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Particle effects on the amplitude modulation are investigated in this study based on observational data with various mass loading acquired from long-term measurements of aeolian sandstorms in high-Reynolds-number ($Re_{\tau }\sim O(10^6)$) near-neutral atmospheric surface layers. In both particle-laden and unladen flows, in addition to the positive top–down modulation behaviour in the logarithmic region, a significant modulation effect that exists for some specific motions is also found for the single-point amplitude modulation. The most energetic turbulent motions exhibit the strongest modulation effect, and the modulating signals do not change with the small-scale motions being modulated. In particle-laden flows, the length of the most energetic structure is almost constant, thus the scales of the modulating signal and carrier signal are hardly affected by particles. However, the addition of particles changes the distribution of energy between multi-scale turbulent motions. The kinetic energy of the large-scale component is less enhanced than the total kinetic energy by particles. This leads to a reduced energy proportion of the large-scale component and an augmented one of the small-scale component. Moreover, the particles produce a large damping in the degree of the amplitude modulation and move down the positions of the modulating signals and carrier signals corresponding to the strongest inter-layer modulation, but the damping is weakened with the wall-normal distance due to the decreased mass loading. This study may provide a more general insight into the modulation mechanism between multi-scale turbulent motions and the effect of particles on turbulence.
The spacetime structure is that of a manifold with a Lorentz metric and associated affine connection.
We introduce in §2.1 the concept of a manifold and in §2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in §2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined by a vector field gives the Lie derivative defined in §2.4; another differential operation which depends only on the manifold structure is exterior differentiation, also defined in that section. This operation occurs in the generalized form of Stokes’ theorem.
The connection is introduced in §2.5, defining the covariant derivative and the curvature tensor. The connection is related to the metric on the manifold in §2.6; the curvature tensor is decomposed into the Weyl tensor and Ricci tensor, which are related to each other by the Bianchi identities.
The induced metric and connection on a hypersurface are discussed in §2.7, and the Gauss–Codacci relations are derived. The volume element defined by the metric is introduced in §2.8, and used to prove Gauss’ theorem.
We experimentally explored the effect of single-sidewall cooling on Rayleigh–Bénard (RB) convection. Canonical RB was also studied to aid insight. The scenarios shared tank dimensions and bottom and top wall temperatures; the single sidewall cooling had the top wall temperature. Turbulence was explored at two canonical Rayleigh numbers, $Ra=1.6\times 10^{10}$ and $Ra=2\times 10^9$ under Prandtl number $Pr=5.4$. Particle image velocimetry described vertical planes parallel and perpendicular to the sidewall cooling. The two $Ra$ scenarios reveal pronounced changes in the flow structure and large-scale circulation (LSC) due to the sidewall cooling. The density gradient induced by the sidewall cooling led to asymmetric descending and ascending flows and irregular LSC. Flow statistics departed from the canonical case, exhibiting lower buoyancy effects, represented by an effective Rayleigh number with effective height dependent on the distance from the lateral cooling. Velocity spectra show two scalings, $\varPhi \propto f^{-5/3}$ Kolmogorov (KO41) and $\varPhi \propto f^{-11/5}$ Bolgiano (BO59) in the larger $Ra$; the latter was not present in the smaller set-up. The BO59 scaling with sidewall cooling appears at higher frequencies than its canonical counterpart, suggesting weaker buoyancy effects. The LSC core motions allowed us to identify a characteristic time scale of the order of vortex turnover time associated with distinct vortex modes. The velocity spectra of the vortex core oscillation along its principal axis showed a scaling of $\varPhi _c \propto f^{-5/3}$ for the single sidewall cooling, which was dominant closer there. It did not occur in the canonical case, evidencing the modulation of LSC oscillation on the flow.