Numerical approach to simulating turbulent flow of a viscoelastic polymer solution

In this paper, we present two new numerical algorithms for updating the equations of motion for a viscoelastic fluid that can be described by the finite extensible nonlinear elastic polymer model with the closure proposed by Peterlin (so called FENE-P model) in a transient calculation. In particular, our algorithms address two difficulties found in earlier formulations. First, the polymer extension, represented by the trace of the conformation tensor, can numerically exceed the finite extensible length causing the restoring spring force to change sign and the calculation to rapidly diverge. In our formulations, we have redefined the conformation tensor so that this possibility no longer exists. Secondly, the conformation tensor must remain symmetric and positive definite at all times for the calculation to remain stable. The accumulation of numerical errors can cause loss of this property, leading to the growth of Hadamard instabilities [J. Non-Newtonian Fluid Mech. 60 (1995) 53]. We present two matrix decompositions that enable us to construct the conformational tensor in a manner that ensures positive definiteness. Numerical tests of the new algorithms show significant departures from other approaches that rely on filtering to remove the instabilities.