Turbulent Natural Convection of Non-Newtonian Fluids in Enclosed Spaces

**Principal Investigator:**

Sahin Yigit, Josef Hasslberger, Markus Klein

**Affiliation:**

Numerical Methods in Aerospace Engineering, Bundeswehr University Munich

**Local Project ID:**

pn56di

**HPC Platform used:**

SuperMUC and SuperMUC-NG of LRZ

**Date published:**

**Introduction**

This project focuses on the modelling and physical understanding of threedimensional turbulent natural convection of non-Newtonian fluids (i.e. where the strain rate dependence of shear stresses is nonlinear in nature) in enclosures. This topic has wide relevance in many engineering applications such as preservation of canned foods, polymer and chemical processing, biochemical synthesis, solar and nuclear energy, thermal energy storages. Therefore, the flow and heat transfer knowledge of more complex than Newtonian fluids (fluids like water, air where viscous stress is directly proportional to strain rate) is essential from an engineering perspective since the non-Newtonian character of fluids can also be very useful for designing new adaptive thermal management systems.

**Methods**

Direct Numerical Simulations (DNS) of three-dimensional turbulent Rayleigh-Bénard convection of yield stress fluids obeying a Bingham model and inelastic shear thinning / thickening fluids obeying power-law model in a cubical enclosure have been performed under Dirichlet boundary conditions. The simulation configuration is schematically shown in Fig.1, which demonstrates that the differentially heated horizontal walls are subjected to constant wall temperature boundary conditions. The bottom wall is taken to be at higher temperature than the top wall (i.e. *T _{H} > T_{C}*) and all the other walls are considered to be adiabatic (i.e. the temperature gradient in the wall normal direction vanishes at the wall). No-slip and impermeability conditions are specified for all walls. The enclosure is taken to be cubic (i.e.

To solve the non-linear set of governing equations in a finite-volume framework, the open-source CFD package OpenFOAM has been utilised. The pressure-velocity coupling has been addressed by the use of PIMPLE algorithm. Convective and diffusive fluxes are evaluated by second-order centered difference schemes. Temporal advancement has been achieved by the second-order Crank-Nicolson scheme with constant time-stepping. It has been ensured that the Courant number is always sufficiently below unity so that the underlying physics is captured with sufficient temporal resolution.

**Results**

The distributions of non-dimensional temperature iso-surfaces and apparently unyielded regions (AURs are shown as grey regions) are shown in Fig. 2 for different Bingham number (i.e. *Bn*) at *Ra=10 ^{8}*, and

This also leads to the weakening of convection with increasing *Bn*, which can be substantiated from Fig. 2b where the size and the probability of finding AURs increases with increasing *Bn*, and this gives rise to a reduction in the mean Nusselt number (see Fig. 3). For very large values of Bingham number, the boundary layer thickness becomes of the order of the enclosure size *H*. At that stage, the fluid flow does not influence the thermal transport and heat transfer takes place principally due to conduction, which is reflected in the unity mean Nusselt number.

It is worth noting that in the context of bi-viscosity regularisation, the flow does not stop in a true sense in AURs but the flow within the islands of AURs is too weak to influence the heat transfer rate and thus the exact shape and size of AURs do not affect the heat transfer rate and the mean Nusselt number.

**Ongoing Research / Outlook**

After obtaining experience with yield stress fluids, the next step was the investigation of Prandtl number (*Pr*) effects near active walls on the velocity gradient and flow topologies. This is because liquids like non-Newtonian fluids typically have high *Pr* (i.e. from 10^{2} to 10^{3}) [1]. This is important as the modelling strategy for turbulent natural convection of gaseous fluids may not be equally well suited for the simulations of turbulent natural convection of liquids with high values of *Pr*.

As it can be seen in Fig.4 (left column) the expected and well-known teardrop shape of the joint PDF of velocity gradient tensor invariants *Q ^{+}* and

**References and Links**

[1] S. Yigit, J. Hasslberger, N. Chakraborty and M. Klein, Effects of Rayleigh-Bénard convection on spectra of viscoplastic fluids, Int. J. Heat and Mass Transfer, 147 (2020), 118947.

[2] S. Yigit, J. Hasslberger, M. Klein and N. Chakraborty, Near wall Prandtl number effects on velocity gradient invariants and flow topologies in turbulent Rayleigh Bénard convection, Scientific Reports, 10, 14887, 2020.

**Scientific Contact**

Univ.-Prof. Dr.-Ing. (habil) Markus Klein

Institute of Applied Mathematics and Scientific Computing

of the Department of Aerospace Engineering

Bundeswehr University Munich

Werner Heisenberg Weg 39, D-85577 Neubiberg (Germany)

e-mail: markus.klein [@] unibw.de

**NOTE: **This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2020 (2021)" (ISBN 978-3-9816675-4-7)

*Local project ID: pn56di*

*September 2021*