ENGINEERING AND CFD
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. TH > TC) 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. H = W = L).
Figure 1: Schematic diagram of the simulation domain. © Uni BW
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=108, and Pr=320. It can be seen from Fig. 2a that the iso-surfaces become less deformed and thermal plumes from the bottom wall become less apparent as Bn increases. These are the indications of the relative weakening of buoyancy force due to the augmented flow resistance as a result of increased yield stress.
Figure 2: a) Distributions of iso-surfaces of non-dimensional temperature, b) three-dimensional AURs for different values of Bn at Ra=108and Pr=320 [1]. © Uni BW
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.
Figure 3: Temporal variation of Nu with nondimensional simulation time for hot (i.e. Nuhw) and cold (i.e. Nucw) wall for different Bn at Ra=108 and Pr=320. The time averaged mean Nusselt number is shown with a blue dashed line [1]. © Uni BW
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 102 to 103) [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.
Figure 4: Left column: Contours of joint probability density functions of Q+ and R+ invariants on a logarithmic scale in the bulk region and at the heated and cooled boundaries for Ra=108 in the case of a) Pr=1, b) Pr=320 [2]. Right column: a) Non-dimensional temperature iso-surfaces θ= 0.4, θ= 0.6, shown in blue and yellow colour, respectively. b) Path lines coloured by nondimensional vertical velocity magnitude for different Pr at Ra=107 [2]. © Uni BW
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 R+ changes near active walls depending on Pr. This can be explained by looking at Fig.4 (right column): isolated plumes drive the convection process in the Pr > 1 case, whereas frequent roll ups in the Pr = 1 are indicative of a large-scale circulation [2].
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