## Abstract

The drag coefficient of a sphere placed in a non-stationary flow is studied experimentally over a wide range of Reynolds numbers in subsonic and supersonic flows. Experiments were conducted in a shock tube where the investigated balls were suspended, far from all the tube walls, on a very thin wire taken from a spider web. During each experiment, many shadowgraph photos were taken to enable an accurate construction of the sphere's trajectory. Based on the sphere's trajectory, its drag coefficient was evaluated. It was shown that a large difference exists between the sphere drag coefficient in steady and non-steady flows. In the investigated range of Reynolds numbers, the difference exceeds 50%. Based on the obtained results, a correlation for the non-stationary drag coefficient of a sphere is given. This correlation can be used safely in simulating two-phase flows composed of small spherical particles immersed in a gaseous medium.

## 1. Introduction

For evaluating the motion of a solid object in a gaseous medium, one has to know the drag coefficient of the considered object. It is therefore not surprising that significant research has been carried out and published on this matter during the past century. In experimental investigations aimed at finding the drag coefficient of a solid sphere moving through a fluid, various techniques were employed. These different techniques were used for evaluating the drag coefficient over a wide range of Reynolds and Mach numbers. Among the frequently used methods, one should mention freely falling spheres in a liquid or air, spheres placed in wind tunnels, spheres mounted on flying aircraft, spheres towed in water channels and spheres flying in aero-ballistic range. In many of these experiments, the relative velocity of the sphere was constant or almost constant. Based on the results from experiments conducted using the above-mentioned techniques, a ‘standard drag coefficient’ curve has been derived for a sphere. For low subsonic speeds, one curve represents the sphere drag coefficient over a large range of Reynolds numbers; this curve is available in almost all fluid mechanics textbooks (see Olson 1980). For high subsonic and supersonic flows, the flow Mach number appears as a parameter (Bailey & Hiatt 1972; Bailey & Starr 1976; Henderson 1976). This detailed and accurate knowledge available for the drag coefficient of a sphere is limited to the case where the sphere's relative velocity is constant, i.e. to a steady flow condition. However, in many engineering applications, the sphere's motion through the fluid in which it is immersed is not steady, i.e. it experiences acceleration or deceleration. Examples of this are the nozzle flow of a solid propellant rocket or flows behind shock/blast waves propagating into a dusty gas, volcanic eruptions, etc. Using the standard drag curve for such flows is, at least, questionable. To have a reliable simulation of a non-stationary two-phase flow (solid particles immersed in a gaseous medium), an accurate knowledge of the particle drag coefficient is essential. Measurements of sphere drag coefficients for non-stationary flows were conducted during the past four decades. Among the published results, one should mention the work of Buckley (1968), Selberg & Nicholls (1968), Rudinger (1970), Karanfilian & Kotas (1978), Outa *et al*. (1981), Sommerfeld (1985), Igra & Takayama (1993), Boiko *et al*. (1997), Tano *et al*. (2003) and Sun *et al*. (2004). Selberg & Nicholls (1968) used a shock tube for accelerating single particles and photographed their displacement from which the particle trajectory was reconstructed and the drag coefficient was deduced. The particle trajectory, which serves as the basis for obtaining the particle velocity and acceleration, was obtained through a curve fit to the observed particle displacements. Because the curve fit was derived from only five observed particle positions, it is obvious that the deduced particle velocity and acceleration are only a rough estimate of the real values and so is the particle drag coefficient, *C*_{D}. Furthermore, in the technique used, by injecting particles into the shock tube channel, one observes many particles on the recorded film. In such a case, some particles are flying in the wake produced by particles flying ahead (closer to the incident shock wave), and thereby experiencing a drag force which is different from that