The phenomenon of the ‘coil-stretch’ (C-S) transition, wherein a long-chain polymer initially in a coiled state undergoes a sudden configuration change to become fully stretched under steady elongational flows, has been widely recognized. This transition can display intricate hysteresis behaviours under specific critical conditions, giving rise to unique rheological characteristics in dilute polymer solutions. Historically, microscopic stochastic models and Brownian dynamics simulations have shed light on the underlying mechanisms of the transition by uncovering bistable configurations of polymer chains. Following the initial work by Cerf (J. Chem. Phys., vol. 20, 1952, pp. 395–402), we introduce a continuum model in this study to investigate the C-S transition in a constant uniaxial elongational flow. Our approach involves approximating the unfolding process of the polymer chain as an axisymmetric deformation of an elastic particle. We make the assumption that the particle possesses uniform material properties, which can be represented by a nonlinear, strain-hardening constitutive equation to replicate the finite extensibility of the polymer chain. Subsequently, we analytically solve for the steady-state deformation using a polarization method. By employing this reduced model, we effectively capture the C-S transition and establish its specific correlations with material and geometric properties. The hysteresis phenomena can be comprehended through a force-balance analysis, which involves comparing the externally applied viscous forces with the intrinsic elastic responsive forces. We demonstrate that our model, while simple, unveils rich elastohydrodynamics of the C-S transition.

Direct numerical simulations of hypersonic turbulent Couette flows are performed for top-wall Mach numbers of 6, 7 and 8, inspired by non-reactive high-enthalpy wind tunnel free-stream conditions, with the goal of analysing the physical processes driving the sub-filter-scale (SFS) stresses to inform development of large-eddy simulation techniques for hypersonic wall-bounded flows. Semi-local scaling laws collapse mean profiles and second-order turbulent statistics well, in spite of the strong wall-normal gradients of temperature and density. On the other hand, the SFS shear stresses exhibit an unexpected profile characterized by a region of pronounced shear stress deficit, which becomes more pronounced for higher Mach numbers. Instantaneous visualizations suggest that the SFS shear stress deficit is induced by the counter-gradient resolved momentum transport driven by residual velocity motions at the interface between high-density low-speed streaks being ejected away from the wall, and low-density high-speed ones replacing the displaced fluid, qualifying this as a compressibility effect. It is shown that the SFS shear stresses are primarily driven by second-order interactions between residual velocities, in spite of their triply nonlinear nature. This, in turn, motivated a statistical quadrant analysis revealing the presence of a SFS shear stress deficit, that is, SFS processes driving momentum transport of the resolved field towards the top wall. Additionally, higher-Reynolds-number simulations reveal that there is an upper limit of spatial filter widths to resolve large-scale structures, and such deficit is observable at any Reynolds number when a reasonable spatial filter is applied.

This work is a numerical and experimental study of a rectangular thin plate undergoing stall flutter at Mach 0.8. This constitutes one of the first studies of this kind where three-dimensionality is fully implemented in a numerical simulation including the test-section effects characterizing wind-tunnel experiments. In order to break down the fluid–structure interaction to its main driving phenomena, an aerodynamic model is proposed that is based on computationally inexpensive steady-state simulations. Two types of dynamic instability are observed in the numerical simulations; Flutter by mode coalescence is promoted at zero flow incidence, however, high bending precludes this from happening for higher values of angle of attack. Stall flutter is instead a nonlinear one-degree type of instability. Both of these instability mechanisms can be explained in terms of hysteretic behaviour of the pressure distribution, which becomes more pronounced at high angles of attack, when a large separation region is formed. Tests were conducted employing titanium alloy plates in order to survive the aerodynamic loads characterizing the wind-tunnel initial transient. However, due to wall interference, high bending was promoted so that the internal stress exceeded the yield values before flutter could be measured. Numerical simulations were in general agreement with the experiment in terms of both amplitude and oscillation frequency.

