We present an experimental investigation of entrainment and the dynamics near the turbulent/non-turbulent interface in a dense gravity current. The main goal of the study is to investigate changes in the interfacial physics due to the presence of stratification and to examine their impact on the entrainment rate. To this end, three-dimensional data sets of the density and the velocity fields are obtained through a combined scanning particle tracking velocimetry/laser-induced fluorescence approach for two different stratification levels with inflow Richardson numbers of $\mathit{Ri}_{0}=0.23$ and $\mathit{Ri}_{0}=0.46$, respectively, at a Reynolds number around $\mathit{Re}_{0}=3700$. An analysis conditioned on the instantaneous position of the turbulent/non-turbulent interface as defined by a threshold on enstrophy reveals an interfacial region that is in many aspects independent of the initial level of stratification. This is reflected most prominently in matching peaks of the gradient Richardson number $\mathit{Ri}_{g}\approx 0.1$ located approximately $10{\it\eta}$ from the position of the interface inside the turbulent region, where ${\it\eta}=({\it\nu}^{3}/{\it\epsilon})^{1/4}$ is the Kolmogorov scale, and ${\it\nu}$ and ${\it\epsilon}$ denote the kinematic viscosity and the rate of turbulent dissipation, respectively. A possible explanation for this finding is offered in terms of a cyclic evolution in the interaction of stratification and shear involving the buildup of density and velocity gradients through inviscid amplification and their subsequent depletion through molecular effects and pressure. In accordance with the close agreement of the interfacial properties for the two cases, no significant differences were found for the local entrainment velocity, $v_{n}$ (defined as the propagation velocity of an enstrophy isosurface relative to the fluid), at different initial stratification levels. Moreover, we find that the baroclinic torque does not contribute significantly to the local entrainment velocity. Comparing results for the surface area of the convoluted interface to estimates from fractal scaling theory, we identify differences in the interface geometry as the major factor in the reduction of the entrainment rate due to density stratification.

We report on effects of mean shear on the turbulent entrainment process, focusing in particular on their relation to small-scale processes in the proximity of the turbulent/non-turbulent interface (TNTI). Three-dimensional particle tracking velocimetry (3D-PTV) measurements of an axisymmetric jet are compared to data from a direct numerical simulation (DNS) of a zero-mean-shear (ZMS) flow. First, conditional statistics relative to the interface position are investigated in a pseudo-Eulerian view (i.e. in a fixed frame relative to the interface position) and in a Lagrangian view. We find that in a pseudo-Eulerian frame of reference, both vorticity fluctuations and mean shear contribute to the vorticity jump at the boundary between irrotational and turbulent regions. In contrast, the Lagrangian evolution of enstrophy along trajectories crossing the entrainment interface is almost exclusively dominated by vorticity fluctuations, at least during the first Kolmogorov time scales after passing the interface. A mapping between distance to the instantaneous interface versus conditional time along the trajectory shows that entraining particles remain initially close to the TNTI and therefore attain lower average enstrophy values. The ratio between the rate of change of enstrophy in the two frames of references defines the local entrainment velocity ${v}_{n} = - (\mathrm{D} {\omega }^{2} / \mathrm{D} t)/ (\partial {\omega }^{2} / \partial {\hat {x} }_{n} )$, where ${\omega }^{2} $ is enstrophy and ${\hat {x} }_{n} $ is the coordinate normal to the TNTI. The quantity ${v}_{n} $ is decomposed into mean and fluctuating components and it is found that mean shear enhances the local entrainment velocity via inviscid and viscous effects. Further, the analysis substantiates that for all investigated flow configurations the local entrainment velocity depends considerably on the geometrical shape of the interface. Depending on the surface shape, different small-scale mechanisms are dominant for the local entrainment process, i.e. viscous effects for convex shapes and vortex stretching for concave shapes, looking from the turbulent region towards the convoluted boundary. Moreover, turbulent fluctuations display a stronger dependence on the shape of the interface than mean shear effects.

