Reynolds number effect on the flow statistics and turbulent–non-turbulent interface of a planar jet

Effect of Reynolds number on the stability of a single flexible tube predicted by the quasi-steady model in tube bundles

This paper exploits the turbulent flow control method using streamwise travelling waves (Quadrio et al. J. Fluid Mech., vol. 627, 2009, pp. 161–178) to study the effect of Reynolds number on turbulent skin-friction drag reduction. Direct numerical simulations (DNS) of a turbulent channel flow subjected to the streamwise travelling waves of spanwise wall velocity have been performed at Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=200$ to 1600. To the best of the authors’ knowledge, this is the highest Reynolds number attempted with DNS for this type of flow control. The present DNS results confirm that the effectiveness of drag reduction deteriorates, and the maximum drag reduction achieved by travelling waves decreases significantly as the Reynolds number increases. The intensity of both the drag reduction and drag increase is reduced with the Reynolds number. Another important finding is that the value of the optimal control parameters changes, even in wall units, when the Reynolds number is increased. This trend is observed for the wall oscillation, stationary wave, and streamwise travelling wave cases. This implies that, when the control parameters used are close to optimal values found at a lower Reynolds number, the drag reduction deteriorates rapidly with increased Reynolds number. In this study, the effect of Reynolds number for the travelling wave is quantified using a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$. No universal constant is found for the scaling parameter ${\it\alpha}$. Instead, the scaling parameter ${\it\alpha}$ has a wide range of values depending on the flow control conditions. Further Reynolds number scaling issues are discussed. Turbulent statistics are analysed to explain a weaker drag reduction observed at high Reynolds numbers. The changes in the Stokes layer and also the mean and root-mean-squared (r.m.s.) velocity with the Reynolds number are also reported. The Reynolds shear stress analysis suggests an interesting possibility of a finite drag reduction at very high Reynolds numbers.

The temporally developing self-similar turbulent jet is fundamentally different from its spatially developing namesake because the former conserves volume flux and has zero cross-stream mean flow velocity whereas the latter conserves momentum flux and does not have zero cross-stream mean flow velocity. It follows that, irrespective of the turbulent dissipation's power-law scalings, the time-local Reynolds number remains constant, and the jet half-width $\delta$ , the Kolmogorov length $\eta$ and the Taylor length $\lambda$ grow identically as the square root of time during the temporally developing self-similar planar jet's evolution. We predict theoretically and confirm numerically by direct numerical simulations that the mean centreline velocity, the Kolmogorov velocity and the mean propagation speed of the turbulent/non-turbulent interface (TNTI) of this planar jet decay identically as the inverse square root of time. The TNTI has an inner structure over a wide range of closely spatially packed iso-enstrophy surfaces with fractal dimensions that are well defined over a range of scales between $\lambda$ and $\delta$ , and that decrease with decreasing iso-enstrophy towards values close to $2$ at the viscous superlayer. The smallest scale on these isosurfaces is approximately $\eta$ , and the length scales between $\eta$ and $\lambda$ contribute significantly to the surface area of the iso-enstrophy surfaces without being characterised by a well-defined fractal dimension. A simple model is sketched for the mean propagation speeds of the iso-enstrophy surfaces within the TNTI of temporally developing self-similar turbulent planar jets. This model is based on a generalised Corrsin length, on the multiscale geometrical properties of the TNTI, and on a proportionality between the turbulent jet volume's growth rate and the growth rate of $\delta$ . A prediction of this model is that the mean propagation speed at the outer edge of the viscous superlayer is proportional to the Kolmogorov velocity multiplied by the $1/4$ th power of the global Reynolds number.

