Derivation of significant flow quantities and data assimilation

Abstract

Based on 2D2C, 2D3C or 3D3C PIV or 2D and 3D3C LPT field measurement data with and with-out temporal resolution many significant flow quantities can be derived which are important for the understanding and characterization of the investigated flow field and validation of CFD methods. With gridded data from PIV or scattered data from LPT several one-, two- and multi point statistics of the measured values (velocities and accelerations) can be calculated in a plane or volume. According to Reynolds (triple) decomposition statistical flow quantities are mean, (periodic) and fluctuation components of all three velocity vector components. The respective Reynolds stress tensor and higher order statistics are further measures for characterizing unsteady and turbulent flows. Important for the reliability of both statistics is a local convergence study and a precise uncertainty quantification. Recently, a versatile functional binning procedure has been introduced which optimizes the convergence speed of 3D LPT statistics by using the whole track- and uncertainty information (instead of single particle sampling) and arbitrarily shaped bins and weights allowing for adaptation to mean flow gradients. On instantaneous gridded 3D PIV data vector field operators with finite differencing schemes can be applied in order to derive 3D shear-, normal-strain- and vorticity vector fields. Other vortex detection schemes are the Q- or 2-criterion. Furthermore, a joint PDF of the local flow topologies derived from the invariants P, Q and R of the (time-resolved) velocity gradient tensor Aij(t) can be estimated. By solving the pressure -Poisson-equation 3D (or 2D) pressure fields can be estimated as well. The solvers are based on (time-series) of instantaneous PIV results and use certain assumptions, boundary conditions and various integration schemes, depending on the available flow data. For scattered 3D LPT results single time-step techniques for data assimilation schemes have been established using solenoidal constraints or a full incompressible Navier-Stokes-regularization. For the latter velocity and acceleration (material derivative) of the particle tracks are used as input values (left side of the momentum equation) for e.g. FlowFit, VIC +/#. Those schemes provide a continuous functional representation of the assimilated velocity vector field with-out additional spatial filtering (on the basis of 3D radial basis functions or B-Splines). Analytical derivatives can be computed for Aij(t) and 3D pressure fields are integrated by solving the Poisson equation implicitly using non-linear optimization solvers (e.g. L-BFGS).