on the propeller. Further, results from both freeflight model maneuvering data and full scale trial maneuvering data, have failed to provide a quantitative determination of the scaling differences between model and full scale maneuvers. A number of possible sources for these differences between model scale and full scale exist and include, Reynolds number scaling issues, experimental errors, differences in the wetside geometry, variations in mass and moments of inertia, as well as variations in the distance (BG) between the center of buoyancy and the center of gravity.
A determination of the reasons underlying the observed maneuvering differences are complicated by the unsteady fluid dynamics which govern a majority of the vehicle motions. The fluid dynamics for a submarine are characterized by relatively large, timevarying combined anglesofattack (α) and sideslip angles (β). The maneuvers also appear to be strongly influenced by forced unsteady fluid mechanics. The maneuvering vehicle is characterized by pitch and yaw angular rates (q, r) as well as angular accelerations which yield significant values for the reduced frequency parameter (k). Consistent with standard conventions the length constant used to determine (k) was half the boat length (L/2). To analytically estimate the reduced frequency of the maneuvering submarine, representative cases were approximated as harmonic motions.^{1} To simplify the analysis, the period of the motion (τ) was calculated using a sawtooth wave which is dependent on only the pitch and yaw angular rates. The frequency (f) for the harmonic motion was calculated as 1/τ. For simplicity, the mean amplitude α_{m} was taken to be zero in all cases and only pure pitching or yawing conditions were calculated. Oscillation amplitudes of 5, 10 and 20 degrees, and RCM pitch and yaw rates of 0.5, 2 and 4 deg/sec were assumed. The estimates for the pitch and yaw rates were intentionally conservative so as to underestimate, rather than overestimate, the reduced frequency (k). This approximation is shown schematically in Figure 1. As shown, if the oscillation amplitude α_{ω} is halved, the period is halved and the corresponding frequency is doubled. Figure 2 summarizes the range of reduced frequencies as a function of speed (U) obtained from this analysis.
Clearly, based on this analysis a majority of maneuvers have large reduced frequencies (k≫0.1). This indicates that the vehicle motion drives the development of the unsteady separated flow field. The vortical flow field, in turn, gives rise to the forces generated on the hull during these maneuvers. Further, under these conditions, Reynolds number is known to have only a second order effect on the flow field development. For a limited range of data, experiments support this contention.^{2}

α_{ω}=5 deg 
α_{ω}=10 deg 
α_{ω}=20 deg 
Gentle Maneuver 0.5 deg/sec 
k=0.67/U U=7 k=0.1 
k=0.33/U U=7 k=0.05 
k=0.17/U U=7 k=0.02 
Intermediate 2.0 deg/sec 
k=3.34/U U=7 k=0.48 
k=1.67/U U=7 k=0.24 
k=0.84/U U=7 k=0.12 
Severe Maneuver 4.0 deg/sec 
k=6.7/U U=7 k=0.96 
k=3.35/U U=7 k=0.48 
k=1.67/U U=7 k=0.24 
Figure 2. Reduced frequencies (k) for the RCM during maneuvering.
Accordingly, development of simulation methodologies capable of representing unsteady separated flows is of paramount importance if new maneuver simulation tools are to be developed. Prior work indicates that, across an extremely broad range of parameters, both steady and unsteady fluid mechanics can be modeled using recursive neural networks. Techniques have also been addressed for integrating these unsteady fluid dynamic RNN simulations with mechanical actuators to demonstrate the ease with which adaptive control systems might be produced.
Recursive neural network simulations of unsteady boundary layer development, dynamic stall and dynamic reattachment for threedimensional unsteady separated flow fields have been described.^{3}^{−}^{7} Further, techniques have been examined for integrating these recursive neural network (RNN) reference models within adaptive control systems. Overall, the results clearly demonstrated that RNNs are a highly effective technique for the timedependent simulation of threedimensional unsteady separated flow fields.^{3}^{−}^{7} Operationally, the unsteady flowfield wing interactions could be predicted for any time period over which the motion history was a known function (a few milliseconds to tens of seconds). Consistent with numerous experimental