Turbulent spots in hypervelocity flow

Turbulent spots in hypervelocity flow

Joseph S. Jewell ${ }^{1}$ ・ Ivett A. Leyva ${ }^{2} \cdot$ Joseph E. Shepherd ${ }^{3}$

Abstract

The turbulent spot propagation process in boundary layer flows of air, nitrogen, carbon dioxide, and air/carbon dioxide mixtures in thermochemical nonequilibrium at high enthalpy is investigated. Experiments are performed in a hypervelocity reflected shock tunnel with a 5-degree half-angle axisymmetric cone instrumented with flush-mounted fast-response coaxial thermocouples. Timeresolved and spatially demarcated heat transfer traces are used to track the propagation of turbulent bursts within the mean flow, and convection rates at approximately 91,74 , and $63 \%$ of the boundary layer edge velocity, respectively, are observed for the leading edge, peak, and trailing edge of the spots. A simple model constructed with these spot propagation parameters is used to infer spot generation rates from observed transition onset to completion distance. Spot generation rates in air and nitrogen are estimated to be

^[1]approximately twice the spot generation rates in air/carbon dioxide mixtures.

1 Introduction

Transition from laminar to turbulent flow in boundary layers occurs in most instances through the genesis, growth, and propagation of isolated local turbulence patches, known as turbulent spots. Emmons (1951) was the first to propose that laminar boundary layers break down through the convergence of spots, after observations of a water-table analogy to air flow. Spot formation and propagation has been studied extensively in subsonic and transonic flows, notably theoretical and experimental investigations of spot propagation and growth by Narasimha (1957, 1985), Dhawan and Narasimha (1958), Chen and Thyson (1971), AbuGhannam and Shaw (1980), and Clark et al. (1994).

The first turbulent spots in a supersonic boundary layer were detected by James (1958) on free-launched projectiles using spark shadowgraphs with a conical light field, characterizing both propagation speed and growth rate for freestream Mach numbers from 2.7 to 10. James was able to surmise that the difference between the turbulent-spot propagation process in subsonic and supersonic flow was likely to be small. Fischer (1972) surveyed available supersonic and hypersonic spot studies and showed a relationship between the spreading angle of turbulent disturbances and the Mach number. Since then, a number of spot studies in supersonic and hypersonic flows have been carried out, with reviews included in Mee (2002) and Fiala et al. (2006). Recent cold-flow hypersonic work also includes experiments at Mach 5-14 by Casper et al. (2016) and an analysis of Mach 6 spot pressure fluctuations by Casper et al. (2014). Turbulent spots have generally been found to

Fig. 1 Schematic depiction of a triangular turbulent spot, with the velocities at the leading and trailing edges labeled as $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$, and the spreading angle labeled as $\alpha$, the spreading angle with respect to the velocity vector at the boundary layer edge. Both $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$ are some fraction of $U_{\mathrm{e}}$. Not depicted is $U_{\mathrm{te}}$, the spot peak heating velocity
have roughly triangular shape in planform, as depicted in Fig. 1, with the velocity of the leading and trailing edges each some fraction of the velocity at the boundary layer edge, $U_{\mathrm{e}}$. The rate at which the spot grows laterally as it progresses downstream along the surface and additional fluid is entrained is described by a spreading angle $\alpha$. As Schubauer and Klebanoff (1955) observe: “The triangular shape of the spot with vertex pointing downstream may be accounted for by the fact that the downstream end does not have the time that the upstream end has in which to grow laterally.” As $U_{\mathrm{le}}$ tends to be larger than $U_{\mathrm{te}}$, the spot grows longitudinally as it progresses downstream at a rate controlled by the difference between the two velocities, as well as $\alpha$.

2 Turbulent spot observations

All measurements presented here are made in T5, the freepiston driven reflected-shock tunnel at the California Institute of Technology. T5 is designed to simulate flow conditions and aerothermodynamics of hypervelocity vehicles at total enthalpies up to $25 \mathrm{MJ} / \mathrm{kg}$ and freestream velocities of up to $6000 \mathrm{~m} / \mathrm{s}$. During each T5 experiment, pistoncompressed helium/argon gas in the driver section ruptures a scored, stainless steel primary diaphragm. Following the primary diaphragm rupture, a shock wave propagates in the shock tube, reflects off the end wall, breaks the secondary diaphragm, and re-processes the test gas, which is then expanded through a converging-diverging contoured nozzle to $\sim$ Mach 5.5 in the test section (Hornung 1992). Experiments were performed on a 5-degree half-angle smooth cone, which is more fully described in Jewell (2014), at zero angle of attack in air, nitrogen, carbon dioxide, and air/ carbon dioxide mixtures.

A method of presenting time-resolved (within the mean flow) and spatially demarcated heat flux data has been developed and implemented, which allows the presentation of a “movie” of heat flux over the entire instrumented surface of the cone during the test time by interpolating the signals from each of 80 thermocouples of a design first used by Sanderson and Sturtevant (2002) with $\sim 1 \mu \mathrm{~s}$ response time as characterized by Marineau and Hornung (2009), sampled at 200 kHz with data reduced as described in Jewell (2014). Figure 2 depicts four frames from the results for shot 2698 , over a total time of 0.3 ms during which a turbulent spot is clearly seen to propagate downstream. A similar surface heat flux method (with thin film gauges rather than thermocouples) was previously used to visualize spots in subsonic flow by Anthony et al. (2005) and in supersonic flow by Fiala et al. (2006).

