Relations of POD modes and Lyapunov exponents to the nonlinear dynamic states in flow over oscillating tandem cylinders

Nonlinear dynamic states of flow field are analyzed using the Proper Orthogonal Decomposition (POD) to extract spatial coherent structures, and the temporal development of these spatial structures is further investigated by looking into the largestLyapunov exponents (LLEs) of the temporal coefficients of the POD modes. Additionally, the LLE of vorticity history at a specific point in the wake is also calculated to compare with the LLEs of the modes in the wake region. The flow field used for the analysis is flow around a two-tandem cylinder system with the downstream cylinder oscillating transversely. The numerical simulation is based on an immersed-boundary method for flow with the Reynolds number 100. Three states of typical nonlinear responses of this system are analyzed: lockin, transitional, and quasiperiodic. The results reveal that the paired POD modes can appear in the energetic modes, although this pattern fades in the transitional state. All the LLEs of the energetic modes in the lock-in state are zero, which means that the energetic modes in the lock-in state are stable. For the transitional state, after a long period of time, the LLEs of the first two energetic modes are close to zero, while the others are positive, showing that only the first two modes are relatively stable and that weak chaotic motions exist in the other modes. The LLEs of all the energetic modes in the quasiperiodic state are no larger than zero after a long period of time, resulting in asymptotical stableness of the energetic modes

Time-domain simulation of ultrasound propagation with fractional Laplacians for lossy-medium biological tissues with complicated geometries

Simulations of ultrasound wave propagation inside biological tissues have a wide range of practical applications. In previous studies, wave propagation equations in lossy biological media are solved either with convolutions, which consume a large amount of memory, or with pseudo-spectral methods, which cannot handle complicated geometries effectively. The approach described in the paper employed a fractional central difference method (FCD), combined with the immersed boundary (IB) method for the finite-difference, time-domain simulation. The FCD method can solve the fractional Laplace terms in Chen and Holm’s lossy-medium equations directly in the physical domain without integral transforms. It also works naturally with the IB method, which enables a simple Cartesian-type grid mesh to be used to solve problems with complicated geometries. The numerical results agree very well with the analytical solutions for frequency power-law attenuation lossy mediaThis research is partly supported by the U.S. Army under a cooperative Agreement No. W911NF-14-2-007