Multiscale resultant, principal, perturbation, cascade, and reference Poiseuille-Hagen flows with exposed, hidden, and dual perturbations are constructed and computed in trigonometric, hyperbolic, and elliptic structures, which are invariant with respect to nonlinear algebraic and differential operators. Algebra and differentiation of the invariant cos, sech and cn structures are treated both theoretically and symbolically. Reduction of the invariant cn structures to the invariant cos and sech structures is also considered. As the first application, the invariant cn structures are used to compute series solutions of an asymmetric Hamiltonian system of the fourth order, which models propagation of solitary waves. The developed algorithm allows to compute the series solutions of initial-value problems both for positive and negative pulsations. Comparison with the conventional Felhlberg-Runge-Kutta method shows that the series solutions are preferable with respect to the numerical ones due to their uniform convergence. As the second application, the invariant cn structures are exploited to integrate the exposed and hidden perturbations of the Poiseuille-Hagen flow into the dual perturbations. The invariant cos, sech and cn structures together with the invariant exp structures are invoked to model multiscale transition of the principal, perturbation, and resultant Poiseuille-Hagen flows. Decomposition of the perturbation flows into the cascade and reference flows and an asymptotic treatment of the series solutions at large Reynolds numbers yield that the invariant cos structures model homogeneous turbulence, the invariant sech structures simulate flashes of turbulence, and the invariant cn structures prototype intermittency of turbulence.