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Register a new userQuantum turbulence in coflow of superfluid ${}^4$He
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We study numerically nonuniform quantum turbulence of coflow in a square channel by the vortex filament model. Coflow means that superfluid velocity $bm{v}_s$ and normal fluid velocity $bm{v}_n$ flow in the same direction. Quantum turbulence for thermal counterflow has been long studied theoretically and experimentally. In recent years, experiments of coflow are performed to observe different features from thermal counterflow. By supposing that $bm{v}_s$ is uniform and $bm{v}_n$ takes the Hagen-Poiseiulle profile, our simulation finds that quantized vortices are distributed inhomogeneously. Vortices like to accumulate on the surface of a cylinder with $bm{v}_s simeq bm{v}_n$. Consequently, the vortex configuration becomes degenerate from three-dimensional to two-dimensional.
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There are two commonly discussed forms of quantum turbulence in superfluid $^4$He above 1K: in one there is a random tangle of quantizes vortex lines, existing in the presence of a non-turbulent normal fluid; in the second there is a coupled turbulent motion of the two fluids, often exhibiting quasi-classical characteristics on scales larger than the separation between the quantized vortex lines in the superfluid component. The decay of vortex line density, $L$, in the former case is often described by the equation $dL/dt=-chi_2 (kappa/2pi)L^2$, where $kappa$ is the quantum of circulation, and $chi_2$ is a dimensionless parameter of order unity. The decay of total turbulent energy, $E$, in the second case is often characterized by an effective kinematic viscosity, $ u$, such that $dE/dt=- u kappa^2 L^2$. We present new values of $chi_2$ derived from numerical simulations and from experiment, which we compare with those derived from a theory developed by Vinen and Niemela. We summarise what is presently known about the values of $ u$ from experiment, and we present a brief introductory discussion of the relationship between $chi_2$ and $ u$, leaving a more detailed discussion to a later paper.
We report on a combined theoretical and numerical study of counterflow turbulence in superfluid $^{4}$He in a wide range of parameters. The energy spectra of the velocity fluctuations of both the normal-fluid and superfluid components are strongly anisotropic. The angular dependence of the correlation between velocity fluctuations of the two components plays the key role. A selective energy dissipation intensifies as scales decrease, with the streamwise velocity fluctuations becoming dominant. Most of the flow energy is concentrated in a wavevector plane which is orthogonal to the direction of the counterflow. The phenomenon becomes more prominent at higher temperatures as the coupling between the components depends on the temperature and the direction with respect to the counterflow velocity.
We develop an analytic theory of strong anisotropy of the energy spectra in the thermally-driven turbulent counterflow of superfluid He-4. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by the numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiment, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade-dominated range and a universal critical regime at large wavenumbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is a part of the theme issue Scaling the turbulence edifice.
In the thermally driven superfluid He-4 turbulence, the counterflow velocity $U_{rm ns}$ partially decouples the normal and superfluid turbulent velocities. Recently we suggested [J. Low Temp. Phys. 187, 497 (2017)] that this decoupling should tremendously increase the turbulent energy dissipation by mutual friction and significantly suppress the energy spectra. Comprehensive measurements of the apparent scaling exponent nexp of the 2nd-order normal fluid velocity structure function $S_2(r)propto r^{n_{rm exp}}$ in the counterflow turbulence [Phys.Rev.B 96, 094511 (2017)] confirmed our scenario of gradual dependence of the turbulence statistics on the flow parameters. We develop an analytical theory of the counterflow turbulence, accounting for a twofold mechanism of this phenomenon: i) a scale-dependent competition between the turbulent velocity coupling by the mutual friction and the $U_{rm ns}$-induced turbulent velocity decoupling and ii) the turbulent energy dissipation by the mutual friction enhanced by the velocity decoupling. The suggested theory predicts the energy spectra for a wide range of flow parameters. The mean exponents of the normal fluid energy spectra $langle m_nrangle_{10}$, found without fitting parameters, qualitatively agree with the observed $n_{rm exp} + 1$ for $Tgtrsim 1.85$K
We report a detailed analysis of the energy spectra, second- and high-order structure functions of velocity differences in superfluid $^4$He counterflow turbulence, measured in a wide range of temperatures and heat fluxes. We show that the one-dimensional energy spectrum $E_{xz} (k_y)$ (averaged over the $xz$-plane, parallel to the channel wall), directly measured as a function of the wall-normal wave-vector $k_y$, gives more detailed information on the energy distribution over scales than the corresponding second-order structure function $S_{2}(delta_y)$. In particular, we discover two intervals of $k_y$ with different apparent exponents: $E_{xz} (k_y)propto k_y^{-m_C}$ for $klesssim k_times$ and $E_{xz} (k_y)propto k_y^{-m_F}$ for $kgtrsim k_times$. Here $k_times$ denotes wavenumber that separate scales with relatively strong (for $klesssim k_times$) and relatively weak (for $kgtrsim k_times$) coupling between the normal-fluid and superfluid velocity components. We interpret these $k$-ranges as cascade-dominated and mutual friction-dominated intervals, respectively. General behavior of the experimental spectra $E_{xz}(k_y)$ agree well with the predicted spectra [Phys. Rev. B 97, 214513 (2018)]. Analysis of the $n$-th order structure functions statistics shows that in the energy-containing interval the statistics of counterflow turbulence is close to Gaussian, similar to the classical hydrodynamic turbulence. In the cascade- and mutual friction-dominated intervals we found some modest enhancement of intermittency with respect of its level in classical turbulence. However, at small scales, the intermittency becomes much stronger than in the classical turbulence.
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