## Abstract

The unsteady inviscid force on cylinders and spheres in subcritical compressible flow is investigated. In the limit of incompressible flow, the unsteady inviscid force on a cylinder or sphere is the so-called added-mass force that is proportional to the product of the mass displaced by the body and the instantaneous acceleration. In compressible flow, the finite acoustic propagation speed means that the unsteady inviscid force arising from an instantaneously applied constant acceleration develops gradually and reaches steady values only for non-dimensional times *c*_{∞}*t*/*R*≳10, where *c*_{∞} is the freestream speed of sound and *R* is the radius of the cylinder or sphere. In this limit, an effective added-mass coefficient may be defined. The main conclusion of our study is that the freestream Mach number has a pronounced effect on both the peak value of the unsteady force and the effective added-mass coefficient. At a freestream Mach number of 0.5, the effective added-mass coefficient is about twice as large as the incompressible value for the sphere. Coupled with an impulsive acceleration, the unsteady inviscid force in compressible flow can be more than four times larger than that predicted from incompressible theory. Furthermore, the effect of the ratio of specific heats on the unsteady force becomes more pronounced as the Mach number increases.

## 1. Introduction

In an incompressible flow, well-established analytical expressions exist for the steady and unsteady (added mass and history) forces on cylinders and spheres in the Stokes and inviscid limits (e.g. Landau & Lifshitz 1987). Analytical and empirical extensions of the quasi-steady, added-mass, and history forces to finite Reynolds numbers have been studied extensively (see Crowe *et al.* 1997; Magnaudet & Eames 2000). In the compressible regime, attention has generally been focused on the steady drag force. Detailed parametrizations of the steady drag force in terms of Mach and Reynolds numbers have been considered (see Bailey & Hiatt 1972 and references therein). However, our understanding of unsteady forces in the compressible regime, arising either from the acceleration of the cylinder or sphere or from the acceleration of the surrounding fluid, is limited.

The earliest fundamental contributions to the study of unsteady forces in the compressible regime appear to be due to Love (1904) and Taylor (1942). Miles (1951) investigated the motion of a cylinder impulsively started from rest based on the acoustic approximation of the velocity potential equation. He treated the cases of transient motion generated by a constant force applied over a finite time interval as well as an impulsively applied velocity. Independently, Longhorn (1952) considered the unsteady motion of a sphere based on the same approximation. The work of Longhorn was considered by Ffowcs Williams & Lovely (1977), who determined analytically the acoustic field produced by a sphere accelerated impulsively from rest. Both Miles and Longhorn pointed out the limitations of the conventional added-mass concept in describing the inviscid force in compressible flows.

These limitations are rooted in the relationship between the added-mass force and the instantaneous acceleration. In an incompressible flow, the added-mass force depends only on the instantaneous acceleration. In a compressible flow, on the other hand, the inviscid force develops on an acoustic time scale *R*/*c*_{∞}, where *R* is the radius of the cylinder or sphere and *c*_{∞} is the speed of sound in the ambient fluid in the farfield. The results of Miles and Longhorn show that under constant acceleration, the inviscid force reaches a constant value for *c*_{∞}*t*/*R*≳10. If an added-mass coefficient is computed based on this constant long-time force, values of 1.0 and 0.5 are recovered for the cylinder and sphere, respectively. These values are consistent with the low-Mach number limit implicit in the acoustic approximation employed by Miles and Longhorn. It should also be noted that in a compressible flow the force evolution in response to constant acceleration is non-monotonic. As a result, at intermediate times the instantaneous force on the cylinder or sphere can be substantially larger than the constant final force.

Other relevant work was performed by Tracey (1988), who extended Miles's analytical work on the motion of an impulsively started cylinder to finite Mach number. Numerical simulations of compressible flow about an accelerating cylinder at finite Mach numbers were performed by Brentner (1993). However, the focus of Brentner's study was on the propagation of acoustic energy as the cylinder accelerated impulsively from rest to a Mach number of 0.4.