experienced by a single particle under similar conditions. Their results were limited to Reynolds numbers within the range 200<*Re*<1700. Buckley (1968) investigated the drag force acting on a particles cloud that was accelerated to a transonic slip flow, using a light-extinction technique. Rudinger (1970) accelerated a particles cloud in a vertical shock tube and deduced the particles' velocity from their trajectories recorded by streak photography. Since it was hard to establish an accurate particle trajectory, which is the basis for obtaining the particles' velocity and acceleration needed for evaluating the particles' *C*_{D}, a different method was adopted. In the new approach, Rudinger (1970) evaluated the particle velocity via the deflection of a light beam passing through the particle cloud. From the ratio of the scattered light intensity before the passage of the incident shock wave to that during the passage of the particle cloud, the suspension density ratio across the shock can be estimated. This in turn, for a steady one-dimensional flow, is equal to the ratio between the shock velocity and the particles' velocity. This, at best, would yield an average velocity of the dust cloud, not the velocity of a specific particle. Furthermore, the assumption that the post-shock flow is steady and one-dimensional is questionable. In order that this assumption will not be grossly violated, Rudinger (1970) was forced to limit his experiments to very weak incident shock waves. His suggested correlation for *C*_{D} was therefore limited to the range 50<*Re*<300. Henderson (1976) proposed three different correlations for *C*_{D}: (i) for a subsonic flow, (ii) for a supersonic flow at Mach numbers *M*_{is}>1.75, and (iii) a linear interpolation equation for the intervening region. While considering the effects of physical variables, such as Reynolds number, Mach number and whether or not the flow is continuum or free molecular flow, on the sphere drag coefficient, no attention was given to whether or not the relative flow is steady. Karanfilian & Kotas (1978) measured the drag force on a sphere experiencing a simple harmonic (unsteady) motion in a liquid at rest. Their measurements were limited to the range 10<*Re*<6000. Outa *et al*. (1981) deduced the particle drag from a streak record of dispersed glass spheres in the air inside a shock tube. Sommerfeld (1985) essentially repeated Rudinger's experiments. He used laser Doppler anemometry to find a drag law describing the particle acceleration behind an incident shock wave, in a vertical shock tube filled with a uniform gas–particle suspension. His proposed correlation for the particle *C*_{D} was limited to the range 50<*Re*<500. An attempt to propose a correlation for the particle drag coefficient over a wide range of Reynolds numbers is found in the paper by Igra & Takayama (1993). In their study, a shock tube was used for inducing a relatively high acceleration of small spheres laid on the shock tube floor. Using double-exposure holography, the spheres' trajectory could be reconstructed accurately while they flew along the 260 mm windows of the tube test section. Based on such trajectories, the sphere drag coefficient was evaluated for a range of 6000<*Re*<101 000. As was the case in previous studies, the results in Igra & Takayama (1993) also suggested that the drag coefficients obtained for a sphere in a non-stationary flow are significantly larger than those obtained in a similar steady flow. Boiko *et al*. (1997) embarked on an ambitious investigation, studying experimentally and theoretically the shock wave interaction and propagation through a dust cloud. The dust volume concentration was within the range of 0.1–3%. Obviously, when dealing with a dust cloud, constructing a particle trajectory is impossible; instead, the frontal boundary of the cloud was reconstructed from the recorded shadowgraphs. For the theoretical part, a correlation relating the particle's drag coefficient to the appropriate flow *Re* and Mach numbers is needed. The required correlation was obtained by repeating experiments with a single particle. These experiments were limited to a narrow range of Reynolds numbers, 1.5×10^{4}<*Re*<2.5×10^{4}. The main findings of this investigation are that a reflected shock wave is formed ahead of the particles cloud and a compression wave arises inside the cloud due to the deceleration of the gas by the particles.

