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Stably stratified sheared flows are ubiquitous in geophysical flows from the ocean to the stars, and the route to turbulence in these flows remains an open question. The article by Lefauve et al. (J. Fluid Mech., vol. 848, 2018, pp. 508–544) is an invitation to this journey. With impressive experimental precision mastered by few teams in the world, the nature of the coherent structure that dominates the flow on the verge of turbulent breakdown is revealed and analysed through one- or two-dimensional modern stability analysis of an experimentally obtained base flow. The effect of confinement is surprisingly strong, advocating for leaving the textbook flows, inhomogeneous in only one direction, for the more complex shores of real flows, now accessible to analysis of multidimensional stability problems. The route explored by Lefauve et al. (2018) renews with the long tradition of the supercritical bifurcation scenario, it revisits the linear stability theory with possibility of resonances, critical layers and more to be imagined, since complex base flows are now available to explore both experimentally and analytically.
The occurrence of secondary flows is investigated for three-dimensional sinusoidal roughness where the wavelength and height of the roughness elements are systematically altered. The flow spanned from the transitionally rough regime up to the fully rough regime and the solidity of the roughness ranged from a wavy, sparse roughness to a dense roughness. Analysing the time-averaged velocity, secondary flows are observed in all of the cases, reflected in the coherent stress profile which is dominant in the vicinity of the roughness elements. The roughness sublayer, defined as the region where the coherent stress is non-zero, scales with the roughness wavelength when the roughness is geometrically scaled (proportional increase in both roughness height and wavelength) and when the wavelength increases at fixed roughness height. Premultiplied energy spectra of the streamwise velocity turbulent fluctuations show that energy is reorganised from the largest streamwise wavelengths to the shorter streamwise wavelengths. The peaks in the premultiplied spectra at the streamwise and spanwise wavelengths are correlated with the roughness wavelength in the fully rough regime. Current simulations show that the spanwise scale of roughness determines the occurrence of large-scale secondary flows.
We consider feedback flow control of the linearised complex Ginzburg–Landau system. The particular focus is on any trade-offs present when placing a single sensor and a single actuator. The work is presented in three parts. First, we consider the estimation problem in which a single sensor is used to estimate the entire flow field (without any control). Second, we consider the full information control problem in which the entire flow field is known, but only a single actuator is available for control. By considering the optimal sensor placement and optimal actuator placement while varying the stability of the system, a fundamental trade-off for both problems is made clear. Third, we consider the overall feedback control problem in which only a single sensor is available for measurement; and only a single actuator is available for control. By varying the stability of the system, similar fundamental trade-offs are made clear. We discuss implications for effective feedback control with a single sensor and a single actuator and compare it to previous placement methods.
Motivated by the persistence of natural carbon dioxide ( $\text{CO}_{2}$ ) fields, we investigate the convective dissolution of $\text{CO}_{2}$ at low pressure (below 1 MPa) in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry’s law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the dimensionless dissolution capacity, $\unicode[STIX]{x1D6F1}$ , which gives the fraction of the gas that can be dissolved into the underlying brine. Analytic approximations of the pure diffusion problem with $\unicode[STIX]{x1D6F1}>0$ show that the diffusive base state is no longer self-similar and that diffusive mass transfer declines rapidly with time. Direct numerical simulations at high Rayleigh numbers show that no constant flux regime exists for $\unicode[STIX]{x1D6F1}>0$ ; nevertheless, the quantity $F/C_{s}^{2}$ remains constant, where $F$ is the dissolution flux and $C_{s}$ is the dissolved concentration at the top of the domain. Simple mathematical models are developed to predict the evolution of $C_{s}$ and $F$ for high-Rayleigh-number convection in a closed system. The negative feedback that limits convection in closed systems may explain the persistence of natural $\text{CO}_{2}$ accumulations over millennial time scales.
