2.1. Experimental Setup
The experimental data in this paper were obtained in the High-Enthalpy Shock Tunnel Göttingen (HEG) of the German Aerospace Center [
53]. The HEG is a free-piston-driven shock tunnel capable of generating flowfield conditions equivalent to hypersonic flight in the atmosphere, in terms of pressure and heat flux loads. Test times are in the order of milliseconds, meaning that the walls of the test model remain cold during the test time unless active wall heating is employed.
A total of seven identical shock tunnel runs with free stream Mach number
and unit Reynolds number
m
were conducted.
Table 1 details the observed free stream conditions. The HEG free stream conditions were derived following a calibration procedure detailed in [
53].
In the interest of allowing the flowfield investigated in this work to be fully reproduced, further detail is provided in
Appendix A. Spatially resolved properties are given therein, based on a RANS solution of the nozzle flow obtained under the experimental conditions measured in the present investigation. By using the dataset from
Appendix A, the spatial distribution of the free stream properties upstream of the conical shock produced by the model can be reproduced within
error.
The investigated model is a
half angle at a
angle-of-attack and with a nose tip radius of
mm. The model is instrumented with a line of 21 Medtherm coaxial type E thermocouples. These sensors are distributed along a streamwise line on the surface of the cone, facing the region probed with the optical techniques to be detailed. Heat flux measurements derived from thermocouple data are used to monitor the state of the boundary layer.
Figure 1 shows the measured heat flux levels across all seven shock tunnel runs. It is verified that the independent runs are able to produce flowfields that are consistently similar. The experimental standard deviations at any given location are similar across all runs. For clarity, they are suppressed for all but one run in
Figure 1. The rise in the heat flux values from approximately 400 mm to a higher plateau downstream of 600 mm, accompanied by larger standard deviations, indicates the transition of the boundary layer to a turbulent state. Due to the strong similarity between all runs, the measurements obtained across the full shock tunnel campaign are combined to build a comprehensive overview of the turbulent boundary layer, as will be further detailed in this work. The figure also compiles the surface heat flux distribution obtained in the computational flowfield analyzed in this work, together with a previous experimental distribution obtained in HEG [
54], which was used as a reference to set up the computations.
Additional instrumentation pertinent to the analyses in this work includes a Z-type high-speed schlieren and a quad-foci Focused Laser Differential Interferometer (FLDI).
The schlieren system uses a Phantom v2012 camera and a Cavilux 640 nm laser source, with the knife edge oriented parallel to the surface of the cone. The images are acquired with a sampling rate of 57 kHz, which provides more than 100 frames within the shock tunnel steady-state time. The visualization area is approximately mm (streamwise × wall-normal) centered around the FLDI probing location, with a spatial resolution of approximately 24 pixels/mm.
The schlieren images are primarily used to estimate the
boundary layer thickness in each run. This is performed by noting that the location of the maximum curvature of the wall-normal density distribution presents some proximity to that of
of the velocity magnitude. High-speed conical laminar boundary layers on a cold wall under different HEG free stream conditions have been computed using the TAU code in past works [
55,
56]. Analyses of these computations have revealed that the locations of maximum curvature in density and
of the velocity magnitude are within
of each other. It will be seen in the computational results in
Section 3 of the present work that this relationship is degraded in the turbulent boundary layer, with a difference of around
. This value gives rise to measurement uncertainty, which should not be neglected, and it is therefore taken into account when analyzing the present results. It will be shown in
Section 3.1 that the obtained boundary layer thickness estimates are still accurate enough for the purposes of this work.
Schlieren is used to estimate the boundary layer thickness as follows. The schlieren knife edge parallel to the model surface yields illumination proportional to the first derivative of the flowfield density along the wall-normal direction. Hence, the differences in pixel intensity along this same direction are representative of the second derivative of density, and the maximum curvature is a peak in these values. A reference schlieren flow topology image is obtained in each run as the average of all frames within the experimental steady-state time. For every column in the image (wall-normal direction), a vector of pixel intensity differences is obtained. The new image containing the distribution of the wall-normal differences in pixel intensity is then smoothed with a moving average of 200 pixels across the columns (streamwise direction). In this final image, the peak value along each column is marked as an approximation of the local boundary layer thickness. Finally, a linear regression is found using least squares considering all the points obtained across the full schlieren field of view. The boundary layer thickness at the probing location is calculated by evaluating the linear fit.
The estimates of the boundary layer thickness will be used in
Section 3 and
Section 4 to non-dimensionalize the FLDI probing positions in both the experimental and computational cases.
Table 2 shows the
boundary layer thickness
obtained in each run in this work using the schlieren method above, together with the relative positions of the FLDI probes, to be detailed next.
