Adverse pressure gradient (APG) turbulent boundary layers (TBL) require an understanding of the details of the pressure gradient, or history effect, to characterize the associated variation of spatiotemporal turbulent statistics. The streamwise-varying mean pressure gradient is reflected in the streamwise developing mean flow field and thus resolvent analysis, which captures the amplification of the Navier–Stokes equations linearized about the turbulent mean, can be used to understand linear amplification in APG TBLs. In particular, by using a biglobal approach in which the amplification is characterized by a temporal frequency and spanwise wavenumber, the streamwise and wall-normal inhomogeneities of the APG TBL can be resolved and related to the APG history. The linear response is able to identify multiscale phenomena, identifying a near-wall peak with $\lambda _{z}^+\approx 100$ for zero pressure gradient TBLs and mild to moderate APG TBLs as well as large-scale modes whose amplification increases with APG strength and Reynolds number. It is shown that the monotonic growth in the turbulent statistics with increasing APG is reflected in the linear growth in the associated resolvent amplification. Collapse in the Reynolds stresses is obtained through an augmented hybrid velocity scale, which replaces the local APG strength measure in the hybrid velocity scale presented in Romero et al. (Intl J. Heat Fluid Flow, vol. 93, 2022, 108885) with a velocity that encapsulates the pressure gradient history. While this resolvent approach is applicable to any APG TBL, it is shown from a scaling analysis of the linearized Navier–Stokes equations that the linear growth observed in the resolvent amplification with the history effect is limited to near-equilibrium APG TBLs.

Turbulent boundary layers on immersed objects can be significantly altered by the pressure gradients imposed by the flow outside the boundary layer. The interaction of turbulence and pressure gradients can lead to complex phenomena such as relaminarization, history effects and flow separation. The angular momentum integral (AMI) equation (Elnahhas & Johnson, J. Fluid Mech., vol. 940, 2022, A36) is extended and applied to high-fidelity simulation datasets of non-zero pressure gradient turbulent boundary layers. The AMI equation provides an exact mathematical equation for quantifying how turbulence, free-stream pressure gradients and other effects alter the skin friction coefficient relative to a baseline laminar boundary layer solution. The datasets explored include flat-plate boundary layers with nearly constant adverse pressure gradients, a boundary layer over the suction surface of a two-dimensional NACA 4412 airfoil and flow over a two-dimensional Gaussian bump. Application of the AMI equation to these datasets maps out the similarities and differences in how boundary layers interact with favourable and adverse pressure gradients in various scenarios. Further, the fractional contribution of the pressure gradient to skin friction attenuation in adverse-pressure-gradient boundary layers appears in the AMI equation as a new Clauser-like parameter with some advantages for understanding similarities and differences related to upstream history effects. The results highlight the applicability of the integral-based analysis to provide quantitative, interpretable assessments of complex boundary layer physics.

Aqueous suspensions of cornstarch abruptly increase their viscosity on raising either shear rate or stress, and display the formation of large-amplitude waves when flowing down inclined channels. The two features have been recently connected using constitutive models designed to describe discontinuous shear thickening. By including time-dependent relaxation and spatial diffusion of the frictional contact density responsible for shear thickening, an analysis of steady sheet flow and its linear stability is presented. The inclusion of such effects is motivated by the need to avoid an ill-posed mathematical problem in thin-film theory and the resulting failure to select any preferred wavelength for unstable linear waves. Relaxation, in particular, eliminates an ultraviolet catastrophe in the spectrum of unstable waves and furnishes a preferred wavelength at which growth is maximized. The nonlinear dynamics of the unstable waves is briefly explored. It is found that the linear instability saturates once disturbances reach finite amplitude, creating steadily propagating nonlinear waves. These waves take the form of a series of steep, shear-thickened steps that translate relatively slowly in comparison with the mean flow.

We present an experimental study on the drag reduction by polymers in Taylor–Couette turbulence at Reynolds numbers ($Re$) ranging from $4\times 10^3$ to $2.5\times 10^4$. In this $Re$ regime, the Taylor vortex is present and accounts for more than 50 % of the total angular velocity flux. Polyacrylamide polymers with two different average molecular weights are used. It is found that the drag reduction rate increases with polymer concentration and approaches the maximum drag reduction (MDR) limit. At MDR, the friction factor follows the $-0.58$ scaling, i.e. $C_f \sim Re^{-0.58}$, similar to channel/pipe flows. However, the drag reduction rate is about $20\,\%$ at MDR, which is much lower than that in channel/pipe flows at comparable $Re$. We also find that the Reynolds shear stress does not vanish and the slope of the mean azimuthal velocity profile in the logarithmic layer remains unchanged at MDR. These behaviours are reminiscent of the low drag reduction regime reported in channel flow (Warholic et al., Exp. Fluids, vol. 27, no. 5, 1999, pp. 461–472). We reveal that the lower drag reduction rate originates from the fact that polymers strongly suppress the turbulent flow while only slightly weaken the mean Taylor vortex. We further show that polymers steady the velocity boundary layer and suppress the small-scale Görtler vortices in the near-wall region. The former effect reduces the emission rate of both intense fast and slow plumes detached from the boundary layer, resulting in less flux transport from the inner cylinder to the outer one and reduces energy input into the bulk turbulent flow. Our results suggest that in turbulent flows, where secondary flow structures are statistically persistent and dominate the global transport properties of the system, the drag reduction efficiency of polymer additives is significantly diminished.

We model transient mushy-layer growth for a binary alloy solidifying from a cooled boundary, characterising the impact of liquid composition and thermal growth conditions on the mush porosity and growth rate. We consider cooling from a perfectly conducting isothermal boundary, and from an imperfectly conducting boundary governed by a linearised thermal boundary condition. For an isothermal boundary we characterise different growth regimes depending on a concentration ratio, which can also be viewed as characterising the ratio of composition-dependent freezing point depression versus the temperature difference across the mushy layer. Large concentration ratio leads to high porosity throughout the mushy layer and an asymptotically simplified model for growth with an effective thermal diffusivity accounting for latent heat release from internal solidification. Low concentration ratio leads to low porosity throughout most of the mushy layer, except for a high-porosity boundary layer localised near the mush–liquid interface. We identify scalings for the boundary-layer thickness and mush growth rate. An imperfectly conducting boundary leads to an initial lag in the onset of solidification, followed by an adjustment period, before asymptoting to the perfectly conducting state at large time. We develop asymptotic solutions for large concentration ratio and large effective heat capacity, and characterise the mush structure, growth rate and transition times between the regimes. For low concentration ratio the high porosity zone spans the full mush depth at early times, before localising near the mush–liquid interface at later times. Such variation of porosity has important implications for the properties and biological habitability of mushy sea ice.