W. W. Zhang

X. G. Zhu, Y. F. Nie, W. W. Zhang

This article studies a direct numerical approach for fractional advection-diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order $(0,1]$. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge-Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank-Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the concrete simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the existing algorithms in the open literature, numerical results finally illustrate the validity and accuracy.

Z. Y. Wang, W. W. Zhang

In recent years, machine learning methods represented by deep neural networks (DNN) have been a new paradigm of turbulence modeling. However, in the scenario of high Reynolds numbers, there are still some bottlenecks, including the lack of high-fidelity data and the convergence and stability problem in the coupling process of turbulence models and the RANS solvers. In this paper, we propose an improved ensemble kalman inversion method as a unified approach of data assimilation and turbulence modeling for separated flows at high Reynolds numbers. The trainable parameters of the DNN are optimized according to the given experimental surface pressure coefficients in the framework of mutual coupling between the RANS equations and DNN eddy-viscosity models. In this way, data assimilation and model training are combined into one step to get the high-fidelity turbulence models agree well with experiments efficiently. The effectiveness of the method is verified by cases of separated flows around airfoils(S809) at high Reynolds numbers. The results show that through joint assimilation of vary few experimental states, we can get turbulence models generalizing well to both attached and separated flows at different angles of attack. The errors of lift coefficients at high angles of attack are significantly reduced by more than three times compared with the traditional SA model. The models obtained also perform well in stability and robustness.