Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on ε1 and ε2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions

Lagrangian coherent structures in flow past a backward-facing step

This paper investigates flow past a backward-facing step (BFS) in a duct at Reynolds number $Re = 5080$ based on step height, mean inflow velocity and fluid kinematic viscosity. The flow configuration matches a combustion experiment conducted by Pitz and Daily in 1983. High-resolution velocity fields are obtained in OpenFOAM by direct numerical simulation, and the flow field analysed by Lagrangian approaches. Trajectories of fluid particles in areas of interest are obtained by high-order numerical integration, and used to compute finite-time Lyapunov exponents (FTLEs) and polar rotation angles. Lagrangian coherent structures (LCSs) are extracted using geodesic theory, including hyperbolic LCSs and elliptic LCSs. We use complementary qualitative and quantitative LCS analyses to uncover the underlying flow structures. Notably, we find that a flow pathway in which fluid particles rarely diverge from adjacent particles is opened and closed by FTLE ridges determined by the periodic shedding of vortices from the BFS. Two dominant vortices with significant Lagrangian coherence, generated respectively by the separated boundary layer and shear layer, are self-sustaining and of comparable strength. Hyperbolic repelling LCSs act as transport barriers between the pathway and cores of the coherent vortices, thus playing a major part in the fluid entrainment process. Interactions between these different geometric regions partitioned by LCSs lead to intrinsic complexity in the BFS flow.</jats:p

Pattern evolution and modal decomposition of Faraday waves in a brimful cylinder

This paper investigates the steady-state pattern evolution of symmetric Faraday waves excited in a brimful cylindrical container when driving parameters much exceed critical thresholds. In such liquid systems, parametric surface responses are typically considered as the resonant superposition of unstable standing waves. A modified free-surface synthetic Schlieren method is employed to obtain full three-dimensional spatial reconstructions of instantaneous surface patterns. Multi-azimuth structures and localized travelling waves during the small-elevation phases of the oscillation cycle give rise to modal decomposition in the form of -basis modes. Two-step surface-fitting results provide insight into the spatiotemporal characteristics of dominant wave components and corresponding harmonics in the experimental observations. Arithmetic combination of modal indices and uniform frequency distributions reveal the nonlinear mechanisms behind pattern formation and the primary pathways of energy transfer. Taking the hypothetical surface manifestation of multiple azimuths as the modal solutions, a linear stability analysis of the inviscid system is utilised to calculate fundamental resonance tongues (FRTs) with non-overlapping bottoms, which correspond to subharmonic or harmonic -basis modes induced by surface instability at the air-liquid interface. Close relationships between experimental observations and corresponding FRTs provide qualitative verification of dominant modes identified using surface-fitting results. This supports the validity and rationality of the applied -basis modes.</p

Incentive Contract Design for Cooperation and Win-Win of Chinese Government and Enterprise in the View of Ecological Civilization

In the view of ecological civilization, the incentive contract between government and enterprise to cooperate and build ecological economy is built by introducing value preference parameter into the classical principal-agent model. Model equilibrium has been analyzed and a numerical example verifies the validity of conclusion. Research results prove that value preference parameter can effectively coordinate interest relationship between government and enterprise and control incentive cost. Value preference parameter of government, marginal output and risk avoidance attitude of enterprise can influence the change of incentive contract

Frequency and distribution of AP-1 sites in the human genome

The AP-1-binding sequences are promoter/enhancer elements that play an essential role in the induction of many genes in mammalian cells; however, the number of genes containing AP-1 sites remains unknown. In order to better address the overall effect of AP-1 on expression of genes encoded by the entire genome, a genome-wide analysis of the frequency and distribution of AP-1 sites would be useful; yet to date, no such analysis of AP-1 sites or any other promoter/enhancer elements has been performed. We present here our study of the consensus AP-1 site and two single-bp variants showing that the frequency of AP-1 sites in promoter regions is significantly lower than their average rate of occurrence in the whole genomic sequence, as well as the frequency of a random heptanucleotide suggesting that nature has selected for a decrease in the frequency of AP-1 sites in the regulatory regions of genes. In addition, genes containing multiple AP-1 sites are more prevalent than those containing only one copy of an AP-1 site, which again may have evolved to allow for greater signal amplification or integration in the regulation of AP-1 target genes. However, the number of AP-1-regulated genes identified in various studies is far smaller than the number of genes containing potential AP-1 sites, indicating that not all AP-1 sites are activated in a given cell under a given condition, and is consistent with the prediction by others that cellular context determines which AP-1 sites are targeted by AP-1

Free-surface topography measurements of liquid layers over a smoothly varying bed

This study introduces a single-camera synthetic Schlieren technique for measuring free-surface topography in fluid layers over a smoothly varying solid bed. By modeling light refraction through weakly deformed air–liquid and weakly varying liquid–solid interfaces, we establish a linear relationship between free-surface gradients and pattern displacements that yields an explicit bed-independent formulation for upward-looking configurations. The proposed framework incorporates coordinate mapping to correct refraction-induced parallax distortion influenced by the bed shape. Validation experiments featuring sinusoidal bed topography and wedge-shaped sloped bed topography achieve accurate spatiotemporal reconstruction of both static capillary meniscus profiles and dynamic water drop ripple evolution. The present method advances experimental capabilities for quantifying interfacial hydrodynamics in multi-layer fluid systems.</p

Twin-field quantum key distribution without optical frequency dissemination

Twin-field (TF) quantum key distribution (QKD) has rapidly risen as the most viable solution to long-distance secure fibre communication thanks to its fundamentally repeater-like rate-loss scaling. However, its implementation complexity, if not successfully addressed, could impede or even prevent its advance into real-world. To satisfy its requirement for twin-field coherence, all present setups adopted essentially a gigantic, resource-inefficient interferometer structure that lacks scalability that mature QKD systems provide with simplex quantum links. Here we introduce a novel technique that can stabilise an open channel without using a closed interferometer and has general applicability to phase-sensitive quantum communications. Using locally generated frequency combs to establish mutual coherence, we develop a simple and versatile TF-QKD setup that does not need service fibre and can operate over links of 100 km asymmetry. We confirm the setup's repeater-like behaviour and obtain a finite-size rate of 0.32 bit/s at a distance of 615.6 km.Comment: 14 pages, 7 figure