2. Compressible wake flow

Relaxing the incompressibility condition presents a more general problem of the wake flow in the compressible regime, where the Mach number becomes an additional (and independent) governing parameter along with the Reynolds number. The free-stream flow Mach number is defined as $M_\infty = U_{\infty }/a_{\infty }$ , where $a_{\infty }$ is the acoustic wave speed. The two-dimensional cylinder wake at supersonic and hypersonic Mach numbers, where the flow is compressible, is very distinct from its incompressible flow regime counterpart. As a representative example, figure 1(b) shows an instantaneous flow density gradient map for a cylinder at $M_\infty = 6$ . Distinction between incompressible and compressible cylinder wake flows are qualitatively very evident in figure 1. For $M_{\infty } \gt 1$ , shock waves appear in the flow and the vortex shedding phenomenon disappears. The near-wake region is characterised by the formation of two shear layers, which are symmetric about the cylinder centreline.

Figure 1. (a) An instantaneous snapshot of cylinder wake flow in water at $M_\infty = 0.01$ and $Re_D = 1.4 \times 10^5$ (Djeridi et al. Reference Djeridi, Braza, Perrin, Harran, Cid and Cazin2003). The shedding vortices from the top and bottom of the cylinder aftbody, with opposing senses of rotation, are visualised by controlled cavitation (Djeridi et al. Reference Djeridi, Braza, Perrin, Harran, Cid and Cazin2003). (b) An instantaneous density gradient map around a circular cylinder at $M_\infty = 6$ and $Re_D = 2.8 \times 10^5$ in air obtained using the optical imaging technique of schlieren (Thasu & Duvvuri Reference Thasu and Duvvuri2022).

The canonical compressible cylinder wake problem has received much less scientific attention as compared with the incompressible problem. The near-wake periodic flow unsteadiness in the incompressible regime has been the subject of several detailed studies (Wu Reference Wu1972; Williamson Reference Williamson1996), which span a period of over a hundred years (Govardhan & Ramesh Reference Govardhan and Ramesh2005), and the unsteadiness mechanisms are reasonably well understood (Abernathy & Kronauer Reference Abernathy and Kronauer1962; Gerrard Reference Gerrard1966; Perry et al. Reference Perry, Chong and Lim1982; Williamson Reference Williamson1996). Whereas, for the supersonic/hypersonic flow regime, it is only within the past decade that the near-wake flow region was discovered to exhibit coherent and periodic oscillations (Schmidt & Shepherd Reference Schmidt and Shepherd2015; Awasthi et al. Reference Awasthi, McCreton, Moreau and Doolan2022; Thasu & Duvvuri Reference Thasu and Duvvuri2022). (A schlieren video of these flow oscillations is available as supplementary material to Thasu & Duvvuri Reference Thasu and Duvvuri2022). The oscillations were found to have a single characteristic frequency. Interestingly, the oscillation Strouhal number, formed using the shear layer length and free-stream velocity, exhibits universal behaviour. At high-supersonic and hypersonic Mach numbers and across a range of Reynolds numbers, the Strouhal number was found to be invariant, taking a value of approximately 0.48 (Schmidt & Shepherd Reference Schmidt and Shepherd2015; Thasu & Duvvuri Reference Thasu and Duvvuri2022).

The nature of cylinder wake unsteadiness in the compressible flow regime is fundamentally very different from the incompressible flow scenario. Driven by experimental observations, the literature proposes the following hypothesis: an aeroacoustic feedback mechanism in the near wake causes and sustains flow oscillations (Schmidt & Shepherd Reference Schmidt and Shepherd2015; Thasu & Duvvuri Reference Thasu and Duvvuri2022). Based on this hypothesis, here we develop a quantitative aeroacoustic model to explain the oscillations observed in the supersonic/hypersonic cylinder wake. The model successfully predicts the oscillation frequencies reported from experiments, and thereby provides a clear physical understanding of the phenomenon. Further, the model also explains the experimentally observed universal behaviour and informs us of flow regimes where deviations from universal behaviour are to be expected. Table 1 summarises the experimental data available in the literature for near-wake oscillations at high Mach numbers. The shear layer length is denoted by $S$ (see figure 1 b), and the two Strouhal numbers are defined as $St_{D} = (f\,D)/U_\infty$ and $St_{S} = (f\,S)/U_\infty$ . Data from table 1 are used for the present model development exercise and for validation of model predictions.

