Coherent features of a turbulent Mach 0.9, Reynolds number $10^{6}$ jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

We present the solution of the idealized steady-state gravity current of height $h$ and density $\unicode[STIX]{x1D70C}_{1}$ that propagates into an ambient motionless fluid of height $H$ and density $\unicode[STIX]{x1D70C}_{2}$ with an upper surface open to the atmosphere (open channel) at high Reynolds number. The current propagates with speed $U$ and causes a depth decrease $\unicode[STIX]{x1D712}$ of the top surface. This is a significant extension of Benjamin’s (J. Fluid Mech., vol. 31, 1968, pp. 209–248) seminal solution for a fixed-top channel ($\unicode[STIX]{x1D712}=0$). In the present case, the determination of $\unicode[STIX]{x1D712}$ is part of the problem. The dimensionless parameters of the problem are $a=h/H$ and $r=\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}$. We show that a control-volume analysis determines $\tilde{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D712}/H$ and $\mathit{Fr}=U/(g^{\prime }h)^{1/2}$ as functions of $a$ and $r$, where $g^{\prime }=(r^{-1}-1)g$ is the reduced gravity. The system satisfies balance of volume and momentum (explicitly), and vorticity (implicitly). We develop two branches (models) of solution for the control-volume balances. The first assumes that the outflow velocity $u_{2}$ is $z$-independent, accompanied by a similarly homogeneous headloss; this model is exactly the counterpart of Benjamin’s solution ($z$ is the vertical coordinate). The second model assumes that $u_{2}$ increases from $0$ in a layer of finite thickness $\unicode[STIX]{x1D702}$ (say) above the $z=h$ interface, and the headloss occurs in this layer only (a vortex wake). The $\mathit{Fr},\tilde{\unicode[STIX]{x1D712}}$ results of these models are in excellent agreement, and this increases the confidence concerning the physical relevance of the analysis. (The solution is obtained numerically.) The predicted flows are in general dissipative, and thus physically energetically valid only for $a\leqslant a_{max}(r)\approx 0.5r$, where non-negative dissipation appears. In this range of validity, the open-surface $\mathit{Fr}(a,r)$ is smaller than Benjamin’s $\mathit{Fr}_{b}(a)$, but the reduction is not dramatic, typically a few per cent. The current with thickness $h=a_{max}H$ is energy conserving. For given $r$, the energy conserving $a_{max}(r)$ provides the maximum height depression $\tilde{\unicode[STIX]{x1D712}}_{max}(r)\approx 0.45r^{1/4}(1-r)$. The largest possible depression is $\tilde{\unicode[STIX]{x1D712}}_{max}\approx 0.25$ at $r\approx 0.20$, and this corresponds to the minimum of $\mathit{Fr}(a,r)/\mathit{Fr}_{b}(a)$ which is approximately 0.75. For $r<0.68$, the propagation is subcritical; for $r>0.68$, the subcritical flow condition reduces the applicability slightly to $a\leqslant a_{\mathit{crit}}\approx 0.345$. For a Boussinesq system with $r\approx 1$, we obtain $\tilde{\unicode[STIX]{x1D712}}\ll 1$, and the present $\mathit{Fr}$, dissipation and $a_{\mathit{crit}}$ results differ only slightly from Benjamin’s classical predictions, as expected.

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around $47^{\circ }$, for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below $90^{\circ }$ can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

Dense suspensions are non-Newtonian fluids that exhibit strong shear thickening and normal stress differences. Using numerical simulation of extensional and shear flows, we investigate how rheological properties are determined by the microstructure that is built under flows and by the interactions between particles. By imposing extensional and shear flows, we can assess the degree of flow-type dependence in regimes below and above thickening. Even when the flow-type dependence is hindered, non-dissipative responses, such as normal stress differences, are present and characterise the non-Newtonian behaviour of dense suspensions.