In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field \(\underline{f}(x,y)\). We denote this random field by \(\delta \underline{f}(x,y)\) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters \(\underline{\theta }\) (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that \(\underline{\theta }\) should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

在本文中，我们考虑了一种将能量最小化技术应用于2D流体图像的版本。两个Navier–Stokes方程根据速度场（u，  v），压力场p和强迫场f来描述2D流体的静态流动。除了这两个Navier–Stokes方程之外，我们还有不可压缩条件来评估这三个参数。在实施此系统时，随机噪声（通常是非高斯噪声）会爬入随机力场\（\ underline {f}（x，y）\）中。我们用\（\ delta \ underline {f}（x，y）\）表示此随机字段具有零均值和非平凡的第二和第三矩。我们假设这两个时刻是已知的，除了一些我们希望估计的未知参数\（\下划线{\ theta} \）（例如均值，方差，协方差，偏度等）。在提出的技术中，我们首先计算平均流体能量的近似位移，该平均能量被定义为速度场的二次函数。然后，能量方法要求\（\下划线{\ theta} \）应该使得由于随机强迫分量而导致的能量平均增加最小。但是，我们应该注意，力场估计的标准统计方法是将速度场作为力场的函数进行计算，然后采用统计矩匹配技术。这种方法假定速度场的空间遍历性。从统计矩匹配的观点来看，这种用于力场估计的方法更为准确，但仅在进行速度测量时才有效。能量最小化的前一种技术不需要任何速度测量。本文讨论了这两种技术，并提供了MATLAB仿真。