Unsteady separation and oscillations in a compressible wake of a cylinder

Fay & Goldburg (1963) first examined unsteady hypersonic wakes behind spheres in a ballistic range and noted a close analogy with incompressible wakes. Using the Roshko number $Ro$, which is independent of velocity, they found a linear relationship between $Ro$ and $Re_{h}$. Here, $Re_{h}$ is the Reynolds number evaluated at 90 degrees measured behind the front stagnation point to represent the flow separation point. The Roshko number $Ro$ is the product of the Reynolds number $Re_{h}$ and the Strouhal number, where the Strouhal number $St$ ($fD/V$) is defined in the usual way. It was found that the hypersonic wake and incompressible wake data were similar differing only in the value of linear slopes and zero intercepts of $Re_{h}$ (the value when the Strouhal number $St$ goes to zero) when $Ro$ was plotted against $Re_{h}$. The values of asymptotic Rayleigh Strouhal number $St^{\ast}$ and the zero intercept $Re_{T}$ values for the hypersonic wake and the incompressible wake were 0.7 $\&$ 0.23 and 3000 $\&$ 200 respectively. Fay & Goldburg (1963) remark: “The fact that $St^{\ast}$ and $Re_{T}$ are so little different at Mach number ($M$) 15 from their values at $M\ll$ 1 is indeed remarkable. For these reasons alone it seems very likely that large-scale vortices are generated behind bluff bodies in hypersonic flow and, as in the incompressible case, are the major disturbances that initiate a turbulent growth of the wake”. This statement is further reinforced when we note that Hayes & Probstein (1959) state, in connection with flow behind blunt bodies at hypersonic speeds, that: “ ….On blunt bodies in particular, local flow velocities are subsonic or moderately supersonic over much of the region of interest. …. the differences noted in hypersonic boundary layers are more of degree than of kind.” This is also consistent with the Mach number independence principle (Hayes & Probstein (1959)) which is often invoked for analysing high Mach number ($M\geq$ 4) flows over blunt bodies including wakes.

Fay & Goldburg (1963) point out that while the drag in incompressible flow is mainly due to the momentum defect caused by vortex shedding, the drag in hypersonic flow is entirely due to the momentum defect caused by the strong bow shock. Goldburg et al. (1965) and Goldburg & Florsheim (1966) further explored the similarity of hypersonic and incompressible blunt body wakes of various two and three-dimensional configurations to determine the proper scaling parameter to be used. Based on their low- speed experiments on spheres and cones (Goldburg & Florsheim (1966)), confirmed that total wake momentum thickness ($\theta$) could be used as a correlating parameter and then Goldburg et al. (1965) extended it to spheres and cones in hypersonic flow ($M=14$) in two ballistic ranges. The Roshko number, when plotted against a representative Reynolds number similar to that used by Fay & Goldburg (1963), and using $\theta$ as the scaling parameter, showed a linear relationship with a zero intercept when the Strouhal number went to zero.

Equation 1 below describes the total wake momentum thickness ($\theta$) and its relation to drag coefficient ($C_{D}$) (see Goldburg et al., 1965, for details) as:

Thus, for a two-dimensional cylinder, the total wake momentum thickness becomes ${\theta}/{D}=0.5\,C_{D}$, where $D$ is the cylinder diameter and $C_{D}$ is cylinder drag coefficient. Gerrard (1965)) has discussed the merits of using total wake momentum thickness $\theta$ as a characteristic length and the Strouhal number based on $\theta$ as a universal Strouhal number as has been used by Goldburg et al. (1965). He further suggests that at higher Reynolds numbers ($Re_{D}\geq 30,000$), Goldburg et al. (1965)’s and Roshko (1961)’s definitions of universal non-dimensional frequency ($St_{\theta}$ and $St_{R}$, respectively) are equally justifiable. It is also interesting to note that in the present case, the ratio of Roshko (1961)’s incompressible wake width $d_{w}$ and total wake momentum thickness $\theta$ is 1.3 which is close to the minimum value of 1.2 quoted by Gerrard (1965) as the critical Reynolds number approaches. Herein we follow the approach of Goldburg et al. (1965) and use the total wake momentum thickness $\theta$ as the characteristic length for analysing high supersonic and hypersonic cylinder wakes.

