We present a new Eulerian framework for the computation of turbulent compressible multiphase channel flows, specifically to assess turbulence modulation by dispersed particulate matter in dilute concentrations but with significant mass loadings. By combining a modified low-dissipation numerical scheme for the carrier gas phase and a quadrature-based moment method for the solid particle phase, turbulent statistics of the fluid phase and fluctuations of the particle phase may be obtained as both are resolved as coupled fields. Using direct numerical simulations, we demonstrate how this method effectively resolves the turbulent statistics, kinetic energy, skin friction drag, particle mass flow rate and interphase drag for moderate-Reynolds-number channel flows for the first time. Validation of our approach to the turbulent particle-free flow and the turbulent particle-laden flow proves the applicability of the carrier flow low-dissipation scheme to simulate relatively low-Mach-number compressible flows and of the quadrature-based moment method to simulate the particle phase as an Eulerian field. This study also rationalises the computed interphase drag modulation and total Reynolds shear stress results using a simplified analytical approach, revealing how the particle migration towards the wall can affect the drag between the two phases at different Stokes numbers and particle loadings. Furthermore, we show the effect of near-wall particle accumulation on the particle mass flow rate. Using our Eulerian approach, we also explore the complex interplay between the particles and turbulent fluctuations by capturing the preferential clustering of particles in turbulence streaks. This interplay leads to turbulence modulations similar to recent observations reported in prior computational works using Lagrangian simulations. Our study extends the applicability of the Eulerian approach to accurately study particle–fluid interactions in compressible turbulent flows by explicitly calculating the energy equations for both the particle phase and the carrier fluid motion. Since the formulation is compressible and includes energy equations for both the particle and carrier flow fields, future studies for compressible flows involving heat and mass transfer may be simulated using this methodology.

We explore the drawing of an axisymmetric viscoelastic tube subject to inertial and surface tension effects. We adopt the Giesekus constitutive model and derive asymptotic long-wave equations for weakly viscoelastic effects. Intuitively, one might imagine that the elastic stresses should act to prevent hole closure during the drawing process. Surprisingly, our results show that the hole closure at the take-up point is enhanced by elastic effects for most parameter values. However, the opposite is true if the tube has a sufficiently large hole size at the inlet nozzle of the device or if the axial stretching is sufficiently weak. We explain the physical mechanism underlying this phenomenon by examining how the second normal stress difference induced by elastic effects modifies the hole evolution process. We also determine how viscoelasticity affects the stability of the drawing process and show that elastic effects are always destabilising for negligible inertia. On the other hand, our results show that if the inertia is non-zero, elastic effects can be either stabilising or destabilising depending on the parameters.

We investigate the motion of weakly negatively buoyant spheres settling in surface gravity waves using laboratory experiments. The trajectories of the settling spheres are tracked over most of the water depth with simultaneous measurements of the background fluid flow. These experiments are conducted in the regime relevant for environmental and geophysical applications where both particle inertia and fluid inertia are important. Using these data, we show that the sphere motion is well described by the kinematic sum of the undisturbed fluid velocity and the particle terminal settling velocity as long as the fluid inertia is not too large. We show how this result can be understood in the context of an ad hoc Maxey–Riley–Gatignol-type equation where the drag on the particle is given by the Schiller–Naumann drag correlation. We also evaluate whether inertial particles experience enhanced settling in waves, finding that measurement uncertainties in the particle terminal settling velocity and the presence of Eulerian-mean flows do not allow the small percentage increase in the settling velocity to be measured. When the fluid inertia becomes large enough, we observe path instabilities caused by particle wake effects in both quiescent and wavy conditions. However, the particle velocity fluctuations associated with the path instabilities are unaffected by the background flow. The minimal influence of the wavy flow on the particle path instabilities is thought to be due to the large-scale separation between the waves and the particle.

