## 1 Introduction  
In cold region tunnels, damages from frost may easily cause cracking, peeling-off and ice-covering of lining systems, which may even lead to more serious consequences, such as tunnel discard. The damages from frost not only cause safety issues for public transportation, but also increase maintenance costs of tunnels during operation. At present, applying heat-insulating layers to tunnel lining surfaces is one of the main methods to prevent damages from frost in tunnels in China, e.g., the Zhegu mountain tunnel, the Daban mountain tunnel, and the Kunlun mountain tunnel. Past studies [1; 2] have shown that damages from frost in tunnels are usually observed near tunnel ends, while rarely being found in the middle of tunnels. Using heat-insulating layers can effectively reduce the influence of cold air on the temperature of surrounding rocks and tunnel linings [3; 4]. It is also found that using heat-insulating layers is able to ensure the temperature of tunnel linings remain above 0 \({}^{\circ}\)C to prevent damages from frost. Thus, applying heat-insulating layers near tunnel ends at certain distances is highly recommended for long tunnels in the cold region. A large number of studies on heat-insulating technologies for cold region tunnels had been carried out in terms of theoretical developments and engineering applications worldwide. Prior to 1990s, [PERSON] et al. [5] proposed a two-zone model (frozen zone and unfrozen zone) for porous mediums (e.g., soils and rocks) taking into account ice-water phase change. Later in 1991, [PERSON] [6] carried out both theoretical and experimental studies on heat transfer in porous mediums. Based on [PERSON]'s research, [PERSON] et al. [7] and [PERSON] et al. [8] proposed a three-zone (frozen zone, freezing zone, and unfrozen zone) model to simulate the freezing process in fine-grained loamy soils. The above-mentioned research results laid a solid foundation for the numerical simulation of the temperature fields in rock mass, taking into account phase change. In recent years, [PERSON] et al. [9] carried out a large number of studies on cold region tunnels used in the Qinghai-Tibet Railway, and proposed a temperature-stress-seepage coupling model, which was later used to analyze the temperature fields of cold region tunnels. In 2002, [PERSON] et al. [10] investigated the influences of different construction seasons, initial temperatures and heat-insulating materials on the freeze-thaw action of tunnel surrounding rocks. Based on the research results, [PERSON] et al. [11] subsequently performed a forecast analysis for the refreezing of the thawed permafrost caused by construction reasons in the Fenghuo Mountain tunnel. In 2008, [PERSON] et al. [1] established a theoretical formula to predict the temperature distributions inside tunnels, which was used as a theoretical basis for designing the length of heat-insulating layers. [PERSON] et al. [12] improved the three-zone model and established a new temperature-seepage coupling model, which was employed in analyzing the temperature fields in the Galongla mountain tunnel [13]. However, it was found that all the above-mentioned studies mainly focused on 2D numerical models of tunnels without taking into account the influence of airflows. Studies on the distribution characteristics of airflow temperature fields in tunnels had been carried out by several researchers in recent years. In 1998, [PERSON] et al. [14] studied the influences of ventilation velocity and external air temperatures on airflow temperatures in the Qinling mountain tunnel based on the heat transfer theory. Two years later, [PERSON] et al. [15] proposed a mathematical model for the heat convection between airflows and surrounding rocks based on turbulence equations, which was used to investigate the temperature distributions of airflows in the Qilian mountain tunnel. They also carefully examined the differences between using turbulence equations and laminar equations.