In cases where there is a three-dimensional flow field, the flow is often regarded as comprising two components, a primary flow and a secondary flow. The primary flow is parallel to the main direction of fluid motion and the secondary flow is perpendicular to this. Such flows are commonly produced by the effect of drag in the boundary layers, and some of the more important situations in which such flows arise are discussed here.

Secondary flows occur where there is a flow around a bend in a pipe and this is illustrated in Figure 1. At the bend, there is a transverse pressure gradient, which provides the centripetal force for the fluid elements to change direction. However, the pressure gradient required for the faster moving fluid near the center of the pipe to follow the curve of the bend is greater than that required for the slower moving fluid near to the wall. This results in the fluid near the center of the pipe moving toward the outside of the pipe and the fluid near the wall moving inwards. (See also Bends, Flow and Pressure Drop In.)

Another common situation in which secondary flows arise is in spinning fluids, for example, in a weather system or a stirred teacup. In these systems, there is a balance between the radial pressure gradient and the centrifugal force (Coriolis force in the case of large weather systems). However, near a boundary, drag on the fluid leads to a lower velocity and the centrifugal or Coriolis force can no longer balance the pressure gradient. This results in a secondary flow of the fluid in the radial direction. Figure 2 illustrates this for the case of a cyclonic weather system. This situation is treated by Acheson (1990) and by Holton (1979). (See also Cyclones.)

Turbulent flows through pipes or channels of noncircular crosssection also give rise to secondary flows. There is a movement of fluid away from the corners along the walls and a movement toward the corners near the bisector of the comer. Examples for rectangular and triangular channels are shown in Figure 3. This phenomenon is known as "secondary flow of the second kind" and is discussed in Prandtl (1952) and Kay and Nedderman (1985). (See also Triangular Ducts, Flow and Heat Transfer In.)

There is also a "third kind" of secondary flow which results from *oscillations in a stationary fluid*. These oscillations may be due to, for example, an oscillating body or ultrasonic waves. If there is a variation in the amplitude, as will be the case with standing sound waves, then a secondary flow is induced which moves in the direction of decreasing amplitude. The theory behind this is dealt with by Schlichting (1968), and for a fluid in which the fluid elements move with velocity u(x) cos ωt, then it can be shown that the secondary flow is:

#### REFERENCES

Acheson, D. J. (1990) *Elementary Fluid Dynamics*, Oxford University Press, New York.

Holton, J. R. (1979) *An Introduction to Dynamic Meteorology*, Academic Press, New York.

Kay, J. M. and Nedderman, R. M. (1985) *Fluid Mechanics and Transfer Processes*, Cambridge University Press, Cambridge. DOI: 10.1016/0142-727X(86)90051-2

Prandtl, L. (1952) *Essentials of Fluid Dynamics*, Hafner Publishing Co., New York.

Schlichting, H. (1968) *Boundary-Layer Theory*, McGraw-Hill, New York.