Study on wake field and noise of underwater solid rocket engine

He, Xiaohui; Liu, Zhongle; Yuan, Zhiyong; Wang, Xianyong

doi:10.21595/jve.2021.21956

Journal of Vibroengineering

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Published: 06 August 2021

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Study on wake field and noise of underwater solid rocket engine

Xiaohui He¹

Zhongle Liu²

Zhiyong Yuan³

Xianyong Wang⁴

^{1, 2, 3}School of Ordnance Engineering, Naval University of Engineering, Wuhan, Hubei, 430033, China

^{4, 1}Jiangnan Industries Group Co., Ltd., Xiangtan, Hunan 411207, China

Corresponding Author:

Xiaohui He

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Abstract

In order to study the wake field and noise of an underwater solid rocket engine, the underwater jet flow field was simulated and calculated using the FLUENT software and the Volume of Fluid (VOF) method, and the radiated noise of the wake was calculated and analysed using the FW-H equation. The numerical simulation results are verified by experiments. The research results show that underwater jet gas bubbles experience processes of “necking-expansion-necking”, and finally reach a state of “dynamic balance”. The radiated noise is mainly affected by the pressure differences of the backpressure, i.e., the larger the pressure difference is, the greater the radiated noise. Moreover, the sound pressure presents periodic pulsations, and the peak value of the sound pressure decreases rapidly with an increase in distance from the receivers. The radiated noise spectrum has the characteristics of a wide frequency band and a low frequency, and the energy is mainly concentrated in the range of the low frequency, which then moves to the high frequency range. The research results provide some reference for the design of an underwater solid rocket engine and a noise prediction for the wake field, and also provide some help for underwater detection and identification.

1. Introduction

The numerical simulation of the flow field of an underwater solid rocket engine focuses on the dynamic tracking of the moving gas-liquid interface [1]. Through the simulation analysis, a detailed jet flow field structure can be obtained. As early as the 1970s, Hoefele and Brimacombe [2] studied the underwater gas jet flow field structure of nozzles, in particular, the underwater gas jet flow field structure from necking-expansion and linear nozzles by combining pressure measurements and high-speed photography. This showed that the frequency of the jet pressure pulsation decreases with an increase in gas injection pressure. Hirt, et al. [3] used the VOF method to study the dynamic problem of free boundaries and track the gas-liquid interface of gas jets. Wang Cheng et al. [4] conducted a coupled numerical solution on the interior and exterior flow fields of unsteady gas bubbles in the tail of missiles launched at depths of 10 m, 30 m and 60 m. The heat transfer between and vaporisation of the gas and water media at high temperatures and high pressures were considered in the calculation, the growth and shedding processes of the gas bubbles were simulated, and the basic flow law of the gas bubbles was revealed. Scardovelli [5] used the direct numerical simulation (DNS) method to study the development of gas bubbles in an underwater gas jet process, but it was difficult to apply the results in engineering practice due to the large number of grids. Zhang et al. [6] numerically simulated a three-dimensional underwater gas jet. Zhong Fengquan [7] used the Navier-Stokes equation for unsteady viscous compressible flows to simulate the variations in a flow field of the direct ignition and launching processes of an underwater rocket. Yi Shuqun [8] used the MAC method to numerically simulate underwater jet flow fields at depths of 15 m and 40 m at high speed and preliminarily determined the characteristics of underwater jets. He Xiaoyan, et al. [9] used the LevelSet method to track the gas-liquid interface and numerically simulate an underwater gas jet in the initial stage at a depth of 30 m with a jet Mach number of 1, and the calculated result was qualitatively consistent with the experimental results. Nguyen [10] used the FLUENT software and the VOF method to simulate the impact jet of gas against liquid, and successfully tracked the gas-liquid interface. However, the simulation time was too short, and the gas jet nozzle was not submerged in water, which was different from the underwater gas jet in the physical mechanism. It shows that the VOF method is only an effective method for tracing the gas-liquid interface. In the same year, Smolianski [11] used the FELSOS method to study the motions of a gas-liquid free interface. Muller [12] used a single-fluid model to simulate the development of underwater gas bubbles. Xiang Min, et al. [13] performed numerical simulations on the wake field of underwater solid engines at a depth of 10 m based on the axisymmetric unsteady compressible N-S equation and the VOF two-phase flow model. The growth process of gas bubbles in the wake field was predicted well and the morphology and evolution processes of the wake field in the early working period of underwater engines were both obtained. Zhang Shuai [14, 15] established an underwater gas-liquid two-phase gas jet numerical simulation model based on the VOF interface tracking method and calculated the unsteady wake flow field of an underwater solid engine under overexpansion at depths of 300 m and 400 m and analysed the flow field structure. Jia Youjun [16] used a high-speed camera to record the growth and variation processes of a wake during the underwater ignition of an engine, obtained the morphology and evolution process of the wake, and analysed the influence of the water environment on variations in the wake structure of the engine. Wang Lili, et al. [17] conducted a numerical study on a solid rocket engine underwater supersonic jet, and the results showed that the interactions between water and the jet gas were the direct cause of backpressure variations, as well as variations in the shock wave motion, the momentum thrust and the pressure differential thrust.

