A new arbitrary Lagrangian-Eulerian (ALE) formulation for Navier-Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterization that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier-Stokes equations, and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.
Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations
将Del Rey Fernandez et al.（2019）的双曲守恒律的熵守恒、曲线、双曲、p-精化算法从可压缩Euler方程推广到可压缩Navier-斯托克斯方程。采用了一种简单而灵活的耦合方法，相邻单元之间采用平面插值算子。通过最小化过程数值近似了曲率体积度量项，并满足离散几何守恒律条件。不同的曲线表面度量用于相邻的接口，以构建接口耦合项，从而本地化的离散几何守恒律约束到每个单独的元素。所得到的方案是熵保守/稳定的，元素保守的，和自由度保持。粘性界面耗散算子的开发，保持了熵的稳定性的基础计划。在粘性激波问题、雷诺数Re= 1，600时的泰勒-格林涡问题和Re = 2，000时的亚音速湍流流过球体的情况下，所得到的数值格式的精度和稳定性与原来的协调格式（达到~p+1收敛）相当。
Lattice Boltzmann Methods for Navier-Stokes Equations in General Orthogonal Coordinates for Efficient Flow Simulations using Nonuniform Clustered Grids
Entropy Stable h/p-Nonconforming Discretization with the Summation-by-Parts Property for the Compressible Euler and Navier-Stokes Equations
本文将Del Rey Fernandez等人在曲线网格上求解可压缩Euler和Navier-斯托克斯方程的熵守恒/稳定算法推广到h/p加密/粗化。开发并行计算算法的主要困难是在并行计算接口之间建立适当的耦合过程。在这里，一个计算简单和有效的方法，使用解耦插值算子的基础上利用。所得到的方案是熵保守/稳定和元素保守的。等熵涡和粘性激波传播的数值模拟证实了该方法的熵守恒/稳定性和精度特性（达到~ p + 1收敛），与原始一致格式[4，35]的性能相当。对Re = 1，600时的泰勒-格林涡和Re = 2，000时通过球体的湍流的模拟显示了非结构网格的整体空间离散的鲁棒性和稳定性。最后，为了证明全离散显式熵稳定算法的熵守恒性质，我们采用松弛Runge-Kutta格式模拟等熵涡的传播，给出了该算法的熵函数随时间的演化.
