Flowing air and water are persistent sculptors, gradually working stone, clay, sand and ice into landforms and landscapes. The evolution of shape results from a complex fluid–solid coupling that tends to produce stereotyped forms, and this morphology offers important clues to the history of a landscape and its development. Claudin et al. (J. Fluid Mech., vol. 832, 2017, R2) shed light on how we might read the rippled and scalloped patterns written into dissolving or melting solid surfaces by a flowing fluid. By better understanding the genesis of these patterns, we may explain why they appear in different natural settings, such as the walls of mineral caves dissolving in flowing water, ice caves in wind, and melting icebergs.

The source mechanism of inertia–gravity waves (IGWs) observed in numerical simulations of the differentially heated rotating annulus experiment is investigated. The focus is on the wave generation from the balanced part of the flow, a process presumably contributing significantly to the atmospheric IGW field. Direct numerical simulations are performed for an atmosphere-like configuration of the annulus and possible regions of IGW activity are characterised by a Hilbert-transform algorithm. In addition, the flow is separated into a balanced and unbalanced part, assuming the limit of a small Rossby number, and the forcing of IGWs by the balanced part of the flow is derived rigorously. Tangent-linear simulations are then used to identify the part of the IGW signal that is rather due to radiation by the internal balanced flow than to boundary-layer instabilities at the side walls. An idealised fluid set-up without rigid horizontal boundaries is considered as well, to investigate the effect of the identified balanced forcing unmasked by boundary-layer effects. The direct simulations of the realistic and idealised fluid set-ups show a clear baroclinic-wave structure exhibiting a jet–front system similar to its atmospheric counterparts, superimposed by four distinct IGW packets. The subsequent tangent-linear analysis indicates that three wave packets are radiated from the internal flow and a fourth one is probably caused by boundary-layer instabilities. The forcing by the balanced part of the flow is found to play a significant role in the generation of IGWs, so it supplements boundary-layer instabilities as a key factor in the IGW emission in the differentially heated rotating annulus.

The sparse identification of nonlinear dynamics (SINDy) is a recently proposed data-driven modelling framework that uses sparse regression techniques to identify nonlinear low-order models. With the goal of low-order models of a fluid flow, we combine this approach with dimensionality reduction techniques (e.g. proper orthogonal decomposition) and extend it to enforce physical constraints in the regression, e.g. energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a high-fidelity solver to project the Navier–Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed measurement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of such an approach is demonstrated on two canonical flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favourably against reduced-order models obtained from a standard Galerkin projection procedure. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.

Turbulent skin-friction drag reduction with superhydrophobic (SH) longitudinal microgrooves and riblets is investigated by direct numerical simulation (DNS), using lattice Boltzmann methods, in channel flow. The liquid/gas interfaces in the SH longitudinal microgrooves were modelled as stationary, curved, shear-free boundaries, with the meniscus shape determined from the solution of the Young–Laplace equation. Interface protrusion angles of $\unicode[STIX]{x1D703}=0^{\circ },-30^{\circ },-60^{\circ },-90^{\circ }$ were investigated. For comparison, the same geometries as those formed by the SH interfaces were also studied as riblets. Drag reductions of up to 61 % and up to 5 % were realized in DNS with SH longitudinal microgrooves and riblets, respectively, in turbulent channel flows at bulk Reynolds numbers of $Re_{b}=3600$ ($Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$) and $Re_{b}=7860$ ($Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$), with arrays of SH longitudinal microgrooves or riblets of size $14\lesssim g^{+0}\lesssim 56$ and $g^{+0}/w^{+0}=7$ on both walls, where $g^{+0}$ and $w^{+0}$ denote the widths and spacings of the microgrooves in base flow wall units, respectively. An exact analytical expression is derived which allows the net drag reduction in laminar or turbulent channel flow with any SH or no-slip wall micro-texture to be decomposed into contributions from: (i) the effective slip velocity at the wall, (ii) modifications to the normalized structure of turbulent Reynolds shear stresses due to the presence of this effective slip velocity at the wall, (iii) other modifications to the normalized structure of turbulent Reynolds shear stresses due to the presence of the wall micro-texture, (iv) modifications to the normalized structure of mean flow shear stresses due to the presence of the wall micro-texture and (v) the fraction of the flow rate through the wall micro-texture. Comparison to DNS results shows that SH longitudinal microgrooves and riblets share a common mechanism of drag reduction in which $100\,\%$ of the drag reduction arises from effects (i) and (ii). The contributions from (iii)–(v) were always drag enhancing, and followed a common scaling with SH longitudinal microgrooves and riblets when expressed as a function of the square root of the microgroove cross-sectional area in wall units. Extrapolation of drag reduction data from DNS to high Reynolds number flows of practical interest is discussed. It is shown that, for a given geometry and size of the surface micro-texture in wall units, the drag reduction performance of micro-textured surfaces degrades with increasing bulk Reynolds number of the flow. Curved SH interfaces at low protrusion angle ($\unicode[STIX]{x1D703}=-30^{\circ }$) were found to enhance the drag reduction by up to 3.6 % compared to flat interfaces, while reducing the instantaneous pressure fluctuations on the SH interfaces by up to a factor of two. This suggests that the longevity of SH interfaces in turbulent flow may be improved by embedding the SH surface within the microgrooves of shallow, scalloped riblets.

