Figure 1 shows the geometric model of the computational domain, which is divided into the cooling air cavity and the mainstream channel. The secondary flow enters the mainstream through the film hole structure in the cooling air cavity. The diameter of the cylindrical hole is D, and the secondary flow inlet, the mainstream inlet and the outlet are set. Considering that the structure has a spanwise periodic flow, the hole spacing (3D) is taken as the periodic length, and one of the holes is taken as the calculation object.

Figure 2 shows the geometric size of the waveform hole, which is the minimum positive period of the cosine function in the spanwise direction (T = 2D). In order to ensure that the opening area of a single waveform hole is equal to that of a standard cylindrical hole, the width of the waveform hole along the flow direction is specified to be 0.4D. In order to facilitate the research, the corresponding phase angle of the trigonometric function at the edge is defined as the initial phase φ, the maximum distance from the equilibrium position in the spanwise direction is the amplitude A, the farthest point away from the equilibrium position along the flow direction is the peak, and the other extreme point is the trough.

Blowing ratio: M = ρ c u c ρ g u g (1)

In the formula: ρ _c is the secondary flow density, kg/m³; ρ _g is the mainstream density, kg/m³; u _c is the secondary flow velocity, m/s; u _g is the mainstream speed, m/s.

Temperature ratio: K = T g T c (2)

In the formula: T _c is the secondary flow temperature, K; while T _g is the mainstream temperature, K.

Adiabatic cooling efficiency: η = T g − T aw T g − T c (3)

In the formula: T _c is the secondary flow temperature, K; T _g is the mainstream temperature, K; T _aw is adiabatic wall temperature, K.

Average spanwise cooling efficiency: η ave = 1 n ∑ i = 1 n η i (4)

In the formula, η _i is the cooling efficiency of the spanwise ith grid, and n is the number of spanwise grids.

The computational domain uses the commercial software ICEM CFD 2020 to divide the structured grid and set the boundary layer and periodic surface, while ensuring that the y + < 3 near the wall, and the grid near the intersection of the secondary flow. In addition, through the verification of grid independence, the numerical calculation is carried out for different numbers of grids under the same working conditions, and the number of grids is finally determined to be 1.2 million. At this time, the calculation error with the number of grids of 2.4 million and 4.8 million is within 3%, indicating that the number of grids used has high accuracy, as shown in Figure 3, where X is the distance between the measurement position and the center of the waveform hole.

The mathematical model of steady-state flow and heat transfer between solid structures and hot or cold fluids is as follows:

Continuity equation: ∇ • ρ U ⃗ = 0 (5)

Momentum equation: ∇ • ρ U U ⃗ = − ∇ p + ∇ • τ + ∂ ∂ x j − ρ u i ⃗ u j ⃗ (6)

Energy equation: ∇ • U ⃗ ρ E + p = ∇ • k f ∇ T + τ • U ⃗ (7)

In the formula, ρ is density, U is speed, p is pressure, τ is shear stress, k _f is the thermal conductivity of the fluid, and T is temperature.

The selection of turbulence model is an important factor affecting the numerical results. For the flow separation and heat transfer details, the boundary layer needs to be solved with high precision. To ensure the accuracy of the solution, the SST k-ω turbulence model is selected in this paper. Both the main inlet and the secondary inlet are velocity inlets, and the outlet is pressure outlet. Specific parameters are shown in Table 1. The fluid domain material is incompressible ideal gas, the turbulence degree is 3%, and the others are adiabatic boundaries without heat exchange with the outside world. The coupled algorithm is used to solve the coupling of pressure and velocity, and convergence is based on the residual value less than 10⁻⁶.

In order to verify the feasibility of the calculation method, the numerical calculation is carried out according to Yhenc’s experiment, and the experimental data are compared with the numerical simulation data in the research of Yuen and Martinez-Botas (2003). Figure 4 shows the comparison of the spanwise average cooling efficiency at the downstream of the standard cylindrical hole when the blowing ratio M = 1 and the incident angle of the cooling air flow is 30°. The error between the experimental data and the simulation results is within 5%, indicating that the adopted calculation method is feasible.

Figure 5 shows that when the blowing ratio M = 1, the temperature ratio K = 1.3 and the opening angle θ = 45°, the shape of the film hole is the velocity vector and temperature distribution of the cylindrical hole, the rectangular hole and the waveform hole at the downstream X/D = 5, respectively. The diameter of the cylindrical hole is D, the length of the rectangular hole is 2D, the width is 0.4D, the initial phase of the waveform hole is 180°, the amplitude is 0.2D, and the width is 0.4D. It can be seen from the figure that kidney-shaped vortices are formed downstream of the film hole center. At this position, the kidney-shaped vortex downstream of the cylindrical hole is the largest, the spanwise cooling airflow coverage is the smallest, the kidney-shaped vortex downstream of the waveform hole is the smallest, the spanwise cooling airflow coverage is the largest, and the downstream of the rectangular hole is between the two.

