https://jupiter.math.nctu.edu.tw/~mclai/papers/Lai_1.pdf

An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity

Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the «consistent mass matrix» has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full «velocity-pressure coupling» is also inherent. We re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized

On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory

Vortex identification: New requirements and limitations

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/27355/0000380.pdf;sequence=1

Numerical simulations of the Rayleigh-Taylor instability

A vortex is intuitively recognized as the rotational/swirling motion of the fluids. However, an unambiguous and universally-accepted definition for vortex is yet to be achieved in the field of fluid mechanics, which is probably one of the major obstacles causing considerable confusions and misunderstandings in turbulence research. In our previous work, a new vector quantity which is called vortex vector was proposed to accurately describe the local fluid rotation and clearly display vortical structures. In this paper, the definition of the vortex vector, named Rortex here, is revisited from the mathematical perspective. The existence of the rotational axis is proved through real Schur decomposition. Based on real Schur decomposition, a fast algorithm for calculating Rortex is also presented. In addition, new vorticity tensor and vector decompositions are introduced: the vorticity tensor is decomposed to a rigidly rotational part and an anti-symmetric deformation part, and the vorticity vector is decomposed to a rigidly rotational vector and a non-rotational vector. Several cases, including 2D Couette flow, 2D rigid rotational flow and 3D boundary layer transition on a flat plate, are studied to demonstrate the justification of the definition of Rortex. It can be observed that Rortex identifies both the precise swirling strength and the rotational axis, and thus it can reasonably represent the local fluid rotation and provide a new powerful tool for vortex dynamics and turbulence research.

H. Aref

Papers

Vortex flow in nature and technology