Capillarity is the familiar manifestation of the cohesion of liquids. Since Laplace (Traité de mécanique céleste, vol. IV, supplément au livre X: Sur l’action capillaire, 1805, pp. 1–65), we know that intense attractive forces between the molecules bridge the small with the large as they shape liquid/vapour interfaces at the macroscopic scale through the concept of surface tension (menisci, drops, bubbles, puddles, liquid rise in tubes, etc. …). We concentrate on situations where liquids ‘disgregate’, following the neologism of Clausius (Phil. Mag., vol. 24 (159), 1862, pp. 81–97), meaning that they fragment by the action of deformation stresses whose intensity competes with that of cohesion forces. Various examples, including explosions, blow-ups, hard and soft impacts and shears applied to liquid jets, sheets and drops are reviewed. They concern applications ranging from liquid propulsion, agricultural spraying, to the formation of ocean spray, raindrops and human exhalations by violent respiratory events. In spite of their diversity, the various modes of fragment production share an ultimate common phenomenology – the ligament dynamics – suggesting that the final stable droplet size distribution can be interpreted from elementary principles.

Convection in a container, heated from below, cooled from above and rapidly rotated around a vertical axis, starts from its sidewall. When the imposed vertical temperature gradient is not sufficiently large for bulk modes to set in, thermal convection can start in the form of wall modes, which are observed near the sidewall as pairs of hot ascending and cold descending plumes that drift along the wall. With increasing temperature gradient, different wall and bulk modes occur and interact, leading finally to turbulence. A recent numerical study by Favier & Knobloch (J. Fluid Mech., 895, 2020, R1) reveals an extreme robustness of the wall states. They persist above the onset of bulk modes and turbulence, thereby relating them to the recently discovered boundary zonal flows in highly turbulent rotating thermal convection. More exciting is that the wall modes can be thought of as topologically protected states, as they are robust with respect to the sidewall shape. They stubbornly drift along the wall, following its contour, independent of geometric obstacles.

A new model for the behaviour of a thick, two-dimensional layer of fluid on the surface of a rotating cylinder is presented, incorporating the effects of inertia, rotation, viscosity, gravity and capillarity. Comparisons against direct numerical simulations (DNS) show good accuracy for fluid layers of thickness of the same order as the cylinder radius, even for Reynolds numbers up to $Re\sim 10$. A rich and complex parameter space is revealed, and is elucidated via a variety of analytical and numerical techniques. At moderate rotation rates and fluid masses, the system exhibits either periodic behaviour or converges to a steady state, with the latter generally being favoured by greater masses and lower rotation rates. These behaviours, and the bifurcation structure of the transitions between them, are examined using a combination of both the low-order model and DNS. Specific attention is dedicated to newly accessible regions of parameter space, including the multiple steady state solutions observed for the same parameter values by Lopes et al. (J. Fluid Mech., vol. 835, 2018, pp. 540–574), where the corresponding triple limit point bifurcation structure is recovered by the new low-order model. We also inspect states in which the interface becomes multivalued – and thus outside the reach of the reduced-order model – via DNS. This leads to highly nonlinear multivalued periodic structures appearing at moderate thicknesses and relatively large rotation rates. Even much thicker films may eventually reach steady states (following complex early evolution), provided these are maintained by a combination of forces sufficiently large to counteract gravity.

This paper presents a reduced-order modelling strategy for Rayleigh–Bénard convection of a radiating gas, based on the proper orthogonal decomposition (POD). Direct numerical simulation (DNS) of coupled natural convection and radiative transfer in a cubic Rayleigh–Bénard cell is performed for an $\text{air}/\text{H}_{2}\text{O}/\text{CO}_{2}$ mixture at room temperature and at a Rayleigh number of $10^{7}$. It is shown that radiative transfer between the isothermal walls and the gas triggers a convection growth outside the boundary layers. Mean and turbulent kinetic energy increase with radiation, as well as temperature fluctuations to a lesser extent. As in the uncoupled case, the large-scale circulation (LSC) settles in one of the two diagonal planes of the cube with a clockwise or anticlockwise motion, and experiences occasional brief reorientations which are rotations of $\unicode[STIX]{x03C0}/2$ of the LSC in the horizontal plane. A POD analysis is conducted and reveals that the dominant POD eigenfunctions are preserved with radiation while POD eigenvalues are increased. Two POD-based reduced-order models including radiative transfer effects are then derived: the first one is based on coupled DNS data while the second one is an a priori model based on uncoupled DNS data. Owing to the weak temperature differences, radiation effects on mode amplitudes are linear in the models. Both models capture the increase in energy with radiation and are able to reproduce the low-frequency dynamics of reorientations and the high-frequency dynamics associated with the LSC velocity observed in the coupled DNS.