Gowen & Perkins (1953) measured drag of cylinders in the Mach number range 0.3 $\leq$ M $\leq$ 2.9 and Reynolds number range $5\times 10^{4}\leq Re_{D}\leq 10^{6}$, and found no significant changes in the value of the drag coefficient in the supersonic Mach number range. For high supersonic and hypersonic Mach numbers (M $\geq$ 4), the continuum flow cylinder drag coefficient is about 1.24 (Koppenwallner (1985)). Then, for a two-dimensional cylinder, the wake momentum thickness becomes ${\theta}/{D}=0.5\times 1.24=0.62$, where $C_{D}$ = 1.24 is the cylinder drag coefficient using the expression in equation 1.

Fay & Goldburg (1963) and Goldburg et al. (1965) use the so called the shoulder Reynolds number which is a representative Reynolds number for their correlations. This has a somewhat complicated expression involving flight (freestream) velocity, initial launch tube pressure of the ballistic range before firing, and a characteristic length. It is worth noting that the shoulder Reynolds number is evaluated at (inviscid) separation point which is taken to be at 90 degrees downstream of the front stagnation point of the sphere (behind the bow shock) when the flow is parallel to the freestream. Fay & Goldburg (1963) do not make clear as to why post-shock flow conditions are not used to evaluate the shoulder Reynolds number. On the other hand, Goldburg et al. (1965) do state that for cones, the shoulder Reynolds number was evaluated using flow conditions behind the cone shock.

Now, for blunt bodies in hypersonic flows, it is a well- established convention to use the Reynolds number based on flow conditions behind the bow shock as a similarity parameter (Potter (1967); Lukasiewicz (1973) and Koppenwallner (1985)). Using the normal shock relations across the bow shock, it is easy to relate the freestream Reynolds and Strouhal numbers to their respective post shock values:

It should be pointed out that some uncertainty exists in evaluating viscosity $\mu$ using the data of Schmidt & Shepherd (2015) and Thasu & Duvvuri (2022) especially the freestream values because of very low static temperatures. Using the Sutherland’s law and the power law with index $n$ = 0.666 for air (see White, 1991, pp. 27-29) and a reference temperature of 273.15 K, the power law overestimates the viscosity $\mu_{1}$ by 47$\%$ in the case of Schmidt & Shepherd (2015) and by 57$\%$ in the case of Thasu & Duvvuri (2022). This discrepancy reduces to less than 1$\%$ and 4$\%$ respectively for post shock values of viscosity $\mu_{2}$. Using the Lennard-Jones potential expression from Hirschfelder et al. (1949) to evaluate the viscosity did not change the results significantly. Appendix A shows these changes of viscosity with temperature in more detail.

Based on the above considerations, we now re-examine the cylinder oscillations data of Schmidt & Shepherd (2015), Thasu & Duvvuri (2022), and Awasthi et al. (2022). Figure 2 shows the data in terms of the Roshko number $Ro_{2\theta}\,(=St_{2\theta}\times Re_{2\theta})$ based on total wake momentum thickness ($\theta$) plotted against the post-shock Reynolds number, $Re_{2\theta}$. The data show a linear relationship with a zero intercept of $Re_{2T}$ = 1.85 $\times 10^{3}$ with a slope of 1.24, based on least square fit similar to that of Goldburg et al. (1965). The data of Awasthi et al. (2022) is omitted from the figure because the near wake in their case was turbulent.

Some important inferences follow from this analysis. Firstly, it appears that the wake momentum thickness ($\theta$) and the Strouhal number based on $\theta$ can be used as similarity parameters for supersonic and hypersonic wakes. Secondly, using the Roshko number $Ro$, it is possible to characterise the threshold Reynolds number, $Re_{T}$, at which the wake unsteadiness first appears leading to oscillations and eventual vortex shedding. The cylinder wake data being considered here are consistent with the data from Goldburg et al. (1965) for spheres and cones. Thirdly, use of both Strouhal number and Reynolds number as similarity parameters based on post-shock flow conditions for analysis of wakes of blunt bodies in high supersonic ($M_{\infty}\geq$ 4) and hypersonic flows can be justified.

