When Porous Media Meets Turbulence Gauss Centre for Supercomputing e.V.


When Porous Media Meets Turbulence

Principal Investigator:
Xu Chu, Bernhard Weigand

Institut für Thermodynamik der Luft- und Raumfahrt, Universität Stuttgart

Local Project ID:

HPC Platform used:
Hazel Hen and Hawk of HLRS

Date published:

The SFB1313 Slogan: “Porous media are everywhere”

Porous media are everywhere. In fact, during the COVID-19 pandemic, we have all grown accustomed to wearing FFP2 masks on a daily basis. The porous, nonwoven filtering material, consisting of polymeric fibres that carry a strong electrostatic charge, is able to filter at least 95% of airborne particles. This technology, together with vaccines, is the key to fight the COVID-19 pandemic.

By definition, porous media are materials with a void space where fluids can pass through. Thus, the void space or “pore” is a distinct feature of porous media. Researchers take advantage of the additional contact surfaces created by the pore to achieve objectives in much larger scales. Advanced manufacturing technologies, such as additive manufacturing, have made it easier to develop complex materials with porous media. Additive manufacturing is becoming a new key technology in the era of industry 4.0. “Interesting perspectives are opening up, for example, in the production of gas turbines,” says Siemens AG. It is not a coincidence that 3D-printed turbine blades are already working in the latest GE9X turbofan engine from General Electric. The freedom and flexibility of additive manufacturing makes it possible to optimally design porous media structures within the turbine blade and many other applications [Terzis et al., 2019].

Beyond Darcy’s law

Turbulence in porous media is a controversial issue. As most porous materials considered for use in conventional engineering applications present very small pores and the fluid velocity is relatively small, the dominant regime is obviously the Darcy regime with very low flow velocities, leading to largely laminar flows. Darcy’s law, named after Henry Darcy, (1856) states a simple proportionality relationship between the instantaneous flux through a porous medium, the permeability of the medium, the dynamic viscosity of the fluid, and the pressure drop over a given distance.

However, high velocity fluid flow through porous media can lead to turbulent flow within the pores. Several applications exist where the Reynolds numbers in pores can be so large that the unsteady inertial effects become important, giving rise to the onset of turbulent flow. Examples include, but are not limited to, packed bed catalysis, gas turbine cooling, and pebble-bed, high-temperature nuclear reactors.

The contribution of the inertial effect can dramatically change the topology of the flow field, leading to formation of impinging jets, shearing, separation, etc. within the pores. These flow features can substantially alter the dispersion characteristics and play critical roles in the transport.

Looking inside a digital twin with HPC

Investigating complex flow patterns at the micro-scale is the key to understanding flow physics and developing models. In experiments, access to detailed, three-dimensional flow field measurements is quite challenging due to the inherent space constraints of the porous media. Moreover, porous media are usually bounded by the walls of the container in which they are placed. Separating porous-media-induced flows from the wall-bounded shear flow needs additional attention. In this case and despite its high demand for computational resources, direct numerical simulation (DNS) becomes a valuable approach to understand this special flow mechanism. DNS is characterized by its ability to resolve until the smallest turbulent length scale of the turbulent eddies (Kolmogorov length scale) and therefore eliminates the uncertainties associated with macroscopic modeling. Although it has been widely considered in various canonical conditions, such as channel, pipe and turbulent boundary layer, it is still not prevalent in the porous media community.

In the DISS (data-integrated simulation science) project, funded by the Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg, later the DFG-SFB-1313 and Cluster of Excellence SimTech, the junior group leader Dr. Xu Chu and Prof. Bernhard Weigand from the Institut für Thermodynamik der Luft- und Raumfahrt (ITLR) of University of Stuttgart studied porous media and the physics of fluids inside porous media. Taking advantage of geometric periodicity in a regular porous media, a carefully chosen representative elementary volume (REV) with periodic boundary conditions is sufficient for DNS instead of a large computational domain.

