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Figure 2.5 Pathlines of flow in a pipe (a) laminar (b) turbulent.

Figure 2.5 Pathlines of flow in a pipe (a) laminar (b) turbulent.

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It is the ratio of fluid momentum (arising from 'inertia forces') to viscous friction which determines whether the flow becomes smooth (i.e. laminar) or turbulent. This ratio is usually characterised by the non-dimensional Reynolds number

Here u is the mean speed of the flow, X is a nominated characteristic length of the system (in this case the diameter of the pipe) and v (= ยก/p; as in (2.10)) is the kinematic viscosity of the fluid. The value of ^ is important for characterising types of fluid flow, e.g. turbulence. For instance in pipes, flow will be usually turbulent if ^ is larger than about 2300 > 2300). Criteria for laminar or turbulent flow in heat transfer are discussed in Chapter 3.

In turbulent flow, random local fluctuations of velocity in three dimensions are imposed on the mean flow. Thus small elements of fluid moving along the pipe also move rapidly inwards and outwards across the pipe, as illustrated by Figure 2.5(b). Since fluid does not slip at the pipe surface (Section 2.4), the mean speed near the surface is smaller than the average and the mean speed near the centre of the pipe is correspondingly larger. Therefore the effect of the sideways motions of the fluid elements is to carry fluid of larger velocity outwards, and fluid of smaller velocity inwards. This transfer of momentum by elements of fluid is much larger than the corresponding transfer by molecular motions described in Section 2.4 because an element of fluid may move significantly across the pipe in a single jump. In this case with water as the fluid, the mean free path of a molecule in the liquid is of the order of nanometres.

This transfer of momentum from the fluid to the walls constitutes a sizeable friction force opposing the motion of the fluid. Thus the presence of turbulence increases friction as compared with laminar flow; it is important to appreciate this characteristic of turbulent flow as, for instance, in the aerodynamics of wind turbines.

If the walls of the pipe are hotter than the incoming fluid, then these rapid inward and outward motions transfer heat rapidly to the bulk of the fluid. An element of cold fluid can jump from the centre of the pipe, pick up heat by conduction from the hot wall, and then carry it much more rapidly back into the centre of the pipe than could molecular conduction. Thus turbulence increases heat transfer, as discussed in more detail in Section 3.4. and as applicable in the design of active solar systems.

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