title: Numerical Modeling of Non-Isothermal Viscoelastic Fluids
Creators: name: Meburger, Stefanie
Publisher: name: UNSPECIFIED
Year: 2021

description: A numerical solution procedure for non-isothermal viscoelastic ﬂuids is developed, based on the Finite Volume method on general unstructured meshes. Stable solutions in experimentally relevant conditions, i.e. at high ﬂuid elasticity, are ensured with the root conformation approach. The temperature dependence of the Oldroyd-B type constitutive equations is accounted for with the time-temperature superposition principle with WLF and Arrhenius shift factor. The energy equation considers energetic and entropic energy conversion with a constant partitioning factor. The veriﬁcation by comparison to an analytical solution of a simpliﬁed set of equations proves the correct solution of the numerical framework. Detailed comparison of the numerical solution to experimental data from the literature provides an insight into the advantages and shortcomings of the chosen thermo-rheological model. Evaluation of the diﬀerences between simulation and experimental data in a Weissenberg number range from Wi = 4 − 14.8 shows that the suggested thermal and rheological model are suitable for a good qualitative agreement to experimental data, without encountering any stability issues at high Weissenberg numbers. The comparison with the more complex exponential PTT model with regard to the reproduction of experimental data suggests that the chosen rheological model has less inﬂuence on the considered temperature ﬁeld than the thermal model, even though the diﬀerences in the stress ﬁelds can be large. At all considered Weissenberg numbers and wall temperatures, viscous dissipation, which is regarded as the main source for temperature changes in the ﬂow, is reproduced. The magnitude of the heat production by viscous dissipation is smaller than in the experimental setup, especially in regions of low velocity gradients, where the deviations between simulation results and experimental data are highest. Comparison of results to numerical studies from the literature shows good qualitative and, in case the same ﬂuid is analyzed, also good quantitative agreement. Small deviations are present, which suggest that the stabilization approach inﬂuences the computed ﬂow ﬁeld. The employed energy equation accounts for energetic and entropic energy conversion mechanisms, the partitioning between both is achieved with a constant factor. In natural ﬂow, the conversion changes locally with the ﬂow situation, so that pure energy and pure entropy elasticity provide the bounds in which the solution can be located. Computations for these two limiting cases are performed and the ﬂow ﬁelds compared in two diﬀerent ﬂow regimes, one of low viscosity and low temperature and one of high viscosity and high temperature. In both regimes the deviation between pure entropic and pure energetic energy conversion, and thus the importance of energy elastic eﬀects, increases with increasing elasticity of the ﬂuid. An additional dependence on the thermal boundary conditions is found, yet of diﬀerent nature in the two considered ﬂow regimes: in case of the ﬂuid of low viscosity and low temperature, energetic eﬀects are found to increase with heating of the walls. The magnitude of the deviations and thus the importance of energetic eﬀects is only very small. High deviations and a high inﬂuence of the energetic eﬀects is encountered in the ﬂuid of high viscosity and temperature. Unlike at the low viscosity, the deviations are found to increase with decreasing energy supply and thus are highest for the coolest wall. An alternative Oldroyd-B type constitutive equation with explicit temperature dependent terms that account for the thermal history of the ﬂuid is proposed and investigated in diﬀerent ﬂow regimes and ﬂow geometries. An analysis of the alternative model in comparison to the ’normal’ Oldroyd-B model and experimental data in the low viscosity regime suggests that also an explicit inclusion of temperature in the constitutive equation does not alleviate the deviations between simulation and experimental data. In case of the high viscosity and high temperature ﬂuid, deviations between the two constitutive equations are found in all ﬁeld variables, and are most pronounced for temperature and stress. The predicted peak values diﬀer signiﬁcantly, and the diﬀerence grows with growing elasticity of the ﬂuid and magnitude of the temperature diﬀerence between (initial) ﬂuid temperature and wall temperature. The highest deviations are found at the coolest walls. Diﬀerences in the temperature ﬁelds are found to be especially pronounced in regions where the ﬂuid is expanded, irrespective of the considered ﬂow geometry.