Secondary flow

In fluid dynamics, a secondary flow is a relatively minor flow superimposed on the primary flow, where the primary flow usually matches very closely the flow pattern predicted using simple analytical techniques and assuming the fluid is inviscid. (An inviscid fluid is a theoretical fluid having zero viscosity.)

The primary flow of a fluid, particularly in the majority of the flow field remote from solid surfaces immersed in the fluid, is usually very similar to what would be predicted using the basic principles of physics, and assuming the fluid is inviscid. However, in real flow situations, there are regions in the flow field where the flow is significantly different in both speed and direction to what is predicted for an inviscid fluid using simple analytical techniques. The flow in these regions is the secondary flow. These regions are usually in the vicinity of the boundary of the fluid adjacent to solid surfaces where viscous forces are at work, such as in the boundary layer.

Examples of secondary flows

Wind near ground level

The basic principles of physics and the Coriolis effect satisfactorily explain that the direction of the wind in the atmosphere is parallel to the isobars. Measurements of wind speed and direction at heights well above ground level confirm that the speed of the wind matches that predicted by considerations of gradient flow, and the direction of the wind is indeed parallel to the isobars in the region. However, from ground level up to heights where the influence of the earth’s surface can be neglected, the wind speed is less than predicted by the barometric pressure gradient, and the wind direction is partly across the isobars rather than parallel to them. This flow of air across the isobars near ground level is a secondary flow. It does not conform to the primary flow, which is parallel to the isobars.

At heights well above ground level there is a balance between the Coriolis effect, the local pressure gradient, and the velocity of the wind. This is balanced flow. Closer to the ground the air is not able to accelerate to the speed necessary for balanced flow. Interference by the surface of the ground or water, and by obstructions such as terrain, waves, trees and buildings, cause drag on the atmosphere and prevent the air from accelerating to the speed necessary to achieve balanced flow. As a result, the wind direction near ground level is partly parallel to the isobars in the region, and partly across the isobars in the direction from higher pressure to lower pressure.

As a result of the slower wind speed at the earth’s surface, in a region of low pressure the barometric pressure is usually significantly higher at the surface than would be expected, given the barometric pressure at mid altitudes, due to Bernoulli's principle. Hence, the secondary flow toward the center of a region of low pressure is also drawn upward by the significantly lower pressure at mid altitudes. This slow, widespread ascent of the air in a region of low pressure can cause widespread cloud and rain if the air is of sufficiently high relative humidity.

In a region of high pressure (an anticyclone) the secondary flow includes a slow, widespread descent of air from mid altitudes toward ground level, and then outward across the isobars. This descent causes a reduction in relative humidity and explains why regions of high pressure usually experience cloud-free skies for many days.

Tropical cyclones

The primary flow around a tropical cyclone is parallel to the isobars – and hence circular. The closer to the center of the cyclone, the faster is the wind speed. In accordance with Bernoulli's principle where the wind speed is fastest the barometric pressure is lowest. Consequently, near the center of the cyclone the barometric pressure is very low. There is a strong pressure gradient across the isobars toward the center of the cyclone. This pressure gradient provides the centripetal force necessary for the circular motion of each parcel of air. This strong gradient, coupled with the slower speed of the air near the earth’s surface, causes a secondary flow at surface level toward the center of the cyclone, rather than a wholly circular flow.

Even though the wind speed near the center of a tropical cyclone is very fast, at any point on the earth’s surface it is not as fast as it is above that point away from the retarding influence of the Earth's surface. The slower speed of the air at the earth’s surface prevents the barometric pressure from falling as low as would be expected from the barometric pressure at mid altitudes. This is compatible with Bernoulli's principle. The secondary flow at the Earth's surface is toward the center of the cyclone but is then drawn upward by the significantly lower pressure at mid and high altitudes. As the secondary flow is drawn upward the air cools and its pressure falls, causing extremely heavy rainfall over several days.

