A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

Determination of epsilon for Omega vortex identification method

In the present paper, epsilon（ε） in the Omega vortex identification criterion（ Ω method） is defined as an explicit function in order to apply the Ω method to different cases and even different time steps for the unsteady cases. In our method, e is defined as a function relating with the flow without any subjective adjustment on its coefficient. The newly proposed criteria for the determination of ε is tested in several typical flow cases and is proved to be effective in the current work. The test cases given in the present paper include boundary layer transition, shock wave and boundary layer interaction, and channel flow with different Reynolds numbers.

POD analysis on vortical structures in MVG wake by Liutex core line identification

The Liutex core line method, first combined with the snapshot proper orthogonal decomposition (POD), is utilized in a supersonic micro-vortex generator (MVG) wake flow at Ma = 2.5 and Reθ = 5 760 to reveal the physical significance of each POD mode of the flow field. Compared with other scalar-based vortex identification methods, the Liutex core line identification is verified to be the most appropriate approach that is threshold-free and provides full information of a fluid rotation motion. Meanwhile, the Liutex integration is employed to quantitatively track the evolution of the vortices in MVG wake and is applied to the determination of the effective control section of the MVG wake for the optimization study of MVG design. The physical mechanism of each POD mode for multi-scale and multi-frequency vortical structures is investigated by using Liutex core line identification to give some revelations. For the mean mode (mode 0) indicating the time-averaged velocity flowfield of the MVG wake flow, a pair of primary counter-rotating streamwise vortices and another pair of secondary vortices is uniquely identified by two pairs of Liutex core lines with Liutex magnitude. In contrast, mode 1 is featured by a fluctuated roll-up motion of streamwise vortex, and the streamwise component of the MVG wake is demonstrated to be dominant in terms of the total kinetic energy contribution. Meanwhile, a dominant shedding frequency of St = 0.072 is detected from the temporal behavior of mode 2, which has the organized arc-shaped vortex structures shedding from MVG induced by the K-H instability. Additionally, mode 4 subjects to low-frequency oscillations of the wall vortices and thus takes a relatively lower frequency of St = 0.044.