Drag Measurement in Unsteady Compressible Flow Part 2: Shock Wave Loading of Spheres and Cones

 

 

B.W. Skews; M.S. Bredin; M. Efune

Professor. FSAIMechE, Director Flow Research Unit, University of the Witwatersrand, South Africa. E-mail: beric.skews@wits.ac.za

 

 

---

ABSTRACT

The measurement of aerodynamic drag encountered by spheres and cones exposed to strongly transient flows such as those due to shock wave impact is undertaken. A specially designed stress wave drag balance is used to record the forces during the transit of the wave over the body and for a short time thereafter. Post-shock Reynolds numbers were varied over the range from 2 x 10^s to 6 x 10^s. Numerical simulations were run in order to help interpret the drag variations.
For the sphere the maximum drag occurs just before the shock wave reaches the equator whereas for the cones it occurs when the shock reaches the cone base. The post-shock drag variations are shown to be largely due to wave reflections within the shock tube. Good agreement is obtained between the measured and numerically predicted peak drag values. Comparisons are also made with published steady state drag values, predominantly obtained from incompressible data, as well as steady state compressible values from numerical simulations at the test post-shock Mach numbers.

---

 

 

1. Introduction

The background to this study has been given in part 1 of this work. particularly relating to drag determination in unsteady flows. It is established that the forces acting on a body in an unsteady flow field can be significantly different from those obtained in steady flows at the same Reynolds and Mach numbers. The latter data has generally come from wind or water tunnel facilities where the body is held in a force balance and the conditions in the uniform oncoming flow changed. There is, however, very little data on the effects that flow unsteadiness may have on the forces, notwithstanding that in real practical flow situations there are generally flow variations with time.

There is very limited data available on the dynamic drag variation on a sphere. The sphere data has mostly been obtained from measuring the motion of an unconstrained body after having been exposed to plane shock wave passage. Only post-shock effects could be established since body movement is too small during the diffraction phase as the shock traverses the body.

The first work to evaluate the shock-induced drag coefficient on a sphere in the sub-millisecond range was by Bredin and Skews¹. This is expanded on in the current paper. There is one similar experimental study using spheres which is comparable to this work. Tanno et al.² suspended a sphere from a thin wire in a vertical shock tube and then impacted it with a Mach 1.22 shock from below. They used an accelerometer within the sphere as the sensing device, and following data smoothing and deconvolution extracted the drag.

Only one example was found in the literature regarding the measurement of unsteady drag on shock-loaded bodies other than spheres. Tamai et al.³ suspended models in the same facility as Tanno et al.² under the same test conditions. Model shapes were cones with smooth and coarse surface finish, a double cone, a sphere, a 2:3 ellipsoid and a cylinder. Drag forces were found to reach a maximum when the incident shock wave reaches just before the equator and then decreases to a minimum value. Depending on body shape, the minimum value is negative and is maintained for a few hundred microseconds.

No information was found in the literature for cones under steady flow conditions at the flow Mach numbers found in the current study. Available data is for incompressible flows or for Mach numbers in the transonic regime. Numerical simulation was thus used to obtain some comparative data. One of the advantages of using cones is that the separation point remains fixed at the base and thus, unlike spheres, would be expected to be less sensitive to flow noise and Reynolds number effects under steady flow conditions. Flow separation conditions for a sphere in unsteady flow conditions is an unexplored field.

Under steady flow conditions the free stream density, ρ, and velocity, V, are unambiguous and thus a reliable drag coefficient may be defined in the conventional way as:

where A is the body frontal area. In fully unsteady flow, however, the choice of reference conditions is arbitrary. These can then be taken as the instantaneous values of density and velocity as indicated in part 1 of this study, so that Reynolds number and dynamic pressure also vary as a function of time, or in the normal shock wave impingement case as the pseudo-steady conditions behind the moving normal shock. In this paper in order to avoid the ambiguity of choice of reference conditions the actual drag values will generally be compared between measurement, simulation, and data from the literature.

The stress wave drag balance used in this work consists of a 3 m long, 8 mm diameter brass rod positioned along the axis of a special shock tube and fitted with semi-conductor strain gauges and housed within a windshield. The raw strain data is smoothed and deconvoluted using the unit impulse response function, obtained from calibration, to extract the forces acting on the body as described in part 1. The balance has a response time of better than 20 us, a test time of 1.5 ms, and a force measurement error of less than 15%.

In the tests described below flow conditions will frequently be defined in terms of the incident shock Mach number moving into stationary air at ambient conditions (nominally 0.83 bar, 20 °C). The corresponding post shock flow Mach numbers and Reynolds number per meter are given in Figure 1.

