Flow at Large Reynolds Number: Effects of Viscosity

Abstract

Introduction In this chapter the discussion of the flow of a viscous incompressible fluid of uniform density will be continued. The values of the kinematic viscosity for air and water are so small that the Reynolds numbers for most of the flow systems of importance, whether in nature or in technology or in the laboratory, are very much larger than unity. A Reynolds number of 10 3 is attained in air at 20°C when UL has the very modest value 150 cm 2 /sec, and in water when UL is only 10 cm 2 /sec, where U and L are representative values for the velocity variations and the distances over which they occur in the flow system concerned. Such small values of the product UL are so readily and so often exceeded that flow at large Reynolds number must be regarded as the standard case. The largeness of R = UL / v has its implications for the relative importance of the various terms in the equation of motion, as was seen in §4.7. Provided the non-dimensional quantities |Du′/ Dt ′| and |∇ 2 u′| are both of order unity over most of the flow field (which would of course exclude some simple flow fields, such as steady unidirectional flow in a tube, in which the fluid acceleration is zero everywhere), R is a measure of the ratio of the magnitudes of inertia and viscous forces acting on the fluid; and a flow field for which R ≫ 1 is presumably one in which inertia forces are much greater than viscous forces over most of the field.