The objective of this work is to extend the results of Miles and Longhorn to finite freestream Mach numbers. We will investigate the effect of Mach number on the non-monotonic evolution and asymptotic long-time constant value of the unsteady force in response to a sudden constant acceleration. To this end, we solve numerically the Euler equations in a frame of reference attached to the cylinder and sphere, prescribe their motion and compute the drag coefficient. We thus take an approach similar to that of Brentner, but our goal is the determination of forces and not the computation of the acoustic field. We restrict our attention to the subcritical Mach number regime in this article.

## 2. Numerical method

The numerical method solves the Euler equations in integral form cast in a frame of reference attached to the cylinder or the sphere. Spatial discretization is based on the flux-difference splitting method of Roe (1981) and the weighted essentially non-oscillatory reconstruction described by Haselbacher (2005). The discrete equations are integrated in time using the four-stage Runge–Kutta method. The basic methodology employed in this work has been applied to several unsteady compressible flows and demonstrated good agreement with theory and experimental data (see Haselbacher *et al.* 2007).

For the cylinder, a two-dimensional hexahedral grid of O-type topology is used with 386 cells around the circumference. Relative to the cylinder radius *R*, the radial grid spacing adjacent to the cylinder surface is Δ*r*/*R*=1.624×10^{−2}, thus producing cells with aspect ratios of nearly unity. The radial stretching of grid cells is adjusted such that each layer of cells consists of approximately square cells to minimize internal wave reflections. We have employed grids consisting of up to 1 760 160 cells to assess the grid independence of our solutions. The results shown below were obtained on a grid of 110 010 cells.

For the sphere, a hexahedral grid consisting of six blocks is used. Each block contains 100×100 cells on the sphere surface and 320 cells in the radial direction. Relative to the sphere radius *R*, the radial grid spacing on the surface is Δ*r*/*R*≈2.9×10^{−2}. As for the cylinder grid, the radial stretching of cells is adjusted such that each layer consists of approximately cuboid cells. The results shown below were obtained with a very fine grid of 19 200 000 cells.

For both cylinder and sphere computations, the characteristic boundary conditions of Poinsot & Lele (1992) are applied at the outer boundary, located at 200*R*. We have verified that the combination of characteristic boundary conditions and the large distance to the outer boundary eliminates wave reflections that could affect the determination of the unsteady force on the cylinder or sphere.

## 3. Results

Our objective is to extract the time-dependent inviscid force on a cylinder or sphere in response to suddenly imposed acceleration at finite Mach numbers. To accomplish this, the cylinder or sphere is first held fixed and a steady-state solution is obtained at the chosen farfield Mach number *M*_{∞,0}. At some time *t*_{0}, we impose on the cylinder or the sphere a constant acceleration *a* in the direction opposite to that of the ambient flow. We maintain the constant acceleration for a finite time interval *t*_{f}−*t*_{0} and remove it thereafter. The duration of acceleration, non-dimensionalized in terms of the acoustic time scale, is chosen to be *c*_{∞}(*t*_{f}−*t*_{0})/*R*=20, which is sufficient for the inviscid force to reach a constant value. The instantaneous relative Mach number increases linearly during the period of acceleration and reaches a value *M*_{∞,0}+*δ* by the end of the interval, where *δ*=*a*(*t*_{f}−*t*_{0})/*c*_{∞}.