Soft intertwined channel systems are frequently found in fluid flow networks in nature. The passage geometry of these systems can deform due to fluid flow, which can cause the relationship between flow rate and pressure drop to deviate from the Hagen–Poiseuille linear law. Although fluid–structure interactions in single deformable channels have been extensively studied, such as in Starling's resistor and its variations, the flow transport capacity of an intertwined channel with multiple self-intersections (a ‘hydraulic knot’), is still an open question. We present experiments and theory on soft hydraulic knots formed by interlinked microfluidic devices comprising two intersecting channels separated by a thin elastomeric membrane. Our experiments show flow–pressure relationships similar to flow limitation, where the limiting flow rate depends on the knot configuration. To explain our observations, we develop a mathematical model based on lubrication theory coupled with tension-dominated membrane deflections that compares favourably with our experimental data. Finally, we present two potential hydraulic knot applications for microfluidic flow rectification and attenuation.

This work investigates the compressible turbulence induced by Richtmyer–Meshkov (RM) instability using high-resolution Navier–Stokes simulations. Special attention is paid to the characteristics of RM turbulence including the mixing width growth, the turbulent kinetic energy (TKE) decay, the mixing degree, inhomogeneity and anisotropy. Three distinct initial perturbation spectra are designed at the interface to reveal the initial condition imprint on the RM turbulence. Results show that cases with initial large-scale perturbations present a stronger imprint on statistical characteristics and also a quicker growth of mixing width, whereas cases with small-scale perturbations present a faster TKE decay, greater mixing level, higher isotropy and homogeneity. A thorough analysis on the inter-scale energy transfer in RM turbulence is also presented with the coarse-graining approach that exposes the two subfilter-scale (SFS) energy fluxes (i.e. deformation work and baropycnal work). A strong correlation between the nonlinear model of baropycnal work (Fluids, 4(2), 2019) and the simulation results is confirmed for the first time, demonstrating its potential in modelling the RM turbulence. Two primary mechanisms of baropycnal work (the straining and baroclinic generation processes) are explored with this nonlinear model. The evolutions of two SFS energy fluxes exhibit distinct behaviours at various filter scales, in different flow regions and under various flow motions (strain and rotation). It is found that all three cases share the common inter-scale energy transfer dynamics, which is important for modelling the RM turbulence.

We present a systematic simulation campaign to investigate the pairwise interaction of two mobile, monodisperse particles submerged in a viscous fluid and subjected to monochromatic oscillating flows. To this end, we employ the immersed boundary method to geometrically resolve the flow around the two particles in a non-inertial reference frame. We neglect gravity to focus on fluid–particle interactions associated with particle inertia and consider particles of three different density ratios aligned along the axis of oscillation. We systematically vary the initial particle distance and the frequency based on which the particles show either attractive or repulsive behaviour by approaching or moving away from each other, respectively. This behaviour is consistently confirmed for the three density ratios investigated, although particle inertia dictates the overall magnitude of the particle dynamics. Based on this, threshold conditions for the transition from attraction to repulsion are introduced that obey the same power law for all density ratios investigated. We furthermore analyse the flow patterns by suitable averaging and decomposition of the flow fields and find competing effects of the vorticity induced by the fluid–particle interactions. Based on these flow patterns, we derive a circulation-based criterion that provides a quantitative measure to categorize the different cases. It is shown that such a criterion provides a consistent measure to distinguish the attractive and repulsive arrangements.

Exact steady eastward-moving vortex-dipole solutions, the equatorial modons, are constructed in the asymptotic limit of low divergence and small temperature variations in the thermal rotating shallow water (TRSW) model on the equatorial beta-plane. This regime is known to be relevant for the tropical atmosphere. The model itself is a generalization, allowing for horizontal temperature gradients, of the classical rotating shallow water model. The asymptotic modons can carry a temperature anomaly and exist also on the inhomogeneous temperature background. The modon configurations are then used to initialize numerical simulations, in order to check whether such coherent structures can exist in the full TRSW model. The results show that this is, indeed, the case. The parameter regimes and limitations on the structure of the temperature anomaly inside, in order for the modons to persist, are established. It is also shown that the modons keep their coherence even while evolving on the background of meridionally inhomogeneous temperature fields, or while interacting with sharp temperature fronts. A general scenario of disaggregation of the modons, if they are out of the stability domain, is exhibited and analysed.

