Particle-turbulence interaction of high Stokes number irregular shape particles in accelerating ﬂow: a rocket-engine model

Metal particles in solid propellants enhance rocket engines performance. An interaction of particles with a high Reynolds number turbulent gas ﬂow accelerating to a nozzle, has not been characterized thoroughly. We study the particle-turbulence interactions in a two-dimensional model of a rocket engine. Two-phase particle image/tracking velocimetry provides the ﬂow velocity simultaneously with the velocities of irregularly shaped inertial particles ( d p ∼ 320 µ m, Stokes St ∼ 70, particle Reynolds number Re p ∼ 300). We reveal the local augmentation of turbulent ﬂuctuations in the particle wakes (up to 5 particle diameters downstream the particle). Despite the low mass fraction, the large response time of the particles leads to an increase of turbulent kinetic energy (TKE) everywhere in the chamber. The increase of local particle mass fraction near the nozzle, due to the mass conservation and converging streamlines, compensates for the dampening eﬀect of the strong mean ﬂow acceleration and further augments TKE at the nozzle inlet. Furthermore, this is accompanied by unexpectedly isotropic ﬂuctuations in the proximity of the nozzle. The phenomenon of the isotropic, strongly enhanced turbulence in the proximity of the engine nozzle achievable with the low mass fraction of high St, Re p particles, can be used to improve the design of solid propellant rocket engines.

where d p is the particle diameter, V p is the particle velocity (bold symbols denote vectors), ν is the fluid kinematic viscosity, and U is a so-called "undisturbed fluid velocity at the position of the particle", which is practically estimated as an interpolation of the surrounding fluid velocity to the position of the particle (e.g. Meller and Liberzon, 2015). The particle relaxation time τ p for small and relatively heavy particles, ρ p ρ f and Re p < 1 is defined as: (2) However, for higher Re p a non-linear drag force correction is required (

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In this work, we study experimentally the effect of large, heavy, and irreg-59 ularly shaped particles on the TKE in a simplified model of a rocket engine. 60 We reproduce the key features of the mean flow: a) acceleration towards the 61 nozzle; b) the shape of the chamber and the converging type of flow through a 62 small nozzle throat; and c) particle sizes that correspond to metal agglomerates 63 reported in the literature. In this flow, there are competing effects of acceler- dimensional velocity field as a proxy of the two-dimensional axisymmetric flow 97 field in a cylindrically shaped rocket engine.

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The chamber was positioned vertically with the main flow direction and the 99 particle motion aligned with the gravitational acceleration (Fig. 1a). The air 100 was supplied by a blower through a converging channel (750 mm above the 101 measurement region). The measurement volume is 1L t × 1.8L t , the lowest edge   (Wadell, 1935;Riley, 1941), the square root of 148 the ratio of the inscribed and circumscribed circles of the particle. The particle 149 distribution cut off below 250 µm because the particle was separated from finer 150 particles with a 250 µm sieve. We sieved small batches (about 10 gram) in a 151 controlled manner to verify fine particles were removed after sieving. The mean 152 sphericity of 56 particles, examined under a microscope, is Ψ m = 0.81 ± 0.1.

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The microscopic images are shown in Fig. 2c, emphasizing random shapes, sharp 154 edges, rough surfaces, cavities, and protrusions.

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In table 1, we present the flow and particle parameters (some are given as 156 the range of values in the chamber) for the two experiments. Using PIV data c) Figure 2: a) Probability density function (PDF) of particle effective diameters dp defined as a diameter of an equivalent sphere and normalized by the effective mean diameter dp,m of 320 µm b) PDF of reciprocal of sphericity Ψ −1 normalized by the mean shericity Ψ −1 m c) alumina particle's picture under a microscope. mean particle velocity, were 30 to 40 ms. The ratio of particle response time to 161 the residence time, τ p /τ r , is between 2 and 4, for the two Reynolds number runs. 162 This ratio explains that particles leave the chamber before they can respond to 163 the air streamwise velocity. 164 Furthermore, we estimate the response of the particles to the spatial acceleration of streamwise velocity using the acceleration time scale, τ a : and the ratio of scales, τ a /τ p . Mean slip due to acceleration of the flow is 165 expected when τ a /τ p > 1. In our flow, however, τ a /τ p 1 is everywhere in 166 the chamber (table 1). Thus, these particles can be characterized in general as 167 "unresponsive" (Hardalupas et al., 1989). It is also important to mention that 168 when τ a /τ p = 1, there is a mean slip for finite-size particles due to the shear 169 across the particle diameter.