Regarding noise research, because the gas-liquid flow field of underwater gas injections is much more complex than that of a gas jet field, the noise mechanism of underwater gas injections is also different from that of gas injections. When gas is discharged underwater at low speed, the noise caused by the volume vibrations of the bubbles can be obtained by theoretical calculations. However, when the gas is injected at high speed, the interactions between the bubbles will produce a low and medium frequency noise and an irregular acoustic signal, which is difficult to obtain by theoretical calculations, however, it can be obtained by experiment.

Gavigan [18] carried out a water tunnel experiment on the noise generated from a gas jet in a turbulent wake. When the gas was injected into the moving turbulent field, the noise emitted by the jet flow was mainly controlled by the exhaust volume and the size of the exhaust pipe. In the turbulence caused by the jet, the collapse and disappearance of bubbles might be the main mechanism of the noise generated from an underwater jet. Zhang Wenping, et al. [19] carried out an experimental study on underwater exhaust noise in a static pool with a diesel engine and compared the spectral characteristics and laws of underwater exhaust noise in the frequency domain between pulsating exhaust and stable exhaust. Tam, et al. [20] analysed the jet noise laboratory database of the NASA Langley Research Center and found that the spectral structure of nozzle jets was similar. This conclusion is applicable to supersonic and subsonic jets, as well as rectangular and elliptical jets. Stanley, et al. [21] used direct numerical simulations to study the structure and evolution process of plane jet vortexes under the condition of a low Reynolds number, revealed the formation and development process of shearing layers through experiments, and concluded that the large-scale vortexes of jet flow field structures were anisotropic, while small-scale vortexes were isotropic. Ma Xianguo [22] conducted theoretical and experimental research on steam underwater jet noise and found that the main sound source of steam underwater jet noise was caused by the pressure pulsation wave generated during the collapse of the bubbles. Niklas et al. [23] used a large eddy simulation method to study the relationship between the jet noise of gas with a high Mach number and the nozzle. Li and Gao [24] and Groschel et al, [25] found that the nozzle determined the structure and noise characteristics of a jet. Wang Chunxu et al. [26] conducted experimental measurements and research on an underwater free jet field noise, studied the spectrum structure of underwater radiated noise, and analysed the influence of radiated noise from an expanding nozzle, a straight nozzle and a contracting nozzle. Hao Zongrui [27] conducted an experimental study on the noise characteristics of underwater gas injections, and the results showed that when a bubble flow transformed into a jet flow, the corresponding peak sound pressure at 2 kHz increased non-linearly. Xin Junhua [28] simulated the straight and underwater jet flow fields of a nozzle and an underwater jet noise with the help of the FLUENT software. The results showed that the radiation energy is mainly concentrated in the low frequency domain but after a velocity increase, the main contribution of the jet noise moved to the high frequency band.

At present, the simulation of underwater flow fields of solid rocket engines basically focuses on depths up until 100 m, but the simulation of an underwater engine at a depth greater than 100 m is rarely discussed. In addition, research on underwater jet noise mechanisms remains at the Lighthill acoustic simulation theoretical level [29], but there is still a lack in good experimental equipment and methods for studying the underwater jet noise characteristics of solid rocket engines at a high water depth (≥ 100 m). Based on the abovementioned two aspects, it is necessary to conduct research on the simulation of solid rocket engines at a high water depth (≥ 100 m) and the characteristics of underwater jet noise.

2. Mathematical model and numerical calculation method

This paper is based on the unsteady axisymmetric $N$ - $S$ equation, the standard $k$ - $ε$ two-equation turbulence model and the VOF multiphase flow model. Due to the complex multiphase flow involving underwater ignition and the working process of a rocket engine, the calculation model needs to be appropriately simplified. It is assumed to be a two-dimensional axisymmetric model, taking the underwater ignition gas as ideal gas and taking water as a viscous incompressible medium. Heat transfer between the gas and liquid phases is considered in the calculation, but the phase transition is not considered. The gas enters through the convergence section of the nozzle, the external water flow field is calculated as being static, and the nozzle wall is set to an adiabatic wall. Setting the following boundary conditions, which were determined according to experimental data and experience: total temperature, total pressure and temperature changes upon entrance of the nozzle gas, the engine was put at a set depth under water, at an initial water temperature of 280 K.

2.1. Mathematical model

The continuity equation of fluid motion is as follows:

1

\frac{\partial ρ_{m}}{\partial t} + \nabla ∙ (ρ_{m} {\bar{v}}_{m}) = 0 .

In this equation, the density of the mixture is $ρ_{m} = \sum_{k = 1}^{n} a_{k} ρ_{k}$ , $a_{k}$ is the volume fraction of the $k$ phase, $ρ_{k}$ is the density of the $k$ phase, and the average speed is ${\bar{v}}_{m} = \frac{\sum_{k = 1}^{n} a_{k} ρ_{k} {\bar{v}}_{m}}{ρ_{m}}$ .

