Numerical study on mist-assisted film cooling performance under supersonic condition with discrete coolant injection

Abstract

This study investigates mist-assisted film cooling on a flat plate in a supersonic crossflow (mainstream Ma = 2.0) through numerical simulations under the mist particle diameter of 5 μm. The cooling performance of cylindrical hole, merged hole and sister hole structure is systematically compared at the cooling jet Mach number ranging from 0.4 to 1.4 and mist concentration ranging from 0 to 5%. The effects of shock system, kidney vortex pair and mist particles distribution on film cooling performance are analyzed. Results demonstrate that increasing the cooling jet Mach number could intensify the shear layer effect, shock waves interaction and kidney vortices, promoting both the jet lift-off and mist particle lift-off phenomena, thus reducing the cooling performance enhancement obtained by the mists injection at the near-hole region. Increasing the mist concentration could primarily improve the cooling performance at the more downstream region where the cooling capacities of the air jets decay rapidly and more mist particles diffuse onto the wall surface. Results also indicate that proper management of vortex structures and expansion wave impingement location enable more effective mist transport to protect the wall surface. Among three configurations, sister holes demonstrate superior overall cooling performance in both air-only and mist-assisted conditions, particularly at a higher jet Mach number. Under Ma_c = 1.4 and 5% mist concentration, sister holes achieve a 40% enhancement and merged holes achieve a 16% enhancement in cooling effectiveness compared to cylindrical holes.

Introduction

With ongoing technological advancements, future aero-engine components need to withstand extremely high thermal loads from combustion gases exceeding 2200 K. As an essential active cooling technology, film cooling can effectively cool the aero-engine components and has been extensively implemented in the thermal protection systems¹. In engineering applications, the film cooling performance is governed by numerous factors. Qin et al.² investigated the effects of streamwise pressure gradient and wall curvature through combined experimental and numerical approaches. Li et al.³ conducted a systematic investigation on the sensitivity of the cooling effectiveness to the film hole row layout. Zhang et al.⁴ summarized relevant investigations on novel passive and active cooling strategies. The passive strategies include the use of upstream ramps and trenched configurations, while active strategies involve the implementation of flow modulation devices such as synthetic jet actuators and plasma actuators. Jia et al.^5,6 proposed a spiral-channel hole that enhances cooling effectiveness through swirling flow and established an optimization criterion for hole arrangements based on superellipse theory along coupled temperature gradients.

With the considerable progress in film cooling technology of aero-engines, the cooling potential of pure air has almost been fully exploited. With the continual rise in hot gas temperature after the combustion chamber, the limited heat dissipation capability of pure air can hardly meet the cooling performance requirements of the hot-end components⁷. Due to the rapid dissipation of cooling capacity during the mixing process between the coolant film and high-temperature mainstream, a sharp decline in cooling effectiveness can be observed downstream of film holes. Substantial enhancements to conventional film cooling techniques are therefore imperative.

Li and Wang^8,9,10 demonstrated that the addition of water mist with 2% mass flow rate ratio results in a 30–50% enhancement in adiabatic film cooling effectiveness. Generally, the augmentation of film cooling performance via mist injection originates from two key mechanisms. Firstly, the latent heat of water evaporation and the increased specific heat of the mist-air mixture collectively enhance the heat absorption capacity of coolant. Secondly, a protective vapor layer is generated near the wall after the mist evaporation, the volume expansion caused by the phase change from liquid water to vaporous water also exerts a repulsive force against the hot gas.

Research on mist-assisted film cooling has yielded significant advances in heat transfer mechanisms and parametric optimization. Rao et al.¹¹ adopted the wet-bulb temperature instead of the dry-bulb temperature to enhance the evaluation consistency of the film cooling effectiveness. Pabbisetty and Prasad¹² experimentally confirmed that the mist particles could enhance both the cooling effectiveness and cooling film coverage rate. The performance of mist-assisted film cooling is heavily affected by some mist-specific parameters, including the mist diameter, mist concentration and mist velocity. These mist-specific parameters play a pivotal role in the liquid-phase change process and particle transport. Jiang et al.¹³ established a nonlinear relationship between mist concentration and the cooling film performance, a critical mist concentration value was obtained. Dwivedi and Sarkar¹⁴ found that smaller mist particles could follow the cooling air streamlines, whereas larger ones penetrate into the mainstream. Jeong et al.¹⁵ claimed that the influences of mainstream turbulence intensity on both the mist particle flow characteristics as well as the evaporation process are highly dependent on the mist diameter. Jaiswal et al.¹⁶ found that 112.3% cooling effectiveness enhancement was gained compared with that of the air cooling case under the mist diameter of 5 μm and the mist concentration of 2%. Wang et al.^17,18 developed a dedicated experimental platform to optimize the numerical simulation methodology and analyze the effects of the mist diameter and the initial distribution of mist particles on cooling performance.

Similar to the pure air film cooling, mist-assisted film cooling performance is also governed by the film hole arrangement, hole shapes and the resulted kidney vortex pairs. Thus, a systematic study on the effects of geometric parameters is also needed to provide critical insights for optimizing mist-assisted film cooling performance. Zhang et al.¹⁹ proposed a serrate-type trenched hole structure to enhance mist-assisted film cooling performance. Ragab and Wang²⁰ found that the cooling effectiveness was enhanced by less than 40% with mist injection for cylindrical holes, fan-shaped holes performed better cooling effectiveness enhancement attributed to the superior lateral coverage of the cooling film. Zhou et al.²¹ found that the distinctive anti-kidney vortex structure of console holes results in substantially enhanced cooling film coverage and the later diffusion of mist particles. Huo et al.²² claimed that the staggered arrangement of film holes could enhance the mist-assisted film cooling performance. In addition, mist-assisted film cooling performance under actual operating conditions was also influenced by deposition, erosion and the upstream ramped surface. Handique et al.²³ pointed out some positive effects of mist particle deposition on film cooling effectiveness. Besides, an upstream ramp was used to improve the cooling performance in the near-hole region.

