Numerical method

The film-cooling problem was modeled by the compressible Navier-Stokes and energy equations that are valid for a thermally and calorically perfect gas. The effects of turbulence were modeled by using the two-equation realizable k−ε model. The following algorithms were invoked in Fluent. Since only steady-state solutions were of interest, the SIMPLE algorithm was used. The fluxes at the cell faces representing advection and diffusion were interpolated by using second-order upwind differences. For all computations, iterations were continued until all residuals for all equations plateau to ensure that convergence to steady state has been reached. At convergence, the normalized residuals were always less than 10^–6 for the continuity equation and the three components of the velocity, less than 10^–8 for the energy, and less than 10^–5 for the turbulence quantities.

Computational domain and grids

The computational domain of the interest includes a spanwise period of the mainstream region whose pitch is 3 trunk hole diameters, a single trunk-branch hole and the plenum supplying film flow as shown in Figure 3. The surface grids are shown in Figure 4. The mainstream tunnel is treated as the separated three regions in the streamwise direction when the grids are generated. The unstructured grids are generated in the middle domain joining to the film cooling holes, but the structured grids are generated in the other two domains. The grids become longer in the positive y axis direction. The first layer grids close to the cooled surface is 0.1 mm in positive y axis direction, resulting in the corresponding y+ from 1 to 10 according to the various locations. The grids of the film cooling holes and the plenum are generated respectively. The grid number of the whole domain is about 1.2 × 10⁶.

Figure 3:

Computational domain.

Figure 4:

Grids of surfaces.

Parameter definition and boundary condition

The blowing ratio was calculated by the following equation

Where uc was the averaged velocity based on the mass flow and cross-section area of the hole entrance vertical to the center axis of the trunch hole and uloc was the local velocity of mainstream at the film hole exit. This shows that the same blowing ratios of the trunk-branch hole and the cylindrical hole have the same mass flow.

Local film cooling effectiveness was calculated as

The dimensionless temperature was calculated by as

Where Taw was the adiabatic wall temperature, T∞ was the gas temperature, Tc was the secondary (heated) flow temperature, T was the temperature of the fluid after mixing between the coolant and the gas.

An area-averaged velocity magnitude of 15 m/s was given at the inlet of the mainstream tunnel and the corresponding Reynolds number based on the trunk hole diameter was 9,100. The inlet of the plenum was given the various mass flow rates according to the blowing ratios. The side walls of the whole domain were treated as the periodic boundary conditions. The turbulence intensities of the inlets were considered to be 0.4 %, 5 %, 10 % and 20 %. The temperature of the coolant was maintained 30 K above the mainstream according to the test conditions and the cooled surface was treated as the isothermal surface. The relatively density ratio of the coolant to the mainstream was 0.91. The calculated cases were considered at the blowing ratios of 0.5, 1.0, 1.5 and 2.0.

Comparison of calculated data and experimental data

The numerical results and the experimental data in the literature [2] are shown in Figure 5. The calculated spanwise averaged film cooling effectiveness with a standard cylindrical hole is consistent with the experimental data for the same flow cases and geometries. The spanwise averaged film cooling effectiveness is calculated as

Where n was the number of the spanwise grids at the same location of x/d and the ηi was the film cooling effectiveness of the local spanwise grid. The biggest difference between the experimental data and numerical result is less than 10 %. This indicates that the realizable k−ε turbulence model and the generated grids can be used for calculating film cooling effectiveness. Figure 5 also shows that the effectiveness with the double-outlet holes is much higher than the effectiveness with the cylindrical hole from 200 % to 300 %.

Figure 5:

Comparison of numerical result and experimental data (M = 1.0).

Effect of turbulence on the Surface contour of film cooling effectiveness

Turbulence intensity is a parameter that cannot be ignored in turbine passages. Film cooling characteristics are significantly affected by presence of free-stream turbulence. Figure 6 shows the comparison of the two dimensional contours of the film cooling effectiveness for the low and the high turbulence at the lower blowing ratio of 0.5. The film cooling effectiveness near the hole is weakly affected by the different turbulence and the effectiveness is reduced by the high turbulence beyond x/d=5. The coolant at the low blowing ratio possesses low momentum. The lower momentum jets stay closer to the surface and are weaker than the mainstream. With increased free-stream turbulence, large fluctuations in the mainstream break down the weak jets and reduce film coverage.

Figure 6:

Contours of the film cooling effectiveness with the various turbulence intensities at M=0.5.

At the blowing ratio of 1.5, the surfaces downstream of the hole outlets for the high turbulence are more effectively covered with the film flow as shown in Figure 7(a) and 7(b). The effectiveness contours for both two turbulences deviate from the mainstream direction to the spanwise direction. The film cooling effectiveness on the bad cooled surface is provided by 30 % as the turbulence intensity increased from 0.4 % to 20 %. This shows that the high turbulence results in the uniform effectiveness. The jets are stronger at the high blowing ratio, possessing stronger momentum than the mainstream at the injection location. The jets lift off from the surface and weakly reattached downstream. With increased free-stream turbulence, large fluctuations in the mainstream reduce the length of the jets penetrating into the mainstream and strengthen the jets spread laterally. The differences of the film flow between the low and high blowing ratios cause the different effect of the turbulence on the film cooling effectiveness.

Figure 7:

Contours of the film cooling effectiveness with the various turbulence intensities at M=1.5.

