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A review of various exactly solvable models on the determination of theenergy spectra $E (k) $ of 3D-velocity field, induced by chaotic vortex linesis proposed. This problem is closely related to the sacramental questionwhether a chaotic set of vortex filaments can mimic the real hydrodynamicturbulence. 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. Theknowledge of $\mathcal{P}(\{\mathbf{s}(\xi,t)\})$ is identical to the fullsolution of the problem of quantum turbulence and, in general, $\mathcal{P}$ isunknown. In the paper we discuss several models allowing to evaluate spectra inthe explicit form. This cases include standard vortex configurations such as astraight line, vortex array and ring. Independent chaotic loops of variousfractal dimension as well as interacting loops in the thermodynamic equilibriumalso permit an analytical solution. We also describe the method of an obtainingthe 3D velocity spectrum induced by the straight line perturbed with chaotic 1DKelvin waves on it.

Energy spectrum of the 3D velocity field, induced by vortex tangle