The surface area of turbulent/non-turbulent interfaces (TNTIs) is continuously produced and destroyed via stretching and curvature/propagation effects. Here, the mechanisms responsible for TNTI area growth and destruction are investigated in a turbulent flow with and without stable stratification through the time evolution equation of the TNTI area. We show that both terms have broad distributions and may locally contribute to either production or destruction. On average, however, the area growth is driven by stretching, which is approximately balanced by destruction by the curvature/propagation term. To investigate the contribution of different length scales to these processes, we apply spatial filtering to the data. In doing so, we find that the averages of the stretching and the curvature/propagation terms balance out across spatial scales of TNTI wrinkles and this scale-by-scale balance is consistent with an observed scale invariance of the nearby coherent vortices. Through a conditional analysis, we demonstrate that the TNTI area production (destruction) is localized at the front (lee) edge of the vortical structures in the interface proximity. Finally, we show that while basic mechanisms remain the same, increasing stratification reduces the rates at which TNTI surface area is produced as well as destroyed. We provide evidence that this reduction is largely connected to a change in the multiscale geometry of the interface, which tends to flatten in the wall-normal direction at all active length scales of the TNTI.

We study passive scalar fluctuations convected by statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient. In order to elucidate the parameter dependence of small-scale statistics of scalar fluctuations, we conduct direct numerical simulations of passive scalar turbulence with 59 different combinations of Reynolds number and Schmidt number. For all the cases, we compute time-average statistics of various quantities, which include the scalar derivative skewness and flatness, the ratio of parallel-to-perpendicular scalar-gradient variances, and the anisotropy parameter recently proposed (Hill, Phys. Rev. Fluids, vol. 2, 2017, 094601). Notably, the degree of small-scale anisotropy of passive scalar fluctuation is characterised by a universal function of the Péclet number $Pe_{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}}=u^{\prime }\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x1D705}$, where $u^{\prime }$ is the root mean square velocity, $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ the Taylor microscale of scalar fluctuation, $\unicode[STIX]{x1D705}$ the mass diffusivity. In the definition of the Péclet number, the use of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$, rather than the Taylor microscale of velocity fluctuation, is key to collapsing the data of different Reynolds and Schmidt numbers. When the Péclet number is low, large-scale anisotropic scalar structures emerge irrespective of the Reynolds number. These structures are elongated along the direction of the uniform mean scalar gradient, and their size is significantly larger than the integral length scale of velocity fluctuation.

Exact budget equations for the second-order structure function tensor $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle$, where $\unicode[STIX]{x1D6FF}u_{i}$ is the difference of the $i$th fluctuating velocity component between two points, are used to study the two-point statistics of velocity fluctuations in inhomogeneous turbulence. The anisotropic generalised Kolmogorov equations (AGKE) describe the production, transport, redistribution and dissipation of every Reynolds stress component occurring simultaneously among different scales and in space, i.e. along directions of statistical inhomogeneity. The AGKE are effective to study the inter-component and multi-scale processes of turbulence. In contrast to more classic approaches, such as those based on the spectral decomposition of the velocity field, the AGKE provide a natural definition of scales in the inhomogeneous directions, and describe fluxes across such scales too. Compared to the generalised Kolmogorov equation, which is recovered as their half-trace, the AGKE can describe inter-component energy transfers occurring via the pressure–strain term and contain also budget equations for the off-diagonal components of $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle$. The non-trivial physical interpretation of the AGKE terms is demonstrated with three examples. First, the near-wall cycle of a turbulent channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=200$ is considered. The off-diagonal component $\langle -\unicode[STIX]{x1D6FF}u\unicode[STIX]{x1D6FF}v\rangle$, which cannot be interpreted in terms of scale energy, is discussed in detail. Wall-normal scales in the outer turbulence cycle are then discussed by applying the AGKE to channel flows at $Re_{\unicode[STIX]{x1D70F}}=500$ and $1000$. In a third example, the AGKE are computed for a separating and reattaching flow. The process of spanwise-vortex formation in the reverse boundary layer within the separation bubble is discussed for the first time.

