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Register a new userA numerical study of the Navier-Stokes transport coefficients for 2D granular hydrodynamics
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L\\'idia Almaz\\'an
Publication date
2012
fields
Physics
and research's language is
English
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A numerical study is presented to analyze the thermal mechanisms of unsteady, supersonic granular flow, by means of hydrodynamic simulations of the Navier-Stokes granular equations. For this purpose a paradigmatic problem in granular dynamics such as the Faraday instability is selected. Two different approaches for the Navier-Stokes transport coefficients for granular materials are considered, namely the traditional Jenkins-Richman theory for moderately dense quasi-elastic grains, and the improved Garzo-Dufty-Lutsko theory for arbitrary inelasticity, which we also present here. Both solutions are compared with event-driven simulations of the same system under the same conditions, by analyzing the density, the temperature and the velocity field. Important differences are found between the two approaches leading to interesting implications. In particular, the heat transfer mechanism coupled to the density gradient which is a distinctive feature of inelastic granular gases, is responsible for a major discrepancy in the temperature field and hence in the diffusion mechanisms.
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The Navier--Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture of inelastic hard spheres are determined from the Boltzmann kinetic equation. A normal or hydrodynamic solution to the Boltzmann equation is obtained via the Chapman--Enskog method for states near the local version of the homogeneous time-dependent state. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. As expected, they are given in terms of the solutions of a set of coupled linear integral equations. In addition, in contrast to previous results obtained for low-density granular mixtures, there are also nonzero contributions to the first-order approximations to the partial temperatures $T_i^{(1)}$ and the cooling rate $zeta^{(1)}$. Explicit forms for the diffusion transport coefficients, the shear viscosity coefficient, and the quantities $T_i^{(1)}$ and $zeta^{(1)}$ are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. The above transport coefficients are given in terms of the coefficients of restitution, concentration, and the masses and diameters of the components of the mixture. The results apply in principle for arbitrary degree of inelasticity and are not limited to specific values of concentration, mass and/or size ratios. As a simple application of these results, the violation of the Onsager reciprocal relations for a confined granular mixture is quantified in terms of the parameter space of the problem.
Discrete Boltzmann model (DBM) is a type of coarse-grained mesoscale kinetic model derived from the Boltzmann equation. Physically, it is roughly equivalent to a hydrodynamic model supplemented by a coarse-grained model for the relevant thermodynamic non-equilibrium (TNE) behaviours. The Navier-Stokes (NS) model is a traditional macroscopic hydrodynamic model based on continuity hypothesis and conservation laws. In this study, the two models are compared from two aspects, physical capability and computational cost, by simulating two kinds of flow problems including the thermal Couette flow and a Mach 3 step problem. In the cases where the TNE effects are weak, both the two models give accurate results for the hydrodynamic behaviour. Besides, DBM can provide more detailed non-equilibrium information, while the NS is more efficient if concern only the density, momentum, energy and their derived quantities. It is concluded that, if the TNE effects are strong or are to be investigated, the NS is insufficient while DBM is a good choice. While in the cases where the TNE effects are weak and only the macro flow fields are to be studied, the NS is more preferable.
We introduce a model of interacting singularities of Navier-Stokes, named pin,cons. They follow a Hamiltonian dynamics, obtained by the condition that the velocity field around these singularities obeys locally Navier-Stokes equations. This model can be seen of a generalization of the vorton model of Novikov, that was derived for the Euler equations. When immersed in a regular field, the pin,cons are further transported and sheared by the regular field, while applying a stress onto the regular field, that becomes dominant at a scale that is smaller than the Kolmogorov length. We apply this model to compute the motion of a dipole of pin,cons. When the initial relative orientation of the dipole is inside the interval (0, pi/2), a dipole made of pin,con of same intensity exhibits a transient collapse stage, following a scaling with dipole radius tending to 0 like (tc - t) power 0.63. For long time, the dynamics of the dipole is however repulsive, with both components running away from each other to infinity.
The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future biomedical and technological applications. These microswimmers move autonomously through aqueous media where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than to shear resistance. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.
A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$alpha$ (LANS-$alpha$) equations is developed where the variation in the parameter $alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$alpha$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-$alpha$ simulations using a fixed $alpha$. In order to evaluate the applicability of the dynamic LANS-$alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $alpha$ changes in the wall normal direction. Near a solid wall, the length scale $alpha$ is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, $alpha$ approaches an almost constant value. Furthermore, the behavior of the subgrid scale stresses in the near wall region is captured accurately by the dynamic LANS-$alpha$ model. The dynamic LANS-$alpha$ model has the potential to extend the applicability of the LANS-$alpha$ equations to more complicated anisotropic flows.
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