  We present an optimization-based method to efficiently calculate accurate
nonlinear models of Taylor vortex flow. We use the resolvent formulation of
McKeon & Sharma (2010) to model these Taylor vortex solutions by treating the
nonlinearity not as an inherent part of the governing equations but rather as a
triadic constraint which must be satisfied by the model solution. We exploit
the low rank linear dynamics of the system to calculate an efficient basis for
our solution, the coefficients of which are then calculated through an
optimization problem where the cost function to be minimized is the triadic
consistency of the solution with itself as well as with the input mean flow.
Our approach constitutes, what is to the best of our knowledge, the first fully
nonlinear and self-sustaining, resolvent-based model described in the
literature. We compare our results to direct numerical simulation of Taylor
Couette flow at up to five times the critical Reynolds number, and show that
our model accurately captures the structure of the flow. Additionally, we find
that as the Reynolds number increases the flow undergoes a fundamental
transition from a classical weakly nonlinear regime, where the forcing cascade
is strictly down scale, to a fully nonlinear regime characterized by the
emergence of an inverse (up scale) forcing cascade. Triadic contributions from
the inverse and traditional cascade destructively interfere implying that the
accurate modeling of a certain Fourier mode requires knowledge of its immediate
harmonic and sub-harmonic. We show analytically that this finding is a direct
consequence of the structure of the quadratic nonlinearity of the governing
equations formulated in Fourier space. Finally, we show that using our model
solution as an initial condition to a higher Reynolds number DNS significantly
reduces the time to convergence.