The pressure–strain-rate correlation and pressure fluctuations in convective and near neutral atmospheric surface layers are investigated. Their scaling properties, spectral characteristics, the contributions from the different source terms in the pressure Poisson equation and the effects of the wall are investigated using high-resolution (up to $2048^{3}$ ) large-eddy simulation fields and through spectral predictions. The pressure–strain-rate correlation was found to have the mixed-layer and surface-layer scaling in the strongly convective and near neutral atmospheric surface layers, respectively. Its apparent surface-layer scaling in the moderately convective surface layer is due to the slow variations of the mixed-layer contribution, and is an inherent problem for single-point statistics in a multi-scale surface layer. In the strongly convective surface layer the pressure spectrum has an approximate $k^{-5/3}$ scaling range for small wavenumbers ( $kz\ll 1$ ) due to the turbulent–turbulent contribution, and does not follow the surface-layer scaling, where $k$ and $z$ are the horizontal wavenumber and the distance from the surface respectively. The pressure–strain-rate cospectrum components have a $k^{-1}$ scaling range, consistent with our prediction using the surface layer parameters. It is dominated by the buoyancy contribution. Thus the anisotropy in the surface layer is due to the energy redistribution caused by the density fluctuations of the large eddies, rather than the turbulent–turbulent (inertial) effects. In the near neutral surface layer, the turbulent–turbulent and rapid contributions are primarily responsible for redistribution of energy from the streamwise velocity component to the vertical and spanwise components, respectively. The pressure–strain-rate cospectra peak near $kz\sim 1$ , and have some similarities to those in the strongly convective surface layer for $kz\ll 1$ . For the moderately convective surface layer, the pressure–strain-rate cospectra change signs at scales of the order of the Obukhov length, thereby imposing it as a horizontal length scale in the surface layer. This result provides strong support to the multipoint Monin–Obukhov similarity recently proposed by Tong & Nguyen (J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348). We further decompose the pressure into the free-space (infinite domain), the wall reflection and the harmonic contributions. In the strongly convective surface layer, the free-space contribution to the pressure–strain-rate correlation is dominated by the buoyancy part, and is the main cause of the surface-layer anisotropy. The wall reflection enhances the anisotropy for most of the surface layer, suggesting that the pressure source has a large coherence length. In the near neutral surface layer, the wall reflection is small, suggesting a much smaller source coherence length. The present study also clarifies the understanding of the role of the turbulent–turbulent pressure, and has implications for understanding the dynamics and structure as well as modelling the atmospheric surface layer.
Flexural-gravity wave characteristics are analysed, in the presence of a compressive force and a two-layer fluid, under the assumption of linearized water wave theory and small amplitude structural response. The occurrence of blocking for flexural-gravity waves is demonstrated in both the surface and internal modes. Within the threshold of the blocking and the buckling limit, the dispersion relation possesses four positive roots (for fixed wavenumber). It is shown that, under certain conditions, the phase and group velocities coalesce. Moreover, a wavenumber range for certain critical values of compression and depth is provided within which the internal wave energy moves faster than that of the surface wave. It is also demonstrated that, for shallow water, the wave frequencies in the surface and internal modes will never coalesce. It is established that the phase speed in the surface and internal modes attains a minimum and maximum, respectively, when the interface is located approximately in the middle of the water depth. An analogue of the dead water phenomenon, the occurrence of a high amplitude internal wave with a low amplitude at the surface, is established, irrespective of water depth, when the densities of the two fluids are close to each other. When the interface becomes close to the seabed, the dead water effect ceases to exist. The theory developed in the frequency domain is extended to the time domain and examples of negative energy waves and blocking are presented.
A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.
We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.
The paper investigates experimentally the global wake dynamics of a simplified three-dimensional ground vehicle at a Reynolds number of $Re\simeq 4.0\times 10^{5}$ . The after-body has a blunt rectangular trailing edge leading to a massive flow separation. Both the inclination (yaw and pitch angles) and the distance to the ground (ground clearance) are accurately adjustable. Two different aspect ratios of the rectangular base are considered; wider than it is tall (minor axis perpendicular to the ground) and taller than it is wide (major axis perpendicular to the ground). Measurements of the spatial distribution of the pressure at the base and velocity fields in the wake are used as topological indicators of the flow. Sensitivity analyses of the base pressure gradient expressed in polar form (modulus and phase) varying ground clearance, yaw and pitch are performed. Above a critical ground clearance and whatever the inclination is, the modulus is always found to be large due to the permanent static symmetry-breaking instability, and slightly smaller when aligned with the minor axis of the base rather than when aligned with the major axis. The instability can be characterized with a unique wake mode, quantified by this modulus (asymmetry strength) and a phase (wake orientation) which is the key ingredient of the global wake dynamics. An additional deep rear cavity that suppresses the static instability allows a basic flow to be characterized. It is shown that both the inclination and the ground clearance constrain the phase dynamics of the unstable wake in such way that the component of the pressure gradient aligned with the minor axis of the rectangular base equals that of the basic flow. Meanwhile, the other component related to the major axis adjusts to preserve the large modulus imposed by the instability. In most cases, the dynamics explores only two possible opposite values of the component along the major axis. Their respective probability depends on the geometrical environment of the wake: base shape, body inclination, ground proximity and body supports. An expression for the lateral force coefficients taking into account the wake instability is proposed.
