Secondary flow

# Secondary flow

Discussion

Encyclopedia
In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, 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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

.

### Wind near ground level

The basic principles of physics and the Coriolis effect
Coriolis effect
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...

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
Balanced flow
In atmospheric science, balanced flow is an idealisation of atmospheric motion. The idealisation consists in considering the behaviour of one isolated parcel of air having constant density, its motion on a horizontal plane subject to selected forces acting on it and, finally, steady-state...

. 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
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

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. This is compatible with Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

. As a result, 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
Relative humidity
Relative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. It is defined as the partial pressure of water vapor in the air-water mixture, given as a percentage of the saturated vapor pressure under those conditions...

.

In a region of high pressure (an anticyclone
Anticyclone
An anticyclone is a weather phenomenon defined by the United States' National Weather Service's glossary as "[a] large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere"...

) 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
Tropical cyclone
A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...

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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

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
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

. 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.

A tornado is a violent, dangerous, rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. They are often referred to as a twister or a cyclone, although the word cyclone is used in meteorology in a wider...

es and dust devil
Dust devil
A dust devil is a strong, well-formed, and relatively long-lived whirlwind, ranging from small to large . The primary vertical motion is upward...

s display localised vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

flow. Their fluid motion is similar to tropical cyclone
Tropical cyclone
A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...

s but on a much smaller scale so that the Coriolis effect
Coriolis effect
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...

is not significant. The primary flow is circular around the vertical axis of the tornado or dust devil. As with all vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

flow, the speed of the flow is fastest at the core of the vortex. In accordance with Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

. 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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

.

There is a pressure gradient from the perimeter of the bowl or cup toward the center. This pressure gradient provides the centripetal force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

necessary for the circular motion of each parcel of water. The pressure gradient also accounts for a secondary flow of the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

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.

### 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 force
Centripetal force
Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

s are necessary for the curved path of each parcel of water, and this centripetal force is provided by the pressure gradient.

The primary flow around the bend is vortex
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

flow – fastest speed where the radius of curvature is smallest and slowest speed where the radius is largest. 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
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

.

There is also a secondary flow in the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

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. 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. This motion is called helicoidal flow
Helicoidal flow
Helicoidal flow is the cork-screw-like flow of water in a meander. It is one example of a secondary flow.Helicoidal flow is a contributing factor to the formation of slip-off slopes and river cliffs in a meandering section of the river...

.

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. 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
Meander
A meander in general is a bend in a sinuous watercourse. A meander is formed when the moving water in a stream erodes the outer banks and widens its valley. A stream of any volume may assume a meandering course, alternately eroding sediments from the outside of a bend and depositing them on the...

or a point bar
Point bar
A point bar is a depositional feature of streams. Point bars are found in abundance in mature or meandering streams. They are crescent-shaped and located on the inside of a stream bend, being very similar to, though often smaller than towheads, or river islands.Point bars are composed of sediment...

or, eventually, an oxbow lake
Oxbow lake
An oxbow lake is a U-shaped body of water formed when a wide meander from the main stem of a river is cut off to create a lake. This landform is called an oxbow lake for the distinctive curved shape, named after part of a yoke for oxen. In Australia, an oxbow lake is called a billabong, derived...

.

### Turbomachinery

Secondary flows are important in understanding the performance of turbine
Turbine
A turbine is a rotary engine that extracts energy from a fluid flow and converts it into useful work.The simplest turbines have one moving part, a rotor assembly, which is a shaft or drum with blades attached. Moving fluid acts on the blades, or the blades react to the flow, so that they move and...

s and other turbomachinery
Turbomachinery
Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy from a rotor to a fluid...

.

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.
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 c1 and of magnitude

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

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