# Dynamic Stall Mechanisms of Pitching Airfoil: IDDES Study Across Different Mach Numbers
## 1 Introduction  
Dynamic stall remains a critical aerodynamic phenomenon limiting the maximum flight speed and maneuverability of conventional single-rotor helicopters, primarily due to transient flow separation and stall vortex under unsteady conditions [1]. Over decades, extensive efforts have been devoted to investigating its complex flow mechanisms, using experimental and computational approaches to characterize its evolutionary stages and impacting factors.  
Experimental studies by [PERSON] et al. [2, 3, 4, 5] established foundational insights through oscillating airfoil tests, categorizing dynamic stall into four types based on maximum angle of attack (AoA): no stall, stall onset, light stall, and deep stall [6]. [PERSON] andRaffel [7, 8] further divided the process into five stages using time-resolved particle image velocimetry (TR-PIV), identifying primary and secondary instabilities during stall development. These studies clearly revealed the evolutionary stages and typical aerodynamic events during dynamic stall.  
Compressibility effects, a critical factor in rotorcraft applications, were systematically reviewed by [PERSON] et al. [9], who demonstrated that even at a Mach number of 0.2, supercritical flows significantly alter stall characteristics. Increased Mach numbers reduce lift overshoot amplitudes and advance stall onset [5, 10]. Ames Research Center designed schlieren-based systems and observed shockwave formation and supersonic regions at Mach > 0.4 [11, 12]. [PERSON] et al. [13, 14] experimentally and numerically investigated dynamic stall suppression using combustion-powered actuation on a modified VR-12 airfoil across a Mach number range of 0.2 to 0.5. Nevertheless, experimental studies at Reynolds numbers (_Re_) representative of full-scale rotors (_Re_\(\geq\) 1 \(\times\) 10\({}^{6}\)) remain sparse. Systematic comparisons have revealed significant discrepancies in dynamic stall characteristics between low-Re (<1 \(\times\) 10\({}^{6}\)) and high-Re (\(\geq\)1 \(\times\) 10\({}^{6}\)) conditions [15, 16, 17]. While [PERSON] et al. [18] and [PERSON] et al. [19] explored high-Re conditions (2.0-4.0 \(\times\) 10\({}^{6}\)), they focused on aerodynamic loads and omitted detailed flowfield analyses.  
Computational fluid dynamics (CFD) has emerged as a numerical tool, with turbulence modeling strategies evolving from Reynolds-averaged Navier-Stokes (RANS) to hybrid RANS-LES frameworks. [PERSON] et al. [20, 21, 22] proposed detached eddy simulation (DES) methods, balancing accuracy and computational cost. Recent studies by [PERSON] et al. [23] using SU2 solver highlighted compressibility-induced changes in stall regime at Mach numbers of 0.2-0.4. [PERSON] et al. [24, 25] compared RANS, detached DES (DDES), and improved DDES (IDDES), emphasizing three-dimensional flow features. [PERSON]'s group [16, 26, 27, 28, 29, 30, 31] advanced implicit LES (ILES) for low-to-medium Reynolds flows, revealing effects of compressibility [31] and airfoil geometry [32, 33] on dynamic stall mechanisms: thin airfoils experience leading-edge laminar separation bubble bursting, whereas thicker airfoils encounter trailing-edge separation interference. Nevertheless, these conclusions are constrained to low Mach numbers (_Ma_ < 0.1) and _Re_\(\leq\) 1.0 \(\times\) 10\({}^{6}\).  
Despite significant advancements, our understanding of dynamic stall mechanisms in compressible, high-Reynolds-number flows remains incomplete, particularly at high Mach numbers relevant to helicopter rotor operations. This study employs sliding grid and IDDES methods to simulate dynamic stall at different Mach numbers. Aerodynamic behavior is analyzed through airload hysteresis, stall regime, vortex structures, and aerodynamic damping characteristics. In the paper, numerical methods and their validation is introduced in Section 2. This is followed by results and discussions on airfoil dynamic stall characteristics at two Mach numbers of 0.283 and 0.5, respectively. Finally, key findings are concluded.