Here we provide a discussion regarding the applicability of a family of traditional heat transfer correlation based models for several (unit level) heat transfer problems associated with flight heat transfer estimates and internal flow heat transfer associated with an experimental simulation design (Dobranich 2014). Variability between semi-empirical free-flight models suggests relative differences for heat transfer coefficients on the order of 10%, while the internal annular flow behavior is larger with differences on the order of 20%. We emphasize that these expressions are strictly valid only for the geometries they have been derived for e.g. the fully developed annular flow or simple external flow problems. Though, the application of flat plate skin friction estimate to cylindrical bodies is a traditional procedure to estimate skin friction and heat transfer, an over-prediction bias is often observed using these approximations for missile type bodies. As a correction for this over-estimate trend, we discuss a simple scaling reduction factor for flat plate turbulent skin friction and heat transfer solutions (correlations) applied to blunt bodies of revolution at zero angle of attack. The method estimates the ratio between axisymmetric and 2-d stagnation point heat transfer skin friction and Stanton number solution expressions for sub-turbulent Reynolds numbers %3C1x10 4 . This factor is assumed to also directly influence the flat plate results applied to the cylindrical portion of the flow and the flat plate correlations are modified by
Here we derive approximate physics-based models for wall pressure fluctuation frequency spectra for compressible boundary layer flows. The formulation is based upon approximate analytical solutions to Lighthill’s acoustic equation where very simple representations of the mean-turbulent tensor source term have been applied. The solutions to Lighthill’s equation provide an estimate for the wall pressure fluctuation whereby a Fourier transform for the pressure fluctuation is obtained and used as a surrogate for the actual spectral density. The finite nature of the domain is reflected by our choice of sine or cosine transform. The resulting pressure fluctuation spectrum is explicitly a function of Mach number in terms of both magnitude and spectral curve shape. The resulting expressions exhibit reasonable agreement with Goody’s recommended1 semi-empirical model in terms of spectral peak and high frequency behavior. The low frequency behavior follows a linear behavior as opposed to the "classical" quadratic rate but passes through the zero intercept. Since compressible low frequency pressure fluctuation behavior is not necessarily correctly described by zero power at zero frequency, we examine modifications of the simplified model which yield finite power at zero frequency. We derive this modification by ignoring the very near wall behavior of the source i.e. the no-slip condition where the Mach number must be low. The consequences of ignoring the very near wall (small scale) behavior will be to provide a poor representation of high frequency behavior, but since our concern in this development is focused on compressible flow and the outer region boundary layer region we will consider this approximation acceptable. A composite model is then derived that provides reasonable estimates for frequency spectrum for supersonic boundary layer flow and shows good agreement with relevant experiments, especially for low frequency. This type of approximate analytical formulation may be of value where more complete simulation based procedures are not appropriate.
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 ωnb 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(-klat 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 klong≈0.36-0.28 that is approximately 3x larger than the “traditional” (low speed, large Reynolds number and zero pressure gradient) of klong≈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 klat≈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.
Boundary-layer transition and stability data were obtained at Mach 10 in the Arnold Engineering Development Complex (AEDC) Hypervelocity Wind Tunnel 9 on a 1.5-m long, 7-deg cone at unit Reynolds numbers between 1.8 and 31 million per meter. A total of 24 runs were performed at angles-of-attack between 0 and 10-deg on sharp and blunted cones with nose radii between 5.1 and 50.8-mm. The transition location was determined with coaxial thermocouples and temperature sensitive paint while stability measurements were obtained using high-frequency response pressure sensors. Mean flow and boundary layer-stability computations were also conducted and compared with the experiment. The effect of angle-of-attack and bluntness on the transition location displays similar trends compared to historical hypersonic wind tunnel data at similar Mach and Reynolds numbers. The N factor at start of transition on sharp cones increases with unit Reynolds number. Values between 4 and 7 were observed. The N factor at start of transition significantly decreases as bluntness increases and is successfully correlated with the ratio of transition location to entropy layer swallowing length. Good agreement between the computed and measured spatial amplification rates and most amplified 2nd mode frequencies are obtained for sharp and moderately blunted cones. For large bluntness, where the ratio of transition to entropy swallowing length is below 0.1, 2nd mode waves were not observed before the start of transition on the frustum.