Ash particle deposition in a high-pressure turbine stage was numerically investigated using steady Reynolds-averaged Navier-Stokes (RANS) and unsteady Reynolds-averaged Navie-Stokes (URANS) methods. An inlet temperature profile consisting of Gaussian nonuniformities (hot streaks) was imposed on the vanes, with vane cooling simulated using a constant vane wall temperature. The steady case utilized a mixing plane at the vane–rotor interface, while a sliding mesh was used for the unsteady case. Corrected speed and mass flow were matched to an experiment involving the same geometry, so that the flow solution could be validated against measurements. Particles ranging from 1 to 65 μm were introduced into the vane domain, and tracked using an Eulerian–Lagrangian tracking model. A novel particle rebound and deposition model was employed to determine particles' stick/bounce behavior upon impact with a surface. Predicted impact and capture distributions for different diameters were compared between the steady and unsteady methods, highlighting effects from the circumferential averaging of the mixing plane. The mixing plane simulation was found to generally under predict impact and capture efficiencies compared with the unsteady calculation, as well as under predict particle temperature upon impact with the blade surface. Quantitative impact and capture efficiency trends with the Stokes number are discussed for both the vane and blade, with companion qualitative distributions for the different Stokes regimes.

## Introduction

As the aviation industry expands and intensifies its operations in regions with high concentrations of particulate, the problem of deposition in gas turbines is of increasing importance to operators and original equipment manufacturers (OEMs). Small particles such as sand, sea salt, and other airborne pollutants are ingested by the engine, heated as they make their way through the combustor or cooling passages, and then deposit on turbine component surfaces. This phenomenon is not limited to aircraft engines, as power generation turbines encounter particulate as a product of the combustion of synthetic fuels, which are produced by the liquefaction or gasification of coal. Deposition has several detrimental effects on the performance and life of turbine components. Bons [1] found that small-scale deposition increases the surface roughness of turbine parts, thus increasing heat transfer to the components and degrading aerodynamic performance. Dunn et al. [2] determined that large-scale deposition can clog film cooling holes, resulting in a significant reduction in part-life and even failure in extreme cases. Deposition near film cooling holes has also been shown to have a detrimental effect on film cooling effectiveness [3]. The first stage of the turbine is especially vulnerable to deposition, as it is exposed to particulate at the highest temperatures directly downstream of the combustor.

Several studies have numerically investigated erosion and deposition in both compressor and turbine stages over a range of particle sizes, using various computational methods. Tabakoff et al. [4] numerically and experimentally tracked ash particles with diameters from 2.5 to 135 μm through an axial turbine. Larger particles, with higher Stokes numbers, were found to deviate from the flow more often and cause erosion. These particles were also determined to be centrifuged out toward the blade tip more quickly than the smaller particles, due to their mass. Hamed et al. [5] investigated turbine blade erosion caused by 30–1500 μm particles, and found that the leading edge and pressure surface of the vane were regions of high impact efficiency and significant erosion. It was also concluded that larger particles were more likely to strike both the vane and blade. Using an unsteady simulation with deforming mesh, Suzuki et al. [6] investigated erosion in a compressor stage with 165 μm alumina particles. It was found that the blade tip region was especially susceptible to erosion, and particles were observed to have multiple impacts on both the rotor and stator. Ghenaiet [7] studied erosion in a two-stage gas turbine using a frozen-rotor multiple reference frame model. The injected particles ranged from 1 to 150 μm, and were found to impact the trailing portion of vanes and leading edge region of rotor blades in each stage. Also utilizing a steady model, Corsini et al. [8] performed numerical simulations of erosion in a draft fan for a range of particle sizes (0.75–135 μm). Larger particles were observed to have more localized impact loci near the leading edge and tip of the blade, while smaller particles exhibited more distributed impact locations. Moving away from periodic calculations, Yang and Boulanger [9] investigated erosion of an axial fan by performing steady and unsteady calculations of the entire annulus. A mixing plane was employed in the steady simulations, and was found to over predict erosion by up to 100% when compared to the unsteady case.

While the previous studies all focus on erosion, there have also been several numerical studies of deposition on turbomachinery components. Barker et al. [10] adapted the critical viscosity model developed by Tafti and Sreedharan [11] to include the effects of shear on particle detachment. Compared to experimental results in Ref. [12], they determined that the deposition model predicted initial deposition relatively well, but was not valid after large-scale deposits begin to form. Borello et al. [13] produced numerical predictions of deposition in a subsonic compressor using Thornton and Ning's model [14]. Their simulation was able to achieve a measure of qualitative agreement with experiments, with deposits forming on the pressure surface near the leading edge, as well as on the suction surface near the hub endwall. Suman et al. [15] conducted a numerical study of deposition of microparticles (0.25–2 μm) on a transonic compressor blade, and discussed the effects of various fluid dynamic phenomena on impact distributions. In a later paper, Borello et al. [16] modified the critical viscosity model to account for particles striking existing deposit (as opposed to a clean surface), and applied the model to particle-laden flow around a single film cooling jet as well as a full cooled nozzle guide vane.

