Structure formations: vortices and clusters

When an instability arises, structures emerge. The characterization of structures is achieved by means of correlation functions or structure factors (their Fourier transforms):

where $V=L^d$ is the volume of the system.

If the quantities $a$ and $b$ in the above formulas are components $i$ and $j$ of a vector $\mathbf{a}$ (e.g. the components $u_i$ and $u_j$ of the velocity vector $\mathbf{u}$), then the functions $G_{ij}$ and $S_{ij}$ become isotropic tensor and can be decomposed in two scalar isotropic functions:

It is immediate to verify that, if the vector $\mathbf{a}$ is decomposed into $(d-1)$ components $a_\perp$ perpendicular to $\mathbf{k}$ and one component $a_\parallel$ parallel to $\mathbf{k}$, then

or, if $\mathbf{a}$ is decomposed into $(d-1)$ components $a_\perp'$ perpendicular to $\mathbf{r}$ and one component $a_\parallel'$ parallel to $\mathbf{r}$, then

We use in the rest of this section the subscript $\perp$ and $\parallel$ to indicate the structure factors and correlation functions relative to the velocity vector decomposed in parallel and perpendicular components, i.e. $\mathbf{a}=\mathbf{u}$.

In the isotropic tensor case (the one defined just above) it is often convenient to subtract the self-correlation equilibrium part from the correlation function and from the structure factor, i.e. defining:

where $A(t)=(1/V)\int d\mathbf{r}'\langle (a_i(\mathbf{r}',t))^2 \rangle$, e.g. if $\mathbf{a} \equiv \mathbf{u}$ then $A(t) \equiv T(t)/mn$.

The further assumption of incompressibility (which plays an important role in the following), states that $S_\parallel^+(k,t)=0$ or, equivalently:

$$G_\perp^+(r,t)=G_\parallel^+(r,t)+\frac{r}{d-1}\frac{\partial}{\partial r}G_\parallel^+(r,t)$$ (5.19)

The study of correlation functions in cooling granular gases has been carried along by several authors. In particular the group of van Noije and co-workers has developed two lines of reasoning: a study of the ring kinetic theory [210] and a study of fluctuating hydrodynamics [212]. The first can be considered more fundamental, as it copes with kinetic degrees of freedom, while the second is based on the hydrodynamics description of the fluctuations, and therefore is less general.

Correlation functions from ring kinetic theory

The correlation function is directly related to the the pair correlation function $g_{12}=P_{12}-P_1P_2$ (discussed in paragraph 2.2.7), by the relation:

$$\begin{multline} G_{ij}^+(\mathbf{r},t)=G_{ij}(\mathbf{r},t)-T(t)\delta_{ij}\del... ...}(\mathbf{r}+\mathbf{r}',\mathbf{v}_1,\mathbf{r}',\mathbf{v}_2,t) \end{multline}$$

The authors [210] have calculated, using the second of the ring kinetic equations 2.96, the structure factor for the correlation of shear mode fluctuations:

$$S_\perp^+(k,t)=\frac{T(t)}{mn} \frac{\exp [2 \gamma_0 \tau(1-k^2 \xi_\perp^2) ]-1}{1-k^2\xi_\perp^2}$$ (5.20)

with $\xi_\perp=\sqrt{\eta/mn\omega\gamma_0}$ is the same correlation length appearing in the dispersion relations of the previous paragraph. This expression for the structure factor is identical to the energy spectrum $E(k)$ in the theory of $2D$ or $3D$ homogeneous turbulence in incompressible fluids.

The expression for the correlation functions indicates that $G_\parallel(r,t) \sim -(d-1) G_\perp(r,t) \sim 1/R^d$. The authors have performed 2D molecular dynamics simulations at low solid fraction ($\phi=0.05$) obtaining a good verification of their results.

Correlation functions from fluctuating hydrodynamics

At the hydrodynamic level ( $kl_0 \ll 1$) the correlation functions and structure factors can be obtained from the so-called fluctuating hydrodynamics [137]. This method consists in the study of coupled linear Langevin equations for the values of the hydrodynamic fields, obtained considering the linear transport of linearized hydrodynamics equation as the systematic (dissipative) part of the equations, while the noise is given by fluctuations of fluxes (momentum and heat):

$$\frac{\partial}{\partial \tau} \delta \tilde{\mathbf{a}}(\mathbf{... ...)\delta \tilde{\mathbf{a}}(\mathbf{k},\tau)+\tilde{\mathbf{f}}(\mathbf{k},\tau)$$ (5.21)

where the vector $\delta \tilde{\mathbf{a}}$ and the matrix $\tilde{\mathcal{M}}$ have been defined in the previous paragraph. The noise $\tilde{\mathbf{f}}$ is given by internal fluctuations of the fluxes. The fluxes (or currents) $\mathcal{P}$ and $\mathbf{q}$ are considered fluctuating around their average values:

where the fluctuating part of the currents is assumed to be a white Gaussian noise, local in space, with correlations given by appropriate fluctuation-dissipation relations (they can be found on the last chapter of [137]):

