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In this paper, large-eddy simulation of the interaction between a shock wave and the hypersonic turbulent boundary layer in a compression corner with a fixed 34° deflection angle at Ma = 6 for different Reynolds number cases is conducted. For investigating the effects of the Reynolds number for hypersonic cases, three cases where the free-stream Reynolds numbers are 14000, 20000, and 30000/mm are selected. The averaged statistics, such as the mean velocity, the skin friction, the heat flux, and the wall pressure, are used in this paper. The flow structures in the compression ramp including the shock wave and interaction region are discussed. The decomposition of the mean skin-friction drag for the flat flow is extended to be used in the compression corner. In addition, the turbulent kinetic energy is studied through the decomposition of the mean skin-friction drag for the flat-plate region and the corner region. It is found that higher Reynolds numbers would increase the turbulent kinetic energy by turbulent dissipation at the interaction region, while higher Reynolds numbers would decrease the turbulent kinetic energy by turbulent dissipation after reattachment. In addition, it is also found that the turbulent kinetic energy is larger with a higher Reynolds number and higher turbulent kinetic energy inhibits the movement from the separation point to the inflection point (x = 0 mm), which deduces larger separation bubbles. [ABSTRACT FROM AUTHOR]

In the impinging region of opposing jets, strong mixing and significant energy dissipation are observed, but the mixing parameters invariably change with the opposed impinging strength (OIS). In this paper, the ratio of the turbulent kinetic energy (TKE) intensity at the theoretical impinging point to the nozzle exit is defined as the opposed impinging strength. To examine the mixing properties of opposed impinging jets (OIJs) in a limited field under various OIS, a renormalization group k-ε turbulence model is employed to calculate three-dimensional OIJs under various OIS. The nozzle exit diameter is set to 0.6 m, and the inlet velocity is between 0.08 and 8 m/s, so the simulations are performed at Re between 4.8 × 104 and 4.8 × 106. This work focuses on the radial and vertical jets produced after impinging as well as the distribution of the TKE, flow field, and vortices. A thorough investigation reveals that although the OIS of the jets is primarily determined by the degree of jet development, it increases with the Reynolds number. A low OIS results in less mixing in the surrounding water and relatively unconstrained jet generation; however, it also results in limited energy extraction from the fluid. Once the OIS is high, there is more mixing in the surrounding water, and more energy is also lost during impinging. The distribution of vortices in the vortex field is not only influenced by the OIS but is also very closely related to the scale of the limited mixing field. [ABSTRACT FROM AUTHOR]

This paper presents an experimental study on the effects of the Reynolds number (Resj = 300, 600, and 900) and porosity (ϕ = 20%–85%) on synthetic jet vortex rings impinging onto a porous wall. Laser-induced fluorescence and particle image velocimetry are used to acquire flow information qualitatively and quantitatively. When Resj is low (Resj = 300), ϕ plays a key role in determining the formation of transmitted vortex rings downstream. For the first time, a row of individual small-scale vortex rings that form at the lowest porosity (ϕ = 20%) have been observed in the synthetic jet/porous wall interaction. As Resj increases to 900, the triggered Kelvin–Helmholtz instability promotes the vorticity cancellation at a low porosity (ϕ = 30%), and thus contributes to the formation of a transmitted vortex ring. It is concluded that the vorticity cancellation is the dominant factor affecting the generation of a transmitted vortex ring. Time-averaged characteristics indicate that for a low Resj, the incoherence of the vortex ring is mainly due to the viscous effects. However, for a high Resj, it is the transition that leads to a significant enhancement in the turbulent kinetic energy. Measurements of flow macroscopic parameters show that the loss of the momentum flux exhibits a linear relationship with ϕ for all Resj, while the loss of the kinetic energy transport is nonlinearly dependent on ϕ. Incorporating ϕ, this study presents a more comprehensive similarity parameter, ϕln(Resj2 d h * 3 ), to characterize the synthetic jet/porous wall interaction. [ABSTRACT FROM AUTHOR]