Table 1. Strouhal number data from earlier experiments.

3. Aeroacoustic feedback mechanism

We begin with a brief description of the key flow features (see figure 2). Given the flow symmetry about the cylinder centreline, only the top half of the cylinder and flow are depicted in the figure. A steady bow shock wave forms upstream of the cylinder, and the flow downstream of the shock wave in region 2 is subsonic. The subsonic fluid accelerates as it moves around the cylinder, attains Mach 1 at the sonic line and further accelerates to supersonic Mach numbers as the flow expands around the cylinder. Further downstream the flow separates from the cylinder surface (due to the limitation on the maximum turn angle of supersonic flows) and generates a separation shock wave. Flow separation on the top and bottom surfaces of the cylinder results in the formation of symmetric supersonic shear layers on either side of the centreline. The region of intersection of the two shear layers is referred to as the ‘neck’ of the wake (marked in figure 1 b). The shear layers and the cylinder surface enclose two regions of subsonic recirculating flow with opposing senses of rotation. Downstream of the neck the flow turns parallel to the free stream through tail shock waves that are generated at the neck region.

Figure 2. A schematic illustration of the flow structure over the top half of a supersonic/hypersonic cylinder. Flow structure in the bottom half is symmetric (about the cylinder centreline). Shock waves are shown in red, the stagnation streamline in blue and recirculation region in dashed grey curves.

4. An analytical model

By considering the feedback loop to be linear, and matching the wavespeed to wavelength ratio (i.e. frequency) between downstream-propagating vortical disturbances and upstream-propagating acoustic disturbances, the following expression can be obtained for the disturbance frequency $f$ (Powell Reference Powell1953, Reference Powell1964; Rossiter Reference Rossiter1964; Heller et al. Reference Heller, Holmes and Covert1971):

(4.1) \begin{equation} f = \left (\frac {1}{S}\right )\left (\frac {m-\phi }{\dfrac {1}{a_{{r}}}+\dfrac {1}{kU_{i}}}\right ). \end{equation}

Here, $U_{i}$ is the flow velocity downstream of the separation shock wave (see figure 2), $kU_{i}$ is the propagation speed of vortical disturbances (with $k$ being a constant), $a_{{r}}$ is the speed of the acoustic waves that propagate upstream along the shear layer of length $S$ , integer $m$ is the mode number for the oscillations and $\phi$ is the phase difference between the vortical and acoustic disturbances at the neck region. It is noted that, by construction, $\phi$ accounts for the total phase difference between the vortical and acoustic disturbances at both the ends, i.e. the separation point and neck region (Rossiter Reference Rossiter1964). The flow oscillation time scale is taken to be the same as the vortical and acoustic disturbance time scale, i.e. $f$ also denotes the wake oscillation frequency. The Strouhal number $St_D$ of wake oscillations can then be written as

(4.2) \begin{equation} St_D = \frac {fD}{U_\infty } = \left (\frac {D}{S}\right )\left (\frac {m-\phi }{\dfrac {U_\infty }{a_{{r}}}+\dfrac {U_\infty }{kU_{i}}}\right ). \end{equation}

4.2. Recirculation zone

In the recirculation region, the flow velocities are relatively very low, and hence the region is regarded as stagnant. The acoustic speed in this region, denoted as $a_{{r}}$ , is estimated by determining the average local temperature $T_{{r}}$ within the recirculation region. The recovery temperature is a good estimate for $T_{{r}}$ since it accounts for viscous losses in the shear layer (Anderson Reference Anderson1984; Kumar et al. Reference Kumar, Sasidharan, Kumara and Duvvuri2024). The recovery temperature is given by

