The purpose of this research was to provide further understanding of turbulent dynamics and heat transfer mechanisms in accelerating flows with thermophysical variations and pressure drops in micron tubes. Direct numerical simulations were conducted to investigate the turbulence to supercritical pressure ${\rm CO}_2$ in heated micron tubes with inner diameter $99.2~\mathrm {\mu }$m. In general, the turbulent heat transfer enhancement/deterioration at supercritical pressure is dominated by variations in thermophysical properties, buoyancy and thermal acceleration; however, the mechanism differs in micron tubes ($d^* < 100\ \mathrm {\mu }$m). The results showed that the pressure drop and scale effect made significant contributions to the development of turbulence flows heated at supercritical pressure in micron tubes, leading to the prominent property change and flow acceleration in the inlet fully developed turbulent flow. The deviation on temperature distribution because of pressure changes was non-negligible. The primary contribution of the acceleration was the decay of a boundary layer, which significantly suppressed the production of turbulence and decreased heat transfer. The acceleration had stabilizing effects on the ejection and sweep motions of the turbulent flow. The high-speed fluid contributed to a new disturbance scenario of the flow with a larger spanwise wavenumber superimposed on existing perturbations. The high-speed streak width in the quasilaminar region was approximately $150$–$160\nu /u_\tau$ in accelerating flow. In the micron tubes, the Reynolds stress events of quadrant Q4 contributed 60 % of the Reynolds stress, greater than those of quadrant Q2.

In this paper the multiscale dynamics of streamwise-rotating channel turbulence is studied through direct numerical simulations. Using the generalized Kolmogorov equation, we find that as rotation becomes stronger, the turbulence in the buffer layer is obviously reduced by the intense spatial turbulent convection. On the contrary, in other layers, the turbulence is strengthened mainly by the modified production peak, the intense spatial turbulent convection and the suppressed forward energy cascades. It is also discovered that under a system rotation, small- and large-scale inclined structures have different angles with the streamwise direction, and the difference is strengthened with increasing rotation rates. The multiscale inclined structures are further confirmed quantitatively through a newly defined angle based on the velocity vector. Through the budget balance of Reynolds stresses and the hairpin vortex model, it is discovered that the Coriolis force and the pressure–velocity correlation are responsible for sustaining the inclined structures. The Coriolis force directly decreases the inclination angles but indirectly induces inclined structures in a more predominant way. The pressure–velocity correlation term is related to the strain rate tensor. Finally, the anisotropic generalized Kolmogorov equation is used to validate the above findings and reveals that the multiscale behaviours of the inclined structures are mainly induced by the mean spanwise velocity gradients.

We study the application of Taylor's frozen hypothesis to the pressure fluctuations in turbulent channels by means of spatio-temporal data from direct numerical simulations with large computational domains up to the friction Reynolds number $R{e_\tau } = 2000$. The applicability of the hypothesis is quantitatively verified by comparing the wavenumber and Taylor (frequency) premultiplied spectra of the pressure fluctuations at each distance y from the wall. Using the local mean velocity $U(y)$ for the hypothesis leads to a large difference between both spectra with a value of $O(50\,{\%})$ for its maximum from the wall to $y/h \approx 0.2$, where h is the channel half-depth. Alternatively, the convection velocity of the pressure fluctuations ${C_p}(y)$, originally defined by Del Álamo & Jiménez (J. Fluid Mech., vol. 640, 2009, pp. 5–26) as a function of y, is investigated and adopted for the hypothesis. It is nearly equal to $U(y)$ from ${y^ + } = 20$ to the channel centre, where ${y^ + } = y{u_\tau }/\nu $, in which ${u_\tau }$ and $\nu $ represent the friction velocity and kinematic viscosity, respectively. For ${y^ + } \le 20$, ${C_p}(y)$ is almost constant with a value of around $12{u_\tau }$. Applying ${C_p}(y)$ for the hypothesis decreases the difference between both spectra down to a value of $O(10\,{\%})$ for its maximum from the wall to $y/h \approx 0.2$. Finally, the difference between the pressure wavenumber and frequency premultiplied spectra near the wall is reduced further via applying a wavenumber-dependent convection velocity. This wavenumber-dependent convection velocity is the linear combination of the convection velocities of the turbulent structures associated with the pressure field weighted by their relative contributions to the pressure variance.

