Gas phase method

The flowfield inside the IPS is simulated using an in-house CFD code named MAP (Multi-purpose Advanced Prediction code for fluid dynamics). The detailed description and validation of the code can be found in (Ning, 2014). It solves the integral form of governing equations discretized in space using the cell-centered finite-volume method. The advective fluxes are evaluated using the LDFSS scheme (Edwards, 1997) coupled with MUSCL interpolation to obtain high-order spatial accuracy. The diffusive fluxes are solved using traditional central differencing. The one-equation Spalart-Allmaras turbulence model with near-wall correction (Ning and Xu, 2001) is used for turbulent flows, which is discretized and solved in a coupled manner with the mean flow equations. Message Passing Interface (MPI) is used to parallelize the code. The discretized system is solved using the matrix-free Gauss-Seidel algorithm (Luo et al., 1998).

Concerning the boundary conditions, total pressure, total temperature, and flow angles are specified at the inflow boundary, whereas static pressure, considering radial equilibrium, is specified at the outflow boundary. Then, the 1D non-reflecting boundary condition method is applied. Non-slip and adiabatic conditions are employed for the solid walls. A mixing plane model based on flux conservation (Du and Ning, 2016) is used at interfaces between upstream mesh blocks and downstream mesh blocks with different pitches.

Particle phase method

In particle modeling, the discrete particle model was used, in which each sand particle is treated individually as a mass point. Neither the effect of the sand particle phase on the gas phase nor the effects between sand particles are considered. The drag force due to gas is regarded as the most important force, and other forces such as buoyancy force, pressure gradient force, Basset force, virtual mass force, Saffman force, and Magnus force are neglected (Lei, 2020).

The equation of motion of a sand particle in a relative rotating coordinate system is established as:

where mp is the mass of the sand particle, W→p is the relative velocity vector, D→ is the drag force vector, Ω→ is the rotation angular velocity vector of the coordinate system, and X→p is the position vector of the sand particle. The drag force vector D→ is computed as:

where U→ is the relative velocity vector between the sand particle and air, ρg is the density of air, Cd is the drag coefficient, and S is the characteristic area (taken as πd2/4, where d is the diameter of the sand particle).

When a sand particle collides on a solid wall, it is assumed unbreakable and its rebound velocity vector is computed using empirical correlations given by Niu (2017). Restitution ratios of normal direction en and tangential direction et, and their standard deviations σ(en), σ(eτ) are written in the following form:

where Vn,rebound is the normal velocity after rebound, Vn,impact is the normal velocity before rebound, Vt,rebound is the tangential velocity after rebound, Vt,impact is the tangential velocity before rebound, θ is the angle between the incident velocity vector and the wall-normal direction in radian. Values of empirical parameters. αi, γi,εi and λi of different particle diameters are shown in Table 2. Furthermore, linear interpolation or constant extrapolation is used for particle diameter within or out of range [100 μm, 700 μm].

An irregular particle drag coefficient model is used to predict a particle’s trajectory more accurately than the simple spherical drag model. The Corey Shape Factor (CSF) proposed by Connolly et al. (2020) is applied to characterize the shape of a particle. CSF is bound to 0 < CSF ≤ 1, while CSF = 1 indicates an aspect ratio of 1 in all three orthogonal planes (sphere), and CSF close to 0 indicates a very thin disk or plate. Their particle shape corrections are expressed as follows:

where Cd∗ is the drag coefficient of a spherical particle given by the following correlation:

where Reynolds number of spherical particle Rep∗ and Reynolds number of irregular particle Rep are associated by Cshape and fshape. Rep is calculated as:

where μg is the viscosity of air.

Irregular particles undoubtedly have lighter mass than spherical particles with the same maximum outer diameter. A new parameter Z is defined as the mass ratio of irregular particle to spherical particle. Apparently, Z = 0 when CSF = 0, and Z = 1 when CSF = 1. Statistical data shows that for sand particles, Z is in the range of 0.50 to 0.53 (Duffy and Shattuck, 1975b), while the averaged CSF is about 0.55 (Connolly et al., 2020). Thus, we propose the following approximation, which can be seen as a first-order linear fitting of Z(CSF):

Using this correction, the mass of a sand particle is calculated by:

In this paper, the solid wall material is taken as an aluminum alloy. Sand particles are selected as quartz, and the density is 2,650 kg/m³, satisfying both the requirements of diameter (coarse Arizona road dust, known as AC-Coarse sand) and CSF distribution (Connolly et al., 2020), which is listed in Table 3. During computations, sand particles are randomly released at the inlet plane with the axial velocity equaling 80% of the gas velocity. When a sand particle passes through the interface, its radial position remains unchanged while its circumferential position is reset randomly. The vaneless scavenge flow path domain's length is twice the original length, and its sector angle is 2 degrees.

During the experiment, the uncertainty of separation efficiency due to the incomplete collection of sand particles (Barone et al., 2015; Xu and Chen, 2021) and the ambiguity of sand particle’s diameter (Duffy and Shattuck, 1975a) would be about 1%. As for computation, in addition to errors due to the discretization error of the governing equation and the error of the empirical model, there are also random errors caused by random parameters, such as the diameter distribution of sand particles, release position, and rebound effects. Considering the impact of these random factors, more than five calculations are conducted to get the mean value and standard deviation for each set of calculation input parameters.

