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The critical Reynolds number for the onset of turbulence is best stated in terms of the width of the flow’ and depends on the shape of the laminar velocity profile; it is typically O( lo2) for profiles with inflection points and 6(103) or more for profiles of single curvature. Thus, boundary layers in falling pressures are a good deal more stable than those that are suffering a pressure rise; similarly, jets and wakes are also very unstable. When turbulence appears, as originally observed and described by Reynolds for pipe flow, the flow field may be expressed as V v and P p , where the lower-case quantities represent the fluctuating components about the mean (with zero average) and V and P are the mean components; V = ( V I , U2, U3}T and v = (u1, u2, ~ 3 in ) an ~ (XI, x2, x3)-frame.

If the cylinder is in a conduit of complex geometry, m’ may be determined analytically, numerically or experimentally, and the added mass per unit length expressed by m’ m* = - = C,pA. 3 and by others (Chen 1987; Gibert 1988; Naudascher & Rockwell 1994). In many 36 SLENDER STRUCTURES AND AXIAL FLOW cases the approximation is made that the added mass in quiescent (stagnant) and flowing fluid is the same, although this is not rigorous. Such an approximation is definitely shaky if the flow is grossly unsteady or accelerating.

If it is zero, then there is no net energy flow in and out of the system, which must therefore be conservative; otherwise, the system is nonconservative. 40)+ - see Chapter 3. Viewed another way, this means that it is not possible to derive these forces from a potential; like dissipative forces, for instance, they are nonconservative, at least for this set of boundary conditions. The second observation suggests that, for u # 0, the results from either method (a) or (b) must be wrong. 3. 1. In method (a) the extra mass, Me,is included in the equation of motion via a Dirac delta function, while in (b) it is accounted for in the boundary conditions.

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