The temperature of the gas exiting the combustor and entering the turbine is generally not uniform, due to the discrete fuel injectors that make up the combustor. The behavior and effects of these nonuniform temperature profiles (also known as “hot streaks”) on turbine vane and rotor performance have been studied in great detail. Examples include Shang and Epstein [17], Povey et al. [18], Ong and Miller [19], and Qingjun et al. [20]. These inlet temperature nonuniformities can also affect deposition rates and patterns on turbine vanes, shown in practice by Chambers [21]. There have been limited studies on this subject. Casaday et al. [22] computationally investigated the effects of hot streaks on ash deposition in an uncooled turbine vane passage. Hot streaks were modeled as Gaussian distributions upstream of the vanes, with varied clocking position. The critical viscosity sticking model was used to predict deposition, and they found that increased surface temperatures correlated with increased levels of deposition. Casaday et al. [23] used the deposition model in Ref. [10] to investigate the effects of a nonuniform inlet temperature profile on external deposition. It was found by comparing to experiments that the model could be tuned to produce more representative results. Prenter et al. [24] extended this work by studying the effects of both hot streaks and slot film cooling on nozzle guide vane deposition, experimentally and computationally. Each of the above studies found that nonuniform inlet temperature profiles have a first-order effect on external deposition.

The objective of the current study is to investigate deposition in a high pressure turbine stage at realistic engine conditions, as well as to compare the predictions obtained by both steady and unsteady methods. It is the natural extension of a recent study by Zagnoli et al. [25] that explored particle impacts using the same geometry to compare differences between steady and unsteady calculation methods. This study adds the physical realities of cooling and hot streaks. While the literature review above has detailed several papers on deposition in compressor stages and erosion in turbine stages, to the authors' knowledge there are no publicly available unsteady numerical studies of deposition in a full turbine stage. The boundary conditions used for the model were developed by Haldeman et al. [26], who tested the effects of inlet temperature profiles and cooling on blade aerothermodynamic performance. Additionally, pressure measurements from the experiment are used to validate the flow solution of the current study. A novel particle–surface interaction model recently developed by Bons et al. [27] was implemented to predict particle rebound behavior and deposition. Trends in particle trajectories and impact locations are emphasized so that the results may be generalized and interpreted independent of a chosen deposition model as well.

## Methods

### Governing Equations and Solution Methods.

The steady and unsteady forms of the continuity, momentum, and energy equations were numerically solved using the commercial finite volume code fluent. The working fluid is a compressible gas with properties determined using temperature dependent polynomials. The effects of turbulence were simulated using the k–ω shear stress transport (SST) turbulence model within the Reynolds-averaged Navier–Stokes (RANS) and unsteady Reynolds-averaged Navier–Stokes (URANS) frameworks. This model relates the Reynolds stresses to the mean velocity gradients and the turbulent (or eddy) viscosity νt by the Boussinesq hypothesis, and is described in further detail in Ref. [28]. The k–ω SST model was chosen because of its ability to predict bulk flow-field and heat transfer quantities in the context of a turbine stage, as well as more nuanced effects due to complex secondary flows, as demonstrated by several past studies [2933]. A density-based solver was utilized in the implicit formulation, with convective fluxes determined using Roe's flux-difference splitting scheme.

For the steady calculations, a mixing plane was employed at the vane–rotor interface to account for the rotation of the blades. In mixing plane simulations, total pressure and total temperature at the exit of the vane domain are circumferentially averaged in radial bands and passed as the inlet boundary condition to the rotor domain. Static pressure from the rotor domain is communicated upstream and acts as an outlet boundary for the vane domain. These two boundary conditions vary as the flow solution is computed, and thus also converge simultaneously with the rest of the flow. In the unsteady simulations, a rotating mesh with a sliding interface is used for the rotor domain. In this formulation, all flow properties are passed directly through the interface as the rotation of the domain fully accounts for the rotation of the rotors. The physical time step in the unsteady simulations was chosen such that one period (a blade passing a full vane passage) would be divided by 20 time steps. This corresponds to Δt = 5 × 10−6 s, and resulted in residuals converging within 25 subiterations.

### Computational Domain and Boundary Conditions.

The computational domain in both the steady and unsteady simulations consists of two full vane passages and three full rotor passages. The actual number of vanes and rotors in the experiment were 24 and 38, respectively, which unfortunately does not provide a convenient ratio for periodic calculations. Thus, a 24–36 vane/rotor count was assumed so that the 2 to 3 periodic configurations could be used. This small approximation allows for a very significant reduction in computational cost. The main inlet to the domain (the vane inlet) was located approximately 0.5 vane axial chords upstream of the vane leading edge, and the main outlet (rotor outlet) was placed approximately 0.3 vane axial chords downstream of the rotor trailing edge. Periodic boundaries in both the vane and rotor domains were used to simulate the full annular configuration. A schematic of the computational domain is shown in Fig. 1.