The experimental FLDI setup employed in this work is a quad-foci FLDI, with four independent probes in a arrangement along the perpendicular streamwise and wall-normal directions. The streamwise pairs are used to obtain convection velocity estimates. The wall-normal splitting allows measurements of velocity and frequency spectra at two distances from the model wall simultaneously in each shock tunnel run.
The main characteristics of the quad-foci FLDI are listed in
Table 3, in which
is the laser wavelength,
is the maximum beam width at the field lenses and
d is the distance between the field lenses and the focus of the system. The separation between the interferometric pairs is denoted by
and measured by analyzing the response of the system to a weak lens, as described in [
33,
57]. The separation between the independent FLDI probes in the streamwise direction,
, is measured using the weak blast wave approach described in [
58]. Finally, the separation between independent FLDI probes in the wall-normal direction,
, is measured by means of direct imaging using the schlieren camera with a semi-transparent stopper at the focus of the FLDI.
In order to allow simultaneous schlieren measurements in every run, the FLDI is used with an angle of
with respect to the spanwise direction. This angle is considered when calculating the convection velocities. Nonetheless, the
yaw represents a maximum streamwise difference of only
mm between the right and left edges of the intersection between the FLDI axis and boundary layer under the conditions investigated in this work. Therefore, the angle will be disregarded in the interpretation of the frequency spectra in
Section 4.
The splitting and recombination of beams for interferometry is obtained using a pair of Sanderson prisms [
59], which is calibrated using the lens approach detailed in [
33,
57]. The Sanderson prism is oriented such as to split the interferometric pair in the streamwise direction. The selected interferometry distance
seen in
Table 3 is chosen so as to maximize the frequency bandwidth of the instrument, while still presenting a sufficient signal-to-noise ratio, based on previous HEG tests under similar conditions. The Nyquist frequency of the FLDI in the flowfield conditions investigated in this work is estimated as approximately 10 MHz.
The FLDI laser source is an Oxxius LCX-532S DPSS. The beam intensity is detected using a Thorlabs DET36A2 photodetector of nominal bandwidth 25 MHz. The photodetectors are connected with
termination to an SRS SR445A DC-350 MHz preamplifier with
amplification. The resulting signals are recorded on an AMOtronics transient recorder with DC coupling and a 100 MHz sampling rate. Conversion of the recorded voltage into the FLDI phase difference is performed following [
60]. Prior to each shock tunnel run, the FLDI is adjusted to half the maximum output value, for optimal sensitivity.
The duplication of the FLDI probe in the streamwise direction is achieved using a
Wollaston prism, as detailed in [
58]. In the wall-normal direction, a second pair is obtained using the combination of a non-polarizing beamsplitter cube and a convergent lens of focal distance 400 mm near the laser source, as shown in
Figure 2. A half-waveplate is used before the streamwise splitting to adjust the beams to similar power levels, such that all instruments present similar signal-to-noise ratios. The optical setup is such that the two FLDI probes that are closer to the model wall are slightly more powerful than the other two, but the powers of each streamwise pair of probes are identical.
The attention given in [
58] to the production of parallel FLDI probes aiming at reliable convection velocity correlations is retained here. Therefore, the additions illustrated in
Figure 2 are conceived such that all beams cross the center axis of the instrument at the focal distance of the field lens, where the Sanderson prism is located.
In all shock tunnel runs, the FLDI setup is positioned 825 mm downstream of the cone tip, measured along its axis. As highlighted in
Figure 1, this location is approximately 200 mm downstream of the boundary layer transition region. This distance is chosen such that a turbulent boundary layer with well-developed features is probed.
In the wall-normal direction, the FLDI position is varied between runs, to compose a broad picture of the spectra and convection velocity distributions. The locations are listed in
Table 2, comprising 5 stations fully inside the boundary layer, 4 stations between one and two times the boundary layer thickness and another 5 above this. Measurement of the wall-normal locations is performed by imaging a semi-transparent stopper at the focus of the FLDI with the schlieren camera, using additional lenses for improved resolution and a calibration target to provide a dimensional reference. The associated level of uncertainty (see
Table 2), while not negligible, is considered tolerable when using the measured quantities as approximate wall-normal probing locations.
Frequency spectra from the FLDI measurements are calculated using Welch’s method with segments of points and overlap, on a 2 ms time window during steady state. The velocity estimates are obtained through cross-correlation between the streamwise-separated pairs of FLDI probes, using the same 2 ms time window, but divided into 20 segments of ms each with no overlap. This is done so that small fluctuations within the steady state are detectable, and the experimental uncertainty may be calculated.