Tornadoes and dust devils

Tornadoes and dust devils display localised vortex flow. Their fluid motion is similar to tropical cyclones but on a much smaller scale so that the Coriolis effect is not significant. The primary flow is circular around the vertical axis of the tornado or dust devil. As with all vortex flow, the speed of the flow is fastest at the core of the vortex. In accordance with Bernoulli's principle where the wind speed is fastest the air pressure is lowest; and where the wind speed is slowest the air pressure is highest. Consequently, near the center of the tornado or dust devil the air pressure is low. There is a pressure gradient toward the center of the vortex. This gradient, coupled with the slower speed of the air near the earth’s surface, causes a secondary flow toward the center of the tornado or dust devil, rather than in a purely circular pattern.

The slower speed of the air at the surface prevents the air pressure from falling as low as would normally be expected from the air pressure at greater heights. This is compatible with Bernoulli's principle. The secondary flow is toward the center of the tornado or dust devil, and is then drawn upward by the significantly lower pressure several thousands of feet above the surface in the case of a tornado, or several hundred feet in the case of a dust devil. Tornadoes can be very destructive and the secondary flow can cause debris to be swept into a central location and carried to low altitudes.

Dust devils can be seen by the dust stirred up at ground level, swept up by the secondary flow and concentrated in a central location. The accumulation of dust then accompanies the secondary flow upward into the region of intense low pressure that exists outside the influence of the ground.

Circular flow in a bowl or cup

When water in a circular bowl or cup is moving in circular motion the water displays vortex flow – the water at the center of the bowl or cup spins at relatively high speed, and the water at the perimeter spins more slowly. The water is a little deeper at the perimeter and a little more shallow at the center, and the surface of the water is not flat but displays the characteristic depression toward the axis of the spinning fluid. At any elevation within the water the pressure is a little greater near the perimeter of the bowl or cup where the water is a little deeper, than near the center. The water pressure is a little greater where the water speed is a little slower, and the pressure is a little less where the speed is faster, and this is consistent with Bernoulli's principle.

There is a pressure gradient from the perimeter of the bowl or cup toward the center. This pressure gradient provides the centripetal force necessary for the circular motion of each parcel of water. The pressure gradient also accounts for a secondary flow of the boundary layer in the water flowing across the floor of the bowl or cup. The slower speed of the water in the boundary layer is unable to balance the pressure gradient. The boundary layer spirals inward toward the axis of circulation of the water. On reaching the center the secondary flow is then upward toward the surface, progressively mixing with the primary flow. Near the surface there may also be a slow secondary flow outward toward the perimeter.

The secondary flow along the floor of the bowl or cup can be seen by sprinkling heavy particles such as sugar, sand, rice or tea leaves into the water and then setting the water in circular motion by stirring with a hand or spoon. The boundary layer spirals inward and sweeps the heavier solids into a neat pile in the center of the bowl or cup. With water circulating in a bowl or cup, the primary flow is purely circular and might be expected to fling heavy particles outward to the perimeter. Instead, heavy particles can be seen to congregate in the center as a result of the secondary flow along the floor.^[1]

River bends

Water flowing through a bend in a river must follow curved streamlines to remain within the banks of the river. The water surface is slightly higher near the concave bank than near the convex bank. (The concave bank has the greater radius, and the convex bank has the smaller radius.) As a result, at any elevation within the river the water pressure is slightly higher near the concave bank than near the convex bank. There is a pressure gradient from the concave bank toward the convex bank. Centripetal forces are necessary for the curved path of each parcel of water, and this centripetal force is provided by the pressure gradient.^[1]

The primary flow around the bend is vortex flow – fastest speed where the radius of curvature is smallest and slowest speed where the radius is largest.^[2] The higher pressure near the concave bank is accompanied by slower water speed, and the lower pressure near the convex bank is accompanied by faster water speed, and all this is consistent with Bernoulli's principle.

There is also a secondary flow in the boundary layer along the floor of the river bed. The boundary layer is not moving fast enough to balance the pressure gradient and so its path is partly downstream and partly across the stream from the concave bank toward the convex bank, driven by the pressure gradient.^[3] The secondary flow is then upward toward the surface where it mixes with the primary flow or moves slowly across the surface, back toward the concave bank.^[4] This motion is called helicoidal flow.

