The Boussinesq-Rayleigh series for two-dimensional flows away from boundaries

Abstract

The Boussinesq-Rayleigh series solutions of the unsteadyNavier-Stokes equations are computed symbolically in twodimensions. For finite Reynolds numbers, a nonlinear system ofdifferential recurrent relations admits two formal solutions: ageneral solution for flows forced by the dynamic pressure and ageneral solution for freestreams. For generating functions, whichare bounded together with their derivatives, the absoluteconvergence of the series solutions is shown symbolically byconverting the differential recurrent relations into tensorrecurrent relations and using the comparison and ratio tests. Atriangular structure of three-dimensional tensors of derivativesemployed in the tensor recurrent relations is obtained byinduction. A detailed examination of four basic forced flows andfour basic freestreams shows that the formal series solutions awayfrom boundaries are nonlinear superpositions of the Stokes flow, the Bernoulli flow, the Couette flow, and the Poiseuille flow thatare unsteady, two-dimensional continuations of the classicalsolutions at high Reynolds numbers. A tensor algorithm fornumerical evaluation and continuation of the series solutions isimplemented by parallel computing. Emergence of multi-scalecoherent structures of the Poiseuille flow at high Reynoldsnumbers is tackled by using multivalued contours of the streamfunction.