We investigate experimentally the effects of micro-grooves on the development of a zero pressure gradient turbulent boundary layer at two different values of the friction Reynolds number. We consider both the well-known streamwise aligned riblets as well as wavy riblets, characterized by a sinusoidal pattern in the mean flow direction. Previous investigations by the authors showed that sinusoidal riblets yield larger values of drag reduction with respect to the streamwise aligned ones. We perform new particle image velocimetry experiments on wall-parallel planes to get insights into the effect of the sinusoidal shape on the near-wall organisation of the boundary layer and the structures responsible for the friction drag reduction and the turbulence generation. Conditional averages, aimed at identifying the topology of the low-speed streaks in the turbulent boundary layer, reveal that the flow is highly susceptible to wall manipulation. This is particularly evident in the cases that are associated with greater values of drag reduction. The results suggest a fragmentation and/or weakening of the streaks in the sinusoidal cases, that is triggered by the larger values of the wall-normal vorticity found at the streaks’ edges. The results are also confirmed by applying the variable interval spatial averaging events eduction technique. The turbulent kinetic energy budget also shows that the sinusoidal geometry significantly attenuates the turbulence production, hence supporting the idea of the manipulation of the turbulence regeneration cycle.

We report an experimental study of the formation and evolution of laminar thermal structures generated by a small heat source, with a focus on their correlation to the thermal boundary layer and effects of heating time $t_{heat}$. The experiments are performed over the flux Rayleigh number ($Ra_f$) range $2.1\times 10^6 \leq Ra_f \leq 3.6\times 10^{7}$ and the Prandtl number ($Pr$) range $28.6 \leq Pr \leq 904.7$. The corresponding Rayleigh number ($Ra= t_{heat}\,Ra_{f}/\tau _0\,Pr$) range is $900 \leq Ra \leq 4\times 10^{4}$, where $\tau _0$ is a diffusion time scale. For thermal structures generated by continuous heating (i.e. starting plumes), their formation process exists three characteristic times that are well reflected by changes in the thermal boundary layer thickness. These characteristic times, denoted as $t_{emit}$, $t_{recover}$ and $t_{static}$, correspond to the moments when the plume emission begins and completes, and when the thermal boundary layer becomes quasi-static, respectively. Their $Ra_f$–$Pr$ dependencies are found to be $t_{emit}/\tau _0\sim Ra_f^{-0.41}\,Pr^{0.41}$, $t_{recover}/\tau _0\sim Ra_f^{-0.48}\,Pr^{0.48}$ and $t_{static}/\tau _0\sim Ra_f^{-0.49}\,Pr^{0.33}$, respectively. Thermal structures generated by finite $t_{heat}$ exhibit similar evolution dynamics once $t_{heat} \ge t_{emit}$, with the accelerating stage behaving like starting plumes and the decay stage like thermals (i.e. a finite amount of buoyant fluids). It is further found that their maximum rising velocity experiences a transition in the $Ra$-dependence from $Ra$ to $(Ra\ln Ra)^{0.5}$ at $Ra \simeq 6000$; and their maximum acceleration reaches the value of starting plumes at $t_{heat}\simeq t_{recover}$, and remains unchanged for larger $t_{heat}$. In particular, the maximum rising velocity for the cases with $t_{heat} = t_{recover}$ follows a scaling relation $Ra_f^{0.37}\,Pr^{-0.37}$, in contrast to the relation $Ra_f^{0.48}\,Pr^{-0.48}$ for starting plumes. This study provides a more comprehensive understanding of laminar thermal structures, which are relevant to a range of processes in nature and laboratory systems such as Rayleigh–Bénard convection.