The momentum equation of the fluid motion is as follows:

2

\frac{\partial ρ_{m} {\bar{v}}_{m}}{\partial t} + \nabla ∙ (ρ_{m} {\bar{v}}_{m} {\bar{v}}_{m}) = - \nabla p + \nabla ∙ [μ_{m} (\nabla {\bar{v}}_{m} + \nabla v_{m}^{- T})] + ρ_{m} g .

In this equation, $μ_{m}$ is the viscosity of the mixture, $μ_{m} = \sum_{k = 1}^{n} a_{k} μ_{k}$ .

The energy equation of the fluid motion is as follows:

3

\frac{\partial}{\partial t} \sum_{k = 1}^{n} (a_{k} ρ_{k} E_{k}) + \nabla ∙ \sum_{k = 1}^{n} [a_{k} {\bar{v}}_{k}] (ρ_{k} E_{k} + p) = \nabla ∙ (k_{e f f} \nabla T),

E_{k} = \{\begin{array}{l} h_{k} - \frac{p}{p_{k}} + \frac{v_{k}^{2}}{2}, & C o m p r e s s i b l e p h a s e, \\ h_{k}, & I n c o m p r e s s i b l e p h a s e, \end{array}

where $h_{k}$ is the enthalpy of the $k$ phase, $k_{e f f} = k + k_{t}$ , $k_{t}$ is the heat conductivity of turbulence.

The turbulence equation for fluid motion ( $K$ - $E p s i l o n$ ) is:

4

\frac{\partial}{\partial t} (ρ_{m} k) + \nabla ∙ (ρ_{m} k v) = \nabla ∙ [(μ_{m} + \frac{μ_{t}}{σ_{k}}) \nabla k] + P_{k} - ρ (ε - ε_{0}) + S_{k},

5

\frac{\partial}{\partial t} (ρ_{m} ε) + \nabla ∙ (ρ_{m} ε v) = \nabla ∙ [(μ_{m} + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + \frac{1}{T_{e}} G_{1 ε} P_{ε} - G_{2 ε} f_{2} (\frac{ε}{T_{e}} - \frac{ε_{0}}{T_{0}}) + S_{ε},

where $v$ is the mean velocity, $μ_{t}$ is the dynamic viscosity, $G_{1 ε}$ , $G_{2 ε}$ , $σ_{k}$ , $σ_{ε}$ are model coefficients, $P_{k}$ , $P_{ε}$ are production terms, $f_{2}$ is the damping function, $S_{k}$ and $S_{ε}$ are user-specified source terms and $T_{0}$ is the definition of a specific time-scale.

2.2. Noise equation

The sound pressure at the monitoring point X was calculated according to the free space Green’s function by the FW-H (Kirchhoff – Ffowcs Williams and Hawkings) equation. The FW-H equation is as follows:

6

p^{'} (x, t) = p_{T}^{'} (x, t) + p_{L}^{'} (x, t) + p_{Q}^{'} (x, t) .

The single pole term is as follows:

7

p_{T}^{'} (x, t) = \frac{1}{4 π} ((\frac{\partial}{\partial t}) \int_{S}^{} {[\frac{Q}{r (1 - M_{r})}]}_{r e t} d S) .

The dipole term is as follows:

8

p_{L}^{'} (x, t) = \frac{1}{4 π} ((\frac{\partial^{2}}{(\partial_{x_{i}}) (\partial_{x_{j}})}) \int_{V}^{} {[\frac{L_{i}}{r (1 - M_{r})}]}_{r e t} d S) .

The quadrupole term is as follows:

9

p_{Q}^{'} (x, t) = \frac{1}{4 π} ((\frac{\partial^{2}}{(\partial_{x_{i}}) (\partial_{x_{j}})}) \int_{V}^{} {[\frac{T_{i j}}{r (1 - M_{r})}]}_{r e t} d V) .

Q = ρ_{0} U_{i} n_{i}, U_{i} = (1 - \frac{ρ}{ρ_{0}}) v_{i} + \frac{ρ u_{i}}{ρ_{0}}, L_{i} = P_{i j} n_{i} + ρ u_{i} (u_{n} - v_{n}),

P_{i j} = (p - p_{0}) δ_{i j} - σ_{i j}, T_{i j} = ρ u_{i} u_{j} + δ_{i j} [(p - p_{0}) - c_{0}^{2} (p - p_{0})] - σ_{i j},

where $u_{i}$ is the fluid velocity component in the direction of $i$ , $u_{n}$ is the fluid velocity component perpendicular to the surface, $v_{i}$ is the surface velocity component in the direction of $i$ , $v_{n}$ is the surface velocity component perpendicular to the surface, $n_{i}$ is the surface normal vector, $σ_{i j}$ is the viscous stress tensor, $ρ_{0}$ is the far-field density, $P_{i j}$ is the compression stress tensor, and $T_{i j}$ is the Lighthill stress tensor.

Fig. 1Contours of gas phase volume fractions at different instants