Computation input parameters

Sand particles’ diameter distribution is crucial to gain accurate computation results because total separation efficiency is essentially the weighted mean of separation efficiency of different diameters. A straightforward way to determine particles’ diameters is by randomly setting each particle’s diameter according to a given probability distribution. However, the results would converge to a specific value only if a sufficient number of particles were simulated, which would be beyond the available computation resources. Another easier but reasonable way is to limit the particle diameters to finite representative values and then calculate total separation efficiency as the weighted mean of separation efficiency at these representative diameters. Computation results using mid-values of subintervals as representative diameters conform to the experiment data very well (as shown in Figure 8) while using the upper bound or the lower bound of subintervals as representative diameters would produce about an error of 3% in separation efficiency for AC coarse sand.

Figure 8.

The separation efficiency calculated with different numbers of sand particles.

Results in Figure 8 also show that the separation efficiency can’t be predicted accurately even when the number of particles reaches 10⁶ using the random method, while 10⁴ particles seem enough for the representative method to get an adequately accurate result. Hence, the representative method using the mid-value of the subintervals and particle number of 10⁴ is adopted hereafter.

Figure 9 shows that as a representative value of CSF varies from 0.01 to 1.0 (nondimensionalized to [−1,1], the same hereafter), the particle's inertia increases, and thus separation efficiency rises. Both results of the random and representative methods using CSF = 0.5 agree well with the experiment data.

Figure 9.

The variation of separation efficiency for some input parameters.

While mean particle release velocity magnitude varies from zero to twice the gas velocity magnitude and particle release velocity direction randomly varies from axial to radial with the average direction of axial both have little impact on separation efficiency, particle release position does have a significant effect on separation efficiency, as shown in Figures 7 and 10. The non-uniformity of separation efficiency mainly occurs along the radial direction, affected by the relative position of the scavenge flow path and the core flow path. The higher separation efficiency would be achieved when the sand particle was released closer to the shroud. Besides, there is a certain degree of non-uniformity in the circumferential direction near the leading edge due to the existence of the pre-swirl blades. The difference in separation efficiency between different release positions is as high as 33.35%, as shown in Figure 7, indicating special attention should be paid.

Figure 10.

Separation efficiency distribution when sand particles are released at different radial positions.

The influence of the bullet-shaped intake device's existence has been proven to be small, as shown in Figure 11. Although the distribution of sand particles at the blade's leading edge deviates from a random distribution in Figure 12, the final calculated separation efficiency is only about 1% higher. Also, the length of the scavenge flow path of the IPS is varied to identify its effects on separation efficiency. When the length of the scavenge flow path is taken as 0.6∼2 times its physical length, there is no significant difference in the flowfield (as shown in Figure 13). Thus, no significant impact on separation efficiency has been observed.

Figure 11.

Influence of some parameters on separation efficiency.

Figure 12.

Trajectory of sand particle considering intake device.

Figure 13.

Meridional averaged flowfield with different scavenge flow path length.

To evaluate the deviation of separation efficiency caused by the variation of SCR, computations with SCR varying from 0.1 to 0.24 (nondimensionalized to [−1,1] in Figure 9) are carried out. During simulation, the mass flow rate of the core flow path is kept constant while the mass flow rate of the scavenge flow path is changed. Results show that the separation efficiency positively correlates with SCR, but the variation range is insignificant. The suction capacity of the core flow path becomes weaker when SCR is increased. Thus, more sand particles move into the scavenge flow path, resulting in higher separation efficiency. The variation range of the separation efficiency is below 1% when SCR deviates ±5% near its design value (16.9%).

Computation methods

The turbulence model plays an important role in determining the pattern of the flowfield, especially in the separation region. The improved Spalart-Allmaras model, SST model (Menter, 1994), and k−ω model (Wilcox, 2008) are applied separately in the flowfield computation. The meridional averaged flowfield and the size of the separation region are compared in Figure 14. There is a significant difference in the width of the separation region simulated by RANS with different turbulence models. Still, the separation efficiency has no significant deviation, as shown in Figure 11.

Figure 14.

Meridional averaged flowfield simulated using different turbulence models.

The particle–wall collision model is another critical part of the particle’s trajectory simulation. Calculations using models proposed by Niu (2017) and Tabakoff et al. (1996), with and without considering the effect of the randomness of rebound, are performed separately. The empirical relationship of collision velocity recovery coefficient eV, incident angle recovery coefficient eβ and their standard deviations provided by Tabakoff et al. (1996) is expressed as follows:

where V is the velocity of particles relative to the wall, β is the angle between the relative incident velocity vector and the wall in the angular system, the subscript i represents status before collision, and r represents status after collision.

The separation efficiency calculated based on the above two different probability collision models only deviates within 1%, as shown in Figure 11. If the average collision model (without considering standard deviation) is used, the calculation accuracy will degrade, as pointed out by Hamed et al. (1995). Specifically, the separation efficiency of sand particles with a diameter bigger than 200 μm is significantly higher. When the sand particle’s diameter reaches 1,000 μm, the separation efficiency is about 4% higher. Because AC coarse sand is actually relatively fine (having an average diameter of 40 μm, as shown in Table 3), the separation efficiency is not significantly affected eventually. Different considerations of collision models lead to no significant deviation in separation efficiency, as shown in Figure 7.

Changing the sector angle of the scavenge flow path computation domain from 2° to 360° doesn’t significantly impact the flowfield and separation efficiency when the mixing plane method is used between interfaces. To further consider the influence of circumferential non-uniformity, such as the wake of the pre-swirl blade, the sector angle of the scavenge flow path computation domain is set to be the same as the pitchwise angle of the pre-swirl blade (20°) and direct interpolation method is used to transfer the flowfield data, resulting in 1% reduction of separation efficiency (as shown Figures 7 and 11). Thus, the data transfer method between the interfaces has little impact on separation efficiency in this IPS simulation.