Steady, single-phase transport characterized by high eddy diffusivity in axial direction is a special case of vessel transport phenomena which has not been studied systematically so far. In the present paper, experiments are reported for a Gorter-Mellink duct, at zero net mass flow, covering turbulent eddy transport in cold He4pressurized beyond its thermodynamic critical pressure above 4 K. Axial transport in vertical direction is induced by gravitational buoyancy. A new phenomenological equation is proposed on the basis of quantum fluid convection postulated originally for He II below the lambda point (Goner and Mellink, 1949). Our data agree reasonably with the equation.