Level cut Gaussian random field models for transitions from laminar to turbulent flow

Herein, we develop a model for phenomena characterized by random fluctuations between one of two states; these two states can be interpreted as “off” and “on” or 0 and 1. Random phenomena of this type are abundant in nature and can exhibit temporal and/or spatial dependence. For example, the arrival of customers in a queue or vehicles at an intersection are examples of time-dependent random fluctuations, while the presence of random imperfections within material microstructure exhibits spatial dependence. Further, the transition from laminar to turbulent flow over the surface of a structure is believed to involve random fluctuations between laminar and turbulent flow in both time and space. The proposed model is a random field with piecewise constant samples defined by cuts of a Gaussian random field that exceed a specified level set and can be used to model temporally and spatially varying random fluctuations between one of two states. The model can be calibrated to available information, which we assume consists of (1) the marginal probability that the state of the system is “on” and (2) the average number and density of fluctuations between states that occur within a bounded region. Numerous examples are considered to illustrate the calibration of the level cut Gaussian random field to represent these time and spatially varying random fluctuations, including the formation and propagation of localized bursts of turbulent flow over the surface of a flat plate.

► We develop a space/time random field to model localized bursts in turbulent flow.
► The model is defined by space/time dependent cuts of a Gaussian random field.
► For calibration, we utilize outcrossing rates, number of maxima, and a time shift.
► Efficient numerical algorithms are developed to create samples of the random field.