In modern supersonic regimes, cooling challenges are more severe due to shock waves, strong adverse pressure gradients and high-speed shear layers that degrade conventional film cooling performance by promoting jet lift-off, accelerating film dissipation and inducing complex wave-boundary layer interactions. Though mist-assisted film cooling has demonstrated substantial potential in subsonic environments, its application under supersonic conditions introduces additional complexities arising from compressibility effects and two-phase dynamics.

To establish a foundation for understanding mist-assisted cooling, it is necessary to first review pure air film cooling behavior in supersonic flows. Shan et al.²⁴ numerically compared the film cooling performance between supersonic and subsonic conditions at identical blowing ratios. Results showed an average 45% improvement in film cooling effectiveness under supersonic conditions. Peng and Jiang²⁵ found that oblique shock impingement increases local static pressure, leading to an adverse pressure gradient and a recirculation zone. The Mach number within the cooling film and the cooling effectiveness were decreased. This conclusion was further supported by Sun et al.²⁶, who observed that upstream shock impingement generates substantial pressure fluctuations and vortex structures that intensify the coolant-mainstream mixing.

Besides, mist-assisted film cooling also introduces additional complexities compared with pure air cooling due to complex mist-air interactions. The presence of shock waves also significantly alters the mist particle evaporation, mist transport and mist deformation behaviors. Huang and Zhang²⁷ found that larger particles lead to more pronounced shock wave attenuation and more delayed evaporation process, the higher mist particle concentration can enhance the aerodynamic drag, pressure gradient and energy dissipation. Xiao et al.²⁸ found that mist particle breakup is suppressed in supersonic flow than in subsonic flow. Further analysis by Li et al.²⁹ showed that larger mist-gas velocity differences enhance the interphase drag and change the gas-phase velocity field and pressure distribution.

To date, investigations on mist-assisted supersonic film cooling are still limited, leaving the spatial distribution of mist particles and their interaction with complex wave systems poorly understood. Moreover, the interplay between compressibility and two-phase dynamics introduces additional complexities, making parameters such as mist concentration and blowing ratio more influential than in subsonic conditions. To address these issues, this numerical study investigates the effects of jet flow features on mist-assisted supersonic film cooling performance for three hole configurations (cylindrical, merged and sister holes). Simulations are conducted with mist concentrations of 0–5%, a fixed particle diameter of 5 μm and cooling air jet Mach numbers (Ma_c) of 0.4–1.4. The analysis focuses on the interaction between the mainstream and cooling jet, the wave system induced by the jet and shock generator, and the evolution of vortical structures. Subsequently, the spatial distribution and evaporation characteristics of mist particles are compared across different hole types, Ma_c and mist concentrations. Finally, the cooling effectiveness improvement due to mist injection is evaluated and discussed.

Computational method

Geometrical model

Figure 1 illustrates the computational configuration, which consists of the mainstream channel with a wedge-shaped shock wave generator (WSW). The geometry of the mainstream channel is based on the experimental setup by Juhany et al.³⁰, with modification to incorporate three inclined film cooling holes instead of the original tangential slot. The channel dimensions are 100D (length) × 6D (height) × 9D (width), where D represents the diameter of the cylindrical hole (D = 5.00 mm). A 20D flow development section is positioned upstream of the coolant injection location. The coolant is inclined injected through three discrete holes arranged in one row with a lateral (z-direction) pitch of 3D. The WSW with a 4° wedge angle is mounted on the top wall at x = 0, spanning the full channel width in the z-direction.

Fig. 1

Schematic view of the mainstream channel (a) front view (b) bottom view.

This study investigates the kidney vortex formation and cooling film coverage rate in supersonic mist-assisted film cooling under three cooling hole geometries (Fig. 2), namely cylindrical hole (CH), merged hole (MH) and sister hole (SH). All configurations maintain the same lateral spacing, inclination angle and total cross-sectional area. The merged hole configuration is formed by two cylindrical holes (0.73D) with a center-to-center spacing of 0.55D. The sister hole configuration comprises one upstream sub-hole (0.60D) and two downstream sub-holes (0.57D), with the streamwise distance between the primary hole and each sub-hole of 1.5D and spanwise distance between two sub-holes of 0.8D.

Fig. 2

Schematic views of three discrete film cooling hole types (a) cylindrical hole (CH) (b) merged hole (MH) (c) sister hole (SH).

Governing equations

The numerical framework implemented in Ansys Fluent employs a finite volume approach coupled with the discrete phase model (DPM) to account for the low-volume-fraction mist particles. The discrete phase effect is introduced by incorporating source terms into the governing equations of mass, momentum and energy of the continuous phase.

Continuous phase

The governing equations, i.e., the Reynolds-Averaged Navier–Stokes (RANS) equations, for the steady-state continuous phase are expressed as,

Continuity equation

$$\frac{\partial }{{\partial x_{j} }}\left( {\rho u_{j} } \right) = S_{m}$$

(1)

Momentum equation

$$\frac{\partial }{{\partial x_{j} }}\left( {\rho u_{i} u_{j} } \right) = - \frac{\partial p}{{\partial x_{i} }} + \frac{\partial }{{\partial x_{j} }}\left( {\tau_{ij} - \rho \overline{{u^{\prime}_{i} u^{\prime}_{j} }} } \right) + \rho \overline{{g_{i} }} + F_{m}$$

(2)