In all cases considered, the non-dimensional acceleration is chosen carefully to satisfy the following two competing requirements. First, we limit the value of *α* such that *δ*, the change in Mach number, is kept small. This allows interpretation of the resulting time-dependent force on the cylinder or sphere to be at a fixed or frozen Mach number of *M*_{∞,0}. Accordingly, we drop the subscript 0 and simply write *M*_{∞} in the following. Figure 1 shows the time evolution of the non-dimensional force on a cylinder for several values of the non-dimensional acceleration *α* starting from *M*_{∞}=0.3 for a ratio of specific heats *γ*=1.4. Here, the force (per unit width) is non-dimensionalized as , where *m*_{f} is the mass of the fluid displaced per unit width of the cylinder. It can be seen that provided *α* is sufficiently small, the non-dimensional force is observed to be independent of the actual value of *α*. With increasing magnitude of acceleration, e.g. for *α*=1.2×10^{−3}, the increase in relative Mach number over the duration of acceleration has a significant influence on the net force and therefore the result can no longer be considered to correspond to a frozen Mach number of 0.3. At even higher accelerations, the instantaneous Mach number exceeds the critical value of approximately 0.398 and the effect of locally supersonic flow around the cylinder results in a steady increase in the force. Clearly, *α* should be maintained sufficiently small to extract the time-dependent force at a frozen Mach number. The second requirement is that the rate of acceleration must not be too small, for otherwise the resulting force will be very weak with low signal-to-noise ratio. In all the cases considered here, a range of acceleration satisfies the two competing requirements. Provided *α* is chosen to lie within this range, the resulting appropriately non-dimensionalized force is independent of *α*.

Note that the farfield Mach number range investigated here is limited to values below the critical Mach numbers of approximately 0.398 and 0.6 for the cylinder and sphere, respectively. Below the critical farfield Mach number, the steady flows around the cylinder and the sphere remain subsonic everywhere. Therefore, the steady-state inviscid drag force is identically zero before the application of the acceleration as well as after the removal of the acceleration following the decay of transients.

### (a) Effect of Mach number

In an incompressible flow, a cylinder or sphere with a constant acceleration *a* experiences an inviscid force of magnitude *C*_{M}*m*_{f}*a* opposite to the direction of acceleration, where *C*_{M} is the added-mass coefficient. For a cylinder *C*_{M}=1 and for a sphere *C*_{M}=0.5. The added-mass force is realized instantaneously upon the application of acceleration due to the infinite acoustic propagation speed implicit in the incompressibility assumption. The force is proportional to the applied instantaneous acceleration and ceases to exist once the acceleration is removed. As described in §1, once compressibility effects become important, the force is no longer dependent on only the instantaneous acceleration.

#### (i) Cylinder

In figure 2*a* we plot the time-dependent force on the cylinder as a function of non-dimensional time *τ*=*c*_{∞}(*t*−*t*_{0})/*R* for farfield Mach numbers ranging from 0 to 0.39. Guided by incompressible results, the dimensional force *F* per unit cylinder width has been non-dimensionalized as . Also plotted in the figure is the theoretical result of Miles (1951), corresponding to the limit *M*_{∞}→0. The non-monotonic time evolution of the inertial force in response to constant acceleration is clearly visible. After approximately 12 acoustic time units, the non-dimensional force reaches a constant value of 1.0, which corresponds to the added-mass coefficient of a cylinder in incompressible flow. In other words, for *τ*≳12 the acoustic disturbance arising from the sudden onset of acceleration has radiated sufficiently far away that the near field is approximated well by an incompressible potential flow. At intermediate times, however, the inviscid force is observed to be substantially larger. For instance, Miles's solution yields a peak value of at *τ*≈3.1.

The non-dimensional perturbation pressure (difference between the instantaneous pressure and the initial steady pressure distributions, normalized to range between minus and plus one) is plotted in figure 3 for *M*_{∞}=0.2 at the four different times marked by black circles in figure 2*a*. From the compressible form of Bernoulli's equation, the perturbation pressure can be shown to have two contributions. The first contribution arises from the time dependence of the velocity potential. The second contribution is due to changes in the square of the velocity. In figure 3, the influence of the first contribution dominates at early times and a fore–aft asymmetry can be seen clearly. At later times, with the second contribution becoming increasingly important, the perturbation pressure increases and appears to be more fore–aft symmetric.

