3.1. Model verification

For verification of the stability analysis, the results of the present method are compared with those of Fedorov & Tumin (Reference Fedorov and Tumin2011) for a perfect gas. Figure 2 overlays the present results (symbols) against figure 10 from that paper (solid lines). The flow is a perfect gas with constant specific heats with $Ma=4.2$ , Prandtl number of 0.72, Sutherland constant of 110.4 K and an adiabatic wall. Excellent agreement is seen for both growth rates (a) and phase speeds (b), where $c_{r}$ is the real part of the complex phase speed $c={\it\omega}/{\it\alpha}$ .

Figure 2.

Comparison of present stability analysis (symbols) with the results of Fedorov & Tumin (Reference Fedorov and Tumin2011) (solid lines). Flow is a perfect gas with $Ma=4.2$ , $T_{o}^{\ast }=300~\text{K}$ , $Re=2000$ , adiabatic wall. Waves are 2D with ${\it\beta}=0$ ; (a) growth rates; (b) phase speeds.

As a second validation case, a higher-enthalpy flow is selected in which the variation of specific heats is significant and the base flow is not in equilibrium. Figure 3 compares our spatial growth rates with those from the PSE-Chem portion of the STABL software package (Johnson et al. Reference Johnson, Seipp and Candler1998; Wright et al. Reference Wright, Candler and Bose1998; Johnson Reference Johnson2000). The PSE-Chem computations were conducted using 300 points in the wall-normal direction, and convergence was checked by doubling the number of points, which affected the growth rates by at most 1 %. In performing the PSE-Chem analysis, the options for PSE marching and non-parallel effects were turned off so as to make as direct a comparison as possible with our results. However, chemical and vibrational non-equilibria were retained in the PSE-Chem calculation, with the chemistry model consisting of five species: $\text{N}_{2}$ , $\text{O}_{2}$ , NO, N and O. The free-stream concentrations of these species were set to 78 % $\text{N}_{2}$ and 22 % $\text{O}_{2}$ by mole, which matches the present linear stability calculation.

Figure 3.

Comparison of spatial growth rates from STABL PSE-Chem (symbols) and present LST analysis (solid lines). (a) Low enthalpy, $P_{e}^{\ast }=5~\text{kPa}$ , $Ma=5$ , $T_{w}^{\ast }=300~\text{K}$ , $T_{e}^{\ast }=T_{v,e}^{\ast }=300~\text{K}$ . (b) High enthalpy, $P_{e}^{\ast }=20~\text{kPa}$ , $Ma=5$ , $T_{w}^{\ast }=300~\text{K}$ , $T_{e}^{\ast }=T_{v,e}^{\ast }=1500~\text{K}$ . Only 2D waves of ${\it\beta}=0$ are included.

Figure 4.

High-enthalpy comparison between current method and STABL software: $P_{e}^{\ast }=20~\text{kPa}$ , $Ma=5$ , $T_{w}^{\ast }=300~\text{K}$ , $T_{e}^{\ast }=T_{v,e}^{\ast }=2500~\text{K}$ . (a) Comparison of boundary layer profiles at $Re=1500$ ( $x^{\ast }=1.2~\text{m}$ ) from STABL Navier–Stokes solver (symbols) and boundary layer code (solid lines). Profiles are velocity (▿), translational/rotational temperature (▫) and vibrational temperature (♢). (b) Comparison of spatial growth rates from STABL PSE-Chem (symbols) and present LST analysis (solid lines). Only 2D waves of ${\it\beta}=0$ are included.

Figure 3(a) is a relatively low-enthalpy flow that is representative of the free flight of a slender body. At the edge of the boundary layer both translational and vibrational temperatures are 300 K. Figure 3(b) is a higher-enthalpy flow that is representative of a shock tunnel, with $T_{e}^{\ast }=T_{v,e}^{\ast }=1500~\text{K}$ . The base flows corresponding to figure 3 are given in figure 1 for $Re=1500$ . Very close agreement between the two methods is seen, though slightly higher growth rates are predicted by the present model based on conventional linear stability theory (LST). The small differences are greater near the leading edge of the plate and may be caused by the weak shock that develops through the viscous–inviscid interaction: this effect is included in the PSE-Chem analysis, but is excluded from the present LST model. This explanation is consistent with the findings of Chang et al. (Reference Chang, Malik and Hussaini1990), who showed that, when the shock is close to the boundary layer edge, there is a slight stabilization of both the first and second modes. Differences in transport properties also contribute to the discrepancies in growth rates.

This chemically frozen stability analysis might be expected to perform poorly for the very high-enthalpy conditions that are achievable in reflected shock tunnel facilities. For instance, in a uniform volume of air at 20 kPa, the enthalpy of chemical equilibrium air first exceeds that of chemically frozen air by 10 % at approximately 2600 K. This temperature can be significantly exceeded in shock tunnel facilities. However, we have found that, for slender bodies in air, the stability characteristics at such high temperatures can still be predicted reasonably well under the assumption of a chemically inert gas.

Figure 4 shows mean boundary layer profiles and stability calculations for a high-enthalpy boundary layer with an edge temperature of 2500 K and Mach number of $Ma=5$ . This corresponds to a stagnation enthalpy of approximately $15~\text{MJ}~\text{kg}^{-1}$ , which is close to the maximum stagnation enthalpy achievable in reflected shock tunnel facilities like the T5 facility at Caltech. For this case, the peak temperature within the boundary layer exceeds 3500 K, which is well above the limit at which dissociation of $\text{O}_{2}$ becomes significant. For instance, the chemically reacting Navier–Stokes simulations reveal that at $Re=1500$ the mass fraction of atomic oxygen reaches a peak value of 0.72 % in the interior of the boundary layer. The mean boundary layer profiles (figure 4 a) are in reasonable agreement with the Navier–Stokes simulations from STABL, although the boundary layer code produces noticeably greater temperatures by neglecting chemistry. The difference in peak temperature is large at the high Reynolds number shown, but is much smaller at lower Reynolds numbers, where the chemical reactions have not progressed as far towards equilibrium.

