## Abstract

Droplet deposition on a turbine cascade is important for the turbine system performance but its experimental analysis is difficult. Thus, the simulation of a droplet liquid phase in a gas flow field was studied to optimize turbine cascade design. However, the computational fluid dynamics (CFD)-based approach for such droplet problems requires enormous costs. Thus, the application of CFD simulation in the turbine blade's early-stage design is challenging, requiring iterative optimization for adjusting design with performance prediction. Therefore, this study proposed an analytical prediction method, having a reasonable cost and moderate accuracy, as an alternative to the whole multi-phase numerical simulation approach. The proposed method predicts droplet motion using the outline of the turbine blade and gas–liquid physical properties. Furthermore, the approach was validated by performing a three-dimensional Eulerian–Lagrangian simulation with low-pressure turbine blade T106. It was found that the droplet trajectories in the turbine cascade are governed by Stokes number. Furthermore, the streamlines of the gas flow were characterized by the shape of the turbine blade. The proposed model reproduced droplet trajectories obtained using CFD within an error margin of 10%. Consequently, it was concluded that the proposed analytical model is a promising approach to predict the droplet trajectory in a turbine cascade, obtained by the three-dimensional numerical simulation at a low cost.

## 1 Introduction

A fundamental phenomenon, wherein a droplet collides with a solid wall, occurs in many types of industrial equipment. Steam turbines, used to generate electricity at thermal, geothermal, and nuclear power plants, are the most representative of such equipment. In nuclear and geothermal power generation systems, steam turbines operate via wet steam flows, and the droplets in the steam flow are mixed in many steam turbine stages. However, droplet impact causes damage to the machine and severe degradation of its performance. In the turbine cascade of a steam turbine, fog particles are generated by the condensation, and they pass through the turbine cascade. Subsequently, these particles are deposited on the surface of the turbine blade and casing wall, which may produce liquid finger and film on these surfaces. Furthermore, atomization produces large and small droplets, which impact the turbine blade on the backward side of the turbine blade rows. These phenomena lead to wet loss, leading to decreased turbine efficiency as wetness increases. Recently, with the increasing demand for renewable energy, the efficiency of steam turbines for hydrothermal geothermal resources has become increasingly critical. Thus, it is important to control the steam flow to efficiently discharge the liquid drain from the turbine. Numerical simulations for steam and gas turbines have been conducted [1–6] to understand the motion of droplets or solid particles in gas flow. Hussein and Tabakoff [1] simulated solid-particle flows around a turbine blade in an entrained gas flow. Subsequently, several Eulerian–Lagrangian numerical simulations were conducted for gas turbines [1–3] and steam turbines [4–6]. Thereafter, theoretical modeling [7] and numerical simulations [8] were also conducted to clarify the icing process and rain erosion of wind turbines [9]. Hence, there is a high demand to comprehend the dynamics of droplets and particle flows in diverse applications.

Deposited droplets on the turbine blade form a liquid bulk, such as a liquid finger, sheet, or ice, which may significantly affect the gas flow field. However, the direct numerical modeling of such droplet motion is not intrinsic due to the complexity of the relevant multiple processes involved. These include turbulence, condensation, and evaporation, momentum exchange between droplets and gas or steam flow, and interaction between droplets and blade surface. Moreover, an enormous numerical cost is incurred to perform such a direct numerical simulation. In addition, predicting the particle motion near a solid wall is particularly challenging owing to the requirement of a high spatio-temporal resolution to capture the complex droplet behavior near the wall using experiments and numerical simulations. Therefore, although the qualitative explanations of each droplet and gas flow behavior are simple, their quantitative discussion is limited. Consequently, only simple systems, such as pipe flow [10] and curve duct [11], have been recently examined.

This study investigated the behavior of flying droplets in a turbine cascade to determine the amount and position of the liquid phase. The analysis performed is expected to aid in the efficient design of steam and gas turbine systems, because sand particles and droplets enter from an intake impact on the turbine blade, causing erosion and icing. Thus, it is important to understand the basic behavior of droplet motion in the turbine cascade.

