The authors show that the Reynolds-averaged representation of finite-d
imensional dynamical systems with quadratic nonlinearity can be closed
in the limit of infinite moment. The closure is posed for problems wi
th algebraic moment hierarchies, and tested on regular ensembles of th
e viscous Burgers equation and ensembles whose individual members are
chaotic solutions of the Lorenz equations. The solvability of the aver
aged system of equations and the equivalence of direct simulation and
Reynolds-averaged computation, using a simple one-dimensional model pr
oblem, are demonstrated.