The non-dimensionalized average heat flux gauge signals were examined for each of the 24 experiments referenced in Table 1 as Stanton number vs. Reynolds number plots, based on the distance of the gauge from the tip of the cone, as well as heat flux contour plots similar to Fig. 2 for shot 2698. In each case, the boundary layer is on average observed to be laminar over most of the cone (low heatflux regions are illustrated in Fig. 2) for the majority of the test time, with some cases showing incipient or intermittent transition near the end of the cone. Therefore, the observed spots are propagating as isolated turbulent patches within a surrounding flow field that is mainly laminar. Indeed, this relative isolation is the feature that enables tracking each event as a distinct spot over multiple gauges.

The initiating events for these isolated spots are unknown, but may be due to particulate impact on the cone’s boundary layer, which is examined computationally in Fedorov (2013) and experimentally in Jewell et al. (2017). Another possible source of initiation is the nonlinear breakdown of second mode instabilities deriving from freestream acoustic disturbances (Fedorov 2003; Sivasubramanian and Fasel 2012, 2015), which have been observed within similar transitional boundary layers in T5 using recently implemented optical methods for observing low-amplitude, high-frequency density fluctuations, as described in Parziale et al. (2012a, 2012b). Similar events have been observed in other high-enthalpy facilities via schlieren videography (Laurence et al. 2016). A survey of experimental results included in Fiala et al. (2006) did not find significant differences in propagation parameters between artificially induced and naturally occurring turbulent spots. Therefore, it may reasonably be assumed that velocity measurements of the spots observed in the present study and tabulated in Table 1 are relevant for the transition process in T5.

To quantitatively analyze spots, the signals from each of the eight rays of thermocouples mounted on the

Fig. 2 Four frames from shot 2698 covering a total time of 0.3 ms , during which a turbulent spot is observed near the tip and propagates downstream. This example was chosen because the spot is wellisolated and happens to be approximately centered on the developed cone surface. Thermocouple locations are indicated by small circles

Fig. 3 A spot observed as a localized region of increased heat flux typically lasting for $20-30$ time steps $(0.1-0.15 \mathrm{~ms})$ over an individual gauge, during shot 2700 along one ray of gauges between TC16 and TC80, tracked in successive heat transfer (in $\mathrm{MW} / \mathrm{m}^{2}$ ) versus time plots. The steady test time for this experiment begins at 1.0 ms
cone are examined individually for localized regions of elevated heat flux. Turbulent spots are seen on these heat transfer traces as well-ordered excursions above the baseline heat flux with roughly triangular form, typically passing over the gauge in about 0.1 ms , propagating downstream over the series of gauges at some fraction of the boundary layer edge velocity, and growing spatially. One example of a spot tracked down one ray of sensors, observed during shot 2700, is presented in Fig. 3. The spot is seen first on thermocouple (TC) 16 at about 1.7 ms after the trigger time, which is 0.7 ms into the steady test time for this experiment. The spot subsequently appears and passes over each thermocouple shown in the sequence, eventually passing over TC80 at the end of the cone. The ramps in the signal between 1.5 and 2.0 ms indicate arrival and departure at the gauge of the spot’s leading and trailing edges, respectively, and are indicated on the plot for TC80 at the bottom. Typically, each turbulent spot is observed for $20-30$ time steps $(0.1-0.15 \mathrm{~ms})$ over an individual gauge, which is sufficient to resolve it from the mean flow, but not finely grained enough to identify internal features of the spot.

Fig. 4 Leading edge, trailing edge, and peak heating velocities for the spot observed during shot 2700, which is depicted in Fig. 3. The modeled value for the slope of each set of points is taken as the velocity of each location on the spot

Clear leading edge, trailing edge (taken as the time of first heat transfer $5 \%$ above the smoothed signal, and the time of return to within $5 \%$, respectively, with a moving average window size of 0.015 ms ) and peak heat transfer signals for the spot are observed on most of the thermocouples. Where the trailing edge or peak is indistinct, no time is recorded. Each of these three features is tagged in each subsequent trace, and the resulting timestamps correlated with the position on the cone of each thermocouple to produce plots such as Fig. 4. The modeled value for the slope of each set of points is taken as the velocity of each location on the spot, and the $95 \%$ confidence interval and standard error for each slope is calculated using the Matlab Statistics Toolbox.

The 29 leading edge, trailing edge, and peak heating velocities inferred from experiments described in Jewell (2014) and Jewell and Shepherd (2014) are shown graphically in Fig. 5 plotted against the boundary layer edge velocity $U_{\mathrm{e}}$. The measured velocities increase linearly with edge velocity, as expected, but no clear trend (independent of edge velocity) is observed for varying gas mixtures.

