B • TURBULENT FLOW 



it had a larger viscosity than the same fluid flowing in a small pipe. 

 Furthermore, at high rates of flow the viscosity appears to be larger than 

 at low rates of flow. 



Our fictitious fluid properties are governed by an intimate connection 

 to the flow itself, and the apparent anomalies arise because of this fact. 

 As we know, the transporting agents are the turbulent motions, and taken 

 as a whole their velocities are proportional to some velocity characterizing 

 that of the mean flow. This connection between transporting motions and 

 the flow field stands in marked contrast to the independence of molecular 

 motions which do the transporting in laminar flow. 



In every case we must refer back to the turbulence mechanisms in 

 order ultimately to understand any kind of property with which we have 

 endowed the fluid. Phenomenological theories have been employed for 

 this purpose, of which one of the better known examples is the mixing 

 length theory proposed by Prandtl. Ideally, of course, we should like to 

 use the fundamental equations of motion for this purpose, but so far this 

 has not been possible. These questions are discussed in more detail in 

 Art. 10. 



CHAPTER 2. GENERAL HYDRO DYNAMICAL 



EQUATIONS FOR THE TURBULENT MOTION 



OF A COMPRESSIBLE FLUID 



B,4. Equations of Continuity and Momentum. The procedure 



introduced by Reynolds [S] and Lorentz [4], whereby equations of motion 

 and energy balance for an incompressible turbulent flow are obtained, is 

 well known. Briefly the turbulent motion is regarded as consisting of the 

 sum of a mean part and a fluctuating part, and the sum is introduced 

 into the Navier-Stokes equations. The resulting equations give consider- 

 able insight into the character of turbulent motions and serve as a basis 

 for attacking mean flow problems and also for analyzing the turbulence 

 into harmonic components. 



We now follow the same procedure for compressible turbulent flow. 

 The purpose in doing this is primarily to investigate the coupling be- 

 tween the mean motion and its fluctuations, and to establish the general 

 fundamental equations from which some general properties become appar- 

 ent. Later on, these equations may be simplified by approximations retain- 

 ing the significant terms applicable in a particular problem, such as the 

 customary boundary layer approximations. 



The additional difficulties encountered in compressible turbulent flow 

 are two-fold : First, the hydrodynamical equations are nonlinear, with the 

 nonlinear terms not only containing the velocity components and their 



(80) 