As indicated by figure 2*a*, the qualitative behaviour of the unsteady force remains the same at all subcritical Mach numbers. The asymptotic long-time constant value and the peak value of the non-dimensional force increase with *M*_{∞}. For both the peak and asymptotic values of , the effect of Mach number is substantial, as can be seen from figure 4*a*. The peak instantaneous force at *M*_{∞}=0.39 is approximately 2.4 times as large as that predicted from incompressible theory. The asymptotic long-time constant value of the non-dimensional force can be seen to increase steadily from a value of unity at *M*_{∞}→0 to approximately 2.04 at *M*_{∞}=0.39. This steady value can be considered as an effective added-mass coefficient for compressible flow.

The ratio of peak to long-time steady force for the different Mach numbers is also shown in figure 4*a*. It can be seen that this ratio is nearly constant at approximately 1.17–1.19 for the range of Mach numbers considered. With increasing Mach number, the time at which reaches a peak increases. For example, for *M*_{∞}=0.39, the peak occurs at *τ*≈5.26. Correspondingly, the approach to a constant value is also slightly delayed. Interestingly, once the acceleration is turned off, the return to zero force occurs in a manner similar to when the acceleration is first applied.

It is convenient to differentiate the non-dimensional force presented in figure 2*a* and define a kernel as . The resulting response kernels for the different Mach numbers are presented in figure 2*b*. The kernel can be interpreted readily in terms of the non-dimensional inviscid force per unit width on a cylinder subjected to an impulsive jump in the relative velocity of *u*_{0} given as *K*_{cy}(*τ*=*tc*_{∞}/*R*)=*F*/(*m*_{f}(*c*_{∞}*u*_{0}/*R*)). In fact, this is the form in which Miles had presented his result for *M*_{∞}→0. As shown by Miles, in this limit the kernel satisfies the integral equation(3.1)The asymptotic solutions to the integral equation for small and large *τ* were also obtained by Miles as(3.2)

The non-monotonicity seen in figure 2*a* translates to the kernel being positive for a short duration, then becoming negative and slowly approaching zero. This behaviour has interesting implications. The response to an impulsive jump in cylinder velocity is an initial non-dimensional force of unit magnitude (*K*_{cy}(*τ*→0)→1), which decays rapidly with time. Initially, the force is opposite to the direction of cylinder acceleration. But after some time as *K*_{cy} changes sign, the inviscid hydrodynamic force on the cylinder is along the direction of impulsive acceleration. The time at which peaks in figure 2*a* corresponds to the zero-crossing time in figure 2*b*. The slower approach to steady state with increasing *M*_{∞} is clearly visible.

#### (ii) Sphere

The time evolution of the non-dimensional force on the sphere for varying *M*_{∞} is plotted in figure 5*a*. The non-dimensional force is defined to be , where *m*_{f} is the mass of the fluid displaced by the sphere. The analytical result of Longhorn (1952) corresponding to the limit *M*_{∞}→0 is also shown in the figure. In this limit, after the initial transient, the long-time value of non-dimensional force settles at 0.5 consistent with the added-mass coefficient of a sphere in incompressible flow. As for the cylinder, the approach to the constant force is non-monotonic. The peak value of approximately 0.6 is reached at a non-dimensional time of *τ*≈1.6.

The effect of Mach number on the peak value of the unsteady force coefficient and the effective added-mass coefficient is shown in figure 4*b*. The effect of Mach number is again substantial. The long-time asymptotic value for *M*_{∞}=0.5 is approximately 0.97 and a peak value of approximately 1.2 is reached at *τ*≈3.1. As before, the approach to steady state is delayed with increasing Mach number. However, compared with the cylinder, the steady state is approached more rapidly.

