  Three-dimensional numerical simulations in a square duct were conducted to
investigate entrance lengths of normal fluid and superfluid flows in a thermal
counterflow of superfluid $^4$He. The two fluids were coarse-grained by using
the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model and were coupled through
mutual friction. We solved the HVBK equations by parameterizing the coefficient
of the mutual friction to consider the vortex line density. A uniform mutual
friction parameter was assumed in the streamwise direction. Our simulation
showed that the entrance length of the normal fluid from a hot end becomes
shorter than that of a single normal fluid due to the mutual friction with the
parabolically developed superfluid flow near the hot end. As the mutual
friction increases, the entrance length decreases. Same as that, the entrance
length of the superfluid from a cold end is affected by the strength of the
mutual friction due to the parabolically developed normal fluid flow near the
cold end. Aside from the entrance effect, the realized condition of a
tail-flattened flow is discussed by parameterizing the superfluid turbulent
eddy viscosity and the mutual friction.
