Quick
The force a flowing fluid exerts on a body in the flow direction is called drag.
The components of the pressure and wall shear forces in the direction normal to the flow tend to move the body in that direction, and their sum is called lift.
Equations
| (Eq3) | | FD = | ∫ | | dFD = | ∫ | | (−P cos θ + τw sin θ ) dA |
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Drag force
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| (Eq4) | | FL = | ∫ | | dFL = | ∫ | | (−P sin θ + τw cos θ ) dA |
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Lift force
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Drag coefficient
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Lift coefficient
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Nomenclature
Details
It is a common experience that a body meets some resistance when it is forced to move through a fluid, especially a liquid. It is difficult for a person to walk in deep water because of the much greater resistance it offers to motion compared to air. A fluid may exert forces and moments on a body in and about various directions. The force a flowing fluid exerts on a body in the flow direction is called drag. The drag force can be measured directly by simply attaching the body subjected to fluid flow to a calibrated spring and measuring the displacement in the flow direction (just like measuring weight with a spring scale). More sophisticated drag-measuring devices, called drag balances, use flexible beams fitted with strain gages to measure the drag electronically.
Drag is usually an undesirable effect but can also be desirable, depending on the application. But, like friction, usually it is sought to be minimized. Reduction of drag is closely associated with the reduction of fuel consumption in automobiles, submarines, and aircraft, improved safety and durability of structures subjected to high winds, and reduction of noise and vibration. But in some cases drag produces a very beneficial effect and it is sought to be maximized. Friction, for example, is a "life saver" in the brakes of automobiles. Likewise, it is the drag that makes it possible for people to parachute, for pollens to fly to distant locations, and for us all to enjoy the waves of the oceans and the relaxing movements of the leaves of trees.
A stationary fluid exerts only normal pressure forces on the surface of a body immersed in it. A moving fluid, however, also exerts tangential shear forces on the surface because of the no-slip condition caused by viscous effects. Both of these forces, in general, have components in the direction of flow, and thus the drag force is due to the combined effects of pressure and wall shear forces in the flow direction. The components of the pressure and wall shear forces in the direction normal to the flow tend to move the body in that direction, and their sum is called lift.
For two-dimensional flows, the resultant of the pressure and shear forces can be split in two components: one in the direction of flow, which is the drag force, and another in the direction normal to flow, which is the lift, as shown in Figure 1. For three-dimensional flows, there is also a side force component in the direction normal to the page that tends to move the body in that direction.
The fluid forces also may generate moments and cause the body to rotate. The moment about the flow direction is called the rolling moment, the moment about the lift direction is called the yawing moment, and the moment about the side force direction is called the pitching moment. For bodies that possess symmetry about the lift-drag plane such as cars, airplanes, and ships, the side force, the yawing moment, and the rolling moment are zero when the wind and wave forces are aligned with the body. What remains for such bodies are the drag and lift forces and the pitching moment. For axisymmetric bodies aligned with the flow, such as a bullet, the only force exerted by the fluid on the body is the drag force.
The pressure and shear forces acting on a differential area dA on the surface are PdA and τwdA, respectively. The differential drag force and the lift force acting on dA in two-dimensional flow are:
| (Eq1) | dFD = −P dA cos θ + τw dA sin θ |
and:
| (Eq2) | dFL = −P dA sin θ + τw dA cos θ |
where θ is the angle that the outer normal of dA makes with the positive flow direction. The total drag and lift forces acting on the body are determined by integrating Eq1 and Eq2 over the entire surface of the body:
| (Eq3) | | FD = | ∫ | | dFD = | ∫ | | (−P cos θ + τw sin θ ) dA |
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and:
| (Eq4) | | FL = | ∫ | | dFL = | ∫ | | (−P sin θ + τw cos θ ) dA |
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These are the equations used to predict the net drag and lift forces on bodies when flow is simulated on a computer using computational fluid dynamics. However, when experimental analyses are performed, Eq3 and Eq4 are not practical since the detailed distributions of pressure and shear forces are difficult to obtain by measurements. Fortunately, this information is often not needed. Usually what is needed is the resultant drag force and lift acting on the entire body, which can be measured directly and easily in a wind tunnel.
Eq1 and Eq2 show that both the skin friction (wall shear) and pressure, in general, contribute to the drag and the lift. In the special case of a thin flat plate aligned parallel to the flow direction, the drag force depends on the wall shear only and is independent of pressure since θ = 90°. When the flat plate is placed normal to the flow direction, however, the drag force depends on the pressure only and is independent of wall shear since the shear stress in this case acts in the direction normal to flow and theta; = 0° as shown. If the flat plate is tilted at an angle relative to the flow direction, then the drag force depends on both the pressure and the shear stress.
The wings of airplanes are shaped and positioned specifically to generate lift with minimal drag. This is done by maintaining an angle of attack during cruising, as shown. Both lift and drag are strong functions of the angle of attack. The pressure difference between the top and bottom surfaces of the wing generates an upward force that tends to lift the wing and thus the airplane to which it is connected. For slender bodies such as wings, the shear force acts nearly parallel to the flow direction, and thus its contribution to the lift is small. The drag force for such slender bodies is mostly due to shear forces (the skin friction).
The drag and lift forces depend on the density ρ of the fluid, the upstream velocity V, and the size, shape, and orientation of the body, among other things, and it is not practical to list these forces for a variety of situations. Instead, it is found convenient to work with appropriate dimensionless numbers that represent the drag and lift characteristics of the body. These numbers are the drag coefficient CD and the lift coefficient CL and they are defined as:
and:
where A is ordinarily the frontal area (the area projected on a plane normal to the direction of flow) of the body. In other words, A is the area that would be seen by a person looking at the body from the direction of the approaching fluid. The frontal area of a cylinder of diameter D and length L, for example, is A = LD. In lift calculations of some thin bodies, such as airfoils, A is taken to be the planform area, which is the area seen by a person looking at the body from above in a direction normal to the body. The drag and lift coefficients are primarily functions of the shape of the body, but in some cases they also depend on the Reynolds number and the surface roughness. The term (1/2)ρV2 in Eq5 and Eq6 is the dynamic pressure.
The local drag and lift coefficients vary along the surface as a result of the changes in the velocity boundary layer in the flow direction. Interest usually lies with the drag and lift forces for the entire surface, which can be determined using the average drag and lift coefficients. Therefore, correlations for both local (identified with the subscript x) and average drag and lift coefficients are presented. When relations for local drag and lift coefficients for a surface of length L are available, the average drag and lift coefficients for the entire surface can be determined by integration from:
and:
When a body is dropped into the atmosphere or a lake, it first accelerates under the influence of its weight. The motion of the body is resisted by the drag force, which acts in the direction opposite to motion. As the velocity of the body increases, so does the drag force. This continues until all the forces balance each other and the net force acting on the body (and thus its acceleration is zero. Then the velocity of the body remains constant during the rest of its fall if the properties of the fluid in the path of the body remain essentially constant. This is the maximum velocity a falling body can attain and is called the terminal velocity. The forces acting on a falling body are usually the drag force, the buoyant force, and the weight of the body.