We differentiate the non-dimensional force presented in figure 5*a* and define a kernel as . The resulting response kernels are presented in figure 5*b* as a function of Mach number. The inviscid force on a sphere subjected to an instantaneous jump in relative velocity is then given by *m*_{f}(*c*_{∞}*u*_{0}/*R*)*K*_{sp}, where *u*_{0} is the jump in relative velocity. In the limit *M*_{∞}→0, Longhorn obtained(3.3)The approach to steady state is oscillatory, but the rapid exponential decay masks the oscillatory behaviour. The oscillatory nature of the kernel can be discerned from the computational results in figure 5*b*, at least for the higher values of *M*_{∞}. It may be conjectured that *K*_{cy} is also oscillatory. (Such behaviour can be seen in figure 2 for *M*_{∞}=0.39.)

### (b) Mach number expansion

Provided the flow around the cylinder or sphere is irrotational, the velocity field can be expressed in terms of a velocity potential *ϕ*. In the compressible regime, the governing equation for the velocity potential can be expressed as(3.4)where the speed of sound *c* is given by(3.5)with *c*_{0} denoting the stagnation speed of sound. It is understood that . We non-dimensionalize equation (3.4) with *R* as the length scale, *M*_{∞}*c*_{0} as the velocity scale and *T* as a time scale to be specified below. Denoting the non-dimensional quantities by a tilde, the resulting equation can be expressed as(3.6)Immediately after the sudden onset of a constant acceleration *a*, irrespective of its magnitude, the appropriate time scale will be such that the propagation of acoustics and the associated time derivative terms are both important. However, after the acoustics have propagated sufficiently far away from the cylinder or sphere, the appropriate time scale for the variation of the ambient flow is *T*=*M*_{∞}*c*_{0}/*a*. Then it can be deduced that compared with the next three terms in equation (3.6), all of which involve time derivatives, scale as , *α* and *α*, respectively. Provided that the non-dimensional acceleration satisfies the condition , the three terms can be ignored and the resulting equation is(3.7)

Thus, after an initial transient, the compressible potential flow around the cylinder can be considered quasi-steady and expressed in terms of the Janzen–Rayleigh expansion (see Oswatitsch 1956) to as(3.8)where the time dependence enters only through the instantaneous freestream velocity *U*_{∞} and Mach number *M*_{∞}. The above expansion can be substituted into the compressible form of the Bernoulli equation to obtain a Mach number expansion for pressure, which can be integrated around the cylinder to obtain the force. An analytic expansion for the effective added-mass coefficient can then be obtained as(3.9)which is also plotted in figure 4*a*. Unfortunately, with the present numerical simulation it is not possible to go to much smaller Mach numbers and the difference between the above expansion and the numerical results increases with increasing *M*_{∞}. Nevertheless, the above simple theory supports the qualitative behaviour of increasing added-mass effect at finite Mach number.

The role of the specific-heat ratio *γ* can now be examined. In the limit *M*_{∞}→0, equation (3.6) reduces to(3.10)This is the fundamental equation underlying the works of Miles and Longhorn. It indicates that in the limit *M*_{∞}→0 the flow and force evolution in response to infinitesimal acceleration will be independent of *γ*. On the other hand, if we assume *M*_{∞} to be finite, then we find that at short times the velocity potential is given by the complete equation (3.6) and the leading-order term including the specific-heat ratio is *O*(*M*_{∞}). At long times, however, equation (3.7) applies and the leading-order term depending on the specific-heat ratio scales as . Thus, the effect of the specific-heat ratio on the effective added-mass coefficient can be expected to be weak at small values of *M*_{∞}.

These theoretical results are corroborated by our computations. In figure 6, we present the effect of varying *γ* for three different values of *M*_{∞} on the unsteady force for a suddenly accelerated cylinder. As with the previously presented results, the acceleration is small enough that quasi-steady conditions are maintained. It can be seen clearly that for *M*_{∞}=0.2, the effect of *γ* is indiscernible. For *M*_{∞}=0.39, however, the effect of *γ* is noticeable for both the peak and steady-state values of the force coefficient.

## 4. Discussion

In the following, we discuss some specific questions related to the results presented in §3. We focus, in particular, on the relevance of these results to predict particle motion using force laws.