The flow of a thin film coating the underside of an inclined substrate is studied. We measure experimentally spatial growth rates and compare them to the linear stability analysis of a flat film modelled by the lubrication equation. When forced by a stationary localized perturbation, a front develops that we predict with the group velocity of the unstable wave packet. We compare our experimental measurements with numerical solutions of the nonlinear lubrication equation with complete curvature. Streamwise structures dominate and saturate after some distance. We recover their profile with a one-dimensional lubrication equation suitably modified to ensure an invariant profile along the streamwise direction and compare them with the solution of a purely two-dimensional pendent drop, showing overall a very good agreement. Finally, those different profiles agree also with a two-dimensional simulation of the Stokes equations.

We study direct numerical simulations of turbulence arising from the interaction of an initial background shear, a linear background stratification and an external body force. In each simulation the turbulence produced is spatially intermittent, with dissipation rates varying over orders of magnitude in the vertical. We focus analysis on the statistically quasi-steady states achieved by applying large-scale body forcing to the domain, and compare flows forced by internal gravity waves with those forced by vertically uniform vortical modes. By considering the turbulent energy budgets for each simulation, we find that the injection of potential energy from the wave forcing permits a reversal in the sign of the mean buoyancy flux. This change in the sign of the buoyancy flux is associated with large, convective density overturnings, which in turn lead to more efficient mixing in the wave-forced simulations. The inhomogeneous dissipation in each simulation allows us to investigate localised correlations between the kinetic and potential energy dissipation rates. These correlations lead us to the conclusion that an appropriate definition of an instantaneous mixing efficiency, $\unicode[STIX]{x1D702}(t):=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+\unicode[STIX]{x1D700})$ (where $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D712}$ are the volume-averaged turbulent viscous dissipation rate and fluctuation density variance destruction rate respectively) in the wave-forced cases is independent of an appropriately defined local turbulent Froude number, consistent with scalings proposed for low Froude number stratified turbulence.

Kaneda’s (J. Fluid Mech., vol. 107, 1981, pp. 131–145) Lagrangian renormalized approximation was extended to single-time spectral closure under two assumptions: (i) Markovianization and (ii) the Lagrangian velocity response function is expressed by $G(k,\unicode[STIX]{x1D70F})=\exp (-C_{1}(k)\unicode[STIX]{x1D70F}-C_{2}(k)\unicode[STIX]{x1D70F}^{2}/2)$. The unknown functions $C_{1}(k)$ and $C_{2}(k)$ are theoretically derived to be consistent with the exact short-time behaviour of $G(k,\unicode[STIX]{x1D70F})$ and the asymptotic short-time behaviour of assumed exponential form of $G(k,\unicode[STIX]{x1D70F})$, i.e. the present closure is derived from the Navier–Stokes equation without introduction of any adjustable parameters and it can calculate the statistical quantities by theory. The results show that the present closure has good agreement with direct numerical simulation for single- and two-point statistics.

We present a general simulation approach for fluid–solid interactions based on the fully Eulerian reference map technique. The approach permits the modelling of one or more finitely deformable continuum solid bodies interacting with a fluid and with each other. A key advantage of this approach is its ease of use, as the solid and fluid are discretized on the same fixed grid, which greatly simplifies the coupling between the phases. We use the method to study a number of illustrative examples involving an incompressible Navier–Stokes fluid interacting with multiple neo-Hookean solids. Our method has several useful features including the ability to model solids with sharp corners and the ability to model actuated solids. The latter permits the simulation of active media such as swimmers, which we demonstrate. The method is validated favourably in the flag-flapping geometry, for which a number of experimental, numerical and analytical studies have been performed. We extend the flapping analysis beyond the thin-flag limit, revealing an additional destabilization mechanism to induce flapping.