Figure 6 shows the comparison of film cooling efficiency of three different shapes of film holes at different positions along the flow direction when the blowing ratio M = 1, the temperature ratio K = 1.3 and the opening angle θ = 45°. It can be seen from the diagram that the cooling effect of the film hole with cylindrical hole structure in the spanwise direction is poor, and the cooling air fails to diffuse effectively in the spanwise direction under the suppression of the mainstream, and the adhesion effect is poor. Compared with the rectangular hole, the waveform hole structure weakens the energy of the kidney vortex, and the cooling airflow diffuses more evenly in the spanwise direction, and the cooling effect is more significant.

As shown in Figure 7, when the blowing ratio M = 1, the temperature ratio K = 1.3, and the opening angle θ = 45°, the shape of the film hole is the comparison of the spanwise average cooling efficiency of the cylindrical hole, the rectangular hole and the waveform hole at different downstream positions. It can be seen from the figure that compared with the cylindrical hole, the spanwise average cooling efficiency of the rectangular hole and the waveform hole at the same measurement position is greatly improved. The spanwise average film cooling efficiency of the rectangular hole structure is increased by about 100%, and the spanwise average film cooling efficiency of the waveform hole structure is increased by about 130%. Compared with the rectangular hole, the waveform hole structure has a better spanwise cooling effect downstream of the film hole, especially when 5 < X/D < 15, the cooling effect of the waveform hole structure is more significant.

Figure 8 shows the velocity vector and temperature distribution of waveform holes with different initial phases at downstream X/D = 5 when blowing ratio M = 1, temperature ratio K = 1.3, opening angle θ = 45°, and amplitude A = 0.2D. It can be seen from the figure that a kidney-shaped vortex is formed downstream of the film hole outlet. When the initial phase ϕ = 0°, the trough is at the center of the film hole, the kidney vortex generated at the convergence of the secondary flow and the mainstream reaches the maximum, and the coverage of the cooling air flow in the spanwise direction is the smallest. When the initial phase ϕ = 90°, the peak and trough are at Y/D = ±0.5. It can be seen that the vortex at the downstream of the trough is significantly larger than that at the downstream of the peak. The cooling effect of the cooling air flow at Y/D > 0 is better than that at Y/D < 0. When the initial phase ϕ = 180°, the kidney vortex generated at the convergence of the secondary flow and the mainstream is the smallest, and the coverage of the cooling air flow in the spanwise direction reaches the maximum.

Figure 9 shows the comparison of film cooling efficiency at different positions along the flow direction of the wavy holes with different initial phases when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the amplitude A = 0.2D. It can be seen from the figure that when the initial phase ϕ = 0°, the cooling air flow is unevenly distributed in the spanwise direction, and the cooling effect is the worst. When the initial phase ϕ = 90°, due to the peaks and troughs on both sides of the center of the film hole, the spanwise pressure difference is uneven, which makes the secondary flow spread faster to the downstream side of the trough. When the initial phase ϕ = 180°, the distribution of the cooling airflow in the spanwise direction is more uniform, and the cooling effect is the best.

Figure 10 shows that when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the amplitude A = 0.2D, the mass flow rate at the outlet of the waveform hole with different initial phases is compared. It can be seen from the figure that with the increase of the initial phase ϕ, the flow rate of the cooling gas flowing from the trough gradually decreases, and the flow rate of the cooling gas flowing from the peak gradually increases. This is mainly due to the friction and viscosity between the wall surface of the film hole and the cooling gas, resulting in a slow flow rate of the fluid near the hole wall.

