## 1 Introduction  
In order to ensure the normal operation of the equipment, the vehicle uses cooling water to absorb the heat generated by the operation of the equipment in the process of underwater movement, and the cooling water is discharged to form a thermal jet [1; 2; 3; 4], which is mixed with the environmental fluid for heat exchange. The thermal jet floats up to form infrared characteristics on the water surface, which can be easily detected by infrared detection equipment. Domestic and foreign scholars have conducted numerous experimental and simulation studies on the characteristics of thermal jets. In [5], the authors used the similarity model of underwater vehicles in the thermal jet experiment to measure parameters under certain constraints (the Reynold number of fluid is lower than the normal value) and observed the turbulent flow field of the thermal jet through PIV technology. In [6; 7], the authors experimentally studied the thermal characteristics of turbulent jet impingement on water surfaces at different Reynold numbers. An infrared thermal image of the water surface was obtained by a high-resolution infrared detector. In [8], the authors studied the vertical temperature distribution of seawater in the actual marine environment and carried out numerical calculations on the thermal jets caused by the appendages and propellers of underwater vehicles in the temperature-stratified environment. In [9], [PERSON] established and gradually improved the laboratory underwater vehicle thermal jet discharge simulation experimental platform and carried out thermal jet discharge simulation experiments in the laboratory in water with a uniform and stratified temperature. The formation mechanism, the means of detection and key influencing factors of thermal jets were explored. [PERSON] et al. compared the flow characteristics of thermal jets and non-thermal jets under the condition of a Froude number with the same density through the discharge experiment. The experimental results show that the thermal jet flow track was farther, and the flow field range was wider in the lenti environment [10]. According to the similarity principle, the authors carried out the infrared detection experiment with a scaled underwater vehicle model under the condition of realizing geometric similarity, heat flux similarity and temperature field similarity, and the authors studied the influence of different experimental conditions on the infrared detection of thermal jets in [11]. [PERSON] et al. [12] analyzed the characteristics of different types of thermal jets through experiments, focusing on the difference of thermal jet temperature changes. [PERSON] et al. [13] experimentally observed the mixing process of thermal jets and environmental fluid through a high-speed camera, laser imaging and ultrasonic velocimetry. The mixing processes of jets can be divided into fully turbulent regime and semi-turbulence regime, and the semi-turbulence regime can be further divided into the momentum-dominated regime and the buoyancy-dominated regime. It was concluded that the fluid flow is controlled by the interactions among inertia, viscosity and the buoyancy of fluids. [PERSON] et al. [14] analyzed the flow state of a unilateral thermal jet in a cross-flow environment under an unsteady N-S equation, and the results show that the ratio of the jet velocity to the environmental water flow velocity affected the initial temperature of the jet. [PERSON] et al. [15] studied the influence of the non-orthogonality of grids on numerical calculation solutions of thermal jet flow and used a hexahedral grid element and a prism grid element to divide and compare. The results show that the prismatic grid element increased the computational instability compared with the hexahedral grid element. Although the non-orthogonal correction method can improve calculation stability, the convergence time increased. [PERSON] et al.