Three-Dimensional Coupled Dynamics of Two-Fluid Model in Superfluid ${}^4$He: Deformed Velocity Profile of Normal Fluid in Thermal Counterflow

The coupled dynamics of the two-fluid model of superfluid ${}^4$He is numerically studied for quantum turbulence of the thermal counterflow in a square channel. We combine the vortex filament model of the superfluid and the Navier-Stokes equations of normal fluid. Simulations of the coupled dynamics show that the velocity profile of the normal fluid is deformed significantly by superfluid turbulence as the vortices become dense. This result is consistent with recently performed visualization experiments. We introduce a dimensionless parameter that characterizes the deformation of the velocity profile.