Direct numerical simulation of incompressible turbulent boundary layers and planar jets at high Reynolds numbers initialized with implicit large eddy simulation

There has been a need for improved prediction methods for low pressure turbine (LPT) blades operating at low Reynolds numbers. This is known to occur when LPT blades are subjugated to high altitude operations causing a decrease in the inlet Reynolds number. Boundary layer separation is more likely to be present within the flowfield of the LPT stages due to increase in the region adverse pressure gradients on the blade suction surface. Accurate CFD predictions are needed in order to improve design methods and performance prediction of LPT stages operating at low Reynolds numbers. CFD models were created for the flow over two low pressure turbine blade designs using a new turbulent transitional flow model, originally developed by Walters and Leylek (2004, “A New Model for Boundary Layer Transition Using a Single Point RANS Approach,” ASME J. Turbomach., 126(1), pp. 193–202). Part I of this study applied Walters and Leylek’s model to a cascade CFD model of a LPT blade airfoil with a light loading level. Flows were simulated over a Reynolds number range of 15,000–100,000 and predicted the laminar-to-turbulent transitional flow behavior adequately. It showed significant improvement in performance prediction compared to conventional RANS turbulence models. Part II of this paper presents the application of the prediction methodology developed in Part I to both two-dimensional and three-dimensional cascade models of a largely separated LPT blade geometry with a high blade loading level. Comparisons were made with available experimental cascade results on the prediction of the inlet Reynolds number effect on surface static pressure distribution, suction surface boundary layer behavior, and the wake total pressure loss coefficient. The kT-kL-ω transitional flow model accuracy was judged sufficient for an understanding of the flow behavior within the flow passage, and can identify when and where a separation event occurs. This model will provide the performance prediction needed for modeling of low Reynolds number effects on more complex geometries.

Direct numerical simulations (DNSs) are used to examine a turbulent boundary layer with large magnitudes of adverse and favourable pressure gradients, thus involving separation and reattachment. The present DNS datasets have been obtained from the simulations of a pressure-induced turbulent separation bubble (Abe 2017 J. Fluid Mech. 833 563–98). Three values of the inlet Reynolds number ( 600 and 900) are used. These magnitudes increase abruptly with pressure gradients; the maximum values are attained near reattachment (i.e. 2070 and 2915). Also used are the datasets with varying pressure gradient for a fixed inlet Reynolds number (). Particular attention is given to effects of Reynolds number and pressure gradient on the momentum transport. In particular, we examine the Reynolds shear stress () with the use of the transport equation, quadrant analysis and instantaneous fields and discuss how this quantity is generated in the present flow and is associated with mean streamwise velocity (). It is shown that the magnitude of increases in the APG and separated regions, while it decreases near top of the bubble due to the convex curvature when the bubble is large. For all three Reynolds numbers, the peaks are obtained in the outer region in which exhibits an inflection point. The similarity to a free shear flow in is convincing in the separated shear layer where the effect of the inlet Reynolds number (i.e. the active motion in an incoming boundary layer) is still observed in The maximum is attained after reattachment due to the convective nature of the present flow and depends essentially on the pressure gradient. It is also shown that large-scale structures of streamwise velocity fluctuation become more pronounced with moving downstream and contribute significantly to the momentum transport, which yield a large departure from the log law of mean streamwise velocity.

