Numerical investigation of the flow noise of an underwater vehicle

Flow noise is an important issue of underwater vehicles. In this paper, the ability of different numerical methods to predict the flow noise of the DARPA SUBOFF underwater vehicle model is investigated and compared. It is found that the front of the command deck and the tail fins are the main noise sources. Numerical results show that delayed detached eddy simulation is a suitable computational model for flow noise simulations.


Introduction
The acoustic performance of underwater vehicles directly determines their survivability and is one of the most important issues in the marine field.Hydrodynamic noise is one of the main noise sources of underwater vehicles [1], which mainly includes flow noise and air bubble noise.The FW-H equation [2] predicts the flow noise based on the flow field, so the accuracy of the flow noise is directly related to the flow field simulated by the numerical methods.
Reynolds-averaged Navier-Stokes (RANS) method solves the averaged physical quantities in the N-S equation.This method requires less computational cost and has been most widely used in engineering, but it has limited ability to predict the sound field.The large eddy simulation (LES) method has a high computational accuracy but a large computational cost.The hybrid RANS/LES methods, represented by delayed detached eddy simulation (DDES), adopt the RANS method near the wall and adopt the LES method in the detached region, which obtains considerable accuracy with moderate computational cost.The hybrid RANS/LES methods have been used for more and more applications in recent years [3][4][5].
In 1989, the David Taylor Research Center (DTRC) proposed the DARPA SUBOFF underwater vehicle model [6] and carried out a series of experiments.The DARPA SUBOFF program provided detailed data on the hydrodynamics and acoustics of the underwater vehicle that can be used to validate the accuracy of numerical methods.Bull [7] numerically simulated the wake of the underwater vehicle model by using different turbulence models.Zeng et al. [8] investigated the radiated noise at characteristic points of the DARPA SUBOFF model based on the RANS method and FW-H equation.Numerical simulations were carried out to obtain the distribution of the noise sources and to analyse the characteristics of the acoustic directivity at different frequencies.Wang et al. [9] studied the noise source distribution and radiated noise at characteristic points of the DARPA SUBOFF model based on the LES method and FW-H equation.The radiation intervals of flow noise are found to increase with the frequency, and the quadrupole noise radiation intervals are more than the dipole noise radiation intervals at the same frequency.Özden et al. [10] numerically investigated the flow field and radiated noise for the DARPA SUBOFF underwater vehicle model and a marine propeller based on the URANS method and FW-H equation.
At present, there are limited studies on the flow noise of underwater vehicles using the hybrid RANS/LES method, and the ability of different numerical simulation methods to predict the flow noise of underwater vehicles needs to be investigated.

Computational method
In this paper, the DARPA SUBOFF underwater vehicle model is firstly simulated by the RANS method, and then simulated by URANS, DDES, and LES methods, respectively.
The governing equations for the FW-H equation: where () is the monopole source; ()

Geometry
The DARPA SUBOFF model is composed of a hull, a command deck, and four tail fins, as shown in Figure 1.The model length is 4.356 m, the diameter of the hull is 0.508 m, and the command deck length is 0.368 m.The distance from the tail fins to the bow is 4.007 m.

Computational mesh
The computational domain is cylindrical with a radius of five times the hull radius.The distance from the inlet to the bow of the underwater vehicle is the same as the model length, and the distance from the outlet to the aft of the underwater vehicle is four times the boat length.Figure 2 shows the structured computational mesh.The dimensionless wall distance y + is taken as 50.

Boundary conditions
The boundary condition of the inlet is velocity inlet, where the flow speed is 5.93 knots (3.05039 m/s).
The boundary condition of the underwater vehicle is defined as no slip.The boundary condition of the outlet is the pressure outlet.

Steady numerical simulation
The turbulence model used for the steady numerical simulation is the RNG k-ε turbulence model.Figure 3 shows the velocity contours on the 13 cross profiles along the flow direction.The command deck causes a very long wake that affects the flow field all the way to the tail fins.In the tail, the tail fins cause more drastic changes in the flow field.It is shown that the velocity contour distribution is denser in the wake of the command deck and the tail fins compared to other regions.

Unsteady numerical simulation
unsteady numerical simulation, URANS, DDES, and LES methods are adopted, respectively, while the FW-H equations are used for flow noise prediction.The time step of the unsteady numerical simulation is taken as  The spectra can be divided into three frequency intervals within 0 to 10,000 Hz.The first frequency interval is from 0 to 3500 Hz, the maximum value occurs around 1000 Hz, and the sound pressure level in this frequency interval is greater than the values in the other frequency intervals.The second frequency interval is from 3500 to 6500 Hz, and the maximum value in this frequency interval occurs around 5000 Hz.The third frequency interval is from 6500 to 10000 Hz, and the maximum value in this frequency interval occurs around 8500 Hz.The spectra of SPL simulated by DDES and LES methods show similar characteristics in different frequency intervals.
Figure 5 shows the time derivative of surface pressure dp/dt of the underwater vehicle, which can be used to determine the strength of acoustic sources.The numerical simulation results show that the front of the command deck and the tail fins are the main noise sources.

Comparison of different computational methods
Tables 1, 2 and 3 show the simulation results of URANS, DDES and LES methods compared to the reference values of the DARPA SUBOFF project.As can be seen from the tables, the simulation results predicted by the LES method have larger relative errors to the reference values than the other computational methods.This may be because the dimensionless wall distance y + used in this study does not satisfy the requirement of the LES method for an approximation of 1.The computational results of URANS and DDES reveal that both methods have a good prediction accuracy of OASPL.Since the DDES method predicts the total resistance and friction resistance more accurately, it is a suitable computational method for flow noise simulations.

Conclusion
In this work, the DARPA SUBOFF underwater vehicle model is firstly simulated by the RANS method, and then simulated by URANS, DDES, and LES methods, respectively.The front of the command deck and the tail fins are found to be the main noise sources.By comparing the numerical simulation results of different computational models with the reference values of the DARPA SUBOFF project, it is found that DDES is a suitable computational model for flow noise simulations.

Figure 1 .
Figure 1.Geometry of the DARPA SUBOFF underwater vehicle model.

Figure 2 .
Figure 2. Computational mesh of the DARPA SUBOFF underwater vehicle model.

Figure 3 .
Figure 3. Distribution of velocity contours on the cross profiles.
point below the underwater vehicle is selected as the feature point with a coordinate (2.178, -2, 0), and the bow of the boat is defined as the origin of the coordinate axis.The spectra of sound pressure level (SPL) at the feature point are shown in Figure4.

Figure 4 .
Figure 4. Spectra of SPL at the feature point.

Figure 5 .
Figure 5.Time derivative of surface pressure dp/dt on the surfaces.

Table 1 .
Comparison of simulated and experimental values of total resistance.

Table 2 .
Comparison of simulated and experimental values of frictional resistance.

Table 3 .
Comparison of simulated and experimental values of OASPL.