We study mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyse the behaviour of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyse a double-gyre synthetic velocity field. Second, a Rayleigh–Bénard instability (the Horton–Rogers–Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh–Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated and convection shutdown. The interface compression model correctly predicts the behaviour of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behaviour of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behaviour between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

Onset of natural convection due to transient diffusion in porous media has attracted considerable attention for its applications to CO$_{2}$ sequestration. Stability analyses typically investigate the onset of such convection using an initial value problem approach in which a perturbation is introduced to the concentration field at an initial time $t=t_{p}$. This leads to debate concerning physically appropriate perturbations, the critical time $t_{c}$ for linear instability and the counter-intuitive notion of an optimal initial time $t_{p}$ that maximizes perturbation growth. We propose an alternate approach in which transient diffusion is continuously perturbed by small porosity variations. With this approach, instability occurs immediately ($t_{c}=0$) without violating any physical constraints, such that the concepts of initial time $t_{p}$ and critical time $t_{c}$ become irrelevant. We also argue that the onset time for nonlinear convection is a more physically relevant parameter, and show that it can be predicted using a simple asymptotic expansion. Using the expansion, we explore the onset of nonlinear convection due to porosity perturbations that vary sinusoidally in the horizontal and vertical directions, and show there are optimal wavelengths that minimize the onset time. Finally, we find simple relationships for these wavelengths as functions of perturbation magnitude. These show that even small porosity perturbations, typically considered negligible in previous literature, are sufficient to trigger nonlinear convection and thereby influence the rate of CO$_{2}$ dissolution within time scales comparable to previous analyses.

High-speed X-ray phase-contrast imaging of the cavitating flow developing within an axisymmetric throttle orifice has been conducted using high-flux synchrotron radiation. A white X-ray beam with energy of 6 keV was utilized to visualize the highly turbulent flow at 67 890 frames per second with an exposure time of 347 ns. The working medium employed was commercial diesel fuel at flow conditions characterized by Reynolds and cavitation numbers in the range of 18 000–35 500 and 1.6–7.7, respectively. Appropriate post-processing of the obtained side-view radiographs enabled the detailed illustration of the interface topology of the arising vortical cavity. In addition, the visualization temporal and spatial resolution allowed the correlation of the prevailing flow conditions to the periodicity of cavitation onset and collapse, to the magnitude of the underlying vortical motion, as well as to the local turbulence intensity.

In this paper, we investigate both experimentally and theoretically the dynamics of a liquid plug driven by a cyclic periodic forcing inside a cylindrical rigid capillary tube. First, it is shown that, depending on the type of forcing (flow rate or pressure cycle), the dynamics of the liquid plug can either be stable and periodic, or conversely accelerative and eventually leading to plug rupture. In the latter case, we identify the sources of the instability as: (i) the cyclic diminution of the plug viscous resistance to motion due to the decrease in the plug length and (ii) a cyclic reduction of the plug interfacial resistance due to a lubrication effect. Since the flow is quasi-static and the forcing periodic, this cyclic evolution of the resistances relies on the existence of flow memories stored in the length of the plug and the thickness of the trailing film. Second, we show that, contrary to unidirectional pressure forcing, cyclic forcing enables breaking of large plugs in a confined space although it requires longer times. All the experimentally observed tendencies are quantitatively recovered from an analytical model. This study not only reveals the underlying physics but also opens up the prospect for the simulation of ‘breathing’ of liquid plugs in complex geometries and the determination of optimal cycles for obstructed airways reopening.