We present here a comprehensive derivation for the speed of a small bottom-heavy sphere forced by a transverse acoustic field and thereby establish how density inhomogeneities may play a critical role in acoustic propulsion. The sphere is trapped at the pressure node of a standing wave whose wavelength is much larger than the sphere diameter. Due to its inhomogeneous density, the sphere oscillates in translation and rotation relative to the surrounding fluid. The perturbative flows induced by the sphere’s rotation and translation are shown to generate a rectified inertial flow responsible for a net mean force on the sphere that is able to propel the particle within the zero-pressure plane. To avoid an explicit derivation of the streaming flow, the propulsion speed is computed exactly using a suitable version of the Lorentz reciprocal theorem. The propulsion speed is shown to scale as the inverse of the viscosity, the cube of the amplitude of the acoustic field and is a non-trivial function of the acoustic frequency. Interestingly, for some combinations of the constitutive parameters (fluid-to-solid density ratio, moment of inertia and centroid to centre of mass distance), the direction of propulsion is reversed as soon as the frequency of the forcing acoustic field becomes larger than a certain threshold. The results produced by the model are compatible with both the observed phenomenology and the orders of magnitude of the measured velocities.

Classical viscous fingering (VF) instability, the formation of finger-like interfacial patterns, occurs when a less viscous fluid displaces a more viscous one in porous media in immiscible and fully miscible systems. However, the dynamics in partially miscible fluid pairs, exhibiting a phase separation due to its finite solubility into each other, has not been largely understood so far. This study has succeeded in experimentally changing the solution system from immiscible to fully miscible or partially miscible by varying the compositions of the components in an aqueous two-phase system (ATPS) while leaving the viscosities relatively unchanged at room temperature and atmospheric pressure. Here, we have experimentally discovered a new topological transition of VF instability by performing a Hele-Shaw cell experiment using the partially miscible system. The finger formation in the investigated partially miscible system changes to the generation of spontaneously moving multiple droplets. Through additional experimental investigations, we determine that such anomalous VF dynamics is driven by thermodynamic instability such as phase separation due to spinodal decomposition and Korteweg convection induced by compositional gradient during such phase separation. We perform the numerical simulation by coupling hydrodynamics with such chemical thermodynamics and the spontaneously moving droplet dynamics is obtained, which is in good agreement with the experimental investigations of the ATPS. This numerical result strongly supports our claim that the origin of such anomalous VF dynamics is thermodynamic instability.

We present a fluid-mechanical explanation of the formation of sedimentary wedges deposited at ice-stream grounding zones. We model both ice and till as layers of viscous fluid spreading under gravity into an inviscid ocean. To test the fundamentals of our theory, we perform a series of laboratory experiments in which we find that a similar wedge of underlying, less viscous fluid accumulates spontaneously around the grounding line. We formulate a simple local condition relating wedge slopes, which determines wedge geometry. It expresses a balance of fluxes of till either side of the grounding line and involves upstream and downstream gradients of till thicknesses as well as the upper surface gradient of the ice. It shows that a wedge will form, that is the upstream till thickness gradients are positive, when the flux of till driven by the glaciostatic pressure gradient of the overlying ice is greater than the flux of till ahead of the grounding zone. This is related to the unloading of the till as the ice sheet crosses the grounding line.

Viscous streaming flows generated by objects of constant curvature (circular cylinders, infinite plates) have been well understood. Yet, characterization and understanding of such flows when multiple body length scales are involved has not been looked into in rigorous detail. We propose a simplified setting to understand and explore the effect of multiple body curvatures on streaming flows, analysing the system through the lens of bifurcation theory. Our set-up consists of periodic, regular lattices of cylinders characterized by two distinct radii, so as to inject discrete curvatures into the system, which in turn affect the streaming field generated due to an oscillatory background flow. We demonstrate that our understanding based on this system, and in particular the role of bifurcations in determining the local flow topology, can be then generalized to a variety of individual convex shapes presenting a spectrum of curvatures, explaining prior experimental and computational observations. Thus, this study illustrates a route towards the rational manipulation of viscous streaming flow topology, through regulated variation of object geometry.

