TM WOo 377 



Equation (A=82) is a xoathematical statement that the mean horizontal 

 inertial acceleration of a water element is balanced by the mean horizontsil 

 pressure gradient^ horizontal gradients of the mean horizontal dynamic 

 pi-essure^ and the vertical gradient of the mean eddy stress. Equation (A-83) 

 indicates that the vertical mean pressure gradient is balanced by the gravi- 

 tational force tcpon the water element^ the horizontal gradient of the eddy 

 stresses^ and the vertical gradient of the vertical dynamic pressure. 



Comparing equations (A=82) and (A=83) with equations (A-7I) and (A-72), 

 it is evl.dent that the Reynolds stresses are interpreted as real stresses on 

 the fluid elements and that they act in addition to the forces in^jarted by 

 the pressure and pure viscous stresses. 



The Reynolds stresses have normal as well as tractive or tangential com- 

 ponents,, An interp2retation of the normal stress can be made in a similar 

 manner Consider a horizontal element ds normal to the vertical axis. Pmrther 

 assume uTsO $ allowing only a turbiolent component normal to the element. Then 

 -■ft?*} den otes the average flow of momentum through the surface element, namely 

 " C-Put'jJo'di " This flow of momentum causes a inaction on the surface^ and, when 

 integrated over the time T and aversjed^, it is sensed as a dynamic presstire. 



Thus, the products of time variable advective quantities can be envisaged 

 as real forces acting trpon the particles of water. 



The formulation of Reynolds stresses throws considerable light on the 

 mechanism of turbulence^ whereas the mathematical analysis is not readily 

 soluble. In general, no analytical method is known whereby the Reynolds 

 stresses can be expressed in terms of the mean velocities and their derivatives; 

 so, for the most part, equations of the form of (A'=82) and (A-83) remain intrac- 

 table. One can try, however, to solve the equations expeilmentallyj i.e., one 

 can measure certain qtantities in the equations experimentally and conjecture 

 magnitudes of the unmeasurable quantities, thxis providing an estimate of the 

 validity of the equations themselves and a judgment on the assumptions made to 

 develop themo 



Kinetic Energy Relations -=• To derive the eqtiation expi^ssing the flvDC of 

 mean energy in the waves, consider the general momentum equations (A-80) and 

 (A-81), neglecting the term ^j^"-"*^^ in equation (A-80). First multiply equatioBS 

 (A=8o) and (A-81) by f a and 1?^, respectiTOly. Since w is defined as identically 

 zero, the result is a si33gle equation given by; 



By definition the dynamic viscosity 'V ~ fV. The term 



is equivalent to the time rate of change of the kinetic energy of the mean 



A-=l6 