(with $\eta'$ the bulk viscosity, defined in paragraph 2.3.9) so that the (derived) fluctuation-dissipation relations for the noise $\tilde{\mathbf{f}}$ are given by:

$$\frac{1}{V} \langle \tilde{\mathbf{f}}_a(\mathbf{k},\tau)\tilde{\... ...{f}}_b(-\mathbf{k},\tau') \rangle=\tilde{\mathcal{C}}_{ab}(k)\delta(\tau-\tau')$$ (5.24)

with

$$\tilde{\mathcal{C}}= \begin{pmatrix}0 & 0 & 0 & 0 \ 0 & \gamma_... ..._0k^2\xi_\parallel^2/n & 0 \ 0 & 0 & 0 & 4\gamma_0k^2\xi_T^2/dn \end{pmatrix}$$ (5.25)

(as it can be seen, different components of noise are uncorrelated, and in the equation for the density there is no noise).

The results of the analysis is reviewed in the following table:

The structure factor for the shear modes is identical to that derived from the ring kinetic equations (Eq. (5.22)). The factors calculated for the longitudinal velocity components and for the density fluctuations are obtained in the dissipative range ( $kl_0 \ll \gamma_0$), in the absence of propagating modes (all eigenvalues of the linear stability matrix are real).

The above expressions for the structure factors yields the following important conclusions:

• The shear structure factor at the largest wavelength slowly decays as $S_\perp(k,t) \sim \exp(-2\gamma_0\xi_\perp^2k^2\tau)$ and this represents vorticity diffusion on the scale $\tau$, with a diffusivity $\gamma_0 \xi_\perp^2=\eta/(\omega m n)$ and the typical length scales of vortices that grows like $L_v(t) \sim \sqrt{\tau}$, independently of the degree of inelasticity.

• The longitudinal (heat) structure factor at the largest wavelength follows a similar law, $S_\parallel(k,t) \sim \exp(-2\gamma_0\xi_\parallel^2k^2\tau)$, so that its diffusivity $\gamma_0\xi_\parallel^2$ is much larger than that of vorticity; the correlation length grows as $L_\parallel(t) \sim \xi_\parallel\sqrt{\gamma_0 \tau} \sim \sqrt{\tau/\epsilon}$ (it depends on inelasticity).

• The structure factor relative to density fluctuations reveals that the density instability (clustering) is driven through a coupling to the unstable heat mode, which is, in the dissipative range, a pure longitudinal velocity mode. This coupling is rather weak, which explains why structures in the flow field appear long before density clusters appear; the initial growth of the typical cluster length scale is similar to that of the longitudinal fluctuations, i.e. $L_{cl} \sim L_\parallel \sim \sqrt{\tau/\epsilon}$.

Incompressible and compressible flows

An important assumption can be used to simplify the study of structures in cooling granular gases, that of incompressibility. This is a fairly realistic assumption for what concerns ordinary elastic fluids, so we can consider it applicable also to granular gases for small inelasticity.

In terms of hydrodynamics fields, it states that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$, and in terms of fluctuations modes it implies that $u_\parallel(\mathbf{k},t) \simeq 0$. The consequence of this is that $S_\parallel \equiv 0$, that the density structure factor $S_{nn}$ does not evolve in time and that the temperature fluctuations $\delta T(\mathbf{k},t)$ simply decay as kinetic modes, staying spatially homogeneous. With this assumption is also easier to calculate the spatial correlation functions $G(\mathbf{r},t)$. It has been found that these correlation functions exhibits power law tails $G_\parallel (r,t) \sim G_\perp (r,t) \sim r^{-d}$ [212].

Moreover there is a difference between the two correlation functions: $G_\perp$ has a negative minimum in correspondence with $L_v(t)$ (this can be identified as the mean diameter of vortices), while $G_\parallel$ is always positive.

If the incompressibility assumption is relaxed (this is necessary at high inelasticity), the behavior of the correlation functions at large $r$ changes. It happens that an exponential cut-off superimposes over the power law tail $r^{-d}$, at a typical distance $\xi_\parallel$. The power law behavior becomes an intermediate behavior, observable if $\xi_\parallel$ and $\xi_\perp$ are well separated. In the compressible regime [213], the correlation function of the density fluctuations shows a typical length scale $L_{cl} \sim \sqrt{\tau/\epsilon}$ which agrees with the prediction of the structure factor.