The maximum size and lifetime of an acoustically nucleated cavitation bubble is inversely proportional to the driving frequency and has achieved a limit of approximately 10 MHz. Smaller cavitation bubbles that are critical to microscopic applications require shorter lifetimes that correspond to higher oscillation frequencies. Here, we demonstrate that acoustic cavitation in the 100 MHz range and beyond can be achieved through wave propagation in a solid rather than in a liquid. The cavitation bubble is nucleated at a nano-sized fracture on a glass substrate, and its expansion is driven by a leaky Rayleigh wave, while the inertial collapse is induced by a trailing shock wave. As both waves travel at different velocities, the time interval between these two events is a function of the distance to the source. In this way, we demonstrate experimentally control of the lifetime of the bubbles in a range between 6 and 80 ns, corresponding to oscillation frequencies between 13 and 166 MHz. Our results agree with finite-volume fluid–structure interaction simulations.

Three-dimensional (3-D) wake transitions of a steady flow past two side-by-side circular cylinders are investigated through Floquet analysis and direct numerical simulations (DNS) over the gap-to-diameter ratio $g^*$ up to 3.5 and Reynolds number ${\textit {Re}}$ up to 400. When the flows behind two cylinders form in-phase and anti-phase wakes at large $g^*$, the wake transition is similar to the isolated cylinder counterpart, with the critical ${\textit {Re}}$ for the onset of 3-D transition (${\textit {Re}}_{cr-1}$) happens at around 180. At small $g^*$, 3-D transition becomes interestingly complex due to the distinct characteristics formed in base flows. The ${\textit {Re}}_{cr-1}$ suddenly drops to around 60–100 and forms distinct variation trends with $g^*$. Precisely, the ${\textit {Re}}_{cr-1}$ of the single symmetric wake (SS, $g^*\lessapprox 0.25$) is more than half of the isolated cylinder counterpart due to the large length scale of the SS wake. Only mode A is detected in SS. In the asymmetric single wake (ASS, $g^* \approx 0.25\unicode{x2013}0.6$) and flip-flop wake (FF, $g^* \approx 0.6\unicode{x2013}1.8$), the 3-D transition develops at ${\textit {Re}} \approx 103\unicode{x2013}60$ and 75–60, respectively. The decrease in ${\textit {Re}}_{cr-1}$ with increasing $g^*$ is because of the increased level of wake asymmetry in ASS and irregular vortex shedding in FF. Floquet analysis predicts two new unstable modes, namely mode A$'$ and subharmonic mode C$'$, of ASS. Both modes are transient features in 3-D DNS and the flow eventually saturates into a new 3-D mode, mode ASS. The gap flow of mode ASS is distinctly characterised by its time-independent spanwise waviness structure that is deflected towards different transverse directions with a long wavelength of about $14$ cylinder diameters. The 3-D mode of the FF is irregular both temporally and spatially. Variations of ${\textit {Re}}_{cr-1}$ with $g^*$, the characteristics and the physical mechanisms of each 3-D mode are discussed in this study.

The scaling law of the structure function of Richtmyer–Meshkov (RM) turbulence is investigated both numerically and theoretically. High-fidelity simulations with a minimum-dispersion, adaptive-dissipation scheme are first performed. Results show that the mixing width experiences an exponential growth and the turbulent kinetic energy has a visible $-3/2$ spectrum. The scalar field exhibits a greater degree of intermittency than the velocity field, and also the small-scale statistics suffer a larger influence of large scales. Visible differences in the scaling law of the structure function among the RM turbulence and other types of turbulence are observed, which reveal the unique characteristic of RM turbulence. A phenomenological theory, which gives the spatial and temporal scaling laws of the structure functions of velocity and scalar of RM turbulence, is developed for the first time by introducing an external agent. The spatial scaling exponents of structure functions from simulation deviate from the Kolmogorov exponents, but are quite close to the RM-modified anomalous exponents. This demonstrates the validity of the present phenomenological theory. The temporal scaling exponents of structure functions first meet the RM-modified anomalous exponents, and then approach the Kolmogorov–Obukhov–Corrsin non-intermittent ones.

