# Hydrodynamics of granular gases: clustering, universality and importance of subsonic convective waves

by Mathias Hummel

Date of Examination:2016-10-26

Date of issue:2017-01-16

Advisor:Dr. Marco Giacomo Mazza

Referee:Prof. Dr. Stephan Herminghaus

Referee:Prof. Dr. Eberhard Bodenschatz

Referee:Prof. Dr. Klaus Kassner

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## Abstract

### English

In this thesis we study a hydrodynamic Ansatz for granular gases. The structure of this thesis is the following. We start with a brief introduction into the kinetic theory of granular gases in Chapter 2 and we introduce the coefficient of restitution as the important parameter to describe the dissipative nature of granular particles. There we show how to derive the NS equations for granular gases which are the fundamental set of equations we study here. We discuss the more realistic problem of viscoelastic particles in Chapter 2 and compare the standard model for viscoelastic spheres with a simple alternative model. If the system-size is larger than a critical size a transition from the homogeneous cooling state (HCS) into an inhomogeneous cooling state (ICS) appears, which is characterized by the formation of dense clusters. However, in reality the coefficient of restitution depends on the impact velocity. The related linear stability analysis is given at the end of Chapter. In Chapter 3 we derive the dimensionless, quasi-conservative form of the granular NS equations. We discuss the pair correlation functions we have implemented in our numerical solver and discuss in Section 3.4 the granular dimensionless numbers: Mach, Reynolds and Prandtl numbers. The necessity of studying large system sizes and of characterizing fluctuations in regions of sharp gradients in temperature and density, developing into supersonic flow, without ambiguities motivates us to tackle the continuous hydrodynamic equations beyond perturbative schemes. In Chapter 4 we describe in detail our solver to perform DNS. We solve the integrated form of the NS equations on a lattice grid with the help of a finite volume method. The integration is performed via Gaussian integration points and the values at such points are reconstructed with a seventh order weighted essentially nonoscillatory (WENO) method. To solve the coupled equations we implemented an operator splitting method between the convective part and the diffusive part of the equations. At the surface of the finite volumes it is important to calculate the fluxes within the finite volume framework. For the convective fluxes, including the dissipative part, we use a MUSCL-Hancock Method (MHM) and the multi-stage (MUSTA) approach to solve the ensuing Riemann problem. The diffusive flux is solved with the diffusive generalized Riemann problem (dGRP) flux. Our DNS allow us to study fluctuations in a freely cooling granular gas inaccessible to molecular dynamics simulations. In Chapter 5 we show the results of the study of such fluctuations. We show that there is a universal power law relation between the amount of density fluctuations of a granular gas and the Mach number of the system. This implies that there is no typical timescale inside a freely cooling granular gas and that we can describe the age of a granular gas in a universal way just by means of the Mach number. In Chapter 6 we step away from the global description of the granular gas and follow a local description of a granular gas. We start with a common heuristic argument for the origin of the clustering inside a freely cooling granular gas. We prove that this heuristic does not hold and show a new understanding for the origin of the clustering inside a freely cooling granular gas. Convective subsonic velocities are nearly conserved while the temperature drops, eventually this leads to a transition to shock waves which are the only reason for clustering. Starting from DNS, we compare our results with MD simulations and present analytical arguments that confirm our finding. We want to stress the point that our findings demonstrate that the textbook understanding of the clustering in freely cooling granular gases is erroneous. In Chapter 7 we study the free cooling in the more realistic model where particles are treated as viscoelastic spheres. The enormous decrease of granular temperature during the cooling process leads to a drastic decrease of the energy dissipation what makes the assumption of a constant coefficient of restitution unrealistic. For a realistic granular gas the slope of Haff’s is 5/3 and it has been shown that the clustering is a transient phenomenon. Here we show that the disappearance of clusters in granular gases is in principle a finite-size effect. As the granular temperature decreases the critical system size increases until it reaches the size of the system, upon which the density inhomogeneities dissolve and the system returns to a homogeneous state. However, the strong dependence of the time of homogenization on system size implies that for any amount of dissipation a system size can be found where the gas will remain heterogeneous at any realistic time scale. Our findings imply that in an infinite system, a freely cooling granular gas of viscoelastic spheres will not become homogeneous again. With our DNS we explore the frontiers of the NS approach. In the outlook Chapter 8 we describe a number of studies which are designed to demonstrate the range of applicability of our solver. Here we show results from driven granular systems. We discuss the topic of how to devise a protocol to homogenize a granular gas starting from an inhomogeneous state, Here we also demonstrate that the granular Leidenfrost effect and convective rolls are reproduced by our solver, which, to the best of the authors knowledge, has not been observed in hydrodynamic simulations before. Similar to what has been done before in experiments and in MD simulations we derive a phase diagram which matches rather well the results from MD simulations and experiments. This proves that even in a situation where the assumptions which lead to the NS equations do not hold anymore, we can still find results which are physical. Finally, in the outlook we demonstrate the effect of a central force on a granular gas, thereby we show possible interesting future projects. With our contribution we hope to advance the research on the hydrodynamics of granular gases and contribute important insights into the fundamental properties of granular systems that have been unknown up to this point.**Keywords:**granular media; granular gas; Navier-Stokes Equations; direct numerical simulation; freely cooling granular gas; clustering in granular gas