Detection of second-mode instabilities on a flared cone in Mach 6 quiet flow with linear array focused laser differential interferometry

Abstract

In this work, an enhanced version of the focused laser differential interferometer (FLDI) is used to measure second-mode instabilities in the boundary layer on a flared cone in Mach 6 quiet flow. A diffractive optical element added to the FLDI system provides six beams that pass through a single Wollaston prism, allowing for multiple points of measurements without the need for additional Wollaston prisms. This technique, linear array-FLDI (LA-FLDI), is used for six simultaneous measurements of second-mode instabilities in a hypersonic boundary layer. This is the first time that this variation of FLDI has been used for second-mode instability measurements and demonstrates that the inclusion of a diffractive optical element into the traditional FLDI and splitting of beam power does not preclude such usage. Thus, LA-FLDI appears to have significant potential for increasing the efficiency of measurements of high-speed boundary layers by enabling multiple measurements at different locations within a single facility run. The velocity of the second-mode wavepacket is estimated and a bispectral analysis is presented showing that harmonics are associated with quadratically phased coupled interactions.

Graphic abstract

Appendix 1: Uncertainty analysis for correlation-based velocity

The approach taken for uncertainty analysis here utilizes standard quadrature relations (Taylor 1997). For independent measured variables X and Y, with uncertainties $\varDelta X$ and $\varDelta Y$ , respectively, the propagated uncertainty in their summation or difference that yields Z is given by:

$$\begin{aligned} \varDelta Z = \sqrt{\varDelta X^2 +\varDelta Y^2} \end{aligned},$$

(4)

whereas for the case that measurements X and Y are multiplied or divided yielding Z, the resulting uncertainty is:

$$\begin{aligned} \varDelta Z = Z\sqrt{(\varDelta X/X)^2 +(\varDelta Y/Y)^2} \end{aligned}$$

(5)

For the beam pair separation distance S, the uncertainty is taken as being equal to the diameter of the beam at the focus D, which should be a conservative estimate. Then, the lag time resulting correlation analysis $t_\mathrm{corr}$ has an uncertainty determined by the time resolution, which is in turn dictated by the sampling frequency $f_s$. Thus, for the velocity v determined by $v=S/t_\mathrm{corr}$, the uncertainty is estimated using the relations in Eq. 4 and Eq. 5 as:

$$\begin{aligned} \varDelta v = v\sqrt{(D/S)^2 +(1/f_st_\mathrm{corr})^2} \end{aligned}$$

(6)

Note that the expression suggests that larger beam spacing and higher sampling frequencies produce smaller uncertainty in the resulting velocity, which agrees with considerations in the primary text. For the value of $\varDelta v/v =$ 13% discussed in the text, the beam diameter was taken as 80 $\mu$m, the sampling frequency was 2 MHz, and the spacing of the beam pairs was taken as 4.67 mm based on measurements. Jitter was determined to be a non-factor based on the data acquisition hardware used here and the sampling frequency used. Thus, it was omitted from the uncertainty analysis.