In all the shock tube experiments described so far, the diameter of the spheres used was very small in comparison with the dimension of the shock tube test section. Therefore, the time to complete the shock wave diffraction over the sphere was negligibly small and, during most of the investigated time, the sphere acceleration was due mostly to the relative velocity existing between the sphere and the post-shock gas flow. This was not the case in the studies carried out by Tano *et al*. (2003) and Sun *et al*. (2004), where a large sphere (80 mm in diameter) was tested inside a 300 mm×300 mm cross-section shock tube. In this study, the sphere drag coefficient was evaluated experimentally and numerically during its interaction process with the incident shock wave. During the process starting when the incident shock wave collides with the sphere and continues while it diffracts over it, the sphere experiences a steep increase in the applied drag force. This, almost instantaneous, rise in the drag coefficient reaches *C*_{D}=10. Thereafter, the drag coefficient drops down quickly as the shock wave moves to the rear side of the sphere. Once the shock wave propagated downstream from the sphere, the drag coefficient reaches a low and stable value (see figs 5 and 6 of Sun *et al*. (2004)). This stable value is the drag experienced by the sphere during its non-stationary motion in the post-shock gas flow. The duration of the very quick rise and decline in the sphere's drag is approximately equal to the time taken by the shock wave to travel a distance equal to the sphere's diameter. For spheres tested in the present work, where the largest diameter was 6.5 mm, the time of shock diffraction over the sphere was negligibly small.

Igra & Takayama's (1993) work provided a significant improvement to the previously available data regarding the drag coefficient of a sphere in a non-stationary flow because the evaluation of the sphere's trajectory was based on a large number of recorded sphere locations taken during a relatively long time (optical field of view of 260 mm×150 mm). It also covered a much wider range of Reynolds numbers than in the previously published works. However, it suffered from two basic setbacks. Before the arrival of the incident shock wave, the investigated small spheres were placed on the shock tube floor. When they started their motion, after the incident shock wave had passed them, they were partly submerged inside the boundary layer developed along the shock tube walls. It was argued by Igra & Takayama (1993) that, during the time when the particle starts its motion, the prevailing boundary-layer thickness is very small and most of the investigated particles were out of this layer. Nevertheless, one could not blindly ignore the effect of the boundary layer and the friction between the shock tube wall and the particle on the evolved sphere's trajectory and thereby on the deduced *C*_{D}. It is crystal clear that starting the particle motion when it is far away from all walls would yield more reliable results. The second disadvantage lies in the following facts.

In all of the Igra & Takayama (1993) results, the post-shock gas velocity was subsonic. To have a reliable correlation for

*C*_{D}in a non-steady flow, one should include a significant amount of both subsonic and supersonic post-shock flows.To obtain the particle trajectory, many runs had to be repeated with the same initial conditions but photographing the sphere at different times during its motion. This was needed since, in each run, only one photograph was taken. Even with a high repeatability, such a procedure is a source for unavoidable errors in the construction of the sphere's trajectory.

Later, Rodriguez *et al*. (1993) investigated both the steady and unsteady drag coefficients of a sphere initially free-falling in a vertical shock tube, using a rapid camera shadowgraph technique. Their work investigated the drag coefficient of a sphere, but, in each experiment, 20–30 particles were dropped down the vertical shock tube in order to ensure that at least one of them would be present in the field of view during its interaction with the incident shock wave. No correction was made to account for the wake interference between the falling particles. More recently, Suzuki *et al*. (1999) developed a technique enabling the injection of a spherical particle into the middle of a horizontal shock tube test section, just before the incident shock wave reached the injected sphere. A triple-exposure photographic technique was used for recording the particle displacement caused by the incident shock wave. The test section field of view in their experiments was only 200 mm in the flow direction and the incident shock wave Mach number was within the range of 1.10≤*M*_{is}≤1.40, resulting in the subsonic post-shock flow in all their experiments.

The significant achievement of the present experimental study described subsequently is the method used for the drag force measurement by tracing the trajectory of a single spherical particle suspended in mid-air by a fine filament (spider web) from the roof of a shock tube. Multiple shadowgraphs (several tens) were acquired at each shot, eliminating the serious repeatability-related inaccuracy encountered in former studies where only a single or a few shadowgraphs were recorded per shot. Moreover, by employing multiple-sphere shots (up to four different particles), the repeatability-associated inaccuracy was further reduced. In nine experiments, the post-shock flow was subsonic and in seven supersonic. In four of the subsonic experiments, two different spheres were tested simultaneously and, in one, three different spheres were tested simultaneously. In one of the supersonic experiments, two different spheres were tested simultaneously; in two experiments three different spheres were tested simultaneously; and, in one run, four different spheres were tested simultaneously. Furthermore, in the presently reported results, the field of view was 300 mm long, enabling a very accurate construction of the investigated particle trajectory during each test.

