2.1 Experimental set-up

The experiments were conducted in a supersonic blowdown wind tunnel at Nanjing University of Aeronautics and Astronautics. The incoming boundary-layer condition is matched to a recent experimental study of Zhuang et al. (Reference Zhuang, Tan, Huang, Liu and Zhang2018a ), featuring a free-stream Mach number of $M_{0}=2.83$ , a thickness of $\unicode[STIX]{x1D6FF}_{0}=11.6$  mm and a Reynolds number ( $\unicode[STIX]{x1D6FF}_{0}$ based) of $Re_{\unicode[STIX]{x1D6FF}_{0}}=9.14\times 10^{4}$ . The actual incoming flow parameters of this study are summarised in table 1. A ramp-on-plate model with a width of $W=150$  mm and a deflection angle of $\unicode[STIX]{x1D717}=25^{\circ }$ is employed, resulting in a separation bubble and a separation–reattachment shock wave system (see figure 1). The IC-PLS visualisation technique, details of which will be given in the next subsection, is deployed at the mid-plane of the test model. To avoid complexity in the vicinity of the separation bubble, this study focuses on TNTIs after reattachment. Thus, the region of interest, indicated by the yellow box in figure 1, features a rectangular area of height and width $1.6\unicode[STIX]{x1D6FF}_{0}$ and $3\unicode[STIX]{x1D6FF}_{0}$ , respectively, located $1.5\unicode[STIX]{x1D6FF}_{0}$ behind the corner, downstream of the separation bubble (Settles, Fitzpatrick & Bogdonoff Reference Settles, Fitzpatrick and Bogdonoff1979; Loginov, Adams & Zheltovodov Reference Loginov, Adams and Zheltovodov2006).

Figure 1.

(a) Illustration of the experimental facilities. IC-PLS images are acquired at the streamwise-vertical ( $X$ – $Y$ ) centreplane. One sidewall and half of the shock waves are omitted for clarity. Drawing not to scale. (b) Image of the experimental model.

Table 1.

Main aerodynamic parameters of the incoming flow. ( $M_{\infty }$ , $Re_{\unicode[STIX]{x1D6FF}_{0}}$ , $U_{\infty }$ , $\unicode[STIX]{x1D6FF}_{0}$ , $\unicode[STIX]{x1D702}_{0}$ , $\unicode[STIX]{x1D706}_{0}$ , $T_{stag.}$ , $P_{stag.}$ and $\unicode[STIX]{x1D719}_{stag.}$ stand for the free-stream Mach number, the Reynolds number based on the incoming boundary-layer thickness, the free-stream speed, the incoming boundary-layer thickness, the Kolmogorov microscale, the Taylor microscale calculated near the interface, the stagnation temperature, the stagnation pressure and the relative humidity in stagnated air).

2.2 Ice-cluster-based planar laser scattering (IC-PLS) technique

The IC-PLS technique is a contactless flow visualisation technique characterised by high temporal resolution and has been applied successfully in supersonic airflow-related studies (Smith & Smits Reference Smith and Smits1995; Poggie et al. Reference Poggie, Erbland, Smits and Miles2004). While expanding in the nozzle, the temperature of airflow falls substantially, and water molecules deposit inside and accumulate into small ice clusters, which act as tracers (Wegener & Pouring Reference Wegener and Pouring1964). A thin laser sheet is introduced to the plane of interest and illuminates a slice of the flow field. The flow pattern at this cross-section is illustrated by laser scattering of ice clusters. Although this method is not new, with developments on each of the components its spatial resolution has improved significantly (Zhuang et al. Reference Zhuang, Tan, Li, Sheng and Zhang2018c ). This provides a solid base for studies on the geometry of the TNTI, which requires data at various length scales.

A digital single-lens reflex camera (Canon 1Dx Mark II) is used in the experiment, resulting in images with a spatial resolution of $35.1~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ , or $2.81~\unicode[STIX]{x1D702}_{0}~\text{pixel}^{-1}$ . The Beamtech Vlite-380 pulsed laser (Nd:YAG laser with a wavelength of 532 nm and a pulse duration of 10 ns) together with a laser light arm and an optical lens provide the illuminating laser sheet. The short pulse duration indicates a short exposure time and thus freezes the flow effectively. The sampling frequency of the system is 7 Hz, which implies that acquired images are temporally uncorrelated. This technique offers an accessible scale range covering nearly three orders of magnitude, which allows the TNTI to be measured approximately at both scale ends of the turbulent boundary layer (that is, the levels of $\unicode[STIX]{x1D6FF}_{0}$ and $\unicode[STIX]{x1D702}_{0}$ ). Such a wide range provides a good chance of observing whether any self-similarity exists.

