Quadratic Gappy-POD Applied to the Acoustic Near-Field of High Mach Number Jets

The gappy-POD proposed by Everson and Sirovich (J. Opt. Soc. America, Vol. 12, 1995) is derived to include higher-order terms not previously considered in the error minimization problem. The mathematical framework is provided for all available heterogeneous forms of the technique. The first of these forms applies the gappy sensor set to the eigenvectors and is referred to as the XX-topos form, whereas the second form uses the POD expansion coefficients and is referred to as the XX-chronos form; the naming convention follows the language proposed by Aubry (Theor. Comp. Fluid Dyn., Vol. 12, 1991). A third form, the YX form, is also evaluated and is shown to correspond to the original gappy-POD framework proposed by Everson and Sirovich. The quadratic gappy-POD forms are then scrutinized using spatially and temporally resolved schlieren images of sound waves emanating from the near-field of a Mach 3 jet; these sound waves are Mach waves generating by turbulence in the shear layer being convected at supersonic speeds relative to the ambient field. It is shown that quadratic terms are needed when an extremely limited number of sensors are available and performs best with the XX-chronos form of the technique. A computer program is provided in the Appendix that computes the heterogeneous XX-chronos form of the gappy-POD using conventional kernels to include quadratic terms.