Article outline

This paper presents a numerical study of the flow of a fluid with temperature-dependent kinematic viscosity in a channel containing a heated cavity. The problem is motivated by the fact that viscosity variations caused by temperature changes may significantly affect the structure of the flow, the intensity of circulation inside the cavity, and the behavior of integral flow characteristics such as the flow rate. Such effects are especially important for thermoviscous and anomalously thermoviscous fluids, for which the viscosity is not constant but depends on the temperature field. In particular, the paper considers both a monotonic temperature dependence of viscosity and a non-monotonic dependence with a pronounced maximum.

The main objective of the study is to determine how different models of temperature-dependent viscosity influence the unsteady dynamics of the flow rate and the corresponding phase trajectories in a three-dimensional channel flow with a heated cavity. The geometry consists of a straight channel with a cavity located on one of its walls. The cavity walls are heated, which leads to a non-uniform temperature distribution and, consequently, to spatial variations in the kinematic viscosity. These variations affect the velocity field and may change the interaction between the main flow in the channel and the recirculating flow inside the cavity.

The numerical simulations are performed in a three-dimensional formulation using the OpenFOAM software package. The governing equations include the incompressibility condition, the momentum equations, and the energy equation. A modified version of the simpleFoam solver is used. The solver is supplemented with the temperature equation and with a viscosity model in which the kinematic viscosity is treated as a function of temperature. Two viscosity laws are analyzed. The first one describes a monotonic decrease of viscosity with increasing temperature. The second one describes an anomalous non-monotonic dependence, in which the viscosity increases up to a certain temperature range and then decreases.

The main quantity used to characterize the flow dynamics is the flow rate through the inlet cross section of the channel. The time series of the flow rate Q(t) is recorded during the calculation and then processed in order to construct phase trajectories in the (Q, Q̇) plane. The derivative of the flow rate is evaluated using finite-difference schemes of different orders of accuracy. Three-point, four-point, and five-point approximations are considered. The five-point scheme is used in the subsequent analysis because it has the highest order of approximation among the schemes considered. In addition, smoothing of the time signals is applied in order to reduce the influence of small-scale numerical oscillations and to reveal the main features of the phase trajectories more clearly.

The results show that the dynamic behavior of the flow rate depends noticeably on the chosen model of temperature-dependent viscosity. In the case of monotonic viscosity variation, the flow rate exhibits more complex oscillatory behavior, which is reflected in the more complicated structure of the phase trajectories. For the anomalous non-monotonic viscosity model, the phase trajectories are more compact and more regular in the considered calculations. This indicates that the form of the viscosity-temperature relation can substantially affect the unsteady response of the flow system.

The study also emphasizes the importance of the three-dimensional formulation. A three-dimensional calculation makes it possible to analyze the spatial structure of the flow in the cavity region, including the formation of circulation zones and the interaction between the cavity flow and the main channel flow. Therefore, the analysis of the flow rate alone is not sufficient for a complete description of the system; it should be supplemented by the consideration of the spatial velocity field.

The obtained results demonstrate that temperature-dependent viscosity is an important factor controlling the dynamics of flows in channels with complex geometry. The use of phase trajectories provides a convenient tool for studying unsteady flow regimes and for comparing the influence of different viscosity models. The proposed approach may be useful for further numerical investigations of thermoviscous and anomalously thermoviscous fluids in channels, cavities, and other configurations where heat transfer and viscosity variations are strongly coupled.