Reynolds and Mach number effects in compressible turbulent channel flow

In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, on the particle statistics is examined over the largest range to date (from $R_{{\it\lambda}}=88$ to 597), at small, intermediate and large Kolmogorov-scale Stokes numbers $St$. We first explore the effect of preferential sampling on the single-particle statistics and find that low-$St$ inertial particles are ejected from both vortex tubes and vortex sheets (the latter becoming increasingly prevalent at higher Reynolds numbers) and preferentially accumulate in regions of irrotational dissipation. We use this understanding of preferential sampling to provide a physical explanation for many of the trends in the particle velocity gradients, kinetic energies and accelerations at low $St$, which are well represented by the model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251). As $St$ increases, inertial filtering effects become more important, causing the particle kinetic energies and accelerations to decrease. The effect of inertial filtering on the particle kinetic energies and accelerations diminishes with increasing Reynolds number and is well captured by the models of Abrahamson (Chem. Engng Sci., vol. 30, 1975, pp. 1371–1379) and Zaichik & Alipchenkov (Intl J. Multiphase Flow, vol. 34 (9), 2008, pp. 865–868), respectively. We then consider particle-pair statistics, and focus our attention on the relative velocities and radial distribution functions (RDFs) of the particles, with the aim of understanding the underlying physical mechanisms contributing to particle collisions. The relative velocity statistics indicate that preferential sampling effects are important for $St\lesssim 0.1$ and that path-history/non-local effects become increasingly important for $St\gtrsim 0.2$. While higher-order relative velocity statistics are influenced by the increased intermittency of the turbulence at high Reynolds numbers, the lower-order relative velocity statistics are only weakly sensitive to changes in Reynolds number at low $St$. The Reynolds-number trends in these quantities at intermediate and large $St$ are explained based on the influence of the available flow scales on the path-history and inertial filtering effects. We find that the RDFs peak near $St$ of order unity, that they exhibit power-law scaling for low and intermediate $St$ and that they are largely independent of Reynolds number for low and intermediate $St$. We use the model of Zaichik & Alipchenkov (New J. Phys., vol. 11, 2009, 103018) to explain the physical mechanisms responsible for these trends, and find that this model is able to capture the quantitative behaviour of the RDFs extremely well when direct numerical simulation data for the structure functions are specified, in agreement with Bragg & Collins (New J. Phys., vol. 16, 2014a, 055013). We also observe that at large $St$, changes in the RDF are related to changes in the scaling exponents of the relative velocity variances. The particle collision kernel closely matches that computed by Rosa et al. (New J. Phys., vol. 15, 2013, 045032) and is found to be largely insensitive to the flow Reynolds number. This suggests that relatively low-Reynolds-number simulations may be able to capture much of the relevant physics of droplet collisions and growth in the adiabatic cores of atmospheric clouds.

A recent assessment of available direct numerical simulation (DNS) data from turbulent boundary layer flows (Schlatter & Örlü,J. Fluid Mech., vol. 659, 2010, pp. 116–126) showed surprisingly large differences not only in the skin friction coefficient or shape factor, but also in their predictions of mean and fluctuation profiles far into the sublayer. While such differences are expected at very low Reynolds numbers and/or the immediate vicinity of the inflow or tripping region, it remains unclear whether inflow and tripping effects explain the differences observed even at moderate Reynolds numbers. This question is systematically addressed by re-simulating the DNS of a zero-pressure-gradient turbulent boundary layer flow by Schlatteret al. (Phys. Fluids, vol. 21, 2009, art. 051702). The previous DNS serves as the baseline simulation, and the new DNS with a range of physically different inflow conditions and tripping effects are carefully compared. The downstream evolution of integral quantities as well as mean and fluctuation profiles is analysed, and the results show that different inflow conditions and tripping effects do indeed explain most of the differences observed when comparing available DNS at low Reynolds number. It is further found that, if transition is initiated inside the boundary layer at a low enough Reynolds number (based on the momentum-loss thickness)${\mathit{Re}}_{\theta } \lt 300$, all quantities agree well for both inner and outer layer for${\mathit{Re}}_{\theta } \gt 2000$. This result gives a lower limit for meaningful comparisons between numerical and/or wind tunnel experiments, assuming that the flow was not severely over- or understimulated. It is further shown that even profiles of the wall-normal velocity fluctuations and Reynolds shear stress collapse for higher${\mathit{Re}}_{\theta } $irrespective of the upstream conditions. In addition, the overshoot in the total shear stress within the sublayer observed in the DNS of Wu & Moin (Phys. Fluids, vol. 22, 2010, art. 085105) has been identified as a feature of transitional boundary layers.