We call thick those films for which the disjoining pressure and thermal fluctuations are ineffective. Water films with thickness $h$ in the $1{-}100~\unicode[STIX]{x03BC}\text{m}$ range are thick, but are also known, paradoxically, to nucleate holes spontaneously. We have uncovered a mechanism solving the paradox, relying on the extreme sensitivity of the film to surface tension inhomogeneities. The surface tension of a free liquid film is lowered by an amount $\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ over a size $a$ by chemical or thermal contamination. At the same time this spot diffuses (within a time $a^{2}/D$, with $D$ the diffusion coefficient of the pollutant in the substrate), the Marangoni stress $\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}/a$ induces an inhomogeneous outward interstitial flow which digs the film within a time $\unicode[STIX]{x1D70F}_{0}\sim \sqrt{\unicode[STIX]{x1D70C}ha^{2}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}$, with $\unicode[STIX]{x1D70C}$ the density of the liquid. When the Péclet number $Pe=a^{2}/D\unicode[STIX]{x1D70F}_{0}$ is larger than unity, the liquid substrate motion reinforces the surface tension gradient, triggering a self-sustained instability insensitive to diffusional regularisation. Several experimental illustrations of the phenomenon are given, both qualitative and quantitative, including a precise study of the first instants of the unstable dynamics made by controlled perturbations of a Savart sheet at large $Pe$.

A liquid foam is a dispersion of gas bubbles in a liquid matrix containing surface-active agents. Its flow involves the relative motion of bubbles, which switch neighbours during a so-called topological rearrangement of type 1 (T1). The dynamics of T1 events, as well as foam rheology, have been extensively studied, and experimental results point to the key role played by surfactants in these processes. However, the complex and multiscale nature of the system has so far impeded a complete understanding of the mechanisms involved. In this work, we investigate numerically the effect of surfactants on the rheological response of a 2D sheared bubble cluster. To do so, a level-set method previously employed for simulation of two-phase flow has been extended to include the effects of surfactants. The dynamical processes of the surfactants – diffusion in the liquid and along the interface, adsorption/desorption at the interface – and their coupling with the flow – surfactant advection and Laplace and Marangoni stresses at the interface – are all taken into account explicitly. Through a systematic study of the Biot, capillary and Péclet numbers that characterise the surfactant properties in the simulation, we find that the presence of surfactants can affect the liquid/gas hydrodynamic boundary condition (from a rigid-like situation to a mobile one), which modifies the nature of the flow in the volume from a purely extensional situation to a shear. Furthermore, the work done by surface tension (the 2D analogue of the work by pressure forces), resulting from surfactant and interface dynamics, can be interpreted as an effective dissipation, which reaches a maximum for a Péclet number of order unity. Our results, obtained at high liquid fraction, should provide a reference point, with which experiments and models of T1 dynamics and foam rheology can be compared.

Buoyancy–thermocapillary convection in a layer of volatile liquid driven by a horizontal temperature gradient arises in a variety of situations. Recent studies have shown that the composition of the gas phase, which is typically a mixture of vapour and air, has a noticeable effect on the critical Marangoni number describing the onset of convection as well as on the observed convection pattern. Specifically, as the total pressure or, equivalently, the average concentration of air is decreased, the threshold of the instability leading to the emergence of convective rolls is found to increase rather significantly. We present a linear stability analysis of the problem which shows that this trend can be readily understood by considering the transport of heat and vapour through the gas phase. In particular, we show that transport in the gas phase has a noticeable effect even at atmospheric conditions, when phase change is greatly suppressed.