We derive linear instability and nonlinear stability thresholds for a problem of thermal convection in a bidispersive porous medium with a single temperature when Darcy theory is employed in the micropores whereas Brinkman theory is utilized in the macropores. It is important to note that we show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection. The coincidence of the linear and nonlinear stability boundaries is established under general thermal boundary conditions.

The structure and propagation of lock-release bottom gravity currents in a linearly stratified ambient with the presence of a submerged canopy are investigated for the first time using large-eddy simulations. The canopy density (i.e. the solid volume fraction), the strength of ambient stratification and the canopy height are varied to study their respective effects on the gravity current. Both denser canopies and stronger ambient stratification tend to switch the horizontal boundary along which the current propagates from the channel bed towards the canopy top (i.e. the through-to-over flow transition). It is found that the dilution of the current density is enhanced by denser canopies but is weakened by stronger ambient stratification. The non-monotonic relationship between front velocity and canopy density proposed by Zhou et al. (J. Fluid Mech., vol. 831, 2017, pp. 394–417) in homogeneous environments is also observed in stratified environments. However, as the ambient stratification is strengthened, the present study shows a shift of the turning point (beyond which increasing canopy density leads to faster current propagation) towards sparser canopies, accompanied by a more pronounced recovery of the front velocity. This is the combined action of three stratification-induced mechanisms: the promotion of through-to-over flow transition (less canopy drag), the upward displacement of current nose in a stably stratified water column (more buoyancy gain) and the weakening of current dilution (less buoyancy loss). Under stronger ambient stratification, the propagation of gravity currents shows a lower sensitivity to the retarding effect of the submerged canopy.

We present a solution for the scattered field caused by an incident wave interacting with an infinite cascade of blades with complex boundary conditions. This extends previous studies by allowing the blades to be compliant, porous or satisfy a generalised impedance condition. Beginning with the convected wave equation, we employ Fourier transforms to obtain an integral equation amenable to the Wiener–Hopf method. This Wiener–Hopf system is solved using a method that avoids the factorisation of matrix functions. The Fourier transform is inverted to obtain an expression for the acoustic potential function that is valid throughout the entire domain. We observe that the principal effect of complex boundary conditions is to perturb the zeros of the Wiener–Hopf kernel, which correspond to the duct modes in the inter-blade region. We focus efforts on understanding the role of porosity, and present a range of results on sound transmission and generation. The behaviour of the duct modes is discussed in detail in order to explain the physical mechanisms behind the associated noise reductions. In particular, we show that cut-on duct modes do not exist for arbitrary porosity coefficients. Conversely, the acoustic far-field modes are unchanged by modifications to the boundary conditions. We apply our solution to a cascade of perforated plates and see that a fractional open area of 1 % is sufficient to significantly attenuate backscattering. The solution is essentially analytic (the only numerical requirements are matrix inversion and root finding) and is therefore extremely rapid to compute.