This study investigates the strong influence of a splitter plate on two- and three-dimensional wake transitions of a circular cylinder. Direct numerical simulations and Floquet analyses are conducted over a parameter space including Reynolds numbers (Re) of 10–480 and non-dimensional plate lengths (L/D) of 0–6. With the increase in L/D, the critical Re for the onset of vortex shedding (Recr2D) increases monotonically. The delayed onset of vortex shedding with elongation of the body is physically explained. The critical Re for the onset of three-dimensionality (Recr3D) and the three-dimensional wake instability modes and structures are also significantly altered by the splitter plate. Compared with an isolated cylinder, the Recr3D for L/D = 1 is significantly reduced via a long wavelength mode, whereas the Recr3D for L/D = 2–6 is significantly increased via other modes. For each L/D, with increasing Re over the wake transition process, the spanwise wavelength of the wake structure gradually decreases, and the wake structure becomes increasingly chaotic. The strong influence of the splitter plate on the formation of the primary vortices and three-dimensional wake structures alter the hydrodynamic characteristics strongly. In particular, optimal lift reduction is achieved at L/D ∼ 1. In addition, the existence/absence of a hysteresis effect at the onset of three-dimensionality is identified by three methods. Among which, the method involving the Landau equation may be contaminated by initial transients induced by stable Floquet modes and may thus lead to a false conclusion on the existence/absence of hysteresis.

Sub-convective wall pressure fluctuations play a critical role in vibroacoustic and noise analyses of vehicle structures as they serve as the primary forcing function. However, measuring these fluctuations is challenging due to their weak pressure magnitudes, typically $10^{-3}{-}10^{-5}$ of convective fluctuations. This study introduces a non-intrusive measurement technique using an array of multi-pore Helmholtz resonator sensors to capture sub-convective fluctuations with high resolution. The array features large-area, spanwise-oriented sensors arranged linearly for optimal sampling. Results provide a continuous streamwise wavenumber–frequency spectrum, resolving sub-convective fluctuations with sufficient range and accuracy. Convergence analysis indicates that long sampling durations, $\mathcal{O}(10^6 \delta ^*/U_\infty )$, $\delta^*$ is the displacement thickness of the boundary layer. $U_\infty$ is the freestream velocity are necessary to capture true sub-convective levels. Comparisons with four existing wall pressure models, which account for sensor area averaging, reveal discrepancies in predicted levels, convection speed relations and convective ridge characteristics. Notably, the measured data align most closely with the Chase (1980, J. Sound Vib., vol.70, pp. 29–67) model at convective peak levels and in the sub-convective domain. However, the observed roll-off at wavenumbers exceeding the convective wavenumber decays more slowly than predicted, giving the convective ridge an asymmetric profile about the convective line. These findings underscore the need for improved wall pressure models that incorporate frequency-dependent convective speed relations, ridge asymmetry, and more accurate sub-convective levels. Further validation using a microphone array from Farabee & Geib (1991) confirms the accuracy of our measurements, which indicate sub-convective pressure levels lower than reported previously.

For Stokes waves in finite depth within the neighbourhood of the Benjamin–Feir stability transition, there are two families of periodic waves, one modulationally unstable and the other stable. In this paper we show that these two families can be joined by a heteroclinic connection, which manifests in the fluid as a travelling front. By shifting the analysis to the setting of Whitham modulation theory, this front is in wavenumber and frequency space. An implication of this jump is that a permanent frequency downshift of the Stokes wave can occur in the absence of viscous effects. This argument, which is built on a sequence of asymptotic expansions of the phase dynamics, is confirmed via energetic arguments, with additional corroboration obtained by numerical simulations of a reduced model based on the Benney–Roskes equation.

This paper explores the construction of quadratic Lyapunov functions for establishing the conditional stability of shear flows described by truncated ordinary differential equations, addressing the limitations of traditional methods like the Reynolds–Orr equation and linear stability analysis. The Reynolds–Orr equation, while effective for predicting unconditional stability thresholds in shear flows due to the non-contribution of nonlinear terms, often underestimates critical Reynolds numbers. Linear stability analysis, conversely, can yield impractically high limits due to subcritical transitions. Quadratic Lyapunov functions offer a promising alternative, capable of proving conditional stability, albeit with challenges in their construction. Typically, sum-of-squares programs are employed for this purpose, but these can result in sizable optimisation problems as system complexity increases. This study introduces a novel approach using linear transformations described by matrices to define quadratic Lyapunov functions, validated through nonlinear optimisation techniques. This method proves particularly advantageous for large systems by leveraging analytical gradients in the optimisation process. Two construction methods are proposed: one based on general optimisation of transformation matrix coefficients, and another focusing solely on the system’s linear aspects for more efficient Lyapunov function construction. These approaches are tested on low-order models of subcritical transition and a two-dimensional Poiseuille flow model with degrees of freedom nearing 1000, demonstrating their effectiveness and efficiency compared with sum-of-squares programs.