Evidence of correlation between leading edge, trailing edge, and peak heating spot velocities and the boundary layer edge properties is sought using reverse-stepwise linear regression (Draper and Smith 1998) as implemented in Matlab. The predictor parameters chosen to seek correlation with the three spot velocity measurements are the edge conditions: pressure $P_{\mathrm{e}}$, velocity $U_{\mathrm{e}}$, temperature $T_{\mathrm{e}}$; and shot number (which is chosen as a dummy variable and not expected to have any effect). The $p$ value required

Fig. 5 Leading edge, trailing edge, and peak heating velocities for all spots during the present campaign. The velocities increase proportionally with $U_{\mathrm{e}}$
to remove a parameter from the regression is 0.05 . Under this criterion, the spot velocities are found to have a linear dependence only upon the calculated velocity at the boundary layer edge, $U_{\mathrm{e}}$. Therefore, the measured spot velocities are normalized by $U_{\mathrm{e}}$ to find appropriate values for $C_{\mathrm{le}}, C_{\mathrm{m}}$, and $C_{\mathrm{te}}$ which are compared with past spot experiments and computations in Sect. 5 and used as an input in the spot propagation simulation discussed in Sect. 3. There is substantial uncertainty in defining the precise leading and trailing edges of the spot at each thermocouple, which is estimated at 0.01 ms (twice the sampling period of 0.005 ms ) for the leading edge and peak, and more conservatively 0.02 ms (four times the sampling period) for the trailing edge, which is less sharply defined. The uncertainty in the position of each thermocouple is estimated as 0.1 mm . These uncertainties are incorporated into a least-squares estimate of the slope (velocity) using the method of York et al. (2004), and the resulting standard errors are represented by uncertainty bars in Figs. 5 and 6. The uncertainty in the computed velocity decreases as the number of thermocouples used to observe the spot increases. The number ranges from 3 to 9 in the present study. For random, uncorrelated measurement noise, the velocity uncertainty determined from the standard error of the linear regression slope is proportional to $1 / \sqrt{(N)}$. The velocity results of all 29 isolated spot examples are combined using a linear regression means-weighted by the inverse standard error calculated for each example, which also permits the calculation of an overall parameter confidence interval for the average normalized velocities $\left(C_{\mathrm{le}}, C_{\mathrm{m}}\right.$ and $\left.C_{\mathrm{te}}\right)$. The normalized velocities are presented in Fig. 6. The leading edges propagate downstream at a velocity of $0.91 U_{\mathrm{e}}(95 \%$

Fig. 6 Leading edge and trailing edge velocities for all spots normalized by $U_{\mathrm{t}}$, with average normalized velocities $\left(C_{\mathrm{le}}, C_{\mathrm{m}}\right.$ and $\left.C_{\mathrm{te}}\right)$ depicted as lines with uncertainty zones
confidence interval $0.88-0.94$ ), the peaks at $0.74 U_{\mathrm{e}}(95 \%$ confidence interval $0.70-0.78$ ), and the trailing edges at $0.63 U_{\mathrm{e}}(95 \%$ confidence interval $0.59-0.66)$. In this figure, the points representing $C_{\mathrm{m}}$ are omitted for clarity, although the mean value and $95 \%$ confidence interval for that measure are given.

The velocity results for all 29 spots are presented in Table 1 along with other relevant tunnel and boundary layer parameters for each test. Note that there are two entries for several shots, indicating that two individual spots were observed during these experiments. They are differentiated by time of first observation during the test, and the number of the first thermocouple their signal appears on. The normalized velocity coefficient uncertainties in this table are the standard errors as determined by the fitting process.

3 Turbulent spot model and simulations

3.1 Background

Key variables determining the transition zone length, or the distance from transition onset to completion, $\Delta x_{\mathrm{Ti}}$, are the spot generation, breakdown, or initiation rate $n$, spreading angle $\alpha$, and leading- and trailing-edge velocities $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$ (Kimmel 1993). The spot generation rate $n$ may depend upon the receptivity of the boundary layer and the disturbance environment of the facility; Narasimha (1985) surveyed subsonic data and found that $n$ generally increases with freestream turbulence levels. Mee and Tanguy (2015) developed a procedure using a simple spot propagation model to infer turbulent spot initiation rates by coupling experimentally measured
transition zone length with three assumptions about spot growth and development; namely, that the spot spreading angle and leading- and trailing-edge propagation speeds are known. A similar model is used for the present analysis.

The present model is based upon the algorithm developed in Jewell (2008) for a flat plate, which was in turn an implementation in Matlab of the spot propagation process outlined in Clark (1993). In all of these approaches, the turbulent spots are modeled, following Narasimha (1985), as triangular in planform, and assumed to be generated randomly in time and $y$-position at the line defined by a particular $x$-displacement (or in a defined distribution within a small band around a particular $x$-displacement) from the edge or tip of the test article. The program as implemented in the present study models the stochastic nature of spot production by generating a random value for $y$ within the physical extent of the simulated surface, at which a spot is generated each time increment $\Delta t$, where $\Delta t$ is the inverse of the spatial spot generation rate $n$ (which is varied as an input) divided by the width of the surface. The generated spots are deterministically propagated as described below, with the leading and trailing edges advancing by $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$, respectively, and the leading edge points spreading by $\alpha$.

