# Bacon, a Lightweight Deep Reinforcement Learning Benchmark Library for Flow Control
## 3 Environments
### Vortex  
#### 3.7.1 Physics  
We consider the resolution of a dynamical system modeling the non-linear, vortex-induced vibrations of a rigid circular cylinder. The flow motion is governed by the incompressible Navier-Stokes equations, whereas the cylinder motion is a simple translation governed by a linear mass-damper-spring equation affected by the fluid loading. This is modeled by coupled amplitude equations derived in [11; 35] after dominant balance arguments as follows:  
\[\begin{split}\partial_{t}A&=\lambda\bigg{(}\frac{1} {\text{Re}_{*}}-\frac{1}{\text{Re}}\bigg{)}A-\mu A|A|^{2}+\alpha Y\\ \partial_{t}Y&=(-\omega_{*}\gamma+\text{i}(\omega_{ *}-\omega_{*}))Y+\frac{\beta}{\omega_{*}m}A\end{split} \tag{28}\]  
where \(A\) and \(Y\) are unknown slow, time-varying, complex amplitudes modeling, respectively, the flow disturbances and the cylinder center of mass, \(\text{Re}=50\) is the Reynolds number, \(\text{Re}_{*}=46.6\) is the threshold of instability of the steady cylinder, \(\omega_{*}=0.74\) is the frequency of the marginally stable eigenmode classically computed from the flow past a fixed cylinder [36], \(\omega=1.1\) is the dimensionless natural frequency of the cylinder in vacuum, \(\gamma=0.023\) is the structural damping coefficient, and \(m=10\) is the ratio of the solid to the fluid densities. The coefficients \(\lambda\), \(\mu\), \(\alpha\), and \(\beta\) in (30) are analytically computable from an asymptotic analysis of the coupled flow-cylinder system, their numerical value being taken from [35] as:  
\[\lambda=9.153+3.239\text{i},\quad\mu=308.9-1025\text{i},\quad\alpha=0.03492+0.01472\text{i},\quad\beta=1. \tag{29}\]  
The ability of the model to reproduce the physics of vortex-induced vibrations has been assessed from the study of the non-linear limit cycles, i.e., the periodic, synchronized orbits reached by the system in large periods of time whose analytical expressions are reported in [35]. Of particular importance is the simultaneous existence of multiple stable cycles (either a single limit cycle or three over specific ranges of frequencies), which is shown to trigger a complex hysteretic behavior in the lock-in regime. Since all limit cycle solutions are periodic, the mean mechanical energy averaged over a period is zero, and the mean work received from the fluctuating lift force is entirely dissipated by structural damping. As discussed in [35], it follows that the leading-order mean dissipated energy is a simple quadratic function of the displacement amplitude, meaning that only the upper limit cycle (the one limit cycle yielding the largest displacement amplitude) is of practical interest for the energy extraction problem, that is, in cases where one seeks to leverage such vortex-induced vibrations to generate electrical energy, for instance, by having the oscillation of the cylinder periodically displace a magnet within a coil.  
In essence, it can be inferred that there must exist an optimal structural parameter setting for which the dissipated energy is maximum. On the one hand, the flow-cylinder system must be synchronized for the cylinder displacement amplitude to be large. On the other hand, the energy tends to zero in the limit \(\gamma\gg 1\), where the work received from the lift force is limited by the low amplitude of the displacement, and in the limit \(\gamma\ll 1\), where the displacement is self-limited. As evidenced in [11], the problem shown is that the optimum lies at the edge of a discontinuity, corresponding to parameter settings where the system undergoes a transition from a hysteretic to a non-hysteretic regime. This has important consequences for the application since small inaccuracies in the structural parameters of small external flow disturbances may tip the system outside the hysteresis zone and lead to convergence to cycles of lower energy, resulting in a dramatic drop in the harnessed energy.