非定常黏性涡中的精确物质环量面
A Zero-Flux Material Cylinder in the Unsteady Kiknadze--Krasnov Vortex
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中文总结 AI 辅助
本文为非定常黏性涡推导出精确物质环量面，该面兼具拉格朗日径向划分、峰值涡量位置和零传递环量边界，可作为数值误差评估基准。
中文摘要 AI 辅助
针对含源的非定常Kiknadze--Krasnov涡，推导出了一个精确的物质环量面。对于s>1的单一分布模态，缩放半径x=βr²满足D x/D t=4νβ[(s-1)-x]，因此x=s-1定义了一个物质圆柱面，适用于任何容许的时间相关应变。该圆柱面同时包含最大分布涡量。其包围的环量ΓP(s,s-1)是常数，因为黏性环量传递2πνr ∂_rω_z在该处恒为零。因此，一个运动表面同时提供了拉格朗日径向划分、峰值涡量位置和零传递环量边界。已知的稳态源-应变驻点圆柱面作为极限情况被恢复。完整的速度和压力场、容许参数范围、多模态限制以及径向尺度动力学也被推导出来，并直接从不可压缩Navier--Stokes方程进行了验证。该表面可作为精确基准，用于评估在给定应变下物质表面追踪和环量输运的数值误差。
英文摘要
In viscous flow, the circulation enclosed by a deterministic material loop generally changes through vorticity diffusion. It is shown that the source-bearing unsteady Kiknadze--Krasnov vortex possesses a distinguished cylindrical material surface on which this viscous circulation transfer vanishes identically. For a single distributed mode, the scaled squared radius $x=β(t)r^2$ satisfies \[ \frac{\mathrm D x}{\mathrm D t} = 4νβ(t)\big[(s-1)-x\big], \] where $β(t)$ is the inverse squared radial scale, $ν$ is the kinematic viscosity, and $s$ is the incomplete-gamma shape parameter. For $s>1$, the cylinder $x=s-1$ is transported exactly by the radial flow and preserves the inside/outside ordering of fluid trajectories. The same factor $(s-1)-x$ governs the radial vorticity gradient and the material circulation-transfer rate. Consequently, the magnitude of the distributed axial vorticity $|ω_z|$ attains its annular maximum on this material cylinder, $\partial_rω_z=0$ there, and the enclosed circulation remains exactly $ΓP(s,s-1)$ for every admissible prescribed strain history. Neighbouring material cylinders continue to exchange circulation viscously, while the distinguished moving cylinder carries a fixed fraction of the distributed circulation throughout the admissible evolution. In the steady source--strain limit, the surface reduces to the radial stagnation cylinder. The complete velocity and pressure fields, exact radial-scale evolution, admissible parameter ranges, and qualifications for multiple radial scales are also derived. The resulting solution-specific material-circulation invariant provides an exact benchmark for numerical material-surface tracking, vortex-core evolution, and viscous circulation transport, with the principal analytical results additionally supported by machine-checked formal verification.
发表机构
- Arab Academy for Science, Technology and Maritime Transport（阿拉伯科学、技术和海运学院）
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