The initial-value problem for three-dimensional disturbances in plane Poiseuille flow of helium II

Abstract

The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.

Effect of vortex line distribution in superfluid plane Poiseuille flow instability

“…total length of quantized superfluid vortex lines per unit volume; subscript s denotes superfluid. The mutual friction term in the governing equations in Godfrey et al (2001) and Bergström (2008) posses a sign different from that in Tough (1982). In this work the sign as in Tough (1982) and Tilley & Tilley (2005) is followed, because mutual friction will decelerate the normal fluid as reported in Hall & Vinen (1956a,b), which is more appropriate as also discussed in Sooraj & Sameen (2011).…”

“…Therefore a bimodal Gaussian distribution is assumed for the superfluid vortex line distribution, having the peaks at points of zero relative velocity between the two fluids. The assumed distribution with standard deviation σ is written as (Godfrey et al 2001;Bergström 2008) …”

“…The strength of the mutual friction term is characterized by parameter f max . A given combination of σ and f max depicts a situation of superfluid flow at some particular temperature (Bergström 2008).…”

“…The pressure gradient is chosen such that Re dP/dx = −2, as in the case of classical 720 R1-3 fluid (Barenghi, Donnelly & Vinen 1983;Donnelly 1991;Bergström 2008). Since U s is assumed inviscid, the superfluid velocity is constant across the channel width.…”

“…The action of superfluid vortices did change the velocity profile, but was considered to be independent of Reynolds number, Re, as also reported in Sooraj & Sameen (2011), which need not be true. Bergström (2008) studied the transient growth of three-dimensional disturbances in plane Poiseuille flow of He-II using non-modal analysis. A substantial increase in transient growth of threedimensional perturbations, in comparison with classical fluids, was observed in normal fluid due to the action of mutual friction from the superfluid.…”