Energy equation

$$\begin{aligned} \frac{\partial }{{\partial x_{j} }}\left( {\rho u_{j} h_{tot} } \right) & = \frac{\partial }{{\partial x_{j} }}\left( {\lambda \frac{\partial T}{{\partial x_{j} }} - \rho \overline{{u_{j}^{\prime } h}} } \right) \\ & \quad + \frac{\partial }{{\partial x_{j} }}\left[ {u_{i} \left( {\tau_{ij} - \rho \overline{{u_{i}^{\prime } u_{j}^{\prime } }} } \right)} \right] + S_{e} \\ \end{aligned}$$

(3)

where u_i and u_j denote the time-averaged component of velocity, $u_{i}^{\prime }$ and $u_{j}^{\prime }$ represent the time-varying component of velocity, h_tot is the total enthalpy, τ is the molecular stress tensor and λ is the thermal conductivity. S_m, F_m and S_e represent the coupling forces corresponding to the mass, momentum and energy transferred from the discrete phase to the continuous phase. The species transport equation for the water evaporation is given as,

$$\frac{\partial }{{\partial x_{i} }}(\rho \mu {}_{i}C_{j} ) = \frac{\partial }{{\partial x_{i} }}\left( {\rho D_{j} \frac{{\partial C_{j} }}{{\partial x_{i} }} - \rho \overline{{\mu_{i}^{\prime } C_{j}^{\prime } }} } \right) + S_{j}$$

(4)

where C_j is the mass fraction of each species in the mixture and S_j is its source term. D_j is the diffusion coefficient which characterizes the molecular diffusion capacity.

The present study employs the Realizable k-ε turbulence model developed by Shih et al.³¹. To accurately simulate the behavior of supersonic compressible flow and mist-air two-phase flow, the compressibility correction term (Y_M), buoyancy term (G_b) and source terms (S_k and S_ε) are also considered,

$$\begin{aligned} \frac{\partial }{\partial t}(\rho k) + \frac{\partial }{{\partial x_{j} }}(\rho ku_{j} ) & = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{k} }}} \right)\frac{\partial k}{{\partial x_{j} }}} \right] \\ & \quad + G_{k} + G_{b} - \rho \varepsilon - Y_{M} + S_{k} \\ \end{aligned}$$

(5)

$$\begin{aligned} \frac{\partial }{\partial t}(\rho \varepsilon ) + \frac{\partial }{{\partial x_{j} }}(\rho \varepsilon u_{j} ) & = \frac{\partial }{{\partial x_{j} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{\varepsilon } }}} \right)\frac{\partial \varepsilon }{{\partial x_{j} }}} \right] \\ & \quad + \rho C_{1} S_{\varepsilon } + \rho C_{2} \left( {\frac{{\varepsilon^{2} }}{{k + \sqrt {\nu \varepsilon } }}} \right) \\ & \quad + C_{1\varepsilon } \left( {\frac{\varepsilon }{k}} \right)C_{3\varepsilon } G_{b} + S_{\varepsilon } \\ \end{aligned}$$

(6)

where G_k represents the turbulent kinetic energy generated by mean velocity gradients. σ_k and σ_ε are the turbulent Prandtl numbers for turbulent kinetic energy k and dissipation rate ε, respectively.

Discrete phase

The Lagrangian method is applied to track the water mist distribution and simulate the accompanying heat, mass and momentum exchange with the surrounding air. The motion of these discrete mist particles is expressed as,

$$\frac{{{\text{d}}u_{{\text{p}}} }}{{{\text{d}}t}} = F_{D} (u - u_{{\text{p}}} ) + {\text{g}}\frac{{{\varvec{\rho}}_{{\text{p}}} - {\varvec{\rho}}}}{{{\varvec{\rho}}_{{\text{p}}} }} + F_{{\text{x}}}$$

(7)

where u and ρ are the air velocity and density, u_p and ρ_p are the mist particle velocity and density, respectively. F_x denotes the additional forces per unit mass imposed on mist particles. The drag force F_D is,

$$F_{D} = \frac{3}{4}\frac{{\mu C_{D} Re_{{\text{p}}} }}{{\rho_{{\text{p}}} d_{{\text{p}}}^{2} }}$$

(8)

where d_p, μ, C_D and Re_p represent the particle diameter, dynamic viscosity, drag coefficient and relative Reynolds number, respectively. Among them, C_D and Re_p have the following relationship,

$$C_{D} = \frac{24}{{Re_{{\text{p}}} }}\;{\text{for}} \;Re_{{\text{p}}} \le 1$$

(9)

$$C_{D} = \frac{24}{{Re_{{\text{p}}} }}\left( {1 + \frac{1}{6}Re_{{\text{p}}}^{2/3} } \right)\;{\text{for}} \;1 \le Re_{{\text{p}}} \le 400$$

(10)

$$Re_{{\text{p}}} = \frac{{\rho d_{{\text{p}}} |u_{{\text{p}}} - u|}}{\mu }$$

(11)

The energy balance equation in mist particles is,

$$m_{{\text{p}}} c_{{\text{p}}} \frac{{{\text{d}}T_{{\text{p}}} }}{{{\text{d}}t}} = hA_{{\text{p}}} (T_{\infty } - T_{{\text{p}}} ) + \frac{{{\text{d}}m_{{\text{p}}} }}{{{\text{d}}t}}h_{{{\text{fg}}}}$$

(12)

where m_p, c_p, A_p and T_p are the mass flow rate, specific heat capacity, surface area and temperature of the particle, respectively. T_∞ is the temperature of the surrounding continuous phase, dm_p/dt and h_fg are the particle evaporation rate and latent heat of vaporization, respectively. The convective heat transfer coefficient h is given by the correlation of Ranz et al.³². The evaporation rate dm_p/dt is given as a function of the concentration difference,

$$\frac{{{\text{d}}m_{{\text{p}}} }}{{{\text{d}}t}} = - A_{{\text{p}}} k_{{\text{c}}} (C_{{\text{s}}} - C_{\infty } )$$