The Jewell (2008) algorithm is adapted for conical flow by performing the calculation on a developed cone surface, as shown in Fig. 7. In this case $U_{\mathrm{e}}$ is nonparallel with respect to the coordinate system of the developed surface, but emanates radially from the origin. Both $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$ are some fraction of $U_{\mathrm{e}}$. As $U_{\mathrm{le}}$ tends to be larger than $U_{\mathrm{te}}$, the spot grows longitudinally as it progresses downstream. The rate at which it grows laterally is controlled by $\alpha$ and the angle of the cone. The spreading angle is assumed to vary inversely with edge Mach number for each case with the theoretical relationship found by Doorley and Smith (1992) ${ }^{1}$ in the data reported by Fischer (1972).
$\alpha \approx \frac{0.27}{\mathrm{M}_{\mathrm{e}}}$
This relationship is given in Eq. 1 and was also used in the Mee and Tanguy (2015) model. $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$ are calculated from multiplying the boundary layer edge velocity by $C_{\mathrm{le}}$ and $C_{\mathrm{te}}$, which are normalized experimental means, as shown in Fig. 6. The $95 \%$ confidence intervals for the three $C$ values are comparable to or smaller than past reported uncertainties for these measures [cf. Mee (2002), who reported $\pm 10 \%$ for similar conditions, and Clark et al. (1994), who reported

^[1]

Table 1 Individual spots observed during the present test campaign $(n=29)$ from 24 tests

Each of these spots was isolated enough to distinguish as a distinct phenomenon on the cone during the steady test time, and was observed to propagate downstream over three or more heat transfer gauges, permitting the measurement of leading edge, trailing edge, and peak heating velocities. For tests 2737, 2809-2818, the balance of the composition was air. There were various extents of nonequilibrium and dissociation in all cases; the computed boundary layer edge conditions (species, temperature and pressure) are tabulated in Jewell and Shepherd (2014)
up to $\pm 4 \%$ for much slower conditions]. $U_{\mathrm{e}}$ and $\mathrm{M}_{\mathrm{e}}$ are computed as described in Jewell (2014).

3.2 Flat plate and conical simulations and theory

Narasimha (1957) introduced a universal intermittency curve, Eq. 2, where $\gamma$ is the intermittency, or fraction of the test time that the flow at a given $x$-displacement beyond the transition onset location, $x_{\mathrm{Tr}}$, is turbulent. Eq. 2 applies where $x \geq x_{\mathrm{Tr}}$. For $x \leq x_{\mathrm{Tr}}, \gamma=0$.
$\gamma_{\text {plate }}(x)=1-\exp \left[-\frac{n \sigma}{U_{\mathrm{e}}}\left(x-x_{\mathrm{Tr}}\right)^{x}\right]$
This concept, which was derived for the propagation of spots on a flat plate, is further developed in Narasimha (1985) and has been found to hold for both incompressible
(e.g., Clark et al. 1994) and compressible (e.g., Mee and Tanguy 2015) transitional boundary layers. The relationship between $\gamma$ and $x$ depends upon edge velocity, the nondimensional spot growth (or propagation) parameter $\sigma$ of Emmons (1951), and the spot generation parameter $n$, which is the number of spots generated per unit length and time across $x=x_{\mathrm{Tr}} . \sigma$ incorporates both lateral and streamwise growth, and is commonly taken to be, as in Vinod and Rama (2004),
$\sigma=\left[\frac{1}{U_{\text {te }}}-\frac{1}{U_{\text {le }}}\right] U_{\mathrm{e}} \tan \alpha$
The universal intermittency curve for axisymmetric conical flow,
$\gamma_{\text {cone }}(x)=1-\exp \left[-\frac{n \sigma}{U_{\mathrm{e}}} x_{\mathrm{Tr}}\left(\ln \frac{x}{x_{\mathrm{Tr}}}\right)\left(x-x_{\mathrm{Tr}}\right)\right]$

Fig. 7 Schematic depiction of a triangular turbulent spot on the developed surface of a cone, with the velocities at the leading and trailing edges labeled as $U_{\mathrm{le}}$ and $U_{\mathrm{te}}$, and the angle at which additional fluid is entrained in the spot as it moves labeled as $\alpha$, the spreading angle with respect to the velocity at the boundary layer edge, $U_{\mathrm{e}}$. In this coordinate system, the boundary layer edge velocity acting on the spots has a transverse component

Fig. 8 Flat plate and cone intermittency $\gamma$ and curves for $n=4 \times 10^{6}$ computed at shot 2776 conditions, compared with the relevant universal intermittency curves. Excellent agreement is obtained
was derived by Cebeci and Smith (1974), where again $\gamma=0$ for $x \leq x_{\mathrm{Tr}}$.

In Fig. 8, $\gamma_{\text {plate }}$ and $\gamma_{\text {cone }}$ from Eqs. 2 and 4 are compared with propagating-spot simulation results using the edge conditions of shot 2776 and $\alpha$ as defined in Eq. 1, with $n=4 \times 10^{6}$ spots $/ \mathrm{m} / \mathrm{s}$. Excellent agreement is found between theory and the simulations. It is also seen that at any given $x$-displacement, intermittency is higher for the plate than the cone, which indicates that spots, on average, take longer to merge with their neighbors in the conical than in the planar boundary layer, lengthening the transition region for conical cases.

