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By R. A. Wallis

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Dp/dx = 0. An­ other solution exists for fully developed flow in a pipe. In a turbulent boundary layer, the turbulent stresses which provide the turbulent mixing are much greater than the viscous stresses. I n an attempt to represent the turbulent stresses, the first equation of motion is empirically modified to include a term, e, called the " a p p a r e n t " kinematic viscosity due to turbulent shear. Hence, U du te +V du 1 dp . +{v+e) Ty=- £ . d 2u W* / 0 ox (3 3) · p Despite many attempts, no successful solution of this equation has been obtained.

5. Turbulent boundary layer profiles In the special case of zero velocity gradient (see eq. 39) When Cf is known as a function of x, this expression can be evaluated. F. 40 BOUNDARY LAYER AND SKIN FRICTION RELATIONS Following a procedure similar to that adopted in Reference 6, it can be shown that eq. 5. Turbulent Skin Friction Ludwieg and Tillmann< n > have shown that turbulent skin friction is a function of the shape parameter, H, and the Reynolds number, Re, and they have suggested the relation Cf = 0-246 X 1 0 - 0 - 6 7 8 ^ - 0 .

33 LAMINAR SKIN FRICTION On account of the inability of a laminar layer to persist far into a region of increasing pressure without separating, it is usually sufficient, for engineering purposes, to assume that separation occurs just downstream of the point where the pressure begins to rise. Hence the use of the above relation­ ships will be limited to positive values of λ and Λ ; some useful relations are given graphically in Fig. 2. 10 1 0Ο8 λ 6h θ θ δ δ δ* Ιθ04 ————— 1 ΟΌ"> Fig. 2. 14) where r is the local radius and x is the distance along the surface.

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