By: Zongxian Liang

Objective
Apply Proper Orthogonal Decomposition and Galerkin projection to flow field with moving airfoil to achieve reduced order modeling in the form of ordinary differential equation which can further invoke control theory into flow control of MAV design with high accuracy and efficiency.

Preliminary Results
The effect of pressure gradient is considered by two algorithms of POD of the pressure field. In the first algorithm the pressure POD is based on the form in which the full pressure is the sum of mean pressure and fluctuating pressure, similar to the expression of the POD of velocity field. In the second algorithm it doesn’t invoke the mean pressure. The decomposition of velocity and pressure into mean and fluctuation are commonly used in turbulence study in which the suitable average of fluctuating velocity and pressure is zero, i.e. . The POD of fluctuating velocity measures the turbulent kinetic energy while the inner product of fluctuating pressure doesn’t have the straight forward physical meaning. Thus, it is necessary to re-inspect the effectiveness of decomposition in realistic applications, especially for low Reynolds number flow.

A 2-D stationary membrane airfoil is staying in a flow field with free stream velocity U=1(Figure 1). The velocity boundary on Г1, Г3 and Г4 are of Dirichlet boundary condition. That on Г2 is of homogeneous Neumann boundary condition. The pressure on four boundaries is of homogeneous Neumann boundary. The Reynolds number is 200. The angle of attack of airfoil is 30 degree. The snapshots are taken after the vortex shedding in the flow field reaches steady state and the vortex street in the wake is in periodic pattern (60 frames per period).
The eigenvalue spectrum from POD is shown in Figure 3. Fourteen modes and thirteen modes are achieved at accuracy threshold of 10-6 from Algorithm 1 (A1 for short) and Algorithm 2 (A2), respectively. Despite of the first mode from A2, the rest 12 modes in the curve of normalized eigen values follows a very simliar trend which appears in the 14 modes of A1. Moreover, by comparing these two curves it can be found that the eigenvalues from A2 go to zero faster than that from A1. In other words, we can use less modes to represent the original pressure fields. For better understanding the similarity and difference between Algorithm 1 amd 2, the mean pressure (A1 only) and several pressure mode contours are illustrated in Figure 4. Although not exactly same, the contour of mean pressure from A1 is pretty close to the first mode from A2. The position of local extremum of mode 1 from A1 are close to that of mode 2 from A2 but the signs are switched. Mode 2 of A1 and mode 3 of A2 are analogical to the prior modes, so on so forth.
For this vortex dominant case, first four modes nearly gain over 98% energy of the flow field. The Galerkin projections for both algorithms are conducted and the prediction errors[31] are compared with a case which neglects the pressure term It is shown in Figure 2 that the omission of pressure term can augment the prediction error over 20 times. The curves of prediction errors from Algorithm 1 and 2 coincide. The projection errors are also computed but are not shown here since the maximum error is at 10-5 order of magnitude. This value is reasonable since we employ 32 modes to reconstruct the original snapshots. The time histories of amplitude of first four modes of velocity from A1 are shown in Figure 5. The modes from the reduced-order simulation match the projected one almost exactly. Algorithm 2 doesn’t change any curves from Algorithm 1 since the predition errors for two algorithms are close. To investigate the effect on accuracy, a four-mode simulation is conducted and the the predition error of A2 has a very small advantage than that of A1. This is because the effect of pressure term is small.

Final Goal
The POD-Galerkin projection method will be applied to moving airfoil. The extension from two-dimensions to three-dimensions has been done by Dr.Wan. It is straight forward to add a third dimension in current two-dimensional code. The next step is to make the Galerkin projection accurately represent the history of basis functions in pitch and plunge case where the wing-wake interaction is little. The final goal is to apply this methodology to hovering flight in where there exists lots of wing-wing interaction, wing-wake interaction and induced flow phenomenon.