Gyarmathy [12] estimated a two-dimensional streamline of the steam flow around the turbine blade using the inlet angle of the steam flow and meridional chord length, whose streamline is close to the shape of the pressure surface. Young and Yau [6] also calculated the two-dimensional steam flow to predict the motion of fog droplets around the turbine blade. In both studies, analytical models were derived and validated using Young's simulated inertial deposition rates. Although such an analytical approach can successfully capture the deposition rate at a fog droplet diameter smaller than 1.5 *µ*m, it can only detect tiny droplets and regions near the pressure surface. In an actual steam turbine operation, the typical Reynolds number is greater than 10^{5} [13], and droplets of various sizes beyond a few micrometers are observed [14]. However, the earlier proposed analytical approach [6] did not cover these ranges to investigate the practical range of the Reynolds and Stokes numbers in the turbine cascade.

An experiment and a large numerical simulation can, in principle, determine the impact point of a droplet and the damage rate to the surface of the turbine blade, overcoming the limitations of the analytical approach. However, in practice, the cost of such approaches can be excessively high for the engineering purpose, i.e., optimal design of turbine blades. For example, Zhang et al. [15] performed direct Monte Carlo simulations of the particles in a straight pipe flow. Tracking all individual particles successfully simulated the deposition process on the wall in detail but required enormous computational costs for practical use. Thus, they proposed a simplified model to mimic the solution of expensive direct simulation by assuming equivalent Markov motion of particle in the turbulent freestream. The consistency of solutions between the direct simulation and its approximated model results was good enough to justify the simplified model as a pragmatic prediction model in the early stages of turbine development. The establishment of such a pragmatic model for the earlier stages of turbine development is demanded for the engineering of the turbine cascade.

Hence, the scope of this study is the introduction of an analytical approach for predicting the motion of droplet particles in the fluid flow. To simplify the problem, we focus on the fluid motion as a gas. Secondary factors such as evaporation and condensation, which characterize steam behaviors, were not involved. First, an analytical model for droplet trajectories was proposed using the gas flow streamlines calculated from the outline of the turbine blade. Second, the droplet trajectories around low-pressure turbine blades T106 were calculated using the prediction model; subsequently, three-dimensional (3D) numerical simulations were performed to validate the proposed model. Finally, the validation results of the model using 3D numerical simulation results were presented and the accuracy and limitations of the proposed model were discussed.

## 2 Analytical Modeling

### 2.1 Droplet Trajectory.

Dring and Suo [16] examined a theoretical model for predicting the particle trajectory for a pure swirl flow and investigated the deviation of particles from the gas flow streamline. This study used a similar approach to discuss the derivation of a droplet from a gas flow streamline. The gas flow streamlines in the linear turbine cascade were approximated using the parabola, hyperbola, or ellipse functions.

This study considered a droplet trajectory on the gas flow field in polar and Cartesian systems, as shown in Fig. 1. Gas flow velocity *W* and initial radius *r*_{0}(*θ*) of the streamline function as angle *θ* or time *t*. The range of *θ* was set from 0 deg to 90 deg, and *r*(*θ*) is the radius of curvature at each *θ* or *t*, which is consistent with the droplet trajectory; *V* is the velocity of the droplet relative to the gas flow.

The proposed analytical model for predicting the particle trajectories was based on the following assumptions:

The gas flow field is an incompressible irrotational field and does not include a stagnation point or secondary flow.

Droplets in the gas flow do not deform; the shape of the droplet remains spherical.

The inertia of the droplets is limited only along the radial direction.

The gas flow velocity

*W*is constant along the gas flow streamline.