(13)

where C_s and C_∞ are the vapor concentrations at the particle surface and in the continuous phase, respectively, k_c is the mass transfer coefficient obtained from the Sherwood number correlation. When the mist particle temperature reaches the phase change point, the evaporation process is started and is expressed as,

$$\frac{{{\text{d}}m_{{\text{p}}} }}{{{\text{d}}t}} = - \frac{A}{{C_{{\text{p}}} }}\left( {\frac{\lambda }{d}} \right)\left( {2.0 + 0.46Re_{{\text{p}}}^{0.5} } \right)ln\left( {1 + C_{{\text{p}}} \frac{{(T_{\infty } - T)}}{{h_{{{\text{fg}}}} }}} \right)$$

(14)

Mist particle tracking

To clarify the mechanism by which turbulent dissipation affects particle tracking, a stochastic method is adopted for the simulation. The source term, which serves as an information transfer medium at the interface between the continuous air phase and the discrete mist particle phase, is as follows³³,

Mass source term

$$S_{m} = \frac{{\Delta m_{p} }}{{m_{p,o} }}\frac{{\dot{m}_{p,o} }}{dV}$$

(15)

Momentum source term

$$F_{m} = \sum {\left( {\frac{{18\mu C_{D} Re_{D} }}{{24\rho_{p} d_{p}^{2} }}(u_{p,i} - u_{i} )} \right)} \dot{m}_{p} \Delta t$$

(16)

Energy source term

$$\begin{aligned} S_{e} & = \frac{1}{dV}\frac{{\dot{m}_{p,o} }}{{m_{p,o} }}\left[ {(m_{p,in} - m_{p,out} )( - H_{lat,ref} + H_{pyrol} )} \right. \\ & \quad \quad \left. { - m_{p,out} \int_{{T_{ref} }}^{{T_{p,out} }} {C_{{P_{p} }} dT} + m_{p,in} \int_{{T_{ref} }}^{{T_{p,in} }} {C_{{P_{p} }} dT} } \right] \\ \end{aligned}$$

(17)

Besides, Watchers et al.³⁴ proposed that the behavior of water particles after impacting on a wall is determined by its wall-normal Weber number. Notably in this study, the Weber number remains below 10 in all cases due to the 5 µm initial particle diameter and low wall-normal velocity, thus the reflective boundary condition is selected to determine the behavior of mist particles after impacting the wall.

Boundary conditions

The thermophysical properties of the ideal air and ideal water vapor are calculated by using the NIST Standard³⁵. Pressure-far-field boundary condition is set at both the mainstream inlet and film cooling hole inlet. The corresponding aerodynamic parameters are shown in Table 1. At the inlet of the mainstream channel and cooling holes, the continuous phase only contains ideal air. The mainstream static pressure P_s,∞ is 100 kPa and the mainstream Mach number Ma_∞ is 2.00. The static pressure of the cooling air P_s,c is 1.5 times that of the mainstream to prevent the mainstream air from entering the film holes. The static temperature of the mainstream is 373.15 K which is the starting temperature of liquid water evaporation. The static temperature of the cooling air is 284.15 K which is the boiling point of liquid water under the mainstream static pressure. The blowing ratio M is defined as,

$$M = \frac{{\rho_{{\text{c}}} U_{{\text{c}}} }}{{\rho_{\infty } U_{\infty } }}$$

(18)

where ρ_c, U_c and ρ_∞, U_∞ are the density and area-averaged velocity of the cooling air and mainstream air, respectively. The total pressure P_t and total temperature T_t of both the mainstream and the coolant are determined via the isentropic relations,

$$P_{{\text{t}}} = P_{{\text{s}}} \left( {1 + \frac{\gamma - 1}{2}Ma^{2} } \right)^{{\frac{\gamma }{\gamma - 1}}}$$

(19)

$$T_{{\text{t}}} = T_{{\text{s}}} \left( {1 + \frac{\gamma - 1}{2}Ma^{2} } \right)$$

(20)

where γ is the adiabatic coefficient. The liquid water mist with a diameter of 5 μm is evenly injected into three film holes with the same velocity of the cooling air. Under each Ma_c, the mist mass concentration m_mist is varied from 0 to 5%.

Table 1 Brief summary of aerodynamic parameters.

A low turbulence intensity of 1% is set for both the mainstream inlet and cooling hole inlet. Pressure outlet boundary condition is applied at the mainstream channel exit. For the mainstream channel, periodic boundary condition is imposed in the spanwise direction. All solid walls are treated as adiabatic with no-slip conditions. The adiabatic film cooling effectiveness is defined as,

$$\eta = \frac{{T_{{{\text{r}}\infty }} - T_{{{\text{aw}}}} }}{{T_{{{\text{r}}\infty }} - T_{{{\text{rc}}}} }}$$

(21)

where T_r∞ and T_rc are the recovery temperatures of the mainstream and cooling air, respectively. T_aw represents the adiabatic wall temperature. The recovery temperature is defined as,

$$T_{{\text{r}}} = T_{{\text{s}}} \left( {1 + r\frac{\gamma - 1}{2}Ma^{2} } \right)$$

(22)

where the recovery factor r in turbulent flow is defined as,

$$r = Pr^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}}$$

(23)

The boundary layer thickness, as a precondition of supersonic film cooling research, also has a huge influence^36,37 on the flow interactions between the mainstream air and the coolant, such as the development of the mixing/shear layer and vortices. Figure 3 illustrates the near-wall velocity ratio at 1.0D upstream of the jet hole. The boundary layer thickness δ is about 0.42 times the diameter of cylindrical hole, about 0.57 times the diameter of each side of the merged hole and about 0.7 times the diameter of the upstream sub-hole of the sister hole combination. Therefore, δ is smaller than the film hole diameter under all three hole types and its effects are limited in this study.