(4.3) \begin{equation} T_{{r}} = \left [\frac {1+\sqrt {Pr}\left (\frac {\gamma -1}{2}\right )M_{i}^2}{1+\left (\frac {\gamma -1}{2}\right )M_{i}^2}\right ]T_0\,, \end{equation}

where $T_0$ is the stagnation temperature, $Pr$ is the Prandtl number, $\gamma$ is the ratio of specific heats for the fluid and $M_{i}$ is the Mach number downstream of the separation shock wave. With $R$ as the gas constant, $a_{{r}}$ is then written as

(4.4) \begin{equation} a_{{r}} = \sqrt {\gamma R T_{{r}}}\,. \end{equation}

In passing, we note that in the cavity flow literature, $a_r$ is estimated using the static temperature (Rossiter Reference Rossiter1964) or stagnation temperature (Heller et al. Reference Heller, Holmes and Covert1971).

4.3. Shear layer instability

The propagation speed of the vortical disturbances ( $kU_{i}$ ) is estimated by modelling the compressible shear layer as a two-dimensional mixing layer, with supersonic flow on the top and stagnant fluid on the bottom. Extensive literature is available on compressible mixing layers, including studies on the growth rate of shear layer thickness and the convection speed of disturbances (Bogdanoff Reference Bogdanoff1983; Papamoschou & Roshko Reference Papamoschou and Roshko1988; Elliott & Samimy Reference Elliott and Samimy1990; Hall et al. Reference Hall, Dimotakis and Rosemann1993; Murray & Elliott Reference Murray and Elliott2001; Pantano & Sarkar Reference Pantano and Sarkar2002). Flow instabilities in mixing layers consist of three families of waves, labelled as ‘Kelvin–Helmholtz’, ‘supersonic’ and ‘subsonic’ instability waves (Oertel Sen Reference Oertel Sen1979, Reference Oertel Sen1983; Tam & Hu Reference Tam and Hu1989). Propagation speeds of these waves can formally be obtained through linear stability analysis of the mixing layer (Tam & Hu Reference Tam and Hu1989). For the present purpose, however, a simple vortex train model of these instability waves (Oertel Sen et al. Reference Oertel Sen, Seiler, Srulijes and Hruschka2016) is used to obtain reasonably accurate estimates for the propagation speeds. The vortex train model gives the propagation (or convection) speeds $w_{\textit{sup}}$ , $w_{\textit{KH}}$ , $w_{\textit{sub}}$ of supersonic, Kelvin–Helmholtz, subsonic instability waves, respectively, as

(4.5) \begin{align} &\qquad \frac {w_{\textit{sup}}}{U_{i}} = \frac {1}{1+\alpha } \nonumber \\ \frac {w_{\textit{KH}}}{U_{i}} &= \frac {1+\alpha \left (\dfrac {w_{\textit{sup}}}{U_{i}}\right )}{1+\alpha } = \frac {1+2\alpha }{\left (1+\alpha \right )^2}\nonumber\\ \frac {w_{\textit{sub}}}{U_{i}} &= \frac {1-\alpha \left (\dfrac {w_{\textit{sup}}}{U_{i}}\right )}{1+\alpha } = \frac {1}{\left (1+\alpha \right )^2}. \end{align}

Here, $\alpha = a_{i}/a_{{r}}$ is the ratio of sound speeds between the supersonic flow side and the stagnant flow side of the shear layer (see figure 2). It is noted that all three non-dimensional propagation speeds ( $w_{\textit{sup}}/U_{i}$ , $w_{\textit{KH}}/U_{i}$ , $w_{\textit{sub}}/U_{i}$ ) are solely a function of $\alpha$ .

Figure 3. A map of dominant instability waves in the [ $M_{i}$ , $\alpha$ ] parameter space. Square and circle markers correspond to experimental data given in table 1.