In figure 4(b), the chemically frozen stability predictions from the present method are compared with the chemically reacting results from STABL. The agreement is reasonably good, especially considering that we have neglected chemical reactions and are using different transport property models. These results suggest that, despite the high temperatures in the interior of the boundary layer, the present chemically frozen analysis can be used to study the stability of flows of air over slender bodies in high-enthalpy shock tunnels without incurring serious errors. This finding can be explained in part by the analysis of Fujii & Hornung (Reference Fujii and Hornung2003), which showed that, in air at 1000–3000 K, the absorption of sound waves caused by thermochemical non-equilibrium occurs at frequencies much lower than those of second-mode disturbances in hypervelocity boundary layers. For blunt geometries, however, the extremely high temperatures and low flow velocities that occur behind the bow shock may lead to more significant chemical effects and the frozen analysis may no longer be adequate.

3.2. Mode characteristics and terminology

To demonstrate the mode characteristics and introduce some terminology, figure 5 shows the disturbance spectrum for a high-Mach-number, low-enthalpy flow. Spatial growth rates (a) and phase speeds (b) are shown for $Ma=4.5$ and $Re=\sqrt{U_{e}^{\ast }x^{\ast }/{\it\nu}_{e}^{\ast }}=2000$ with a cold free stream ( $T_{e}^{\ast }=65.15~\text{K}$ ) and an adiabatic wall. This same case was computed by Ma & Zhong (Reference Ma and Zhong2003), and our results are identical to theirs to within plotting accuracy.

Figure 5.

Spatial growth rates (a) and phase speeds (b) for low-enthalpy flow with $Ma=4.5$ , $Re=2000$ , $T_{e}^{\ast }=65.15~\text{K}$ , $P_{e}^{\ast }=728~\text{Pa}$ , adiabatic wall. Flow conditions modelled after Ma & Zhong (Reference Ma and Zhong2003). Only 2D waves with ${\it\beta}=0$ are shown.

Figure 6.

Spatial growth rates (a) and phase speeds (b) for high-enthalpy flow with $Ma=4.5$ , $Re=2000$ , $T_{e}^{\ast }=1500~\text{K}$ , $T_{w}^{\ast }=300~\text{K}$ , $P_{e}^{\ast }=10~\text{kPa}$ . Only 2D waves with ${\it\beta}=0$ are shown.

The disturbance spectrum features an infinite sequence of modes that ‘cut in’ as the frequency increases. By ‘cut in’, it is meant that these modes transition from the continuous spectrum to the discrete spectrum. Each mode cuts in with a dimensionless phase speed of $1+1/Ma$ , which is the speed of fast acoustic waves in the free stream, hence these modes are termed ‘fast modes’ and designated $F_{1}$ , $F_{2}$ , etc. in order of increasing cut-in frequency. Each additional mode that cuts in is characterized by an additional zero in the real part of the pressure eigenfunction, with mode $F_{n}$ having $n-1$ zeros. There is also one mode that has a phase speed of $1-1/Ma$ at ${\it\omega}=0$ ; this mode is termed the ‘slow mode’ and labelled $S_{1}$ .

At the locations where the phase speeds of the fast and slow modes cross, there is a possibility of synchronization, which may lead to increased growth rate and instability. At these crossing points, the phase speeds of the slow and fast modes are equal and their eigenfunctions are nearly identical. This synchronism has been analysed by Fedorov & Khokhlov (Reference Fedorov and Khokhlov2001), who proposed a reduced-order model of the dispersion relation in the vicinity of the synchronism region, which captures the topology of the spectrum that is found there. These synchronous interactions between fast and slow modes are frequently called the ‘Mack modes’ and are labelled Mack’s second mode, third mode, etc. in order of increasing frequency. As discussed by Fedorov & Tumin (Reference Fedorov and Tumin2011), the Mack modes are not ‘modes’ in the mathematical sense as are modes $F_{n}$ and $S_{1}$ , but since this terminology is in common use it will be used throughout in this paper.

Lastly, we point out that the lowest-frequency unstable region in figure 5, which was labelled the ‘first mode’ by Mack, does not involve a synchronism between fast and slow acoustic modes. This instability is the compressible counterpart to the Tollmien–Schlichting waves that are found in incompressible flows, and its instability characteristics depend differently on Mach number, Reynolds number and wall temperature compared to the acoustic instabilities.

Figure 7.

Effect of wall cooling on the spatial growth rate (a) and phase speed (b). For all cases, $Ma=5$ , $Re=2000$ , $T_{w}^{\ast }=300~\text{K}$ , $P_{e}^{\ast }=10~\text{kPa}$ . Lines with symbols designate an additional mode that appears when $T_{w}^{\ast }\ll T_{e}^{\ast }$ . Only 2D waves with ${\it\beta}=0$ are shown.

For comparison with the low-enthalpy adiabatic case given in figure 5, the stability diagram for a high-enthalpy flow with a cold wall is shown in figure 6. Here the Mach number is again 4.5 and the Reynolds number is $Re=2000$ , but the free-stream temperature is now 1500 K and the wall temperature is 300 K. These conditions are representative of a moderate-enthalpy flow in a reflected shock tunnel. By comparing figures 5 and 6, it is seen that the maximum growth rate is more than a factor of 2 greater for the cold-wall case, which is consistent with the well-known fact that the second-mode instability is destabilized by wall cooling. Also the first-mode instability is absent in the case with a cold wall, which again is well-known behaviour for hypersonic flows. An additional difference is that in figure 5 the mode $S_{1}$ is unstable while in figure 6 the mode $F_{1}$ is unstable. This behaviour has also been observed by Fedorov & Khokhlov (Reference Fedorov and Khokhlov2001), who demonstrated that the acoustic instabilities are associated with a pair of branch points in the complex plane whose location depends on the Reynolds number, Mach number, frequency, wall temperature and other parameters. As the frequency is increased, either $F_{1}$ or $S_{1}$ can be unstable depending on whether the dispersion curve passes above or below these branch points. Further discussion regarding this behaviour is available from Fedorov & Tumin (Reference Fedorov and Tumin2011).