From figure 2, the slope of the straight line is seen to be 1.24, so that

Considering further the data in figure 2, we can interpret that below $Re_{2\theta}=1.85\times 10^{3}$ (or $Re_{1D}=1.12\times 10^{4}$) oscillations will not occur as $St_{2\theta}$ = 0 then. This would imply that there is a threshold Reynolds number below which the wake will not exhibit unsteadiness or oscillations. This may be verified when we note that most of the well-known cylinder near wake investigations done in the past at high supersonic and hypersonic flows have all been at (adiabatic or near adiabatic wall) flow conditions wherein the post-shock Reynolds numbers generally range between $10^{3}\leq Re_{2D}\leq 10^{4}$. For example, see the data presented in Table 3 of Park et al. (2010). It is also seen from the same paper (figure 12 in their paper) that in all the cylinder wake experiments (except for the low supersonic Mach number experiments of Walter & Lange (1953), the base pressure (and hence drag) is nearly constant irrespective of Mach number and Reynolds number. It appears, therefore, that oscillations and subsequent vortex shedding is very much Reynolds number dependent as in incompressible flow. The reason, perhaps, is that hypersonic blunt body experiments quoted above were conducted at moderate to low freestream Reynolds numbers which result in stronger viscous effects in the wake (Dewey Jr (1965)) so that oscillations such as those observed by Schmidt & Shepherd (2015) or Thasu & Duvvuri (2022) were not seen or were too weak to be detected. This also seems to be the case in experiments of Nagata et al. (2019) wherein tests on spheres in a ballistic range at transonic and low supersonic Mach numbers in the Reynolds number range between $10^{3}$ and $10^{5}$ (based on freestream conditions and sphere diameter) showed little sign of oscillations for Reynolds numbers less than $10^{5}$. Further, Bashkin et al. (1998), in their numerical study, also note that when $Re_{1D}\geq 10^{4}$, at $M=5$, ‘flow restructuring’ and non-uniformity in the recirculating flow occurs including oscillations in velocity and temperatures as well as possibility of secondary separation. This is also in line with the observation of Hinman & Johansen (2017) that when both Mach number and Reynolds number ranges such as those in the experiments of Schmidt & Shepherd (2015) and Thasu & Duvvuri (2022) fall in the range wherein unsteadiness and eventual transition to turbulence in the near wake are possible.

These Reynolds number dependent phenomena would therefore suggest that the oscillations noted by Schmidt & Shepherd (2015) and corroborated by Thasu & Duvvuri (2022) are due to the relatively high Reynolds number of their experiments and a precursor to eventual transition to turbulence and vortex shedding. At present this is only a conjecture and further experimental confirmation at both higher and lower Reynolds numbers as well as at high supersonic and hypersonic Mach numbers is definitely warranted.

As discussed above, Fay & Goldburg (1963)and Goldburg et al. (1965) appear to be the first to use the Roshko number and the Strouhal number to analyse the wakes of blunt bodies at hypersonic speeds. Goldburg et al. (1965) then showed that the Strouhal number, based on the total wake momentum thickness ($\theta$) was independent of the shape of the body producing the wake. They based their conclusions on their experiments on spheres and cones. In using the Roshko number, these authors used a Reynolds number based on flight/freestream conditions. Since the Roshko number is independent of velocity, it is instructive here to analyse the Schmidt & Shepherd (2015) and Thasu & Duvvuri (2022) data along with those of Goldburg et al. (1965) to see if any such correlation exists for cylinder wakes.

Figure 3 shows these results in terms of freestream conditions unlike in figure 2, where we have used the post-shock Reynolds number as similarity parameter.Since we have used Reynolds number based on freestream conditions here, it is possible to estimate the error bars for Schmidt & Shepherd (2015) and Thasu & Duvvuri (2022) data as shown. It is seen that the cylinder data correlates reasonably well with the hypersonic sphere and cone data of Goldburg et al. (1965). The threshold Reynolds number $Re_{1T}$ is seen to be $6.21\times 10^{3}$ and the slope is 0.25. Note that this is not too different from that of Goldburg et al. (1965) value of 0.229, a difference of about 9%. It is also noteworthy that all the data of Goldburg et al. (1965) and some data of Schmidt & Shepherd (2015) are below $Re_{1\theta}=10^{5}$, while all the data of Thasu & Duvvuri (2022) are at $Re_{1\theta}>10^{5}$. We can then write:

Further elucidation of the Strouhal number based on total wake momentum thickness $\theta$ can be seen from figure 4, where $St_{1\theta}$ is shown plotted against $Re_{1\theta}$. An exponential curve is fitted through all the data points up to $Re_{1\theta}=10^{5}$. We make the following observations. All the data of Goldburg et al. (1965) are clustered close to the curve and they are all within $Re_{1\theta}=10^{5}$. There is a larger scatter in the sphere than the cone data. Note also that the lower Reynolds number data of Schmidt & Shepherd (2015) are closer to the curve as well as the sphere data. The high Reynolds number data of Schmidt & Shepherd (2015) and all the data of Thasu & Duvvuri (2022) show large deviations from the fitted curve and its asymptotic value beyond $Re_{1\theta}=10^{5}$. It is therefore suggested that data at Reynolds numbers less than $Re_{1\theta}=10^{5}$ may indicate laminar (regular) oscillations, while those greater than $Re_{1\theta}=10^{5}$ may exhibit irregular oscillations. This possibility is all the more likely when we note the observations by Chapman et al. (1958).

