  A review of various exactly solvable models on the determination of the
energy spectra $E (k) $ of 3D-velocity field, induced by chaotic vortex lines
is proposed. This problem is closely related to the sacramental question
whether a chaotic set of vortex filaments can mimic the real hydrodynamic
turbulence. The quantity $<\mathbf{v(k)v(-k)}>$ can be exactly calculated,
provided that we know the probability distribution functional $%
\mathcal{P}(\{\mathbf{s}(\xi,t)\})$ of vortex loops configurations. The
knowledge of $\mathcal{P}(\{\mathbf{s}(\xi,t)\})$ is identical to the full
solution of the problem of quantum turbulence and, in general, $\mathcal{P}$ is
unknown. In the paper we discuss several models allowing to evaluate spectra in
the explicit form. This cases include standard vortex configurations such as a
straight line, vortex array and ring. Independent chaotic loops of various
fractal dimension as well as interacting loops in the thermodynamic equilibrium
also permit an analytical solution. We also describe the method of an obtaining
the 3D velocity spectrum induced by the straight line perturbed with chaotic 1D
Kelvin waves on it.