## 2. Experimental facilities

Experiments were conducted in the 80×80 mm cross-section, multi-phase variable inclination shock tube of the University of Provence. This shock tube, in its horizontal position, and the diagnostics used are shown in figure 1. The total length of the tube is 3.75 m, 75 cm is the driver's length and 3 m is the driven section length. The driver pressure can be raised up to 20 bars and the driven section pressure can be reduced down to 0.5 mbar to result in a maximum pressure ratio of approximately 40 000. Consequently, experiments with the incident shock wave Mach number within the range 1.1<*M*_{is}<5 can be easily conducted. This in turn allows the present investigations to cover both subsonic and supersonic post-shock flow cases. As was done in previous experiments (Igra & Takayama 1993; Rodriguez *et al*. 1993; Suzuki *et al*. 1999), here too the sphere drag coefficient, *C*_{D}, was deduced from its trajectory. To obtain an accurate reconstruction of the investigated sphere's trajectory, the windows in the presently used test section had a field of view of 80×300 mm (300 mm in the flow direction). The spatial locations of the sphere during its motion, induced by the post-shock flow, were recorded using a shadowgraph technique coupled with a high-speed camera and a stroboscopic Nanolite flash lamp that was synchronized with the rotating-drum camera. This optical facility enables photographing every 70 μs during each experiment (i.e. having approx. 60 and approx. 35 photos per run in the subsonic and supersonic cases, respectively), thereby ensuring the accurate construction of the tested sphere's trajectory. PCB piezoelectric transducers were used for recording pressure histories, deducing the incident shock wave Mach number and triggering optical and recording facilities. Recording the post-shock pressure history is essential since we limited the reconstruction of the sphere's trajectory to the duration in which a uniform post-shock flow prevails. During this period of time, the post-shock flow properties can be easily predicted using the Rankine–Hugoniot shock relations.

The diameter (*ϕ*), the material density (*ρ*_{p}) and the type of spheres used in the present experiments are listed in table 1.

A crucial point in the experiments was the problem of how to keep the tested spheres away from the test section walls. After checking various options, the following method was chosen. The tested sphere was suspended from the tube ceiling, close behind the entrance to the test section, on a wire taken from a spider web. The spider wire was strong enough to keep the small sphere suspended in the air until the arrival of the incident shock wave. This technique, although very delicate, is made possible thanks to the sticky substance that covers the spider wire. Furthermore, the spider wire quickly accelerates to the post-shock gas flow velocity before disintegrating. Figure 2*a*–*c* contains three shadowgraphs taken, respectively, 36 μs before the shock collision with the sphere, 34 μs after the incident shock wave hit the sphere and 104 μs after its interaction with the 1.92 mm diameter nylon sphere suspended on the spider wire. As can be seen in figure 2, the process of the spider wire acceleration and disintegration is completed very quickly after it collides, head-on, with the incident shock wave. Disturbances produced by the spider wire and its breaking were hardly noticeable. On the other hand, given that in the considered case the post-shock flow is supersonic, the detached shock wave from the sphere is easily noted in figure 2*c*.

