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In general, steady fluid flow (quasi-steady for turbulent flow) 

 manifests itself in two distinct types of motion, laminar at low Reynolds num- 

 bers and turbulent at high Reynolds numbers. Whereas laminar flow proceeds 

 in a regular pattern of streamlines, turbulent flow advances in a haphazard 

 combination of mixing motions. Although the variations in velocity and pres- 

 sure of turbulent flow follow a random course, they may be considered to fluc- 

 tuate about some mean value as shown in Figure 1 . 



The analytic treatment of turbulent flow consists in separating the 

 fluid motion into a mean flow and a superposed fluctuation flow. Then the 

 mean and fluctuation velocities and 

 pressures are substituted into the 

 Navier-Stokes equations for viscous 

 flow and into the equation of conti- 

 nuity. After appropriate time aver- 

 ages are formed, the resulting equa- 

 tions in terms of mean quantities 

 resemble those for laminar flow, with 

 the exception of additional terms in- 

 volving averages of various products 

 of the fluctuation quantities. These 

 fluctuation terms represent the mixing 

 motion of turbulence and act as appar- 

 ent stresses (Reynolds stresses) with- 

 in the fluid. Since an analytical 

 description of the Reynolds stresses 

 has not been formulated, a direct solu 

 tion of the differential equations for 

 turbulent flow is virtually impossible. 



The theory of the flow of viscous fluids serves as a useful guide, however, 

 even though empirical or semi-empirical methods may be followed in many prac- 

 tical cases. 



At relatively high Reynolds numbers, the viscous effects of fluids 

 like air or water are confined mostly to a narrow region or band next to the 

 solid surface. The laminar flow through this region, or boundary layer as it 

 is better known, acts in the same manner as fully developed viscous flow in 

 pipes, inasmuch as it becomes turbulent after a critical value of some Reyn- 

 olds number is exceeded. The equations of motion for the turbulent boundary 

 layer are found analytically from the Navier-Stokes equations for turbulent 

 flow in a way analogous to that for the laminar boundary layer — that is, by 



Figure 1 - Velocity Fluctuations of 

 Turbulent Flow at a Fixed Position 