The above-mentioned differences between figures 5 and 6 are well known, but we also observe two new features in figure 6 that have not been widely reported in the literature. The first is an abrupt change in slope of the second-mode unstable region at approximately ${\it\omega}=0.5$ , which leads to a wider range of unstable frequencies for the cold-wall case. This behaviour is also visible in a few of the stability calculations reported by Klentzman & Tumin (Reference Klentzman and Tumin2013), which involved high-enthalpy flows of oxygen. The second new feature is that the mode labelled $F_{1}$ ceases to exist for frequencies greater than approximately ${\it\omega}=0.62$ , but a new related mode (dashed line) appears at ${\it\omega}=0.54$ . These two modes have nearly the same phase speed over the frequency range $0.54<{\it\omega}<0.62$ for which they both exist. Note that a similar phenomenon involving the disappearance of one mode and the appearance of another occurs for $F_{1}$ at ${\it\omega}=0.32$ , but the effect is hardly visible at the scale shown in the figure.

3.3. Shape of dispersion curve

We first investigate the sharp change in slope of the dispersion curve observed in figure 6. Figure 7 shows the growth rates and phase speeds for flow at $Ma=5$ and $Re=2000$ . Three cases are shown with free-stream temperatures of 300, 1000 and 1500 K and with the wall temperature fixed at 300 K in all cases. As the ratio $T_{w}^{\ast }/T_{e}^{\ast }$ decreases, a kink appears in the high-frequency end of the growth rate curve and the width of the unstable region increases. For each case, the frequency at which the kink occurs is marked by a vertical dashed line. From the intersection between these vertical lines and the phase speed diagram below, one can see that the change in slope of the growth rate curve occurs precisely when the dimensionless phase speed falls below $1-1/Ma$ . That is, the change in shape of the dispersion curve occurs when the unstable modes travel supersonically relative to the free stream. Supersonic modes can exist for all wall temperature conditions, but they only become unstable when the wall is highly cooled, usually (but not necessarily) below the free-stream temperature. This behaviour was also encountered by Chang et al. (Reference Chang, Vinh and Malik1997) in their study of chemically reacting, high-enthalpy flow over a wedge, and the existence of supersonic unstable modes was also briefly mentioned by Mack (Reference Mack1969, Reference Mack1987) for the inviscid case.

The eigenfunctions corresponding to two values of ${\it\omega}$ from figure 7 are shown in figure 8 with $T_{e}^{\ast }=1500~\text{K}$ . In each case, the locations of the critical layer $c_{r}=\bar{U}$ and the sonic lines $c_{r}=\bar{U}\pm \bar{a}$ are indicated by horizontal dashed lines. For ${\it\omega}=0.35$ (just before the kink in the dispersion curve), the phase speed is above $c_{r}>1-1/Ma$ and only one sonic line exists, but for ${\it\omega}=0.45$ (just after the kink) there are two sonic lines because $c_{r}<1-1/Ma$ . Between these two lines the disturbance is travelling subsonically relative to the fluid, but outside of these lines the disturbance travels supersonically.

Figure 8.

Eigenfunctions corresponding to two points along the dispersion curve in figure 7, for $Re=2000$ , $T_{e}^{\ast }=1500~\text{K}$ , $T_{w}^{\ast }=300~\text{K}$ , $Ma=5$ : (a)  ${\it\omega}=0.35$ , subsonic mode; (b)  ${\it\omega}=0.45$ , supersonic mode.

From figure 8 one can see that for ${\it\omega}=0.45$ the eigenfunctions decay very slowly in the free stream. This is highlighted by the contours of the temperature disturbance given in figure 9. In the interior of the boundary layer, the structure is similar for ${\it\omega}=0.35$ and 0.45, and a large peak in temperature is visible at the critical layer. For ${\it\omega}=0.35$ , the waves decay rapidly in the free stream, but for ${\it\omega}=0.45$ the second sonic line acts as a turning point, leading to oscillatory waves that are radiated into the free stream. The eigenfunction behaviour in the free stream is similar for the pressure and velocity fluctuations.

Figure 9.

Contours of temperature fluctuations at two points along the dispersion curve in figure 7 for $Re=2000$ , $T_{e}^{\ast }=1500~\text{K}$ , $T_{w}^{\ast }=300~\text{K}$ , $Ma=5$ : (a)  ${\it\omega}=0.35$ , subsonic mode; (b)  ${\it\omega}=0.45$ , supersonic mode.

This behaviour can be anticipated from the compressible Rayleigh equation for the pressure (see Lees & Lin Reference Lees and Lin1946; Lees & Reshotko Reference Lees and Reshotko1962; Mack Reference Mack1969)

(3.1) $$\begin{eqnarray}\frac{\text{d}^{2}\hat{p}}{\text{d}y^{2}}-\left(\frac{2\bar{U}^{\prime }}{\bar{U}-c}-\frac{\bar{T}^{\prime }}{\bar{T}}\right)\frac{\text{d}\hat{p}}{\text{d}y}-{\it\alpha}^{2}\left(1-\frac{Ma^{2}(\bar{U}-c)^{2}}{\bar{T}}\right)\hat{p}=0.\end{eqnarray}$$

In the free stream this equation reduces to

(3.2) $$\begin{eqnarray}\frac{\text{d}^{2}\hat{p}}{\text{d}y^{2}}-{\it\alpha}^{2}[1-Ma^{2}(1-c)^{2}]\hat{p}=0,\end{eqnarray}$$

which has the solution

(3.3) $$\begin{eqnarray}\hat{p}=\exp ({\it\alpha}\sqrt{1-Ma^{2}(1-c)^{2}}y).\end{eqnarray}$$

Here the branch of the square root is chosen such that the real part of the exponential’s argument is negative, so the pressure is bounded. Since $|c_{i}|\ll |c_{r}|$ and $|{\it\alpha}_{i}|\ll |{\it\alpha}_{r}|$ , the pressure is mainly exponentially decaying if $c_{r}>1-1/Ma$ with slight oscillations arising from the imaginary parts of ${\it\alpha}$ and $c$ . Conversely, for $c_{r}<1-1/Ma$ the solution is mainly oscillatory, with slight damping caused by the imaginary parts of ${\it\alpha}$ and $c$ . Therefore, slowly decaying eigenfunctions can always be expected for supersonic modes at high Reynolds numbers.

