Common modal decomposition techniques for flow-field analysis, data-driven modelling and flow control, such as proper orthogonal decomposition and dynamic mode decomposition, are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid–structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of proper orthogonal decomposition and dynamic mode decomposition using direct numerical simulations of two canonical flow configurations at Mach 0.5, i.e. the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that application of LMA to steady non-uniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents. We examine the mathematical link between finite-time Lyapunov exponents and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.

When a volatile drop impacts on a superheated solid, air drainage and vapour generation conspire to create an intermediate gas layer that delays or even prevents contact between the liquid and the solid. In this article, we use high-speed synchronized reflection interference and total internal reflection imaging to measure the short-time dynamics of the intermediate gas film and to probe the transition between levitation and contact. We observe that the substrate temperature strongly affects the vertical position of the liquid–gas interface and that the dynamic Leidenfrost transition is influenced by both air and vapour drainage (i.e. gas drainage), and evaporation, the latter giving rise to hitherto unreported vertical oscillations of the gas film that can trigger liquid–solid contact. We first derive scaling relations for the height of the gas film trapped under the drop's centreline, called the dimple height, and the minimum film thickness at short times. The former is set by a competition between gas drainage and liquid inertia, similarly as for isothermal impacts, while the latter strongly depends on the vapour production. The gas pressure, at the location where the minimum thickness is reached, is determined by liquid inertia and vapour production and ultimately balanced by the increasing interfacial curvature, determining the minimum thickness. We show that, in the low impact velocity limit, the transient stability of the draining gas film remarkably makes dynamic levitation less demanding than static levitation. We characterize the vertical gas film oscillations by measuring their frequency and monitoring their occurrence in the parameter space spanned by surface temperature and impact velocity. Finally, we model the occurrence of these oscillations and account for their frequency through a hydrodynamic mechanism.

Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF$_6$ interface with controllable initial conditions are performed to examine the dependence of perturbation growth on initial spectra. We deduce and demonstrate experimentally that the amplitude development of each mode is influenced by the mode-competition effect from quasi-linear stages. It is confirmed that the mode-competition effect is closely related to initial spectra, including the wavenumber, the phase and the initial amplitude of constituent modes. By considering both the mode-competition effect and the high-order harmonics effect, a nonlinear model is established based on initial spectra to predict the amplitude growth of each individual mode. The nonlinear model is validated by the present experiments and data in the literature by considering diverse initial spectra, shock intensities and density ratios. Moreover, the nonlinear model is successfully extended based on the superposition principle to predict the growths of the total perturbation width and the bubble/spike width from quasi-linear to nonlinear stages.