Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> C. Strouhal Number (Reduced Frequency Parameter) <s> Results of an investigation of the global aerodynamic coefe cients and e ow characterization around an Apollostyle capsule model at low speed are presented. Flow features were ascertained quantitatively with laser Doppler velocimetry, digital particle image velocimetry, and hot-wire measurements, as well as qualitatively by e ow visualizations. The study revealed a meandering vortex ring around the capsule, well-organized streamwise vortices whose maximum vorticity increased in the recirculating wake before decaying farther downstream, as well as a complex array ofe ow separation and reattachmentfeatures. A numberofparticularitiesin the e owe eld have been identie ed as potential candidates for triggering the onset of vehicle instability. Based on these e ndings, a modeling approach is proposed. <s> BIB001 </s> Survey of Blunt Body Dynamic Stability in Supersonic Flow <s> C. Strouhal Number (Reduced Frequency Parameter) <s> A novel method of calculating dynamic stability derivatives using Computational Fluid Dynamics is presented. This method uses a non-linear, reduced-frequency approach to simulate the response to a forced oscillation using a single frequency component at the forcing frequency. This provides an order of magnitude improvement in computational e ciency over similar time-dependent schemes without loss of generality. The reduced-frequency approach is implemented with an automated Cartesian mesh scheme. This combination of Cartesian meshing and reduced-frequency solver enables damping derivatives for arbitrary flight condition and geometric complexity to be e ciently and accurately calculated. The method is validated for 3-D reference missile and aircraft dynamic test configurations through the transonic and high-alpha flight regimes. Comparisons with the results of time-dependent simulations are also included. <s> BIB002
Strouhal number (Sr, St, or k), also known as the "reduced frequency parameter," is a nondimensional scaling parameter calculated using the relationship below:  where ω is the angular velocity of the oscillatory motion (in rad/s), is the characteristic length (diameter for blunt bodies), and is the free stream flow velocity. Note that a factor of 2 in the denominator is occasionally used in the literature. This scaling parameter captures the relative magnitude of the frequency content of a body's oscillatory motion to the mean rate at which the free stream flow passes over the body. As the interactions of the resulting flow structure and the aftbody of a vehicle are critical to dynamic stability, matching of k during sub scale testing of dynamic behavior is important for estimating limit cycle behavior and identifying any possible resonance frequencies that may be responsible for excitation of pitching oscillations. Wang et al  - BIB001 studied the spectral characteristics of the flow structure at various locations around an Apollo capsule at low speed. Of particular interest were the oscillations of the rear stagnation point because of their possible influence on unsteady loads acting on the aftbody of the vehicle. Throughout the wake, frequency signatures were detected corresponding to reduced frequencies of 0.08 and 0.16. The authors theorize that these structures were related to the shedding and meandering of vortices in the near wake. Other spectral signatures were found in the flow, but their physical significance could not be identified. Although their studies were done at subsonic and transonic speeds, they postulate that the flow structures identified would be similar enough to be used for characterizing the dynamics at higher Mach numbers. Murman BIB002 used this same idea to develop a computational method for predicting damping with reduced computational cost. Experimentally, a few studies that considered the effect of the reduced frequency parameter found a correlation between its value and the observed damping. The BLDT configuration of the Viking vehicle was found to be highly sensitive to reduced frequency at various Reynolds numbers and a Mach number of 1.76. Interestingly, their results indicate the strongest influence at a Strouhal number of 0.016 (Fig. 28 , note the authors utilize the factor of two in the denominator in their definition), an order of magnitude less than the dominant frequencies found in the studies by Wang  . Similarly, Teramoto  cited the Strouhal number of the Muses-C capsule as 0.03. This may indicate that the frequency content of the flow features in the wake of a blunt body is sensitive to changes in vehicle geometry or that extending the incompressible measurements to higher Mach numbers should be avoided.