*a*in the polar coordinate system and mass

*m*of a droplet can be expressed as

*v*is the volume of a spherical droplet and

*ρ*is the density of the liquid. Furthermore, the unit vectors in the polar coordinate system are

_{L}*i*and

_{r}*i*. Following the formulations by Dring and Suo [16], the two types of forces acting on a droplet are drag

_{θ}*D*and buoyancy

*B*, which are expressed as

*C*is the drag coefficient,

_{D}*A*is the cross-sectional area of the particle, and

_{C}*ρ*is the gas density. The velocity of droplet

_{G}*V*is divided into

*V*and

_{r}*V*, and the relations are expressed as

_{θ}*V*between $r(t)\theta \u02d9$ and

_{θ}*W*is neglected, and the equations are expressed as

*c*is a constant. Hence,

*W*must satisfy the following equation:

*r*=

*r*(

*t*) was replaced by

*r*=

*r*(

*θ*) using the following equations:

*r*=

*r*(

*θ*), can be derived using the dimensionless variable

*f*=

*r*(

*θ*)/

*r*

_{0}(

*θ*):

*A*can be derived as follows:

_{C}*D*is the diameter of the droplet, and

_{L}*αβr*

_{0}(

*θ*) in Eq. (22) can be rewritten as

*C*is a function of the particle Reynolds number Re

_{D}*, which is defined as*

_{P}*μ*is the gas viscosity,

_{G}*W*

_{In}is the inlet velocity of the gas flow, and the inlet Reynolds number Re

_{In}is defined as follows:

*is less than 1, the expression for*

_{P}*C*can be obtained from the Stokes' law [17]:

_{D}*is less than 1, Eq. (28) can be used to solve the droplet trajectory with Stokes' law, which is consistent with pure swirl flow [16]. However, when Re*

_{P}*> 1, Eq. (28) is invalid because Stokes' law is violated. Furthermore, for the range 1 < Re*

_{P}*< 1000, the Wen–Yu drag model [18] for the correlation of*

_{P}*C*is used, which is expressed as

_{D}*C*as follows:

_{D}This equation becomes slightly more complicated than Eq. (28); however, it requires a lower computational cost in comparison to the computational fluid dynamics (CFD) model.

Initial conditions of *f* and *f*′ were set at *f*(0) = 1 and *f*′(0) = 0, and the radius of gas flow streamline *r*_{0}(*θ*) was calculated using the outline of the turbine blade as described in the next section. Based on the above initial conditions, the ordinary differential equation expresses the trajectory of the droplets.

As evident from Eq. (29), the St number becomes small with the increase in *r*(*θ*). In the case of a turbine blade where *r*(*θ*) becomes more prominent toward the trailing edge, the St number is small at the trailing edge, and *W*, at large *θ*, is not strictly equal to *W*_{In}. Consequently, the curve of the droplet trajectory near the trailing edge may be underestimated, which is a limitation of the proposed model.

### 2.2 Gas Flow Streamline.

In contrast to the direct numerical simulation, the gas flow streamlines of the proposed analytical model were obtained with uncertainty, which was attributed to the gas flow passing between the pressure and suction surfaces in the turbine cascade. The curvature of the suction surface at the front of the turbine blade was very steep, whereas that of the pressure surface was gentle. Based on assumption (i), the gas flow streamlines near the turbine blade surface follow their respective surfaces. Accordingly, the actual streamlines in the turbine cascade in the pitch-wise direction were assumed to lie between the two turbine blade surfaces. The point cloud of the pressure and suction surfaces was taken as the initial radius *r*_{0}(*θ*) to solve Eqs. (28) and (31). Thus, the two streamlines obtained by the suction and pressure surfaces were the upper and lower limits of the streamline range. Therefore, the droplet trajectories, including the error bar, could be predicted without expensive numerical simulations, based on the gas flow streamline characterized by the outline of the turbine blade.

