We develop saddle-point approximations of the probability density function (PDF) for the surface elevation in nonlinear, irregular seas, without any assumptions regarding uni-directionality of the wave field or narrow-bandedness of the spectrum. A new, closed-form approximation of the PDF at third order is derived, accounting for both skewness and kurtosis. The approximate PDF matches known asymptotics of the third-order positive tail exactly and is free of unphysical oscillations in the negative tail, which widely plague existing PDFs describing nonlinear seas. A normalisation procedure is introduced to enforce unit integral, zero mean and unit variance. We further show that the input cumulants can be adjusted to likewise yield prescribed skewness and kurtosis. Comparison of the new third-order PDF against challenging datasets stemming from irregular wave theory, a fully nonlinear wave model as well as laboratory experiments establishes accuracy. In the Appendix we additionally develop a novel perturbation method, which is used to derive the full form of the asymptotic positive and negative tails at fourth and fifth order, including (previously unknown) prefactors. Collectively, it is shown that even-order solutions inevitably involve oscillatory negative tails, whereas odd orders admit non-oscillatory negative-tail solutions.
Transversely segregated longitudinal flow bands, or stripes, can form in rapid thin dense granular flows due to the formation of a series of counter-rotating secondary vortices whose axes of rotation are aligned with the flow direction (Pearse et al. 2026 J. Fluid Mech. 1032 , A4). The formation of the secondary vortices is still a subject of active research, but it is likely due to a combination of dilatation and second-normal-stress differences. This paper assumes the existence of the secondary flow and investigates how large and small particles are able to segregate to form stable compositional stripes. In the absence of diffusion, the secondary flow rotates the sharp interface between large particles (above) and small particles (below), around to the vertical, where it breaks to form a series of pure large- and pure small-particle bands that extend through the flow depth, and are separated by a series of breaking-size-segregation waves. Diffusion can compete against this overturning effect. In particular, there is a region of parameter space where the diffusion and the secondary recirculation balance to create a stable wavy interface between large and small particles, which can suppress the development of surface compositional differences. In this regime, a depth-averaged theory for segregation is derived that captures the three-dimensional time-dependent development of the particle-size distribution from a uniformly inversely graded inflow into a series of transversely segregated stripes. The solution is in good quantitative agreement with the evolution of the surface particle-size distribution observed in experiments.
In this work, we describe our experimental observations of trapped waves and guided jet waves in schlieren images of jets operating at jet Mach numbers upper M Subscript j Baseline equals 0.9 M j = 0.9 $M_{\!j}=0.9$ and upper M Subscript j Baseline equals 0.983 M j = 0.983 $M_{\!j}=0.983$ produced by circular nozzles. Results from a chevron nozzle are also included as a special case. We establish that: (i) several families of trapped waves are detectable in our experimental images, including higher order axisymmetric and helical modes; (ii) the dispersion relationship of these waves is in agreement with the vortex sheet model of Tam & Hu ( J. Fluid Mech. , 1989, vol. 201, pp. 447–483) and (iii) the introduction of chevrons does not significantly affect the presence or intensity of the first several peaks associated with the trapped wave modes. Finally, we observe that all detected trapped waves have subsonic group velocities with respect to the quiescent stream.
We consider the generation of extreme waves of flexible features in deep-water surface gravity waves, in the context of the fully nonlinear wave model, i.e. Euler equations of the free surface. This is achieved through a recently developed GenWave approach, coupled with the high-order spectral method to solve the Euler equations. Through an iterative refinement procedure, GenWave progressively refines the initial phases associated with a given wave spectrum, which leads to extreme-wave generation at specified design time and location, with their probability of exceedance quantified. In this study, we consider two types of extreme waves, enabled by the flexibility of GenWave. The first type is a spatially and temporally localised rogue wave, and second is a spatially localised extreme-wave group propagating at a constant speed. The two generated extreme-wave profiles are compared with their prototypes – the Peregrine soliton and regular soliton (or bright soliton) in the context of the nonlinear Schrödinger equation (NLS), with reasonable agreement. For the persistent group, comparison with linear evolution and the NLS identifies nonlinear persistence, faster propagation and bound-wave content beyond the leading-order NLS description.
This study investigates the aerodynamic noise control mechanisms of a forced rotating circular cylinder at a subcritical Reynolds number ( upper R e equals 4.7 times 10 Superscript 4 R e = 4.7 × 10 4 $Re = 4.7 \times 10^4$ ) using large-eddy simulation coupled with the Ffowcs Williams–Hawkings acoustic analogy, validated by wind tunnel experiments. Five representative rotation ratios, alpha equals 0 α = 0 $\alpha =0$ , 0.5 0.5 $0.5$ , 1 1 $1$ , 2 2 $2$ and 4 4 $4$ , are analysed to relate wall-pressure fluctuations and wake organisation to the radiated noise. At alpha equals 0.5 α = 0.5 $\alpha =0.5$ , the natural shedding tone separates into two components, upper S t 01 almost equals 0.18 S t 01 ≈ 0.18 $St_{01}\approx 0.18$ and upper S t 02 almost equals 0.23 S t 02 ≈ 0.23 $St_{02}\approx 0.23$ . Their three-dimensional (3-D) structures exhibit spanwise phase dislocation and reduced coherence, explaining the decrease in tonal radiation despite locally enhanced wall-pressure fluctuations. Coherent shedding partially recovers at alpha equals 1 α = 1 $\alpha =1$ , whereas alpha equals 2 α = 2 $\alpha =2$ gives the strongest tonal-noise reduction among the investigated cases, accompanied by weakened wake fluctuations, spanwise coherence and wall–wake correlation. At alpha equals 4 α = 4 $\alpha =4$ , the classical Kármán mode is suppressed and localised near-wall helical structures are associated with a broadband elevation near upper S t almost equals 0.1 S t ≈ 0.1 $St\approx 0.1$ . The results show that rotation controls noise through changes in both unsteady-loading amplitude and 3-D source organisation.