This effect is illustrated by comparing plotting the spot outline in Cartesian coordinates in Fig. 9, with

Fig. 9 Path of a single centered spot with spreading angle $\alpha=2.5^{\circ}$ generated at $x_{\mathrm{Tr}}=0.389$, for a plate or cylinder (top) and an axisymmetric cone with $5^{\circ}$ half-angle (bottom). While the growth profile of a spot in both geometries is similar, the increasing total surface area of the cone with respect to $x$ means that each spot covers a smaller circumferential fraction of the surface, as shown in Fig. 10

Fig. 10 Path of a single centered spot in angular coordinates with spreading angle $\alpha=2.5^{\circ}$ generated at $x_{\mathrm{Tr}}=0.389$, for a cylinder (top) and an axisymmetric cone with $5^{\circ}$ half-angle (bottom), both 1 m long with circumference of 0.220 m at the transition point
polar coordinates in Fig. 10. Both of these figures depict the outer envelope of a single spot with spreading angle $\alpha=2.5^{\circ}$ generated along the centerline of the cone or cylinder at $x_{\mathrm{Tr}}=0.389$ as it grows while propagating down the surface. Figure 9 presents the spot envelope in Cartesian coordinates for a flat plate (or cylinder) in the top plot, and for a cone in the bottom plot. The physical growth of the spot is similar in both cases-in fact, due to the nonparallel edge velocity acting at the edges of the spot in the conical

Fig. 11 Flat plate and cone intermittency $(\gamma)$ curves for several spot generation rates $n$ computed at the conditions of shot 2776

Fig. 12 Flat plate and cone intermittency $(\gamma)$ curves for several spot generation rates $n$ computed at the conditions of shot 2776. These curves have been transformed by the Narasimha (1957) linearizing function $F(\gamma)$
case, the lateral extent of the spot is actually slightly greater by the end of the 1 m cone than it as at the end of the 1 m flat plate (or equivalently, 1 m cylinder).

However, in the conical case, the surface area also increases with $x$, which is not the case for a plate or cylinder. Figure 10 presents the spot’s outer envelope in angular coordinates for a cylinder (with circumference of 0.220 m , equal to that of the cone at the transition point) in the top plot, and for a cone in the bottom plot. In angular terms, the turbulent spot on the cylinder covers nearly twice the circumferential fraction of the surface by the end of the body compared to the turbulent spot on the cone. This accounts for the difference in intermittency seen in Fig. 8.

The time-averaged intermittency at each $x$ for the edge conditions of shot 2776 over each of five full plate and cone simulations is presented in Fig. 11, and linearized with the $F(\gamma)$ of Narasimha (1957) in Fig. 12. The distance $\Delta x_{\mathrm{Tr}}$ from $x_{\mathrm{Tr}}$ that it takes to reach $\gamma \approx 1$ (here taken as $\gamma=0.99$ ) decreases with increasing $n$, and for all cases the transition length is shorter for the flat plate case than the conical case with the same $n$, which is the same behavior seen in Fig. 8.

4 Correlation with experiments to infer spot generation rate

For each experiment recorded in Jewell (2014) and Jewell and Shepherd (2014) for which both transition onset and transition completion are clearly observed, a set of simulations are run varying the spot generation parameter $n$, with spots generated randomly along the band defined by $x=x_{\mathrm{Tr}}$ at a rate defined by the product of the circumference of that band and the input $n$. For each case, a total of 10 ms of test time is run (discarding the first $2\left(1-x_{\mathrm{Tr}}\right) / U_{\mathrm{c}} \mathrm{~ms}$ to allow time for the spots to fully develop), with a time step

Fig. 13 One frame from the 10 ms conical spot simulation for shot 2740 with $n=6 \times 10^{6}$ spots $/ \mathrm{m} / \mathrm{s}$

Fig. 14 Experimentally measured distance between transition onset and transition completion for shot 2740. The contour plot (bottom) presents the average heat flux during the test time over the surface of the cone. On the Re vs. St plot (top), a line is fit through the transitional region, and the transition onset location $x_{\mathrm{Tr}}$ is taken as the intersection of the fit with the laminar correlation. The transition completion location $x_{\mathrm{Tr} / \text { step }}$ is taken as the intersection of the fit with the White and Christoph (1972) turbulent correlation. The distance between the two points is $\Delta x_{\mathrm{Tr}}$
sufficiently small to resolve the motion of individual spots. One frame from such a simulation is presented in Fig. 13, taken from a computation at the conditions of shot 2740 with a simulated spot generation rate of $n=6 \times 10^{6}$ spots/ $\mathrm{m} / \mathrm{s}$. The laminar region upstream of $x_{\mathrm{Tr}}$, where $\gamma=0$, a region with distinct spots, and a region of merged spots, where $\gamma \approx 1$, are all clearly discernible. These regions are interpreted as laminar, intermittent/transitional, and fully turbulent flow, respectively.

The transition length $\Delta x_{\mathrm{Tr}}$ is defined as the distance between the spot generation location and the location where the spots have merged such that $\gamma=0.99$. The value of $\Delta x_{\mathrm{Tr}}$ is taken from the linearized curves $F(\gamma)$ as in Fig. 12, and

Fig. 15 Spot generation rates for the present data compared with the results from Mee and Tanguy (2015)

Fig. 16 Computed and experimental transition distance $\Delta x_{\mathrm{Tr}}$ for shot 2740
compared with the experimentally determined transition length, shown in Fig. 14. This measured transition length is shown with the predicted transition length for a range of $n$ values in Fig. 16. As in Mee and Tanguy (2015), $n$ is taken at the point where the computed transition length curve matches the measured transition length. The results for all 17 cases are tabulated in Table 2 along with relevant experimental parameters, and are plotted against unit Reynolds number, along with the replotted data of Mee and Tanguy (2015), in Fig. 15.