The two-dimensional space spanned by the velocity gradient invariants Q and R is expanded to three dimensions by the decomposition of R into its strain production −1/3sijsjkski and enstrophy production 1/4ωiωjsij terms. The {Q; R} space is a planar projection of the new three-dimensional representation. In the {Q; −sss; ωωs} space the Lagrangian evolution of the velocity gradient tensor Aij is studied via conditional mean trajectories (CMTs) as introduced by Martín et al. (Phys. Fluids, vol. 10, 1998, p. 2012). From an analysis of a numerical data set for isotropic turbulence of Reλ ~ 434, taken from the Johns Hopkins University (JHU) turbulence database, we observe a pronounced cyclic evolution that is almost perpendicular to the Q–R plane. The relatively weak cyclic evolution in the Q–R space is thus only a projection of a much stronger cycle in the {Q; −sss; ωωs} space. Further, we find that the restricted Euler (RE) dynamics are primarily counteracted by the deviatoric non-local part of the pressure Hessian and not by the viscous term. The contribution of the Laplacian of Aij, on the other hand, seems the main responsible for intermittently alternating between low and high intensity Aij states.

This paper presents an analysis of flow properties in the proximity of the turbulent/non-turbulent interface (TNTI), with particular focus on the acceleration of fluid particles, pressure and related small scale quantities such as enstrophy, ω2 = ωiωi, and strain, s2 = sijsij. The emphasis is on the qualitative differences between turbulent, intermediate and non-turbulent flow regions, emanating from the solenoidal nature of the turbulent region, the irrotational character of the non-turbulent region and the mixed nature of the intermediate region in between. The results are obtained from a particle tracking experiment and direct numerical simulations (DNS) of a temporally developing flow without mean shear. The analysis reveals that turbulence influences its neighbouring ambient flow in three different ways depending on the distance to the TNTI: (i) pressure has the longest range of influence into the ambient region and in the far region non-local effects dominate. This is felt on the level of velocity as irrotational fluctuations, on the level of acceleration as local change of velocity due to pressure gradients, Du/Dt ≃ ∂u/∂t ≃ −∇ p/ρ, and, finally, on the level of strain due to pressure-Hessian/strain interaction, (D/Dt)(s2/2) ≃ (∂/∂t)(s2/2) ≃ −sijp,ij > 0; (ii) at intermediate distances convective terms (both for acceleration and strain) as well as strain production −sijsjkski > 0 start to set in. Comparison of the results at Taylor-based Reynolds numbers Reλ = 50 and Reλ = 110 suggests that the distances to the far or intermediate regions scale with the Taylor microscale λ or the Kolmogorov length scale η of the flow, rather than with an integral length scale; (iii) in the close proximity of the TNTI the velocity field loses its purely irrotational character as viscous effects lead to a sharp increase of enstrophy and enstrophy-related terms. Convective terms show a positive peak reflecting previous findings that in the laboratory frame of reference the interface moves locally with a velocity comparable to the fluid velocity fluctuations.

We report an analysis of small-scale enstrophy ω2 and rate of strain s2 dynamics in the proximity of the turbulent/non-turbulent interface in a flow without strong mean shear. The techniques used are three-dimensional particle tracking (3D-PTV), allowing the field of velocity derivatives to be measured and followed in a Lagrangian manner, and direct numerical simulations (DNS). In both experiment and simulation the Taylor-microscale Reynolds number is Reλ = 50. The results are based on the Lagrangian viewpoint with the main focus on flow particle tracers crossing the turbulent/non-turbulent interface. This approach allowed a direct investigation of the key physical processes underlying the entrainment phenomenon and revealed the role of small-scale non-local, inviscid and viscous processes. We found that the entrainment mechanism is initiated by self-amplification of s2 through the combined effect of strain production and pressure--strain interaction. This process is followed by a sharp change of ω2 induced mostly by production due to viscous effects. The influence of inviscid production is initially small but gradually increasing, whereas viscous production changes abruptly towards the destruction of ω2. Finally, shortly after the crossing of the turbulent/non-turbulent interface, production and dissipation of both enstrophy and strain reach a balance. The characteristic time scale of the described processes is the Kolmogorov time scale, τη. Locally, the characteristic velocity of the fluid relative to the turbulent/non-turbulent interface is the Kolmogorov velocity, uη.

