﻿ 密度分层对热射流流动的影响预测模型
 舰船科学技术  2021, Vol. 43 Issue (11): 14-19    DOI: 10.3404/j.issn.1672-7649.2021.11.003 PDF

Effect of density stratification on hot jet flow
MAO De-long, WANG Kang-shuo, SHA Jiang, HE Lin, REN Hai-gang, WANG Yi-hao, WANG Ben
System Engineering Research Institute, Beijing 100094, China
Abstract: This study attempts to investigate the effect of seawater density stratification on the thermal jet flow behind the underwater vehicle. The effects of temperature and salinity on the density of seawater along the depth direction are studied, and the relevant numerical simulation is carried out, and the correction coefficient of density stratification effect is introduced to modify the numerical simulation of seawater without density stratification. Previous results showed that with the increase of depth, the salinity of seawater increases, the temperature decreases, and finally the density increases. Compared calculation results of fitting formula in this study to the numerical simulation results, the error of fitting formula is less than 6%, and the error of most data points is less than 1%, which meets the requirements of prediction.
Key words: underwater vehicle     thermal wakes     seawater density     numerical simulation     correction factor
0 引　言

 图 1 40° N海水密度分层的典型变化规律 Fig. 1 Typical variation law of seawater density stratification at 40°N latitude

1 数学模型

 $\begin{split}&\frac{{\partial \left( {\rho {u_i}} \right)}}{{\partial t}} + \frac{{\partial \left( {\rho {u_i}{u_j}} \right)}}{{\partial {x_j}}} = - \frac{{\partial p}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left( {\mu \frac{{\partial {u_i}}}{{\partial {x_j}}}} \right) + f_i\text{，} \\ &\left( {i, j = 1,2, 3} \right)\text{，}\end{split}$ (1)
 $\frac{{\partial \left( {{u_j}} \right)}}{{\partial {x_j}}} = 0 {\text{。}}$ (2)

 $\frac{{\partial T}}{{\partial t}} + \frac{{\partial T}}{{\partial {x_j}}} = \frac{\partial }{{\partial {x_j}}}\left( {\alpha \frac{{\partial T}}{{\partial {x_j}}}} \right),\qquad\left( {i, j = 1, 2,3} \right) ,$ (3)

 $\rho=\rho_{\mathcal{R e f}}\left[1-\beta\left(T-T_{\text {Ref }}\right)\right] {\text{，}}$ (4)

 $G=\rho_{R{\rm{ef}}}[1-\beta(T-T_{R{\rm{ef}}})]g{\text{，}}$ (5)

 $F={\rho }_{R\text{ef}}g{\text{。}}$ (6)

 ${F}_{\text{i}}={\rho }_{R\text{ef}}[\beta (T-{T}_{R\text{ef}})]g {\text{。}}$ (7)

 $\beta^{\prime}=\frac{\rho(h)}{\rho_{\text {Ifof }}} \beta {\text{。}}$ (8)

2 CFD模拟 2.1 计算区域与边界条件 2.1.1 计算区域

 图 2 计算区域 Fig. 2 Computational region
2.1.2 边界条件

2.2 工况计算

 $\varphi=\frac{\left(\rho_{50}-\rho_{0}\right)}{\rho_{0}} {\text{。}}$ (9)

 图 3 t=16min，xy截面温度云图 Fig. 3 xy section temperature nephogram in 16 min

3 密度分层对热射流影响的仿真分析 3.1 密度分层影响的定性说明

 图 4 15.01℃等值面 Fig. 4 15.01 °C isosurface

 图 5 t=4 min，热射流15.01℃等值面 Fig. 5 Hot jet 15.01 °C isosurface at 4 min

 图 6 t=4 min，热射流15.01℃等值面 Fig. 6 Hot jet 15.01 °C isosurface at 4 min

$\varphi$ 的定义可知， $\varphi$ 值越大，则某一固定深度处的海水密度 $\rho_{\rm{Ref}}$ 越大。式 ${F}_{\text{i}}={\rho }_{R\text{ef}}[\beta (T-{T}_{R\text{ef}})]g$ ，说明此时热射流所受浮力与重力的合力 $F_i$ 也越大。因此 $\varphi$ 值越大，热射流漂浮的速度越快，相同时刻热射流的高度H也越高。