In the context of transition in shear layers after separation from a forward-facing step geometry, Chapman et al. (1958) point out that at Reynolds numbers (based on the interaction length and freestream conditions) below $1\times 10^{5}$, the separation is of the laminar type and between Reynolds numbers $1\times 10^{5}$ and $2.5\times 10^{5}$ it is of the transitional type. And above the Reynolds number of $2.5\times 10^{5}$ the separation remains transitional. Since the separation here is of free interaction type, we assume that the process is at least approximately, similar in the wake in the present instance. Then, the Reynolds number based on the total momentum thickness ($\theta$) is of the same order or equivalently $2\times 10^{5}$ based on shear layer length and freestream conditions, then half of Schmidt & Shepherd (2015) and all of Thasu & Duvvuri (2022) data are greater than this (critical) Reynolds number of $Re_{1\theta}=10^{5}$, and fall into the transitional range. A further comment relating to this is given in the discussion of Kendall (1962)’s paper below. Referring to figure 4 again, an asymptotic value for the curve (shown by a dashed line) can still be identified at $St_{1\theta}=0.25$. It is interesting to note that this Strouhal number – Reynolds number curve resembles the low speed $St-Re$ curve of figure 4 of Roshko (1954) which delineates various flow regimes of wake behind a cylinder.

Although Goldburg et al. (1965) did not claim that $St_{\theta}$ is a ‘universal’ Strouhal number, Gerrard (1965) and Bearman (1967) have labelled it as such. While Gerrard (1965) says that $St_{\theta}$ may qualify as a ‘universal’ Strouhal number at high enough Reynolds number, Bearman (1967) disagrees with this, because in an incompressible flow, $St_{\theta}$ shows dependency on the base pressure. Be that as it may, it does seem to correlate the cylinder as well as sphere and cone data reasonably well for laminar wakes at hypersonic speeds wherein the base pressure is fairly independent of both Mach and Reynolds number.

As mentioned in the Introduction, at the time this paper was nearing completion, the authors came across a very detailed experimental investigation of wakes behind cylinders and spheres at a supersonic Mach number of 3.7 and Reynolds numbers, $Re_{1D}$ ranging from $0.95\times 10^{4}$ to $4.2\times 10^{4}$ by Kendall (1962) conducted at the Jet Propulsion Laboratory of Caltech, USA (Technical Report No. 32-363). This investigation was conducted in a supersonic wind tunnel in which the freestream turbulence level was very low ($<$ 0.1%). This paper is not referred to either in Schmidt & Shepherd (2015) or Thasu & Duvvuri (2022). It is also not cited by Awasthi et al. (2022).

In his study, Kendall (1962) conducted extensive hot wire measurements and delineated four regimes of the wakes behind cylinders based on increasing Reynolds numbers $Re_{1D}$. They are, (a) laminar steady wake; (b) laminar wake with large amplitude oscillations; (c) irregular fluctuations, which eventually break down to develop into (d) a turbulent wake. Kendall (1962) then suggests a threshold Reynolds number $Re_{1D}$ = $1.2\times 10^{4}$, below which oscillations do not occur. This is the same as $Re_{1T}$ that we defined earlier and below which oscillations do not occur. It is thus consistent with and confirms our analysis.

Interestingly, Kendall (1962) further remarks that a possible reason for the origin of these oscillations may be three-dimensional disturbances as a result of finite aspect ratio of cylinders although in his experiments, most of the time, the aspect ratio varied from 12 to 72. This is more than the aspect ratio of 10 suggested by Dewey Jr (1965) to achieve reasonably two-dimensional flow. In comparison, the aspect ratio in Schmidt & Shepherd (2015)’s experiments varied from 2.85 to 40. The aspect ratio in Thasu & Duvvuri (2022)’s experiments was quite small, between 2.4 and 4. It is therefore quite possible that larger cylinder data of Schmidt & Shepherd (2015) and all of Thasu & Duvvuri (2022) data are affected by spanwise flow disturbances.