 

 

2. Sphere Drag

A single 50 mm diameter aluminium sphere was tested at shock Mach numbers ranging from 1.08 to 1.31, corresponding to Reynolds numbers from 100 000 to 600 000 and flow Mach numbers of 0.132 to 0.401. Typical results for Reynolds number, drag, and coefficient of drag are given below. The Reynolds number is determined from measurements taken 626 mm up-stream from the sphere and the results at the sphere predicted using the method of characteristics as described in part 1. This thus represents an equivalent instantaneous free-stream Reynolds number at the sphere position. Calculating Reynolds number from the instrumentation taken at the sphere position gives similar results except for the spike induced from the reflected shock wave.

The drag extracted for this case after deconvolution is given in figure 3.

 

 

 

 

The instantaneous drag coefficient determined from the actual drag at any time and the corresponding instantaneous flow density and velocity is given in figure 4.

 

 

 

 

 

 

 

 

There are three distinct stages evident. The primary stage is that from when the shock first hits the sphere to the point at which the initial peak returns to some form of steady state value. During this stage the shock reflection and diffraction over the sphere are predominant. This will be referred to as the diffraction phase and occurs between 0.1 to 0.4 ms for the above test. Just before the end of this stage the drag becomes negative.

The shock wave reflecting off the sphere propagates upstream and to the outer wall of the shock tube from where it is reflected back on to the sphere. This causes high amplitude, high frequency drag fluctuations between 0.4 and 0.6 ms. This is the secondary stage and more detail of the flow will become evident from the CFD simulations discussed later.

The third phase is from 0.6 ms to the end of the test. These are large fluctuations and the reasons for them are not entirely clear since at that stage the main wave system is well attenuated in the region of the sphere. They may be related to wake and separation behaviour and vortex shedding. More detailed investigation would be needed to obtain complete clarity. An interesting investigation in this regard comes from tests done by Tyler and Salt⁴. A sphere was held by an electromagnet at the top end wall of a shock tube and released just before shock arrival. The data exhibited periodic discontinuities in acceleration thereby indicating significant abrupt changes in drag.

The coefficient of drag curve (figure 4) is compared with that from the standard drag curve for a sphere at the same mean post-shock Reynolds number. This is for incompressible flows and the separate effects of compressibility will be discussed in the next section on cones. Also shown is the mean drag coefficient for times after the diffraction phase is over. That it is higher than the standard drag curve correlates with other shock tube tests using unconstrained spheres as discussed in part 1.

Drag curves for eleven tests are overlaid in figure 5 in order to see the relationship with change in shock Mach number. The cunes corresponding to the weakest and the strongest shock are shown with different line styles. The direction of increasing shock strength is shown with arrows. It is clear how the initial force increases with increasing shock strength, as expected, and how the increase in drag at 0.8 ms occurs first with the faster moving stronger shocks. The amplitude of this wave also decreases as the incident shock strength decreases.

An examination of the drag during the diffraction phase is given in figure 6. The initial steep drag rise results because the pressure behind the wave reflected off the sphere and the sphere surface is significantly higher than the stationary air surrounding the remainder of the surface. The maximum measured drag, shown with crosses, is compared to the time at which the incident shock passes the sphere equator, shown with circles. The maximum consistently occurs before the maximum area of the sphere is exposed to the shock. It should be noted that the reflected shock propagates outwards from the sphere in a spherical manner and decreases in strength as it propagates. Consequently the reflected pressure apparently has reduced somewhat by the time the shock wave reaches the centre of the sphere so that the effect of the lower pressure overtakes the effect of increased area.

Interpretation of the shape of the experimental drag curves is conveniently done by comparing it with numerical predictions. Figure 7 shows the drag curve predicted using the Fluent⁵ CFD code for three incident shock Mach numbers, with the density contour plots at specific instants of time given in figure 8.

 

 

The first major point to note is the strong similarity between the experimental and numerical drag curves. A number of points are of interest in comparing the drag curves and the contour plots. A few particular features are marked.

The first point of interest (A, in figure 7) is the peak pressure as discussed above. The drag then drops off quite sharply as the shock diffracts around the back of the cylinder raising the pressure there. The diffracted wave then reflects off the sting support, close to the axis of symmetry raising the pressure still further (figure 8). Originally when it was noted that there was a period of negative drag this was thought to be due to this focusing effect on the rear surface, but as noted the lowest drag occurs substantially later where there is a combination of this focussing and the imploding shock reflected off the walls of the tube. This occurs between 0.4 and 0.42 ms corresponding to points B and C on figure 7. A similar significant delay before the appearance of negative drag occurred in Tanno's² tests at Mach 1.22. In their case the ratio of tube to sphere diameter was larger and they did not explore the effects of tube confinement. Thus the delays in the appearance of negative drag, significantly beyond a time when the diffracted shock encloses the rear of the sphere, should not be specifically ascribed to shock focussing. On the other hand the magnitude of the negative drag relative to the maximum was much lower in the tests of Tanno et al. than in the present tests so the effects of confinement on the wave pattern cannot be ignored. To account for the significantly slower decrease in drag from the maximum compared to the rapid rise as the shock passes from one side of the sphere to the other, other mechanisms need to be considered. The unsteady growth of the boundary layer may be one such factor which would need further investigation.