It is interesting to note that the two $50 \% \mathrm{CO}_{2}$ results from the present data, shown in Fig. 15, have spot generation rates that are only about one-half of that measured for the air and $\mathrm{N}_{2}$ results. This may indicate that the spot formation process or receptivity to disturbances is inhibited

Fig. 17 Experimental transition distance $\Delta x_{\mathrm{Tr}}$ compared with $\Delta x_{\mathrm{Tr}}$ computed via the mechanistic process described in Sect. 3 for shot 2740 with $C_{\mathrm{te}}=0.5$, which was the value used by Mee and Tanguy (2015). Compared with the computations in Fig. 16, the matching value of $n$ for the observed transition length drops by $3.4 \times 10^{6}$ spots/ $\mathrm{m} / \mathrm{s}$. This accounts for most of the difference between the present computations and those of Mee and Tanguy (2015)
for $\mathrm{CO}_{2}$-containing boundary layers compared with air and $\mathrm{N}_{2}$. It is also possible that there are more disturbance sources, such as particulate matter, in the non- $\mathrm{CO}_{2}$ experiments, but all of the results analyzed in this section and reported in Table 2 used the more stringent cleaning procedure described in Jewell et al. (2017). No other studies of turbulent spots in $\mathrm{CO}_{2}$ were found in the literature. None
of the pure $\mathrm{CO}_{2}$ experiments in the present study with clear transition onset had transition completion by the end of the cone, so these $50 \% \mathrm{CO}_{2}$ cases are the best available data on the effect of $\mathrm{CO}_{2}$ on spot formation in the boundary layer.

Mee and Tanguy (2015) found a dependence of $n$ on unit Reynolds number for air results, and the present air and $\mathrm{N}_{2}$ data show a similar (albeit weak) dependence. While Mee’s results for $n$ are significantly lower than the present data, roughly $70 \%$ of the discrepancy can be accounted for by the fact that the Mee and Tanguy (2015) model used $C_{\text {te }}=0.50$ as given in Mee (2002), while the present model uses $C_{\text {te }}=0.63$ as measured in Sect. 2 (both models used $C_{\text {te }} \approx 0.90$, and the method of calculating $\alpha$ was also identical). This means that Mee’s simulated spots grow more rapidly in the longitudinal dimension, and therefore each spot covers more of the surface, resulting in higher intermittency $\gamma$ for a given spot generation rate $n$.

This effect is demonstrated by running one set of the present simulations (conditions matching shot 2740) with $C_{\text {te }}=0.50$ instead of $C_{\text {te }}=0.63$. These results are shown in Fig. 17. Compared with the results in Fig. 16, the matching value of $n$ for the observed transition length drops from $7.36 \times 10^{6}$ spots $/ \mathrm{m} / \mathrm{s}$ to $3.96 \times 10^{6}$ spots $/ \mathrm{m} / \mathrm{s}$. The value at the same unit Reynolds number for the Mee and Tanguy (2015) curve fit is $2.40 \times 10^{6}$ spots $/ \mathrm{m} / \mathrm{s}$. While the $C_{\text {te }}$ disparity accounts for most of the difference in the $n$ results, the spot generation rate would not necessarily be expected to be identical in these two data sets in any case, as they

Table 2 Spot generation rates computed via the mechanistic process described in Sect. 3 for transition onset-completion length ( $\Delta x_{\mathrm{Tr}}$ ) during the present test campaign, taken from 17 tests for which clear transition onset $\left(x_{\mathrm{Tr}}\right)$ and transition completion ( $x_{\mathrm{TrComp}}$ ) locations were observed

The calculated spot spreading angle $(\alpha)$ and boundary layer edge unit Reynolds number (unit Re) are included along with other relevant parameters

Table 3 Results from past spot propagation studies, based on supersonic and hypersonic experiments (Zanchetta and Hillier 1996; Fiala et al. 2006; Mee 2002; Clark et al. 1994) and computations [Krishnan and Sandham (2006), two results from Jocksch and Kleiser (2008),

Jocksch (2009) and Sivasubramanian and Fasel (2010)] reported for a range of conditions and presented together with the results from the present study

${ }^{a}$ Value not reported
${ }^{\mathrm{b}}$ Calculated by the present authors from other reported values
${ }^{\text {c }}$ Spot “wing tip” convection velocity
were acquired in different facilities with different noise profiles and disturbance sources, as well as on different geometries: flat plates for Mee and a cone in the present work.

5 Comparison with past turbulent spot studies

5.1 Experimental

In addition to the results from Mee and Tanguy (2015) for the spot generation rate $n$, the present experimental results for $C_{\mathrm{le}}, C_{\mathrm{m}}$ and $C_{\mathrm{te}}$ compare well with other experimental supersonic and hypersonic results at similar and disparate boundary layer edge conditions, presented as four cases designated “Exp.” in Table 3.