Detailed observations of the velocities of Jovian vortices exist at only one height in the atmosphere, so their vertical structures are poorly understood. This motivates this study that computes stable three-dimensional, long-lived planetary vortices that satisfy the equations of motion. We solve the anelastic equations with a high-resolution pseudo-spectral method using the observed Jovian atmospheric temperatures and zonal flow. We examine several families of vortices and find that constant-vorticity vortices, which have nearly uniform vorticity as a function of height, and horizontal areas that go to zero at their tops and bottoms, converge to stable vortices that look like the Great Red Spot (GRS) and other Jovian anticyclones. In contrast, the constant-area vortices proposed in previous studies, which have nearly uniform areas as a function of height, and vertical vorticities that go to zero at their tops and bottoms, are far from equilibrium, break apart, and converge to constant-vorticity vortices. Our late-time vortices show unexpected properties. Vortices that are initially non-hollow become hollow (i.e. have local minima of vertical vorticity at their centres), which is a feature of the GRS that cannot be explained with two-dimensional simulations. The central axes of the final vortices align with the planetary spin axis even if they align initially with the local direction of gravity. We present scaling laws for how vortex properties change with the Rossby number and other non-dimensional parameters. We prove analytically that the horizontal mid-plane of a stable vortex must lie at a height above the top of the convective zone.

This experimental study investigates the fluid–structure–surface interactions of a flexibly mounted rigid plate in axial flow, focusing on flow-induced vibration (FIV) response and vortex dynamics of the system within a reduced velocity range of $U^*=0.29\unicode{x2013}8.73$, corresponding to a Reynolds number range of $Re=518\unicode{x2013}15\,331$. The plate, with one and two degrees of freedom (DoFs) for pitching and plunging oscillations, is examined at various submerged heights near the free surface. Results show that the plate exhibits divergence instability at low reduced velocities in both 1DoF and 2DoF systems. As the flow velocity surpasses a critical reduced velocity, periodic limit-cycle oscillations (LCOs) occur, increasing in amplitude until a second critical reduced velocity is reached. Beyond this point, LCOs are suppressed, and the plate experiences an increased static divergence angle with further flow velocity increase. The proximity to the free surface significantly influences the FIV response, with decreasing submerged heights leading to reduced LCO amplitudes and a shift of instabilities to higher reduced velocities. Vortex dynamics are analysed using time-resolved volumetric particle tracking velocimetry and hydrogen bubble flow visualisation. The analysis reveals disruptions in the symmetric flow field near the free surface, causing elongation and fragmentation of vortices in the wake of the plate, as well as vortex coupling. Proper orthogonal decomposition (POD) identifies dominant coherent structures, including leading-edge and trailing-edge vortices, captured in the first and second paired modes. On the other hand, higher POD modes capture the interaction of vortices in the wake and near the free surface.

Turbulent flows over porous lattices consisting of rectangular cuboid pores are investigated using scale-resolving direct numerical simulations. Beyond a certain threshold which is primarily determined by the wall-normal Darcy permeability, ${{\mathsf{K}}_y}$, near-wall turbulence transitions from its canonical regime, marked by the presence of streak-like structures, to another marked by the presence of Kelvin–Helmholtz-like (K–H-like) spanwise-coherent structures. The threshold agrees well with that previously established in studies where permeable-wall boundary conditions had been used as surrogates for a porous substrate (Gómez-de Segura & García-Mayoral, J. Fluid Mech., vol. 875, 2019, pp. 124–172). In the smooth-wall-like regime, none of the investigated substrates demonstrate any reduction in drag relative to a smooth-wall flow. At the permeable surface, a notable component of the flow is that which adheres to the pore geometry and undergoes modulation by the turbulent scales of motions due to the interaction mechanism described by Abderrahaman-Elena et al. (J. Fluid Mech., vol. 865, 2019, pp. 1042–1071). Its resulting effect can be quantified in terms of an amplitude modulation (AM) using the approach of Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337). This pore-coherent flow component persists throughout the porous substrate, highlighting the importance of a given substrate's microstructure in the presence of an overlying turbulent flow. This geometry-related aspect of the flow is not accounted for when continuum-based models for a porous medium or effective representations of them, such as wall boundary conditions, are used. The intensity of the AM effect is enhanced in the K–H-like regime and becomes strengthened with larger permeability. As a result, structured porous materials may be designed to exploit or mitigate these flow features depending upon the intended application.