Proper EMA-balance (balance of kinetic energy, linear momentum and angular momentum), pressure-robustness and Re-semi-robustness (Re: Reynolds number) are three important properties of Navier–Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum in an appropriate sense; pressure-robustness means that the velocity errors are independent of the pressure; Re-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust reconstruction framework and certain suggested reconstruction operators in Linke and Merdon [Comput. Methods Appl. Mech. Eng.311 (2016) 304–326], we propose a reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a suitably redefined discrete energy. The lowest order case is the Bernardi–Raugel element on general shape-regular meshes. Some numerical comparisons with exactly divergence-free methods, the original pressure-robust reconstruction methods and the EMAC method are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

SUMMARY This paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor-Green vortex at Re = 1600 is considered. The results are compared with those obtained using a pseudo-spectral solver, converged on a 5123 grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 2883, the fourth-order accurate discontinuous Galerkin method (DGM) solution ( p = 3) is already very close to the pseudo-spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 3843, the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under-resolved grid (resolution of 1923). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth-order DGM is also compared with an energy conserving fourth-order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 7683 DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a Reynolds number increase transition from self-oscillations close to single-frequency ones to the temporally chaotic regime in the flow in a cylindrical channel driven by a spatially periodic force with four half-periods is experimentally investigated. The parameter ε proportional to the mean rate of the kinetic energy dissipation in unit mass per unit time associated with perturbations in the fluid is used as a basic characteristic of self-oscillations. The Reynolds number dependence ε(Re) for single frequency self-oscillations is considered theoretically. [ABSTRACT FROM AUTHOR]

There have been numerous studies concerning the possibility of self-similar scaling laws in fully developed turbulent shear flows, driven over the past half-century or so by the early seminal work of Townsend (1956, The Structure of Turbulent Shear Flow. Cambridge University Press). His and nearly all subsequent analyses depend crucially on a hypothesis about the nature of the dissipation, ϵ, of turbulence kinetic energy, k. It has usually been assumed (sometimes implicitly) that this is governed by the famous Kolmogorov relation ϵ = Cεk3/2/L, where L is a length scale of the energy-containing eddies and Cϵ is a constant. The paper by Dairay et al. (J. Fluid Mech. vol. 781, 2015, pp. 166-195) demonstrates, however, that, in the specific context of an axisymmetric wake, there can be regions where ϵ has a different behaviour, characterised by a C that is not constant but depends on a varying local Reynolds number (despite the existence of a -5/3 region in the spectra). This leads to fundamentally different scaling laws for the wake. [ABSTRACT FROM AUTHOR]

We discuss how the projection of geophysical equations of motion onto an exponential grid allows the determination of realistic values of parameters at a moderate cost. This allows us to perform many simulations over a wide range of parameters, thereby leading to general scaling laws of transport efficiency that can then be used to parametrize the turbulent transport in general climate models for Earth or other planets. We illustrate this process using the equation describing heat transport in a dry atmosphere to obtain the scaling laws for the onset of convection as a function of rotation. We confirm the theoretical scaling of the critical Rayleigh number, Ra c ∼ E − 4 / 3 , over a wide range of parameters. We have also demonstrated the existence of two regimes of convection: one laminar regime extending near the convection onset, and one turbulent regime occurring as soon as the vertical Reynolds number reaches a value of 10 4 . We derive general scaling laws for these two regimes, both for the transport of heat and the dissipation of kinetic energy, and values of anisotropy and temperature fluctuations. [ABSTRACT FROM AUTHOR]

