Stereoscopic PIV measurements of a turbulent boundary layer with a large spatial dynamic range

Abstract

The flow in a streamwise/wall-normal plane of a turbulent boundary layer at moderate Reynolds number (Re _θ = 2,200) is characterized using two stereo PIV systems just overlapping in the streamwise direction. The aim is to generate SPIV data for near-wall turbulence with enough spatial dynamic range to resolve most of the coherent structures present in the flow and to facilitate future comparisons with direct numerical simulations. This is made possibly through the use of four cameras with large CCD arrays (4,008 px × 2,672 px) and through a rigorous experimental procedure designed to minimize the impact of measurement noise on the resolution of the small scales. For the first time, both a large field of view [S _x ; S _y ] = [2.6δ; 0.75δ] and a high spatial resolution (with an interrogation window size of 13.6⁺) have been achieved. The quality of the data is assessed through an analysis of some of the statistical results such as the mean velocity profile, the rms and the PDF of the fluctuations, and the power spectra.

Appendix

This appendix provides the detailed methodology employed to compute the effective cut-off wavenumber kc _exp of the stereo PIV longitudinal power spectra. This wavenumber was defined as the wavenumber at which the signal-to-noise ratio of the experiment is equal to one.

Following Foucaut et al. (2004), the PIV power spectrum E _PIV(k) can be expressed as

$$ E_{\rm PIV}(k)=(E_{\rm flow}(k)+E_{\rm noise}(k))\left({\rm sin}_c \left(k{\frac{L_{\rm IW}}{2}}\right)\right)^2 $$

(6)

where:

• E _flow(k) is the exact spectrum of the flow.

• E _noise(k) is the PIV noise (including both recording and processing (at given IW size) sources of noise)

• the squared cardinal sine function $\left({\rm sin}_c(k{\frac{L_{\rm IW}}{2}})\right)^2$ is the PIV transfer function and represents the filtering over a square interrogation window.

At SNR = 1, by definition E _flow(kc _exp) = E _noise(kc _exp), and one gets

$$ E_{\rm PIV}(kc_{\rm exp})=(2E_{\rm flow}(kc_{\rm exp}))\left({\rm sin}_c \left(kc_{\rm exp}{\frac{L_{\rm IW}}{2}}\right)\right)^2 $$

The spectrum of the flow is a priori unknown; however, in the present contribution, we propose to use as a reference the spectrum from a DNS of channel flow [see Herpin et al. (2007) for a presentation of the simulation; its characteristics are close to the ones of Kim et al. (1987b)'s]. As the PIV experiment and the DNS are at different Reynolds numbers (Re _τ,DNS = 600 and δ⁺ _PIV = 1,400), they are compared using the inner scaling from Perry et al. (1986):

$$ \tilde{k}=ky \quad \tilde{E}(\tilde{k}) =E(\tilde{k})/(u_\tau^2)=E(k)/(u_\tau^2y) $$

where u _τ is the friction velocity and y is the wall normal position at which the spectrum is computed. The non-dimensional PIV and DNS spectra read

$$ \begin{aligned} \tilde{k}&=k_{\rm PIV}\ y_{\rm PIV} \quad \tilde{E}_{\rm PIV}(\tilde{k})\\ &=E_{\rm PIV}(\tilde{k})/(u_{\tau,{\rm PIV}}^2)=E_{\rm PIV}(k_{\rm PIV})/(u_{\tau,{\rm PIV}}^2\ y_{\rm PIV})\\ \tilde{k}&=k_{\rm DNS}\ y_{\rm DNS} \quad \tilde{E}_{\rm DNS}(\tilde{k})\\&=E_{\rm DNS}(\tilde{k})/(u_{\tau,{\rm DNS}}^2)=E_{\rm DNS}(k_{\rm DNS})/(u_{\tau,{\rm DNS}}^2\ y_{\rm DNS}) \end{aligned} $$

The PIV and DNS non-dimensional spectra for the u component at y ⁺ = 100 are shown in Fig. 12, for a PIV IW size of 32 px × 32 px; 1,936 velocity fields were used to compute the PIV spectrum, and five time steps were used to compute the DNS spectrum (this corresponds to the spectrum presented in Sect. 4 in Fig. 11a). A very good agreement can be noted at low and intermediate wavenumbers, supporting the validity of the scaling employed. The disparity in the high wavenumbers domain denotes the influence of the PIV noise.

Fig. 12

PIV and DNS non-dimensional spectra

Now, assuming that the non-dimensional spectrum of the flow is equal to the non-dimensional spectrum of the DNS, the non-dimensional PIV spectrum at $\tilde{k}c_{\rm exp}$ can be expressed as

$$ \tilde{E}_{\rm PIV}\left(\tilde{k}c_{\rm exp}\right)=2\ \tilde{E}_{\rm DNS} \left(\tilde{k}c_{\rm exp}\right)\left({\rm sin}_c \left(\tilde{k}c_{\rm exp} {\frac{L_{\rm IW}}{2}}{\frac{1}{y_{\rm PIV}}}\right)\right)^2 $$

(7)

The multiplication factor $\left({\rm sin}_c\left(\tilde{k}c_{\rm exp}{\frac{L_{\rm IW}}{2}}{\frac{1} {y_{\rm PIV}}}\right)\right)^2$ represents the low-pass filtering effect of PIV due to the averaging over the interrogation window. Equation 7 indicates that $\tilde{k}c_{\rm exp}$ can be retrieved as the wavenumber at which $\tilde{E}_{\rm PIV}(\tilde{k})$ intersects with $2\tilde{E}_{\rm DNS}(\tilde{k})\left({\rm sin}_c \left(\tilde{k}c_{\rm exp}{\frac{L_{\rm IW}}{2}}{\frac{1}{y_{\rm PIV}}}\right)\right)^2.$ This procedure is illustrated in Fig. 13. The non-dimensional cut-off wavenumber of the PIV spectrum is found to be $\tilde{k}c_{\rm exp}=11.8\pm0.5,$ corresponding to a minimum resolvable spatial variation $SV_{\rm min}^+={\frac{2\Pi y^+_{PIV}}{\tilde{k}c_{\rm exp}}}=53\pm2$ : a vortex of diameter d ⁺ = 53 will be resolved with a signal-to-noise ratio of 1 by the PIV measurements. Note that these values of $\tilde{k}c_{\rm exp}$ and SV ⁺_min  = d ⁺ were computed from the power spectra of the u component only (E ₁₁). The same methodology can be applied on E ₂₂ and E ₃₃ as well, to derive a global cut-off wavenumber $\tilde{k}c_{\rm exp}$ and a global minimum resolvable spatial variation SV ⁺_min (see Sect. 4).

Fig. 13

Methodology employed to retrieve kc _exp