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

The slamming wave force and pressure variabilities for monopile wave impacts are studied as functions of wave breaking shape and transverse perturbations on the breaking wave front. The impacting wave topology is characterized as slosh, flip-through, $\varOmega$, overturning and fully broken. Fifty test repetitions are conducted for each type of wave impact to assess the variability of force impulse, force and pressure. The results for the unperturbed cases show that the slamming force is highest among the nominal slosh, flip-through and $\varOmega$ tests, and that the slamming force variability is highest for the first two. We demonstrate that the slamming force and pressure variabilities decrease notably after selecting and regrouping the tests by similar crest heights and temporal slopes measured at an upstream wave gauge. The group representing $\varOmega$ wave impacts shows the largest mean slamming force and peak pressure, and their variability is the highest among all groups. Further, the effect of lateral perturbations on the pressure, force and impulse variabilities is investigated. Due to the perturbations, the slamming pressure variability for the wave impacts in which the wave front hits the monopile surface increases significantly. The variability of the slamming force is also increased for the perturbed impacts; however, it is smaller than the slamming pressure variability. The force impulse variability shows a low sensitivity to perturbations, and its magnitude is smaller than that of the force variability. Finally, the slamming pressure using fifteen pressure sensors for five selected events is studied. For these tests, oscillations at frequencies associated with structural or bubble oscillations are seen. Further, the air entertainment is documented through video recordings.

An experimental study in a wind tunnel is presented to explore the wake of a floating wind turbine subjected to harmonic side-to-side and fore–aft motions under laminar inflow conditions. The wake recovery is analysed as a function of the frequency of motion $f_p$, expressed by the rotor-based Strouhal number, $St = f_p D / U_{\infty }$ ($D$ is the rotor diameter, $U_{\infty }$ the inflow wind speed). Our findings indicate that both directions of motion accelerate the transition to the far-wake compared with the fixed turbine. The experimental outcomes confirm the computational fluid dynamics results of Li et al. (J. Fluid Mech., vol. 934, 2022, p. A29) showing that sideways motions lead to faster wake recovery, especially for $St \in [0.2, 0.6]$. Additionally, we find that fore–aft motions also lead to better recovery for $St \in [0.3, 0.9]$. The recovery is closely linked to nonlinear spatiotemporal dynamics found in the shear layer region of the wake. For both directions of motion and $St \in [0.2, 0.55]$, the noisy wake dynamics lock in to the frequency of the motion. In this synchronised-like state, sideways motions result in large coherent structures of meandering, and fore–aft movements induce coherent pulsing of the wake. For fore–aft motion and $St \in [0.55, 0.9]$, the wake shows a more complex quasiperiodic dynamic, namely, a self-generated meandering mode emerges, which interacts nonlinearly with the excitation frequency $St$, as evidenced by the occurrence of mixing components. The coherent structures grow nonlinearly, enhance wake mixing and accelerate the transition to the far-wake, which, once reached, exhibits universal behaviour.

Dense suspensions of solid particles in viscous liquid are ubiquitous in both industry and nature, and there is a clear need for efficient numerical routines to simulate their rheology and microstructure. Particles of micron size present a particular challenge: at low shear rates, colloidal interactions control their dynamics while at high rates, granular-like contacts dominate. While there are established particle-based simulation schemes for large-scale non-Brownian suspensions using only pairwise lubrication and contact forces, common schemes for colloidal suspensions generally are more computationally costly and thus restricted to relatively small system sizes. Here, we present a minimal particle-based numerical model for dense colloidal suspensions that incorporates Brownian forces in pairwise form alongside contact and lubrication forces. We show that this scheme reproduces key features of dense suspension rheology near the colloidal-to-granular transition, including both shear thinning due to entropic forces at low rates and shear thickening at high rates due to contact formation. This scheme is implemented in LAMMPS, a widely used open source code for parallelised particle-based simulations, with a runtime that scales linearly with the number of particles, making it amenable for large-scale simulations.