The second peak in drag occurs when the wall shock engulfs the front of the sphere at 0.52 ms (point D). More complex interactions occur thereafter, including a further short period of negative drag.

Because of the uncertainty of CFD code predictions for transient compressible separation phenomena on curved surfaces, direct comparison between CFD and experiment will not be done for the sphere at this stage, except to note the broad agreement between figures 4 and 7. More detailed comparisons will be done for cones in the next section, where the separation point is always at the edge of the base of the cone.

Even in steady flows similarity parameters accounting for both Mach and Reynolds number effects have not been documented. Curves presented in Morrison⁶ give some indication of the difficulty. Nevertheless the very regular variation of the drag with shock Mach number evident in figures 5 and 6 suggest that the curves may be collapsed by suitable non-dimensionalisation. Preliminary attempts have shown some success, but then only for the diffraction phase. Figure 9 shows the variation of the maximum instantaneous drag coefficient with incident shock Mach number. If, as a first approximation, the drag is taken to be inversely proportional to Mach number, and the drag coefficient is multiplied by (M-l) then it is found that the diffraction phase of the drag variation collapses to a significant extent as shown in Figure 10, including the negative drag portion of the curve. Further simplification would also require modification of the time scale. Such analysis might be useful for cases where there is an input shock wave followed by steady post shock flow, but is usefulness in more general unsteady flows will be limited.

 

 

 

 

These tests, and all others ever done in the sub-millisecond range, were done with bodies exposed to a plane shock wave and therefore have a step change in flow velocity. However, the experimental rig was designed to produce a range of more general unsteady flows. This adds considerably to the complexity of the flow but offers a wide range of unsteady flows that could potentially be explored. Such work is still in its infancy and initial results of some preliminary tests are presented here.

Consider the case shown in figure 11 which was obtained by modifying the shock tube with some gaps along its length and changing the test section position. It consists of an initial compression wave followed by two shocks, a short steady flow region and then a further compression.

 

 

The measured drag variation is shown in figure 12 and is completely consistent with the previous findings. As the initial compression engages the sphere the drag increases followed by a drop as the wave passes over the rear of the body. This is followed by typical peaks due to the diffraction of the two shock waves. At the time of the trailing compression there is again a general increase followed by a decrease in the drag.

 

 

3. Cone Drag

Four different aluminium cones were tested having half-vertex angles of 15. 20, 25, and 30 degrees respectively, denoted as cones 1,2,3. and 4. They all had a base diameter of 50 mm. Plane shock wave loading was employed over an incident shock wave Mach number range of 1.12 to 1.31 with corresponding post-shock Reynolds numbers between 2 x 10⁵ and 6 x 10⁵.

Numerical predictions for the drag at three Mach numbers are given in figure 13 for cone 1, followed by a series of density contour plots showing the evolution of the flow for M = 1.5, in figure 14. Consider the Mach 1.5 curve. The drag starts to rise at 0.14 ms. which is the time at which the incident shock wave arrives at the apex of the cone. Immediately after the Shockwave impacts the cone, a bow-shaped reflected shock forms and propagates upstream as illustrated in figure 14. Closer inspection shows that the type of reflection is Mach reflection. The region between the reflected wave and the cone experiences a very high pressure while the downstream portion of the cone is still at atmospheric pressure. The increase in drag is due to this large pressure differential.

 

 

 

 

The drag reaches a maximum when the incident shock reaches the base of the cone. At this point, the shock wave starts to diffract around the base forming a vortex. As the incident shock wave propagates further inward onto the axis behind the base of the cone, it impacts and reflects off the cone support (corresponding to the SWDB of the experimental apparatus). At the same time, the first reflected shock wave from the cone has reached the wall of the shock tube and been reflected.

The drag starts to decrease less rapidly at approximately 0.34 ms before rising slightly again at 0.44 ms. At 0.47 ms (Point B) the drag drops once more. This behaviour is consistent with the vortex being shed from the cone.