## 3 Numerical Model and Setup

To validate the analytical model, this study simulated the gas flow field and droplet flow around the low-pressure turbine linear cascade T106 [19] using Open Source Field Operation and Manipulation (OpenFOAM). The governing equations of mass, momentum, and total energy for compressible one-phase flow were solved using rhoCentralFOAM [20–22], a density-based compressible flow solver utilizing central-upwind schemes [20]. The advection term was solved by the second-order linear-upwind scheme, and the time integration was solved by the second-order Crank–Nicolson scheme. Furthermore, the dynamic viscosity was fixed as a constant, and turbulent viscosity was calculated using the *k*–*ω* shear stress transport (SST) detached eddy simulation (DES) model [23]. DES uses a hybrid approach to solve the Reynolds-averaged Navier–Stokes equation near the walls and large eddy simulation model in other regions. Effective viscosity is defined as the sum of the dynamic and turbulent viscosities. The Lagrangian particle tracking method was used to calculate the motion of particles in the gas flow based on a one-way coupling manner. Consequently, the effect of forces based on drag, buoyancy, and pressure gradient for a spherical particle was calculated. We used the standard drag [24] and lift models [25,26] for the spherical particles.

Table 1 lists the design configuration of T106 and aerodynamic specifications for the numerical simulations. The geometry of the computational system and the initial and boundary conditions for the numerical simulation are shown in Fig. 2. The numerical conditions correspond to Hoheisel's experimental state [19]. The chord length *C* of T106 was fixed at a value of 100 mm, and the span length of the blade was 20 mm, which was shorter than that of the actual T106.

Aerodynamic specifications and numerical condition | |
---|---|

Chord length, C (mm) | 100 |

Blade span (mm) | 20 |

Inlet angle (deg) | 37.7 |

Inlet Mach number | 0.3 |

Outlet Mach number | 0.6 |

Outlet static pressure (Pa) | 1 × 10^{5} |

Outlet Reynolds number, Re_{Out} | 1 × 10^{5}, 5 × 10^{5} |

Inlet temperature (K) | 300 |

Diameter of particle (µm) | 1, 10, 50, 100 |

Stokes number at Re_{Out} = 1 × 10^{5} | 4 × 10^{−4}, 4 × 10^{−2}, 1, 4 |

Stokes number at Re_{Out} = 5 × 10^{5} | 2 × 10^{−3}, 2 × 10^{−1}, 5, 20 |

Aerodynamic specifications and numerical condition | |
---|---|

Chord length, C (mm) | 100 |

Blade span (mm) | 20 |

Inlet angle (deg) | 37.7 |

Inlet Mach number | 0.3 |

Outlet Mach number | 0.6 |

Outlet static pressure (Pa) | 1 × 10^{5} |

Outlet Reynolds number, Re_{Out} | 1 × 10^{5}, 5 × 10^{5} |

Inlet temperature (K) | 300 |

Diameter of particle (µm) | 1, 10, 50, 100 |

Stokes number at Re_{Out} = 1 × 10^{5} | 4 × 10^{−4}, 4 × 10^{−2}, 1, 4 |

Stokes number at Re_{Out} = 5 × 10^{5} | 2 × 10^{−3}, 2 × 10^{−1}, 5, 20 |

The inlet Mach number was fixed at 0.3, the velocity was approximately 100 m/s at a uniform distribution, and the angle of the inlet flow was fixed at 37.7 deg. Furthermore, the outlet boundary was set under continuous-flow conditions. In addition, the temperature at the inlet boundary was set at 300 K, and the outlet static pressure *P _{s}* was 1.0 × 10

^{5}. Periodic boundaries were set at the top and bottom boundaries in the pitch-wise direction. Moreover, the surface of T106 was a nonslip boundary, whereas the other boundaries were slip boundaries.

_{Out}is defined as

*V*is the gas velocity at the outlet boundary. This study used different dynamic viscosities

_{G}*μ*to change Re

_{G}_{Out}, with values of 1 × 10

^{5}or 5 × 10

^{5}. In addition, the number of calculation cells was approximately 3 million. A boundary layer mesh with ten cells was generated around T106, and the boundary layer velocity was represented using the logarithmic wall function. Furthermore, the turbulence intensity of inlet gas flow was set at 5%. The mesh parameters for

*N*= 3 million where

*N*is the number of grid cells are summarized in Table 2 with

*y*+ maximum values of 11 and 8 on T106 and other walls, respectively.

