Steady and unsteady prediction methods of dynamic stability derivatives are presented in the unified framework of the unsteady Navier-Stokes equations with the k-w turbulence equations. New approach does not require any modification of the governing equations except addition of non-inertial force terms. The present methods are applied to compute the pitch-damping coefficients using the lunar coning and the zero-spin coning motions in the Cartesian coordinate frame.
A three-dimensional turbulent flow solver is developed. As a spatial discretization method, an improved HLLE(HLLE+) scheme is proposed and its dissipation mechanism is analyzed. The HLLE+ scheme, which successfully eliminates the erronous dissipation and the instability of the Godunov-type schemes, accurately resolves the viscous effects without grid dependency. Three k-w turbulence models using linear and non-linear turbulent eddy viscosity models are implemented in an implicit multigrid method. The detailed techniques of implementation are presented and discussed. A freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The Wilcox k-w, k-w Shear-Stress Transport, and Wilcox-Durbin+ models are first tested for a flat-plate flow and the results are in good agreement with the theoretical correlations. Numerical results for unseparated and separated transonic flows show that the WD+ model using the weakly non-linear eddy viscosity is in better overall agreement with experiment. Especially for the separated airfoil flows, the present multigrid Diagonalized-ADI method gives good convergence without any robustness problem.
To compute the pitch- and the roll-damping moments for the Basic Finner using the unsteady prediction method, a dual-time stepping algorithm combined with the multigrid DADI method is applied. The computed coefficients show good agreement with the experimental data. The linearity of the angular rates and the variation with Mach numbers are examined for both the pitch- and the roll-damping moment coefficients. Through analysis of the pressure distributions at various Mach numbers, the large variations of roll-damping moment coefficient in the transonic region are explained in detail.
Next, the steady prediction method using the lunar coning and zero-spin coning motions is verified for the ANSR and the Basic Finner. The results for the pitch-damping and the Magnus moment coefficients are in good agreement with the PNS data, range data, and the results using the unsteady method. The results show that the steady approach using the unified governing equations in the Cartesian coordinate frame can be successfully applied to predict the pitch-damping coefficients.
Lastly, predictions for SOC-family configurations, such as the SOCBT and the SOCF, are performed in order to examine the effect of the afterbody geometry. The verified methods are applied to predict the static aerodynamic coefficients and the pitch-damping coefficients. The predictions show that the expansion and recompression due to the boattail reduce the pitch-damping moments significantly. On the other hand, the presence of the flare improve the static and dynamic stability of the projectile. The longitudinal developments of the pitching moment and the pitch-damping moment are examined in detail to elucidate the geometrical effect.
