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The acoustic receptivity of Tollmien–Schlichting (TS) waves due to two-dimensional sharp-edged rectangular bumps and gaps in a compressible boundary layer was investigated by direct numerical simulations. The conclusions were based on a new procedure proposed for obtaining the receptivity amplitudes which appeared to be more robust and accurate than previous ones. The procedure is particularly important for the correct evaluation of the receptivity of gaps. The receptivity amplitudes for gaps were smaller than those for bumps, except for the nominally zero height/depth roughness element, where, consistent with a linear behaviour, they had the same absolute value. The procedure also revealed in detail the behaviour in the region downstream of the roughness element where the TS wave is formed (the formation region). This region extends for approximately 
$50\delta ^*_{b}$, regardless of bump height or gap depth. For bumps, the receptivity scaled superlinearly with bump height while for the gaps it scaled sublinear with depth. This behaviour is associated with the different velocity profiles caused by bumps and gaps in the formation region. We also discussed issues regarding comparison with experiments. Investigation of the effect of compressibility confirmed that, in the subsonic regime, the receptivity reduces with Mach number. Finally, we addressed the receptivity scaling with the acoustic wave amplitude. It was found that the receptivity scales linearly with the acoustic wave amplitude in a range for which experiments available in the literature indicated a superlinearly scaling. Reasons for these conflicting results are discussed.
Given a countable group G and two subshifts X and Y over G, a continuous, shift-commuting map 
$\phi : X \to Y$ is called a homomorphism. Our main result states that if G is locally virtually nilpotent, X is aperiodic, and Y has the finite extension property, then there exists a homomorphism 
$\phi : X \to Y$. By combining this theorem with the main result of [1], we obtain that if the same conditions hold, and if additionally the topological entropy of X is less than the topological entropy of Y and Y has no global period, then X embeds into Y.
The motion and deformation of a neutrally buoyant drop in a rectangular channel experiencing a pressure-driven flow at a low Reynolds number has been investigated both experimentally and numerically. A moving-frame boundary-integral algorithm was used to simulate the drop dynamics, with a focus on steady-state drop velocity and deformation. Results are presented for drops of varying undeformed diameters relative to channel height (
$D/H$), drop-to-bulk viscosity ratio (
$\lambda$), capillary number (
$Ca$, ratio of deforming viscous forces to shape-preserving interfacial tension) and initial position in the channel in a parameter space larger than considered previously. The general trend shows that the drop steady-state velocity decreases with increasing drop diameter and viscosity ratio but increases with increasing 
$Ca$. An opposite trend is seen for drops with small viscosity ratio, however, where the steady-state velocity increases with increasing 
$D/H$ and can exceed the maximum background flow velocity. Experimental results verify theoretical predictions. A deformable drop with a size comparable to the channel height when placed off centre migrates towards the centreline and attains a steady state there. In general, a drop with a low viscosity ratio and high capillary number experiences faster cross-stream migration. With increasing aspect ratio, there is a competition between the effect of reduced wall interactions and lower maximum channel centreline velocity at fixed average velocity, with the former helping drops attain higher steady-state velocities at low aspect ratios, but the latter takes over at aspect ratios above approximately 1.5.
Experimental and numerical observations in turbulent shear flows point to the persistence of the anisotropy imprinted by the large-scale velocity gradient down to the smallest scales of turbulence. This is reminiscent of the strong anisotropy induced by a mean passive scalar gradient, which manifests itself by the ‘ramp–cliff’ structures. In the shear flow problem, the anisotropy can be characterised by the odd-order moments of 
$\partial _y u$, where 
$u$ is the fluctuating streamwise velocity component, and 
$y$ is the direction of mean shear. Here, we extend the approach proposed by Buaria et al. (Phys. Rev. Lett., 126, 034504, 2021) for the passive scalar fields, and postulate that fronts of width 
$\delta \sim \eta Re_\lambda ^{1/4}$, where 
$\eta$ is the Kolmogorov length scale, and 
$Re_\lambda$ is the Taylor-based Reynolds number, explain the observed small-scale anisotropy for shear flows. This model is supported by the collapse of the positive tails of the probability density functions (PDFs) of 
$(\partial _y u)/(u^{\prime }/\delta )$ in turbulent homogeneous shear flows (THSF) when the PDFs are normalised by 
$\delta /L$, where 
$u^{\prime }$ is the root-mean-square of 
$u$ and 
$L$ is the integral length scale. The predictions of this model for the odd-order moments of 
$\partial _y u$ in THSF agree well with direct numerical simulation (DNS) and experimental results. Moreover, the extension of our analysis to the log-layer of turbulent channel flows (TCF) leads to the prediction that the odd-order moments of order 
$p (p \gt 1)$ of 
$\partial _y u$ have power-law dependencies on the wall distance 
$y^{+}$: 
$\langle (\partial _y u)^p \rangle /\langle (\partial _y u)^2 \rangle ^{p/2} \sim (y^{+})^{(p-5)/8}$, which is consistent with DNS results.
