  One of the principal barriers in developing accurate and tractable predictive
models in turbulent flows with a large number of species is to track every
species by solving a separate transport equation, which can be computationally
impracticable. In this paper, we present an on-the-fly reduced order modeling
of reactive as well as passive transport equations to reduce the computational
cost. The presented approach seeks a low-rank decomposition of the species to
three time-dependent components: (i) a set of orthonormal spatial modes, (ii) a
low-rank factorization of the instantaneous species correlation matrix, and
(iii) a set of orthonormal species modes, which represent a low-dimensional
time-dependent manifold. Our approach bypasses the need to solve the
full-dimensional species to generate high-fidelity data - as it is commonly
performed in data-driven dimension reduction techniques such as the principle
component analysis. Instead, the low-rank components are directly extracted
from the species transport equation. The evolution equations for the three
components are obtained from optimality conditions of a variational principle.
The time-dependence of the three components enables an on-the-fly adaptation of
the low-rank decomposition to transient changes in the species. Several
demonstration cases of reduced order modeling of passive and reactive transport
equations are presented.