Fig. 3

Velocity ratio within the region of 0.0 < y/D < 1.0 at 1.0D upstream of the jet hole.

Meshing and grid independence study

Figure 4 shows the mesh distribution in different regions. To enhance numerical accuracy, approximately 20 layers of boundary layer meshes are deployed on wall surfaces with a controlled stretch ratio of 1.1. The mesh configuration ensures a non-dimensional wall distance y⁺  < 1.0 on both the hole surfaces and the upper/lower walls of the mainstream region, satisfying the resolution requirements for enhanced wall treatment.

Fig. 4

Established mesh on the computational domain.

Figure 5a presents the spanwise-averaged adiabatic film cooling effectiveness for the CH configuration under Ma_c = 1.0 (M = 0.860) and m_mist = 5%. The differences between 4.0 million and 5.0 million nodes become negligible, with the variation at all regions less than 0.5%. Consequently, the grid resolution of 4.0 million nodes is selected for the CH configuration. Following the same meshing methodology, both the MH and SH configurations adopt 4.0 million grid nodes and their specific grid independence results are not shown in the figure any more. The Grid Convergence Index (GCI) method³⁸ is employed to evaluate the grid under Ma_c = 1.4 (M = 1.205) and m_mist = 5%. GCI error is defined as,

$${\text{GCI}}_{12} = \frac{{F_{s} \left| \varepsilon \right|}}{{r^{p} - 1}}$$

(24)

where the safety factor F_s = 1.25, r denotes the grid refinement ratio (set to 2), and the relative error ε is defined as,

$$\varepsilon = \frac{{f_{2} - f_{1} }}{{f_{1} }}$$

(25)

where f₁ and f₂ are the numerical solutions obtained from grids with normalized spacings of 1 and 2, respectively. The order of convergence p is defined as,

$$p = {{\ln \left( {\frac{{f_{3} - f_{2} }}{{f_{2} - f_{1} }}} \right)} \mathord{\left/ {\vphantom {{\ln \left( {\frac{{f_{3} - f_{2} }}{{f_{2} - f_{1} }}} \right)} {\ln \left( r \right)}}} \right. \kern-0pt} {\ln \left( r \right)}}$$

(26)

where f₃ is defined as the solution at a normalized grid spacing of 4. Based on these solutions, the Richardson extrapolation value is calculated as,

$$f_{h = 0} \cong f_{1} + \frac{{f_{1} - f_{2} }}{{r^{p} - 1}}$$

(27)

Fig. 5

Mesh independence study of mist-assisted film cooling models (a) mesh independence analysis in CH case (b) GCI error analysis.

Figure 5b presents the predicted overall-averaged film cooling effectiveness $\overline{\overline{\eta }}$ on the flat plate under three film cooling hole configurations, with 4.0 million nodes normalized as the grid spacing 1. The predicted $\overline{\overline{\eta }}$ values demonstrate progressive convergence toward the Richardson extrapolation when increasing the node density. The Grid Convergence Index (GCI₁₂) is below 0.8% in all configurations, confirming the adequate mesh resolution and solution reliability.

Validation of turbulence model

The RANS equations are solved using ANSYS Fluent 2023. Spatial discretization utilizes the AUSM scheme for convective fluxes and a second-order upwind method for other terms. The calculation convergence requires residuals below 1 × 10^–4 for the continuity equation and 1 × 10^–5 for the momentum, turbulence and energy equations. The experimental work from Kanda et al.³⁹ is used to validate the numerical method for supersonic film cooling featured with tangential injection and an upstream shock generator. As shown in Fig. 6, all three turbulence models yield comparable η results, and accurately capture the detrimental effect of shock impingement though the computed decrease is more pronounced than that in the experimental data. The Realizable k-ε turbulence model provides relatively more accurate predictions in the region where 30 < x/h_inj < 120 (h_inj is the height of the jet slot). Specifically, the research by Dwivedi and Sarkar¹⁴ demonstrated that in subsonic mist-assisted film cooling, the Realizable k-ε model exhibits superior capability in resolving the lateral spreading of jets and provides accurate predictions of cooling effectiveness. Additionally, the study by Xu et al.⁴⁰ on transonic turbine film cooling proved that the Realizable k-ε model provide comparable prediction accuracy to the RSM model in capturing shock waves and other supersonic flow structures.

Fig. 6

Validation of turbulence models for film cooling effectiveness predictions under supersonic flow conditions.

Figure 7 compares the prediction of the Realizable k-ε model with the experimental data from Zhao and Wang⁴¹ for mist-assisted film cooling with cylindrical holes at M = 0.6. From a quantitative perspective, the model tends to overpredict η along the centerline but underpredict the spanwise-averaged values, with a mean relative error of less than 20%. From a qualitative perspective, the predicted effectiveness distribution matches the observed trend of the experimental data, characterized by a sharp η decay immediately downstream the coolant injection and a progressive η attenuation driven by the turbulent mixing. Based on the above considerations, the Realizable k-ε turbulence model is ultimately employed in this study.

Fig. 7

Numerical results with Realizable k-ε turbulence model and experimental results.

Results and discussions

Flow features of continuous phase air

The phase change process of water mist is pivotal for boosting supersonic film cooling performance. The movement and distribution of water mist are greatly influenced by supersonic flow behaviors and flow interactions between the mainstream and cooling air. Thus, it is imperative to firstly explore the flow characteristics of the continuous phase air, such as flow distribution at the hole exit, wave system and vortex structure.