Numerical investigations are performed on the convection heat transfer of supercritical pressure fluid flowing through vertical mini tube with inner diameter of 0.27 mm and inlet Reynolds number of 1900 under various heat fluxes conditions using low Reynolds number k-ε turbulence models due to LB (Lam and Bremhorst), LS (Launder and Sharma) and V2F (v2-f). The predictions are compared with the corresponding experimentally measured values. The prediction ability of various low Reynolds number k-ε turbulence models under deteriorated heat transfer conditions induced by combinations of buoyancy and flow acceleration effects are evaluated. Results show that all the three models give fairly good predictions of local wall temperature variations in conditions with relatively high inlet Reynolds number. For cases with relatively low inlet Reynolds number, V2F model is able to capture the general trends of deteriorated heat transfer when the heat flux is relatively low. However, the LS and V2F models exaggerate the flow acceleration effect when the heat flux increases, while the LB model produces qualitative predictions, but further improvements are still needed for quantitative prediction. Based on the detailed flow and heat transfer information generated by simulation, a better understanding of the mechanism of heat transfer deterioration is obtained. Results show that the redistribution of flow field induced by the buoyancy and flow acceleration effects are main factors leading to the heat transfer deterioration.

Mixing induced through the life-cycle of Kelvin–Helmholtz (KH) billows is studied for a range of low and intermediate Reynolds numbers using direct numerical simulations (DNS). The amount of stirring, and therefore mixing, is significantly controlled by the process of vortex pairing of two KH billows. For low Reynolds numbers, vortex pairing of the billows is complete in the pre-turbulent stage or early stages of turbulence, generating a high amount of stirring. At higher Reynolds numbers, vortex pairing is suppressed by the growth of three-dimensional instabilities, and the amount of stirring is significantly reduced. For single KH billows, as the Reynolds number increases, there is a transition in the characteristics of the mixing, similar to the laboratory measurements of Breidenthal (J. Fluid Mech., vol. 109, 1981, pp. 1–24) and Koochesfahani & Dimotakis (J. Fluid Mech., vol. 170, 1986, pp. 83–112). The transition in mixing is associated with the growth and sustainability of three-dimensional motions at sufficiently high Reynolds numbers. We examine this ‘mixing transition’ and the influence of vortex pairing on it by examining the flow properties at different stages and the exchange between the energy partitions. As the Reynolds number increases, three-dimensional motions develop over a wider range of length scales, and smaller scale eddies form. However, this does not necessarily result in a greater amount of mixing. The maximum total amount of mixing induced over the lifetime of a KH instability, for billows both with and without vortex pairing, occurs when the large-scale eddies that cause the stirring are the most energetic. The mixing efficiency reveals a non-monotonic dependence on the Reynolds number.

Direct numerical simulations (DNS) of temporally developing turbulent boundary layers are performed with a wide range of Reynolds numbers based on the momentum thickness $Re_{\theta } = 2000$ – $13\,000$ for investigating the Reynolds number dependence of the turbulent/non-turbulent interface (TNTI) layer. The grid spacing in the DNS is determined carefully such that small-scale turbulent motions near the TNTI are well resolved. The outer edge of the TNTI layer, called the irrotational boundary, is detected with vorticity magnitude. The mean thicknesses of the TNTI layer, $\delta _{TNTI}$ , turbulent sublayer, $\delta _{TSL}$ , and viscous superlayer, $\delta _{VSL}$ , are found to be approximately $15\eta _{TI}$ , $10\eta _{TI}$ and $5\eta _{TI}$ , respectively, where $\eta _{TI}$ is the Kolmogorov scale taken in the turbulent region near the TNTI layer. The mean curvature of the irrotational boundary is also characterized by $\eta _{TI}$ . The shear parameter and the shear-to-vorticity ratio show that the mean shear effects near the TNTI layer are not significant for both large and small scales. The anisotropy tensors of Reynolds stress and vorticity suggest that the turbulence under the TNTI layer tends to be isotropic at high $Re_{\theta }$ , for which $\eta _{TI}/\delta \sim Re_{\theta }^{-3/4}$ is valid with the boundary layer thickness $\delta$ . The surface area of the irrotational boundary is consistent with the fractal analysis of the interface, where the fractal dimension $D_f$ is found to be $2.14$ – $2.20$ . The present results suggest that the mean entrainment rate per unit horizontal area normalized by the friction velocity varies slowly as $Re_{\theta }^{(3/4)(D_f-2)}$ for $Re_{\theta } \geq 4000$ .

