Theory and applications of the vortex-surface field

We review the progress on the theory and applications of the vortex-surface field (VSF). The VSF provides a systematic Lagrangian-based framework for the identification, characterization, and modeling of flow structures. From the theoretical perspective, the vorticity is simplified to a scalar field as the VSF. The VSF isosurface is a vortex surface consisting of vortex lines. By introducing the virtual circulation-preserving velocity, the Helmholtz theorem and the Ertel theorem are extended to some non-ideal flows, so that the VSF evolution equation can be expressed in a Lagrangian conservation form. Thus the VSF isosurfaces of the same threshold at different times have strong coherence, facilitating the tracking of vortex surfaces. As a general flow diagnostic tool, the numerical VSF solution can be constructed in arbitrary flow fields by solving a pseudo-transport equation driven by the frozen, instantaneous vorticity. In addition, the local optimization method and the boundary-constraint method can further improve the smoothness and convergence of VSF solutions. Then the two-time method is developed for calculating the temporal evolution of VSFs. From post-processing of large-scale database of numerical simulations, the VSF elucidates mechanisms in the flows with essential vortex dynamics, such as turbulence and transition. For example, the VSF reveals the complex network of tangling vortex tubes in isotropic turbulence, consistent with the vorticity equation and dynamics. In addition, Eulerian vortex-identification criteria cannot identify complete vortex tubes, so the visual "breakdown" of worm-like structures were often reported in the literature. The universal framework of the VSF evolution illustrates an ordered evolution process of coherence structures in transitional flows, in which it is no need to ad hoc change the structure-identification method and isocontour threshold. Moreover, the quantitative VSF study elucidates the interaction between the vortex surface and the shock wave, flame, or electromagnetic field in multi-physics coupled flows. Based on the characterization of VSFs, we seek robust statistical features, and then determine and model correlations between the features and critical quantities to be predicted in engineering applications. For example, we develop a predictive model of the skin-friction coefficient in boundary-layer transition based on the small-scale inclination angle in experimental images. Some open problems in the VSF study are discussed, including the regularization for the breakdown of virtual conservation theorems at vorticity nulls and the speedup of the numerical construction and evolution of VSFs.