The compressible turbulent flows through a square duct including an adiabatic bottom-wall with three Mach numbers, M = 1,1.5,2 are investigated using the direct numerical simulation. The secondary flows, which are caused by the turbulent anisotropic effects, are drastically changed with respect to the number and flow pattern of the secondary circulation due to the dependency of the Mach number and the existence of an adiabatic wall. At high Mach number case, the secondary flow near the adiabatic wall grows and is strengthened. Thus, due to the strong secondary flow, the heating fluid near the bottom wall is transported to the upper region. As increasing Mach number, the heat is generated near the adiabatic wall and the fluid temperature rises in the whole region except those near the isothermal walls. The strong secondary flow at high Mach number exerts an influence on the distribution of the mean quantities of the streamwise velocity, density and temperature. From the viewpoint of the budget of the streamwise vorticity, the Reynolds shear stress plays a primary part in producing the secondary flow, and the contribution of the difference between the normal stresses is weakened as the Mach number increases unlike the incompressible turbulent flow through a square duct.