When a body is placed in a moving fluid, a force is exerted on the body by the moving fluid. Similarly when a body moves within a fluid at rest then also a force is exerted on the body by the fluid. Motion of submarines, torpedos and aeroplanes are examples of this kind.
Consider a body placed in a fluid which is moving at a velocity U. The moving fluid will exert a force F on the body which can be resolved into two components namely the drag force exert a Fd in the direction of motion and the lift force FL normal to the direction of motion.
When the fluid is in motion the drag force acts on the body tending to move the body or drag the body in the direction of motion of the fluid. On the contrary when the body moves in the fluid, the drag force acts as a resistance to the motion of the body.
What Causes Drag and Lift Forces?
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Fig. 17.2 shows a body placed within a fluid which is moving horizontally with a velocity U. Consider an elemental area da on the surface of the body. Let p be the pressure intensity and τ the shear stress intensity acting on the elemental area. Let the tangent to the elemental area be at θ with the direction of motion of the fluid.
The shape of the body and its position in the fluid influence the relative contribution of the pressure and the shear drags. For instance, when a plate is kept in a vertical position at right angles to a moving fluid, the drag force is entirely a pressure drag and the shear drag is ignorable (See Fig. 17.3).
If the plate is placed horizontally as in Fig. 17.4, the drag force is entirely a shear drag and the pressure drag is ignorable.
This is explained in greater detail below:
We will now consider bodies of three shapes, a flat circular disc, a sphere and a stream lined body. When any of these bodies is placed in a moving fluid, a separation region of low pressure called a wake is formed behind the body. This region is also characterized by large sized irregular eddies which results in a large loss of energy. See Fig 17.5. The wake zone is very large in the case of the disc and it is slightly less for the sphere whereas it is very little for the stream lined body.
For the disc facing the fluid in motion, the shear stresses are normal to the direction of flow and do not contribute to drag. The drag is entirely due to the large pressure difference between the approaching fluid and the fluid in the wake zone. Hence the drag in this case is basically pressure drag.
For the sphere the wake zone is smaller and as in the previous case a drag force exists due to the pressure difference between the approaching fluid and the fluid in the wake zone. There will also be a drag force due to shear stresses acting along the contact faces. Hence in this case the drag is partly a pressure drag and partly a shear drag.
In the case of the streamlined body the wake zone is too small. Fluid pressure does not contribute to drag. Shear stresses along the contact surfaces are parallel to the direction of fluid motion and hence the drag is practically a shear drag.
General Equation for the Force Exerted on a Body Placed in a Moving Fluid:
The force exerted by a fluid on a body depends on the dimension of the body, the density of the fluid, the viscosity of the fluid, the elastic modulus of the fluid, the velocity of the fluid and the acceleration due to gravity. If F is the force exerted on the body then with the usual notations-
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F = f (l, ρ, μ, E, U, g)
There are seven quantities in the relation above involving three fundamental units. Hence we can frame 7 – 3 = 4 dimensionless constants.
These constants are:
In many of the cases we handle, the Mach’s number is less than 0.3 and in this condition the effect of elasticity of the fluid on the drag and lift forces is ignorable. We also find when the body is fully submerged in the fluid, the gravitational effect on the drag and lift forces is ignorable. Hence with the above approximations.
Drag on a Sphere:
The drag force on a sphere immersed in a moving fluid is a function of the Reynold’s number DUρ/μ. The relation of the drag force with Reynold’s number is different at different ranges of the Reynold’s number.
Drag Force on a Cylinder:
Suppose a cylinder of diameter D and length I is placed in a stream so that the length of the cylinder is at right direction of motion of the stream.
(i) When the Reynold’s number is between 1000 and 3,00,000.
In this range the drag coefficient is more or less constant and equal to 1.2.
(ii) When the Reynold’s number is between 3,00,000 and 5,00,000. In this range the drag coefficient drops from 1.2 to 0.3.
Karman Vortex Trail:
When a fluid flows past a cylinder and as the Reynold’s number DUρ/μ exceeds 30, two vertices are developed and are washed away downstream. Again two more vertices are developed which are also washed away downstream. In this manner a series of pairs of vortex are formed which move downstream with a small velocity.
The path or trail of vortices occurring on the downstream of the cylinder is called Karman Vortex Trail or Karman Street. There are two possible configurations of vortices namely- (i) Symmetrical configurations and (ii) Staggered configuration. See Fig. 17.13.
The configuration attains a stable condition according to Karman, when the ratio a/L = 0.281. The velocity of the vortex trail is given by –
Ut = 0.355γ/a
The quantity fD/U is called Strouhal Number. The Strouhal number varies from 0.16 to 0.198 as the Reynold’s number changes from 100 to 10,000.
