Experimental determination of drag coefficient for volcanic materials: Calibration and application of a model to Popocatépetl volcano (Mexico) ballistic projectiles



[1] This study shows the results of experiments performed in a subsonic wind tunnel to measure drag coefficients (Cd) for volcanic particles emitted by Popocatépetl volcano (Mexico). The results indicate that volcanic projectiles move in a range of Reynolds number (Re) values where Cd is independent of Re. Drag coefficients of volcanic fragments were found between the values of spheres and cubes with low values. In this study, initial conditions of the ballistic equations were calibrated with the “launching” kinetic energy from ballistics expelled by Popocatépetl volcano during the eruptive event of December 17, 1998. The maximum range of the projectiles calculated with the experimentally measured drag coefficient is in better agreement with the field and observational data reported here than using the values of geometrical bodies as in previous studies. Our study can be useful for definition of safety zones around volcanoes to protect people and infrastructure adequately.

1. Introduction

[2] 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. VBP represent a hazard to life, property and air navigation due to their impact energies and high temperatures. 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.

[3] Several studies describe the movement of VBP [Sherwood, 1967; Fudali and Melson, 1972; Wilson, 1972; Self et al., 1980; Steinberg and Lorenz, 1983; Fagents and Wilson, 1993; Waitt et al., 1995; Bower and Woods, 1996]. Most of them calculate initial velocities of the projectiles during explosive events taking into account the maximum distance reached from the crater. For this task, they used a model based on the main forces acting on the VBP: gravity and drag force. The latter is proportional to the drag coefficient (Cd), whose value is determined experimentally. Unfortunately, the existing Cd values were only for geometrical bodies and no data existed for volcanic particles.

[4] Some VBP studies considered a constant Cd: Sherwood [1967], Self et al. [1980], and Bower and Woods [1996] used Cd = 1; Steinberg and Babenko [1978] and Steinberg and Lorenz [1983] used Cd = 0.65 for subsonic flow and Cd = 1.25 for supersonic flow; Fudali and Melson [1972] used Cd = 0.8. Wilson [1972], Self et al. [1974], Nairn and Self [1978], and Fagents and Wilson [1993] used Cd values of cylinders after the free-fall experiments of pyroclasts carried out by Walker et al. [1971]. Waitt et al. [1995] claimed that initial velocities for VBP calculated with those data were exaggerated taking into account the expected pressure conditions during volcanic explosions. Instead, they used Cd values for spheres, and the calculated VBP velocities decreased. However, no experiments or observations confirmed such low Cd values for VBP.

[5] Paucity in experimental data for VBP induces uncertainty in the results' accuracy. For instance, ranges calculated with Cd = 0.65 [Steinberg and Lorenz, 1983] are 130–180% of those calculated with Cd = 1 [Bower and Woods, 1996]; but using Cd values for spheres [Waitt et al., 1995] resulting ranges are 250% of those calculated using Cd = 1. Therefore, experimental studies to determine Cd for volcanic particles are crucial.

[6] This study reports drag coefficient values measured in a subsonic wind tunnel using actual volcanic particles. The results were calibrated with observational data from Popocatépetl volcano (Mexico).

2. Drag Coefficient: Experiments and Determination

[7] Experimental results indicate that Cd varies with two dimensionless parameters that indicate flow regime: Reynolds (Re) and Mach (Ma) numbers. Re is important for Ma < 1 and relates the relative importance of viscous versus inertial force and is defined by Re = vD/μ where v is the VBP velocity, D diameter and μ is the kinematical viscosity of the fluid. Mach number (Ma = 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.

[8] Measurements of Cd corresponding to different volcanic particles were performed in a C2-00 Armfield Technical Education (England) subsonic wind tunnel. This tunnel allows the variation of wind velocity up to 26 m/s (Ma ∼ 0.08) and directly measure the drag force experienced by an object placed at the center of the test section. Cd is calculated from the wind velocity in the tunnel (u) and the drag force (Fa) measured in the experiment using equation (1):

equation image

where (A) is the cross-sectional area of the volcanic particle, and ρa is air density.