On the floor of the river bed the secondary flow sweeps sand, silt and gravel across the river and deposits the solids near the convex bank, in similar fashion to sugar or tea leaves being swept toward the center of a bowl or cup as described above.^[1] River bends often have a convex bank which is shallow and made up of sand, silt and gravel; and a concave bank which is steep and heavily eroded. This process can lead to formation of a meander or a point bar or, eventually, an oxbow lake.

Turbomachinery

Secondary flows are important in understanding the performance of turbines and other turbomachinery.^[5]^[6]

Many types of secondary flows occur in turbomachinery, including inlet prerotation (intakes vorticity), tip clearance flow (tip leakage), flows at off-design performance (e.g. flow separation), and secondary vorticity flows.^[7] Although secondary flows occur in all turbomachinery, it is particularly considered in axial flow compressors because of the thick boundary layers on the annulus walls.

For such axial-flow compressors, consider a set of guide vanes with an approach velocity c1. The velocity profile will be non-uniform due to friction between the annulus wall and the fluid. The vorticity of this boundary layer is normal to the approach velocity $c_{1}$ and of magnitude

$w_{1}={\frac {dc_{1}}{dz}}$

Where z is the distance to the wall. As the vorticity of each blade onto each other will be of opposite directions, a secondary vorticity will be generated. If the deflection angle, e, between the guide vanes is small, the magnitude of the secondary vorticity is represented as

$w_{s}=-2e\left({\frac {dc_{1}}{dz}}\right)$

This secondary flow will be the integrated effect of the distribution of secondary vorticity along the blade length.

Notes

^ ^a ^b ^c Bowker, Kent A. (1988). "Albert Einstein and Meandering Rivers". Earth Science History. 1 (1). Retrieved 2016-07-01.
^ In the absence of secondary flow, bend flow seeks to conserve angular momentum so that it tends to conform to that of a free vortex with high velocity at the smaller radius of the inner bank and lower velocity at the outer bank where radial acceleration is lower. Hickin, Edward J. (2003), "Meandering Channels", in Middleton, Gerard V. (ed.), Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432 ISBN 1-4020-0872-4
^ Near the bed, where velocity and thus the centrifugal effects are lowest, the balance of forces is dominated by the inward hydraulic gradient of the super-elevated water surface and secondary flow moves toward the inner bank. Hickin, Edward J. (2003), "Meandering Channels", in Middleton, Gerard V. (ed.), Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432 ISBN 1-4020-0872-4
^ Journal of Geophysical Research, Volume 107 (2002)
^ Formation of Secondary Flows in Turbines
^ Secondary Flow Research at the University of Durham
^ Brennen, C.E., Hydrodynamics of Pumps, retrieved 2010-03-24

References

Dixon, S.L. (1978), Fluid Mechanics and Thermodynamics of Turbomachinery pp 181–184, Third edition, Pergamon Press Ltd, UK ISBN 0-7506-7870-4

External links

[Einstein-1] Bowker, Kent A. (1988). "Albert Einstein and Meandering Rivers". Earth Science History. 1 (1). Retrieved 2016-07-01.

[2] In the absence of secondary flow, bend flow seeks to conserve angular momentum so that it tends to conform to that of a free vortex with high velocity at the smaller radius of the inner bank and lower velocity at the outer bank where radial acceleration is lower. Hickin, Edward J. (2003), "Meandering Channels", in Middleton, Gerard V. (ed.), Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432 ISBN 1-4020-0872-4

[3] Near the bed, where velocity and thus the centrifugal effects are lowest, the balance of forces is dominated by the inward hydraulic gradient of the super-elevated water surface and secondary flow moves toward the inner bank. Hickin, Edward J. (2003), "Meandering Channels", in Middleton, Gerard V. (ed.), Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432 ISBN 1-4020-0872-4

[4] Journal of Geophysical Research, Volume 107 (2002)

[5] Formation of Secondary Flows in Turbines

[6] Secondary Flow Research at the University of Durham

[7] Brennen, C.E., Hydrodynamics of Pumps, retrieved 2010-03-24

[1]

[2]

[3]

[4]

[5]

[6]

[7]