The experiment in Ref. [26] was run at laboratory conditions, that is, at lower temperatures and reduced rotational speeds, while matching corrected speed and corrected mass flow of the machine. As temperature has been shown as a critical parameter in deposition, the temperature and pressure have been scaled up to engine relevant conditions in this study (while still matching the corrected speed and mass flow). Matching these corrected parameters ensures that the flow and blade Mach numbers match those in the experiment. In this manner, it is possible to study deposition at engine-relevant conditions while still being able to compare to the measurements made by Haldeman et al. for validation of the flow solution. The main inlet total pressure was set to a value corresponding to a compressor overall pressure ratio (OPR) of 14. The stage exit static pressure was then set by matching the stage pressure ratio of 3.79 reported in Ref. [26]. The values of these boundary conditions are listed in Table 1 along with other boundary conditions to be discussed.

To simulate the temperature nonuniformities present at the turbine inlet in actual engines, Gaussian total temperature profiles were superimposed on a base total temperature to represent hot streaks. The inlet profile for this study had a mass-averaged temperature of Tt,avg = 1650 K, a maximum-to-average temperature ratio of Tt,max/Tt,avg = 1.07, and was created using the following expression:
$Tt=A exp(−(y−y0)22σy2−(z−z0)22σz2)+C$
(1)

with A = 165, σy = σz = 0.012, and C = 1600. The terms y0 and z0 represent the hot streak center locations, which are clocked to midpassage for this study. The resulting inlet total temperature profile is shown in Fig. 2. The maximum-to-average temperature ratio was chosen based on the ratios investigated in previous studies [19,18,34]. Velocity and turbulence profiles at the inlet of a turbine are also very nonuniform; however, these nonuniformities were not included in these simulations. Studying these effects on particle trajectories is subject to future work, where the current study should make for an interesting comparison.

It was not computationally feasible to use a grid that included discrete film cooling holes, as the unsteady simulations are quite expensive even with the current cell count, and a significant increase in this number would be required to adequately resolve the holes. As an alternative, the vane walls were set to a constant (cooler) temperature to simulate the effects of cooling without the considerable increase in computational cost. As fluid moves past the cool surface, a thermal boundary layer develops and yields cooler fluid in the vane wake, which is typical of the actual cooled case. While any hydrodynamic effects of the coolant being ejected over the vane surface are lost, the resulting nonuniform temperature profile at the exit of the vane and inlet to the rotor still provides an interesting condition to study (especially within the context of deposition). The vane wall temperature was set to 0.71Tt,avg as reported in Ref. [26], corresponding to approximately 1170 K. The rotors were not cooled in this study and their walls were set as adiabatic, along with the hub and case walls. A full list of the boundary conditions is provided in Table 1.

### Mesh.

A fully structured mesh was developed using multiple blocks of hexahedral cells, with refined boundary layer regions near all vane, rotor, and endwall surfaces. A grid independence study was conducted by generating a fine mesh and then coarsening it to create a medium mesh. Steady midspan vane and blade pressure traces were found to be within 3% between the two meshes, and thus the medium mesh was utilized for all computations reported in this study (steady and unsteady). The details of each mesh are listed in Table 2 and the medium mesh is shown in Fig. 1.

### Solution Validation.

The accurate prediction of deposition requires that particles be delivered to the surface in a realistic manner. This in turn relies on the accuracy of the particle tracking model and the flow solution. To evaluate the latter, midspan pressure traces from the steady mixing plane simulations are plotted against experimental measurements (from Ref. [26]) in Fig. 3. Because the flow and blade Mach numbers are matched, it is possible to compare this scaled-up simulation to the experiment run at laboratory conditions. Except for a few locations, there is a reasonable agreement between the simulations and the experiment, with all predicted pressures within 10% of the measured value. The simulation predicted a sharp rise in pressure on the blade suction surface just after the suction peak, which seems to be confirmed by the data. The sudden change in pressure on the vane suction surface indicates the presence of a shock. Time-averaged blade pressures from the unsteady simulation are also plotted, and show similar agreement with the experimental data. Secondary flows such as rotor tip leakage flow (and vortex), and the passage vortex were identified in both the steady mixing plane and unsteady sliding mesh simulations. The prediction of these complex flows provides further confidence that the models produce a representative flow field through which to track particulate.

### Particle Injection and Tracking.

Particle tracking was conducted using fluent's built-in discrete phase model (DPM). This model calculates the trajectory of particles in the Lagrangian frame by balancing the flow-induced forces on the particle with its inertia. The governing equation of motion can be written as [35]
$dupdt=FD(u−up)+g(ρp−ρ)ρp+F$
(2)
where $F$ is an additional acceleration term, $FD(u−up)$ is the drag force per unit particle mass, and
$FD=18μρpdp2CDRed24$
(3)
In the above equations, ρp, dp, and $up$ are the particle's density, diameter, and velocity, with ρ and μ denoting the fluid's density and viscosity. The relative particle Reynolds number is defined as
$Red≡ρdp|up−u|μ$
(4)
and a spherical drag coefficient from Ref. [36] is assumed
$CD=a1+a2Red+a3Red2$
(5)
The additional acceleration term $F$ is used to account for “fictitious” forces that occur in the rotating frame, such as centrifugal and Coriolis forces. The rotating forces per unit mass are
$F=2ω×ur+ω×ω×r$
(6)

where $ω$ is the rotational speed, $ur$ is the fluid velocity in the rotating frame, and $r$ is the position vector.