A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$ , where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$ , and the initial centreline velocity, $\overline{U}_{1}(0)$ . A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.
Electrodynamic fluidization is a technique to generate suspensions of electrically conducting particles using electric forces to overcome their weight. An analysis of electrodynamic fluidization is presented for a monodisperse aerosol of non-coalescing particles of infinite electrical conductivity and negligible inertia suspended in a gas in the gap between two horizontal plate electrodes. A DC voltage is applied between the electrodes that charges the particles initially deposited on the lower electrode and leads to a vertical electric force that lifts the particles and pushes them upwards across the gap. The direction of this force reverses when the particles reach the upper electrode, pushing them downwards until they fall onto the lower electrode and repeat the cycle. Stationary distributions of particles are computed for given values of the applied voltage and the number of suspended particles per unit electrode area. Interparticle collisions play a role when the second of these parameters is of the order of the inverse of the particle cross-section or larger. The electric field induced by the charge of the particles opposes the field due to the applied voltage at the lower electrode and thus sets an upper bound to the number of particles that can be suspended for a given voltage. This bound is attained in the normal operation of a fluidization device, in which there is an excess of particles deposited at the lower electrode, and is computed as a function of the applied voltage. The predictions are compared to experimental results in the literature. A linear stability analysis for dilute aerosols with negligible collision effects shows that the stationary solution becomes unstable when the deposition threshold is approached with a number of suspended particles per unit electrode area larger than a certain critical value. A hydrodynamic instability appears near the lower electrode, where the electric force on a localized accumulation of charged particles leads to an upward gas flow that helps carrying the particles away from the electrode and increases the amplitude of the initial particle accumulation. The instability gives rise to electrohydrodynamic plumes whose dynamics involves collisions, mergers and generation of new plumes.
The stability of stably stratified vortices is studied by local stability analysis. Three base flows that possess hyperbolic stagnation points are considered: the two-dimensional (2-D) Taylor–Green vortices, the Stuart vortices and the Lamb–Chaplygin dipole. It is shown that the elliptic instability is stabilized by stratification; it is completely stabilized for the 2-D Taylor–Green vortices, while it remains and merges into hyperbolic instability near the boundary or the heteroclinic streamlines connecting the hyperbolic stagnation points for the Stuart vortices and the Lamb–Chaplygin dipole. More importantly, a new instability caused by hyperbolic instability near the hyperbolic stagnation points and phase shift by the internal gravity waves is found; it is named the strato-hyperbolic instability; the underlying mechanism is parametric resonance as unstable band structures appear in contours of the growth rate. A simplified model explains the mechanism and the resonance curves. The growth rate of the strato-hyperbolic instability is comparable to that of the elliptic instability for the 2-D Taylor–Green vortices, while it is smaller for the Stuart vortices and the Lamb–Chaplygin dipole. For the Lamb–Chaplygin dipole, the tripolar instability is found to merge with the strato-hyperbolic instability as stratification becomes strong. The modal stability analysis is also performed for the 2-D Taylor–Green vortices. It is shown that global modes of the strato-hyperbolic instability exist; the structure of an unstable eigenmode is in good agreement with the results obtained by local stability analysis. The strato-hyperbolic mode becomes dominant depending on the parameter values.
Taylor–Couette flows have been widely studied in part due to the enhanced mixing performance from the variety of hydrodynamic flow states accessible. These process improvements have been demonstrated despite the traditionally limited injection mechanisms from the complexity of the Taylor–Couette geometry. In this study, using a newly designed, modified Taylor–Couette cell, axial mass transport behaviour is experimentally determined over two orders of magnitude of Reynolds number. Four different flow states, including laminar and turbulent Taylor vortex flows and laminar and turbulent wavy vortex flows, were studied. Using flow visualization techniques, the measured dispersion coefficient was found to increase with increasing $Re$ , and a single, unified regression is found for all vortices studied. In addition to mass transport, the vortex structures’ stability to radial injection is also quantified. A dimensionless stability criterion, the ratio of injection to diffusion time scales, was found to capture the conditions under which vortex structures are stable to injection. Using the stability criterion, global and transitional stability regions are identified as a function of Reynolds number, $Re$ .
