Symmetry

Flexible grid regulation necessitates Francis turbines to operate at heads of 120–180 m (compared to the rated head of 154.6 m), breaking the designed rotational symmetry and inducing hydraulic instabilities that threaten structural integrity and operational reliability. This study presents extensive field measurements of pressure pulsations in a 600 MW prototype Francis turbine operating at heads of 120–180 m and loads of 20–600 MW across 77 operating conditions (7 head levels × 11 load points). We strategically positioned high-precision piezoelectric pressure sensors at three critical locations—volute inlet, vaneless space, and draft tube cone—to capture the amplitude and frequency characteristics of symmetry-breaking phenomena. Advanced signal processing revealed three distinct mechanisms with characteristic pressure pulsation signatures: (1) Draft tube rotating vortex rope (RVR) represents spontaneous breaking of axial symmetry, exhibiting helical precession at 0.38 Hz (approximately 0.18 f_n, where f_n = 2.08 Hz) with maximum peak-to-peak amplitudes of 108 kPa (87% of the rated pressure p_rated = 124 kPa) at H = 180 m and P = 300 MW, demonstrating approximately 70% amplitude reduction potential through load-based operational strategies. (2) Vaneless space rotor-stator interaction (RSI) reflects periodic disruption of the combined C₂₄ × C₁₃ symmetry at the blade-passing frequency of 27.1 Hz (N_r × f_n = 13 × 2.08 Hz), reaching peak amplitudes of 164 kPa (132% p_rated) at H = 180 m and P = 150 MW, representing the most severe symmetry-breaking phenomenon. (3) Volute multi-point excitation exhibits broadband spectral characteristics (4–10 Hz) with peak amplitudes of 146 kPa (118% p_rated) under small guide vane openings. The spatial amplitude hierarchy—vaneless space (164 kPa) > volute (146 kPa) > draft tube (108 kPa)—directly correlates with the local symmetry-breaking intensity, providing quantitative evidence for the relationship between geometric symmetry disruption and hydraulic excitation magnitude. Systematic head-dependent amplitude increases of 22–43% across all monitoring locations are attributed to effects related to Euler head scaling and Reynolds number variation, with the vaneless space demonstrating the highest sensitivity (0.83 kPa/m, equivalent to 0.67% p_rated/m). The study establishes data-driven operational guidelines identifying forbidden operating regions (H = 160–180 m, P = 20–150 MW for vaneless space; H = 160–180 m, P = 250–350 MW for draft tube) and critical monitoring frequencies (0.38 Hz for RVR, 27.1 Hz for RSI), providing essential reference data for condition monitoring systems and operational optimization of large Francis turbines functioning as flexible grid-regulating units in renewable energy integration scenarios.

This paper explores the integrability of the Akbota equation with various types of solitary wave solutions. This equation belongs to a class of Heisenberg ferromagnet-type models. The model captures the dynamics of interactions between atomic magnetic moments, as governed by Heisenberg ferromagnetism. To reveal its further physical importance, we calculate more solutions with the applications of the logarithmic transformation, the M-shaped rational solution method, the periodic cross-rational solution technique, and the periodic cross-kink wave solution approach. These methods allow us to derive new analytical families of soliton solutions, highlighting the interplay of discrete and continuous symmetries that govern soliton formation and stability in Heisenberg-type systems. In contrast to earlier studies, our findings present notable advancements. These results hold potential significance for further exploration of similar frameworks in addressing nonlinear problems across applied sciences. The results highlight the intrinsic role of symmetry in the underlying nonlinear structure of the Akbota equation, where integrability and soliton formation are governed by continuous and discrete symmetry transformations. The derived solutions provide original insights into how symmetry-breaking parameters control soliton dynamics, and their novelty is verified through analytical and computational checks. The interplay between these symmetries and the magnetic spin dynamics of the Heisenberg ferromagnet demonstrates how symmetry-breaking parameters control the diversity and stability of optical solitons. Additionally, the outcomes contribute to a deeper understanding of fluid propagation and incompressible fluid behavior. The solutions obtained for the Akbota equation are original and, to the best of our knowledge, have not been previously reported. Several of these solutions are illustrated through 3-D, contour, and 2-D plots by using Mathematica software. The validity and accuracy of all solutions we present here are thoroughly verified.

This study investigates the conformable nonlinear Schrödinger equation (NLSE) with self-phase modulation (SPM) and Kudryashov’s generalized refractive index, crucial for pulse propagation in optical fibers. By applying the modified simplest equation method, we derive several novel soliton solutions and investigate their dynamic behavior within the NLSE framework enhanced with a conformable derivative. The governing conformable NLSE also exhibits symmetry patterns that support the structure and stability of the constructed soliton solutions, linking this work directly with symmetry-based analysis in nonlinear wave models. Furthermore, various graphs are presented through 2D, 3D, and contour plots. These visualizations highlight different soliton profiles, including kink-type, wave, dark, and bell-shaped solitons, showcasing the diverse dynamics achievable under this model, influenced by SPM and Kudryashov’s generalized refractive index. The influence of the conformable parameter and temporal effects on these solitons is also explored. These findings advance the understanding of nonlinear wave propagation and have critical implications for optical fiber communications, where managing pulse distortion and maintaining signal integrity are vital.