Figure 7 showed that cases which develop a supersonic unstable mode also experience a discontinuity in the dispersion curve. As the frequency increases, the fast mode (labelled $F_{1}^{+}$ in the figure) disappears and a new mode (labelled $F_{1}^{-}$ ) appears. To further investigate this effect, the real and imaginary parts of the complex phase velocity are plotted for mode $F_{1}$ in figure 10. The data are exactly the same as in figure 7, except that now the real and imaginary parts of the phase speed are plotted with ${\it\omega}$ as a parameter along the curves. Also shown in the figure are three thick black lines which mark the branch cuts corresponding to the continuous spectra; the two nearly horizontal branches are the acoustic branch cuts, while the vertical branch near $c_{r}=1$ (which is actually two overlapping branches) corresponds to vorticity and entropy waves. Further discussion about the location of these branch cuts and methods for their computation are available from Balakumar & Malik (Reference Balakumar and Malik1992). It should be noted that the continuous spectrum is a (weak) function of ${\it\omega}$ , and the spectra plotted in figure 10 are evaluated at a single value of ${\it\omega}=0.4$ .

Figure 10.

Map of real and imaginary parts of phase speed in the complex plane, for $Ma=5$ and $Re=2000$ . Solid black lines correspond to the continuous spectra evaluated at ${\it\omega}=0.4$ . Lines with symbols designate an additional mode that appears when $T_{w}^{\ast }\ll T_{e}^{\ast }$ .

At ${\it\omega}=0$ , the mode $F_{1}^{+}$ is a neutral wave with phase speed $c_{r}=1+1/Ma$ , which originates at the branch point of the fast acoustic branch cut. As indicated by the arrow in figure 10, as the frequency increases, the mode’s phase speed reduces and it becomes damped. As the mode crosses the branch cut at $c_{r}\sim 1$ , the growth rate suffers a small jump while the real part of ${\it\alpha}$ remains nearly continuous. This phenomenon is discussed further by Fedorov & Khokhlov (Reference Fedorov and Khokhlov2001), who showed that the synchronism between the eigenvalue and the vorticity/entropy waves of the continuous spectrum causes the mode $F_{1}$ to be especially receptive to entropy spots and vortical free-stream disturbances at this condition. Additional analysis of the receptivity to entropy disturbances is available from Fedorov & Tumin (Reference Fedorov and Tumin2003), who solved the initial value problem of a temperature spot located at the edge of a boundary layer.

As the frequency is increased further, the mode becomes unstable. For the case with $T_{e}^{\ast }=300~\text{K}$ , the mode becomes stable again before the phase speed falls below $1-1/Ma$ , so the mode passes below the slow acoustic branch point in the complex plane. This is typical of boundary layers that are adiabatic or have wall temperatures greater than the free-stream temperature. In contrast, for the cases with $T_{e}^{\ast }=1000$ and 1500 K, the mode remains unstable for $c_{r}<1-1/Ma$ and therefore passes above the branch point of the slow acoustic waves.

Because the mode with $T_{e}^{\ast }=1000$ or 1500 K passes above the branch point, it must eventually cross the slow acoustic branch cut. When this happens, the mode coalesces with the branch cut and ceases to be a discrete mode; however, a new mode emerges from the other side of the branch cut. This new mode emerges from the branch cut at a slightly lower frequency than the one at which the coalescing mode disappears. The modes on the top and bottom of the branch cut are distinguished by the labels $F_{1}^{+}$ and $F_{1}^{-}$ , similar to the notation used by Fedorov & Tumin (Reference Fedorov and Tumin2011) to describe modes crossing the branch cut at $c_{r}\sim 1$ .

This crossing of the branch cut results in a synchronism between the instability mode and acoustic disturbances in the free stream. This is similar to the synchronism with vorticity/entropy waves that occurs when the branch cut at $c_{r}=1$ is crossed (Fedorov & Khokhlov Reference Fedorov and Khokhlov2001; Fedorov et al. Reference Fedorov, Ryzhov, Soudakov and Utyuzhnikov2013), which has been found to produce increased receptivity to vorticity and entropy spottiness. The synchronism identified here is expected to cause a similar effect, but with enhanced receptivity to acoustic disturbances in the free stream rather than vorticity/entropy disturbances. However, unlike the vorticity/entropy synchronism, which occurs upstream of the lower neutral branch, the synchronism with free-stream acoustic waves takes place downstream of the upper neutral branch. As a result, modes excited in this manner by free-stream acoustic waves are unlikely to experience amplification. Nevertheless, the synchronism can still affect the downstream development of amplified waves, which may be important during the nonlinear stages of transition.

3.4. Effect of Reynolds and Mach numbers

In figure 7, it was shown that the supersonic unstable modes only appear when the wall is sufficiently cold. In this section we investigate also how the Reynolds and Mach numbers influence these supersonic modes. Figure 11(a) compares the spatial growth rates for a high-enthalpy boundary layer at several different Reynolds numbers. For low enough Reynolds numbers, the mode remains subsonic throughout the entire unstable region, but for higher $Re$ the ‘tail’ on the dispersion curve appears and grows considerably larger as the Reynolds number is further increased.

Figure 11.

(a) Effect of Reynolds number on the shape of the dispersion curve for a cold-wall case: $Ma=5$ , $T_{e}^{\ast }=1500~\text{K}$ , $T_{w}^{\ast }=300~\text{K}$ , $P_{e}^{\ast }=20~\text{kPa}$ . (b) The $N$  factor curve corresponding to (a). The heavy black line is the complete $N$ factor curve. The dot-dashed red line includes only values of ${\it\omega}<0.4$ . Only 2D waves with ${\it\beta}=0$ are included.

As can be noted from figure 11(a), at higher Reynolds numbers the supersonic modes contribute significantly to the area under the dispersion curve, which is related to the $N$ factor distribution. This brings into question whether the supersonic modes significantly influence the $N$ factors. Figure 11(b) shows the $N$ factor envelope curves for a cold-wall case with $Ma=5$ and $T_{e}^{\ast }=1500~\text{K}$ . Two envelope curves are included in this figure: the dashed red line excludes growth rates having frequencies above ${\it\omega}=0.4$ , while the solid black line is the full $N$ factor curve. The difference between the two curves then indicates the contribution of supersonic modes. Clearly, the supersonic modes do not contribute much to the $N$ factor until the Reynolds number exceeds approximately 1500. However, at this point the $N$ factor is already fairly high, reaching the level of 5–10 at which transition is typically observed experimentally. This suggests that the supersonic modes do not significantly increase the level of amplification, despite their influence on the shape and behaviour of the dispersion curve.