The spectral/hp element solver Nektar++ [Cantwell et al., 2011, Chu et al., 2019, Pandey et al., 2020, Chu et al., 2020a] is used to numerically resolve the complex geometrical structures. The solver framework allows arbitrary-order spectral/hp element discretisations with hybrid shaped elements. The spectral-accurate discretisation combined with meshing flexibility is optimal to deal with complex porous structures and to resolve the interface region. The solver reaches over 60% parallel efficiency on over 100,000 cores on both the Hazel Hen and the HAWK HPC systems at HLRS. A DNS-based microscopic analysis of turbulent flow in a regular porous media was conducted with the aim to link the macroscopic statistics and qualitative observations with microscopic pore-scale signatures of the same. The regular porous media are comprised of square cylinders in a staggered array. The pore Reynolds number ranges from 500 to 1500. A reduced order modeling method—Proper orthogonal decomposition (POD)—is employed to distinguish the energy-conserving structures. The results support the pore scale prevalence hypothesis (PSPH). However, energetic coherent structures are observed in case of a sparsely-packed porous medium.

The onset of chaos in porous media

Compared with the recent progress in the turbulent range, even less knowledge exists referring to the transitional range in porous media. The topological complexity limits the in-depth qualitative analysis of appearing instabilities in this range. The onset of turbulence remains to be explained. For that purpose, we utilized the BiGlobal stability analysis and DNS to explain the onset of turbulence in the porous media and transition mechanism. The stability analysis indicates the first unstable Reynolds number at around Re = 90. Two unstable modes are captured by the linear stability analysis: a two-dimensional oscillatory mode and a three-dimensional stationary mode. In addition, counter-rotating mushroom vortices are captured in the wake area, which is a consequence of the lift-up effect. The mushroom vortices in the wake area induce streamwise vortices in the downstream direction.

Driven by big data

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from the high-fidelity multi-scale simulations and advanced measurement techniques. However, the analysis of these data strongly relies on domain expertise and statistical analysis. Machine learning presents a wealth of techniques to extract information from data that can be translated into knowledge about the underlying physics [Wang et al., 2020]. The ITLR is attempting to combine the merit of machine learning algorithms to the state of the art high-performance computing. Physics-informed, data-driven methods are aimed at contributing to the upscaling from microscale to the macroscale.

In the post-pandemic time, the digitalization of the industry and society as well as the involvement of artificial intelligence bring both new challenges and new opportunities. We are confident that our research can contribute to these challenges.


C. D. Cantwell, S. J. Sherwin, R. M. Kirby, and P. H. Kelly. From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements. Computers & Fluids, 43(1):23–28, 2011.

X. Chu, B. Weigand, and V. Vaikuntanathan. Flow turbulence topology in regular porous media: From macroscopic to microscopic scale with direct numerical simulation. Physics of Fluids, 30(6):065102, 2018.

X. Chu, G. Yang, S. Pandey, and B. Weigand. Direct numerical simulation of convective heat transfer in porous media. International Journal of Heat and Mass Transfer, 133:11–20, 2019.

X. Chu, W. Wang, G. Yang, A. Terzis, R. Helmig, and B. Weigand. Transport of turbulence across per- meable interface in a turbulent channel flow: interface-resolved direct numerical simulation. Transport in Porous Media, 135(2), 2020a.

X. Chu, Y. Wu, U. Rist, and B. Weigand. Instability and transition in an elementary porous medium. Physical Review Fluids, 5(4):044304, 2020b.

S. Pandey, X. Chu, B. Weigand, E. Laurien, and J. Schumacher. Relaminarized and recovered turbulence under nonuniform body forces. Physical Review Fluids, 5(10):104604, 2020.

A. Terzis, I. Zarikos, K. Weishaupt, G. Yang, X. Chu, R. Helmig, and B. Weigand. Microscopic velocity field measurements inside a regular porous medium adjacent to a low Reynolds number channel flow. Physics of Fluids, 31(4):042001, 2019.

W. Wang, X. Chu, A. Lozano-Dur´an, R. Helmig, and B. Weigand. Information transfer between turbulent boundary layer and porous media. 2020. doi: arXiv:2011.01305.

Scientific Contact

Dr.-Ing. Xu Chu
University of Stuttgart
Institute of Aerospace Thermodynamics
Pfaffenwaldring 31, D-70569 Stuttgart (Germany)
e-mail: xu.chu [@] itlr.uni-stuttgart.de

Local project ID: PoroDNS

February 2021

Tags: HLRS Universität Stuttgart CSE