[9] Samples of pyroclastic material expelled by Popocatépetl volcano were selected. Several volcanic fragments were observed in the field and, after visual inspection, six general types of shapes were distinguished (from angular to round shapes and from rough to smooth textures, see Table 1). A sample of every type was randomly picked. These samples were ejected during different explosive events, and were collected at an average distance of 3.3 km north of the volcano's crater, which is not the maximum range. The volcanic particles' cross-sectional area (A) was measured according to the position in which they were placed in the wind tunnel using image processing software (Table 1). Their equivalent diameter (DE) is the diameter of a sphere with same cross-sectional area (DE =equation image). Fragments II and VI are pumice whereas the rest are lithic particles.

Table 1. Cross-Sectional Area (A) and Equivalent Diameter (DE) of Six Volcanic Samples and Three Geometric Objects Used in the Experimentsa
SampleShapeTextureA, cm2DE, cmCd
  • a

    Cd values are the constant values to which Cd tend, measured with Re ∼104 (see Figure 1).

Iangularsmooth surface11.623.850.71
Frontal cube  27.145.871.06
Vertex cube  43.377.430.77
Circular cylinder  22.455.351.23
Sphere  32.376.420.51

[10] Cd for every particle as a function of Re (calculated using DE) is shown in Figure 1. Cd values of the smallest samples (I, II, III, and IV) describe an abrupt decrease at Re ∼ 2 × 104. This decrease is very similar to the case of spheres and cylinders [e.g., Hoerner, 1965] and is caused by the flow-regime change at the boundary layer (from laminar to turbulent flow). The Re value where this change occurs is called Re critical. The Re critical value for VBP (∼2 × 104) is lower than that for spheres and cylinders, possibly due to the rough texture of the VBP. Because of their size, the Res of the largest particles (V and VI) are greater and thus, this strong decrease is not observed.

Figure 1.

Measured drag coefficient values as a function of Re for different volcanic samples from Popocatépetl volcano.

[11] In Figure 1 a general tendency to constant values is observed (little variation of Cd values) after the abrupt decrease. This behavior is also similar to the case of spheres and cylinders with Re > Re critical. It is noticeable the little variation of the final value of Cd for larger Res. Typical VBP (D > 10 cm) moving at v > 20 m/s, Re > 105 > Re critical, are in a range of Res where the Cd is independent from Re and is only a function of Ma. In the case of incompressible flows it can be considered constant. As a consequence, Cd values are also independent from VBP' diameter at a given velocity. For this reason, the results obtained for the fragments used in these experiments are also valid for larger VBP in subsonic flows.

[12] The identified constant Cd values of every sample are shown in Table 1. The range of Cd (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. Nevertheless, because the range of the VBP depends on Cd, in order to establish safety zones by calculating the maximum possible range of VBP during volcanic explosions, the lowest Cd value should be considered.

[13] During vulcanian, subplinian and plinian volcanic eruptions, VBP are expelled with very high velocities, and so the flow can be considered compressible (Ma > 0.7) [Fudali and Melson, 1972; Wilson, 1972; Steinberg and Lorenz, 1983; Fagents and Wilson, 1993; Waitt et al., 1995]. Because of this, it is also important to determine Cd as a function of Ma. Due to a lack in experimental data for Cd of VBP in compressible flows, it is necessary to adopt existing Cd data for geometrical bodies, suitable for maximum range calculation of VBP in compressible flows. For that end, Cd measured data for volcanic particles can be compared with Cd values for geometrical bodies under similar conditions.

[14] Table 1 shows Cd values for spheres, cylinders and cubes (frontal and with the apex facing the flow) at Re ∼104 in incompressible flow, according to measurements made in this study. Comparing Cd values for volcanic particles and geometrical bodies, the four smallest Cd values for VBP (significant for maximum range calculation) fall between the Cd values of the sphere and cube with the apex facing the flow.

