We present a study of the thermal conductivity of a partly covered inner tube in a double-walled carbon nanotube with varied covering ratios using a non-equilibrium molecular dynamics method.Our results show that the thermal conductivity of the inner tube changes non-linearly and non-monotonically with the increasing coverage ratio,forming a V-shaped curve.Minimal conductivity occurs at the coverage ratio of 58％,with its value being 69％ of the maximal conductivity,which appears in the full coverage case.We analyze three mutually competitive mechanisms that result in this thermal conductivity behavior with the assistance of a phonon spectrum calculation.