Applied Fluid Mechanics for Engineers by Schobeiri Meinhard T

Author:Schobeiri, Meinhard T.
Language: eng
Format: epub
Tags: -
Publisher: McGraw-Hill Education
Published: 2014-01-26T16:00:00+00:00

9.1.2 Correlations, Length, and Time Scales

As we saw in Chap. 8, the Reynolds-averaging procedure has created an apparent stress tensor, , with nine components from which, for symmetric reasons, six are distinct. Thus, the creation of this tensor has added six more unknowns to Navier-Stokes equations. In order to find additional equations to close the equation set that consists of continuity, momentum, and energy balances, we need to construct additional equations. This is done by multiplying the i-th component of the Navier-Stokes equation with the j-th one. Thereby we expect to find turbulence models that establish relations between the new equations and the set of equations mentioned above. It should be pointed out that this purely mathematical manipulation does not represent any new conservation law with a physical background. However, it helps in providing additional tools that are necessary for turbulence modeling. In this context, correlations are indispensable tools for providing additional insight into turbulence. As we know from Navier-Stokes equations, the second-order tensor is the mean product of the fluctuation components at a single point in space; it is called a single-point correlation. It does not give any further information about the turbulence structure, such as the length and time scales of eddies. We obtain this information from a two-point correlation. It is a second-order tensor of the mean product of fluctuation components at two different points in space and time, namely (x,t) and (x + r, t + τ). For a purely spatial correlation with τ = 0, the same fluctuating quantity is measured at two different spatial positions, x and x + r. Figure 9.2 shows the position of the fluctuation components for a single-point correlation and several two-point correlations. For a general two-point correlation, we construct the second-order tensor:

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