The DPM takes the Eulerian–Lagrangian approach, where the particle tracking is separated into two steps. In the first step, the flow solution is computed absent of any particulate. A large amount of dispersed particulate is then injected and tracked in the second step, where each trajectory is predicted by computing the local forces, as described above, at each step along the particle's path. This trajectory is stored for each particle.

The particulate investigated in this study is the Jim Bridger power station (JBPS) sub-bituminous coal fly ash studied experimentally by Ai [37]. This ash has a size distribution ranging from 0 to 70 μm, with a mass mean diameter of 13.4 μm. With such a large range, the natural question of which diameters to study arises. Several previous papers (e.g., Ref. [6]) involving unsteady particle tracking investigated only a single diameter, because of the computational cost associated with having to solve the unsteady flow field with each new diameter injected. Another method is to inject particles with a distribution of diameters representative of the size distribution of the particulate. The challenge with this method, however, is that a much larger number of the smaller particles are required to make up the mass fractions compared to the larger particles. As a result, a statistically significant number of larger particles will generally be accompanied by an unfeasibly larger amount of smaller particles.

An alternative procedure is implemented in this study, which allows the investigation of a range of diameters while avoiding the issues mentioned above. Ten diameters that span the range of the distribution are chosen, and a statistically significant number of each diameter is injected. The specific number of diameters chosen is not important, using more sizes does increase the accuracy of the method however. When particles are predicted to impact or deposit on a surface, a user defined function (UDF) keeps track of the separate sizes. This enables the calculation of the number impact efficiency and number capture efficiency for each diameter, defined by
$εN,i=Nimp,iNinj,i, κN,i=Ndep,iNinj,i$
(7)
where Nimp,i, Ndep,i, and Ninj,i are the number of impacts, deposits, and particles injected at each diameter i, respectively. These number efficiencies can then be converted into mass efficiencies using the mass fraction at each diameter fm,i
$εm,i=εN,ifm,i, κm,i=κN,ifm,i$
(8)

In this manner, mass-based impact and capture efficiencies are determined using a representative range of diameters, with a statistically significant number of particles at each diameter. The fact that this method is implemented in a single flow solution (as opposed to a separate flow solution for each diameter) is a considerable reduction in computational cost and time.

The diameters chosen for this study are 1, 3, 5, 10, 15, 25, 35, 45, 55, 65 μm, and are shown against the cumulative size distribution in Fig. 4. For both the steady and unsteady simulations, 20,000 particles at each of the chosen diameters were injected, for a total of 200,000 particles. This injection was implemented over a full period in the unsteady case, to ensure that the position of the rotors at the initial injection time did not bias results. In the steady simulation, all particles were injected at once but required special treatment at the mixing plane. Particle locations at the vane outlet plane were stored and then randomly distributed in the circumferential direction while preserving their radial position. The question of whether to average particle variables (temperature, velocity, etc.) or preserve them was addressed by Zagnoli et al. [25], who found no measureable difference between the two methods. Thus, in this study the particles' properties were preserved, and the new randomly distributed circumferential locations were used as an injection into the rotor domain.

### Coefficient of Restitution and Deposition Model.

A novel particle rebound and deposition model developed by Bons et al. [27] is employed in this study. The model is grounded in physics, but allows for tuning of the temperature sensitivities of various particle properties that are often unknown. The physics included are elastic and plastic deformation, adhesion, and shear removal. With its algebraic formulation, the model is ideal for implementation in computational fluid dynamics calculations such as the ones conducted for this study. The parameters of the model implemented here have been tuned to experimental data of Ai [37], where the same JBPS ash was utilized. Further details about the model are available in Ref. [27] and the properties of the ash are available in Refs. [24] and [37].

## Results

### Steady Mixing Plane Results.

Contours of total temperature in a midspan slice of the vane domain are shown in Fig. 5. The hot streaks which were initially clocked at midpassage remain coherent as they travel through the passage to the vane exit. The effect of the simulated vane cooling is apparent in the cooler temperatures present in the vane wakes, which contribute to the nonuniform exit temperature profile despite some mixing. Figure 6 shows the footprint of the hot streaks and vane cooling at the vane outlet. The reduction in spanwise passage width from the inlet to the outlet is equal from the hub and the case, thus the contours suggest a radial migration of the hot streaks toward the hub. The figure also illustrates how the circumferential variations are lost through the averaging of the mixing plane, resulting in a purely radial profile entering the rotor domain.

