Due to a lack in experimental data for C d of VBP in compressible flows, it is necessary to adopt existing C d data for geometrical bodies, suitable for maximum range calculation of VBP in compressible flows. Because of this, it is also important to determine C d as a function of M a. During vulcanian, subplinian and plinian volcanic eruptions, VBP are expelled with very high velocities, and so the flow can be considered compressible ( M a > 0.7). Nevertheless, because the range of the VBP depends on C d, in order to establish safety zones by calculating the maximum possible range of VBP during volcanic explosions, the lowest C d value should be considered. The range of C d (0.62–1.01 in these samples) depends mainly on the shape and texture of the VBP, and this should be considered in a ballistic study. The identified constant C d values of every sample are shown in Table 1. For this reason, the results obtained for the fragments used in these experiments are also valid for larger VBP in subsonic flows. As a consequence, C d values are also independent from VBP' diameter at a given velocity. In the case of incompressible flows it can be considered constant. Typical VBP ( D > 10 cm) moving at v > 20 m/s, R e > 10 5 > R e critical, are in a range of R es where the C d is independent from R e and is only a function of M a. It is noticeable the little variation of the final value of C d for larger R es. This behavior is also similar to the case of spheres and cylinders with R e > R e critical. In Figure 1 a general tendency to constant values is observed (little variation of C d values) after the abrupt decrease. Measured drag coefficient values as a function of R e for different volcanic samples from Popocatépetl volcano. Mach number ( M a = v/c where c is the fluid's sonic velocity in the fluid) reflects the effects on the flow due to compression of the fluid and is important in compressible flows. R e is important for M a < 1 and relates the relative importance of viscous versus inertial force and is defined by R e = vD/μ where v is the VBP velocity, D diameter and μ is the kinematical viscosity of the fluid. Experimental results indicate that C d varies with two dimensionless parameters that indicate flow regime: Reynolds ( R e) and Mach ( M a) numbers. Drag Coefficient: Experiments and Determination The results were calibrated with observational data from Popocatépetl volcano (Mexico). This study reports drag coefficient values measured in a subsonic wind tunnel using actual volcanic particles. Therefore, experimental studies to determine C d for volcanic particles are crucial. For instance, ranges calculated with C d = 0.65 are 130–180% of those calculated with C d = 1 but using C d values for spheres resulting ranges are 250% of those calculated using C d = 1. Paucity in experimental data for VBP induces uncertainty in the results' accuracy. However, no experiments or observations confirmed such low C d values for VBP. Instead, they used C d values for spheres, and the calculated VBP velocities decreased. claimed that initial velocities for VBP calculated with those data were exaggerated taking into account the expected pressure conditions during volcanic explosions. , Nairn and Self, and Fagents and Wilson used C d values of cylinders after the free-fall experiments of pyroclasts carried out by Walker et al. , and Bower and Woods used C d = 1 Steinberg and Babenko and Steinberg and Lorenz used C d = 0.65 for subsonic flow and C d = 1.25 for supersonic flow Fudali and Melson used C d = 0.8. Some VBP studies considered a constant C d: Sherwood, Self et al. Unfortunately, the existing C d values were only for geometrical bodies and no data existed for volcanic particles. The latter is proportional to the drag coefficient ( C d), whose value is determined experimentally. For this task, they used a model based on the main forces acting on the VBP: gravity and drag force. Most of them calculate initial velocities of the projectiles during explosive events taking into account the maximum distance reached from the crater. Several studies describe the movement of VBP. In order to better assess those hazards and protect the people and infrastructure, it is necessary to establish safety areas using a model to calculate adequately the trajectory of these projectiles. VBP represent a hazard to life, property and air navigation due to their impact energies and high temperatures. Volcanic explosions throw rock fragments known as volcanic ballistic projectiles (VBP) that follow parabolic trajectories modified by a drag force before impacting the Earth's surface.
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