Compared to Figure 8(a) and 8(b), at the location of the near the hole x/d=5, the vortices for the low turbulence is very strong at the location of y/d from 0.5 to 1.0. The high turbulence intensity causes weakly vortices and the vortices are close to the surface. This is because the high turbulence strengthens the lateral spread of the coolant, resulting in the short length of the coolant penetrating in the mainstream. With the increased x/d, the high turbulence causes the weaker vortices and the uniform distributions of the fluid temperature as shown in Figure 9(a) and 9(b). In the range of y/d from 0 to 1, the dimensionless temperature of the fluid is from 0.2 to 0.5 for the high turbulence and from 0.05 to 0.5 for the low turbulence. This means that the high turbulence causes the laterally uniform mixing between the mainstream and the coolant, resulting in the improved film cooling effectiveness as show in Figure 7.

Figure 8:

Velocity vectors normal to the mainstream direction at x/d=5 (M = 0.5).

Figure 9:

Velocity vectors normal to the mainstream direction at x/d=20 (M = 0.5).

Effect of turbulence on the spanwise averaged film cooling effectiveness

Figure 10 shows the spanwise averaged film cooling effectiveness at the various turbulence cases. As shown in Figure 10(a), the turbulence weakly affects the film cooling effectiveness at x/d < 5 when the blowing ratio is 0.5. The film cooling effectiveness is reduced by 5 % to 20 % according to the various x/d from 5 to 30. This trend agrees with the results obtained by Mehendale in 1990. The increased turbulence caused the film flow to dissipate faster into the mainstream. At the low turbulence case, the film flow structures are maintained over a longer distance downstream of injection, producing higher film cooling effectiveness far away from the hole.

Figure 10:

Influence of turbulence intensities on the spanwise averaged film cooling effectiveness.

As shown in Figure 10(b), the film cooling effectiveness for M=1.0 alters weakly when turbulence intensity increases from 0.4 % to 10 %, but the slightly lower film cooling effectiveness is found at Tu=20 %. As the blowing ratios increase from 0.5 to 1.0, the jet momentum becomes stronger and the diffusion of the jets into the mainstream becomes slower. The film flow structures are maintained over a longer distance, producing a very minimal effect of turbulence on film cooling effectiveness.

As shown in Figure 10(c), the higher turbulence for M = 1.5 causes the higher film cooling effectiveness and this trend is reverse to the M=0.5 case. The longer distance is, the more significant the turbulence effect is. As the turbulence intensities increase from 0.4 % to 20 %, the film cooling effectiveness alters slightly at x/d=1, but increases by 35 % at x/d=30. This shows that the film cooling effectiveness for the high turbulence diminishes more slowly in the mainstream direction.

Figure 11(a) shows the effect of turbulence on the spanwise distributions of the film cooling effectiveness at the lower blowing ratio of 0.5. The film cooling effectiveness is reduced by the high turbulence in the range of z/d from –1 to –0.5 at x/d=5 and in the range of z/d from –1.5 to 0.3 at x/d=20. The effect of turbulence on the lateral spread of the film flow becomes significant with the increase of the x/d. The fluctuation in the normal direction for the high turbulence case becomes strong with the increase of x/d as compared to the case of the low turbulence. The fluctuation in the lateral direction is weakly affected by the turbulence. This causes that the high turbulence reduces the value of only the high film cooling effectiveness and do not increases the value of the low film cooling effectiveness at the same x/d.

Figure 11:

Spanwise distributions of the film cooling effectiveness for the different turbulence.

Figure 11(b) shows the effect of turbulence on the spanwise distributions of the film cooling effectiveness at the higher blowing ratio of 1.5. At x/d=5, the increase of the turbulence causes the increase of the film cooling effectiveness about 0.1 in the range of z/d from –1.6 to –1.3 and in the range of z/d from –0.3 to 0.5. While the increase of the turbulence causes the slight decrease of the film cooling effectiveness in the range of z/d from –1.3 and to –0.3. This implies that the lateral uniformity of the film cooling effectiveness is significantly improved by the high turbulence near the hole. At x/d=20, the film cooling effectiveness increases in the whole range when the turbulence increases. This means that the high turbulence causes the film flow laterally uniform coverage and closer to the surface. The jets are stronger at the high blowing ratio, possessing stronger momentum than the mainstream at the injection location. The jets lift off from the surface and weakly reattached downstream. The large fluctuation makes the jets weakly lift off and strongly spread laterally. This causes the better coverage of the film flow for the high turbulence near the hole.

Averaged surface film cooling effectiveness

Figure 12 shows the averaged surface film cooling effectiveness at the various turbulence cases. The averaged surface film cooling effectiveness is calculated as

Figure 12:

Influence of turbulence intensities on the surface averaged film cooling effectiveness.

Where m is the number of the grids downstream the hole starting x/d=1, ηi is the film cooling effectiveness of the local grid downstream the hole.

As the turbulence intensity increases from 0.4 % to 20 %, the effectiveness is slightly reduced at M=0.5 and M=1.0, but is significantly increased by 17 % and 30 % for M=1.5 and M=2.0, respectively. The highest effectiveness is found at M=1.0 for both Tu=0.4 % and Tu=5.0 %, but at M=1.5 for both Tu=10 % and Tu=20 %, respectively. The optimal blowing ratios are improved as the turbulence intensity increases.