The sources of wall-pressure fluctuations in turbulent channel flow are studied using a novel framework. The wall-pressure power spectral density (PSD) $(\unicode[STIX]{x1D719}_{pp}(\unicode[STIX]{x1D714}))$ is expressed as an integrated contribution from all wall-parallel plane pairs, $\unicode[STIX]{x1D719}_{pp}(\unicode[STIX]{x1D714})=\int _{-\unicode[STIX]{x1D6FF}}^{+\unicode[STIX]{x1D6FF}}\int _{-\unicode[STIX]{x1D6FF}}^{+\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6E4}(r,s,\unicode[STIX]{x1D714})\,\text{d}r\,\text{d}s$, using the Green’s function. Here, $\unicode[STIX]{x1D6E4}(r,s,\unicode[STIX]{x1D714})$ is termed the net source cross-spectral density (CSD) between two wall-parallel planes, $y=r$ and $y=s$ and $\unicode[STIX]{x1D6FF}$ is the half-channel height. Direct numerical simulation data at friction Reynolds numbers of 180 and 400 are used to compute $\unicode[STIX]{x1D6E4}(r,s,\unicode[STIX]{x1D714})$. Analysis of the net source CSD, $\unicode[STIX]{x1D6E4}(r,s,\unicode[STIX]{x1D714})$ reveals that the location of dominant sources responsible for the premultiplied peak in the power spectra at $\unicode[STIX]{x1D714}^{+}\approx 0.35$ (Hu et al., AIAA J., vol. 44, 2006, pp. 1541–1549) and the wavenumber spectra at $\unicode[STIX]{x1D706}^{+}\approx 200$ (Panton et al., Phys. Rev. Fluids, vol. 2, 2017, 094604) are in the buffer layer at $y^{+}\approx 16.5$ and 18.4 for $Re_{\unicode[STIX]{x1D70F}}=180$ and 400, respectively. The contribution from a wall-parallel plane (located at distance $y^{+}$ from the wall) to wall-pressure PSD is log-normal in $y^{+}$ for $\unicode[STIX]{x1D714}^{+}>0.35$. A dominant inner-overlap region interaction of the sources is observed at low frequencies. Further, the decorrelated features of the wall-pressure fluctuation sources are analysed using spectral proper orthogonal decomposition (POD). Instead of the commonly used $L^{2}$ inner product, we require the modes to be orthogonal in an inner product with a symmetric positive definite kernel. Spectral POD supports the case that the net source is composed of two components – active and inactive. The dominant spectral POD mode that comprises the active part contributes to the entire wall-pressure PSD. The suboptimal spectral POD modes that constitute the inactive portion do not contribute to the PSD. Further, the active and inactive parts of the net source are decorrelated because they stem from different modes. The structure represented by the dominant POD mode at the premultiplied wall-pressure PSD peak inclines in the downstream direction. At the low-frequency linear PSD peak, the dominant mode resembles a large scale vertical pattern. Such patterns have been observed previously in the instantaneous contours of rapid pressure fluctuations by Abe et al. (2005, Fourth International Symposium on Turbulence and Shear Flow Phenomena, Begel House Inc.).

Open-cell porous materials have been reported as a promising concept for mitigating turbulent boundary-layer trailing-edge noise. This manuscript examines the aeroacoustics of a porous trailing edge to study its noise reduction mechanisms. Numerical investigations have been carried out for a NACA 0018 aerofoil with three different types of trailing edge: a baseline solid trailing edge, a fully porous trailing edge and a blocked-porous variant in which a solid core is added at the symmetry plane. The latter prevents flow interaction between the two sides of the aerofoil. Flow-field solutions are obtained by solving the explicit, transient and compressible lattice-Boltzmann equation, while the Ffowcs-Williams and Hawkings acoustic analogy has been used to compute far-field noise. The porous material is modelled using an equivalent fluid region governed by Darcy’s law, in which the properties of a Ni-Cr-Al open-cell metal foam are applied. The simulation results are validated against reference data from experiments. The regular porous trailing edge reduces noise substantially, particularly at low frequency, whereas the blocked variant retains similar noise characteristics as the solid one. By employing a beamforming technique, the dominant source is found at the trailing edge for the solid and blocked trailing edges, while for the fully porous one, the dominant source is located near the solid–porous junction. The analysis of the scattered sound suggests that the permeability of the porous trailing edge allows for acoustic scattering along the porous medium surface that promotes destructive interference, and in turn, attenuates far-field noise intensity. The spectra and spanwise coherence of surface pressure fluctuations at the trailing edge are hardly affected by the presence of the porous material, which are found to be insufficient to justify the noise reduction. The flow field inside the porous medium is also examined to explain the differences between the fully porous and blocked-porous trailing edges. While the mean velocity components are similar for both, substantial difference is found for the velocity fluctuations. The impedance of the porous medium is computed as the ratio of velocity and pressure fluctuations. Unlike the blocked variant, the impedance in the fully porous trailing edge gradually decreases along the downstream direction, which leads to the distributed noise scattering along the porous medium surface. Additionally, the scattering efficiency at the actual trailing edge location is reduced due to the smaller impedance discontinuity.