Secondary flow

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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 that assume the fluid is inviscid. (An inviscid fluid is a theoretical fluid having zero viscosity.)

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation,

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.

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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.

In physics and fluid mechanics, a boundary layer is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

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.

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 conditions.

In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag forces depend on velocity. Drag force is proportional to the velocity for a laminar flow and the squared velocity for a turbulent flow. Even though the ultimate cause of a drag is viscous friction, the turbulent drag is independent of viscosity.

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 fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler who derived Bernoulli's equation in its usual form in 1752. The principle is only applicable for isentropic flows: when the effects of irreversible processes and non-adiabatic processes are small and can be neglected.

Relative humidity (RH) is the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at a given temperature. Relative humidity depends on temperature and the pressure of the system of interest. The same amount of water vapor results in higher relative humidity in cool air than warm air. A related parameter is that of dewpoint.

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.

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". Effects of surface-based anticyclones include clearing skies as well as cooler, drier air. Fog can also form overnight within a region of higher pressure. Mid-tropospheric systems, such as the subtropical ridge, deflect tropical cyclones around their periphery and cause a temperature inversion inhibiting free convection near their center, building up surface-based haze under their base. Anticyclones aloft can form within warm core lows such as tropical cyclones, due to descending cool air from the backside of upper troughs such as polar highs, or from large scale sinking such as the subtropical ridge. The evolution of an anticyclone depends on a few variables such as its size, intensity, moist-convection, Coriolis force etc.

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 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. The "convex bank" has the smaller radius.) As a result, at any elevation within the river, water pressure is slightly higher near the concave bank than near the convex bank. A pressure gradient results from the concave bank toward the other bank. Centripetal forces are necessary for the curved path of each parcel of water, which is provided by the pressure gradient. [1]

The primary flow around the bend is vortex flow – fastest speed where the radius of curvature of the stream itself is smallest and slowest speed where the radius is largest. [2] The higher pressure near the concave (outer) 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.

A secondary flow results 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] This process can lead to accentuation or creation of D-shaped islands, meanders through creation of cut banks and opposing point bars which in turn may result in an oxbow lake. The convex (inner) bank of river bends tends to be shallow and made up of sand, silt and fine gravel; the concave (outer) bank tends to be steep and elevated due to heavy erosion.

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 ${\displaystyle c_{1}}$ and of magnitude

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

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

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

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

Notes

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., Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432
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., Encyclopedia of Sediments and Sedimentary Rocks, New York: Springer, p. 432
4. Formation of Secondary Flows in Turbines Archived 2007-12-17 at the Wayback Machine
5. Brennen, C.E., Hydrodynamics of Pumps, archived from the original on 2010-03-09, retrieved 2010-03-24

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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