**Mathematical Models in Fluid Mechanics**

**1** 

*Brazil* 

**One Dimensional Turbulent Transfer Using** 

H. E. Schulz1,2, G. B. Lopes Júnior2, A. L. A. Simões2 and R. J. Lobosco2

The mathematical treatment of phenomena that oscillate randomly in space and time, generating the so called "statistical governing equations", is still a difficult task for scientists and engineers. Turbulence in fluids is an example of such phenomena, which has great influence on the transport of physical proprieties by the fluids, but which statistical quantification is still strongly based on *ad hoc* models. In turbulent flows, parameters like velocity, temperature and mass concentration oscillate continuously in turbulent fluids, but their detailed behavior, considering all the possible time and space scales, has been considered difficult to be reproduced mathematically since the very beginning of the studies on turbulence. So, statistical equations were proposed and refined by several authors, aiming to describe the evolution of the "mean values" of the different parameters (see a

The governing equations of fluid motion are nonlinear. This characteristic imposes that the classical statistical description of turbulence, in which the oscillating parameters are separated into mean functions and fluctuations, produces new unknown parameters when applied on the original equations. The generation of new variables is known as the "closure problem of statistical turbulence" and, in fact, appears in any phenomena of physical nature that oscillates randomly and whose representation is expressed by nonlinear conservation equations. The closure problem is described in many texts, like Hinze (1959), Monin & Yaglom (1979, 1981), and Pope (2000), and a general form to overcome this difficulty is

As reported by Schulz et al. (2011a), considering scalar transport in turbulent fluids, an early attempt to theoretically predict RMS profiles of the concentration fluctuations using "ideal random signals" was proposed by Schulz (1985) and Schulz & Schulz (1991). The authors used random square waves to represent concentration oscillations during mass transfer across the air-water interface, and showed that the RMS profile of the concentration fluctuations may be expressed as a function of the mean concentration profile. In other words, the mean concentration profile helps to know the RMS profile. In these studies, the authors did not consider the effect of diffusion, but argued that their

description, for example, in Monin & Yaglom, 1979, 1981).

**1. Introduction** 

matter of many studies.

**Random Square Waves – Scalar/Velocity** 

**and Velocity/Velocity Interactions**

*2Department of Hydraulics and Sanitary Engineering School* 

*of Engineering of São Carlos, University of São Paulo* 

*1Nucleus of Thermal Engineering and Fluids* 
