This paper studies circular formation control of multi-vehicle systems, modeled with unicycle dynamics, in a uniform flowfield with different temporal phase arrangements, while assuring that the trajectories of the vehicles remains bounded within a region. Using the idea of Barrier Lyapunov function in conjunction with temporal phase potential functions, we derive control laws that achieves our objectives, under a mild assumption on the initial states of the vehicles. We further obtain bounds on the various signals in the post-design analysis and show that these depends on the initial conditions and controller gains. Simulations are provided to illustrate the theoretical findings. © 2019 IEEE.