Another interesting finding from Kendall (1962)’s experiments was that the oscillation frequencies varied between 50 Hz to 850 Hz, giving Strouhal numbers as low as 0.001 to 0.016. This appears to be based on peak fluctuation energy measured at a location of 4 diameters behind the base when the wake was still laminar. However, he comments that the fluctuation energy spectrum did not change much along the axis which he attributed to the low Reynolds number and near-stability of the wake. The wake became turbulent only at the highest Reynolds number $Re_{1D}=4.2\times 10^{4}$ of his experiments. Kendall (1962) comments that although in many ways there were similarities with low speed wakes, they could not detect frequencies with Strouhal number of 0.21. Kendall (1962) found that sphere wakes were either completely steady or completely unsteady depending on the threshold Reynolds number. At lower Reynolds numbers it was steady but became highly unsteady and three-dimensional at higher Reynolds numbers ($Re_{1}D>2.5\times 10^{4}$). This is clearly reflected in the relatively large scatter of sphere wake data (compared to cylinder and cone) as seen in figure 4, , especially in the higher Reynolds numbers range. The threshold Reynolds number in the case of spheres was nearly twice that of the cylinder.

As illustrated in the schematic diagram of figure 1, the deceleration of the flow behind the cylinder and the coalescence of the shear layers shed by the separated boundary layer forms a ‘neck’. The neck region at reattachment of the near wake is typical of strong viscous effect usually associated with low Reynolds number and high Mach number hypersonic flow. This is unlike the Chapman isentropic re-compression. Lees & Reeves (1964) and Dewey Jr (1965) have shown that with an initial finite thickness shear layer, the wake neck thickness varies with Reynolds number. Figure 5 shows this variation for the data of Schmidt & Shepherd (2015) along with Dewey Jr (1965). For comparison, figure 5 is drawn on a log-log scale, and it is seen that approximately $\delta_{w}/D\sim Re_{1D}^{-1/2}$ with $\delta_{w}/D=0.61$ at the threshold Reynolds number $Re_{1T}=1.1\times 10^{4}$. Note that this is close to the wake momentum thickness ratio $\theta/D=0.62$. It is also interesting to note that the low Reynolds number data of Schmidt & Shepherd (2015) and Dewey Jr (1965) (which are all laminar) agree quite well below $Re_{1D}=10^{5}$.

It has been known for quite some time (see, Shaw (1956) and Lighthill Rosenhead (1963)) that in incompressible flow ($M$ ¡ 1) behind a cylinder, the eddies shed into the wake and their frequency is associated with acoustic disturbances generated at separation points. This hypothesis is also supported by the experiments of Schmidt & Shepherd (2015) as discussed earlier. Wu et al. (2004) in their study of flow past a cylinder in low Reynolds number incompressible flow note that separation point is not stationary but fluctuates up to an angular displacement of about 3^∘. Fage (1929) provided a remarkably detailed account of the process of separation of flow over a large cylinder at low speeds in the Reynolds number range $1\times 10^{5}$ to $3.5\times 10^{5}$. Further, Fage (1929) showed that, depending on the Reynolds number, the separation process took place over 7.5^∘ and 9^∘ of angular displacement at the lowest and highest Reynolds numbers, respectively. In other words, the separation took place between 76.5^∘ and 84^∘ at the lowest Reynolds number and between 96^∘ and 105^∘ at the highest Reynolds number, where the angular displacement is measured from the front stagnation point. Fage (1929) further showed that the separation process begins at the first inflection point (just after the pressure minimum) and is completed by the second inflection point and during which the pressure is near constant. This is followed by the separating shear layer forming a vortex sheet and growing by entraining the outer fluid into the dead air region. It is during this process that acoustic pulses are generated which are then propagated downstream resulting in oscillating flow and eddies.

Schmidt & Shepherd (2015), however, did not observe any unsteadiness in or movement of separation points in their experiments although their inviscid simulations did show movement of separation points over the surface. Secondly, in their experiments, the separation point varied considerably, from 105^∘ presumably at the lowest Reynolds number ($2\times 10^{4}$) to 120^∘, at the highest Reynolds number ($5\times 10^{5}$), which does indicate some Reynolds number effect. This, however, is not made clear. In contrast, Bashkin et al. (1998)’s numerical simulations showed only small Reynolds and Mach number effect. This angular variation of separation point in Schmidt & Shepherd (2015) experiments with Reynolds number may be similar to what was observed by Fage (1929) in his low speed cylinder experiments. Further, for all the data of Schmidt & Shepherd (2015), Thasu & Duvvuri (2022), and Awasthi et al. (2022), the Bashkin et al. (1998) empirical relation (based on their numerical simulations) predicts a near constant separation angle of 131 deg, while the actual separation angle varied between 8% to 20% (Schmidt & Shepherd (2015)), 9% (Thasu & Duvvuri (2022)), and 5% (Awasthi et al. (2022)) (see Table 2).