Clark (1993) and Clark et al. (1994) studied the propagation of naturally occurring turbulent spots in turbine-representative flows from Mach 0.24 to Mach 1.86 using platinum thin-film heat transfer gauges to track individual spots. Clark characterized turbulent spot leading-edge, trailingedge, and “mean” or centroid velocities, and also measured the spreading angle at several Mach numbers in this range. Clark also examined the propagation of turbulent spots in mild and strong pressure gradients both favorable and adverse. Hofeldt (1996) studied spots in flows from Mach 0.24 to Mach 1.86 using thin-film heat transfer gauges, examining the effect of gas-to-wall temperature ratios as well as the overhang region, in which the turbulent spot’s spatial extent in the downstream direction is greater further from the plate, and observing “becalmed” regions behind
turbulent spots. Hofeldt was able to show that the becalmed region behind a turbulent spot is in fact consistent with the re-establishment of a laminar boundary layer.

Mee and Goyne (1996) performed experiments to detect turbulent spots on a flat plate in free-piston shock tunnel flows of Mach 5.6 to 6.1 at low, mid-range, and high unit Reynolds numbers (unit $\mathrm{Re}_{\mathrm{s}}$ between $1.6 \times 10^{6} \mathrm{~m}^{-1}$ and $4.9 \times 10^{6} \mathrm{~m}^{-1}$ ) using thin-film heat transfer gauges. They were able to detect turbulent spot activity and measure intermittency, and recommended further tests to measure convection speeds and spreading rate. Mee (2002), using the same facility as Mee and Goyne (1996) with new instrumentation, measured the effect of using 2 mm -high boundary layer “trips” behind the leading edge of a flat plate in Mach 5.5 to Mach 6.3 flow and found them to be capable of advancing the transition location. Mee measured a spot growth angle of $3.5^{\circ} \pm 0.5^{\circ}$.

Fiala et al. (2006) measured turbulent spots progressing on a blunt cylindrical body with spherical nose in super/ hypersonic flow (Mach 8.9 free stream; Mach 3.5 at the edge of the boundary layer) using a series of thin-film heat transfer gauges. They were able to detect clear turbulent spot activity and measure intermittency by comparing heat transfer time histories from axial gauges in the intermittent region of the body, and also visualize the passing signals from individual spots with a circumferential array of gauges. Laurence et al. (2012) measured the propagation speed of instability waves in a hypersonic boundary layer on a $7.0^{\circ}$ half-angle cone $\left(h_{\text {res }}=3.4 \quad \mathrm{MJ} / \mathrm{kg}\right.$,

$P_{\text {res }}=28.5 \mathrm{MPa}, M_{\mathrm{e}}=6.34$ ) and found ${ }^{2}$ a normalized propagation velocity $C=0.83 \pm 0.02$.

5.2 Direct numerical simulation

Direct numerical simulation (DNS) studies of spot propagation in supersonic flows have been carried out by Chong and Zhong (2005), Krishnan and Sandham (2006), and Jocksch and Kleiser (2008). Most recently, Sivasubramanian and Fasel (2010) performed DNS of turbulent spot evolution on a cone in Mach 6 cold flow and observed the breakdown of two-dimensional second mode disturbances into a three-dimensional wave packet or spot. The present experimental results for $C_{\mathrm{le}}, C_{\mathrm{m}}$ and $C_{\mathrm{te}}$ may be compared with computational supersonic and hypersonic results at similar and disparate boundary layer edge conditions, presented as four cases designated “Comp.” in Table 3. The computations of Krishnan and Sandham (2006), Jocksch and Kleiser (2008), and Sivasubramanian and Fasel (2010) are most representative of the present conditions and these authors’ spot propagation speeds are reasonably consistent with our experimental results, with the exception of one trailing edge velocity from Jocksch and Kleiser (2008). This case has the $T_{\mathrm{w}} / T_{\mathrm{aw}}$ that most closely matches the present T5 conditions, which range from $0.05-0.12$.

Jocksch (2009) observes that the cold wall $C_{\text {te }}$ is nearly equal to the trailing edge velocity for a linear wave packet at the same conditions, and hypothesizes that the cooled spot is elongated due to boundary layer linear stability properties. However, the Jocksch and Kleiser (2008) cold wall $C_{\text {te }}$ value is much slower than any reported experimental result for this velocity, including the Fiala et al. (2006) and Mee (2002) results, which bracket Jocksch in terms of $T_{\mathrm{w}} / T_{\mathrm{aw}}$ . On the other hand, the Fiala et al. (2006) and Mee (2002) $C_{\text {te }}$ results are quite consistent with the present study. One possible explanation for this discrepancy is the structure of the simulated spots, which for the cold wall case have a long trailing edge that is only minimally elevated in terms of skin friction and Stanton number from the background values (e.g., Jocksch 2009, Figs. 3.51-3.54). It is quite likely that the thermocouples of the present study, and even the thin-film gauges used by Fiala et al. (2006) and Mee (2002), do not fully resolve this long tail if it exists, leading to a systematic over-measurement of $C_{\text {te }}$.