Figure 8 shows the comparison of flow parameters distribution at the hole exit among three film hole types. Under Ma_c = 0.4, mainstream compression leads to higher P_s at the upstream side and consequently higher Ma and M at the downstream side. Due to weak compressibility effects at a lower Ma_c, ρ variations are minimal, resulting in M distributions similar to Ma near the downstream side of the hole. Mist particles closely follow the cooling air due to its small size (5 μm), exhibiting the mist velocity distributions similar to M, particularly in the SH case. Under Ma_c = 1.4, the compression effect of the mainstream can only be transmitted into the film holes through the region where Ma < 1.0. The nonuniformity of the P_s and Ma distribution decreases a bit, but the nonuniform distribution of ρ becomes more obvious. Consequently, the M and mist velocity distributions become more uniform than that under Ma_c = 0.4. These results indicate that coolant jet development within film holes varies significantly with Ma_c, with larger Ma_c promoting more uniform exit distributions.

Fig. 8

Flow parameter distribution of three film hole types at the hole outlet (a) Ma_c = 0.4 (M = 0.344) (b) Ma_c = 1.4 (M = 1.205).

Figure 9 shows the P_s distribution on the symmetry plane under Ma_c = 0.4 (M = 0.344) and 1.4 (M = 1.205). A jet induced bow shock (JS) forms upstream of the film hole due to coolant-mainstream interaction and undergoes multiple reflections along the channel. With increasing Ma_c, both JS angle and intensity increase, enhancing the P_s downstream of the JS shock. In comparison, since the coolant mass flow rate is dispersed into three sub-holes in SH case, the JS shock angle decreases a little bit. A wedge induced shock (WS) generated at the shock generator leading edge also undergoes reflections but becomes negligible beyond x/D = 40. Since wedge geometry and mainstream conditions remain constant, WS propagation is largely unaffected by Ma_c or hole type. Besides, A wedge induced expansion wave (WEW) is generated at the middle edge of the WSW. Additional expansion waves may form downstream of the film hole when the coolant jet contains a supersonic portion.

Fig. 9

P_s distribution on the symmetry plane under Ma_c = 0.4 (M = 0.344) and 1.4 (M = 1.205).

Figure 10 illustrated the supersonic flow features at the near-hole region based on the density gradient on the central plane. Generally, due to the large jet incident angle (30°), the JS shock intensity as well as its effect is much larger than that of the WS shock. In SH cases, dispersed jet concentration lead to reduced JS shock intensity. Increasing Ma_c intensifies jet-mainstream interaction, shifting JS impingement upstream and indicating stronger shocks. The mixing/shear layer is another main flow feature due to the interaction between the separated boundary layers of both the jet and the mainstream on the windward side of the cooling jet. The substantial disparity in aerodynamic parameters generates a pronounced density gradient inside the mixing/shear layer. The mixing/shear layer also develops on the leeward side of the cooling jet between the detached coolant boundary layer and the ambient mainstream. Since the more laterally expanded coolant in MH case, the cooling jet is better attached to the mainstream channel bottom wall at the near-hole region than the CH and SH cases.

Fig. 10

Density gradient magnitude distribution at the near-hole region under Ma_c = 0.4 and 1.4 (a) Ma_c = 0.4 (M = 0.344) (b) Ma_c = 1.4 (M = 1.205).

Figure 11 presents the Mach number distribution on the central plane. Intense mixing/shear at the jet-mainstream interface produces a pronounced Mach number gradient. The impingement of the JS and WS generally decelerates the cooling jet and constricts the lateral spreading of the cooling film, which adversely affects the cooling performance. In contrast, the WEW incident accelerates the jet and promotes lateral spreading of the cooling film, thereby enhancing the cooling effectiveness. With increasing Ma_c, the effect of the JS shock strengthens and a separation bubble is gradually generated which further lifts the cooling film. The above conclusions are similar among three film hole types. In addition, since the dispersed jet distribution in SH case, the mixing/shear layer is much closer to the bottom wall and the Mach number gradient inside the mixing/shear layer starts to decrease much earlier than that in CH and MH cases.

Fig. 11

Ma distribution at the near-bottom wall region coupled with cooling jet streamline (a) CH cases (b) MH cases (c) SH cases.

Figure 12 presents the cooling jet streamline colored by coolant Mach number and the Mach number distribution on several cross-sections downstream of the jet hole. After leaving the jet hole exit, the cooling jet concentrated region tends to lift off from the mainstream channel bottom wall. The cooling jet then gradually falls back toward the wall driven by the mainstream’s dominant pressure. It is evident that with increasing Ma_c, the cooling jet reaches higher y/D position, especially at the cross-section of x/D = 5.0 in CH case. Due to the more laterally expanded cooling jet in MH case, the jet is much closer to the bottom wall at the cross-section of x/D = 5.0. Due to the more dispersed jet intensity in the SH case, the cooling jet could fall back to the bottom wall more rapidly.

Fig. 12

Ma distribution at the near-bottom wall region coupled with cooling jet streamline.

Figure 13 shows the Mach number distribution and tangential velocity vectors on the cross-section of x/D = 5 as well as the cooling jet streamline colored by the mist diameter. In CH cases, with increasing Ma_c, the kidney vortex pair becomes more obvious. More coolant is sucked in and lifted away from the mainstream channel bottom wall due to the increasing effect of the kidney vortex pair. However, in MH cases, the cooling jet concentrated region is located at a more lateral position than that of the kidney vortex pair, hence the enhanced mixing between the cooling jet and the mainstream caused by the kidney vortex pair is weakened. In SH cases, since the cooling jet is dispersed into three sub-holes, the cooling film is more uniformly distributed along the lateral direction. Besides, due to the interlaced arrangement of the upstream sub-hole and two downstream sub-holes, the rotation direction of the kidney vortex pair formed by the upstream sub-hole is opposite to that of the inner side of the kidney vortex pair formed by two downstream sub-holes which lead to the mutual weakening of the upstream and downstream kidney vortex pairs.

Fig. 13

Mach number distribution and tangential velocity vectors on the cross section of x/D = 5 as well as the cooling jet streamline.