Figure 2(a) shows a sample IC-PLS image of a shock wave in this experiment. The image brightness represents the intensity of scattering and thus shows the local flow density if the ice clusters can follow the fluid faithfully. However, due to the hysteresis effect, it will take a certain period for tracers to catch up with disturbed flows. This time period is known as the relaxation time $\unicode[STIX]{x1D70F}_{r}$ , which is an important feature for flow tracers. Considering that the thickness of the shock wave is negligible, the relaxation distance $L=0.47~\text{mm}$ is acquired by measuring the brightness slope after the shock wave (demonstrated in figure 2 b). Then, the relaxation time $\unicode[STIX]{x1D70F}_{r}=1.55~\unicode[STIX]{x03BC}\text{s}$ is obtained from the known incoming flow speed. Also, according to Zhao et al. (Reference Zhao, Yi, Tian and Cheng2009), the typical diameters of ice clusters are approximately 500 nm, which is two orders of magnitude smaller than the projected scale of a single pixel on the sampling plane ( $35.1~\unicode[STIX]{x03BC}\text{m}$ ), and thus is small enough to depict the flow field.

Figure 2.

(a) An IC-PLS sample showing the relaxation of particles while passing through a shock wave. (b) Normalised brightness distribution in the vicinity of a shock wave. The distance between the blue and red lines is a measure of the relaxation length of the particles. (c) A sketch of the particle distribution near the interface between the turbulent region and the non-turbulent region. Brighter areas refer to places with a higher density of particles.

Vortical structures are the basic elements of turbulent flows. According to Samimy & Lele (Reference Samimy and Lele1991), only when the Stokes number is smaller than 0.1 can the particles in compressible turbulent boundary layers trace faithfully. Thus, the following equation is acquired:

(2.1) $$\begin{eqnarray}\displaystyle S_{tk}=\frac{\unicode[STIX]{x1D70F}u}{l}\leqslant 0.1, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is the relaxation time of the particles, $u$ is the speed of flow well away from the obstacle and $l$ is the scale of obstacle. In this test, the relaxation time of ice clusters is $1.55~\unicode[STIX]{x03BC}\text{s}$ , and the value of $u$ can be chosen as $0.1U_{0}$ (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000). Considering that flow in the vortex moves approximately around its centreline, the tracers have to fight against the centrifugal force to enter the core regions of the vortex. This indicates that it is reasonable to treat the vortices in turbulence as obstacles in (2.1); thus, $l$ should be the diameter of a vortex at the intersection plane perpendicular to the centreline, since vortices are tube-like. Thus, the critical value of $l_{c}=0.95$  mm is achieved, and ice clusters cannot follow the flow in vortices with $l$ smaller than 0.95 mm. Considering this critical value is only one order of magnitude smaller than the boundary-layer thickness, it is considered a large value, which means that particles in the turbulent region have a high probability of not following the flow faithfully. Consequently, as shown in figure 2(c), ice clusters accumulate in spaces between vortices and the edge of the turbulent region, indicating that the turbulent region is bounded by areas of high ice-cluster density (bright regions). This provides the basis for using this technique to detect the TNTI.

The turbulent region is bounded by bright bands, which allows the application of the brightness threshold TNTI detection method. This method has been widely used in the detection of the TNTI (Sreenivasan & Meneveau Reference Sreenivasan and Meneveau1986; Prasad & Sreenivasan Reference Prasad and Sreenivasan1989, Reference Prasad and Sreenivasan1990). The normalised brightness threshold value used in this study is 0.16 (0 and 1 refer to black and white, respectively) and no other values are tested since it is a weak factor on the final fractal dimension (Sreenivasan & Meneveau Reference Sreenivasan and Meneveau1986; Prasad & Sreenivasan Reference Prasad and Sreenivasan1990; Sreenivasan Reference Sreenivasan1991; Mathew & Basu Reference Mathew and Basu2002; Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013). The results agree well with visual inspection; samples are shown in the next section.

2.3 Fractal analysis and the box-count method

TNTIs, which are imprints of complex vortices underneath, are highly convoluted. The convolutions exist at a large length-scale range due to the hierarchical structures in high-Reynolds-number turbulence. As a result, the geometry of TNTIs cannot be built up by traditional means without introducing dubious assertions, whereas geometry at various scales is a natural attribute of fractals (Mandelbrot Reference Mandelbrot1982). To test the fractal feature of the TNTI, the box-counting algorithm (Liebovitch & Toth Reference Liebovitch and Toth1989) in the program of FracLac (Karperien, A., FracLac for ImageJ, version 2.5. https://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm. 1999–2013) built for the ImageJ software package (Schneider, Rasband & Eliceiri Reference Schneider, Rasband and Eliceiri2012) is applied. The box-counting algorithm tests the geometry of objects under different observation scales by breaking objects into pieces of box shape on a scale hierarchy and finding the relationship between the number of pieces and the corresponding length scale. In this test, the value of $N(r)$ , the minimum number of boxes needed to cover the target interface, is counted in the scale range $\unicode[STIX]{x1D702}_{0}$ – $\unicode[STIX]{x1D6FF}_{0}$ . The gross streamwise length of the TNTI extracted from different pictures is larger than $100\unicode[STIX]{x1D6FF}_{0}$ , a sufficient length for reliable results.