This is a report on a field experiment in an atmospheric surface layer at heights between 0.8 and 10m with the Taylor micro-scale Reynolds number in the range Reλ = 1.6−6.6 ×103. Explicit information is obtained on the full set of velocity and temperature derivatives both spatial and temporal, i.e. no use of Taylor hypothesis is made. The report consists of three parts. Part 1 is devoted to the description of facilities, methods and some general results. Certain results are similar to those reported before and give us confidence in both old and new data, since this is the first repetition of this kind of experiment at better data quality. Other results were not obtained before, the typical example being the so-called tear-drop R-Q plot and several others. Part 2 concerns accelerations and related matters. Part 3 is devoted to issues concerning temperature, with the emphasis on joint statistics of temperature and velocity derivatives. The results obtained in this work are similar to those obtained in experiments in laboratory turbulent grid flow and in direct numerical simulations of Navier–Stokes equations at much smaller Reynolds numbers Reλ ~ 102, and this similarity is not only qualitative, but to a large extent quantitative. This is true of such basic processes as enstrophy and strain production, geometrical statistics, the role of concentrated vorticity and strain, reduction of nonlinearity and non-local effects. The present experiments went far beyond the previous ones in two main respects. (i) All the data were obtained without invoking the Taylor hypothesis, and therefore a variety of results on fluid particle accelerations became possible. (ii) Simultaneous measurements of temperature and its gradients with the emphasis on joint statistics of temperature and velocity derivatives. These are reported in Parts 2 and 3.

This is part 3 of our work describing experiments in which explicit information was obtained on all the derivatives, i.e. spatial derivatives, ∂/∂xj, and temporal derivatives, ∂/∂t, of velocity and temperature fields (and all the components of velocity fluctuations and temperature) at the Reynolds number Reλ~104.

This part is devoted to the issues concerning temperature with the emphasis on joint statistics of temperature and velocity derivatives, based on preliminary results from a jet facility and the main results from a field experiment. Apart from a number of conventional results, these contain a variety of results concerning production of temperature gradients, such as role of vorticity and strain, eigen-contributions, geometrical statistics such as alignments of the temperature gradient and the eigenframe of the rate-of-strain tensor, tilting of the temperature gradient, comparison of the true production of the temperature gradient with its surrogate. Among the specific results of importance is the essential difference in the behaviour of the production of temperature gradients in regions dominated by vorticity and strain. Namely, the production of temperature gradients is much more intensive in regions dominated by strain, whereas production of temperature gradients is practically independent of the magnitude of vorticity. In contrast, vorticity and strain are contributing equally to the tilting of the vector of temperature gradients.

The production of temperature gradients is mainly due to the fluctuative strain, the terms associated with mean fields are unimportant. It was checked directly (by looking at corresponding eigen-contributions and alignments), that the production of the temperature gradients is due to predominant compressing of fluid elements rather than stretching, which is true of other processes in turbulent flows, e.g. turbulent energy production in shear flows. Though the production of the temperature gradient and its surrogate possess similar univariate PDFs (which indicates the tendency to isotropy in small scales by this particular criterion), their joint PDF is not close to a bisector. This means that the true production of the temperature gradient is far from being fully represented by its surrogate. The main technical achievement is demonstrating the possibility of obtaining experimentally joint statistics of velocity and temperature gradients.

We report the first results of an experiment, in which explicit information on all velocity derivatives (the nine spatial derivatives, ∂ui∂xj, and the three temporal derivatives, ∂ui/∂t) along with the three components of velocity fluctuations at a Reynolds number as high as Reλ~104 is obtained. No use of the Taylor hypothesis was made, and this allowed us to obtain a variety of results concerning acceleration and its different Eulerian components along with vorticity, strain and other small-scale quantities. The field experiments were performed at five heights between 0.8 and 10m above the ground.