[15] Extrapolating this for compressible flows, a weighted average function Cd(Ma) is considered between the Cd values of the sphere and cube. Figure 2 shows the Cd(Ma) functions for spheres and low cubes approximated with polynomials using data from Hoerner [1965], and the weighted average function of these polynomials. This Cd(Ma) starts closely at the lowest measured Cd value (sample VI) and therefore, it reproduces adequately the maximum possible range. Using this Cd(Ma) function, the ballistic equations can be solved using the method described in section 3.

Figure 2.

Drag coefficient as a function of Mach number for spheres and cubes with the apex facing the flow [data from Hoerner, 1965] and a weighted average function of both. This Cd(Ma) starts at the lowest measured Cd value (sample VI) and therefore, it allows to reproduce adequately the maximum possible range.

3. Ballistic Equations

[16] Taking into account gravity and drag force only, the ballistic equation describing the trajectory of VBP expelled from a vent can be expressed in a rectangular coordinate system as follows [Wilson, 1972; Waitt et al., 1995]:

equation image
equation image
equation image
equation image

where x and z are the horizontal and vertical position coordinates respectively, t is time, v = (vx, vz) is the velocity vector of the projectile, u is wind velocity, g is gravity, Cd(∣vu∣) is the drag coefficient expressed as a function of the magnitude of the relative velocity between the ballistic and the wind, ρa(z) is air density as a function of altitude z, A is cross-sectional area of the VBP, m is the mass and ∣vu∣ = equation image. For ellipsoidal VBP the ratio A/m = 3/(2ρbD) where ρb is the ballistic density and D is the geometric mean of three perpendicular diameters. Air density can be calculated using altitude-density data (e.g., U.S. Standard Atmosphere, 1976). Cd(∣vu∣) is calculated from the Cd(Ma) function obtained in this study (Figure 2) and the altitude-air sonic velocity data (e.g., U.S. Standard Atmosphere, 1976).

[17] The ballistic equations mentioned above can be integrated throughout the trajectory using a fourth order Runge-Kutta method as described by Wilson [1972]. The initial position is (0, zo), where zo is the altitude of the vent above sea level, and the initial velocity is (vo cos θo, vo sen θo), where θo is the “launching” angle with respect to the horizontal axis. Trajectory calculation is made during discrete time intervals until z = zf, where zf is the altitude of the landing point. Noticeably, Mastin [2001] incorporated a low drag zone due to the cloud of tephra that move at roughly the same velocity as the large blocks. He used an arbitrary distance from the vent over which drag is reduced.

4. Parameter Calibration With Actual Explosions From Popocatépetl Volcano

[18] On December 21st 1994, Popocatépetl volcano started to erupt and for more than ten years the eruptive activity has been characterized by lava effusion and vulcanian explosions [Delgado-Granados et al., 2001]. During these events, incandescent ballistic fragments have been expelled to various distances from the crater. On December 17, 1998, one of the largest explosions expelled VBP to a maximum distance of 3.7 km from the crater with impact-site altitudes of ∼4000 masl (the altitude of the source is 5000 masl), triggering fires at the volcano's surroundings. The average size of the VBP sampled at this distance was D = 30 cm and density ρb = 2500 kg/m3. Their travel times were measured from videos recorded at the National Center for Disaster Prevention (33 to 40 seconds for VBP with maximum range).

[19] In order to calibrate the initial conditions of the ballistic equation, the “launching” kinetic energy needed by the VBP to reach their maximum range was calculated using the method described above. It was assumed that these VBP were launched with the angle that allows them to reach the maximum range (optimum angle) and minimize their travel times (33 s). Table 2 shows the launching kinetic energy associated to the VBP from Popocatépetl volcano and their maximum range calculated using the drag coefficient values determined in this study. It also shows the results obtained using a constant Cd = 1, and the results obtained by the software called Eject! [Mastin, 2001] for spheres and cubes with the apex facing to the flow. In every case, it was considered the maximum range (calculated using the optimum angle) reached by the VBP in a travel time of 33 s (minimum time observed).

