Simulation of time-dependent viscoelastic channel poiseuille flow at high reynolds numbers

The development of a fully spectral, three-dimensional (in space) and time-dependent simulation for viscoelastic flows is described. For smooth flow geometries, such as the plane Poiseuille flow which is studied here, the method combines the advantages of a high accuracy, exponentially fast convergent with mesh refinement and spectral approximation with those of an efficient numerical implementation well suited for a vector/parallel computer architecture. This approach is shown here to lead to a stable numerical approximation for We ≈ O(1) provided the size of the time step is kept small enough. Three different viscoelastic constitutive equations of the differential type are considered here: the upper convected Maxwell, the Oldroyd-B and the Chilcott-Rallison models. At large enough times, and before a stationary state is reached, numerical instabilities set in depending on the time-step size and initial conditions. Before the onset of the numerical instabilities, the conformation tensor is always observed to lose its positive-definiteness. For the constitutive equations utilized here this is due to the accumulation of numerical error since it can be shown that, if the constitutive equations are integrated in time exactly, the conformation tensor remains always positive definite. For the UCM model, this loss of positive definiteness is associated with the loss of evolutionarity of the governing equations. This is presumed to be responsible for the breakdown of the numerical simulations. Examination of the results of the three-dimensional simulations has shown similar results for the rms values of the velocity field and the Reynolds stress as compared to the Newtonian case. However, sharp boundary layers have been observed for all the components of the conformation tensor, for both their mean and rms values. These results were obtained at a Reynolds number of 5000, for initial perturbations to the steady Poiseuille flow solution in the form of the most unstable eigenvectors with a relative amplitude of the order of 1/2% in the energy norm.