The sweeping jet actuator (SJA), a type of fluid oscillator, has attracted considerable attention in the areas of enhanced heat transfer and flow control. In this study, an unsteady Reynolds-averaged Navier–Stokes numerical investigation (k–ω baseline) was conducted to evaluate the flow mechanisms and heat transfer characteristics of a sweeping jet (SJ) on a confined concave surface. A conventional oscillating jet with a unit aspect ratio was used to generate the SJ. The heat transfer at the target surface and the flow details inside the flow field are discussed in detail to analyze the mechanism of the effect of SJ on impinging cooling. First, an appropriate numerical model was selected based on a comparison with a previous experiment. Subsequently, the time-averaged and time-resolved flow field and heat transfer results were investigated at three target distances for three different Reynolds numbers. The unsteady time-averaged results show that the SJ exhibits better heat transfer performance at higher Reynolds numbers and narrower target spacings than circle jet, which improves by 13% at Re = 30 000, H/D = 1. Subsequently, the turbulent kinetic energy and velocity loss theories were combined to compare the time-averaged and time-resolved flow field details of the two jets. The heat transfer characteristics of the target surface and the flow details inside the flow field in time and space were effectively correlated. Finally, the topology and three-dimensional (3D) vortex structure inside the confined channel were remodeled to better understand the unsteady sweeping process and provide theoretical support for subsequent applications. [ABSTRACT FROM AUTHOR]

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We numerically study the impact of a liquid drop onto no-slip rigid substrates with different wettabilities using a diffuse interface method, aiming to obtain a universal model for the maximal spreading of the impacting drop at moderate Weber numbers. We find that the wettability plays an important role in the maximal spreading and that the ratio of the surface energy to the initial kinetic energy of the drop at the maximal spreading, η, follows η ∼ W e − 1 / 2 at high fixed Reynolds numbers, where We is the Weber number. Taking account of the wettability effect, we obtain a scaling law at high Reynolds numbers from an analysis of energy transformation. This scaling law is compatible with the one derived from the momentum balance at the high impact velocity by Clanet et al. ["Maximal deformation of an impacting drop," J. Fluid Mech. 517, 199–208 (2004)]. Moreover, we attribute it to the presence of a viscous–capillary regime, in which the viscous dissipation of the kinetic energy from the substrate is as significant as the kinetic energy transformed into the surface energy. Accordingly, we identify a new impact parameter, which makes all the numerical results of maximum drop deformation (from the viscous regime to the viscous–capillary regime with Reynolds number up to 104) collapse onto a single curve. Finally, we propose a universal model, the predictions of which are shown to agree well with numerical results for a wide range of Weber and Reynolds numbers. [ABSTRACT FROM AUTHOR]

Here, we present some interesting new results on the linear modal stability of the Couette–Poiseuille flow (CPF), by numerically solving the Orr–Sommerfeld equation within the temporal framework. We provide a mechanistic explanation to the well-known result that the upper plate movement causes progressive stabilization of the CPF. The temporal energy budget reveals appearance of a region of negative energy production with plate movement, which is located near the plate that has a higher relative velocity in the direction of the bulk flow. This has a negative contribution to the integral production term, which, along with the expectation that energy dissipation is nominally constant (to leading order) at a given Reynolds number (Re), reduces the growth rate of kinetic energy, thereby causing flow stabilization with an increase in plate speed. Next, we compare the linear stability portrait of the CPF for positive and negative plate speeds and find that the upper and lower neutral branches in the frequency-Re plane cross over each other (to form a "loop") for a certain range of negative plate speeds. Interestingly, this behavior is not seen for positive plate speeds. At high Re and small plate speeds (positive/negative), we observe that the CPF supports an additional unstable mode manifested in the neutral curves as a bifurcation of the unstable region into primary and secondary regions. The secondary unstable mode is found to be sufficiently distinct in character from the primary mode. These results contribute to the contemporary efforts to better understand the stability of the CPF. [ABSTRACT FROM AUTHOR]

