ON PRESSURE BOUNDARY-CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
ON PRESSURE BOUNDARY-CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
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DOI：
10.1002/fld.1650071008
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发表时间：
1987-10-01
影响因子：
1.8
通讯作者：
SANI, RL
中科院分区：
文献类型：
--
作者：
GRESHO, PM;SANI, RL
在不可压缩流动中，压力是一个有点神秘的量。它不是热力学变量，因为不可压缩流体没有“状态方程”。从某种意义上说，它是一种数学制品——限制速度场保持无散度的拉格朗日乘子；即不可压缩，但它的梯度是一个相关的物理量：每单位体积的力。它以无限的速度传播，以保持流动始终无处不在不可压缩；即，它总是与时变无散度速度场平衡。计算通常也很困难和/或昂贵。虽然压力是由描述质量和动量守恒的控制方程完美定义的（至少达到任意加性常数），但（具有讽刺意味的是）当用泊松方程更直接地表达时，压力就不太好定义了，泊松方程既可以从原始守恒方程导出，也可以用来（或误用）来代替质量守恒方程。这是因为在后一种形式中，还需要直接解决压力边界条件的主题，其正确的规范（在许多方面）至关重要，并构成了这项工作的基础。在此，我们证明，质量和动量守恒的相同原理与连续性论证相结合，得出压力泊松方程的正确边界条件：即，通过在边界处应用动量方程的法向分量简单地导出诺伊曼条件。通常，但并不那么重要，切向动量方程在边界处也满足。
The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence‐free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is alwaysin equilibriumwith a time‐varying divergence‐free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well‐defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.