We performed numerical simulations of a homogeneous swarm of bubbles rising at large Reynolds number, $Re=760$, with volume fractions ranging from 1 % to 10 %. We consider a simplified model in which the interfaces are not resolved, but which allows us to simulate flows with a large number of bubbles and to emphasize the interactions between bubble wakes. The liquid phase is described by solving, on an Eulerian grid, the Navier–Stokes equations, including sources of momentum which model the effect of the bubbles. The dynamics of each bubble is determined within the Lagrangian framework by solving an equation of motion involving the hydrodynamic forces exerted by the fluid accounting for the correction of the fictitious self-interaction of a bubble with its own wake. The comparison with experiments shows that this coarse-grained simulations approach can reliably describe the dynamics of the resolved flow scales. We use conditional averaging to characterize the mean bubble wakes and obtain in particular the typical shear imposed by the rising bubbles. On the basis of the spectral decomposition of the energy budget, we observe that the flow is dominated by production at large scales and by dissipation at small scales and we rule out the presence of an intermediate range in which the production and dissipation are locally in balance. We propose that the $k^{-3}$ subrange of the energy spectra results from the mean shear rate imposed by the bubbles, which controls the rate of return to isotropy.

A dense particle suspension under shear may lose its uniform state to large local density and stress fluctuations, which challenge the mean-field description of the system. Here, we explore the novel dynamics of a non-Brownian suspension under orbital oscillations, where localized density waves along the flow direction appear beyond an excitation frequency threshold and self-organize into a hexagonal pattern across the system. The spontaneous occurrence of the inhomogeneity pattern arises from a coupling between particle advection and the shear-thickening nature of the suspension. Through linear stability analysis, we show that they overcome the stabilizing effects of particle pressure at sufficient particle volume fraction and oscillation frequency. In addition, the long-standing density waves degenerate into random fluctuations when replacing the free surface with rigid confinement. It indicates that the shear-thickened state is intrinsically heterogeneous, and the boundary conditions are crucial for developing local disturbance.

In this article, we present direct numerical simulation results for the expansion of spherical cap bubbles attached to a rigid wall due to a sudden drop in the ambient pressure. The critical pressure drop beyond which the bubble growth becomes unstable is found to match well with the predictions from classical theory of heterogeneous nucleation imposing a quasi-static bubble evolution. When the pressure drop is significantly higher than the critical value, a liquid microlayer appears between the bubble and the wall. In this regime, the interface outside the microlayer grows at an asymptotic velocity that can be predicted from the Rayleigh–Plesset equation, while the contact line evolves with another asymptotic velocity that scales with a visco-capillary velocity that obeys the Cox–Voinov law. In general, three distinctive regions can be distinguished: the region very close to the contact line where dynamics is governed by visco-capillary effects, an intermediate region controlled by inertio-viscous effects away from the contact line yet inside the viscous boundary layer, and the region outside the boundary layer dominated by inertial effects. The microlayer forms in a regime where the capillary effects are confined in a region much smaller than the viscous boundary layer thickness. In this regime, the global capillary number takes values much larger then the critical capillary number for bubble nucleation, and the microlayer height is controlled by viscous effects and not surface tension.

We study the statistically steady states of the forced dissipative three-dimensional homogeneous isotropic turbulence at scales larger than the forcing scale in real separation space. The probability density functions (p.d.f.s) of longitudinal velocity difference at large separations are close to, but deviate from, Gaussian, measured by their non-zero odd parts. The analytical expressions of the third-order longitudinal structure functions derived from the Kármán–Howarth–Monin equation prove that the odd-part p.d.f.s of velocity differences at large separations are small but non-zero. Specifically, when the forcing effect in the displacement space decays exponentially as the displacement tends to infinity, the odd-order longitudinal structure functions have a power-law decay with an exponent of $-$2, implying a significant coupling between large and small scales. Under the assumption that forcing controls the large-scale dynamics, we propose a conjugate regime to Kolmogorov's inertial range, independent of the forcing scale, to capture the odd parts of p.d.f.s. Thus, dynamics of large scales departs from the absolute equilibrium, and we can partially recover small-scale information without explicitly resolving small-scale dynamics. The departure from the statistical equilibrium is quantified and found to be viscosity-independent. Even though this departure is small, it is significant and should be considered when studying the large scales of the forced three-dimensional homogeneous isotropic turbulence.