## 3. Theoretical background

When a solid particle is exposed to a gas flow, its response depends on the relative velocity that exists between the particle and the flow. For a low concentration of solid particles in a suspension (and surely in the case when only a few small spheres are immersed in the flowing gas), one can ignore both the interaction between solid particles and their contribution to the suspension pressure. In such a case, the drag force acting on the solid particle(s) is the sole meaningful force that determines the particle motion. In experiments conducted in shock tubes with relatively small (and therefore light) particles, the particles experience a very large acceleration due to the very fast post-shock gas flow. Until the particles reach the post-shock flow velocity, the relative velocity between the particle and the gas flow changes and the particle motion is non-stationary. Should the particle trajectory be recorded accurately, its drag coefficient could be evaluated as follows. The equation of motion of a solid particle accelerated by the gas flow is(3.1)where **U**_{p}, *ϕ* and *ρ*_{p} are the solid sphere's velocity, diameter and material density, respectively. **U**_{g} and *ρ*_{g} are the gas velocity and density, respectively. It was shown by Igra & Takayama (1993) that, based on equation (3.1), the particle drag coefficient and the appropriate Reynolds number can be expressed as follows:(3.2)(3.3)where *u* and *v* are the components of the velocity vector * U* in the

*x*and

*y*directions, respectively.

*g*is the gravity acceleration and

*μ*

_{g}is the gas viscosity. Similarly, the sphere's Mach number, based on the relative velocity, is(3.4)where

*γ*and

*R*are the gas specific heat ratio and the gas constant, respectively.

In the following, the trajectory of a sphere suspended in the entrance to the shock tube test section will be reconstructed accurately using the optical system described earlier. Each trajectory is a curve fit passing, on average, through 35–60 recorded locations passed by the considered particle during its motion imposed by the incident shock wave. Once the particle trajectory is available, its velocity and acceleration can be easily obtained by the first differentiation of its trajectory (velocity) and the second differentiation yields its acceleration. Substituting the obtained values for the sphere's velocity and acceleration in equations (3.2)–(3.4) provides the sphere's *C*_{D}, Reynolds and Mach numbers.

## 4. Results and discussion

Unlike in the experimental work of Igra & Takayama (1993), where many experiments were repeated in order to reconstruct the sphere's trajectory, in the present case, approximately 35–60 photos with a time difference of 70 μs between successive frames were taken in a single test. Therefore, the obtained temporal locations of the flying sphere were more than needed for an accurate reconstruction of the sphere's trajectory. In the present results, the accuracy of measuring the sphere location, from the recorded shadowgraphs, is within ±1 mm. A summary of all the conducted experiments is given in table 2. It provides information about the tested spheres, the initial and the prevailing conditions in the shock tube. In some experiments, a few particles were tested in the same run. In such cases, the different spheres were suspended either along a line perpendicular to the flow direction for minimizing the wake interference between the tested spheres, or with a significant longitudinal gap between them. The different spheres tested in the same run are marked by letters ‘a’, ‘b’, ‘c’, etc. appearing after the run number in table 2. In all the experiments where the post-shock flow was subsonic, the flying sphere experiences a uniform post-shock flow until the arrival of the reflected shock wave from the tube end wall. (*x*–*t*) diagrams (computed using the random choice method), showing the lines of constant density, appear in figure 3. Numbers appearing in the uniform flow zones indicate the local density in kg m^{−3}. A typical case where the post-shock flow is subsonic is shown in figure 3*a*. In all the experiments where the post-shock flow was supersonic, the flying sphere experiences a uniform post-shock flow until the arrival of the contact surface as shown in figure 3*b*. In evaluating the sphere drag coefficient, only the part of the sphere's trajectory reconstructed inside the uniform post-shock flow zone was used. In this zone, the post-shock flow conditions were derived using the Rankine–Hugoniot shock relations and the measured incident shock wave Mach number. In the following, the procedure used for deducing the drag coefficient of spheres exposed to the shock wave-induced flow is outlined for four different cases. In two cases, the post-shock-induced flow was subsonic and, in the other two, supersonic.

The sample of the obtained shadowgraphs for the subsonic cases is shown in figures 4 (run no. 148, where 14 out of the 60 recorded shadowgraphs are shown) and 5 (run no. 166, where 14 out of the 63 recorded shadowgraphs are shown). The results obtained for the supersonic cases are shown in figures 6 (run no. 142, where 14 out of the 36 recorded shadowgraphs are shown) and 7 (run no. 184, where 12 out of the 15 recorded shadowgraphs are shown). The distance between the two vertical reference lines appearing on all shadowgraphs was 21.3 cm except in run no. 142, where it was 19.7 cm.

