非梯度对流下n维Navier-Stokes解的迭代构造
Iterative Construction of n-Dimensional Navier-Stokes Solutions with Non-Gradient Convection
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中文总结 AI 辅助
通过迭代求解线性扩散问题，构造任意维度下非梯度对流的Navier-Stokes精确周期解，在三维Re=10时收敛，Re=30时发散，并给出未吸收对流场的比例。
中文摘要 AI 辅助
当对流场是纯梯度并可被吸收到压力中时，不可压缩Navier-Stokes方程的精确周期解会出现。我们通过迭代在任意空间维度构造此类之外的解，其中每一步求解一个由前一对流场的横向部分驱动的线性扩散问题。在三维情形下，解可显式获得：在Re=10时，连续迭代以0.57的因子收缩，极限满足相对缺陷为1.0 x 10^-3的温和方程。到Re=30时收缩因子超过1，划定了收敛范围。对于初始场的n维推广，未被压力吸收的对流场部分对每个n>=3均为2sqrt(2)/3。散度和胞动能提供了简单诊断，可在失败发生的步骤处检测到失败。
英文摘要
Exact periodic solutions of the incompressible Navier-Stokes equations arise when the convective field is a pure gradient and can be absorbed into the pressure. We construct solutions outside this class, in arbitrary spatial dimension, by an iteration in which each step solves a linear diffusion problem forced by the transverse part of the preceding convective field. The construction is dimension-independent, and for a cyclic family of initial fields the first two iterates are available in closed form. Carried out numerically in three dimensions, the iteration converges: at Re=10 successive iterates contract with an observed factor 0.57 and the limit satisfies the mild equation with relative defect 1.0 x 10^-3. The observed factor exceeds unity by Re=30, which identifies the practical range of the iteration rather than a proved convergence boundary. We also show what goes wrong when the projection is omitted, as it was in an earlier iteration of the author's: the iterates cease to be divergence-free and an apparent growth appears that belongs to the expansion and not to the flow. For an n-dimensional generalisation of the initial field the fraction of the convective field not absorbed by pressure is 2sqrt(2)/3, independently of n over the range examined. Divergence and cell kinetic energy provide simple diagnostics that detect failure at the step where it occurs.
发表机构
- Schlumberger Cambridge Research（施伦伯格剑桥研究）
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