Flow features of discrete phase water mist

Figure 14 shows the mist diameter distribution on the near-bottom wall plane of y/D = 0.1 coupled with the spatial distribution of water mist particles (colored with mist diameter) as well as the cooling air streamline (in purple color). Compared between the spatial distribution of mist particles and cooling air streamline, the mist particles could diffuse to more lateral positions along their flow direction, and hence the film cooling performance of the regions between adjacent cooling holes is enhanced by the mist particles. Compared to the CH and MH cases, more mist particles can enter the near-bottom wall region all along the x/D direction under Ma_c = 0.4 and enter the near-bottom wall region when x/D > 30 under Ma_c = 1.4 in the SH case.

Fig. 14

Mist diameter distribution on the near-bottom wall plane of y/D = 0.1 coupled with the spatial distribution of water mist particles as well as the cooling air streamline.

Figure 15 shows the mist diameter distribution on the central plane coupled with the spatial distribution of water mist particles (colored with mist diameter) as well as the cooling air streamline (in purple color) under m_mist = 5%. Results show that the mist particles have their own concentrated region which is quite different from the cooling air concentrated region. The water mist particles invade deeply into the mainstream immediately after leaving the film holes (mists lift-off phenomenon) and demonstrates the characteristics of free diffusion and evaporation along the flow direction. Consequently, water mists contribute minimally to bottom wall cooling in the near-hole region. With increasing Ma_c, mist lift-off intensifies, delaying particle return to the wall and indicating that mist diffusion and evaporation processes are strongly influenced by Ma_c.

Fig. 15

Mist diameter distribution on the central plane coupled with the spatial distribution of water mist particles as well as the cooling air streamline.

Since the film cooling performance is the most concerning issue, so it is necessary to clarify the near-wall flow characteristics of water mist particles and find ways to enable water mist particles to complete their evaporation process near the bottom wall as much as possible. Figure 16 is a local magnification of Fig. 15 at the area of 0 < x/D < 30 and 0 < y/D < 3. In CH cases, mist particles remain concentrated above the cooling air jet. The interface between the mist concentrated region and the air concentrated region coincides with the mixing/shear layer. Due to the large Mach number gradient, the mixing/shear layer acts as an “air wall” to resist mist particles from diffusing onto the bottom wall. As the shear layer develops downstream, the Mach number gradient gradually decreases, allowing mist diffusion into the air jet. Mists are more delayed to return to the bottom under higher Ma_c, causing more particles to evaporate in the mainstream and the waste of mists’ cooling potential. In MH cases, the more laterally expanded jet and weakened kidney vortex pairs bring the mist region closer to the wall than in CH cases. In SH cases, due to weakened kidney vortex pairs and dispersed jet intensity, the cooling air jet and the mist particles are most close to the bottom wall. The rapid reduction of Mach number gradient within the mixing/shear layer in SH cases also enables mist to diffuse into the air jet and return to the bottom wall much earlier. It can be concluded that reducing the effect of kidney vortex pair, adopting more dispersed cooling jets and arranging the cooling holes in front of the incident location of the expansion wave are conducive to allowing more mist particles to return to the bottom wall.

Fig. 16

Mist flow features at the near-hole region (a) CH cases (b) MH cases (c) SH cases.

Supersonic film cooling performance of mist-assisted cooling

Figure 17a shows the comparison of spanwise-averaged film cooling effectiveness $\overline{\eta }$ among three hole types under Ma_c = 0.4 (M = 0.344) and Ma_c = 1.4 (M = 1.205). With increasing Ma_c, the jet lift-off phenomenon becomes more obvious, the $\overline{\eta }$ decreases much earlier and a secondary $\overline{\eta }$ peak can be observed when the air jets gradually fall back to the bottom wall. However, the mists lift-off phenomenon is obvious under all Ma_c, the mists yield almost no contributions to $\overline{\eta }$ at the near-hole region. Compared among three hole types under Ma_c = 0.4, the $\overline{\eta }$ difference of three air cooling cases is quite small and the secondary $\overline{\eta }$ peak cannot be observed because of the well wall-attached air jet. Compared among three hole types under Ma_c = 1.4, the $\overline{\eta }$ difference is quite large, the SH case demonstrates a significant cooling advantage at a high Ma_c and the secondary $\overline{\eta }$ peak can also not be observed due to the more dispersed jet intensity and the mutual weakening of kidney vortex pairs. Under the condition of Ma_c = 0.4 and m_mist = 5%, $\overline{\eta }$ of the MH case is about 7% higher than that of CH case, while the SH case shows a $\overline{\eta }$ improvement of approximately 14% over CH case. Under the condition of Ma_c = 1.4 and m_mist = 5%, the MH and SH cases achieve approximately 16% and 40% higher $\overline{\eta }$ than CH case, respectively. The $\overline{\eta }$ variation tendency along the streamwise direction mainly depends on the flow behaviors of the continuous phase air governed by the film cooling hole and Ma_c, the mist particle injection is conducive to improving the cooling performance at the downstream region where the cooling capacity of the cooling air is intensely consumed.

Fig. 17

Spanwise-averaged film cooling effectiveness $\overline{\eta }$ (a) comparison among three hole types (b) CH cases (c) MH cases (d) SH cases.

Figure 17b–d show the $\overline{\eta }$ distribution based on three film hole configurations. With mist injection, $\overline{\eta }$ is enhanced at downstream region. Due to the enhanced jet lift-off and kidney vortex effects, CH and MH cases exhibit optimal film cooling performance at Ma_c = 1.0 and 1.2, respectively. In contrast, SH cases show monotonically increasing $\overline{\eta }$ with increased Ma_c across the entire streamwise direction due to the more dispersed jet intensity and the mutual weakening of adjacent kidney vortex pairs.