While in run no. 148 (figure 4) only one sphere was present in the test section, in run no. 166 (figure 5) two different spheres were employed. One, a nylon sphere (*ϕ*=1.96 mm and *ρ*_{p}=1204 kg m^{−3}), was suspended on a spider wire just at the entrance to the test section (see the frame taken at *t*=0 μs in figure 5), the second sphere, made of polystyrene foam (*ϕ*=6.5 mm and *ρ*_{p}=25 kg m^{−3}), was laid on the shock tube floor far upstream of the nylon sphere, and therefore is not seen in figure 5 until *t*=1109 μs. It is apparent from the shadowgraphs shown in figures 4–7 that the spheres move along horizontal lines, and therefore one can safely neglect the gravity effect and a possible sphere's rotation. As expected, the sphere response to the shock-induced flow is not instantaneous. It is apparent from figure 4 that, for the first 300 μs after the collision between the incident shock wave and the nylon sphere, the sphere hardly moves. The same delay is observed in figure 5 for the nylon ball. The 6.5 mm polystyrene foam ball is significantly lighter than the 1.96 mm nylon ball. Although it was placed far upstream of the nylon ball, it reaches the smaller ball at approximately *t*=1110 μs and soon thereafter passes it (figure 5). As seen in figure 5, shortly before *t*=2500 μs, the large light polystyrene foam ball reaches the end of the field of view and soon thereafter the reflected shock wave from the shock tube end wall enters the field of view. This significant difference in velocity between the two balls, tested in run no. 166, will be seen clearly in the reconstructed sphere's trajectories.

Based on all the recorded shadowgraphs, the sphere's trajectories during experiment nos. 148 and 166 were reconstructed; they are shown in figure 8*a*,*b*, respectively, and those deduced from the shadowgraphs taken during experiment nos. 142 and 184 are shown in figure 9*a*,*b*, respectively. The straight lines appearing in figures 8 and 9 represent the incident shock wave and the shock wave reflected from the shock tube end wall. The appropriate wave's velocity is indicated near these lines. The sphere's velocity can be obtained from the results shown in figures 8 and 9 by either of the following two options.

Executing a curve fit through the recorded points that indicate the different sphere locations during the considered experiment. The first differentiation of this curve fit provides the sphere's velocity.

Conducting numerical differentiation of the sphere's path as represented by the recorded discrete sphere displacements (shown in figures 8 and 9).

Both options were checked. In the cases where the recorded sphere locations exhibit a smooth, monotonic change, the difference between the two results obtained for the sphere's velocity was very small (less than 2%). The results obtained for the sphere's velocities via numerical differentiation of the recorded sphere locations (shown in figures 8 and 9) appear as discrete points in figures 10 and 11, respectively. The line passing through these discrete points is the sphere's velocity obtained by the differentiation of the curve fit to the recorded sphere locations. The straight solid lines in figures 10 and 11 indicate the gas velocity ahead and behind the incident shock wave. The sphere's acceleration, needed for evaluating its drag coefficient, can be obtained by differentiating its velocity curve. Since numerical differentiation yields a very noisy acceleration curve, we used the polynomial curve fit to the recorded sphere locations as a basis for evaluating the sphere's velocity and acceleration. Polynomials of different orders could be fitted to the given set of sphere locations for constructing its trajectory. Choosing a high-order polynomial fit resulted in a relatively noisy acceleration curve. Therefore, a third-order polynomial fit was used for constructing the sphere's trajectory. The values obtained for the sphere's velocity and acceleration (by first and second differentiations of the third-order polynomial fit to the sphere's trajectory) were substituted into equation (3.2) for deducing the sphere drag coefficient. The appropriate sphere's Reynolds number was obtained from equation (3.3). The results obtained for the sphere's *C*_{D} versus Reynolds number, for the subsonic cases (run nos. 148 and 166), are shown in figure 12*a*,*b*, respectively. The results obtained for the cases where the post-shock gas flow is supersonic (run nos. 142 and 184) are shown in figure 13*a*,*b*, respectively. Also shown in figures 12 and 13 are the sphere's Mach number (deduced from equation (3.4)) and the appropriate ‘standard drag curve’. It is clear from the results shown in figures 12 and 13 that, for the covered range of *Re*, appearing in each figure, the standard drag curve is practically constant and the changes in the deduced unsteady sphere drag coefficient are moderate. Therefore, an average value of *C*_{D} was deduced from the results shown in figures 12 and 13 to be associated with the appropriate average value of *Re*. This average value of *C*_{D} appears in table 2 and figure 15, where the results obtained from a specific run appear as a single point. For example, the results shown in figure 12*a* will appear as *C*_{D}=0.6 and *Re*=48 000. Furthermore, it is clearly visible from figures 12 and 13 that the obtained values of *C*_{D} are higher than those obtained for a similar *C*_{D} in a steady flow.