Based on harmonic analysis of the linearised governing equations of compressible inviscid mixing layer flow (Tam & Hu Reference Tam and Hu1989), some key observations of the instability wave characteristics are made here. At low supersonic Mach numbers, only the Kelvin–Helmholtz and subsonic instability waves are active, with the Kelvin–Helmholtz waves dominating the flow. Supersonic instability waves emerge only when $M_{i} \gt 1 + (1/\alpha )$ (Tam & Hu Reference Tam and Hu1989). The growth rates of Kelvin–Helmholtz and supersonic instability waves depend on $M_{i}$ and $\alpha$ . Specifically, as $M_{i}$ increases, the dominance of Kelvin–Helmholtz instability waves decreases while the growth rate of supersonic instability waves steadily rises. The Mach number $M_{i}$ above which supersonic instabilities become dominant is termed the critical Mach number. Subsonic waves are active only when the mixing layer has a finite (but small) thickness. Their growth rates are small, and hence they are considered to be the least unstable of the three wave families (Tam & Hu Reference Tam and Hu1989). Figure 3 shows a map of the [ $M_{i}$ , $\alpha$ ] parameter space, wherein the $M_{i}$ and $\alpha$ values for the experimental data points given in table 1 are estimated using the modelling approach outlined earlier in this paper. The figure clearly shows that Kelvin–Helmholtz instability waves are expected to be the dominant instability waves in the wake shear layers across all the experimental data points considered here. Hence, the propagation velocity of vortical disturbances ( $kU_{i}$ ) in (4.1) and (4.2) is taken to be the convection velocity of the Kelvin–Helmholtz instability waves $w_{\textit{KH}}$ (4.5); we have.

(4.6) \begin{equation} kU_{i} = w_{\textit{KH}} = \frac {1+2\alpha }{\left (1+\alpha \right )^2}\,U_{i}. \end{equation}

It is noted that Rossiter (Reference Rossiter1964) obtains the propagation speed of vortical disturbances in an empirical manner by using experimental data.

The shear layer length $S$ for use in (4.2) is obtained from experimental data given in table 1. The mode number in (4.2) is taken as $m = 2$ . This choice is guided by frequencies observed in experiments (Schmidt & Shepherd Reference Schmidt and Shepherd2015; Thasu & Duvvuri Reference Thasu and Duvvuri2022), which suggest that the second mode frequency is active in the flow. Across different flows in which similar aeroacoustic feedback loops exist, such as cavity flows (Rossiter Reference Rossiter1964; Heller et al. Reference Heller, Holmes and Covert1971), impinging shear layers (Rockwell & Naudascher Reference Rockwell and Naudascher1979) and high-speed double cones (Kumar et al. Reference Kumar, Sasidharan, Kumara and Duvvuri2024), the phase difference $\phi$ is consistently observed to be $0.25$ . This is not entirely surprising since $\phi$ is determined by the local mechanics of vortical disturbance interactions that lead to the generation of acoustic waves. Hence $\phi$ is taken to be 0.25 for the present modelling purposes. With that, the modelling exercise is complete, and $St_D$ can be predicted using (4.2).

4.4. Comparison with experimental data

The Strouhal number predictions from the model are compared against experimental data in figure 4. The model is seen to perform well in predicting the experimental measurements at both $M_{\infty } = 4$ and 6. Hence, this exercise lends clear support to the hypothesis that the aeroacoustic mechanism outlined here drives the near-wake oscillations.

Figure 4. Strouhal number comparison between experimental data and predictions from present model.

5. Universal behaviour

Figure 5. (a) Model prediction for Strouhal number variation with the two governing parameters, $M_\infty$ and $Re_D$ . (b) Variation in propagation speeds of vortical and acoustic disturbances.

We now consider $St_S$

(5.1) \begin{equation} St_S = St_D \left (\frac {S}{D}\right ) = \frac {m-\phi }{\dfrac {U_\infty }{a_{{r}}}+\dfrac {U_\infty }{kU_{i}}}. \end{equation}

Unlike $St_D$ , $St_S$ does not depend on the geometric parameters $S$ , $D$ . The numerator $ (m-\phi )$ in the above equation is a constant, and the first and the second terms of the denominator account for the roles of acoustic waves and vortical disturbances, respectively, in the feedback loop. From (4.3), (4.4), (4.6), it is seen that the denominator in (5.1) depends only on the flow conditions downstream of the separation shock wave (region i) and $\alpha$ , both of which in turn depend on $M_\infty$ and $Re_D$ . Hence we write