Figure 11 shows the comparison of the spanwise average cooling efficiency of the waveform holes with different initial phases at different downstream positions when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the amplitude A = 0.2D. When 0 < X/D < 3 near the wall, the cooling efficiency of the cooling air flow on the hot side wall increases with the increase of the initial phase. This is mainly because the trough is farther away from the same measuring point on the hot side wall surface than other positions on the waveform hole, so the cooling gas flowing from the trough has more sufficient time and distance to be fully adhered to the wall after being suppressed by the mainstream. Due to the existence of viscosity, the average flow rate of the secondary flow flowing from the central area of the film hole is larger. Therefore, when the trough is at the most central position of the waveform hole (ϕ = 0°), the mainstream can suppress more flow of cooling gas in the area of 0 < X/D < 3 downstream of the film hole outlet, and the cooling efficiency in this area will be higher. Similarly, when the trough is at the most marginal position of the waveform hole (ϕ = 180°), the average flow rate of cooling gas from the trough is the smallest, and the cooling efficiency is the lowest in the region of 0 < X/D < 3 downstream of the film hole outlet. When near the wall X/D > 3, the cooling efficiency of the cooling air flow on the hot side wall decreases with the increase of the initial phase. This is mainly because the cooling gas flowing from the trough is first suppressed by the mainstream to the near wall and diffuses in the spanwise direction. Therefore, when the trough is at the most central position of the waveform hole (ϕ = 0°), the cooling gas dissipates too much at the near wall 0 < X/D < 3, resulting in lower cooling efficiency at the near wall X/D > 3. In the same way, when the trough is at the most edge position of the waveform hole (ϕ = 180°), the cooling gas dissipates the least at the near wall 0 < X/D < 3, resulting in the highest cooling efficiency at the near wall X/D > 3.

Figure 12 shows the comparison of the spanwise average cooling efficiency of the waveform holes with different initial phases at different downstream positions when the temperature ratio K = 1.3, the opening angle θ = 45°, the amplitude A = 0.2D, and the blowing ratio M is 0.5, 1, and 2, respectively. It can be seen from the figure that when the blowing ratio M = 0.5, the change of the initial phase of the waveform hole has little effect on the spanwise average cooling efficiency of each measured position on the downstream wall of the film hole outlet, but the overall cooling efficiency increases with the increase of the initial phase ϕ. When the blowing ratio M ≥ 1, the initial phase ϕ of the waveform hole has a great influence on the spanwise average cooling efficiency of the downstream wall of the film hole outlet, and the difference is more obvious when the blowing ratio M = 2.

Figure 13 shows the velocity vector and temperature distribution of the waveform holes with different amplitudes at the downstream X/D = 5 when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the initial phase ϕ = 180°. It can be seen from the figure that the kidney-shaped vortex formed downstream of the center of the waveform hole gradually flattened with the increase of amplitude A, which effectively inhibited the mixing of mainstream gas and cooling gas. In addition, when the amplitude A is 0.6D, two symmetrical vortices are formed on both sides of the kidney-shaped vortex, which weakens the area of the kidney-shaped vortex. In the temperature distribution, when the amplitude A increases from 0.1D to 0.4D, the distribution of the cooling airflow in the spanwise direction gradually tends to be uniform, and the full coverage of the cooling airflow is achieved when A = 0.4D. When the amplitude A increases from 0.4D to 0.6D, although the cooling air flow is fully covered in the spanwise direction, when the amplitude A = 0.6D, two symmetrical vortices at Y/D = ±1.25D aggravate the mixing of the mainstream and the secondary flow, which makes the cooling effect near the wall worse.

Figure 14 shows the comparison of film cooling efficiency at different positions along the flow direction of the wavy holes with different amplitudes when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the initial phase ϕ = 180°. It can be seen from the figure that when 0 < X/D < 5, the smaller the amplitude A, the better the cooling effect of the cooling air flow on the hot side wall. When X/D > 5, as the amplitude A increases, the cooling air flow cools the hot side wall more evenly. However, when the amplitude A increases from 0.4D to 0.6D, although the cooling air flow cools the hot side wall more evenly, at the same measurement position, the spanwise film cooling efficiency is the highest when the amplitude A = 0.4D.

Figure 15 shows the comparison of the spanwise average cooling efficiency of the waveform holes with different amplitudes at different downstream positions when the blowing ratio M = 1, the temperature ratio K = 1.3, the opening angle θ = 45°, and the initial phase ϕ = 180°. It can be seen from the diagram that when the amplitude A increases from 0.1D to 0.4D, the cooling efficiency of the secondary flow to the wall surface gradually increases. This is mainly because as the amplitude A increases, the relative distance between the peak and trough of the waveform hole also increases. Therefore, the velocity difference of the cooling gas flowing from the peak and trough to the same position near the wall surface gradually increases, so the diffusion speed of the cooling gas flowing from the peak in the spanwise direction also increases, which can cool the hot side wall faster and more evenly. When the amplitude A increases from 0.4D to 0.6D, the cooling efficiency of the secondary flow on the wall surface gradually decreases. Although the increase of the amplitude A will accelerate the diffusion of the cooling air flow in the spanwise direction, it also widens the distance between the trough and the hot side wall surface, so that the cooling air flow from the vicinity of the trough has lost too much when it reaches each measurement position, and the acceleration of the cooling air diffusion speed in the spanwise direction cannot compensate for this loss. In the scope of this study, when the amplitude A = 0.4D, the waveform hole structure has the best cooling effect on the hot side wall.