The present paper examines the effect of Reynolds number on turbulence properties in the transition region of a circular jet issuing from a smoothly contracting nozzle. Hot-wire measurements were performed for this investigation through varying the jet-exit Reynolds number Red (≡ Ujd/ν, where Uj, d, and ν are the jet-exit mean velocity, nozzle diameter, and kinematic viscosity) approximately from Red ≈ 4 × 103 to Red ≈ 2 × 104. Results reveal that the rates of the mean flow decay and spread vary with Reynolds number for Red < 104 and tend to become Reynolds-number independent at Red ≥ 104. Even more importantly, the small-scale turbulence properties, e.g., the mean rate of dissipation of kinetic energy (&eh;), the Kolmogorov and Taylor microscales, are found to vary in different forms over the Red ranges of Red > 104 and Red < 104. Namely, the critical Reynolds number appears to occur at Red,cr ≈ 104 across which the jet turbulence behaves distinctly. Two turbulence regimes are therefore identified: (i) developing or partially developed turbulence at Red < Red,cr and (ii) fully developed turbulence at Red ≥ Red,cr. It is suggested that the energy dissipation rate (DR) can be expressed as &eh;∼νUc2/R2 in regime (i) and &eh;∼Uc3/R in regime (ii), where Uc and R are the centerline (or maximum) mean velocity and half-radius at which the mean velocity is 0.5Uc. In addition, the critical Reynolds number appears to vary from flow to flow. [ABSTRACT FROM AUTHOR]

The K-ε model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier-Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number Re has suggested that the two dominant O(Re1/2) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected O(Re-1/2) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the -5/3 law was responsible for the O(Re-1/2) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation. [ABSTRACT FROM AUTHOR]

We present a study of a large-scale energy spectrum and integral invariants in temporally developing grid turbulence at mesh Reynolds numbers of ReM = 10 000 and 20 000 by employing direct numerical simulations (DNSs) in a periodic box. The simulations are initialized with a velocity field that approximates the wakes induced by the bars of conventional square grids. The turbulence statistics obtained in the temporal DNS agree well with those of the previous experiments in both the production and decay regions. The temporally developing grid turbulence also has a so-called non-equilibrium region, which is consistent with its spatially developing counterpart, where the normalized dissipation rate of turbulence kinetic energy (TKE), Cε, increases as the turbulence decays. The decay exponent n of TKE is n = 1.22 at ReM = 20 000 and n = 1.35 at ReM = 10 000, which are close to the values for the Saffman turbulence [i.e., 6/5 for ReM = 20 000 and 6(1 + p)/5 ≈ 1.36 for ReM = 10 000 with p ≈ 0.13 obtained by Cε ∼ tp at large t]. The longitudinal integral length scale and the TKE dissipation rate also exhibit temporal evolutions consistent with the Saffman turbulence for both ReM. The Saffman integral directly evaluated in the grid turbulence tends to be time-independent after the turbulence evolves for about 200 times of characteristic time scale defined by mesh size divided by the mean velocity of a fluid passing the grid. A direct evaluation of the TKE spectrum E(k) shows that E(k) ≈ Lk2/4π2 is valid for a finite range of low wavenumbers. [ABSTRACT FROM AUTHOR]

This article reviews evidence concerning the cornerstone dissipation scaling of turbulence theory: , with Cε=const., ε the dissipation rate of turbulent kinetic energy , and an integral length scale characterizing the energy-containing turbulent eddies. This scaling is intimately linked to the Richardson-Kolmogorov equilibrium cascade. Accumulating evidence shows that a significant nonequilibrium region exists in various turbulent flows in which the energy spectrum has Kolmogorov's −5/3 wave-number scaling over a wide wave-number range, yet Cε∼ RemI/ RenL, with m≈1≈ n, ReI a global/inlet Reynolds number, and ReL a local turbulence Reynolds number. [ABSTRACT FROM AUTHOR]

This study focuses on quantifying hydrodynamic trails produced by freely swimming zooplankton. We combined volumetric tracking of swimming trajectories with planar observations of the flow field induced by Daphnia of different size and swimming in different patterns. Spatial extension of the planar flow field along the trajectories was used to interrogate the dimensions (length and volume) and energetics (dissipation rate of kinetic energy and total dissipated power) of the trails. Our findings demonstrate that neither swimming pattern nor size of the organisms affect the trail width or the dissipation rate. However, we found that the trail volume increases with increasing organism size and swimming velocity, more precisely the trail volume is proportional to the third power of Reynolds number. This increase furthermore results in significantly enhanced total dissipated power at higher Reynolds number. The biggest trail volume observed corresponds to about 500 times the body volume of the largest daphnids. Trail-averaged viscous dissipation rate of the swimming daphnids vary in the range of to and the observed magnitudes of total dissipated power between and , respectively. Among other zooplankton species, daphnids display the highest total dissipated power in their trails. These findings are discussed in the context of fluid mixing and transport by organisms swimming at intermediate Reynolds numbers. [ABSTRACT FROM AUTHOR]