We consider pressure-driven flow of an ion-carrying viscous Newtonian fluid through a non-uniformly shaped channel coated with a charged deformable porous layer, as a model for blood flow through microvessels that are lined with an endothelial glycocalyx layer (EGL). The EGL is negatively charged and electrically interacts with ions dissolved in the blood plasma. The focus here is on the interplay between electrochemical effects, and the pressure-driven flow through the microvessel. To analyse these effects we use triphasic mixture theory (TMT) which describes the coupled dynamics of the fluid phase, the elastic EGL, ion transport within the fluid and electric fields within the microvessel. The resulting equations are solved numerically using a coupled boundary–finite element method (BEM-FEM) scheme. However, in the physiological regime considered here, ion concentrations and electric potentials vary rapidly over a thin transitional region (Debye layer) that straddles the lumen–EGL interface, which is difficult to resolve numerically. Accordingly we analyse this region asymptotically, to determine effective jump conditions across the interface for BEM-FEM computations within the bulk EGL/lumen. Our results demonstrate that ion–EGL electrical interactions can influence the near-wall flow, causing it to become reversed. This alters the stresses exerted upon the vessel wall, which has implications for the hypothesised role of the EGL as a transmitter of mechanical signals from the blood flow to the endothelial vessel surface.

Experiments and large eddy simulation (LES) were performed to study the development of the Rayleigh–Taylor instability into the saturated, nonlinear regime, produced between two gases accelerated by a rarefaction wave. Single-mode two-dimensional, and single-mode three-dimensional initial perturbations were introduced on the diffuse interface between the two gases prior to acceleration. The rarefaction wave imparts a non-constant acceleration, and a time decreasing Atwood number, $A=(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})/(\unicode[STIX]{x1D70C}_{2}+\unicode[STIX]{x1D70C}_{1})$, where $\unicode[STIX]{x1D70C}_{2}$ and $\unicode[STIX]{x1D70C}_{1}$ are the densities of the heavy and light gas, respectively. Experiments and simulations are presented for initial Atwood numbers of $A=0.49$, $A=0.63$, $A=0.82$ and $A=0.94$. Nominally two-dimensional (2-D) experiments (initiated with nearly 2-D perturbations) and 2-D simulations are observed to approach an intermediate-time velocity plateau that is in disagreement with the late-time velocity obtained from the incompressible model of Goncharov (Phys. Rev. Lett., vol. 88, 2002, 134502). Reacceleration from an intermediate velocity is observed for 2-D bubbles in large wavenumber, $k=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D706}=0.247~\text{mm}^{-1}$, experiments and simulations, where $\unicode[STIX]{x1D706}$ is the wavelength of the initial perturbation. At moderate Atwood numbers, the bubble and spike velocities approach larger values than those predicted by Goncharov’s model. These late-time velocity trends are predicted well by numerical simulations using the LLNL Miranda code, and by the 2009 model of Mikaelian (Phys. Fluids., vol. 21, 2009, 024103) that extends Layzer type models to variable acceleration and density. Large Atwood number experiments show a delayed roll up, and exhibit a free-fall like behaviour. Finally, experiments initiated with three-dimensional perturbations tend to agree better with models and a simulation using the LLNL Ares code initiated with an axisymmetric rather than Cartesian symmetry.

Aquatic micro-organisms and artificial microswimmers locomoting in turbulent flow encounter velocity gradients that rotate them, thereby changing their swimming direction and possibly providing cues about the local flow environment. Using numerical simulations of ellipsoidal particles in isotropic turbulence, we investigate the effects of body shape and swimming velocity on particle motion. Four particle shapes (sphere, rod, disc and triaxial ellipsoid) are investigated at five different swimming velocities in the range $0\leqslant V_{s}\leqslant 5u_{\unicode[STIX]{x1D702}}$, where $V_{s}$ is the swimming velocity and $u_{\unicode[STIX]{x1D702}}$ is the Kolmogorov velocity scale. We find that anisotropic, swimming particles preferentially sample regions of lower fluid vorticity than passive particles do, and hence they accumulate in these regions. While this effect is monotonic with swimming velocity, the particle enstrophy (variance of particle angular velocity) varies non-monotonically with swimming velocity. In contrast to passive particles, the particle enstrophy is a function of shape for swimming particles. The particle enstrophy is largest for triaxial ellipsoids swimming at a velocity smaller than $u_{\unicode[STIX]{x1D702}}$. We also observe that the average alignment of particles with the directions of the velocity gradient tensor are altered by swimming leading to a more equal distribution of rotation about different particle axes.