Blade surface temperatures from the steady simulation are plotted in Fig. 7. It is apparent that the highest temperatures on the surface are at more outward radial station than one would predict from the rotor inlet profile in Fig. 6. This is because the static temperature at the rotor inlet, which is affected by the downstream rotor domain in the mixing plane formulation, has its maximum closer to midspan. This coupled with the initial movement of the warmer fluid outward on radius as it enters the blade passage results in the blade surface temperature profile observed. A band of cooler temperature on the suction side just downstream of the leading edge is visible, in the region of the supersonic suction peak.

When investigating particle impact and deposition trends, it is useful to consider the Stokes number, which is defined as the ratio of the particle relaxation time and the characteristic timescale of the flow or an obstacle
$St≡τpτf=ρpdp2Uc18μLc$
(9)

where Uc is the characteristic velocity of the flow away from the obstacle and Lc is the characteristic length scale of the obstacle. This nondimensional parameter provides insight into the likelihood of a particle to follow the flow (St ≪ 1) or follow a ballistic trajectory and strike the obstacle (St ≫ 1). Two Stokes numbers are defined here, each based on the relevant values for the vane and rotor, respectively. The chord of each airfoil is used for Lc, as one could argue that this is the characteristic length that particles are presented with as they move toward and through each passage. The velocity just upstream of each component is taken as the characteristic velocity. The vane and rotor Stokes numbers are listed in Table 3 for each diameter.

Contours of impact and capture efficiency for the vane are presented in Fig. 8 for three different Stokes regimes. The trend in impact efficiency with the Stokes number is clear, with the increased impact efficiency as the nondimensional parameter increases to and above unity. The majority of impacts are located on the aft portion of the pressure surface, and no impacts were found on the suction surface of the vane. The trend with capture efficiency is slightly different, with an initial increase with the Stokes number before dropping to essentially zero at the larger particle sizes (despite the highest impact efficiency at these diameters). This is due to the fact that adhesion scales with $dp2$, while kinetic energy (and elastic strain energy) of the particle scales with $dp3$. The larger particles have enough kinetic energy (and thus elastic strain energy) to overcome the adhesion forces. Figure 9 provides a quantitative perspective of the impact efficiency trend with particle size. Because the deposition model allows for rebounds and multiple impacts, it is possible to obtain an impact efficiency greater than 100%. Unfortunately, this obscures the quantitative trend with Stokes. To resolve this, particles' first impacts were recorded separately during the deposition simulations. This data is included on the plot in Fig. 9 and shows the rise in impact efficiency with Stokes, asymptotic around 100%. The curve reaches this asymptote between 1 < St < 10, suggesting that at least for the vane the chosen characteristic velocity and length scale were reasonable. The average capture efficiencies for the vane are also plotted in Fig. 9. As observed in the contour plots, there is an initial rise in capture efficiency as Stokes increases, followed by a sharp decline for the largest diameters due to their higher kinetic energy. This is especially true for the first impact data, which showed no predicted deposits on the first impact for the six largest diameters. (Note: These data points were set to 10−2 instead of zero to fit on the logarithmic plot.)

The same impact and capture efficiency contours are shown for the blades in Fig. 10. Note that the values for the chosen Stokes regimes have been shifted to more relevant values for the blade domain. A similar trend in impact efficiency is observed, with the increased efficiency with Stokes. While it may be difficult to discern from the summed contours, there is a spatial trend of impacts on the pressure surface with increasing particle size, where the smallest particles impact closer to the hub, the intermediate particles strike near midspan with some impacts near the tip, and the largest particles only impact near the tip on the pressure surface. This pattern is quantified in Fig. 11 where the average impact location as a percentage of the span is plotted against the Stokes number. Tabakoff et al. [4] found a qualitatively similar trend, and attributed the effect to larger particles being centrifuged outward more quickly than their smaller counterparts. The monotonic trend of radial migration is interrupted after St > 4. This is due to the ballistic trajectories of the largest particles, which strike the rotor leading edge and do not travel enough distance through the passage to be centrifuged further. Rebounds from the rotor case also contribute to the trend.

There is an apparent chordwise trend in impact locations with the increased Stokes number. For the low diameter particle range, impacts are spread over the whole pressure surface from just downstream of the leading edge to the trailing edge. This spreading is expected as the smaller particles more readily follow flow streamlines and are easily redirected from their path. The medium particle size range has impacts shifted slightly aft on the pressure surface, as well as focusing near midspan and midchord. The focusing of impact locus is expected as particles become more ballistic at larger diameters. A local convexity on the blade pressure surface is the main contributor to the distinct group of impacts in this region. The impacts of the largest particles are located either aft on the pressure surface near the tip, or on the suction surface near the leading edge. These suction surface impacts are caused by the low velocity of the larger particles, this effect will be discussed in more detail in the unsteady results section. The capture efficiency distributions in Fig. 10 generally mimic the impact patterns with a few subtle differences. For the midsized particle group, there is a lower amount of deposits on the blade tip relative to the rest of the pressure surface. Considering the blade surface temperatures in Fig. 7, which give an idea of the particle temperatures impacting different regions, the tip does not seem cooler than the hub region. Thus, it is likely that the reduced deposits in this region are due to higher particle impact velocity or angle. The model predicted a very low amount of deposits for the largest particles, despite the significant amount of impacts. As in the case of the vane, this is due to the mismatch in scaling of adhesion (scales with $dp2$) versus elastic strain energy (scales with $dp3$).

