50
2 Water at Rest and in Motion
For the bluff bodies, a drag force is not so sensitive to the flow direction,
however a surface roughness is either without effect or it increases drag. Investigations of the effect of surface roughness on cylinders showed that beyond
some large values of Re, depending on the surface roughness, the pressure distribution becomes independent of the Re and is determined by the characteristics
of the surface roughness. For relative roughness (roughness height/body diameter) of about 10- 5 , flow essentially behaves as for a smooth body even though
the boundary layer may be fully rough (Sarpkaya and Isaacson, 1981).
In certain narrow range, rough surface induces turbulence and can thereby
postpone separation of fluid travelling around bluff body. For a circular cylinder, a reduction of drag due to roughness was observed for Reynolds number
between 100,000 and 250,000 (Vogel, 1994). A good example of a positive effect
of roughness for drag reduction during a motion in air is a golf ball.
A second important force acts perpendicular to drag and is called lift, because
the most familiar example of a force of this kind is the upward force which acts
on the wings of aircraft and keep them in the air. In spite its name, lift does not
necessarily act upwards. Structures like aircraft wings, which are designed to
produce lift to move in the air, are called aerofoils, while structures producing
lift in water are known as hydrofoils.
To this point we have assumed that fluid flow is steady. When fluid is accelerating, an additional force is imposed on a stationary body, which is called
inertia force. This force is usually accompanied by a drag force. The total
force imposed on stationary or moving body is a vectorial sum of all forces involved. In the following sections each of these forces and their physical nature
and methods of their determination are examined.
2.6.2 Drag Force
Let us start with a simple example of a circular cylinder with its axis normal
to the uniform flow (Fig. 2.22a). When fluid is assumed to be non-viscid, ideal,
streamlines can be determined and the total force imposed on the cylinder can
be calculated. As is shown in Appendix CA, the resulting force is nil! This is
so called D' Alembert's paradox, which is valid for bodies of any shape. In the
case of a circular cylinder, there are so called stagnation points S, at upstream
and downstream extremities, where the fluid is locally stationary with respect
to the cylinder (Fig. 2.22a). The fluid reaches maximal velocity u = 2uo at the
sides of the cylinder, when B = ±90°, where the pressure drops to the value
p/ Pwg = Const- (2u5!g) , as follows from the Bernoulli equation (see Eq. C.84);
the Const is the total head. The whole pressure distribution is symmetrical
about both the cross- and along-flow axes of the cylinder (Fig. 2.22b).
Pressure at a given point on the cylinder surface can generally be represented
as a summation of ambient pressure Po and dynamic pressure induced by the
interaction of the cylinder with the flow:
2 Water at Rest and in Motion
For the bluff bodies, a drag force is not so sensitive to the flow direction,
however a surface roughness is either without effect or it increases drag. Investigations of the effect of surface roughness on cylinders showed that beyond
some large values of Re, depending on the surface roughness, the pressure distribution becomes independent of the Re and is determined by the characteristics
of the surface roughness. For relative roughness (roughness height/body diameter) of about 10- 5 , flow essentially behaves as for a smooth body even though
the boundary layer may be fully rough (Sarpkaya and Isaacson, 1981).
In certain narrow range, rough surface induces turbulence and can thereby
postpone separation of fluid travelling around bluff body. For a circular cylinder, a reduction of drag due to roughness was observed for Reynolds number
between 100,000 and 250,000 (Vogel, 1994). A good example of a positive effect
of roughness for drag reduction during a motion in air is a golf ball.
A second important force acts perpendicular to drag and is called lift, because
the most familiar example of a force of this kind is the upward force which acts
on the wings of aircraft and keep them in the air. In spite its name, lift does not
necessarily act upwards. Structures like aircraft wings, which are designed to
produce lift to move in the air, are called aerofoils, while structures producing
lift in water are known as hydrofoils.
To this point we have assumed that fluid flow is steady. When fluid is accelerating, an additional force is imposed on a stationary body, which is called
inertia force. This force is usually accompanied by a drag force. The total
force imposed on stationary or moving body is a vectorial sum of all forces involved. In the following sections each of these forces and their physical nature
and methods of their determination are examined.
2.6.2 Drag Force
Let us start with a simple example of a circular cylinder with its axis normal
to the uniform flow (Fig. 2.22a). When fluid is assumed to be non-viscid, ideal,
streamlines can be determined and the total force imposed on the cylinder can
be calculated. As is shown in Appendix CA, the resulting force is nil! This is
so called D' Alembert's paradox, which is valid for bodies of any shape. In the
case of a circular cylinder, there are so called stagnation points S, at upstream
and downstream extremities, where the fluid is locally stationary with respect
to the cylinder (Fig. 2.22a). The fluid reaches maximal velocity u = 2uo at the
sides of the cylinder, when B = ±90°, where the pressure drops to the value
p/ Pwg = Const- (2u5!g) , as follows from the Bernoulli equation (see Eq. C.84);
the Const is the total head. The whole pressure distribution is symmetrical
about both the cross- and along-flow axes of the cylinder (Fig. 2.22b).
Pressure at a given point on the cylinder surface can generally be represented
as a summation of ambient pressure Po and dynamic pressure induced by the
interaction of the cylinder with the flow: