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the appropriate time step for the numerical simulations. Such conditions are representative of a
straight-and-level LCO. Figure 5-1 shows the upper wing surface pressure coefficient, Cp, at
88% span as a function of non-dimensional chord on the down-stroke of the pitch oscillation, (as
indicated by the inset box plot of the airfoil position in the lower portion of the plot), for two
time steps, Δt = 0.00025 (low rate in blue) and 0.000025 (high rate in red) seconds. The inset in
the lower left-hand portion of Figure 5-1, and similar figures to follow, illustrates the wing
motion (horizontal blue line) with respect to the pitch-axis (vertical red line). The starting
position of the wing is also illustrated by the horizontal red line. The flow direction is from left
to right. These time steps are chosen based on the rule of thumb that aerodynamic features of
interest are usually “visible” at non-dimensional time steps of approximately Δt* = 0.01, defined
in Equation 5-1;
Δ *
t = ΔtU
∞
c
where U∞ is free-stream velocity and c is the mean aerodynamic chord.
The computations are both performed for the same physical time (0.625 seconds,
corresponding to 5 cycles of oscillation) by varying the number of iterations for each time step,
and each computation is completed with five Newton sub-iterations. No significant changes in
the upper wing surface Cp results are evident, as seen in Figure 5-1. There is only slight variation
seen in the shock region at 60-75% chord. Therefore, all subsequent calculations are performed
with a physical time step of Δt = 0.00025 seconds and 5 Newton sub-iterations.
Geometric Convergence Study
A geometric convergence study is conducted in order to find the optimal level of
refinement for the F-16 undergoing 8 Hz ± 0.5° pitch oscillations in Mach=0.9 flow at 5,000
feet. Five grids are used for the research presented: three half-span and two full-span models all
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(5-1)