The single and multiple impact efficiencies are plotted for each diameter in Fig. 12, and show a trend with Stokes number similar to that observed for the vane. Considering the first impact data, there is an initial rise in impact efficiency with the Stokes number for the four smallest particle sizes, before an almost constant value is reached. The multiple impact results showed an increase in impact efficiency for only the seven larger particle sizes, as the smallest three diameters had high sticking efficiencies often leading to deposition on first impact. The average capture efficiencies exhibit the same initial rise with increasing Stokes number, due to the rising impact efficiency. Starting with the fourth particle size, the capture efficiency curve begins to deviate from the impact efficiency as larger particles are able to overcome adhesive forces with their higher kinetic energy. Considering the first-impact deposit data, the same trend observed for the vane is apparent, with a sharp drop in first-impact capture efficiency for larger particles. The overall capture efficiency for the largest five particle sizes remains rather level, due to the increased impact efficiency at these diameters.

### Unsteady Results.

Impact and capture efficiencies for the vane from the unsteady case were found to be very similar to those from the steady case. This may be expected, as the vane domain is not significantly affected by the downstream rotors and the flow field remains rather steady. As an example of the similarity, Fig. 13 compares the predicted vane impact efficiency from the steady and unsteady simulations. The two plots are almost identical, with the increasing then asymptotic trend with the Stokes number already discussed. Capture efficiency on the vane surface was also very similar in magnitude and distribution as that of the steady case, and thus is omitted for conciseness.

Figure 14 shows the particle distributions at the vane outlet plane for the three different Stokes regimes. The distributions are representative of both the steady and unsteady simulations, and will help explain some of the differences observed between the two cases. It is clear that the lowest Stokes number range has the most spread out distribution, with most of the outlet plane covered by at least some particulate. This is due to the small particles' ability to follow the flow streamlines through the passage, and thus exit the vane passage in a similarly distributed manner as they entered it. Particles in the two larger Stokes ranges do not exhibit this behavior, but instead are concentrated into bands that coincide with the vane wake. This pattern is formed because larger particles that hit the pressure surface of the vane either rebound at a shallow angle or even strike the vane again, leading to trajectories at the exit that coincide with the vane metal angle. Larger particles that avoid the leading edge around the suction surface (or strike the leading edge and rebound around the suction surface side) are not sufficiently turned by the flow in the passage and often impact the adjacent vane's pressure surface near the trailing edge. These two phenomena result in vane outlet distributions for larger particles that coincide with the vane wake. In the mixing plane, the vane wake is smeared circumferentially by averaging, and particles are redistributed randomly in the circumferential direction. In the unsteady simulation, the vane wakes are preserved and convect into the blade domain. This is one of the major differences between the two simulations, and plays a roll in some of the differences observed in deposition predictions. The smaller particles might be expected to have similar behaviors in the two simulations, as they are well distributed at the vane outlet in both the unsteady and mixing plane cases. The larger particles, however, are congregated in the wake in the unsteady simulation, while in the mixing plane case they are redistributed and convected by fluid with the average velocity.

The impact and capture distributions on the blades from the unsteady case bear many similarities to those in the mixing plane case, as shown in Fig. 15. At the lowest Stokes numbers, particles impact the pressure surface near the hub in a similar fashion to the steady simulation, and are distributed from leading edge to trailing edge in the chordwise direction. Considering the medium Stokes regime results, the impact pattern shows a band of impacts near the hub over most of the chord with a concentration of impacts at approximately midchord, as well as another band of impacts near the tip. The distributions from both the small and medium sized particles show a distinct lack of impacts from the hub to 5–10% span, all along the chord. While it is tempting to attribute this to a secondary flow, the main cause is a geometric one: the rotor hub moves in on radius relative to the hub at vane–rotor interface plane. Nevertheless, the passage vortex was captured by both the steady and unsteady simulations, and its contribution to this pattern cannot be ruled out.

The steady simulation results share all of the above qualitative features. A difference between the two cases, however, is a streak of impacts in the unsteady case that runs diagonally from the center of the hub band (60% wetted distance on PS) across to the end of the tip band near the trailing edge (90% wetted distance). This pattern can be explained by considering the particle distribution at the vane outlet in Fig. 14. The distribution in the center of the figure shows particles primarily congregating in the vane wake, with a distinctive curved shape. Examining unsteady particle tracks confirmed that the particles maintain this shape to some extent while traveling through the blade passage, before impacting the blade and producing the pattern seen in Fig. 15. The mixing plane simulation results do not show this pattern on the blade surface, as the wake is smeared out circumferentially and particles are redistributed randomly in the circumferential direction as well. Thus, it is clear that while the mixing plane simulation can reproduce the general impact trends for this particle size range, the smearing effect of the interface can cause the model to miss out on discrete features that the unsteady simulation is able to capture.