The report consists of three parts. Part 1 is devoted to the description of facilities, methods and some general results. Part 2 concerns accelerations and related matters. Part 3 is devoted to the issues concerning temperature with the emphasis on joint statistics of temperature and velocity derivatives.

Lagrangian auto- and cross-correlation functions of the rate of strain s2, enstrophy ω2, their respective production terms −sijsjkski and ωiωjsij, and material derivatives, Ds2/Dt and Dω2/Dt are estimated using experimental results obtained through three-dimensional particle tracking velocimetry (three-dimensional-PTV) in homogeneous turbulence at Reλ=50. The autocorrelation functions are used to estimate the Lagrangian time scales of different quantities, while the cross-correlation functions are used to clarify some aspects of the interaction mechanisms between vorticity ω and the rate of strain tensor sij, that are responsible for the statistically stationary, in the Eulerian sense, levels of enstrophy and rate of strain in homogeneous turbulent flow. Results show that at the Reynolds number of the experiment these quantities exhibit different time scales, varying from the relatively long time scale of ω2 to the relatively shorter time scales of s2, ωiωjsij and −sijsjkski. Cross-correlation functions suggest that the dynamics of enstrophy and strain, in this flow, is driven by a set of different-time-scale processes that depend on the local magnitudes of s2 and ω2. In particular, there are indications that, in a statistical sense, (i) strain production anticipates enstrophy production in low-strain–low-enstrophy regions (ii) strain production and enstrophy production display high correlation in high-strain–high-enstrophy regions, (iii) vorticity dampening in high-enstrophy regions is associated with weak correlations between −sijsjkski and s2 and between −sijsjkski and Ds2/Dt, in addition to a marked anti-correlation between ωiωjsij and Ds2/Dt. Vorticity dampening in high-enstrophy regions is thus related to the decay of s2 and its production term, −sijsjkski.

The splitting of a smooth gradient into mixed layers separated by thin interfaces is studied in a constant stable solute gradient heated uniformly from below by raising the temperature at its bottom to a prescribed value above the ambient. The phenomenon under study is characterized by the development of a first mixed layer from the bottom up to a certain height; it is shown that its instantaneous thickness is proportional to the thermal expansion coefficient and the instantaneous temperature difference between the bottom heated plate and the undisturbed far fluid and is inversely proportional to the solute gradient. A second layer starts its formation during the growth of the first one and persists above the bottom layer notwithstanding the advancing interface below. Further layer formation occurs, depending on the prescribed Rayleigh number of the system (which is based on the vertical temperature difference and the initial solute gradient). The secondary layers grow with time and the final thickness of all the layers (except the bottom one) is almost equal. Their average final thickness is found to depend mostly on the solute gradient and the fluid properties. A final quasi-steady state is reached in which the depth of the whole convective region is proportional to the expansion coefficient and the vertical temperature difference and is inversely proportional to the initial solute gradient

A comparison is made between a number of properties of a quasi-homogeneous isotropic turbulent field obtained from a direct numerical simulation of the Navier–Stokes equation and its random counterpart with the same energy spectrum. It is demonstrated that some effects in a real flow have a considerable contribution of a kinematic nature (e.g. reduction of nonlinearity), while others are mostly dynamical (e.g. alignment between vorticity and eigenvectors of the rate of strain).

We present results of experiments on a turbulent grid flow and a few results on measurements in the outer region of a boundary layer over a smooth plate. The air flow measurements included three velocity components and their nine gradients. This was done by a twelve-wire hot-wire probe (3 arrays × 4 wires), produced for this purpose using specially made equipment (micromanipulators and some other auxiliary special equipment), calibration unit and calibration procedure. The probe had no common prongs and the calibration procedure was based on constructing a calibration function for each combination of three wires in each array (total 12) as a three-dimensional Chebishev polynomial of fourth order. A variety of checks were made in order to estimate the reliability of the results.

