Superfluid turbulence in a nonuniform rectangular channel

Abstract

In this detailed study of inhomogeneous superfluid turbulence created in a spatially nonuniform flow, we measure the turbulent dissipation produced by thermal counterflow in a nonuniform rectangular channel. The width of the channel increases linearly with position r along the channel length to impose a known spatial nonuniformity on the velocity in the flow direction, V\ensuremath{\propto}Q\ifmmode \dot{}\else \.{}\fi{}/r, where Q\ifmmode \dot{}\else \.{}\fi{} is the heat current. Because the channel widens gradually, this divergence is weak, producing a small deviation from uniform flow. Probes embedded in the channel walls allow access to the local temperature at several positions r. We employ sensitive differential thermometry to measure the temperature difference between any two of these locations to high precision. The single turbulent state we observe is clearly the analog of the T-II state of fully developed, homogeneous turbulence present in uniform flows. Making an approximation of local uniformity, we attempt to extend the theoretical description of turbulence in uniform flows to our results by replacing the uniform velocity with the local velocity in the expression for the temperature gradient. When applied locally, the Schwarz model of homogeneous turbulence, so successful in describing the T-II state in uniform flows, has the wrong dependence on both r and Q\ifmmode \dot{}\else \.{}\fi{} to characterize our results. The Geurst three-fluid model, which extends to situations where a gradient is present in the line density in an otherwise uniform flow, likewise does not agree with our observations when used in the local approximation.We therefore find that even a small perturbation imposed on a uniform flow radically changes the nature of the T-II state, and the inhomogeneous T-II state we observe is markedly different from that of homogeneous turbulence. This rather surprising result points to the need for a proper theory, or direct numerical simulation, of inhomogeneous turbulence in nonuniform flow. By introducing a substantial modification to the Schwarz line density, we identity a modified line density that constitutes a three-parameter fit yielding excellent agreement with our results. Finally, we find that the transition from laminar to turbulent flow occurs at a stationary front. Observations of the location of this front as a function of Q\ifmmode \dot{}\else \.{}\fi{} show that the critical condition for maintaining turbulence is more consistent with a minimum line density than a critical velocity.