A new method preserving the positive definiteness of a second order tensor variable in flow simulations with application to viscoelastic turbulence

We present a new numerical method for the simulation of the flow of complex fluids with internal microstructure that is described by a second order, positive definite, internal structural parameter, so that its positive definiteness is preserved in the simulations. An example of such an internal structural parameter is the conformation tensor, used in the modeling of viscoelastic flows, which characterizes the flow-induced molecular deformation. Numerical methods that guarantee the preservation of the positive definiteness of the conformation tensor help both the physical interpretation of the results and the numerical stability of the simulations. The new method presented here is based on the log-conformation representation of a second order tensor. It is implemented through the application of the Cayley–Hamilton theorem for second order tensors. When necessary, we also use an additional mapping that ensures the boundness of the conformation tensor. This approach has been applied in Direct Numerical Simulations (DNS) of viscoelastic turbulent channel flow, which has received great attention since the mid 1990s with ultimate goal to understand the phenomenon of maximum drag reduction. The algorithm uses a full 3D spectral representation of the spatial dependence for the flow and conformation variables and a second order accurate backward differentiation formula for the integration of the governing equations in time. The key issue for the successful implementation of the proposed scheme is a second order finite difference multigrid diffusion applied to the conformation field. Numerical diffusion has been always added to the hyperbolic evolution equations for the conformation tensor in spectral DNS of viscoelastic turbulent flows in order to remove a-physical high wavenumber instabilities induced due to the chaotic nature of the flow. The main advantage of the finite-difference implementation (as opposed to a spectral implementation as done before) is that it allows for the preservation of the positive definiteness of the conformation tensor.