Parameter | Characteristic/value |
---|---|

Mesh type | Tetrahedral |

Maximum aspect ratio | 20.5 |

Minimum face area | 5.73705 × 10^{−09} (m^{2}) |

Minimum volume | 8.41588 × 10^{−13} (m^{3}) |

Maximum skewness | 1.48922 |

Mesh nonorthogonality | Maximum: 63.377 |

Average: 15.3769 |

Parameter | Characteristic/value |
---|---|

Mesh type | Tetrahedral |

Maximum aspect ratio | 20.5 |

Minimum face area | 5.73705 × 10^{−09} (m^{2}) |

Minimum volume | 8.41588 × 10^{−13} (m^{3}) |

Maximum skewness | 1.48922 |

Mesh nonorthogonality | Maximum: 63.377 |

Average: 15.3769 |

The simulation used eight Intel Xeon Gold 6148 processors (20 cores, 2.4 GHz). For a time span of 1 ms, the elapsed time required to solve the model was approximately 4 h. The convergence criteria tolerance was set at 10^{−10} for the velocity and specific internal energy and at 10^{−6} for the turbulent kinetic energy *k* and specific dissipation rate *ω*. In practice, the residuals of velocity and specific internal energy are of the order of 10^{−12}, while those of *k* and *ω* are approximately 10^{−10} and 10^{−7}, respectively, at convergence.

## 4 Results and Discussion

### 4.1 Theoretical Analysis.

First, we consider the streamlines in the turbine cascade of the gas flow in this study and those in a previous study [12], as shown in Fig. 3. In a previous study, to predict the motion of a droplet that collides with the pressure surface, the steam flow streamline was calculated using the inflow angle of the steam flow and chord length of the turbine blade. The estimated streamline of steam flow was along the shape of the pressure surface of the turbine blade. However, the gap between the actual and estimated streamlines markedly increased at a position far from the pressure surface. In contrast, the proposed streamlines in this study follow two turbine blade surfaces.

Figure 4 shows the calculated droplet trajectories for different diameters and dynamic viscosities. The orbits were drawn considering Eqs. (28) and (31). The black dashed lines representing the pressure and suction surface outlines of T106 indicate the upper and lower bands of the background gas flow streamlines. The case of Re_{Out} = 1 × 10^{5} was examined; when the droplet size was as small as *D _{L}* = 1

*µ*m (St = 4 × 10

^{−4}), the results almost corresponded to the gas flow streamline. However, when the droplet size increased to

*D*= 10

_{L}*µ*m (St = 4 × 10

^{−2}), the droplet trajectories began to deviate from the streamline because the effect of the drag force on the droplet motion decreased. Furthermore, for large droplet sizes of

*D*= 50 and 100

_{L}*µ*m, the droplet motion was almost independent of the gas flow and controlled by inertia.

With the increase in Re_{Out}, the St number at each droplet size increased. Therefore, droplets with small diameters tended to deviate from the gas flow streamline. The curves of the droplet trajectories at each St number calculated using Eq. (31) were slightly smaller than those obtained using Eq. (28). However, the difference between the two curves was not significant within the range of the examined conditions. Moreover, all droplet trajectories were nearly Stokes flows under the target Reynolds number conditions.

### 4.2 Numerical Simulation.

The numerical results were compared with the experimental results [19] to verify the quality of the solution. Figure 5 shows the static pressure distribution *P _{s}* on the turbine blade T106 normalized by the mean of the inlet total pressure

*P*. The numerical and experimental data were measured at the middle span of the T106. To check the convergence of the numerical solution, Fig. 5 shows the results obtained using different numbers of grid cells,

_{t}*N*= 2 million and 3 million. The consistency of the solutions indicates that there is no grid size dependence above

*N*= 2 million. Based on this observation, the result with

*N*= 3 million cells is analyzed for the later sections. The pressure distribution ratio

*P*/

_{s}*P*on the pressure surface at Re

_{t}_{Out}= 5 × 10

^{5}was consistent with the experimental result, whereas that on the suction surface was slightly lower than the one in the experimental result. This disagreement occurs because the pressure distribution varies sensitively with the parameters, and the location of the separation point is governed by parameter optimization [27]. Thus, the deviation from the experimental results near the suction surface may be improved through parameter optimization of the turbulence model or wall function on T106. However, the exemplary flow structure near the blade edge for the prediction of particle trajectory is not within the scope of this study; therefore, calibration was not conducted. Furthermore, the curves of the numerical solutions were not significantly different between the cases with Re

_{Out}= 5 × 10

^{5}and Re

_{Out}= 1 × 10

^{5}. The same trend was reported by Hoheisel [19].

