Bistabilities in two parallel Kármán wakes

Abstract

Bistabilities of two equilibrium states discovered in the coupled side-by-side Kármán wakes are investigated through Floquet analysis and direct numerical simulation (DNS) with different initial conditions over a range of gap-to-diameter ratio ( $g^*= 0.2\text {--}3.5$ ) and Reynolds number ( $Re = 47\text {--}100$ ). Two bistabilities are found in the transitional $g^*-Re$ regions from in-phase (IP) to anti-phase (AP) vortex shedding states. By initialising the flow in DNS with zero initial conditions, the flow in the first bistable region (i.e. bistable IP/FF $_C$ at $g^*= 1.4 \text {--} 2.0$ , where FF $_C$ denotes the conditional flip-flop flow) attains flip-flop (FF) flow, it settles into the IP state by initialising the flow with an IP flow. The second bistability is observed between cylinder-scale IP and AP states at large $g^*$ ( $=$ 2.0–3.5). The transition from the FF $_C$ to IP is dependent on initial conditions and irreversible over the parameter space, meaning that the flow cannot revert back to the FF $_C$ state once it jumps to the IP state irrespective of the direction of $Re$ variations. Its counterpart for the bistable IP/AP state is reversible. We also found that the FF $_C$ flow in the first bistable region is primarily bifurcated from synchronised AP with cluster-scale features, possibly because the cluster-scale AP flow is inherently unstable to FF flow instabilities. It is demonstrated that the irreversible bistability exists in other interacting wakes around multiple cylinders. A good understanding of flow bistabilities is pivotal to flow control applications and the interpretation of desynchronised flow features observed at high $Re$ values.

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