(5.2) \begin{equation} St_S = g\left (M_\infty ,Re_D\right ), \end{equation}

where $g$ indicates functional dependence. The above equation essentially reiterates the fact that the Mach and Reynolds numbers are the two governing parameters for the cylinder wake in the compressible flow regime. The function $g$ constructed using the aeroacoustic model, i.e. by using (5.1), is shown in figure 5(a). For the entire range of $M_\infty$ and $Re_D$ considered here, $M_{\textrm {i}}$ and $\alpha$ are found to be such that Kelvin–Helmholtz waves are dominant throughout. Hence (4.6) is used without any modifications for generating figure 5(a). The Strouhal number $St_S$ shows a considerable dependence on $M_\infty$ at low values of $M_\infty$ , whereas the dependence becomes increasingly weaker at higher $M_\infty$ . However, $Re_D$ has a relatively smaller effect, with $St_S$ showing only a small variation across four orders of $Re_D$ . The model predicts that $St_S$ attains an invariant value of 0.45 at large $M_\infty$ and $Re_D$ . Although the model is rather simple in its nature (i.e. the model neglects the effects of viscosity and non-isentropic behaviour of flow around the cylinder forebody), the $St_S$ prediction is found to be very close to the value of 0.48 reported in the literature (Schmidt & Shepherd Reference Schmidt and Shepherd2015; Thasu & Duvvuri Reference Thasu and Duvvuri2022). This underscores the fact that the model captures the essential physical elements that drive and sustain the oscillations. Further, it is interesting to see that the model predicts that the universal behaviour breaks at lower supersonic Mach numbers, where $St_S$ is expected to be sensitive to $M_\infty$ .

It is noted that the vortical and acoustic disturbance propagation speeds ( $kU_{i}$ and $a_{{r}}$ , respectively) are the velocity scales relevant for the feedback loop. We now consider behaviour of these disturbance propagation speeds when scaled by the free-stream flow velocity, i.e. the quantities $(kU_{i}/U_\infty )$ and $({a_{{r}}}/U_\infty )$ . Figure 5(b) shows the variation of $(kU_{i}/U_\infty )^{-1}$ and $({a_{{r}}}/U_\infty )^{-1}$ with $M_\infty$ and $Re_D$ obtained from the model. Variation is shown for inverse of the scaled propagation speeds since the denominator of (5.1) contains the inverse of the speeds. It is seen that, at high Mach and Reynolds numbers, the disturbance propagation speeds scaled by $U_\infty$ become invariant (and thereby the quantity $[U_\infty /a_{{r}} \,+\, U_\infty /kU_{i}]$ in (5.1) also becomes invariant). Therefore, $U_\infty$ can be treated as the single relevant velocity scale for the feedback loop. Further, it is noted that the length over which both vortical and acoustic disturbances propagate is $S$ , which naturally makes it the relevant length scale for oscillations. From these arguments, we conclude that $U_\infty$ and $S$ are the appropriate velocity and length scales, respectively, to form the Strouhal number (i.e. $St_S$ ). And, when the oscillation frequency $f$ is scaled with $S$ and $U_\infty$ , we should expect to see invariant behaviour at high Mach and Reynolds numbers. This explains the universal behaviour of $St_S$ observed in experiments.

6. Brief conclusions

The aeroacoustic model developed here presents a convincing explanation for the phenomenon of self-sustained flow oscillations in the near-wake region of a cylinder at supersonic/hypersonic Mach numbers. The model also brings out the underlying reason for the universal behaviour of the oscillation frequency observed in experiments, and predicts that the universality will breakdown at lower supersonic Mach numbers. Overall, the findings presented in this paper are broadly relevant to a more general class of non-canonical compressible wake flows. The aeroacoustic mechanisms discussed here could potentially be at play in high-speed wake flows generated by geometries that are more complex than a simple cylinder, like ballutes, oblong bodies and atmospheric re-entry bodies.