Lastly, the effect of the Mach number on the stability of highly cooled flows is considered. Figure 12 shows dispersion curves for several Mach numbers with $T_{e}^{\ast }=1500~\text{K}$ and $T_{w}^{\ast }=300~\text{K}$ . The left panel is for a lower Reynolds number of $Re=1000$ , while the right one is for $Re=2000$ . These plots reveal that the supersonic unstable modes first appear and experience the largest growth rate at approximately $Ma=5$ .

Figure 12.

Dispersion curves for cold-wall flows ( $T_{w}^{\ast }/T_{e}^{\ast }=0.2$ ) for various Mach numbers: (a)  $Re=1000$ ; (b)  $Re=2000$ .

3.5. Vibrational non-equilibrium effects

In this section, the influence of thermal non-equilibrium on the growth of disturbances is examined. Vibrational non-equilibrium primarily enters the stability analysis in two ways. First, the mean profiles of temperature, density and (to a smaller extent) velocity are influenced by the exchange of energy between the translational and vibrational modes. This indirectly affects the disturbances through changes to the mean flow profile. The second effect of vibrational non-equilibrium is the attenuation of sound waves caused by the phase lag between the kinetic energy of molecules and their internal energy modes (Lighthill Reference Lighthill, Batchelor and Davies1956; Clarke & McChesney Reference Clarke and McChesney1964; Vincenti & Kruger Reference Vincenti and Kruger1967; Fujii & Hornung Reference Fujii and Hornung2003).

The first effect of vibrational non-equilibrium, changes to the base flow, can be assessed by conducting a stability analysis for three different base flows: a frozen flow with vibrational energy neglected, a non-equilibrium flow with finite rates of vibrational energy transfer, and an equilibrium flow in which vibrational energy is assumed to be exchanged at an infinite rate. In computing the stability of these three base flows, fully non-equilibrium disturbances are modelled. The resulting stability diagram is shown in figure 13(a). As could be expected, the non-equilibrium solution agrees best with the frozen solution for small Reynolds numbers, and tends to the equilibrium solution for large $Re$ . The corresponding $N$ factors shown in figure 13(b) reveal that the equilibrium model lies closest to the non-equilibrium one, but there is a noticeable discrepancy between them.

Figure 13.

(a) Stability diagrams for $Ma=5$ , $T_{e}^{\ast }=T_{v,e}^{\ast }=1500~\text{K}$ and $P_{e}^{\ast }=20~\text{kPa}$ . Three different models of the mean flow: frozen, non-equilibrium and equilibrium. Disturbances are non-equilibrium. (b) The $N$ factor diagram corresponding to (a). Only 2D waves with ${\it\beta}=0$ are included.

Figure 14.

(a) Stability diagrams for $Ma=5$ , $T_{e}^{\ast }=T_{v,e}^{\ast }=1500~\text{K}$ and $P_{e}^{\ast }=20~\text{kPa}$ . (a) Three different models of the disturbances: frozen, non-equilibrium and equilibrium. Mean flow is non-equilibrium. (b) The $N$ factor diagram corresponding to (a). Only 2D waves with ${\it\beta}=0$ are included.

The influence of acoustic absorption on the disturbance growth rates can be investigated in a similar manner. Three different disturbance models are now applied to a single fully non-equilibrium base flow. The three disturbance models are frozen, thermal equilibrium and full thermal non-equilibrium. The growth rates are compared in figure 14(a) and the $N$ factors in figure 14(b). In this case, the frozen disturbance model is nearly indistinguishable from the non-equilibrium one. This is consistent with the findings of Fujii & Hornung (Reference Fujii and Hornung2003), who showed that, for air at temperatures below approximately 3000 K, the frequency of maximum acoustic absorption is several orders of magnitude lower than the frequency ( ${\sim}\text{MHz}$ ) of second-mode disturbances in hypervelocity boundary layers.

Similar calculations to figures 13 and 14 have been carried out for a wide range of Mach numbers and wall temperature conditions relevant to shock tunnel experiments. In some cases it was found that a non-equilibrium base flow was needed to fully capture the stability characteristics, but the frozen disturbance model never differed from the non-equilibrium one in any significant way. This finding is independent of the mean pressure of the flow, and results given in terms of the Reynolds number, dimensionless wavenumber and dimensionless frequency apply for any pressure. This is because both the viscous time scale and the time scale of vibrational relaxation are inversely proportional to the pressure, meaning that the effects of vibrational non-equilibrium are always the same at a given Reynolds number. The pressure independence of the results can also be deduced from the dimensionless governing equations (2.10), in which none of the parameters depend on the mean pressure.

3.6. Stability trends

This section summarizes the effects of wall temperature and Mach number on the stability characteristics. Although other researchers have conducted similar investigations, these results are included because they provide a complete picture of the instability behaviour over a much wider range of conditions than is considered in other references. Also, previous work (Mack Reference Mack1969, Reference Mack1984; Malik Reference Malik, Arnal and Michel1989b ; Masad et al. Reference Masad, Nayfeh and Al-Maaitah1992) has targeted low-enthalpy conditions, whereas these results are relevant to high-enthalpy flows. The results given here are for a flat plate, but the growth rates and frequencies can be related to those for a sharp cone through the Mangler transform (Malik & Spall Reference Malik and Spall1991), by which it is found that the boundary layer is a factor of $\sqrt{3}$ thinner for a cone than for a flat plate. Therefore, the results presented here can be related to those for a sharp cone by the transformations $Re\rightarrow Re\sqrt{3}$ , ${\it\alpha}\rightarrow {\it\alpha}\sqrt{3}$ and ${\it\omega}\rightarrow {\it\omega}\sqrt{3}$ .

A sequence of simulations was conducted in which the wall temperature was fixed at 300 K while the free-stream temperature was incrementally raised from 70 to 2000 K. Although wall temperature effects are often spoken of in terms of ‘wall cooling’, we choose here to keep the wall temperature fixed and adjust the free-stream temperature so that comparisons with experiments in both high- and low-enthalpy impulse facilities can be made more easily. However, it should be recognized that, in terms of the stability behaviour, raising the free-stream temperature is nearly equivalent to cooling the wall, with differences being caused only by the temperature dependence of transport properties and specific heats, as well as non-equilibrium effects. Accordingly, we refer to cooling of the wall and heating of the free stream interchangeably. These results are also presented in terms of $T_{w}^{\ast }/T_{e}^{\ast }$ rather than the customary $T_{w}^{\ast }/T_{ad}^{\ast }$ because the adiabatic wall temperature depends on one’s assumptions about specific heats, chemistry and transport models, and must in general be simulated. Thus it can be difficult to reproduce results that are reported in terms of $T_{ad}^{\ast }$ .

