  The study is devoted to the development of new effective tools and methods of
ana-lytical hydrodynamics, including problems of existence, smoothness and
structure of laminar and turbulent flows. The main problem is complex
Navier-Stokes equations and turbulence in complex spaces. The necessity of
introducing complex spaces in hydrodynamics is deter-mined by the mechanism of
transition of a laminar flow into a turbulent flow. The author pro-poses a
non-traditional scenario of the transition: the cause of turbulence is in
destruction (cessation of existence) of a laminar flow. The article contains
the mathematical rationale for the necessity of development of the theory of
turbulence in the complex configurational space: the complex configurational
space is the natural area of existence of turbulence. Hydrody-namic flows are
regarded as flows on entropy manifolds that [flows] are supported by the two
symmetries: the symmetry of conservation of general entropy and the symmetry of
duality of impulse representation. The new symmetry has been introduced and
studied: the forminvari-ance of Helmholtz matrix of impulse density. The strict
foundation has been provided for the known fact of chaotic mechanics:
appearance of the new structure (a turbulent flow) is a result of interaction
of two entities: dissipation and vorticity. On the deep level the phenomenology
of turbulence in complex spaces is based on the transition from the mechanics
of a material point to the mechanics of an oriented material point, that
[transition] takes place in a current period of time.