The effect of the initial condition upon the transport dynamics of miscible flowing fluids in a porous medium is investigated under viscosity and density contrasts. Such flows have attracted significant attention due to their importance in many fields of science and engineering, such as $\mathrm {CO}_2$ sequestration and aquifer remediation. Using high-resolution two-dimensional numerical simulations, we illustrate the impact of viscosity and density contrasts on the temporal evolution of the spreading and mixing quantities. We show that such impact depends on the initial shape of the source distribution where the solute is injected and on the intensity of the horizontal background flux. We find that rates of mixing are dependent on whether the solute is more or less viscous than the ambient fluid, a result usually not taken into consideration in studies on gravity fingering. At higher background flux, the effects due to horizontal viscous fingering dominate over gravitational fingering. Our computational analysis also suggests a non-trivial relationship between mixing and the length of the plume's interface under fingering instabilities. Finally, we show how a stratified permeability field can interact with these sources of instabilities and affect the transport behaviour of the plume.

In impermeable media, a hydraulic fracture can continue expanding even without additional fluid injection if its volume exceeds the limiting volume of a hydrostatically loaded radial fracture. This limit depends on the mechanical properties of the surrounding solid and the density contrast between the fluid and the solid. We show that two dimensionless numbers characterize self-sustained fracture growth. The first is a buoyancy factor that compares the total released volume to the volume of a hydrostatically loaded radial fracture to determine whether buoyant growth occurs. The second number is the dimensionless viscosity of a radial fracture when buoyant effects become of order one. Notably, this dimensionless viscosity depends on the rate at which the fluid volume is released, indicating that both the total volume and release history impact self-sustained buoyant growth. We identify six well-defined propagation histories based on these two dimensionless numbers. Their growth evolves between distinct limiting regimes of radial and buoyant propagation, resulting in different fracture shapes. Notably, our findings reveal two growth rates depending on the dominant energy dissipation mechanism (viscous flow versus fracture creation) in the fracture head. For finite values of material toughness, the toughness-dominated limit represents a late-time solution for all fractures in growth rate and head shape (possibly reached only at a very late time). The viscosity-dominated limit can appear at intermediate times. Our three-dimensional simulations confirm the predicted scalings. This contribution highlights the importance of the entire propagation and release history for accurate analysis of buoyant hydraulic fractures.

The present experimental study investigated the dynamics of single- and multi-port gaseous jet diffusion flames exposed to acoustic excitation via a standing wave situated in a closed waveguide at atmospheric pressure. High-speed imaging of the oscillatory flame was analysed via proper orthogonal decomposition (POD), revealing distinct signatures in both mode shapes and phase portraits for transitions in the acoustically coupled combustion process. For Reynolds numbers between 20 and 100, and for low to moderate forcing amplitudes, the flame exhibited sustained oscillatory combustion (SOC) that was highly coupled to the acoustic forcing. Frequency analysis of the temporal POD modes accurately recovered the forcing frequency and its higher harmonics. At higher forcing amplitudes, a multi-frequency response was observed, resulting from a combination of the forcing frequency and much lower frequency oscillations due to periodic lift-off and reattachment (PLOR) of the flame, preceding a transition to flame blow-off (BO). For both single- and triple-jet flames, transitions from SOC to PLOR to BO were characterized by significant alterations in primary modal energetic content, deflection and eventual smearing in phase portraits, and the development of additional frequencies in modal spectra, although transitional behaviour for the triple jet flames involved additional complexity in the dynamics due to its structure. These features provide the potential for the development of reduced-order models that can characterize and predict acoustically coupled combustion behaviour.