Perhaps the most striking difference between the steady and unsteady cases is the very large number of impacts just downstream of the leading edge on the suction surface for the largest Stokes group. While there are still impacts on the pressure surface near the tip, these are swamped in comparison by the suction surface impacts. Note that the contour levels are capped at 1.5% so that comparison with the steady contour results is possible. To explore these suction surface impacts, a midspan slice with finite thickness is extracted and all particles outside of this band are blanked out. The result is Fig. 16, which shows locations of 10 and 45 μm particles at four different time steps, with particles' velocity magnitude indicated. In the 10 μm case, particles are accelerated as they leave the vane passage and are able to deviate to some degree around the rotor blades, preventing suction surface impacts. The larger particles exit the vane domain with a lower velocity, having struck the vane pressure surface multiple times on average. These particles are also in the lower velocity vane wake fluid and thus are not reentrained to the “average” vane exit velocity. Unlike the smaller particles, they are not able to deviate from the oncoming rotor blades and instead the blades effectively “chop” the stream of larger particles as they rotate by. These larger particles are less prone to depositing and thus rebound multiple times on the rotor blade suction surface as the blade continues to rotate toward them. An arrow indicating the magnitude of the rotors' rotational velocity is located below the colorbar in each figure, and shows that the larger particles are indeed slower than the rotating blades while the smaller particles are able to accelerate and avoid the blades.

Suction surface impacts were also observed in the steady mixing plane case for the larger particles; however, the number of impacts is much lower than for the unsteady case. The difference can again be explained by considering the particle distribution at the vane outlet for these large particles. These particles find themselves in the slower moving wake fluid, which in the unsteady simulation is preserved through the interface. Using a simple velocity triangle, it can be shown that fluid with a lower absolute velocity has a lower effective incidence angle for the blade. This is similar to the phenomenon of hot streaks being transported to the rotor pressure surface due to the Kerrebrock and Mikolajczak effect [17]; however, in this case the effect is the opposite with the slower moving fluid directed toward the blade suction surface. It is argued that the larger particles concentrated in the slower moving vane wake fluid have a higher propensity to strike the blade suction surface than in the mixing plane simulation, where particles are redistributed into fluid with an average velocity. When the particles do strike the suction surface, there is a very low rate of sticking, and thus the particles impact multiple times leading to the large amount of impacts shown in the contour. It should be noted that significant suction surface deposits or fouling might not be common in practice, as particles of this size are not usually found in the turbine (large particles are generally broken up by the compressor before reaching the hot section). Additionally, if these large particles were present, there would be a considerable amount of erosion due to their size and number of impacts. Thus, any deposits that were to form in this region would likely be removed by erosion, an effect which is not implemented in the current particle–surface interaction model. Also note that no suction surface impacts were predicted in Ref. [25], as the maximum particle size investigated in that study was 15 μm and the flow conditions were set to laboratory conditions.

Impact and capture efficiencies for the blades from both the steady and unsteady simulations are compared in Figs. 17 and 18. Considering first the impact efficiencies, it is apparent at first glance that the overall trend with the Stokes number is similar between the two cases. Looking closer at the values, the average impact efficiency is almost identical for the first two particle sizes (1 and 3 μm). From the third diameter (5 μm) and larger the unsteady case generally predicted higher impact efficiencies. This is especially true for the largest particles, which experienced multiple impacts on the suction surface. These three particle sizes are included in the vane outlet distribution shown in Fig. 14 for the smallest particles. The individual distributions for each diameter show that the two smallest particle sizes are generally well spread out throughout the passage, while at the 5 μm size particles are beginning to congregate in a tight band. As already discussed, the outlet distributions (and corresponding local flow velocities) for the two smallest particle sizes are a better match to those produced by redistribution at mixing plane interface. It thus makes sense that similar results would be observed for these particle sizes between the two cases. The larger particles, however, have quite different distributions and local flow velocities entering the blade domain, resulting in discrepancies in average impact efficiency values. The capture efficiencies plotted in Fig. 18 reflect these trends, with a close match for the two smallest particle sizes and generally higher values predicted by the unsteady simulation for particles larger than this.

Another important aspect to consider is the effect of rotor inlet temperature, which is a circumferentially averaged profile in the mixing plane case and a nonuniform profile that varies with time in the unsteady case. Figure 19 shows how the blade temperatures can change throughout a vane-passing period. Specifically, this difference is important to deposition mechanics if it affects the particle temperature upon impact, as the particles' mechanical properties are a function of temperature. As such, the average particle impact temperatures for each of the cases are plotted in Fig. 20. The average blade impact temperatures do show some difference between the two cases, with the unsteady case generally higher than the mixing plane temperatures. Differences in average temperature of approximately 50 K for small and midsize particles and up to 125 K for the larger particles are observed, in a regime where the particle yield stress is quite sensitive to temperature. To determine the effect that this discrepancy had on the predictions of deposition by the model employed, a sticking efficiency is calculated by dividing the capture efficiency by the impact efficiency at each diameter. The result is plotted for the blade from the steady and unsteady simulations in Fig. 21, and shows an increased sticking efficiency for the unsteady case over the majority of particle sizes. While temperature is not the only parameter that would cause this increase, it is bound to contribute to it. This comparison provides another example of differences between the predictions obtained from steady and unsteady methods, which can and do affect the overall simulation of deposition in a turbine stage.

