## 5 Numerical Simulation Results  
The validation of the numerical simulation results with model test results at temperature measuring point 1 at Section 3-3, which has a good representative, is shown in Figure 18.  
As shown in Figure 18, the numerical simulation results are basically consistent with the model test results; the two curves both show a change process of steep drop first, then slow drop and finally stabilized. The difference between them mainly occurs at the steep drop stage, where the model test results are always lower than the numerical simulation results, and the maximum reaches 2.5 \({}^{\circ}\)C. The difference in refrigerating capacity may be due to the unstable ambient temperature in the laboratory or the influence of some unpredictable factors; that is, the model test results may have errors in reflecting the development of freezing temperature field. In contrast, the numerical simulation results can reflect the development of the freezing temperature field more reasonably.  
The evolution law of the three-dimensional freezing temperature field is shown in Figure 19 after freezing for 4 min, 28 min, 44 min and 60 min.  
Figure 19 shows that the three-dimensional freezing temperature field has obvious non-uniformity after freezing for 4 min, 28 min, 44 min and 60 min. The frozen walls at both ends of the freezing pipe connect at first, and the frozen walls in the middle of the freezing pipe connect subsequently. After freezing for 4 min, four independent frozen soil columns are formed with the characteristic that both ends are coarser and the middle is thinner. After freezing for 28 min, the two frozen soil columns in the middle connect in turn at Section 3-3 and Section 1-1, then expand along with the negative and positive directions of freezing depth, respectively, and eventually converge at Section 2-2. During this period, the frozen soil columns on both sides also gradually develop following this law. After freezing for 48 min, the frozen wall basically intersects, and the whole frozen wall still presents the law that is thin in the middle and thick at both ends. After freezing for 60 min, the frozen wall completely intersects, and the thickness of the frozen wall meets the design requirements.  
Figure 18: The validation of numerical simulation results with model test results.  
The comparison of the freezing temperature field calculated by numerical simulation and model test is shown in Figure 20 after freezing for 60 min.  
Figure 19: Evolution of three-dimensional freezing temperature field: (**a**) 4 min; (**b**) 28 min; (**c**) 44 min; (**d**) 60 min.  
Figure 20: _Cont._  
Figure 20 shows that after freezing for 60 min, the thickness of the frozen wall at three sections is different; that is, the thickness of Section 1-1, 2-2 and 3-3 at tunnel arch is 45.7 mm, 39.1 mm and 47.3 mm in the model test, while it is 46.3 mm, 40.9 mm and 49.4 mm in the numerical simulation, respectively. The difference between them is 0.6 mm, 1.8 mm and 2.1 mm, and no matter the sections perpendicular or parallel to the length direction of the freezing pipes, the shapes of the frozen wall at each section are all basically consistent in the numerical simulation and the model test, which indicates that the numerical simulation method has high prediction accuracy after introducing the temperature fitting function of the freezing pipe wall.