Scalar image velocimetry (SIV) is the technique to extract velocity vectors from scalar field measurements. The usual technique involves minimising a cost functional, that penalises the deviation from the scalar conservation equation. This approach requires the measured scalar field to be sufficiently resolved and relatively noise free, such that space and time derivatives of the measured scalar field can be accurately evaluated. We quantify these requirements for a synthetic two-dimensional (2-D) turbulent flow field by evaluating the velocity reconstruction accuracy as a function of the temporal and spatial resolution and the noise level. We propose an improved SIV scheme, that reconstructs not only the velocity field but also the scalar field, which does not require approximating the space and time derivatives of the measured scalar field. Improved velocity reconstruction is demonstrated for the 2-D synthetic field. We furthermore apply the scheme to interferograms of the thickness field of a falling soap film, where 2-D turbulence is generated by an array of cylindrical obstacles. The statistics of the reconstructed velocity field are within 10 % of laser Doppler velocimetry measurements.
This experimental study examines ventilated supercavity formation in a free-surface bounded environment where a body is in motion and the fluid is at rest. For a given torpedo-shaped body and water depth ( $H$ ), depending on the cavitator diameter ( $d_{c}$ ) and the submergence depth ( $h_{s}$ ), four different cases are investigated according to the blockage ratio ( $B=d_{c}/d_{h}$ , where $d_{h}$ is the hydraulic diameter) and the dimensionless submergence depth ( $h^{\ast }=h_{s}/H$ ). Cases 1–4 are, respectively, no cavitator in fully submerged ( $B=0$ , $h^{\ast }=0.5$ ), small blockage in fully submerged ( $B=1.5\,\%$ , $h^{\ast }=0.5$ ), small blockage in shallowly submerged ( $B=1.5\,\%$ , $h^{\ast }=0.17$ ) and large blockage in fully submerged ( $B=3\,\%$ , $h^{\ast }=0.5$ ) cases. In case 1, no supercavitation is observed and only a bubbly flow (B) and a foamy cavity (FC) are observed. In non-zero blockage cases 2–4, various non-bubbly and non-foamy steady states are observed according to the cavitator-diameter-based Froude number ( $Fr$ ), air-entrainment coefficient ( $C_{q}$ ) and the cavitation number ( $\unicode[STIX]{x1D70E}_{c}$ ). The ranges of $Fr$ , $C_{q}$ and $\unicode[STIX]{x1D70E}_{c}$ are $Fr=2.6{-}18.2$ , $C_{q}=0{-}6$ , $\unicode[STIX]{x1D70E}_{c}=0{-}1$ for cases 2 and 3, and $Fr=1.8{-}12.9$ , $C_{q}=0{-}1.5$ , $\unicode[STIX]{x1D70E}_{c}=0{-}1$ for case 4. In cases 2 and 3, a twin-vortex supercavity (TV), a reentrant-jet supercavity (RJ), a half-supercavity with foamy cavity downstream (HSF), B and FC are observed. Supercavities in case 3 are not top–bottom symmetric. In case 4, a half-supercavity with a ring-type vortex shedding downstream (HSV), double-layer supercavities (RJ inside and TV outside (RJTV), TV inside and TV outside (TVTV), RJ inside and RJ outside (RJRJ)), B, FC and TV are observed. The cavitation numbers ( $\unicode[STIX]{x1D70E}_{c}$ ) are approximately 0.9 for the B, FC and HSF, 0.25 for the HSV, and 0.1 for the TV, RJ, RJTV, TVTV and RJRJ supercavities. In cases 2–4, for a given $Fr$ , there exists a minimum cavitation number in the formation of a supercavity while the minimum cavitation number decreases as the $Fr$ increases. In cases 2 and 3, it is observed that a high $Fr$ favours an RJ and a low $Fr$ favours a TV. For the RJ supercavities in cases 2 and 3, the cavity width is always larger than the cavity height. In addition, the cavity length, height and width all increase (decrease) as the $\unicode[STIX]{x1D70E}_{c}$ decreases (increases). The cavity length in case 3 is smaller than that in case 2. In both cases 2 and 3, the cavity length depends little on the $Fr$ . In case 2, the cavity height and width increase as the $Fr$ increases. In case 3, the cavity height and width show a weak dependence on the $Fr$ . Compared to case 2, for the same $Fr$ , $C_{q}$ and $\unicode[STIX]{x1D70E}_{c}$ , case 4 admits a double-layer supercavity instead of a single-layer supercavity. Connected with this behavioural observation, the body-frontal-area-based drag coefficient for a moving torpedo-shaped body with a supercavity is measured to be approximately 0.11 while that for a cavitator-free moving body without a supercavity is approximately 0.4.
