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We expand on the method of sequential filtering for calculating the spectra of inhomogeneous fields. Sadek and Aluie [Phys. Rev. Fluids 3, 124610 (2018)] showed that the filtering kernel has to have at least p vanishing moments to extract a power-law spectrum k−α with α<p+2 by low-pass filtering. Here, we show that sequential high-pass filtering allows for extracting steeper spectra with α<2p+3 using the same pth order kernel. For example, the spectrum of a field that is shallower than k−5 can be extracted by sequential high-pass filtering the field using any first-order kernel such as a Gaussian or top-hat. Finally, we demonstrate how the second-order structure function fails to capture spectral peaks because it cannot detect scaling that is too shallow. 

We apply Lagrangian particle tracking to the two-dimensional single-mode Rayleigh–Taylor (RT) instability to study the dynamical evolution of fluid interface. At the onset of the nonlinear RT stage, we select three ensembles of tracer particles located at the bubble tip, at the spike tip, and inside the spiral of the mushroom structure, which cover most of the interfacial region as the instability develops. Conditional statistics performed on the three sets of particles and over different RT evolution stages, such as the trajectory curvature, velocity, and acceleration, reveals the temporal and spatial flow patterns characterizing the single-mode RT growth. The probability density functions of tracer particle velocity and trajectory curvature exhibit scalings compatible with local flow topology, such as the swirling motion of the spiral particles. Large-scale anisotropy of RT interfacial flows, measured by the ratio of horizontal to vertical kinetic energy, also varies for different particle ensembles arising from the differing evolution patterns of the particle acceleration. In addition, we provide direct evidence to connect the RT bubble re-acceleration to its interaction with the transported fluid from the spike side, due to the shear driven Kelvin–Helmholtz instability. Furthermore, we reveal that the secondary RT instability inside the spiral, which destabilizes the spiraling motion and induces complex flow structures, is generated by the centrifugal acceleration. 

We study energy scale transfer in Rayleigh–Taylor (RT) flows by coarse graining in physical space without Fourier transforms, allowing scale analysis along the vertical direction. Two processes are responsible for kinetic energy flux across scales: baropycnal work $$\varLambda$$ , due to large-scale pressure gradients acting on small scales of density and velocity; and deformation work $$\varPi$$ , due to multiscale velocity. Our coarse-graining analysis shows how these fluxes exhibit self-similar evolution that is quadratic-in-time, similar to the RT mixing layer. We find that $$\varLambda$$ is a conduit for potential energy, transferring energy non-locally from the largest scales to smaller scales in the inertial range where $$\varPi$$ takes over. In three dimensions, $$\varPi$$ continues a persistent cascade to smaller scales, whereas in two dimensions $$\varPi$$ rechannels the energy back to larger scales despite the lack of vorticity conservation in two-dimensional (2-D) variable density flows. This gives rise to a positive feedback loop in 2-D RT (absent in three dimensions) in which mixing layer growth and the associated potential energy release are enhanced relative to 3-D RT, explaining the oft-observed larger $$\alpha$$ values in 2-D simulations. Despite higher bulk kinetic energy levels in two dimensions, small inertial scales are weaker than in three dimensions. Moreover, the net upscale cascade in two dimensions tends to isotropize the large-scale flow, in stark contrast to three dimensions. Our findings indicate the absence of net upscale energy transfer in three-dimensional RT as is often claimed; growth of large-scale bubbles and spikes is not due to ‘mergers’ but solely due to baropycnal work $$\varLambda$$ .