Publications

Band limited correlation estimates for A(ξω/U) and B(ηω/U) using Beresh et. al. 2013 data sets

DeChant, Lawrence J.; Smith, Justin S.

Here we discuss an improved Corcos (Corcos (1963), (1963)) style cross spectral density utilizing zero pressure gradient, supersonic (Beresh et. al. (2013)) data sets. Using the connection between narrow band measurements with broadband cross-spectral density, i.e. Γ(ξ ,η ,ω )= Φ (ω) A(ωη/U )exp (-i ωξ/U) we focus on estimating coherence expressions of the form: A (ξω _nb/U) and B (ηω _nb/ U) where ω_nb denotes the narrow band frequency, i.e. the band center frequency value and ξ and η are sensors spacing in streamwise/longitudinal and cross-stream/lateral directions, respectively. A methodology to estimate the parameters which retains the Corcos exponential functional form, A(ξω/U)=exp(-k_lat ηω/U) but identifies new parameters (constants) consistent with the Beresh et. al. data sets is discussed. The Corcos result requires that the data be properly explained by self-similar variable: ξω/U and ηω/U. The longitudinal (streamwise) variable ξω/U tends to provide a better data collapse, while, consistent with the literature the lateral ηω/U is only successful for higher band center frequencies. Assuming the similarity variables provide a useful description of the data, the longitudinal coherence decay constant result using the Beresh et. al. data sets yields a value for the longitudinal constant k_long≈0.36-0.28 that is approximately 3x larger than the “traditional” (low speed, large Reynolds number and zero pressure gradient) of k_long≈0.11. We suggest that the most likely reason that the Beresh et. al. data sets incur increased longitudinal decay which results in reduced coherence lengths is due to wall shear induced compression causing an adverse pressure gradient. Focusing on the higher band center frequency measurements where the frequency dependent similarity variables are applicable, the lateral or transverse coherence decay constant k_lat≈0.7 is consistent with the “traditional” (low speed, large Reynolds number and zero pressure gradient). It should be noted, that the longitudinal/streamwise coherence decay deviates from the value observed by other researchers while the lateral/ cross-stream value is consistent has been observed by other researchers. We believe that while the measurements used to obtain new decay constant estimates are from internal wind tunnel tests, they likely provide a useful estimate expected reentry flow behavior and are therefore recommended for use. These data could also be useful in determining the uncertainty of correlation length for a uncertainty quantification (UQ) analysis.