Flows containing suspended colloidal particles and dissolved solutes are found in a multitude of natural and man-made systems including hydraulic fractures, water filtration systems and microfluidic devices, e.g. those designed for biological or medical applications. In these types of systems, unexpected particle dynamics such as rapid particle transport and focusing has been observed in the presence of local solute gradients due to the cooperating or competing effects of fluid advection and particle diffusiophoresis, the latter driven by local chemical gradients. We develop analytical expressions for the fluid, solute and particle dynamics in long, narrow channels due to the combined influence of pressure-driven channel flow with diffusiophoretic and diffusioosmotic effects. The results confirm a rapid particle focusing effect that can be controlled by manipulating the particle, solute and flow properties, as well as the channel’s geometry and surface chemistry. Thus, we propose a new approach for performing microfluidic zeta potentiometry, as well as techniques for sorting, concentrating and/or capturing particles based on their sizes or zeta potentials. Finally, we demonstrate that diffusioosmotic effects can be used to pump fluid against a pressure gradient.
Turbulent-kinetic-energy (TKE) production $\mathscr{P}_{k}=R_{12}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)$ and TKE dissipation $\mathscr{E}_{k}=\unicode[STIX]{x1D708}\langle (\unicode[STIX]{x2202}u_{i}/x_{k})(\unicode[STIX]{x2202}u_{i}/x_{k})\rangle$ are important quantities in the understanding and modelling of turbulent wall-bounded flows. Here $U$ is the mean velocity in the streamwise direction, $u_{i}$ or $u,v,w$ are the velocity fluctuation in the streamwise $x$ - direction, wall-normal $y$ - direction, and spanwise $z$ -direction, respectively; $\unicode[STIX]{x1D708}$ is the kinematic viscosity; $R_{12}=-\langle uv\rangle$ is the kinematic Reynolds shear stress. Angle brackets denote Reynolds averaging. This paper investigates the integral properties of TKE production and dissipation in turbulent wall-bounded flows, including turbulent channel flows, turbulent pipe flows and zero-pressure-gradient turbulent boundary layer flows (ZPG TBL). The main findings of this work are as follows. (i) The global integral of TKE production is predicted by the RD identity derived by Renard & Deck (J. Fluid Mech., vol. 790, 2016, pp. 339–367) as $\int _{0}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y=U_{b}u_{\unicode[STIX]{x1D70F}}^{2}-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)^{2}\,\text{d}y$ for channel flows, where $U_{b}$ is the bulk mean velocity, $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity and $\unicode[STIX]{x1D6FF}$ is the channel half-height. Using inner scaling, the identity for the global integral of the TKE production in channel flows is $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\text{d}y^{+}=U_{b}^{+}-\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}(\unicode[STIX]{x2202}U^{+}/\unicode[STIX]{x2202}y^{+})^{2}\,\text{d}y^{+}$ . In the present work, superscript $+$ denotes inner scaling. At sufficiently high Reynolds number, the global integral of the TKE production in turbulent channel flows can be approximated as $\int _{0}^{\unicode[STIX]{x1D6FF}^{+}}\mathscr{P}_{k}^{+}\,\text{d}y^{+}\approx U_{b}^{+}-9.13$ . (ii) At sufficiently high Reynolds number, the integrals of TKE production and dissipation are equally partitioned around the peak Reynolds shear stress location $y_{m}:\,\int _{0}^{y_{m}}\mathscr{P}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{P}_{k}\,\text{d}y$ and $\int _{0}^{y_{m}}\mathscr{E}_{k}\,\text{d}y\approx \int _{y_{m}}^{\unicode[STIX]{x1D6FF}}\mathscr{E}_{k}\,\text{d}y$ . (iii) The integral of the TKE production ${\mathcal{I}}_{\mathscr{P}_{k}}(y)=\int _{0}^{y}\mathscr{P}_{k}\,\text{d}y$ and the integral of the TKE dissipation ${\mathcal{I}}_{\mathscr{E}_{k}}(y)=\int _{0}^{y}\mathscr{E}_{k}\,\text{d}y$ exhibit a logarithmic-like layer similar to that of the mean streamwise velocity as, for example, ${\mathcal{I}}_{\mathscr{P}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{P}}$ and ${\mathcal{I}}_{\mathscr{E}_{k}}^{+}(y^{+})\approx (1/\unicode[STIX]{x1D705})\ln (y^{+})+C_{\mathscr{E}}$ , where $\unicode[STIX]{x1D705}$ is the von Kármán constant, $C_{\mathscr{P}}$ and $C_{\mathscr{E}}$ are addititve constants. The logarithmic-like scaling of the global integral of TKE production and dissipation, the equal partition of the integrals of TKE production and dissipation around the peak Reynolds shear stress location $y_{m}$ and the logarithmic-like layer in the integral of TKE production and dissipation are intimately related. It is known that the peak Reynolds shear stress location $y_{m}$ scales with a meso-length scale $l_{m}=\sqrt{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}$ . The equal partition of the integral of the TKE production and dissipation around $y_{m}$ underlines the important role of the meso-length scale $l_{m}$ in the dynamics of turbulent wall-bounded flows.