Pressure-gradient-driven flows through sinusoidal channels have been studied. The analysis was carried out up to the formation of secondary nonlinear states and spanned a range of low and moderate Reynolds numbers. Direct numerical simulations were used to identify and determine the properties of steady as well as non-stationary, two-dimensional (2-D) and three-dimensional secondary flows. Our results indicate the existence of several distinct solution types. Two-dimensional, stationary flows with periodicity determined by the corrugation represent the first type. The second type is associated with the appearance of 2-D oscillatory flows arising from the onset of unstable travelling waves. Such oscillatory solutions are generally out of phase with the wall corrugation but could be in phase in special cases determined by the ratio of the critical disturbance wavelength and the channel corrugation wavelength. Consequently, several distinct types of time-dependent solutions are possible. The third type of solution results from the centrifugal effect caused by wall curvature and leads to three-dimensionalization of the flow through the onset of stationary streamwise vortices. Finally, various states resulting from the interaction of different solution types are possible. We examine those states and present a bifurcation diagram illustrating the formation of some of them. The results presented in this paper might help with the development of small-scale flow measurement and detection devices operating at low and moderate Reynolds numbers, as well as in the use of wall topographies for the intensification of mixing in flows with moderate, subturbulent Reynolds numbers.

The coupled effects of thermodynamic and hydrodynamic instabilities are studied during viscous fingering (VF). We introduced a modified Cahn–Hilliard phase-field model in conjunction with the Korteweg force in the classical VF model and derived consistent governing equations. The free energy of the partially miscible system is described using a modified Flory–Huggins model, which allows us to investigate the temporal evolution of spatial inhomogeneities. The mass flux in the Cahn–Hilliard equations is modified according to modern diffusion theory. The governing equations have been solved through an in-house model implementation using the COMSOL multiphysics software. We successfully demonstrated the transition from the finger-like structures to the droplet formation during spinodal decomposition as demonstrated experimentally in the literature. Our results are also in agreement with earlier numerical results obtained using a classical Landau type mixing energy. We further systematically studied the effects of the Margules parameter (interaction parameter) and the gradient parameter, which is associated to the thermodynamic length scale and capillary number on the VF. Aysmmetric features of the binary mixture are also investigated showing a stronger thermodynamic effect on the system with increasing phase separation and, hence, droplet formation.

For turbulent bubbly flows, multi-phase simulations resolving both the liquid and bubbles are prohibitively expensive in the context of different natural phenomena. One example is breaking waves, where bubbles strongly influence wave impact loads, acoustic emissions and atmospheric-ocean transfer, but detailed simulations in all but the simplest settings are infeasible. An alternative approach is to resolve only large scales, and model small-scale bubbles adopting sub-resolution closures. Here, we introduce a large eddy simulation smoothed particle hydrodynamics (SPH) scheme for simulations of bubbly flows. The continuous liquid phase is resolved with a semi-implicit isothermally compressible SPH framework. This is coupled with a discrete Lagrangian bubble model. Bubbles and liquid interact via exchanges of volume and momentum, through turbulent closures, bubble breakup and entrainment, and free-surface interaction models. By representing bubbles as individual particles, they can be tracked over their lifetimes, allowing closure models for sub-resolution fluctuations, bubble deformation, breakup and free-surface interaction in integral form, accounting for the finite time scales over which these events occur. We investigate two flows: bubble plumes and breaking waves, and find close quantitative agreement with published experimental and numerical data. In particular, for plunging breaking waves, our framework accurately predicts the Hinze scale, bubble size distribution, and growth rate of the entrained bubble population. This is the first coupling of an SPH framework with a discrete bubble model, with potential for cost-effective simulations of wave–structure interactions and more accurate predictions of wave impact loads.

