diffusion terms have been discarded anticipating high Reynolds number 

 applications. 



This system of equations (2.1)-(2.5) is not complete owing to the 

 presence of the Reynolds stress and the flux terms. A hierarchy of turbulence 

 models has been developed to resolve this problem (see, for example. Appendix 

 0). Conventional eddy-viscosity models employ a stress-strain law for 

 time-averaged turbulent flows in a similar form as that for a Newtonian fluid 

 in laminar motion. The Reynolds stress terms are replaced by the products of 

 a mean flow gradient and an eddy viscosity, which is prescribed as a constant 

 or some algebraic function of local flow properties. 



Ti me-Averaged Equations for the Mean Variables 



Assuming hydrostatic pressure distribution (valid when vertical 

 acceleration is negligible compared to the vertical pressure gradient) and 

 employing the eddy-viscosity concept, the basic equations (2.1)-(2.4) can be 

 written for a right-handed coordinate system (x^ ,X2 ,X2)=(x,y,z), where x and y 

 are the horizontal coordinates and z points vertically upward, as: 



- + - + ^ = (26) 



3x dy 3z ^^'°' 



iii + iyi + iiiv+iuw^^^_j_3£^_3_/^ 3u 



3t 3x ay 3z p. 3x 3x I H 3x 



*^K!^W^ kf^i (2.7, 



3y \ n 3y / 3z \ ^ dz 



iv ^ 3UV ^ 3vi ^ 3VW _ ^^ _ ^ 3£ ^ j_ / 9v 



3t 3X 3y 3Z po ay gx \^H g^ 



^^ Nf! ^^ fA,H) (^.8) 



3y \ " 3y / 3Z I V 3z 



21 