## Conclusions

Impact and deposition distributions in a high-pressure turbine stage were evaluated using both a steady mixing plane method and an unsteady sliding mesh calculation. Both simulations predicted very similar results for the vane, with a clear trend of increasing impacts with the increased Stokes number. The majority of impacts were located on the aft portion of the vane pressure surface in both cases. Capture efficiency increased initially with Stokes, but decreased rapidly after St > 1.5 despite the increased impact efficiency. This was attributed to the larger particles' increased kinetic energy and ability to resist adhesive forces. Larger particles were found to rebound on the vane surface multiple times, resulting in a reduced inbound velocity into the rotor domain.

The two methods predicted similar general trends for the blade domain, with some differences. Blade impact distributions in both cases exhibited an outward radial shift with the increased Stokes number due to larger particles centrifuging out toward the blade tip. This phenomenon has been observed in previous studies. Particle impacts and deposits were predicted on the blade pressure surface for the smallest and midsized particles, over much of the chord. The distributions were especially similar for the smallest particle group. For both the steady and unsteady cases, the largest particles were found to strike the suction surface of the blade, due to their low inbound velocity into the blade domain. While the results from each simulation share the characteristics described above, several differences were observed and are summarized below:

1. (1)

The impact and capture distribution on the blade for the medium particle size group exhibited a distinctive pattern in the unsteady case which was not observed for the steady case. The difference is caused by the distribution of particles entering into the blade domain, which in the steady case is redistributed circumferentially while in the unsteady case is preserved in the shape of the vane wake.

2. (2)

The unsteady simulation predicted a much higher percentage of suction surface impacts for the largest particles. This was explained as the particles having a lower absolute velocity in the preserved wake, leading to a lower incidence angle to the blade and thus a tendency toward impacting the suction surface.

3. (3)

The average impact efficiency values predicted by the two methods were almost identical for the two smallest particle sizes, after which the unsteady simulation generally predicted higher efficiencies. This was again attributed to vane outlet distribution, where the two smallest diameters showed locations spread throughout the passage while by the third diameter particles had begun to congregate in the vane wake. In the mixing plane method, particles are redistributed circumferentially, and thus it follows that the two methods would match for the particle sizes where particles are more spread over the blade inlet.

4. (4)

Average particle temperature on impact was also compared, with the unsteady case showing generally higher blade impact temperatures due to the effects of the hot streak. This is a suspected contributor to the increased sticking efficiency calculated for the unsteady case, and is another example of how the circumferential averaging of the mixing plane calculation affects deposition predictions.

In summary, it seems from this comparison that similar trends in particle impact and capture distributions can be obtained from a steady mixing plane calculation, which is a computationally cheaper option than the unsteady alternative. However, several differences were observed between the two methods which were generally caused by the smearing of the vane wake and the redistribution of particles. The authors thus recommend that care be taken when choosing an analysis method for investigating particle behavior within a turbine stage. Unsteady methods should be employed if possible as they more accurately represent the turbine stage environment; however, depending on the size of particles being simulated and their subsequent distribution at the vane exit, a mixing plane simulation may provide sufficiently accurate results.

## Acknowledgment

This work was sponsored by a grant from the U.S. Department of Energy (NETL) with Dr. Seth Lawson as program manager. Computational resources were provided by the Ohio Supercomputer Center.

## Nomenclature

• fm,i =

mass fraction of diameter i

•
• Lc =

characteristic length scale in Stokes number

•
• Ncorr =

corrected speed, $Ncorr=NphysTref/T$

•
• Ndep,i =

number of deposits at diameter i

•
• Nimp,i =

number of impacts at diameter i

•
• Ninj,i =

number of particles injected at diameter i

•
• Pt,in =

inlet total pressure

•
• Pexit =

exit static pressure

•
• St =

Stokes number

•
• Tt,avg =

average inlet total temperature

•
• Tt,max =

maximum inlet total temperature

•
• Tvane =

vane wall temperature

•
• Uc =

characteristic fluid velocity in Stokes number

•
• εm,i =

mass based impact efficiency at diameter i

•
• εN,i =

number based impact efficiency at diameter i

•
• κm,i =

mass based capture efficiency at diameter i

•
• κN,i =

number based capture efficiency at diameter i

•
• μ =

fluid dynamic viscosity

•
• ρ =

fluid density

•
• ρp =

particle density

•
• τf =

characteristic fluid time scale

•
• τp =

particle relaxation time

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