The projection method for the incompressible Navier-Stokes equations: The pressure near a no-slip wall
The projection method for the incompressible Navier-Stokes equations: The pressure near a no-slip wall
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DOI：
10.1016/j.jcp.2014.01.035
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发表时间：
2014-04
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通讯作者：
A. W. Vreman
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文献类型：
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作者：
A. W. Vreman
分析了带无滑移壁面的不可压缩Navier-Stokes方程的显式交错投影方法，并将其用于模拟，以解决当连续性方程被标准压力泊松方程（PPE）取代时，与压力边界条件相关的几个问题。首先，它表明，PPE系统补充与Neumann压力边界条件导出的动量方程可以与Navier-Stokes方程一致，如果它是扩展的要求，在近壁处，u= 0是零，解决方案是足够光滑的。这意味着，它是可能的，制定一个边界条件的标准PPE，而不必诉诸绿色的功能，这是有趣的理论原因。其次，证明了交错投影法与上述PPE系统的交错离散化是等价的。等价性的推导揭示了所谓的PPE悖论，并导致近似的壁值的dupp/dupn，这是不需要的，但隐含的交错投影方法。第三，通过采用交错投影法对槽道湍流进行直接数值模拟，对Navier-Stokes方程解的（近壁）规律性进行了数值分析。通过对近壁区动量方程所有项的数值检验，可以得出结论：对于t> 0，动量方程的三个分量在壁面上都满足（对于短时间，但也在湍流区）。在极限t→ 0，压力梯度在L2范数下收敛到初始压力梯度，这证实了文献中一个有争议的理论结果.即使在最大范数下，压力梯度也会收敛到初始压力梯度。在模拟中观察到的唯一不连续性是切向粘性项的不连续性和t= 0时壁上切向速度的时间导数。因此，数值结果表明，槽道湍流解的规律性比现有理论所声称的更强。
An explicit staggered projection method for the incompressible Navier–Stokes equations with no-slip walls is analyzed and used in simulations to address several issues related to the pressure boundary condition required when the continuity equation is replaced by the standard pressure Poisson equation (PPE),∇ 2 p=∇⋅(− u⋅∇ u+ f). First, it is shown that a PPE system supplemented with a Neumann pressure boundary condition derived from the momentum equation can be made consistent with the Navier–Stokes equations if it is extended with the requirement that∇⋅∇ 2 u= 0 is zero near the wall and the solution is sufficiently smooth. This implies that it is possible to formulate a boundary condition for the standard PPE without the necessity to resort to Green's functions, which is interesting for theoretical reasons. Second, the equivalence is shown between the staggered projection method and the staggered discretization of above PPE system. The derivation of the equivalence sheds light upon the so-called PPE paradox and leads to an approximation of the wall value of∂ p/∂ n, which is not required but implied by the staggered projection method. Third, the (near-wall) regularity of a solution of the Navier–Stokes equations is numerically analyzed by means of Direct Numerical Simulation of turbulent channel flow performed with the staggered projection method. From the numerical inspection of all terms of the momentum equation in the near-wall region, it is concluded that the three components of the momentum equation are satisfied on the wall for t> 0 (for short times, but also in the turbulent regime). In the limit t→ 0, the pressure gradient is observed to converge to the initial pressure gradient in the L 2-norm, which confirms a disputed theoretical result in literature. Even in the maximum norm, the pressure gradient appears to converge to the initial pressure gradient. The only discontinuities observed in the simulations are the discontinuities of the tangential viscous terms and the time derivatives of the tangential velocities on the wall at t= 0. Thus the numerical results indicate that the regularity of the solution for turbulent channel flow is stronger than claimed by existing theory.