We consider fully developed turbulence in straight ducts of non-circular cross-sectional shape, for instance a square. A global friction velocity $\overline{u}_{\unicode[STIX]{x1D70F}}$ is defined from the streamwise pressure gradient $|\text{d}p/\text{d}x|$ and a single characteristic length $h$, half the hydraulic diameter (shapes with disparate length scales, due to high aspect ratio, are excluded). We reason that as the Reynolds number $Re$ reaches high values, outside the viscous region the streamwise velocity differences and the secondary motion scale with $\overline{u}_{\unicode[STIX]{x1D70F}}$ and the Reynolds stresses with $\overline{u}_{\unicode[STIX]{x1D70F}}^{2}$. This extends the classical defect-law argument, associated with Townsend and many others, and is successful in channel and pipe flows. We then posit matched asymptotic expansions with overlap of the law of the wall and the behaviour we assumed in the core region. The wall may be smooth, or have a Nikuradse roughness $k_{S}$ (such that it is fully rough, with $k_{S}^{+}\gg 1$). The consequences include the familiar logarithmic behaviour of the velocity profile, but also the surprising prediction that the skin friction tends to uniformity all around the duct, except near possible corners, asymptotically as $Re\rightarrow \infty$ or $k_{S}/h\rightarrow 0$. This is confirmed by numerical solutions for a square and two ellipses, using a conventional turbulence model, albeit the trend with Reynolds number is slow. The magnitude of the secondary motion also scales as expected, and the skin-friction coefficient follows the logarithm of the appropriate Reynolds number. This is a validation of the mathematical reasoning, but is by no means independent physical evidence, because the turbulence models embody the same assumptions as the theory. The uniformity of the skin friction appears to be a new and falsifiable deduction from turbulence theory, and a candidate for high-Reynolds-number experiments.

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

Consider the dynamics of a healing film driven by surface tension, that is, the inward spreading process of a liquid film to fill a hole. The film is modelled using the lubrication (or thin-film) approximation, which results in a fourth-order nonlinear partial differential equation. We obtain a self-similar solution describing the early-time relaxation of an initial step-function condition and a family of self-similar solutions governing the finite-time healing. The similarity exponent of this family of solutions is not determined purely from scaling arguments; instead, the scaling exponent is a function of the finite thickness of the prewetting film, which we determine numerically. Thus, the solutions that govern the finite-time healing are self-similar solutions of the second kind. Laboratory experiments and time-dependent computations of the partial differential equation are also performed. We compare the self-similar profiles and exponents, obtained by matching the estimated prewetting film thickness, with both measurements in experiments and time-dependent computations near the healing time, and we observe good agreement in each case.

This paper uses matched asymptotic expansions to study the non-localized (which we refer to as global) boundary layer instabilities generated by free-stream acoustic and vortical disturbances at moderate supersonic Mach numbers. The vortical disturbances produce an unsteady boundary layer flow that develops into oblique instability waves with a viscous triple-deck structure in the downstream region. The acoustic disturbances (which for reasons given herein are assumed to have obliqueness angles that are close to a certain critical angle) generate slow boundary layer disturbances which eventually develop into oblique stable disturbances with an inviscid triple-deck structure in a region that lies downstream of the viscous triple-deck region. The paper shows that both the vortically generated instabilities and the acoustically generated oblique disturbances ultimately develop into modified Rayleigh-type instabilities (which can either grow or decay) further downstream.

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

A characteristic feature of axisymmetric jets, and turbulent shear flows in general, is the entrainment of mass across the turbulent/non-turbulent interface (TNTI). The multi-scale nature of the TNTI surface area was recently observed to exhibit power-law scaling with a fractal dimension, $D_{f}$, between $D_{f}=2.3{-}2.4$, inferred from two-dimensional data, in both high Reynolds number boundary layers and the far field of axisymmetric jets. In this paper, we show that the fractal scaling previously observed in the far field of an axisymmetric jet is established at the end of the potential core. Simultaneous measurements of the velocity and scalar fields were obtained and coarse grain filtering was applied over two decades of scale separation, showing that $D_{f}$ evolves to ${\approx}2.35$ at $x/d=4.6$, which is similar to $D_{f}$ found in the far field between $x/d=40{-}60$. This is evidence that scale separation becomes sufficiently developed to achieve scale invariance of the TNTI surface area in the near field of the jet well before self-similarity is established. We also observe that the onset of this geometric scale invariance coincides with the onset of radial homogeneity shown by two-point velocity correlations. Finally, we present a simple theoretical basis for these results using an exact fractal construction based on the Koch curve and applying a coarse-grain filtering analysis.