The geometry of turbulent/non-turbulent interfaces (TNTIs) arising from flows with and without mean shear is investigated using direct numerical simulations of turbulent planar jets (PJET) and shear free turbulence (SFT), respectively, with Taylor Reynolds number of about Reλ≈100. In both flows, the TNTI is preferentially aligned with the tangent to the TNTI displaying convex, where the turbulent fluid nearby tends to have a stronger enstrophy, more frequently than concave shapes. The different flow configurations are reflected in different orientations of the TNTI with respect to the flow direction (and its normal). While the interface orientation with respect to the mean flow direction in PJET has an influence on the velocity field near the TNTI and the enstrophy production in the turbulent sublayer, there is no particular discernible dependence on the interface orientation in SFT. Finally, the intense vorticity structures or “worms,” which are possibly associated with “nibbling” entrainment mechanism, “feel” the local geometry of the TNTI, and it is shown that in PJET, a smaller local radius of these structures arises in regions near the TNTI where the local TNTI faces the mean flow direction.

The experimental analysis of the spatial organization of turbulent/non-turbulent interfaces (TNTI) is an important task in many fields of fluid dynamics, especially to characterize mixing processes. Mixing processes are often associated with the macroscopic motion of coherent flow structures. A classical example for illustration is the turbulent mixing layer visualization by Brown and Roshko (J Fluid Mech 64:775–816, 1974). Today, a question of actual research is if coherent large-scale motions observed in turbulent boundary layer flows have an impact on the structure of the TNTI. As the length of these motions extends over many boundary layer thicknesses and their turbulent energy, and thus significance or impact, raises with Reynolds number, the TNTI detection technique must be accurate at large Reynolds numbers. Furthermore, the technique must be able to resolve the TNTI locally with microscopic spatial resolution and, at the same time, globally over a large macroscopic spatial domain. As the last two points require techniques with a large dynamic spatial range (ratio between largest and smallest scales that can be resolved), only tracer particle-based imaging techniques are suited, as the spatial resolution and field of view (FOV) can both be tuned by adjusting the magnification of the lens and the size and number of camera sensors. In this work, three suited techniques are compared to assess the sensitivity of the TNTI measurement of the method applied. The techniques considered are based on the turbulent kinetic energy, the homogeneity of the non-turbulent flow region, and the particle image density. The effect of bias errors on the TNTI measurement is particularly considered, but the implication of the results for the working range of the various techniques is also outlined. The analysis illustrates exemplary the sensitivity of the intermittency factor and the length of the TNTI with respect to the method applied.

Classical large-eddy simulation (LES) modelling assumes that the passive subgrid-scale (SGS) models do not influence large-scale quantities, even though there is now ample evidence of this in many flows. In this work, direct numerical simulation (DNS) and large-eddy simulations of turbulent planar jets at Reynolds number ReH = 6000 including a passive scalar with Schmidt number Sc = 0.7 are used to study the effect of several SGS models on the flow integral quantities e.g. velocity and scalar jet spreading rates. The models analysed are theSmagorinsky, dynamic Smagorinsky, shear-improved Smagorinsky and the Vreman. Detailed analysis of the thin layer bounding the turbulent and non-turbulent regions – the so-called turbulent/non-turbulent interface (TNTI) – shows that this region raises new challenges for classical SGS models. The small scales are far from equilibrium and contain a high fraction of the total kinetic energy and scalar variance, but the situation is worse for the scalar than for the velocity field. Both a-priori and a-posteriori (LES) tests show that the dynamic Smagorinsky and shear-improved models give the best results because they are able to accurately capture the correct statistics of the velocity and passive scalar fluctuations near the TNTI. The results also suggest the existence of a critical resolution Δx, of the order of the Taylor scale λ, which is needed for the scalar field. Coarser passive scalar LES i.e. Δx ≥ λ results in dramatic changes in the integral quantities. This fact is explained by the dynamics of the small scales near the jet interface.