One-seventh power law

Fig. 19-1. Wind variation with height—measured (solid lines) and one-seventh power law (dashed lines).

This equation is sometimes referred to as the Prandtl one-seventh power law. 

This relation holds provided that the one-seventh power law may be assumed to apply over the whole of the cross-section of the pipe. This is strictly the case only at high Reynolds numbers when the thickness of the laminar sub-layer is small. By combining equations 3.59 and 3.63, the velocity profile is given by ... 

On the assumption that the velocity profile in a fluid in turbulent flow is given by the Prandtl one-seventh power law, calculate the radius at which the flow between it and the centre is equal to that between it and the wall, for a pipe 100 mm in diameter,... 

The Prandtl one-seventh power law gives the velocity at a distance y from the wall, ux, as ... 

The velocity of the fluid may be assumed to obey the Prandtl one seventh power law, given by equation 11.26. If the boundary layer thickness S is replaced by the pipe radius r, this is then given by ... 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer <5, is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by PrandtFs one seventh power law. 

For turbulent flow, the average shear stress may be evaluated, assuming a one-seventh power law profile, as ... 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

Water is flowing through a 150 mm diameter pipe and its flowrate is measured by means of a 50 mm diameter orifice, across which the pressure differential is 2.27 x 104 N/m2. The coefficient of discharge of the orifice meter is independently checked by means of a pitot tube which, when situated at the axis of the pipe, gave a reading of 100 mm on a mercury-under-water manometer. On the assumption that the flow in the pipe is turbulent and that the velocity distribution over the cross-section is given by the Prandtl one-seventh power law, calculate the coefficient of discharge of the orifice meter. 

P13-11b For turbulent flow, the one-seventh-power-law model is usually used to describe the velocity profile, that is. 

Prandtl s one-seventh power law A relationship used to determine the velocity distribution for turbulent flow in pipes carrying fluids given as ... 

---