Figure 15(a) shows the maximum spatial growth rate as a function of Mach number for several values of the ratio $T_{w}^{\ast }/T_{e}^{\ast }$ . Two sets of curves are shown: the dashed lines correspond to the first-mode instability and the solid lines to the second mode. For each curve, the spatial growth rate has been maximized over all frequencies ${\it\omega}$ and all spanwise wavenumbers ${\it\beta}$ . These curves are very similar to those of Mack (Reference Mack1984) and Masad et al. (Reference Masad, Nayfeh and Al-Maaitah1992), except that we consider a wider range of wall temperature conditions and hold $T_{w}^{\ast }/T_{e}^{\ast }$ fixed along each curve rather than $T_{w}^{\ast }/T_{ad}^{\ast }$ . Note that, in this figure and the ones that follow, the neglect of chemistry may not be valid for some of the high-Mach-number cases, but these results are still included for completeness.

Figure 15.

Effect of wall cooling on the maximum spatial growth rates of the first (dashed lines) and second (solid lines) modes for $Re=1500$ , $T_{w}^{\ast }=300~\text{K}$ , $T_{e}^{\ast }=70{-}2000~\text{K}$ , $P_{e}^{\ast }=10~\text{kPa}$ . (a) Maximum growth rates for first and second modes. (b) Maximum growth rate of second mode normalized by ${\it\delta}_{99}$ .

For the first mode, no instability was found for some of the cases having $T_{w}^{\ast }/T_{e}^{\ast }<1$ , so these lines are not included on the figure. For the second mode, a monotonic increase in growth rate is observed as the wall is cooled. Additionally, the second mode becomes unstable at lower Mach numbers as the wall is cooled, with instability occurring at Mach numbers as low as 2.5. Although it is commonly stated that the second mode becomes dominant above approximately $Ma=4$ , this statement is true only when the wall is nearly adiabatic. This result is consistent with that of Mack (Reference Mack, Ashpis, Gatski and Hirsch1993), who also observed second-mode instability at $Ma=3$ .

The destabilizing influence of wall cooling appears to be caused mainly by thinning of the boundary layer. This is illustrated in figure 15(b) in which the spatial growth rate of the second mode is normalized by ${\it\delta}_{99}$ , the boundary layer thickness at which the streamwise velocity reaches 99 % of the free-stream velocity, rather than the Blasius thickness ${\it\delta}=\sqrt{{\it\nu}_{e}^{\ast }x^{\ast }/U_{e}^{\ast }}$ . The maximum second-mode growth rates normalized in this way experience much less variation with both Mach number and wall temperature than in figure 15(a), indicating that the spatial growth rate is nearly inversely proportional to ${\it\delta}_{99}$ . Although it is well known that the frequency and wavenumber of the second mode scale inversely with the boundary layer thickness (Demetriades Reference Demetriades1977; Stetson et al. Reference Stetson, Thompson, Donaldson and Siler1983), the correlation of growth rate with boundary layer thickness has been less widely recognized.

The relationship between the frequency of second-mode disturbances and the boundary layer thickness was first examined experimentally by Demetriades (Reference Demetriades1977) and Stetson et al. (Reference Stetson, Thompson, Donaldson and Siler1983), who both made hot-wire measurements of disturbance fluctuations in the same Mach 8 wind tunnel. Their results showed that the non-dimensional frequency $2f^{\ast }{\it\delta}_{99}/U_{e}^{\ast }$ is nearly constant, decreasing only slightly with Reynolds number and with increased wall cooling. In this relation $f^{\ast }$ is the dimensional frequency in hertz. Figure 16 shows the variation of dimensionless frequency with Mach number and wall temperature ratio at $Re=1000$ . For figure 16(a), the plotted frequency is the most amplified one, i.e. the frequency for which the $N$ factor is largest; this is the frequency that is measured most often in experiments. Figure 16(b) shows the most unstable frequency, that is, the frequency that has the largest spatial growth rate $-{\it\alpha}_{i}$ . For flows with a cold wall, the most unstable and most amplified frequencies follow similar trends, and the most unstable frequency is slightly smaller. This is consistent with the results of Marineau et al. (Reference Marineau, Moraru, Lewis, Norris, Lafferty, Wagnild and Smith2014), who measured both the most unstable and the most amplified frequencies for flow over a sharp cone. However, for an adiabatic wall, the most unstable and most amplified frequencies exhibit very different variations with Mach number.

Figure 16.

Variation of second-mode frequencies with Mach number and wall temperature at $Re=1000$ . For $T_{w}^{\ast }=T_{ad}^{\ast }$ , the edge temperature is $T_{e}^{\ast }=70~\text{K}$ ; for all other cases, $T_{w}^{\ast }=300~\text{K}$ . (a) Most amplified frequency (max $N$ factor). (b) Most unstable frequency (max $-{\it\alpha}_{i}$ ).

For flows with highly cooled walls, figures 15(b) and 16 reveal that the effect of the wall temperature can be nearly scaled out of the problem by normalizing both the frequency and the growth rate by ${\it\delta}_{99}$ . This length scale is preferable to the Blasius boundary layer thickness ${\it\delta}=\sqrt{{\it\nu}_{e}^{\ast }x^{\ast }/U_{e}^{\ast }}$ , which takes into account only the free-stream conditions, whereas the ${\it\delta}_{99}$ length scale includes the physical structure of the boundary layer and encapsulates the change in thickness that occurs when the wall is cooled. These results suggest that by scaling the growth rate and frequency with ${\it\delta}_{99}$ one can generate a single dispersion curve that, to a good approximation, describes a range of wall temperature conditions. An example of such a dispersion curve is given in figure 17. In the left panel the growth rate and frequency are scaled by the usual Blasius boundary layer thickness ${\it\delta}$ , whereas in the right one they are scaled by ${\it\delta}_{99}$ . The latter scaling nearly collapses the dispersion curves onto one another, though for the case with large $T_{w}/T_{e}$ the collapse is not as good. For the cases with smaller $T_{w}/T_{e}$ , however, the collapse is excellent, especially along the lower neutral branch, which contributes most significantly to the $N$ factor.

