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.