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The time scale ratios do not mean that there is insignificant local particle-171 turbulence interaction. Conversely, there is a substantial transfer of momentum 172 between the particulate phase and the turbulent fluctuations of the carrier flow, 173 as will be explained in the following.   175 We follow the procedure previously reported by Khalitov

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In this section we will summarize the main results obtained from the two-  The mass fraction or mass loading ratio, φ, is defined as the ratio of particle

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It should be noted that due to the two round cavities on both sides of the 232 chamber from which particles rebound at high speed, few particles arriving at 233 a large angle to the streamwise direction were excluded from the present data.  respectively) at Re L1 (at the same distances from the nozzle as in Fig 5).  observe somewhat increasing fluctuating components in Fig. 5a-b, however, the 308 most prominent change is in the field of Reynolds stresses in FIg. 5c. Clearly, this increase also affects the production terms shown in Fig. 6a-b. However, because 310 of the strong acceleration, the negative TKE production terms are dominant 311 and increase towards the nozzle entrance. 312 Figure 6: Profiles of turbulent production terms: a) uv ∂U ∂y , b) uv ∂V ∂x , c) u 2 ∂U ∂x , and d) v 2 ∂V ∂y , respectively, for the un-laden case (filled symbols) and laden case (empty symbols) and for Re L1 , at different distances from the nozzle. The symbols and colors legend is the same as in Fig. 4. a relatively small number of heavy particles. Therefore, in the following we 314 present a more insightful, local analysis around the particles and as a function 315 of distance from the particles.

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To reveal the local effects of particles on the turbulent flow, we use condi-317 tional sampling in the following form: we divide the instantaneous PIV/PTV 318 fields in the particle-laden cases, along the centerline, into small control volumes 319 of 20 × 20 mm. From each sub-volume, we conditionally sample the turbulent 320 fluctuations depending on whether the sub-volume in a given flow realization 321 contains particle(s) and marked it as a region B (particles) or C (no particles) 322 (as schematically marked in Fig. 1c). For the sake of reference we compare with (i.e., in the Lagrangian frame of reference attached to the particle center). The 346 normalized r.m.s. of streamwise fluctuations in the particle wake is plotted 347 versus the streamwise distance from the particle, normalized by the particle 348 average diameter, d p (downstream, in the direction of motion of the particle). 349 We observe significantly higher turbulent fluctuation within a region of at least   Root-mean-square of streamwise fluctuations u for the particle laden cases at two Reynolds numbers as a function of distance from the particle, normalized by the corresponding r.m.s of the unladen flow case. The distance is measured along streamwise direction from the origin attached to the particle, xp, normalized by the mean particle diameter.   The combination of the high particle-fluid relative velocity, slow response 386 time, and rapid acceleration of the air mean flow leads to substantial turbu-387 lence argumentation, mostly in the streamwise component. The particle-related 388 mechanism for these Stokes numbers range from St > 75 and the particle 389 Reynolds numbers Re p > 300 were termed in the literature as "vortex shedding" 390 mechanism Hetsroni (1989); Balachndar and Eaton (2010). In the present case, 391 in particular, the particles move slower than the carrier fluid flow. Thus, a tur-392 bulent wake, a region where the flow slows down, is in the direction of relative 393 velocity, which defined as V r = U f − V p . The wake region "downstream" in 394 respect to the particle. It means that the next particle position will be inside 395 the wake of the particle itself at the previous time instant. More detailed local 396 flow around the particles analysis shows that on average, a local reduction of 397 the air flow velocity in the particle wake is pronounced up to five particle diam-398 eters downstream from the particle. We presented the comparison of the local 399 turbulence augmentation in the proximity of the particles and compared it to 400 the turbulence augmentation in the entire region of interest. We also demon-401 strated the peculiar situation of streamline convergence leading to an increase of 402 the local mass fraction, streamwise acceleration, and particle-turbulence inter-403 actions. First, the streamwise average velocity acceleration significantly reduces 404 the streamwise turbulent fluctuations. Second, inertial particles of irregular shape create streamwise fluctuations in their wakes due to the vortex shed-406 ding and compensate the mean flow acceleration effect. In addition to the dra-407 matic increase of TKE, the particle wakes are unexpectedly more isotropic than 408 the surrounding turbulence. Furthermore, the particle wakes are much more 409 isotropic as compared to the unladen flow case with the mean flow acceleration.

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In our experiment, the particle mass fraction is monotonically increasing 411 towards the nozzle due to mass conservation and streamlines convergence (see 412 4). Nevertheless, in the measurement region of interest, the mass fraction is in 413 the two-way coupling regime and we did not observe any clustering of particles.

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Our conclusions are therefore, limited to the dilute two-way coupling regime.

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In respect to the aforementioned particle-flow dimensionless parameters, we 416 have estimated that P a > 10 5 in the entire measurement region. As suggested 417 by Tanaka and Eaton (2008), it falls in the range that predicts an increase in 418 turbulence. The length scales ratio, d p /L (Gore and Crowe, 1989) is lower than 419 0.1 and predicts attenuation, however our particles lead to augmentation. This 420 discrepancy is likely to reflect the fact that the main effect is due to the particle 421 wakes that are five times larger than the particle effective diameter.

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This empirical work does not improve significantly our ability to predict the we demonstrate that irregularly shaped particles, moving more slowly than the 435 surrounding fluid, will create streamwise fluctuations that lead to isotropic tur-436 bulence regions with important consequences for mixing and transport flux. 437 We can infer that both the turbulent mixing and combustion rates could be