The two spheres in run no. 166 had very different characteristics: one (polystyrene foam) was relatively large and very light while the other (a nylon ball) was relatively small and heavy. The polystyrene foam ball experiences a relatively large drag force and therefore it is quickly accelerated towards the prevailing post-shock gas flow velocity. As a result, its relative velocity *u*_{g}−*u*_{p} was smaller than that of the nylon sphere. This explains the lower particle Mach number seen in figure 12*b* for the polystyrene foam ball.

The sphere drag coefficient deduced from the present experiments is based on the total drag force acting on the considered sphere. In the present non-steady flow, this drag is a summation of a few different contributions, as follows: drag due to friction (flow viscosity) and form (pressure distribution on the immersed body); added mass; shock diffraction effect during the incident shock passage over the particle; and history (Basset force). The contribution of the added mass to the total drag coefficient depends on the ratio between the immersed object's material density and the density of the fluid in which the object is immersed. The closer the object material density is to that of the fluid, the larger will be the added mass contribution to the total drag coefficient. Note that, in the present work, the density ratio ranges from one to several hundreds. Moreover, in light of this, it is of interest to note that, in run nos. 142 and 184, spheres of similar diameter were tested: 1.92 mm in run no. 142, 2.2 mm (‘a’) and 2.3 mm (‘d’) in run no. 184. However, there is a significant difference in their material densities; while in run no. 142, *ρ*_{p}=1130 kg m^{−3}, in run no. 184 the sphere's material density is only 25 kg m^{−3}. Therefore, if the added mass contribution in the present experiments would have been of any importance, one would expect to find a significant difference between the drag coefficients obtained in these two runs. Inspecting the results shown in table 2 indicates that this is not the case; the average *C*_{D} for the sphere in run no. 142 is 0.7 while for the two similar spheres ‘a’ and ‘d’, in run no. 184, *C*_{D}=0.74 and 0.71, respectively. It is therefore safe to assume that in the present experiments, the contribution of the added mass to the total drag coefficient is small.

It was mentioned that the results for the sphere's *C*_{D} shown in figures 12 and 13, and those to be shown subsequently, are based on a third-order polynomial fit to the recorded sphere locations. This curve fit was differentiated to obtain the sphere's velocity and its second differentiation yields the sphere acceleration. It is reasonable to ask whether the choice of a third-order polynomial fit is physically justified, as such a choice dictates a linearly declining acceleration. As expected, and observed in the recorded shadowgraphs, after the passage of the incident shock wave, the sphere starts moving; it increases its velocity from zero towards the post-shock flow velocity. The sphere's velocity increases monotonically as is evident from its recorded trajectory. Hence, it is an acceptable and reasonable practice to select the lowest order that provides the requested fit to the available data. In the present case, a third-order polynomial fit provided a very good fit to the recorded sphere locations.

