## 1 Introduction  
The noise sources of a submarine can be categorized into propeller noise, hydrodynamic noise, and mechanical noise. With modern techniques available for the control of propeller noise and mechanical noise, hydrodynamic noise becomes significant, especially when the speed of the submarine is higher than 12 knots [1]. The generation mechanism of the hydrodynamic noise is as follows. First, when the fluid flows around the surface of the submarine, the turbulent fluctuation pressure produces and radiates self-noise, which is known as the flow noise. Second, the turbulent fluctuation pressure excites the elastic shell of the submarine to vibrate and radiate another noise, known as the flow-induced noise. Since the sound radiation efficiency of the flow-induced noise is much higher than that of the flow noise, the latter is usually ignored. Therefore, the hydrodynamic noise only means the flow-induced noise in some cases. Since the flow-induced noise dramatically destroys the submarine's acoustic stealth performance, the reduction of the flow-induced noise is of great significance to enhance the submarine's combat and survival performance [2]. The methods developed for the reduction of the submarine's flow-induced noise are focused on the optimization of line type, active flow control, and passive flow control. In the passive flow control field, the vortex generator (VG) is a practical tool used to suppress the flow separation [3] and inhibit the vibration [4]. The inverted wing with VG highlights the flow physics of how the flow separation can be controlled by VGs [5]. Low-profile VGs have been successfully utilized by at least two other aircrafts [6]. VGs are able to redirect flow and increase lift [7], and even to mitigate shock-induced separation [8]. The supersonic VGs can also reduce flow separation near the throat [9]. A 5% drag reduction can be achieved using proper VG configurations through cumulated effects of flow reattachment [10]. However, some researchers have revealed that, although VGs are simple devices, they still generate drag [11] because stronger vortices are generated by VGs in larger dimensions, which do not necessarily lead to better flow separation control [12]. Thereafter, the micro VG was proposed [13], where shock-induced flow separation can be partially eliminated by the vortices produced from micro VGs [14]. [PERSON] [15] has found that the increase of the control micro VG's height appears to be favorable for closer control location, and thus the mechanism of flow control by the VG has drawn researchers' interests. The results of flow control by the VG were validated by experiments because the capability of numerical calculation is limited by the grid numbers. Examples include [PERSON]'s parametric study of passive VGs using the stereo Particle Image Velocity (PIV) experiment [16], [PERSON]'s experimental study of vane-type VGs [17], [PERSON]'s experimental investigation of low-profile VGs [18], and [PERSON]'s analyses of the experimental study on low profile VGs [19]. With the development of advanced technologies, some models have been proposed to calculate the flow field of VGs. For instance, [PERSON] explored VG control of parameter variations at a low computational cost [20]. The integral boundary layer code XFOIL was used to model the effect of VGs by [PERSON] [21]. [PERSON] used the monotone integrated large-eddy simulation method to investigate the declining angles of the trailing edge of micro-ramp VGs [22]. [PERSON] investigated the airfoil S809's aerodynamic performance, without and with VGs, using simulations [23]. [PERSON] found that the jBAY model captured the VG arrays effect under internal and external flows [24]. [PERSON] investigated an OA209 airfoil with a deployable VG device through numerical simulations [25]. [PERSON] used simulations to investigate the flow over a compression ramp, with and without a micro-ramp VG upstream [26]. On the one hand, the position of the VG is an important parameter.