Decomposition methods in turbulence research

Abstract

Nowadays we have the dynamical velocity vector field of turbulent flow at our disposal coming thanks advances of either mathematical simulation (DNS) or of experiment (time-resolved PIV). Unfortunately there is no standard method for analysis of such data describing complicated extended dynamical systems, which is characterized by excessive number of degrees of freedom. An overview of candidate methods convenient to spatio- temporal analysis for such systems is to be presented. Special attention will be paid to energetic methods including Proper Orthogonal Decomposition (POD) in regular and snapshot variants as well as the Bi-Orthogonal Decomposition (BOD) for joint space-time analysis. Then, stability analysis using Principal Oscillation Patterns (POPs) will be introduced. Finally, the Independent Component Analysis (ICA) method will be proposed for detection of coherent structures in turbulent flow-field defined by time-dependent velocity vector field. Principle and some practical aspects of the methods are to be shown. Special attention is to be paid to physical interpretation of outputs of the methods listed above. In fluid dynamics experimental research the data related to various physical quantities are acquired. The data is evaluated by means of sensor in distinct locations. Classical methods perform measurement in a single point in space (pressure probe, hot wire sensor, LDA). Recently, the spatial methods as PIV, evaluating measured quantities in many points distributed in a measuring plane or even in space simultaneously are used very often. In the presented paper we consider the data acquired in multiple points simultaneously, covering a given measuring zone. The data could be resolved in time as well, meaning that the data acquisition is performed in accordance with the general rules covering a reasonable part of the fluid system response spectrum. The rules to be met include the Nyquist criterion and the autocorrelation functions of the time series, which should be resolved properly, at least in connection with the largest structures characterized by the turbulence integral scale. In practice this means the acquisition frequency of order of kilohertz for common laboratory conditions in air turbulence, for liquids the frequency could be considerably lower. For time-resolved methods the event data acquisition is supposed, the unevenly acquired LDA data is not suitable, and then only temporal statistics could be performed. The resolution in space (i.e. size of interrogation area) and in time (i.e. acquisition period) should be in equilibrium. The same size of structures should be resolved in both domains. The structures of subgrid scales, if present, will produce the data noise, which could not be used for analysis. The spatio-temporal data could be scalars (temperature, concentration) or vectors (velocity vectors with 2 or 3 components).