  We study the influence of two mutual friction models on quantized vortices
and normal fluid using two-way coupled simulations of superfluid $^4$He. The
normal fluid is affected by quantized vortices via mutual friction. A previous
study [Y. Tang, et al. Nat. Commun. 14, 2941 (2023)] compared the time
evolutions of the vortex ring radius and determined that the self-consistent
two-way coupled mutual friction (S2W) model yielded better agreement with the
experimental results than the two-way coupled mutual friction (2W) model whose
model parameters were determined through experiments with rotating superfluid
helium. In this study, we compare the two models in more detail in terms of the
quantized vortex ring propagation, reconnection, and thermal counterflow. We
found that the S2W model exhibits better results than the 2W model on the
microscopic scale near a quantized vortex, such as during quantized vortex ring
propagation and reconnection, although the S2W model requires a higher spatial
resolution. For complex flows such as a thermal counterflow, the 2W model can
be applied even to a low-resolution flow while maintaining the anisotropic
normal fluid velocity fluctuations. In contrast, the 2W model predicts lower
normal fluid velocity fluctuations than the S2W model. The two models show
probability density functions with $- 3$ power-law tails for the normal fluid
velocity fluctuations.
