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A Numerical Investigation of Supersonic Cavity Flow At Mach 2

Numerical simulations were carried out for a two-dimensional shallow cavity with a length-to-depth ratio of 8 in a Mach 2 flow. The simulations were carried out with an in-house code (Eilmer-3) and then compared with the results obtained with the commercially available package FLUENT and experimental data. The standard Wilcox k−ω turbulence model was used to model turbulence in the simulations. While the numerically predicted frequencies generally match experimental data, the dominant frequency does not. Similarly, a difference in pressure level and amplitude between simulation and experiment is observed. Both codes which differ substantially in their structure, predict the flow phenomena in a near identical way, which gives confidence in the validity of the results obtained. Introduction The study of compressible cavity flow is an important research topic in the field of aerodynamics and acoustics. Although the geometry of these cavities is simple, their unsteady fluid dynamic behaviour is complicated and difficult to predict both in subsonic and supersonic flows. These fluid phenomena typically cause unwanted drag, structural noise and vibrations. Cavities are classified based on the length-to-depth ratio (L/D), where L is the length of the cavity and D is the depth. Cavities are called ‘open’ if L/D < 10, ‘closed’ if L/D > 13 and ‘transitional’ for 10 ≤ L/D ≤ 13. In case of open cavities, the boundary layer separates from the leading edge and forms a shear layer which bridges the length of the cavity. This shear layer impinges on the rear wall of the cavity, and a pressure wave is created which will travel upstream inside the cavity, interact with the shear layer and thereby cause a feedback loop. This feedback mechanism produces discrete resonant frequencies [5] [8] [11]. The present study investigates numerically the open cavity resonance and variation of the associated flow field by using two different numerical codes, the in-house Eilmer-3 and the commercial package FLUENT, for a cavity with L/D of 8 in a Mach 2 free stream. Geometry and Flow conditions The cavity investigated here has a geometry as shown in Fig. 1. It comprises of an upstream flat-plate of length l = 184mm, followed by a cavity with a depth D = 4mm and length-to-depth ratio L/D = 8, and another flat plate. The experimental work on this configuration was conducted in a supersonic wind tunnel at Mach 2 and Reynolds number Rel = 4.6× 106. The tunnel test time was approximately 30s and time-resolved measurements were made during a 4s window within this period. In the numerical simulations a flow time of 6ms was used which corresponds to a characteristic time tU∞/D = 763. This was chosen after doing some initial tests to evaluate the times required for flow establishment over the upstream flat plate and also for reaching steady amplitude periodic oscillations in the cavity. It was found that the flow field of interest was fully established after approximately ≈1 ms and that an overall flow time of 6 ms was fully adequate to analyse the flow behaviour. The initial calculations showed that extending the calculations s/D=-46 s/D=0 s/D=1 s/D=9 s/D=10 s/D=16 x y Figure 1: Cavity geometry to larger times did not change the results but only increased the computational effort. The free stream conditions are based on the experimental data and are given in Table 1. Table 1: Free stream conditions M∞ P∞ T∞ Rel k∞ ω∞ − Pa K − m2/s2 1/s 2 25560.9 161 4.6x106 3886 1.9x106 Numerical Details Grid Independence In the present work numerical simulation of a two-dimensional flow over a cavity is carried out using two CFD codes, the inhouse code Eilmer-3 and the commercial CFD package FLUENT. Eilmer-3 is a code developed by Jacobs et al. [4] at the University of Queensland for simulating time accurate compressible flow in two and three dimensions. It is based on a finite-volume formulation of mass, momentum, energy and species conservation equations and is implemented on blockstructured grids. It has capabilities of solving both laminar and turbulent flows. It has a standard Wilcox k − ω turbulence model [10] to compute the turbulence parameters k and ω (Eqns. 1 and 2). The MUSCL scheme is used to obtain second order accuracy in spatial dimensions. The AUSMDV [9] scheme is used to calculate the fluxes. In the present instance, the code is used to predict the behaviour of a turbulent compressible flow in and around a cavity. The viscosity of the fluid (air) is modeled using Sutherland’s formulation. Free stream turbulence parameters are calculated based on equations 1 and 2. The co-ordinate system is non-dimensionalised based on the cavity depth (D), and the wetted distance s, thus s/D = 0 is the top corner of cavity front face and s/D = 1 is the bottom corner of cavity front face. s/D = 9 is the bottom corner of rear face and s/D = 10 is the top corner of the cavity rear face (Fig. 1). The commercially available grid generation software ICEM-CFD was used to create multi-block structured grids. The grid is shown in Fig. 2. k = 3 2 (IturbU∞) 2 (1) where Iturb is the turbulent intensity.