The flow field becomes increasingly difficult to analyse as the number of shock reflections off the shock tube walls and the model support increases. The decrease in drag between point B and point C corresponds to the reflected waves from the tube wall impacting the base of the cone thus creating a force on the cone in the upstream direction. Between point C and point D these reflected waves are travelling over the sides of the cone and the direction of the pressure differential is downstream once more, thus causing the drag to increase. Further fluctuations diminish steadily over time, consistent with the attenuation of the reflected waves.

The shock wave interactions and drag variations on cones 2, 3 and 4 are much the same as described for cone 1. There are some differences though. The shock takes longer to traverse the cones as the vertex angle decreases. Thus the peak is delayed as seen in figure 15. The maximum drag also decreases, presumably because the curved reflected wave is reduced in strength having propagated for a longer time. The first small spike at B, in figure 13, also smoothes out as the half-vertex angle is decreased and it has vanished for cone 4. One aspect of the reflection that is thought to influence the drag in this way is the shape of the reflected wave.

 

 

The mode of reflection of the incident wave off all of the cones is Mach reflection, with the length of the Mach stem increasing as the half-vertex angle decreases. The mode of reflection is not thought to play a critical role in the drag loading.

A typical set of experimental results for cone 1 over a range of Mach numbers is given in figure 16. Maximum drag occurs as the incident shock reaches the cone base. The drag variation is similar to that obtained from the numerical solution but the variations are more marked. Since the agreement will be shown to be very good between the experiment and the simulation during the primary diffraction phase there is no reason to suspect the amplitude of the subsequent pulses. Further simulations would be needed to ensure that the complex wave patterns for the weaker transverse waves are not smeared out in the simulation possibly because the smaller scale structures have not been fully captured by the CFD resolution used.

 

 

An interesting feature is evident in figure 16. The pressure peaks are all in phase below 1.0 ms and then again at 1.5 ms and even for tests at longer times. At 1.25 ms, however, there is apeak in drag for shock Mach numbers above 1.2, whereas for lower Mach numbers there is a drop. This is found to be consistent for all ten Mach numbers tested between Mach 1.1 and Mach 1.3. This suggests a change in the unsteady development of the wake. The other cones also show discrepancies in the same time range but with different details. This is an issue needing further investigation.

A direct comparison between the experimental and numerical results is given in figure 17 and 18 for cones 1 and 3. The drag value determined from published steady state drag coefficients for the appropriate post-shock Reynolds number (incompressible) is included. In addition simulations were run to determine the steady state drag at the correct post-shock Mach and Reynolds number for the given incident shock strength. This then gives a direct measure of the effects of compressibility under steady state conditions.

 

 

 

 

These results show that the average of the post-shock unsteady drag is of the same order of magnitude as the steady state compressible drag. The CFD results mimic the drag variation for the diffraction phase reasonably well. They also tend to show similar oscillations following the diffraction phase but with much reduced amplitude. Experimental flow visualisation and higher resolution simulation of the unsteady flow patterns, particularly relating to the unsteady vortex development and evolution will be required to explore the source of these oscillations in greater detail.

Typical peak drag values from the experiments and the simulations are given in table 1 for all four cones impacted by an incident Mach 1.27 shock wave.

 

 

As shown in figure 16 the maximum unsteady drag values increase sharply with an increase in Mach number. Both the incompressible and compressible steady state drag values also increase as a function of Mach number, though at a slower rate than the unsteady drag as shown in figure 19. Curves for all four cones show the same pattern. This is a dramatic illustration of the effect unsteadiness can have on the drag of a body, accepting that the shock wave case is a severe example.

 

 

4. Conclusion

A stress wave drag balance is used to measure the instantaneous drag variation of spheres and cones exposed to very short duration transient compressible flows. Computational simulations shorv a satisfactory degree of agreement with the experimental results.

 

References

1. Bredin M and Skews BW, The measurement of drag in unsteady compressible flow, 23^rd Int. Symp. on Shock Waves, F Lu (Ed.), Arlington, 2001, 463-471.

2. Tanno H, Itoh K, Saito T, Abe A, and Takayama K, Interaction of a shock with a sphere suspended in a vertical shock tube, Shock Waves 2003, 13, 191-200.         [ Links ]

3.Tamai K, Ogawa T, Ojima H, Falcovitz J, and Takayama K, Unsteady drag force measurements over bodies with various configurations in a vertical shock tube, 24^th Int. Symp. on Shock Waves, Jiang ZL (Ed.), Springer, 2004, 1000-1005.

4.Tyler AL and Salt DL, Periodic discontinuities in the acceleration of spheres in free flight, ASME Pub. 77-WA/FE-3, 1977.

5.FLUENT Inc., http://www.fluent.com/ 1 February 2007.

6.Morrison RB, Design data for aeronautics and astronautics, Wiley, 1962.