Figure 6 shows the pressure distribution profiles and Mach numbers at Re_{Out} = 1 × 10^{5} and 5 × 10^{5} after reaching the steady-state condition in the midspan cross section along the spanwise direction. The pressure and Mach number were similar for the two Reynolds numbers because of the small viscous effects. The inlet and outlet Mach numbers were 0.3 and 0.6, respectively, in both cases. These results indicate that the flow field was almost independent of the Reynolds number in the target range.

In Fig. 7, the gas flow streamlines at Re_{Out} = 1 × 10^{5} are shown. The absolute velocity of the gas flow is represented by the color of the streamlines. The curvature of the gas flow streamlines was consistent with the outline of T106. This observation justifies the use of the turbine blade shape to estimate gas flow streamlines.

The trajectories of particle motion with various droplet sizes at Re_{Out} = 1 × 10^{5} and Re_{Out} = 5 × 10^{5} obtained by Eulerian–Lagrangian simulations are shown in Figs. 8 and 9. The color of each droplet indicates its absolute velocity. The droplet trajectories were uniquely calculated because the injection of droplets started after reaching the steady gas flow. As shown in Fig. 8(a), the particle trajectory with $St\u226a1$ and *D _{L}* = 1

*µ*m followed the gas flow streamline because of the strong drag force from the background gas flow. With the increase in the diameter of the droplets, the particle trajectory deviated from the gas flow streamline. When St was 4 × 10

^{−2}with

*D*= 10

_{L}*µ*m (Fig. 8(b)), the droplets at the trailing edge of the turbine blade moved to the upper side. The relaxation time of the droplets to follow the gas flow increased with the St number. For a small St = 1.1 (Fig. 8(c)) with

*D*= 50

_{L}*µ*m, the effect of the drag on the droplet motion decreased, and most droplets collided with the pressure surface. However, certain droplets passed between the linear turbine cascades. By increasing St = 4 > 1 (Fig. 8(d)) with

*D*= 100

_{L}*µ*m, all major droplets impacted the pressure surface of the turbine blade because droplets in the gas flow moved straight without being affected by the gas flow. The present results are consistent with the results of previous studies [28]. With the increase in the St number, the deviation of the droplet trajectories from the gas flow streamlines became more significant, even under low Reynolds number conditions. As shown in Figs. 8 and 9, the St number is a dominant dimensionless number for droplet motion in gas flow. In the above discussion, it was found that the St number can capture the effect of the droplet diameter as well as the viscous effect on the droplet trajectory.

### 4.3 Comparison of Analytical and Numerical Results.

The droplet trajectories obtained via analytical analysis were compared with the results of the numerical simulation. Figures 10 and 11 show the droplet trajectories calculated analytically and numerically, with a schematic of the turbine blade. The *x*-origin is the leading edge of the turbine blade outline used to predict the droplet flows, and the *z*-coordinate corresponding to the *x*-origin on the numerical trajectories is the *z*-origin. The droplet ejection points are defined at each *x*- and *z*-coordinate. The analytical droplet trajectories were predicted using Eq. (31) from those droplet ejection points. Furthermore, the analytical solution was consistent with the numerical solution when the droplets passed between the red and blue solid lines.

The proposed model is based on the assumption that the growth of the gas flow separation and secondary flow at the surface of the turbine blade is small, as exhibited by the CFD simulations after long-term improvement in the shapes of the turbine blade. Therefore, the proposed model is valid for estimating the streamlines of the gas flow because the droplet trajectories under conditions of low St numbers in Figs. 10(a) and 11(a) agree with the gas flow streamlines.

