# Numerical Methods Comparison of Shock-Induced Separation on Transonic Axisymmetric Hump
## 1 Introduction  
Shock wave/boundary layer interactions (SWBLIs) are prevalent in transonic, supersonic, and hypersonic flows. Since the initial observation of this phenomenon by [PERSON] [1] during experiments on supersonic airfoil flows, the investigation of SWBLIs has received significant emphasis. Through extensive research, it has been recognized that SWBLIs are a significant factor in the performance of high-speed vehicles [2, 3, 4]. Typically, as the gas traverses a shock wave, its pressure increases, exacerbating the adverse pressure gradient within the boundary layer, which facilitates boundary layer separation due to the increased thickness of the layer around the shock. Recent work has developed various supersonic models, such as sidewall [5, 6], cylinder [7] or elliptical cylinder [8], and wedge [9], to uncover the underlying mechanisms of SWBLIs and to establish more accurate simulation methods for predicting aerodynamic characteristics. In supersonic or hypersonic flow fields, the region before the shock wave is supersonic, while the region behind it is subsonic. At transonic speeds, different dynamic characteristics determine the shape, strength, and location of the shock wave in the interaction zone [10]. In general, a normal shock wave with \"\(\lambda\)-structure\" occurring at the foot is often observed in transonic flow, which is due to the thicker boundary layer, which will cause a series of wake compression waves to increase the local boundary-layer-edge pressure and decrease the Mach number before strong normal shock; the border of compression waves and normal shock will combine as \"\(\lambda\)-structure\". Another important thing is the enhancement of adverse pressure gradient inside the boundary layer, which will significantly increase the possibility of flow separation and the flow field structure will be more complex, which makes a large challenge for simulation methods. In [PERSON]'s study [11], the SWBLI of NACA 64A010 airfoil was investigated through experiment and CFD simulation at Mach0.8. The results indicated that CFD methods were inadequate in accurately predicting the flow characteristics due to shock-induced boundary layer separation. In [12; 13], DNS and WMLES methods were applied, respectively, to research the separating features about NACA 0012 airfoil. However, there are a few differences in the position of separation and attachment. Current evidence suggests that the accuracy of CFD methods employed in transonic separation conditions still requires further enhancement. Shock-induced separation is one of the most important phenomena about SWBLI in transonic flow field. The boundary layer separation is generally displayed as a certain size of bubble on the object wall, and the separation bubble will mix with the original separation behind the shock wave in some cases, which could cause larger separation flow and more aerodynamic efficiency loss. [PERSON] [14] developed a simple axisymmetric hump model for comparison with experiment in order to investigate the question about shock-induced separation and expected more CFD methods were verified. In 2019, NASA reconducted the experiment in the Sandia Transonic Wind Tunnel so as to supplement the measurement data [15]. The experiment used particle image velocimetry (PIV), pressure-sensitive paint (PSP), oil-flow visualization, and oil-film interferometry to measure properties of the incoming boundary layer, the shock/separation interaction region, the detached shear layer and reverse flow region, and reattachment. A blind CFD challenge was built for obtaining more precise CFD methods about the experiment, which would publish the results of measurements after the end of the challenge [16; 17; 18]. So many methods were applied in the challenge. [PERSON] [19] used RANS and wall-modeled LES to simulate the flow field of the axisymmetric hump by [PERSON]. In the investigation, one-equation Spalart-Allmaras and two-equation cubic \(k-\varepsilon\) turbulence models are employed and the results of RANS are used as inputs for WMLES.