While the design of the experiment precludes precise measurement of spot spreading angle $\alpha$, approximate bounding values for this parameter have been obtained. For

^[1]example, for shot 2654 , we estimate $2^{\circ}<\alpha<13^{\circ}$. This result brackets the reported value of $3.5^{\circ} \pm 0.5^{\circ}$ of Mee (2002) for similar Mach numbers, as well as the reported value of $6.75^{\circ} \pm 1.0^{\circ}$ of Fiala et al. (2006) for lower Mach numbers. These are the two experimental studies most comparable to the present work in terms of conditions. Both of these values are also consistent with the Mach number-spreading angle relationship, Eq. 1, reported in Doorley and Smith (1992). More precise measurements of spreading angle would be possible with the addition of thermocouples in a more circumferentially dense pattern. The relatively small uncertainty on the spreading angle measurements in Mee (2002) and Fiala et al. (2006) is due to the use of densely packed thin film arrays extending on the test article surface in the direction orthogonal to the flow field. Mee (2002) used seven sensors at 5 mm pitch and Fiala et al. (2006) used 18 sensors at 4 mm pitch. By contrast, the present work uses rows of four sensors at pitches ranging from 19.2 mm for the first row to 82.1 mm for the last row.

6 Conclusions

Time-resolved and spatially demarcated heat transfer traces in a high-enthalpy hypervelocity flow on a 5-degree halfangle cone are measured with surface-mounted thermocouples. Turbulent spots are observed propagating in both heat transfer traces and heat flux “movies” of the developed cone surface. These observations are used to calculate turbulent spot convection rates, which are compared with previous experimental and computational results. Although the present results were obtained at different conditions from past experiments, the normalized spot propagation results for the present hypervelocity Mach $\sim 5\left(U_{\mathrm{e}}=2400-4450 \mathrm{~m} / \mathrm{s}\right.$, $T_{\mathrm{w}} / T_{\mathrm{e}}=0.130-0.243$ ) conditions appear to be generally consistent with past supersonic and hypersonic experiments, as well as with the computational results.

However, the flow conditions in all of the reviewed DNS cases are essentially nonreactive (cold flow with frozen composition), and the ratios of freestream to wall temperature, as well as adiabatic to nonadiabatic wall temperature, in the simulations are far from our experimental conditions. Boundary layer stability for hypervelocity flow has been demonstrated by Bitter and Shepherd (2015) to strongly depend on the free stream to wall temperature ratio. The flow conditions in these T5 tests are designed to simulate hypervelocity atmospheric flight and the flow over the model is hot, partially dissociated gas with some amount of chemical and vibrational nonequilibrium due to the rapid expansion process in the nozzle, as well as heating in the shock layer for blunt cases. The available computational results of spot propagation

in hypersonic flow in the present literature survey simulated much higher wall temperature ratios $T_{\mathrm{w}} / T_{\mathrm{e}}$ and adiabatic wall-temperature ratios $T_{\mathrm{w}} / T_{\mathrm{aw}}$ than actually occur in either reflected shock tunnel experiments or flight (see Table 3).

At present, there are no DNS turbulent spot propagation studies in the literature which fully match the walltemperature ratios which are characteristic of high-enthalpy shock tunnels like T5 and T4. The present results are most directly comparable, in terms of Mach number and wall temperature ratio, to the flat plate T4 results of Mee (2002), and indeed are largely within the uncertainty range of Mee’s measurements. DNS computations with realistic wall-temperature ratios would be quite valuable for comparison with the present experiments.

With the measured parameters, a simple stochastic geometric model for the propagation of turbulent spots has been adapted from Jewell (2008) and used, following Mee and Tanguy (2015), to infer turbulent spot generation rates $n$ from a set of 17 experimentally measured transition onset and completion distances in three different gas mixtures. The results indicate that $n$ is significantly higher in air and $\mathrm{N}_{2}$ boundary layers than for experiments with $50 \% \mathrm{CO}_{2}$-air mixtures. The lower calculated spot generation rate for $\mathrm{CO}_{2}$ mixtures may be due to the reduction in boundary layer instability growth rates due to coupling with vibrational or chemical relaxation processes, but could also depend partly on variations in $C_{\mathrm{te}}$ within the uncertainty bounds of the present measurements. While spot generation rates in air and $\mathrm{N}_{2}$ were higher than those found by Mee and Tanguy (2015), most of the difference is accounted for by differing model inputs for $C_{\text {te }}$, and the Mee results were acquired on a flat plate in a different facility.

The present results represent the first attempt to infer turbulent spot generation rate in T5, as well as the only available data on the effect of $\mathrm{CO}_{2}$ on spot formation in a hypervelocity boundary layer. While ideal flow conditions do not exist in either wind tunnel or flight environments, turbulent spot generation in perfectly quiet, particle-free flows is predicted by DNS to be the outcome of the boundary layer receptivity, wavepacket amplification, and breakdown process (Fedorov 2003; Sivasubramanian and Fasel 2012) and its characterization is therefore important for understanding both receptivity and the region of intermittent turbulence which occurs between transition onset and completion. In particular, predicting the time-resolved heat flux in this region, both for flight and for ground tests, depends upon a model for turbulent spot generation and propagation. As different tunnels have disparate acoustic spectra, particulate contamination properties, and other noise and disturbance sources, information about the spot generation rate in each facility is also important for comparing transition measurements to each other, especially the onset to completion distance.

Acknowledgements The authors would like to thank Bahram Valiferdowsi (California Institute of Technology) and Nick Parziale (Stevens Institute of Technology) for assistance running T5, and Ross Wagnild (Sandia National Laboratories) for help with the program to compute run conditions. Hans Hornung (California Institute of Technology) suggested animating the heat-transfer contours and Thomas Juliano (Notre Dame) provided advice on exporting movies.

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