Figure 18a shows the comparison of spanwise-averaged film cooling effectiveness improvement ratio φ which is defined as,

$$\varphi = \left( {{{\overline{\eta }_{{{\text{mist}}}} - \overline{\eta }_{{{\text{air}}}} } \mathord{\left/ {\vphantom {{\overline{\eta }_{{{\text{mist}}}} - \overline{\eta }_{{{\text{air}}}} } {\overline{\eta }_{{{\text{air}}}} }}} \right. \kern-0pt} {\overline{\eta }_{{{\text{air}}}} }}} \right)$$

(28)

Fig. 18

Spanwise-averaged film cooling effectiveness improvement ratio φ (a) comparison among three hole types (b) CH cases (c) MH cases (d) SH cases.

Due to the mists lift-off phenomenon, φ is small at the near-hole region. Then as the mist particles gradually diffuse onto the bottom wall, φ increases a lot along x/D direction. When Ma_c = 0.4, though the difference of $\overline{\eta }$ is small among three hole types under air cooling scheme, CH cases still yield the least φ and SH cases yield the largest φ. It suggests that, as the effect of jets lift-off phenomenon is insignificant under a small Ma_c, the ability to resist the effect of mists lift-off phenomenon becomes more important. When Ma_c = 1.4, φ of SH cases can still reach the same level with that of CH and MH cases. It suggests that the abilities to resist the effects of mists lift-off and air jets lift-off phenomena are both important under a larger Ma_c.

Figure 18b–d show the φ distribution of three film hole cases respectively. Generally, with increasing m_mist, the difference of φ between Ma_c = 0.4 and Ma_c = 1.4 becomes larger. It indicates that the effect of mists lift-off phenomenon is also enlarged under a larger m_mist. In CH cases, both the air jets and the mist particles are heavily affected by the kidney vortex pair and lift-off phenomenon. The difference of φ between Ma_c = 0.4 and Ma_c = 1.4 is then much larger in MH and SH cases than in CH cases. Results also indicate that, increasing the supersonic film cooling effectiveness by using the mist-assisted cooling scheme is more effective at a lower Ma_c.

Figure 19 illustrates the supersonic film cooling effectiveness η distribution on the bottom wall varied with Ma_c under m_mist = 5%. In Fig. 19a, as both the effects of jets and mists lift-off phenomena are relatively small when Ma_c = 0.4, the air jets and mist particles in three cases can be well attached on the bottom wall. When Ma_c = 1.4, the coolant coverage rate in CH case is lowest among three film hole cases, MH case yield a better coolant coverage rate due to its more laterally expanded cooling jet, SH case again yields the best coolant coverage rate. Figure 19b–d compares the η distributions for the three hole configurations. When Ma_c exceeds 1.0 and 1.2 in CH and MH cases respectively, the low-effectiveness area (η < 0.2) at the downstream region is expanded while the high-effectiveness area (η > 0.7) at the near-hole region is contracted. In contrast, SH cases exhibit the opposite trend that the high-effectiveness region is expanded with the increasing Ma_c. Figure 20 illustrates the η distribution on the bottom wall varied with m_mist under Ma_c = 1.4. It is evident that the η distribution in the near-hole region of x/D < 30 is not obviously affected by m_mist, but the η distribution in more downstream region of x/D > 30 differs a lot with increasing m_mist under all three film hole types.

Fig. 19

Film cooling effectiveness η distribution on the bottom wall varied with Ma_c (a) comparison among three hole types (b) CH cases under m_mist = 5% (c) MH cases under m_mist = 5% (d) SH cases under m_mist = 5%.

Fig. 20

Film cooling effectiveness η distribution on the bottom wall varied with m_mist (a) CH cases under Ma_c = 1.4 (b) MH cases under Ma_c = 1.4 (c) SH cases under Ma_c = 1.4.

Conclusions

In this paper, the mist-assisted film cooling performance is numerically studied under mainstream Mach number of 2.0 and coolant Mach numbers ranging from 0.4 to 1.4. The film cooling performance are also compared among three film cooling hole types (cylindrical hole, merged hole and sister hole) with the mist concentrations ranging from 0 to 5%. The supersonic flow structures and mist distribution patterns are analyzed and compared. The spanwise-averaged film cooling effectiveness as well as the contribution of the water mists to the film cooling effectiveness are also discussed. The principal findings are summarized as follows.

• 1. The sister hole configuration exhibits superior film cooling performance compared to the cylindrical and merged hole designs, as its dispersed jet momentum and mutual suppression of kidney vortex pairs from the sub-holes. Consequently, the sister hole configuration not only demonstrates significantly higher cooling effectiveness under the air-only cooling condition, but also yields a comparable film cooling effectiveness improvement ratio (φ) under the mist-assisted cooling condition, especially under a larger Ma_c.

• 2. Mist particles at the near-hole region are mainly concentrated beyond the cooling air jet and require more time to diffuse to the bottom wall. By suppressing kidney vortex pairs, employing more dispersed coolant jets and positioning cooling holes upstream of the expansion wave impingement region, more mist particles could return to the bottom wall under the supersonic cooling condition, and thereby enhancing the cooling performance.

• 3. Substantial cooling effectiveness improvement is achieved by the mist injection at the downstream region where the cooling capacity of pure air becomes depleted. However, at the near-hole region, the film cooling effectiveness improvement ratio (φ) by the mist injection is found to be significantly influenced by Ma_c. Generally, φ is reduced with increasing Ma_c. The capabilities to resist mists lift-off and jet lift-off phenomena are equally crucial under high Ma_c.

• 4. Within the investigation scope of the mist mass concentration (m_mist) ranging from 0 to 5%, the film cooling effectiveness is consistently enhanced with increasing m_mist. However, φ at the near-hole region is not significantly increased with the increasing m_mist due to the mists lift-off phenomenon, vortex interactions and mixing/shear layer restriction.