Spectral energy transfer in a turbulent channel flow is investigated at Reynolds number $Re_{\unicode[STIX]{x1D70F}}\simeq 1700$ , based on the wall shear velocity and channel half-height, with a particular emphasis on full visualization of triadic wave interactions involved in turbulent transport. As in previous studies, turbulent production is found to be almost uniform, especially over the logarithmic region, and the related spanwise integral length scale is approximately proportional to the distance from the wall. In the logarithmic and outer regions, the energy balance at the integral length scales is mainly formed between production and nonlinear turbulent transport, the latter of which plays the central role in the energy cascade down to the Kolmogorov microscale. While confirming the classical role of the turbulent transport, the triadic wave interaction analysis unveils two new types of scale interaction processes, highly active in the near-wall and the lower logarithmic regions. First, for relatively small energy-containing motions, part of the energy transfer mechanisms from the integral to the adjacent small length scale in the energy cascade is found to be provided by the interactions between larger energy-containing motions. It is subsequently shown that this is related to involvement of large energy-containing motions in skin-friction generation. Second, there exists a non-negligible amount of energy transfer from small to large integral scales in the process of downward energy transfer to the near-wall region. This type of scale interaction is predominant only for the streamwise and spanwise velocity components, and it plays a central role in the formation of the wall-reaching inactive part of large energy-containing motions. A further analysis reveals that this type of scale interaction leads the wall-reaching inactive part to scale in the inner units, consistent with the recent observation. Finally, it is proposed that turbulence production and pressure–strain spectra support the existence of the self-sustaining process as the main turnover dynamics of all the energy-containing motions.
We use high-resolution velocity measurements in a jet-stirred zero-mean-flow facility to investigate the topology and energy transfer properties of homogeneous turbulence over the Reynolds number range $Re_{\unicode[STIX]{x1D706}}\approx 300$ –500. The probability distributions of the enstrophy and strain-rate fields show long tails associated with the most intense events, while the weaker events behave as random variables. The high-enstrophy and high-strain structures are shaped as tube-like and sheet-like objects, respectively, the latter often wrapped around the former. Both types of structures have thickness that scales in Kolmogorov units, and display self-similar topology over a wide range of scales. The small-scale turbulence activity is found to be strongly correlated with the large-scale activity, suggesting that the phenomenon of amplitude modulation (previously observed in advection-dominated shear flows) is not limited to specific production mechanisms. Observing the significant variations in spatially averaged enstrophy, we heuristically define hyperactive and sleeping states of the flow: these also correspond to, respectively, high and low levels of large-scale velocity gradients. Moreover, the hyperactive and sleeping states contribute very differently to the inter-scale energy flux, characterized via the nonlinear transfer term in the Kármán–Howarth–Monin equation. While the energy cascades to smaller scales along the jet-axis direction, a weaker but sizable inverse transfer is observed along the transverse direction; a behaviour so far only observed in spatially developing flows. The hyperactive states are characterized by very intense energy transfers, while the sleeping states account for weaker fluxes, largely directed from small to large scales. This implies that the form of energy cascade depends on the presence (or absence) of intense turbulent structures. These results are at odds with the classic concept of the energy cascade between adjacent scales, but are compatible with the view of a cascade in physical space.
Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence facilitates application of an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second-order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wave–mean flow interaction between the emergent VSHF and the stochastically excited large-scale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results constitute a theory for VSHF formation applicable to interpreting simulations and observations of geophysical examples of turbulent jets such as the ocean’s equatorial deep jets.