Theory and computations are applied to assess the hydrodynamic permeability of cavity- doped hydrogels, central to a variety range of contemporary technological applications. Direct volume-averaging is undertaken in a two-dimensional, Brinkman-hydrodynamic context to test an ensemble-averaging methodology recently proposed for the ion permeability of such media. In two dimensions, the ensemble-averaging integral furnishes a pre-factor $2$ linking the pressure dipole strength of a single inclusion in an unbounded continuum to the effective hydrodynamic permeability of a composite with small inclusion area fraction. The factor is verified by direct computations for dilute simple-square arrays of (cylindrical) inclusions. At area fractions up to the close-packing limit, computations address the hydrodynamic interactions. The theory is shown to accurately predict the effective hydrodynamic permeability of physically relevant composites (Brinkman length of the continuous phase $\ell$ smaller than the inclusion radius $a$) for area fractions $\phi \lesssim {\rm \pi}/ 9 \approx 0.3$. Computations for random ensembles demonstrate that the dilute theory may be extended to higher area fractions by drawing on Rayleigh's self-consistent approximation when the continuous-phase permeability places the continuous-phase flow well into the Darcy regime ($a / \ell \gtrsim 10$). Computations also demonstrate, similarly to Rayleigh theories for scalar diffusion, that microstructural order has a very weak influence on the effective permeability when $\phi \lesssim {\rm \pi}/ 9$ with $\ell / a \lesssim 1$ (Darcy hydrodynamic interactions). Finally, a cursory examination is undertaken of the fluid velocity and its fluctuations arising from shear-viscosity heterogeneity in media with perfectly uniform permeability $\ell ^2$.

An experiment on convective flows induced by the dielectrophoretic force was performed under the microgravity condition provided during a sounding rocket flight. The dielectrophoretic force possesses a non-conservative term that can be seen as resulting from an electric gravity. That gravity can be responsible for an electric Rayleigh–Bénard convection between a hot inner cylinder and a cold outer cylinder when an electric field is applied in the radial direction. Four cells with independent temperature and electric field controls allowed the investigation of eight different values of the electric Rayleigh number relatively close to the onset of the thermo-electric instability. A linear stability analysis is performed to predict the stability threshold and the evolution of the growth rate of the instability. The three-dimensional structure of the flow is captured by simultaneous particle image velocimetry and by shadowgraphy. The amplitude of the instability modes and the time evolution of the flow is analysed, and various methods are proposed to extrapolate the experimental critical value of the electric Rayleigh number for the onset of convection. The measured critical electric Rayleigh number is in agreement with the prediction of the linear stability theory. The comparison of the new experimental results with previous ones from parabolic flight campaigns highlights the importance of long-term microgravity for the achievement of thermal convection at low values of the control parameters.

Microfluidic flow focusing is a versatile method for the production of monodisperse microbubbles for biomedical applications involving ultrasound. Existing studies propose several theoretical models to predict bubble size and production rate as a function of the liquid and gas flow rate. Yet, they typically do not include physical fluid parameters such as density, viscosity and surface tension. Here, we present an exhaustive experimental and numerical investigation of the influence of physical properties of the gas and liquid, and of the channel geometry on bubble size and production rate. We find a particularly strong effect of (i) gas density on the production rate and (ii) liquid viscosity on the bubble size. We further discuss our findings within the context of existing theoretical models to reflect on gaps in our current understanding of the fluid mechanics of bubble formation by flow focusing.

Aerobreakup of drops is a fundamental two-phase flow problem that is essential to many spray applications. A parametric numerical study was performed by varying the gas stream velocity, focusing on the regime of moderate Weber numbers, in which the drop deforms to a forward bag. When the bag is unstable, it inflates and disintegrates into small droplets. Detailed numerical simulations were conducted using the volume-of-fluid method on an adaptive octree mesh to investigate the aerobreakup dynamics. Grid-refinement studies show that converged three-dimensional simulation results for drop deformation and bag formation are achieved by the refinement level equivalent to 512 cells across the initial drop diameter. To resolve the thin liquid sheet when the bag inflates, the mesh is refined further to 2048 cells across the initial drop diameter. The simulation results for the drop length and radius were validated against previous experiments, and good agreement was achieved. The high-resolution results of drop morphological evolution were used to identify the different phases in the aerobreakup process, and to characterize the distinct flow features and dominant mechanisms in each phase. In the early time, the drop deformation and velocity are independent of the Weber number, and a new internal-flow deformation model, which respects this asymptotic limit, has been developed. The pressure and velocity fields around the drop were shown to better understand the internal flow and interfacial instability that dictate the drop deformation. Finally, the impact of drop deformation on the drop dynamics was discussed.

