A Shape Hessian-Based Boundary Roughness Analysis of Navier-Stokes Flow
A Shape Hessian-Based Boundary Roughness Analysis of Navier-Stokes Flow
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基于形状Hessian的纳维-斯托克斯流边界粗糙度分析
DOI：
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发表时间：
2011
影响因子：
1.9
通讯作者：
O. Ghattas
中科院分区：
文献类型：
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作者：
Shan Yang;G. Stadler;R. Moser;O. Ghattas
从偏微分方程系统最优控制理论出发，利用形态微积分分析了边界粗糙度对粘性流体耗散率的影响。为了研究从表面粗糙度形貌到Navier-Stokes流耗散率的映射$mathscr{D}$，采用速度法确定了形状梯度和Hessian的表达式。对于Couette和Poiseuille流动，平坦边界是耗散率泛函的局部极小值。因此，对于小的粗糙度高度，$mathscr{D}$的行为由平壁形状Hessian算子控制，其特征函数显示为傅里叶模态。对于Stokes流，用解析方法确定了形状的Hessian，其特征值随形状扰动的波数线性增长。对于Navier-Stokes流，数值计算了Hessian形状，其特征值与Stokes流的特征值之比仅依赖于Re…
The influence of boundary roughness characteristics on the rate of dissipation in a viscous fluid is analyzed using shape calculus from the theory of optimal control of systems governed by partial differential equations. To study the mapping $mathscr{D}$ from surface roughness topography to the dissipation rate of a Navier–Stokes flow, expressions for the shape gradient and Hessian are determined using the velocity method. In the case of Couette and Poiseuille flows, a flat boundary is a local minimum of the dissipation rate functional. Thus, for small roughness heights the behavior of $mathscr{D}$ is governed by the flat-wall shape Hessian operator, whose eigenfunctions are shown to be the Fourier modes. For Stokes flow, the shape Hessian is determined analytically and its eigenvalues are shown to grow linearly with the wavenumber of the shape perturbation. For Navier–Stokes flow, the shape Hessian is computed numerically, and the ratio of its eigenvalues to those of a Stokes flow depend only on the Re...