  This study introduces novel physics-based scaling laws to estimate the
propulsive performance of synchronously pitching foils in various schooling
configurations at Re=4000. These relations are derived from quasi-steady
lift-based and added mass forces. Hydrodynamic interactions among the schooling
foils are considered through vortex-induced velocities imposed on them,
constituting the ground effect. Generalized scaling equations are formulated
for cycle-averaged coefficients of thrust and power. These equations encompass
both the pure-pitching and induced velocity terms, capturing their combined
effects. The equations are compared to computational results obtained from
two-foils systems, exhibiting foil arrangements over a wide range of parameter
space, including Strouhal number (0.15 \leq St \leq 0.4), pitching amplitude (5
deg \leq \theta_0 \leq 14 deg), and phase difference (0 deg \leq \phi \leq 180
deg). The individual contributions of pure-pitching and induced velocity terms
to propulsive performance elucidate that solely relying on the pure-pitching
terms leads to inadequate estimation, emphasizing the significance of the
induced velocity terms. Validity of the approach is further assessed by testing
it with a three-foil configuration, which displays a collapse. This indicates
that the scaling laws are not only applicable to two-foils systems but also
extend their effectiveness to multi-foil arrangements.
