Mass transport induced by group-forced subharmonic waves (infragravity waves) is investigated in the present study. A theoretical solution for subharmonic waves’ kinematic contributions to fourth-order mass transport and drift velocity has been proposed for any depth and bandwidth for the first time. This model is validated using particle-tracking simulations driven by the flow field generated by the SWASH. The subharmonic-induced mass transport solution is a weighted sum of the subharmonic velocity variance spectrum and velocity skewness bispectrum due to the triad-difference interaction among two primary and one subharmonic components. For narrow-banded waves with long wave group relative to depth, the weightings become independent of spectral components, and the solution is recovered in the time domain. Two mechanisms contributing to mass transport were identified: a forward drift resulting from self-interaction similar to Stokes drift, and a depth-decaying backward drift induced by negative subharmonic velocity skewness due to the anti-phase coupling between subharmonics and wave groups. For narrow-banded waves the forward transport surpasses the backward transport for kh< 0.72, where k is the short wave wavenumber and h is the water depth. For other waves, the critical kh for this phenomenon decreases with increasing wave period and bed slope and decreasing bandwidth. At greater depths or steeper bed slopes, near-surface backward transport predominates over forward transport; at shallower depths or gentler slopes, forward transport is dominant throughout the water column. Although smaller than Stokes transport by short waves, the subharmonic wave-induced mass transport can affect the long-term trajectory of a floating and suspended particle. This study provides the first evidence and insight for the influences of group-forced subharmonics on vertically varying mass transport from the ocean surface to seabed in coastal environments.

Convection in planetary environments is often modelled using stress-free boundary conditions, with diffusion-free geostrophic turbulence scalings frequently assumed. However, key questions remain about whether rotating convection with stress-free boundary conditions truly achieves the diffusion-free geostrophic turbulence regime. Here, we investigated the scaling behaviours of the Nusselt number ($Nu$), Reynolds number (${Re}$) and dimensionless convective length scale ($\ell /H$, where $H$ is the height of the domain) in rotating Rayleigh–Bénard convection under stress-free boundary conditions within a Boussinesq framework. Using direct numerical simulation data for Ekman number $Ek$ down to $5\times 10^{-8}$, Rayleigh number $Ra$ up to $5\times 10^{12}$, and Prandtl number $Pr = 1$, we show that the diffusion-free scaling of the heat transfer $Nu - 1 \sim Ra^{3/2}\, Pr^{-1/2}\, Ek^2$ alone does not necessarily imply that the flow is in a geostrophic turbulence regime. Under the stress-free conditions, ${Re}$ and $\ell /H$ deviate from the diffusion-free scalings, indicating a dependence on molecular diffusivity. We propose new non-diffusion-free scaling relations for this diffusion-free heat transfer regime with stress-free boundary conditions: $\ell /H \sim Ra^{1/8}\, Pr^{-1/8}\, Ek^{1/2}$ and ${Re} \sim Ra^{11/8}\, Pr^{-11/8}\, Ek^{3/2}$. Our findings highlight the need to assess both thermal and dynamic characteristics to confirm geostrophic turbulence.

In turbulent pipe flows, drag-reducing polymers are commonly used to reduce skin-friction drag; however, predicting this reduction in industry applications, such as crude oil pipelines, remains challenging. The skin-friction coefficient ($C_f$) of polymer drag-reduced turbulent pipe flows can be related to three dimensionless parameters: the solvent Reynolds number ($Re_s$), the Weissenberg number ($Wi$) and the ratio of solvent viscosity ($\eta _s$) to zero-shear-rate viscosity ($\eta _0$), denoted as $\beta$. The function that relates these four dimensionless numbers was determined using experiments of various pipe diameters ($D$), flow velocities ($U$) and drag-reducing polyacrylamide solutions. The experiments included measurements of streamwise pressure drop ($\Delta P$) for determining $C_f$, and measurements of shear viscosity ($\eta$) and elastic relaxation time ($\lambda$). This experimental campaign involved 156 flow conditions, each characterised by distinct values for $C_f$, $Re_s$, $Wi$ and $\beta$. Experimental results demonstrated good agreement with the relationship: $C_f^{-1/2} = \widehat {A}\log _{10}(Re_sC_f^{1/2})+\widehat {B}$, where $\widehat {A} = 27.6(Wi \beta )^{0.346}$ and $\widehat {B} = 122/15-58.9(Wi \beta )^{0.346}$. Based on this relationship, onset and maximum drag reduction are predicted to occur when $Wi \beta$ equals $3.76 \times 10^{-3}$ and $3.40 \times 10^{-1}$, respectively. This function can predict $C_f$ of dilute polyacrylamide solutions based on predefined parameters (bulk velocity, pipe diameter, density, solvent viscosity) and two measurable rheological properties of the solution (shear viscosity and elastic relaxation time) with an accuracy of $\pm 9.36$ %.