The first question that arises is: under what conditions will compressibility effects on unsteady forces be important? The relative importance of inviscid (added mass and pressure gradient) and viscous (history) unsteady forces arising from particle acceleration compared to the dominant quasi-steady drag scales as 1/(*ρ*+*C*_{M}) and , respectively, where *ρ* is the ratio of particle density to surrounding fluid density and *C*_{M} is the added-mass coefficient (Bagchi & Balachandar 2002). Thus, in the context of a particle moving in air, owing to the large density ratio encountered in most practical situations, the unsteady forces arising from particle acceleration are typically ignored.

It is important to note that the unsteady forces arising from the acceleration of the surrounding fluid do not follow the above scalings. For example, in the case of a particle injected into an accelerating flow, the ratio of inviscid and viscous unsteady forces to quasi-steady drag can be shown to scale as *Re d*/*L* and , respectively, where *Re* is the Reynolds number based on particle diameter *d* and relative velocity and *L* is the typical length scale of ambient flow variation (Bagchi & Balachandar 2002). Note that the density ratio does not appear in these estimates. The above scaling, although developed for an incompressible flow, will be appropriate even if compressibility effects are important. Thus situations can exist where the motion of a finite-size particle in a rapidly accelerating compressible flow can be influenced by unsteady forces, irrespective of the particle to fluid density ratio. For example, Tedeschi *et al.* (1992) and Thomas (1992) assessed the influence of the history force on the motion of a particle through a shock wave theoretically and numerically using the Basset–Boussinesq–Oseen equation (Crowe *et al.* 1997) and observed the instantaneous history force to be many times larger than the viscous drag force.

The second question is: how would the above results be used in practice? A simple model of the compressible inviscid force arising from the unsteady motion of the particle or the surrounding fluid can be written as follows:(4.1)where D** u**/D

*t*is the ambient fluid acceleration evaluated at the particle position, d

**/d**

*v**t*is the particle acceleration, and

*K*and

*m*

_{f}are chosen appropriately for the cylinder and the sphere, respectively. As cautioned by Miles (1951), Longhorn (1952), Yih (1995) and several others, we refrain from calling the first term on the r.h.s. of the above equation the ‘added-mass force’, since the time-dependent nature of the force does not always reduce to the form of a constant mass multiplying the instantaneous acceleration. In the case of a constant acceleration, for the above integral reduces to a constant times acceleration, thus permitting interpretation in terms of an effective added-mass coefficient, which reduces to the incompressible value as

*M*

_{∞}→0.

We next address the question of how much the incompressible results for the added-mass force are modified by compressibility. As has already been seen in figure 4, the effective added-mass coefficient more than doubles as the Mach number increases in the subcritical range. Furthermore, as pointed out by Miles and Longhorn, the effective influence of the unsteady inviscid force will be stronger if the acceleration is large over a short period of time than if it is small over a long period of time. To explore this behaviour at finite Mach number, consider a problem similar to that described at the beginning of §3: a cylinder or sphere is held fixed in a steady ambient flow of farfield Mach number *M*_{∞}. At time *t*=0, the cylinder or sphere is given a constant acceleration *a* in the direction opposite to the ambient flow until *t*=Δ*t*, when the acceleration is removed. The resulting force on the cylinder or sphere is given by equation (4.1). From this force, the work done by the cylinder or sphere at *t*=*t*^{*} can be expressed as(4.2)Rearranging the integrals we obtain , where(4.3)The work done is equal to the kinetic energy imparted to the fluid due to the acceleration. Thus for *t*^{*}→*∞*, *m*_{f}*ξ* can be interpreted as the added mass due to the fluid whose velocity changed by *a*Δ*t* owing to the acceleration of the cylinder or sphere.