Given a fixed k-uniform hypergraph F, the F-removal lemma states that every hypergraph with few copies of F can be made F-free by the removal of few edges. Unfortunately, for general F, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which F one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when 
$k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all 
$k\gt2$, namely, that the only 
$k$-graphs 
$F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.
We define ‘surface layer’ (SL) as an inertia-dominated turbulence region outside a viscous or roughness surface-adjacent sub-layer (SAS) that is characterised by linear scaling of specific coherence length scales on wall-normal distance, 
$z$. We generalise the mechanisms that underlie the formation of the classical inertial SL in the shear-dominated turbulent boundary layer (TBL) to wall-bounded turbulent flows with zero mean shear. Using particle image velocimetry data from two wind tunnel facilities, we contrast the classical TBL SL with a non-classical shear-free SL generated within grid turbulence advected over an impermeable plate using two grids with different turbulence length scales. Integral-scale variations with 
$z$ and other statistics are quantified. In both shear-dominated and shear-free SLs we observe well-defined linear increases in 
$z$ of the streamwise integral scale of vertical velocity fluctuations. In grid turbulence the shear-free SL initiates just above the SAS that confines friction-generated motions. By contrast, the TBL SL forms with non-zero mean shear rate that extends streamwise coherence lengths of streamwise fluctuations. In both flow classes only the integral scales of vertical fluctuating velocity increase linearly with 
$z$, indicating that the SL is generated by the blockage of vertical fluctuations in the vertical. Whereas the SAS in the TBL is much thinner than in the grid-turbulence flows, the generation of a shear-free SL by the interaction of turbulence eddies and a surface depends on the relative thinness of the SAS. We conclude that the common generalisable SL mechanism is direct blockage of vertical fluctuations by the impermeable surface.
Significant progress has been made in understanding planetary core dynamics using numerical models of rotating convection (RC) in spherical shell geometry. However, the behaviour of forces in these models within various dynamic regimes of RC remains largely unknown. Directional anisotropy, scale dependence and the role of dynamically irrelevant gradient contributions in incompressible flows complicate the representation of dynamical balances in spherical shell RC. In this study, we systematically compare integrated and scale-dependent representations of mean and fluctuation forces and curled forces (which contain no gradient contributions) separately for the three components (
$\hat {r},\hat {\theta },\hat {\phi }$). The analysis is performed with simulations in a range of convective supercriticality 
$Ra_T/Ra_T^{c}=1.2{-}297$ where 
$Ra_T$ and 
$Ra^{c}_T$ are the Rayleigh and critical Rayleigh numbers, respectively and Ekman number 
$E=10^{-3}{-}10^{-6}$, with fixed Prandtl number 
$Pr=1$, along with no-slip and fixed flux boundaries. We have excluded regions from each boundary of the spherical shell, with a thickness equivalent to ten velocity boundary layers, which provides a consistent representation of the bulk dynamics between the volume-averaged force and curled force balance in the parameter space studied. Radial, azimuthal and co-latitudinal components exhibit distinct force and curled force balances. The total magnitudes of the mean forces and mean curled forces exhibit a primary thermal wind balance; the corresponding fluctuating forces are in a quasi-geostrophic primary balance, while the fluctuating curled forces transition from a Viscous–Archimedean–Coriolis balance to an Inertia–Viscous–Archimedean–Coriolis balance with increasing 
$Ra_T/Ra_T^{c}$. The curled force balances are more weakly scale-dependent compared to the forces, and do not show clear cross-over length scales. The fluctuating force and curled force balances are broadly consistent with three regimes of RC (weakly nonlinear, rapidly rotating and weakly rotating), but do not exhibit sharp changes with 
$Ra_{T}/Ra_{T}^{c}$, which inhibits the identification of precise regime boundaries from these balances.