We investigate the coupling effect of buoyancy and shear based on an annular centrifugal Rayleigh–Bénard convection (ACRBC) system in which two cylinders rotate with an angular velocity difference. Direct numerical simulations are performed in a Rayleigh number range $10^6\leq Ra\leq 10^8$, at fixed Prandtl number $Pr=4.3$, inverse Rossby number $Ro^{-1}=20$, and radius ratio $\eta =0.5$. The shear, represented by the non-dimensional rotational speed difference $\varOmega$, varies from $0$ to $10$, corresponding to an ACRBC without shear and a radially heated Taylor–Couette flow with only the inner cylinder rotating, respectively. A stable regime is found in the middle part of the interval for $\varOmega$, and divides the whole parameter space into three regimes: buoyancy-dominated, stable and shear-dominated. Clear boundaries between the regimes are given by linear stability analysis, meaning the marginal state of the flow. In the buoyancy-dominated regime, the flow is a quasi-two-dimensional flow on the $r\varphi$ plane; as shear increases, both the growth rate of instability and the heat transfer are depressed. In the shear-dominated regime, the flow is mainly on the $rz$ plane. The shear is so strong that the temperature acts as a passive scalar, and the heat transfer is greatly enhanced. The study shows that shear can stabilize buoyancy-driven convection, makes a detailed analysis of the flow characteristics in different regimes, and reveals the complex coupling mechanism of shear and buoyancy, which may have implications for fundamental studies and industrial designs.

The lift generation mechanism of leading-edge vortex (LEV) in the case of a pitching and plunging plate is studied using an experimental approach and the improved discrete vortex method in this research. A formation condition of the secondary structure is introduced into the traditional discrete vortex method to compensate for the shortcomings in the simulation of the viscous effect between LEV and plate. The simulation of the secondary structure helps the improved method perform better in flow-field reconstruction and lift prediction. Accordingly, the lift generation mechanism of the LEV and influence of the secondary structure are studied. The lift contribution of the vortex structure is isolated and linearly decomposed into two parts according to sources of flow field: the quasi-potential flow part and the vortex-induced flow part. The vortex lift is defined as the lift contribution of the vortex structure in vortex-induced flow, which gives a new insight into the production of lift of the LEV. The lift generation mechanism through the discrete vortex method is verified and extended in viscous flow through experimental measurement. In addition, a vortex lift indicator based on the reverse flow of the LEV is proposed to examine the change of vortex lift in experimental measurement. The flow mechanism for the decline of vortex lift for different maximum effective angles of attack is revealed based on the vortex lift indicator. Furthermore, for the LEV-dominating flow, the indicator can also be applied in estimating the maximum value and corresponding critical time of overall lift in experiments.

The present work is devoted to the analysis of drop impact on a deep liquid pool, focusing on the high-energy splashing regimes caused by large raindrops at high velocities. Such cases are characterized by short time scales and complex mechanisms, thus they have received very little attention until now. The BASILISK open-source solver is used to perform three-dimensional direct numerical simulations. The capabilities of octree adaptive mesh refinement techniques enable capturing of the small-scale features of the flow, while the volume of fluid approach combined with a balanced-force surface-tension calculation is applied to advect the volume fraction of the liquids and reconstruct the interfaces. The numerical results compare well with experimental visualizations: both the evolution of crown and cavity, the emanation of ligaments, the formation of bubble canopy and the growth of a downward-moving spiral jet that pierces through the cavity bottom, are correctly reproduced. Reliable quantitative agreements are also obtained regarding the time evolution of rim positions, cavity dimensions and droplet distributions through an observation window. Furthermore, simulation gives access to various aspects of the internal flows, which allows us to better explain the observed physical phenomena. Details of the early-time dynamics of bubble ring entrapment and splashing performance, the formation/collapse of bubble canopy and the spreading of drop liquid are discussed. The statistics of droplet size show the bimodal distribution in time, corroborating distinct primary mechanisms of droplet production at different stages.

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.