When a flat plate is held normal to the flow, then also a Karman vortex trail is formed. In this case the Strouhal number remains practically a constant having the value 0.14 for Reynold’s number greater than 1000.
Flow around a Cylinder at Rest:
Consider a fluid flowing past a cylinder as shown in Fig. 17.14.
Let U = approach velocity of the fluid
r = radius of the cylinder
The velocity of the fluid close to the cylinder is affected due to the presence of the cylinder.
Consider the flow of the fluid very close to the cylinder. On one side of the cylinder the fluid flows along the route AMB. On the other side of the cylinder the fluid flows along the route ANB. The velocity of the fluid at any point P on any one of these routes depends on the angle θ that the radius OP makes with the leading radius OA. The velocity is zero at A and B, and is maximum at M and N where θ = 90°. The velocity at any point P is taken as –
vu = 2 U sin θ
since the flow pattern is symmetrical on the two sides of the cylinder.
In this case circulation around the cylinder = 0.
Rotation of a Cylinder in a Fluid:
Suppose a cylinder of radius r is rotated with a uniform angular velocity. Let vp be the peripheral velocity. This velocity is also imparted to the fluid close to the cylinder. The velocity of the fluid at the periphery of the cylinder is equal to vp and the velocity of the fluid decreases at increasing radii. The motion of the fluid may be approximated to a free vortex flow with a constant circulation.
γ = vp 2pr
Circulation γ:
Circulation is the line integral of the velocity along a closed path.
Fig. 17.14 shows a closed line in the zone of a fluid flow. Consider an elemental length dl of the closed curve.
Let,
Vt = component of the velocity along the elemental length of curve
Vt = V cos θ
Where V = Actual velocity
θ = Inclination of the tangent to the curve with the direction of the actual velocity.
∴ Circulation around the closed curve
Circulation in a Free Vortex Flow:
Consider a free vortex flow of a fluid about the axis O (see fig. 17.15). Let v be the velocity at radius r.
For a free vortex flow,
vr – Constant = C (say)
Let v’ be the velocity at any other radius r’
∴ vr = v’r’ = C
Circulation around any circle of radius r’ about the axis O
= v’ .2πr’ = 2π v’r’ = 2π C = constant
Thus circulation around every concentric circle about the axis O
= 2π C = constant
Magnus Effect:
Dynamic Lift:
Consider a cylinder of radius r placed in a fluid having a velocity U. Let the axis of the cylinder be horizontal. Let the cylinder be rotated about its axis at a uniform angular velocity. Let vp be the peripheral velocity of the cylinder.
In Fig. 17.18, at any point P on the surface AMB the velocity of the fluid
= v = 2U sin θ + vp
At any point P on the surface ANB the velocity of the fluid
= v = 2 U sin θ – vp
Thus the velocities of the fluid layers above the cylinder are increased while the velocities of the fluid layers below the cylinder are decreased. By Bernoulli’s theorem, the above changes in velocity give rise to corresponding pressure changes.
The pressure of the fluid layer is increased below the cylinder and decreased above the cylinder. These pressure difference will result in an upward lift force. This phenomenon of production of a lift force on a rotating cylinder placed in a moving mass of fluid is called Magnus Effect.
Lift Force:
Let po be the pressure intensity of the approaching fluid. Let p be the pressure intensity at any point P whose radius vector is at θ with this leading radius OA.
Stagnation Points:
We know, above the cylinder in the route AMB the velocity is increased due to the rotation of the cylinder. But below the cylinder in the route ANB the velocity is decreased. Considering the route ANB at any point P whose radius vector is at θ with the leading radius, the velocity of the fluid is given by –
v = 2U sin θ – vp
When vp is within a certain limit there is a possibility for the fluid velocity to reach zero. When this happens –
2U sin θ – vp = 0
∴ sin θ = vp/2U
This is satisfied for two values of θ corresponding to P and P’ as shown in Fig. 17.20. These points P and P’ where the fluid velocity is zero are called stagnation points. In general, there can be two stagnation points.
For a particular value of vp there will be only one stagnation point which can occur at N corresponding to θ = 90°
For the condition of single stagnation point-
But actual experiments show that Cl varies in a different way. Fig. 17.20 shows the variation of Cl with respect to the ratio [vp/U].
The deviation of the value of Cl from the derived value is due to the following reasons:
(i) The effect of viscosity of the fluid has not been considered.
(ii) The flow around the cylinder due to the rotation of the cylinder has been assumed to be a free vortex flow, this is only an approximation.
(iii) The effect of the length of the cylinder on the flow pattern has not been considered.