The variation of *ξ*(*t*^{*}→*∞*, Δ*t*) for varying Mach numbers is presented in figure 7. For the sphere, Longhorn (1952) showed that in the limit of *M*_{∞}→0, the result reduces to(4.4)At finite Mach numbers, *ξ* was obtained through the numerical integration of equation (4.3) using the kernels presented in figures 2*b* and 5*b*. In the limit of sustained slow acceleration (i.e. Δ*t*→∞, *a*→0), the force on the cylinder or sphere remains constant except for the initial and final transients, and *ξ* approaches the effective added-mass coefficient *C*_{M,eff} presented in figure 4*a*,*b*. However, *ξ* increases as the duration of acceleration is reduced, and in the limit of Δ*t*→0 we observe a doubling of the effective added-mass coefficient. In fact, in the limit of Δ*t*→0, the contribution from the first term in equation (4.2) becomes zero and it can be shown that(4.5)Thus, the combined effect of finite Mach number and impulsive acceleration can intensify the added-mass effect to more than four times of what would be predicted based on incompressible theory.

We now comment on the use of the unsteady inviscid force given in equation (4.1). Consider the motion of a cylinder or sphere in response to an external force *F*_{ext} in a stagnant inviscid compressible fluid. The motion is governed by(4.6)where *m*_{p} is the mass of the cylinder (per unit width) or of the sphere and *m*_{f} is the mass of the fluid displaced by the cylinder (per unit width) or the sphere. If the external force is small and maintained over a long period of time, the resulting acceleration of the cylinder or sphere is given by *F*_{ext}/(*m*_{p}+*m*_{f}*C*_{M,eff}), which reduces to the incompressible result in the limit *M*_{∞}→0. On the other hand, if the external force is large and of very brief duration with a net impulse of *I*, the resulting velocity change due to the impulse is given by *I*/(*m*_{p}+*m*_{f}*C*_{M}) in an incompressible flow. In a compressible flow, however, an asymptotic constant velocity will be reached on an acoustic time scale after the impulse, and if an added-mass coefficient were to be computed in comparison with the above incompressible result, it will be dependent on both *M*_{∞} and the density ratio *ρ*. Similar behaviour can be observed in the results of Tracey (1988). Clearly, as cautioned by other authors, the concept of added mass is fraught with difficulty in compressible flow and it is advantageous to simply consider equation (4.1) as an expression for the unsteady inviscid force.

Finally, we note that in an incompressible flow, the added-mass force has been shown to be independent of the Reynolds number or viscous effects (see Rivero *et al.* 1991; Chang & Maxey 1995; Bagchi & Balachandar 2002; Mougin & Magnaudet 2002; Bagchi & Balachandar 2003; Wakaba & Balachandar 2007). The instantaneous nature of the added-mass force in incompressible flow precludes any interaction with the viscous response to the acceleration. In a compressible flow, the effect of the Reynolds number *Re* on the unsteady inviscid force will depend on the time scales. The acoustic, inertial and viscous time scales are *R*/*c*_{∞}, *R*/*U* and *R*^{2}/*ν*, respectively, where *U* is the characteristic relative velocity and *ν* is the kinematic viscosity of the fluid. The ratio of viscous to acoustic time scales will be proportional to *Re*/*M*_{∞} where *Re*=*UR*/*ν* and thus at sufficiently high Reynolds number the unsteady inviscid force can be expected to be independent of Reynolds number. However, at finite *Re*, quasi-steady and unsteady (history) components of the hydrodynamic force must be taken into account also.

Future work will consider supercritical flows. Under supercritical conditions, shock waves will appear, leading to rotational flow and non-zero steady drag coefficients. We expect the behaviour of the unsteady force coefficient to be qualitatively similar to that for subcritical conditions in that quasi-steady conditions are established for sufficiently small accelerations and long times.

## Acknowledgments

The authors gratefully acknowledge the support by the National Science Foundation under grant no. EAR0609712 and the National Center for Supercomputing Applications under grant no. EAR070006N. The authors thank Dr A.A. Kendoush for helpful discussions.

## Footnotes

One contribution of 11 to a Theme Issue ‘New perspectives on dispersed multiphase flows’.

- © 2008 The Royal Society