3.2 密度分层影响的修正

 $\omega = \frac{{{H_\varphi }}}{{{H_0}}} \text{，}$ (10)

 ${H_\varphi } = \omega {H_0}\text{。}$ (11)

 图 7 $\Delta T=30^{\circ}{\rm C}$ ，不同 $\varphi$ 值的热射流高度曲线 Fig. 7 At $\Delta T=30^{\circ}{\rm C}$ , Heat jet height curves with different $\varphi$ values

 图 8 $\Delta T=30^{\circ}{\rm C}$ ， $\varphi=$ 50%，不同航速V的热射流高度曲线 Fig. 8 At $\Delta T=30^{\circ}{\rm C}$ and $\varphi=$ 50%, Hot jet height curves with different speed V

 图 9 V=0.1 kn， $\varphi=$ 50%，不同温差 $\Delta T$ 热射流高度曲线 Fig. 9 At V=0.1 kn and $\varphi=$ 50%, Hot jet height curve with different temperature difference $\Delta T$

 图 10 $\Delta T=30^{\circ}{\rm C}$ ，V=0.1 kn，不同 $\varphi$ 值的密度分层修正系数ω曲线 Fig. 10 At $\Delta T=30^{\circ}{\rm C}$ and V=0.1 kn, Density stratification correction coefficient ωcurves with different $\varphi$ values

 图 11 $\Delta T=30^{\circ}{\rm C}$ ，V=0.2 kn，不同 $\varphi$ 值的密度分层修正系数ω曲线 Fig. 11 At $\Delta T=30^{\circ}{\rm C}$ and V=0.2 kn, Density stratification correction coefficient ωcurves with different $\varphi$ values

 图 12 $\Delta T=40^{\circ}{\rm C}$ ，V=0.1 kn，不同 $\varphi$ 值的密度分层修正系数ω曲线 Fig. 12 At $\Delta T=40^{\circ}{\rm C}$ and V=0.1 kn, Density stratification correction coefficient ω curves with different $\varphi$ values

 图 13 $\Delta T=30^{\circ}{\rm C}$ ， $\varphi=$ 50%，不同航速V下密度分层修正系数ω曲线 Fig. 13 At $\Delta T=30^{\circ}{\rm C}$ and $\varphi=$ 50%, Density stratification correction coefficient ω curves with different speeds V

 图 14 V=0.1 kn， $\varphi=$ 50%，不同温差 $\Delta T$ 下密度分层修正系数ω曲线 Fig. 14 At V=0.1kn and $\varphi=$ =50%, Density stratification correction coefficient ω curves with different temperature difference $\Delta T$

 $\omega = 1 + a{(1 + V)^b}{\varphi ^c}\Delta {T^d}{{{t}}^e}\text{，}$ (12)

$\varphi=0$ 时，热射流高度不需要修正，即 $\omega=1$ ，因此关联式中第一项为常数1；其次，第二项设置成指数函数的原因在于，随着密度分层值 $\varphi$ 、航速V（kn）、温差 $\Delta T$ （℃）以及时间t（min）的变化，密度修正系数 $\omega$ 均是呈现单调变化规律的，这与指数型函数相符；显然当航速V=0 kn，密度分层值 $\varphi \ne 0$ 时，热射流高度依然需要修正，即 $\omega \neq 1$ 。若航速影响项为Vb，则有 $\omega=1$ ，因此指数函数项出现了 ${(1 + V)^b}$ $\Delta T=0$ 表征热射流与周围环境不存在温差，不会上浮；t=0表征热射流还没有排入海水中，因此温差与时间两项直接写成指数的形式即可。通过Matlab对数值仿真中8个工况，共66个数据点进行拟合，得出密度修正系数 $\omega$ 拟合式如下：

 $\omega = 1 + 2.214\;5{(1 + V)^{2.711\;5}}{\varphi ^{0.784\;5}}\Delta {T^{-0.598\;5}}{{{t}}^{- 0.224\;0}}\text{。}$ (13)

4 结　语

1）随着海水深度的增加，海水的盐度上升，温度下降，这两者均会造成海水密度随深度的增加而增大；

2）所得到的拟合关联式计算结果与数值仿真结果对比，误差在6%以内，且绝大部分数据点的误差都在1%以下。

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