# Stability of the flow between two exactly counter-rotating disks with large aspect ratio

In collaboration with Walker J.S (Urbana Champaign, USA), Foucault E, Pécheux J (LEA, Poitiers).

## Contents

### Object

Figure 1: Configuration and schematic representation of the baseflow.

The flow between two parallel rotating disks besides applications to turbomachinery can also be regarded as a class of fundamental flows. This assertion becomes self-evident as the aspect ratio $\scriptstyle \Gamma\,{\equiv}\, R/H$, (where $\scriptstyle R$ is the disk radius and $\scriptstyle 2 H$ is the inter-disk spacing) is very large. In such case, self-similar solutions valid for disks with infinite radius are a good approximation of the exact solutions of the realistic closed geometry (i.e. finite radius). The self-similar solution is valid in the core region (see figure 1) but restrictions apply in the end-region or when the Reynolds number is above a threshold for which the flow is prone to instability. The Reynolds number is here defined by $\scriptstyle {\rm Re} \,{\equiv}\, \Omega H^2/\nu$, $\scriptstyle \Omega$ being the magnitude of the angular velocity and $\scriptstyle \nu$ the kinematic viscosity. In two different studies [1] and [2], we focus our efforts on the particular case of exactly counter-rotating disks and address the question of the stability of such flows.

### Results

#### Axisymmetric and asymptotic analysis

Figure 2: Critical Reynolds number versus aspect ratio.

In a first study [1], we restricted ourselves to axisymmetric perturbations. In figure 2, the critical Reynolds number for the first bifurcation is plotted as a function of the aspect ratio. It is tempting but erroneous to think that the instability is a shear flow like instability. In such case, as the amplitude of the perturbation is large close to the end region (see figure 3), one would pick a scaling with $\scriptstyle H$ for length and $\scriptstyle v = \Omega R$ for velocity, where $\scriptstyle R = \Gamma H$ stands for the dimensional radius; the critical Reynolds number as defined in the previous section would have scaled as $\scriptstyle Re_c \propto \Gamma^{-1}$. Obviously, this does not fit with the full domain numerical results in figure 2 for which a scaling law is $\scriptstyle {\rm Re_c} \propto \Gamma^{-1/2} \,$. Another explanation has to be found.

 Axisymmetric full domain computation Local asymptotic parallel flow approximation Figure 3: Perturbation of the streamfunction (meridional flow) and azimuthal velocity for $Re=Re_c$. Figure 4: Perturbation of the streamfunction and azimuthal velocity for $Re=Re_c$ as in figure 3 but using the local asymptotic approach. (Note that the abscissa corresponds to one wavelength.)

In the asymptotic analysis for large $\Gamma$, the appropriate scalings lead to the Navier-Stokes equations in a Cartesian frame (i.e. all curvature terms vanish) with an additional centrifugal acceleration. In the core region and to leading order, the base flow is a parallel flow. This allows us to use a Fourier decomposition along the radial direction (see figure 4). The results of the linear stability analysis of this parallel flow are in good agreement with the full domain simulations not only qualitatively (see figure 3 and 4) but also quantitatively (see figure 2) which proves that the key ingredient of the instability is well captured in this simple model. The $\scriptstyle \Gamma^{-1/2}$ scaling and the centrifugal nature of the instability is now understood thanks to this asymptotic analysis. The physical mechanism invoking the coupling between the azimuthal velocity and the centrifugal acceleration is explained in [1].

#### Three-dimensional, fully nonlinear and experimental results

Figure 5: Critical Reynolds number versus aspect ratio.
In a second study [2], the bifurcations and the nonlinear dynamics are numerically and experimentally investigated. A linear stability analysis performed for $\scriptstyle 4 \le \Gamma \le 40$ shows that nonaxisymmetric and axisymmetric modes can be stationary or time dependent in this range. Three-dimensional modes are dominant for $\scriptstyle \Gamma \le 26.5$ while axisymmetric modes are critical for $\scriptstyle \Gamma \ge 26.5$ (see figure 5). The restriction to axisymmetry in the study [1] finds herein some justifications. In the particular case of $\scriptstyle \Gamma =30 \!$, nonlinear computations are performed at Reynolds numbers slightly above threshold and are compared to experimental results, showing the competition between axisymmetric and three-dimensional modes. For such aspect ratio, the linear stability analysis predicts $Re_c=5.19$ for $m=0$ and $Re_c = 5.25$ for $m=12$. For $Re_c=5.75$, $m=0, 1 \$ and $\scriptstyle 9 \le m \le 16$ are all unstable. In figure 6, time evolution of dominant modal energies shows that $m=0$ becomes dominant at $\scriptstyle t \approx 130$ and, for three-dimensional modes, only $m=10$ and its harmonics (not represented) remain and contain $\scriptstyle 10 \%$ of the total fluctuating energy. This result could not be predicted by the linear stability analysis as growth rates for $m=0$ and $10$ are smaller than the one for $m=13$.

 Figure 6: Long time evolution of fluctuating energies $E/E_b$ normalized by the basic state energy at $\Gamma = 30$ and $Re = 5.75$. The $m=13$ mode is dominant at short times but nonlinear simulation reveals that $m=13$ eventually decays, while the $m=0$ and $m=10$ (and its harmonics) modes impose the dynamics at long time. Figure 7: Fully nonlinear simulation at $\Gamma=30$ and Re = 5.75. Strong axisymmetric and weak three-dimensional modes are present. Figure 8: Experiment realized in Poitiers for $\Gamma=30$ and Re=6.5. The view is from above i.e. $(r,\theta)$ plane. Only part of the domain is seen on the picture.

### Conclusions and further work

Different tools have been used to study the stability of the flow between two exactly counter-rotating disks. An asymptotic analysis allowed us to find the mechanism of the dominant axisymmetric instability for large aspect ratio which takes place in the core region and gets stronger as it moves toward the periphery. The linear analysis showed that three-dimensional modes exist and nonlinear computations revealed that their amplitude stays relatively weak with patterns confined very close to the periphery of the disks as shown in figure 7. All these results and observations are in qualitative agreement with an experiment performed in Laboratoire d’Etudes Aérodynamiques de Poitiers (see figure 8). The three-dimensional mode selection in the nonlinear regime is still an open question. The case where the z-symmetry is lost, e.g. imposing different angular rotation rate of the disks, will be an interesting extension of the present work.