We propose new scaling laws for the properties of planetary dynamos. In particular, the Rossby number, the magnetic Reynolds number, the ratio of magnetic to kinetic energy, the Ohmic dissipation timescale and the characteristic aspect ratio of the columnar convection cells are all predicted to be power-law functions of two observable quantities: the magnetic dipole moment and the planetary rotation rate. The resulting scaling laws constitute a somewhat modified version of the scalings proposed by Christensen and Aubert. The main difference is that, in view of the small value of the Rossby number in planetary cores, we insist that the non-linear inertial term, ${\boldsymbol u} \cdot \nabla {\boldsymbol u}$, is negligible. This changes the exponents in the power-laws which relate the various properties of the fluid dynamo to the planetary dipole moment and rotation rate. Our scaling laws are consistent with the available numerical evidence. [ABSTRACT FROM AUTHOR]

A new model of turbulent fibre suspension in pipe flow is developed by deriving the equations of Reynolds averaged Navier-Stokes, turbulence kinetic energy and turbulence dissipation rate with the additional term of the fibres, and the equation of probability distribution function for mean fibre orientation. The equations are solved numerically. The numerical mean velocity is in agreement with the experimental data. The effects of Reynolds number, fibre concentration, and fibre aspect-ratio on the mean velocity, turbulent kinetic energy and turbulent dissipation rate are analysed. The results show that the effect of Reynolds number on the flow behaviour is insignificant. The turbulent kinetic energy and turbulent dissipation rate increase with an increasing fibre concentration and fibre aspect-ratio. © 2012 Canadian Society for Chemical Engineering [ABSTRACT FROM AUTHOR]

Experiments were carried out in a turbulent mixing layer designed to match, as closely as possible, the conditions of the temporally evolving direct numerical simulation of Rogers & Moser (Phys. Fluids, vol. 6, 1994, pp. 903–922). Two Reynolds numbers, based on the local momentum thickness in the self-similar region of the mixing layer, were investigated: ${R}_{\theta } = 1792$ and $2483$. Measurements were also made in the mixing layer in the pre-mixing transition region where ${R}_{\theta } = 432$. The three velocity components and their cross-stream gradients were measured with a small 12-sensor hot-wire probe that traversed the mixing layer. Taylor’s hypothesis was used to estimate the streamwise gradients of the velocity components so that reasonably good approximations of all the components of the velocity gradient tensor would be available. The signal from a single-sensor probe at a fixed position in the high-speed free stream flow provided a reference to the phases of the passage of large-scale, coherent, spanwise-oriented vortices past the 12-sensor probe. The velocity and velocity gradient data were analysed to determine turbulence statistical characteristics, including moments, probability density functions and one-dimensional spectra of the velocity and vorticity fields. Although the velocity statistics obtained from the experiment agree well with those from the direct numerical simulation of Rogers & Moser, there are significant differences in the vorticity statistics. The phase reference from the fixed single-sensor probe permitted phase averaging of the 12-sensor probe data so that the spanwise ‘roller’ vortices could be separated from the small-scale, more random turbulence, as had been previously demonstrated by Hussain & Zaman (J. Fluid Mech., vol. 159, 1985, pp. 85–104). In this manner, the data could be conditionally averaged to examine the spatial distributions, with respect to the roller vortices, of interesting and important characteristics of the turbulence, such as the turbulent kinetic energy production and dissipation rate, enstrophy and vorticity component covariances. [ABSTRACT FROM PUBLISHER]