Table 2. Comparison of Launching Kinetic Energy and Maximum Range Calculated Using Different Cd Valuesa
 Maximum Range, mKinetic Energy, × 106 J
  • a

    Values are for this study, Cd = 1.0 and the Eject! code for spheres and low cubes [Mastin, 2001] for the VBP launched by Popocatépetl volcano on December 17, 1998. Observed minimum travel time (33 s), measured average diameter (30 cm) and optimum “launching” angle were used in every case.

This study36601.8
Cd = 1.030002.5
Eject! sphere47501.1
Eject! low cube34302.6

[20] The maximum range calculated with the Cd reported in this study is in better agreement with the value measured in the field than the results obtained using Cd = 1, and the Eject! code [Mastin, 2001] for spheres and cubes with the apex facing to the flow (Table 2). These results allow stating that if the initial conditions of VBP ejection are calibrated adequately from previous expelled VBP and the topography is considered, the proposed method can be used to calculate the maximum range of the VBP under any explosive scenario of a given volcano [Alatorre-Ibargüengoitia et al., 2006].

[21] Further work is necessary in order to experimentally determine the Cd values for volcanic particles in compressible flows (Ma ≥ 1). For that end, it will be necessary to have a larger collection of samples for every shape and texture type, a range of sizes, and tests for Ma = 1–4. In general, no large changes in Cd would be expected, according to what has been stated in this study. However, the effect of vesicles and rugosity on Cd is intriguing. According to field observations, some VBP explode when impacting the ground. This possible effect of shape and texture of VBP should be elucidated, especially because of its hazard's implications. Cd determination under supersonic conditions would show the importance of the differences between the rough, uneven VBP, and smooth, symmetrical geometric bodies in the transition from subsonic to supersonic flow speeds. It is also important to study the influence of the surrounding environment (expanding gas and air) on the motion and throw range of the VBP by reducing drag at the beginning of the trajectory, especially on big explosions.

5. Conclusions

[22] Experiments using a subsonic wind tunnel allowed the determination of a functional relation Cd(Re) for volcanic fragments launched by Popocatépetl volcano. The results indicate that for these fragments in incompressible flows Re critical ∼ 2 × 104 and for Re of flows in which the VBP moves (Re > 105 > Re critical), the Cd values are independent from Re. The range of Cd values depends mainly on the shape and texture of the VBP, a fact that should be considered in a ballistic study. Nevertheless, in order to calculate the maximum possible range that VBP can achieve during volcanic explosions, the lowest Cd value should be considered. This study found that the lowest Cd values for these fragments are between the values of spheres and cubes (with the apex facing to the flow). In the absence of experimental results for Cd values of VBP in compressible flows, a weighted average function Cd(Ma) was considered between the values for the sphere and low cube.

[23] The kinetic energy of VBP thrown during the explosive event of December 17, 1998 was calculated in order to calibrate the initial conditions of the ballistics expelled by Popocatépetl volcano under its present eruption. The calculated “launching” kinetic energy for projectiles with maximum range resulted in 1.8 × 106 J. The maximum range considering the travel time measured in the video of the explosion for the VBP is in accordance with the field data. Therefore, the Cd values experimentally obtained in this study are far more adequate to describe the movement of the VBP and calculate their maximum ranges, than the values for geometric bodies (spheres, cylinders and cubes) or constant values (Cd = 1.0) used in previous studies. The proposed method can be used to calculate maximum ranges under previously defined explosive scenarios, in order to establish safety zones for adequately protecting people from VBP' effects.


[24] The authors are indebted to Carlos Escalante and Joel Carvajal of the Division of Graduate Studies, Faculty of Engineering (UNAM) who kindly allowed us to use the subsonic wind tunnel for the drag coefficient determination experiments; Miguel A. Alatorre Mendieta and Jerónimo Alatorre for help during the performance of experiments; M. I. Roberto Quaas for access to the video collection of CENAPRED for the observation of explosive events at Popocatépetl volcano, and Instituto de Geofísica, (UNAM) for financial support to M. A. Alatorre-Ibagüengoitia. Comments and criticism from anonymous reviewers greatly improved the manuscript. This study was financially supported by Dirección General de Asuntos del Personal Académico, UNAM (grants IN121201 and IN104905).