Synthetic MAP tomographic reconstruction of turbulent flow fields

Synthetic MAP Tomographic Reconstruction of Turbulent Flow FieldsDavid W. WattUniversity of New Hampshire, Department of Mechanical EngineeringKingsbury Hall, Durham, New Hampshire 03824ABSTRACTTomographic reconstruction from both interferometric and absorption data is a potentially powerful tool forexperimental observations of compressible fluid mechanics, combustion, and heat transfer. In many of these cases,both flow field images and ensemble statistics are desired. The use of an ensemble of noisy tomographic data setsto synthesize image statistics and stabilize individual reconstructions using a Maximum A Posterior! (MAP)reconstruction technique is presented. The MAP technique uses the ensemble mean and variances of the sourcefunction to constrain individual reconstructions of the ensemble. In this paper, we show that by synthesizing themean and variances using preliminary algebraic reconstructions, the reconstruction of the individual realizations canbe improved. The technique is demonstrated using a group of source images generated with a fractal sum of pulses(FSP) technique. The images have similar statistical properties and appearances to the flow field images producedin turbulent jets. The paper discusses a fractal model for turblent mixing field images, the selection of thepreliminary reconstruction technique, and the results of MAP and synthetic MAP reconstructions.1 . INTRODUCTION: INTERFEROMETRIC AND ABSORPTION TOMOGRAPHY FOR FLOW VISUALIZATIONOptical tomography or projection imaging is a family of techniques for obtaining multidimensional scalarmeasurements from a group of projection integrals. The projections are obtained using probe beams passingthrough the field of interest, which causes either an optical phase delay due to the inhomogeneous refractive indexor an attenuation of the beam due to absorption. In either, case the magnitude of the signal is proportional to a lineintegral of the concentration of some scalar quantity such as density or concentration of an absorbing species. Ifthese projections are obtained from a range of viewing angles (see Figure 1 ), the integrals may be inverted to obtainan image of the original scalar distribution, using a procedure known as tomographic reconstruction.The appeal of tomography in experimental fluid mechanics, heat transfer and combustion is that spatially-resolved measurements of density (and by extension pressure or temperature) or concentration can be obtainednon-intrusively. Its advantages in some applications relative to planar imaging techniques are that it can providethree-dimensional scalar measurements and that it is superior for obtaining information normal to the flow direction.The cost of these advantages is a much more complex optical and image acquisition system that requires anadditional computational reconstruction step. Furthermore, the computational reconstruction method, the complexityof field being imaged, and the imaging geometry all affect the accuracy of the measurements. It is common toattempt to enhance the accuracy of tomographic reconstructions by the use of prior information.In the current work, the application of prior information is discussed in the context of an algebraic representationof the projection and reconstruction operations. In this context, the scalar field is represented by a discrete sum oforthogonal basis functions whose amplitudes are denoted by the vector f. The projection measurements aredenoted by the discrete vector R which is related to f using