To begin, we must first be able to express quantities as the sums of their slowly varying components and fluctuating components.
Reynolds decomposition This process is known as
Reynolds decomposition. Temperature can be expressed as: T = \overline{T} + T', where \overline{T}, is the slowly varying component and T' is the fluctuating component. In the above picture, T' can be expressed in terms of the mixing length considering a fluid parcel moving in the z-direction: T' = -\xi' \frac{\partial \overline{T}}{\partial z}. The fluctuating components of velocity, u', v', and w', can also be expressed in a similar fashion: u' = -\xi' \frac{\partial \overline{u}}{\partial z}, \qquad \ v' = -\xi' \frac{\partial \overline{v}}{\partial z}, \qquad \ w' = -\xi' \frac{\partial \overline{w}}{\partial z}. although the theoretical justification for doing so is weaker, as the
pressure gradient force can significantly alter the fluctuating components. Moreover, for the case of vertical velocity, w' must be in a neutrally stratified fluid. Taking the product of horizontal and vertical fluctuations gives us: \overline{u' w'} = \overline{\xi' ^2} \left | \frac{\partial \overline{w}}{\partial z}\right| \frac{\partial \overline{u}}{\partial z}. The eddy viscosity is defined from the equation above as: K_m=\overline{\xi'^2} \left| \frac{\partial \overline{w}}{\partial z}\right|, so we have the eddy viscosity, K_m expressed in terms of the mixing length, \xi'. == See also ==