Figure 17.

Spatial growth rates for several different wall temperature conditions: $Ma=4.5$ , $T_{w}^{\ast }=300~\text{K}$ , $Re=1500$ , $P_{e}^{\ast }=10~\text{kPa}$ . Only 2D waves ( ${\it\beta}=0$ ) are included. (a) Normalized using Blasius thickness ${\it\delta}=\sqrt{{\it\nu}_{e}^{\ast }x^{\ast }/U_{e}^{\ast }}$ . (b) Normalized using ${\it\delta}_{99}$ boundary layer thickness.

The collapse of the scaled dispersion curves in figure 17(b) suggests that one might be able to collapse the $N$ factors in a similar fashion. To achieve this, it is assumed that for a given Mach number the dispersion curves for all wall temperatures are well approximated by a two-parameter function $\mathscr{F}$ of the form

(3.4) $$\begin{eqnarray}-{\it\alpha}_{i}^{\ast }{\it\delta}_{99}=\mathscr{F}\left(Re,\frac{{\it\omega}^{\ast }{\it\delta}_{99}}{U_{e}^{\ast }}\right).\end{eqnarray}$$

This functional dependence follows from the collapsed dispersion curves in figure 17(b) and the fact that the dispersion curve is also a function of the Reynolds number. One can show that the following definitions of the $N$ factor are equivalent:

(3.5) $$\begin{eqnarray}N(F)=\int _{x_{o}^{\ast }}^{x^{\ast }}-{\it\alpha}_{i}^{\ast }(F,x^{\ast })\,\text{d}x^{\ast }=\int _{Re_{o}}^{Re}-2{\it\alpha}_{i}(F,Re)\,\text{d}Re.\end{eqnarray}$$

Here $Re_{o}=\sqrt{U_{e}^{\ast }x_{o}^{\ast }/{\it\nu}_{e}^{\ast }}$ is the Reynolds number at which the frequency $F$ first becomes unstable, ${\it\alpha}_{i}^{\ast }$ is the dimensional spatial growth rate and ${\it\alpha}_{i}$ is the non-dimensional growth rate normalized by the Blasius thickness ${\it\delta}$ . In (3.5), $F$ is the frequency parameter defined by

(3.6) $$\begin{eqnarray}F=\frac{{\it\omega}^{\ast }{\it\nu}_{e}^{\ast }}{U_{e}^{\ast 2}}.\end{eqnarray}$$

Making use of these definitions, one can rewrite the dispersion relation (3.4) in the form

(3.7) $$\begin{eqnarray}{\it\alpha}_{i}\frac{{\it\delta}_{99}}{{\it\delta}}=\mathscr{F}\left(Re,F\frac{{\it\delta}_{99}}{{\it\delta}}\right).\end{eqnarray}$$

By substituting this result into (3.5), one arrives at the functional dependence:

(3.8) $$\begin{eqnarray}N{\it\delta}_{99}/{\it\delta}=\mathscr{F}(Re,F{\it\delta}_{99}/{\it\delta}).\end{eqnarray}$$

The ratio ${\it\delta}_{99}/{\it\delta}$ is constant for a given Mach number and usually falls between approximately 3 and 20, increasing with both Mach number and wall temperature. The functional form of (3.8) suggests that, if one makes measurements of maximum $N$ factors $N_{1}$ and $N_{2}$ for two different wall temperatures, then the maximum $N$ factors and the corresponding most amplified frequencies $F_{1}$ and $F_{2}$ are related by

(3.9a,b ) $$\begin{eqnarray}N_{2}=N_{1}\frac{{\it\delta}_{99,1}/{\it\delta}_{1}}{{\it\delta}_{99,2}/{\it\delta}_{2}},\quad F_{2}=F_{1}\frac{{\it\delta}_{99,1}/{\it\delta}_{1}}{{\it\delta}_{99,2}/{\it\delta}_{2}}.\end{eqnarray}$$

The effectiveness of this scaling is demonstrated by several examples given in figure 18. Figure 18(a) shows the maximum $N$ factors for several different wall temperature conditions, as well as individual $N$ factor curves for the frequency $F=4\times 10^{-4}$ . The $N$ factor at $Re=1500$ is approximately 2.6 times greater for the cold wall than for $T_{w}/T_{e}=3$ , and the frequency $F=4\times 10^{-4}$ is amplified at different locations for each wall temperature condition.

Figure 18.

Maximum $N$ factors for several different wall temperature conditions: $Ma=4.5$ , $T_{w}^{\ast }=300~\text{K}$ , $Re=1500$ , $P_{e}^{\ast }=10~\text{kPa}$ . Only 2D waves ( ${\it\beta}=0$ ) are included. (a) Maximum $N$ factors and individual $N$ factor curves for $F=4\times 10^{-4}$ . (b) Maximum $N$ factors scaled by ${\it\delta}_{99}/{\it\delta}$ and individual $N$ factor curves corresponding to $F({\it\delta}_{99}/{\it\delta})=2\times 10^{-3}$ .

Figure 18(b) shows the same $N$ factor curves, but scaled by the ratio ${\it\delta}_{99}/{\it\delta}$ , which significantly reduces the spread between the curves. Also shown in figure 18(b) are the $N$ factors for the frequencies satisfying $F({\it\delta}_{99}/{\it\delta})=2\times 10^{-3}$ . All of the frequencies scaled in this way are amplified at nearly the same Reynolds number. For both the frequencies and the $N$ factors, the collapse is poorer for large $T_{w}/T_{e}$ . The poorer collapse for large $T_{w}/T_{e}$ may in part be associated with the much smaller free-stream temperatures used in these cases, since the viscosity varies more rapidly at low temperatures.

This manner of collapsing the $N$ factors has been applied over a wide range of wall temperatures and Mach numbers. The result is shown in figure 19. Figure 19(a) reports the maximum $N$ factors at $Re=1500$ without using the scaling of (3.8). This plot includes results only for 2D waves ( ${\it\beta}=0$ ) since these are most amplified for the second mode, but it should be noted that 3D first-mode waves can produce larger $N$ factors for some of the cases with lower Mach numbers and large values of $T_{w}/T_{e}$ . The maximum $N$ factors shown in the figure exhibit the same trends as the growth rates from figure 15, namely, maximum amplification at approximately $Ma=5$ , a systematic increase in growth as the wall is cooled and reduction in the most amplified Mach number as the wall is cooled. This indicates that, although the influences of Mach number and wall temperature have historically been characterized mainly using the maximum spatial growth rates, the $N$ factors behave similarly.