Among the results the most prominent are the experimental confirmation of the strong tendency for alignment between vorticity and the intermediate eigenvector of the rate-of-strain tensor, the positiveness of the total enstrophy-generating term ωiωjsij (sij = ½(∂ui/∂xj+∂uj/∂xi), ωi = εijk∂uj/∂xk) even for rather short records and the tendency for alignment in the strict sense between vorticity and the vortex stretching vector Wi = ωjsij. An emphasis is put on the necessity to measure invariant quantities, i.e. independent of the choice of the system of reference (e.g. sijsij and ωiωjsij) as the most appropriate to describe physical processes. From the methodological point of view the main result is that the multi-hot-wire technique can be successfully used for measurements of all the nine velocity derivatives in turbulent flows, at least at moderate Reynolds numbers.

Precise theoretical relations between statistical characteristics of turbulent velocity and an electrical field are found for homogeneous and isotropic turbulence in the presence of a uniform magnetic field. These include turbulence intensities, correlations and spectra and relations allowing estimation of the degree of anisotropy of a turbulent flow. The theoretical results are verified by experiments with flow of very dilute salted water past a grid in the presence of a uniform magnetic field. The theoretical and the experimental results are in good agreement. Preliminary results of vorticity measurements are also presented.

The dynamics and structure of double-diffusive layers were studied experimentally by heating a linear stable solute gradient from a sidewall in a wide tank. The formation and subsequent development of the layers were investigated by various flow-visualization techniques, namely fluorescent dye, fluorescent particles and shadowgraph. Experiments were performed in order to determine the stability diagram of the flow, following in each experiment the phase trajectory of the system in the phase plane of thermal and solute Rayleigh numbers. The experimentally obtained stability diagram appears to be similar to that obtained numerically by Thangam et al. (1981) and by Hart (1971) for a vertical narrow slot and a steady basic flow. It is shown that if the temperature of the sidewall rises slowly to its prescribed value, the thickness of the initial layers, formed at the onset of instability, is a function of the ambient density gradient and fluid properties only. On the other hand, if the wall temperature rises quickly (almost impulsive heating), the thickness of the initial layers is proportional to the imposed temperature difference, provided that the Rayleigh number, based on this difference, is larger than some critical value which is associated with the first merging of the layers. A criterion for the first merging of the initial layers is obtained, and it is suggested that this merging is due to subsequent instability of the system. The subsequent merging process, following the first merging, is not a simple successive doubling of the layer thickness and in each of five nearly identical experiments a different dependence of the average layer thickness on the instantaneous Rayleigh number is obtained. This indicates that the system of layers behaves chaotically after the first merging. The final thickness of the layers depends on the prescribed lateral temperature difference, and the ratio between the final and the initial thickness of the layers is a linear function of the final Rayleigh number. Flow-visualization experiments indicate that the layers consist of vortices with vertical scale of the layer thickness and various horizontal scales.

When a salinity gradient is heated at a single point, a hot plume is formed with a series of layers around and above it in a ‘Christmas-tree’ flow pattern. Qualitative visual observations of the development of this system are reported. Three kinds of layers are observed: the first kind is formed above the basic plume from a hierarchy of secondary plumes on top of the basic one, the second develops around the upper part of the basic plume, and is driven by the same mechanism as the layers in sidewall-heating experiments, and the third, forming around the lower part of the basic plume, is of the same nature as the second type but also has something in common with double-diffusive intrusions. A characteristic feature for all the three kinds of layers is the existence of systematic shearing motions and vortices. Quantitative estimates for the plume height and the thickness of the layers are obtained.