The difference between numerical and analytical results indicated that the droplet trajectories at small St numbers were approximately in the expected area between red and blue solid lines, regardless of the Re_{Out} number, which was consistent for 80% of the area from the leading to trailing edges. Furthermore, the effect of gas flow separation on the blade surface was limited to the suction pressure surface, and the droplet trajectories tended to fall below the predicted range near the pressure surface. The results indicated that the gas flow streamlines were reasonably predicted by the outlines of the turbine blade because tiny droplets followed the background gas flow motion.

The analytical trajectories with slightly larger St number conditions (St = 4 × 10^{−2}, 2 × 10^{−1}) were consistent with the numerical results, except in the vicinity of the leading edge and later part of the turbine blade cascade. In the case of St = 2 × 10^{−1}, as evident from Fig. 9(b), the droplets near the leading edge moved independently from the gas flow streamline such that the buoyancy force near the wall prevented the droplet from colliding with the wall. As shown in Fig. 10(c), with St = 1, the trajectories were consistent, except near the blade's leading edge. Around the leading edges, the numerical model successfully predicted the detailed gas flow motion, which prevented the particles from colliding with the leading edge of T106; however, the analytical model failed to capture such a flow. The droplets were accelerated in the radial direction at the initial position of the droplet in the analytical prediction, which is different from the assumed initial condition. Our model can also adjust the initial value *f′*(0) to account for the effect of the droplet's radial motion on its trajectory. Therefore, the analytical study makes the precise capture of the droplet behavior near the suction surface challenging.

The droplet straightened when St was markedly larger than 1. The droplet trajectory was well predicted in the first half of the turbine cascade, whereas the gap between the analytical and numerical results became prominent in the latter half of the cascade. This is because a larger diameter of the droplet requires a longer time for momentum exchange between the gas and liquid, which results in a larger *V _{θ}*.

*A*

_{T106}is the area of T106,

*C*

_{ax}is the axial chord length of T106, and

*G*is the gap between the analytically and numerically solved trajectories in the

*z*-direction. The leading edge of T106 was set to

*X*= 0. The analytical and numerical solutions obtained from five different sampling positions at

*X*= 0 (Figs. 8 and 9) deviated within the gap

*G.*The average error rate between the analytically and numerically solved trajectories of the five droplets in the

*z*-direction was less than 10%. In this model, the flow field within the turbine cascade is not resolved, and the initial conditions for the radial direction of the droplets are not precisely zero. Due to these reasons, the predicted droplet trajectories deviate from those obtained by CFD. To improve prediction accuracy, a more accurate gas flow model is required.

Droplet diameter (µm) | Error rate (%) | |||
---|---|---|---|---|

Re_{Out} = 1 × 10^{5} | Re_{Out} = 5 × 10^{5} | |||

Maximum value | Average value | Maximum value | Average value | |

1 | 9 | 4 | 9 | 4 |

10 | 7 | 4 | 16 | 8 |

50 | 27 | 6 | 8 | 2 |

100 | 10 | 3 | 15 | 4 |

Droplet diameter (µm) | Error rate (%) | |||
---|---|---|---|---|

Re_{Out} = 1 × 10^{5} | Re_{Out} = 5 × 10^{5} | |||

Maximum value | Average value | Maximum value | Average value | |

1 | 9 | 4 | 9 | 4 |

10 | 7 | 4 | 16 | 8 |

50 | 27 | 6 | 8 | 2 |

100 | 10 | 3 | 15 | 4 |

Based on the previous discussion, it was elucidated that the droplet trajectories in the turbine cascade can be calculated by linear analysis using initial conditions such as the gas–liquid properties and blade shape at Re_{Out} = 1 × 10^{5}, 5 × 10^{5}. With an increase in the Re number, the droplet diameter corresponding to each St number decreased. Thus, predicting the trajectories of smaller droplets becomes a more critical issue. The St number for droplets a few micrometers in the mainstream approached 1, whereas that for coarse droplets was significantly larger. Therefore, the proposed scheme is beneficial because a similar solution can be obtained with a calculation cost lower than the CFD simulation. However, the proposed scheme is unable to provide an accurate analytical prediction for the later process of the turbine cascade, depending on the St number.

