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Proper orthogonal decomposition and incompressible flow: An application to particle modeling

机译：正确的正交分解和不可压缩流：在粒子建模中的应用

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The proper orthogonal decomposition (POD) is a reduced-order modeling technique that is used to compactly represent unsteady flows. In this paper, we use the POD to capture the parametric variation of a flow with Reynolds number. We study incompressible, axisymmetric, steady flow over spherical particles at various Reynolds numbers in order to give an alternative to correlation-based approaches for predicting the drag on a sphere. In most previous applications of the POD for reduced-order modeling of incompressible flow, the POD modes typically have only described the velocity field; the pressure field was not directly modeled. Since we are interested in drag, which is dependent on the pressure, we formulate the method to directly include the pressure field of an incompressible flow. The POD modes are then derived from numerical flow solutions obtained using an hp-finite element method. A reduced-order model is created by performing a streamwise-upwind-Petrov-Galerkin (SUPG) projection of the incompressible Navier-Stokes equations onto the space spanned by the POD modes. The SUPG approach is taken because when pressure modes are included the Galerkin method fails to give unique solutions for incompressible flow. This is demonstrated for some simple test cases. An efficient numerical implementation is also developed using a Taylor expansion of the SUPG projection of the Navier-Stokes equations. Finally, values of drag are computed from the reduced-order model. Drag can be calculated to within 1.0% of the direct numerical simulations using only a small number of modes while still retaining all of the essential physics around the particle.
机译：适当的正交分解（POD）是一种降阶建模​​技术，用于紧凑地表示非恒定流。在本文中，我们使用POD捕获具有雷诺数的流的参数变化。我们研究了各种雷诺数下球形粒子上的不可压缩，轴对称，稳定的流动，以便为预测基于球的阻力的基于相关性的方法提供替代方法。在POD的大多数先前应用中，对不可压缩流进行降阶建模时，POD模式通常仅描述速度场。压力场没有直接建模。由于我们对取决于压力的阻力感兴趣，因此我们制定了直接包含不可压缩流的压力场的方法。然后从使用hp有限元方法获得的数值流解中得出POD模式。通过执行不可压缩的Navier-Stokes方程的流向上风-Petrov-Galerkin（SUPG）投影到POD模式跨越的空间，可以创建降阶模型。之所以采用SUPG方法，是因为当包含压力模式时，Galerkin方法无法为不可压缩的流动提供独特的解决方案。一些简单的测试案例就证明了这一点。还使用Navier-Stokes方程的SUPG投影的泰勒展开来开发有效的数值实现。最后，从降阶模型计算出阻力值。仅使用少量模式，就可以将阻力计算为直接数值模拟的1.0％以内，同时仍保留了粒子周围的所有基本物理原理。

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