A summary of all the results obtained in the present research is shown in figure 14 in the *C*_{D}−*Re* plane. Three lines appear in this figure. The solid line is a curve fit to the present results. The dotted line is the correlations proposed for *C*_{D} by Igra & Takayama (1993) and the dashed line is the standard drag curve proposed for a similar steady flow. The correlation describing the present results, which appears as a solid line in figure 14, is a third-order polynomial fit, similar to that proposed by Igra & Takayama (1993). This correlation is given in the following equation:(4.1)In this correlation, *Re*_{p} stands for the Reynolds number based on the particle relative velocity.

A clearer picture of the obtained results is achieved when in each run an average value of *C*_{D} is assigned to the appropriate average value of *Re*_{p}. The obtained results are shown in figure 15. Also shown in this figure are the above-mentioned correlations proposed for the sphere's *C*_{D} (ours and that of Igra & Takayama (1993)) and the standard drag curve proposed in the literature for a sphere in a steady flow. It is clear from this figure that there is a significant difference between the drag coefficient of a sphere in a steady flow (the standard drag curve) and that obtained in non-stationary flow conditions. The proposed correlations for the non-stationary drag coefficient suggest values of *C*_{D} which are significantly higher than that obtained in a similar steady flow case for Reynolds numbers within the range of 500<*Re*<10^{4}. Inspection of figure 15 shows that the results obtained for *C*_{D} in this range for subsonic and supersonic flows can be found above and below the correlations given by equation (4.1), indicating that, in the considered range of *M*_{p}, the Mach number of the relative velocity probably does not play a dominant role in the resulting sphere drag coefficient. It may be the case that the relative low number of points in this range does not allow us to conclude definitively on the compressibility effects.

It was suggested by Crowe *et al*. (1963), Selberg & Nicholls (1968), Karanfilian & Kotas (1978) and Temkin & Metha (1982) that the flow unsteadiness contribution to the sphere's drag may be expressed in terms of a non-dimensional parameter . This parameter can be deduced from equation (3.2). In the present experiments, this parameter is quite small, it is within the range of 2.5×10^{−5}≤*Ac*≤3.5×10^{−2} as is evident from table 2. The fact that *Ac* is so small should not be surprising since, in spite of the sphere's high initial acceleration, its diameter is very small and its initial relative velocity is relatively high. Therefore, one could conclude that the parameter *Ac* is not a characteristic parameter for the considered flows. It should be noted that, in Karanfilian & Kotas (1978), the investigated spheres were immersed in liquid and in such a case the added mass contribution to the unsteady drag coefficient is significant (water density is almost three orders of magnitude higher than that of the air). Therefore, it is not surprising that, in their results, *Ac* varies within the range of 0.1≤*Ac*≤0.5; this is significantly higher than the *Ac* values obtained in the present investigation. We therefore believe that, in the considered experiments, the dominant mechanism is the flow establishment around the investigated sphere; similar to the flow establishment around an aerofoil.

## 5. Conclusions

The present results obtained for the sphere drag coefficient, *C*_{D}, strengthen past assessments that a significant difference exists between the values obtained for *C*_{D} in steady and non-steady flows. The correlations proposed in equation (4.1) can be used safely in numerical simulations of dusty flows where the dust concentration is not high, and for Reynolds numbers within the range covered in the present investigation. Furthermore, throughout most of the investigated range of Reynolds numbers, the obtained non-steady values are over 50% higher than those obtained in a similar steady flow case. The gap between the two increases with decreasing Reynolds numbers. Since the drag force in this range is dominated by viscous flow effects, the dynamic drag coefficient obtained in the present study seems to indicate a type of unsteady ‘shear waves’ flow effect. But, at this time, we do not have an appropriate explanation for the significant increase in the sphere drag coefficient in non-steady flows and particularly at the lower range of Reynolds numbers, and we think that this point will be a challenge to researchers in theoretical or computational fluid mechanics.

## Footnotes

- Received May 30, 2007.
- Accepted September 7, 2007.

- © 2007 The Royal Society