The evolution of material lines, $l$, and vorticity, $\omega$, is investigated experimentally through three-dimensional particle-tracking velocimetry (3D-PTV) in quasi-homogeneous isotropic turbulence at $Re_{\lambda }\,{=}\,50$. Through 3D-PTV data the full set of velocity derivatives, $\partial u_{i}/\partial x_{j}$, is accessible. This allows us to monitor the evolution of various turbulent quantities along fluid particle trajectories. The main emphasis of the present work is on the physical mechanisms that govern the Lagrangian evolution of $l$ and $\omega$ and the essential differences inherent in these two processes. For example, we show that vortex stretching is smaller than material lines stretching, i.e. $\langle\omega_{i}\omega_{j}s_{ij}/\omega^{2}\rangle \,{<}\,\langle l_{i}l_{j}s_{ij}/l^{2}\rangle$, and expand on how this issue is closely related to the predominant alignment of $\omega$ and the intermediate principal strain eigenvector $\lambda_{2}$ of the rate of strain tensor, $s_{ij}$. By focusing on Lagrangian quantities we discern whether these alignments are driven and maintained mainly by vorticity or by strain. In this context, the tilting of $\omega$ and the rotation of the eigenframe $\lambda_{i}$ of the rate of strain tensor $s_{ij}$ are investigated systematically conditioned on different magnitudes of strain, $s^{2}$, and enstrophy, $\omega^{2}$. Further, we infer that viscosity contributes through the term $\nu\omega_{i}\nabla^2\omega_{i}$ to ${\rm D}\omega^{2}/{\rm D}t$, whereas ${\rm D}l^{2}/{\rm D}t$ has no diffusive term. This difference plays a key role in defining the mutual orientation between $\omega$ and $\lambda_{i}$. Viscosity thus contributes significantly to the difference in growth rates of $\langle\omega_{i}\omega_{j}s_{ij}\rangle$ and $\langle l_{i}l_{j}s_{ij}\rangle$.

The full set of velocity derivatives, $\partial u_{i}/\partial x_{j}$, is measured experimentally in a Lagrangian way in quasi-homogeneous isotropic turbulence. This is achieved by applying the three-dimensional particle tracking velocimetry (3D-PTV) technique to an electromagnetically forced flow with $\hbox{\it Re}_{\lambda}\,{\thickapprox}\,50$. Checks based on precise kinematic relations show that the technique presented measures the velocity derivatives with good accuracy. In a study on vorticity, characteristic properties of turbulent flows known from direct numerical simulations are reproduced. These are the positive skewness of the intermediate eigenvalue of the rate of strain tensor, $s_{ij}$, $\langle \Lambda_{2}\rangle \,{>}\,0$, the predominance of vortex stretching over vortex compression, $\langle \omega_{i}\omega_{j}s_{ij}\rangle \,{>}\,0$ and the predominant alignment of vorticity, ${\bm \omega}$, with the intermediate principal axis of strain, ${\bm \lambda}_{2}$. Results on the evolution in time of material lines, ${\bm l}$, compared to vortex lines, ${\bm \omega}$, are presented. They show that the nonlinear interaction of vorticity with the surrounding flow assists viscosity in maintaining this predominant ${\bm \lambda}_{2}$-alignment of vorticity. Lagrangian measurements of enstrophy budget terms suggest that there is no pointwise balancing of production and viscous reduction of enstrophy and that the role played by viscosity is of great importance.

This article presents an experimental study of magnetohydrodynamic convection in a tall vertical slot under the influence of a horizontal magnetic field. The test fluid is an eutectic sodium potassium Na22K78 alloy with a small Prandtl number of Pr ≈ 0:02. The experimental setup covers Rayleigh numbers in the range 103 [lsim ] Ra [lsim ] 8×104 and Hartmann numbers 0 < M < 1600. The effect of the magnetic field on the convective heat transport is determined not only by damping as expected from Joule dissipation but also, for magnetic fields not too strong, the convective heat transfer may be considerably enhanced compared to ordinary hydrodynamic (OHD) flow. Estimates of the isotropy properties of the flow by a four-element temperature probe demonstrate that the increase in convective heat transport accompanies the formation of strong local anisotropy of the turbulent eddies in the sense of an alignment of the main direction of vorticity with the magnetic field. The reduced three-dimensional nonlinearities in non-isotropic flow favour the formation of largescale vortex structures compared to OHD flow, which are more effective for convective heat transport. Along with the formation of quasi-two-dimensional vortex structures, temperature fluctuations may be considerably enhanced in a magnetic field that is not too strong. However, above Hartmann numbers M [gsim ] 400 the formerly strongly time-dependent flow suddenly becomes stationary with an extended region of high convective heat transport at stationary flow. Finally, for very high Hartmann numbers the convective motion is strongly suppressed and the heat transport is reduced to a state close to pure heat conduction.