Turbulence closures are essential for predictive fluid flow simulations in both natural and engineering systems. While machine learning offers promising avenues, existing data-driven turbulence models often fail to generalise beyond their training datasets. This study identifies the root cause of this limitation as the conflation of generalisable flow physics and dataset-specific behaviours. We address this challenge using symbolic regression, which yields interpretable, white-box expressions. By decomposing the learned corrections into inner-layer, outer-layer and pressure-gradient components, we isolate universal physics from flow-specific features. The model is trained progressively using high-fidelity datasets for plane channel flows, zero-pressure-gradient turbulent boundary layers (ZPGTBLs), and adverse pressure-gradient turbulent boundary layers (PGTBLs). For example, direct application of a model trained on channel flow data to ZPGTBLs results in incorrect skin friction predictions. However, when only the generalisable inner-layer component is retained and combined with an outer-layer correction specific to ZPGTBLs, predictions improve significantly. Similarly, a pressure-gradient correction derived from PGTBL data enables accurate modelling of aerofoil flows with both favourable and adverse pressure gradients. The resulting symbolic corrections are compact, interpretable, and generalise across configurations – including unseen geometries such as aerofoils and Reynolds numbers outside the training set. The models outperform baseline Reynolds-averaged Navier–Stokes closures (e.g. the Spalart–Allmaras and shear stress transport models) in both a priori and a posteriori tests. These results demonstrate that explicit identification and retention of generalisable components is key to overcoming the generalisation challenge in machine-learned turbulence closures.

Compressible jets impinging on a perpendicular surface can produce high-intensity, discrete-frequency tones. The character of these tones is a function of nozzle shape, jet Mach number, impingement-plate geometry, and the distance between nozzle and plate. Though it has long been recognised that these tones are associated with a resonance cycle, the exact mechanism by which they are generated has remained a topic of some debate. In this work, we present evidence for a number of distinct tone-generation mechanisms, reconciling some of the different findings of prior authors. We demonstrate that the upstream-propagating waves that close resonance can be confined within the jet, or external to it. These waves can be either weak and relatively linear, or strong and nonlinear from their inception. The waves can undergo coalescence or merging, and in some configurations, pairs of waves rather than singletons appear. We discuss both historical and new evidence for multiple distinct processes by which upstream-propagating waves are produced: direct vortex sound, shock leakage, wall-jet-boundary fluctuations, and wall-jet shocklets. We link these various mechanisms to the disparate collection of upstream-propagating waves observed in the data. We also demonstrate that multiple mechanisms can be provoked by a single vortex, providing an explanation as to why sometimes pairs of waves or merging waves are observed. Through this body of work, we demonstrate that rather than being in opposition, the various pieces of past research on this topic were simply identifying different mechanisms that can support resonance.

The paper by Pružina et al. (2025) J. Fluid Mech. 1009, sheds new light on the physical processes responsible for the formation of distinct layers in double-diffusive convection. Towards this end, it discusses direct numerical simulation results within the framework of sorted buoyancy coordinates. In particular, it demonstrates that the eddy diffusivity is negative everywhere, including in the interior of the well-mixed layers. This approach holds promise for analysing other, closely related, flow configurations that give rise to the emergence of pronounced layering features.

Dot array deposition through electrohydrodynamic (EHD) printing is widely used for high resolution and material utilization advantages. However, the conventional printing method is subject to a printing frequency limit known as the capillary frequency of the meniscus oscillation, where the jet directly contacts the substrate. This makes the printing frequency of EHD printing maintain at a low level and that is difficult to improve. In this work, a method for high-frequency EHD printing through continuous pinch-off is proposed. The characteristic frequency is broken through. A model is established to reveal the printing mechanism by combining the Poisson–Nernst–Planck equation and the phase field method. The unreal charge leakage is prevented by constructing a transition function for the fluid’s properties. The stability of the Taylor cone’s deformation and the droplets’ generation is studied. The measurement criterion for printing frequency is determined. The suitable printing height that can prevent the jet from directly contacting the substrate is obtained by investigating its influence on the printing states and frequency. The phase diagram considering the liquid’s conductivity and viscosity is presented to distinguish whether the printing is based on the end-pinching or Rayleigh–Plateau instability. The influence of the conductivity, viscosity, flow rate and printing voltage on the printing frequencies is studied quantitatively. Finally, scaling laws for printing frequency are proposed by theoretical analyses and summarizing the numerical data. This work could be beneficial for further enhancing the printing frequency of EHD printing.