Figure 19.

Effect of Mach number and wall cooling on the maximum $N$ factors for the second mode. Only 2D waves ( ${\it\beta}=0$ ) are considered. In all cases, $Re=1500$ , $T_{w}^{\ast }=300~\text{K}$ and $P_{e}^{\ast }=10~\text{kPa}$ . (a) Maximum $N$ factors. (b) Maximum $N$ factors scaled by ${\it\delta}_{99}/{\it\delta}$ .

Figure 19(b) attempts to collapse the maximum $N$ factors for different wall temperatures by scaling the $N$ factor by ${\it\delta}_{99}/{\it\delta}$ . This scaling is quite successful for the cases with $0.2\leqslant T_{w}/T_{e}\leqslant 1.5$ , but performs poorly for the cases with large $T_{w}/T_{e}$ . In spite of the imperfect collapse of the data, the scaling arguments demonstrated here may be useful for estimating how a change in the wall temperature or stagnation temperature might affect the stability characteristics and transition location in an experiment.

The frequency scaling presented in figure 16 and its variation with wall temperature and Mach number are compared with experimental measurements in table 1. For both experiments and computations, the frequency given is the most amplified one; in the computations only 2D waves with ${\it\beta}=0$ are included since the measured frequencies are believed to be second-mode waves. Measurements in the AEDC Tunnel 9 were made by Marineau et al. (Reference Marineau, Moraru, Lewis, Norris, Lafferty, Wagnild and Smith2014) using PCB pressure transducer arrays. Measurements in Caltech’s T5 reflected shock tunnel were made by Parziale (Reference Parziale2013) using focused laser differential interferometry (FLDI). The FLDI technique was also used in the Hypersonic Ludwieg tube at Braunschweig (HLB) (D. Heitmann, unpublished observations). Further details regarding the operation of this facility are given by Estorf, Wolf & Radespiel (Reference Estorf, Wolf and Radespiel2004) and Heitmann, Radespiel & Knauss (Reference Heitmann, Radespiel and Knauss2011). Measurements in the High Enthalpy shock tunnel at Göttingen (HEG) were made by Laurence et al. (Reference Laurence, Wagner, Ozawa, Schramm and Hannemann2014) using schlieren deflectometry and full-field pulsed-laser schlieren photography. Measurements in AEDC Tunnel B were made using hot wires by Stetson et al. (Reference Stetson, Thompson, Donaldson and Siler1983); these data were more recently re-plotted and analysed by Schneider (Reference Schneider2006).

Table 1.

Comparison of most amplified frequencies from 2D computation ( ${\it\beta}=0$ ) and experiment. Here $Ma$ is the post-shock Mach number; and $Re$ is the Reynolds number for flow over a cone, equal to $\sqrt{3}$ times the value for a plate. The streamwise distance is $s$ and the nose radius is $R_{n}$ . For AEDC Tunnel B, the wall temperature is nearly adiabatic. For all other cases, the wall temperature is ambient.

All of the measured frequencies agree reasonably well with the simulated ones, although the agreement is slightly poorer for the reflected shock tunnels T5 and HEG, where the free-stream conditions are known with less certainty. The agreement between experiment and computation is also generally better at low Reynolds numbers, where the measured disturbances remain in the linear regime. In all cases except HEG, the cones are nominally sharp, and the reported nose radii (which can be determined from table 1) are small enough that the measurement stations are far downstream of the entropy swallowing length. For the HEG data, the cone is slightly blunted with a nose radius of $R_{n}=2.5~\text{mm}$ . According to the swallowing length correlations reported by Stetson (Reference Stetson1980), the measurement station is at about twice the swallowing length, but the local boundary layer thickness may still be influenced by the nose bluntness (Stetson et al. Reference Stetson, Thompson, Donaldson and Siler1984). This may in part explain why the measured frequencies are slightly larger than the sharp-cone predictions. This may also be a factor in the T5 data, where the ratio of the streamwise distance $s$ to the nose radius $R_{n}$ is approximately $s/R_{n}=1260$ and the measured frequencies are again somewhat greater than the sharp-cone prediction. The experiments tabulated here employ an array of different measurement techniques and span a wide range of flow conditions, from low-enthalpy adiabatic flow (AEDC Tunnel B) to high-enthalpy reflected shock tunnels (HEG and T5). Although the Reynolds numbers of the measurements are slightly different from the value $Re=1000$ used in figure 16, the measured data confirm the major features of the trends shown in the figure.

A final illustration of the effects of Mach number and wall temperature on the stability behaviour is given in figure 20. This plot contains the same data from figure 15, but is organized in contour form to better portray the stability boundaries. The contour levels indicate the spatial growth rate, which is optimized over all values of the frequency ${\it\omega}$ and spanwise wavenumber ${\it\beta}$ . For reference, a line (symbols) corresponding to the adiabatic wall condition is also included. Most practical conditions would fall below this line, as the points above it correspond to a heated wall. There is a clear region at low Mach numbers and small values of $T_{w}^{\ast }/T_{e}^{\ast }$ for which no modal instabilities are found. Although this region shrinks slightly as the Reynolds number is increased, it is still present even at rather high Reynolds numbers, $Re>4000$ . Since no modal instabilities are found at these conditions, one might conclude that large-amplitude (nonlinear) disturbances are needed to cause transition when the Mach number is low and the wall is cold. However, recent transient growth calculations have reported elevated levels of non-modal amplification at these conditions (Bitter & Shepherd Reference Bitter and Shepherd2014), which offers an alternative transition mechanism for infinitesimal disturbances.

Figure 20.

Map of unstable regions for first and second modes: $Re=1500$ , $T_{w}^{\ast }=300~\text{K}$ , $T_{e}^{\ast }=70{-}2000~\text{K}$ , $P_{e}^{\ast }=10~\text{kPa}$ . Contour levels are spatial growth rates maximized over all values of ${\it\omega}$ and ${\it\beta}$ for the first mode (dashed) and second mode (solid).