such that the x-dependence in the flow field was known. Despite this fact, only a very small region of the boundary layer showed a balance of the various terms in the integrated boundary layer equation. The skin friction computed from this technique showed up to a 58% increase due to the surface roughness. Various equilibrium parameters were studied and the effect of roughness was investigated. The generated flow was not in equilibrium according to the Clauser (J Aero Sci 21:91–108, 1954) definition due to its developing nature. After a development region, the flow reached the equilibrium condition as defined by Castillo and George (2001), where Λ = const, is the pressure gradient parameter. Moreover, it was found that this equilibrium condition can be used to classify developing APG flows. Furthermore, the Zagarola and Smits (J Fluid Mech 373:33–79, 1998a) scaling of the mean velocity deficit, U _∞δ*/δ, can also be used as a criteria to classify developing APG flows which supports the equilibrium condition of Castillo and George (2001). With this information a ‘full APG region’ was defined.     

Characterization of a system generating a homogeneous isotropic turbulence field by free synthetic jets

A facility inspired by Hwang and Eaton (2004a, b) for generating a homogeneous isotropic turbulence was built, the objective being to study evaporating droplets in the presence of turbulence. Turbulence was produced by the mixing of six synthetic jets, in ambient atmosphere. Combined PIV and LDA techniques were used to measure the statistical turbulence properties. The turbulence produced was found to be homogeneous isotropic with a small mean flow within a domain having an average size of 50 mm × 50 mm × 50 mm. The rms fluctuations were of the order of 0.9 m/s, corresponding to a Taylor Reynolds number of 240 and an integral length scale of about 40 mm. This apparatus proved to be well suited to the study of the evaporation of droplets in a controlled turbulence field.     

Local measurements of turbulent angular momentum transport in circular Couette flow

We report on velocity fluctuations and the fluctuation-driven radial transport of angular momentum in turbulent circular Couette flow. Our apparatus is short (cylinder height to gap width ratio Γ ~ 2) and of relatively high wall curvature (ratio of cylinder radii η ~ 0.35). Fluctuation levels and the mean specific angular momentum are found to be roughly constant over radius, in accordance with previous studies featuring narrower gaps. Synchronized dual beam Laser Doppler Velocimetry (2D LDV) is used to directly measure the r − θ Reynolds stress component as a function of Reynolds number (Re), revealing approximate scalings in the non-dimensional angular momentum transport that confirm previous measurements of torque in similar flows. 2D LDV further allows for a decomposition of the turbulent transport to assess the relative roles of fluctuation intensity and r − θ cross-correlation. We find that the increasing angular momentum transport with Re is due to intensifying absolute fluctuation levels accompanied by a slightly weakening cross-correlation.     

Time resolved flow-field measurements of a turbulent mixing layer over a rectangular cavity

High Reynolds number, low Mach number, turbulent shear flow past a rectangular, shallow cavity has been experimentally investigated with the use of dual-camera cinematographic particle image velocimetry (CPIV). The CPIV had a 3 kHz sampling rate, which was sufficient to monitor the time evolution of large-scale vortices as they formed, evolved downstream and impinged on the downstream cavity wall. The time-averaged flow properties (velocity and vorticity fields, streamwise velocity profiles and momentum and vorticity thickness) were in agreement with previous cavity flow studies under similar operating conditions. The time-resolved results show that the separated shear layer quickly rolled-up and formed eddies immediately downstream of the separation point. The vortices convect downstream at approximately half the free-stream speed. Vorticity strength intermittency as the structures approach the downstream edge suggests an increase in the three-dimensionality of the flow. Time-resolved correlations reveal that the in-plane coherence of the vortices decays within 2–3 structure diameters, and quasi-periodic flow features are present with a vortex passage frequency of ~1 kHz. The power spectra of the vertical velocity fluctuations within the shear layer revealed a peak at a non-dimensional frequency corresponding to that predicted using linear, inviscid instability theory.     

Measurements of mean and fluctuating wall shear stress beneath spanwise-invariant separation bubbles

 Pulsed-wire measurements of wall shear stress have been made beneath two separation bubbles. In one a cross flow was generated by means of a (25°) swept separation line. Fluctuating stresses in orthogonal “streamwise” and cross-flow directions are very nearly equal and independent of at least moderate cross flow velocity. These fluctuations are largely determined by large-scale motions in the outer flow, whereas the mean shear stresses are not. The pdf of the “streamwise” fluctuations is unchanged by the cross flow. When a cross flow is present the pdf of the cross-flow stresses is similar to the “streamwise” pdf. Dependence on Reynolds number is the same in both flows. Received: 10 April 1998/Accepted: 17 July 1998     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.