In this study, we used one-way coupling, assuming that the water droplets in the turbine blade cascade were spherical in shape and sparsely distributed. Validating these model assumptions using experimental results would be challenging because of the limitations of the currently available visualization techniques. Hence, model assumption validation can be explored in a future study.

Furthermore, the current model is designed for a linear turbine cascade. To predict the droplet trajectory in the spanwise direction of the turbine blade, equations of motion that consider the influence of centrifugal forces must be introduced. Such challenges toward a more realistic setting remain for further investigation.

## 5 Conclusion

In this study, an analytical model was proposed to follow the droplet trajectory with a simplified gas flow streamline for an arbitrary blade shape. In order to validate the proposed approach, we performed the 3D numerical simulations, which successfully reproduced Hoheisel's experiment quantitatively.

The 3D simulation showed that the local Mach number and pressure distribution were almost independent of Reynolds number when Re_{Out} exceeded 10^{5}. We also found that the St number was the fundamental control factor of the droplet trajectories. When St number was small, the drag force by the gas flow was strong enough for the droplet to follow the gas flow streamline. When St number was increased to one, the droplets began to deviate from the gas flow streamline and collided with the pressure surface. Moreover, when St was higher than one, the droplet motion was nearly independent of the gas flow; consequently, the droplets moved straight and impacted the surface of T106. We also found that the outline of the turbine blade agreed well with the gas flow streamlines in between the turbine cascades, even in the vicinity of the suction pressure surface.

The simplified analytical model using the outline of the turbine blade was developed to predict the droplets’ motion for various St and Re values instead of the expensive 3D numerical analysis. The proposed model can determine a droplet trajectory based on the approximate shape of the streamlines and successfully reproduce the results of the 3D numerical simulation with good accuracy for practical parameter settings with low computational cost whose average error rate is below 10%.

The limitation of the proposed model came from an extreme case with the inexact treatment of acceleration at the droplet's injection point and very high St conditions (e.g., St > 4) where the gas flow contribution on the droplet flow is significantly reduced. In general, this is not a significant issue, as the linear trajectory of the droplets under high St conditions is intrinsic and therefore not of engineering interest. Despite these limitations, for practical use, the proposed method should be helpful in the optimization of the turbine blade with a reasonable computational cost.

## Acknowledgment

This study was supported by the Earth Simulator project of the Japan Agency for Marine-Earth Science and Technology.

## Funding Data

This article is based on results obtained from a project, JPNP13009, commissioned by the New Energy and Industrial Technology Development Organization (NEDO).

This study was also supported by JSPS KAKENHI (Grant Nos. JP21K14084, JP22H01593, and JP24K17202).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*a*=particle's acceleration

*c*=constant

*f*=dimensionless variable (

*r*(*θ*)*/r*_{0}(*θ*))*i*=unit vector

*k*=turbulent kinetic energy

*m*=mass of a droplet

*r*=radius of streamline function

*t*=time

*v*=volume of a spherical droplet

*A*=cross-sectional area of a particle

*B*=buoyancy force

*C*=chord length

*D*=drag force

*G*=gap between analytical and numerical trajectories in the

*z*-direction*N*=number of cells

*V*=velocity of a droplet relative to gas flow

*W*=gas flow velocity

*A*_{T106}=area of T106

*C*=_{D}drag coefficient

*D*=_{L}droplet size

*P*=_{s}outlet static pressure

*P*=_{t}averaged inlet total pressure

*V*=_{G}gas velocity at the outlet boundary

*W*_{In}=inlet velocity of the gas flow

*x*,*z*=Cartesian coordinate system

- Re =
Reynolds number

- St =
Stokes number

*α*=$(1/2)CD(AC/v)$

*β*=density ratio of gas and liquid

*ɛ*=error rate

*θ*=angle

*μ*=dynamic viscosity

*ρ*=density

*ω*=specific dissipation rate