52
2 Water at Rest and in Motion
we can say that the function j(()), represents an excess of pressure due to
the cylinder presence, normalized by (1/2)Pwu6. In the case of an ideal fluid,
function (2.58) becomes (for derivation see Appendix C.4):
p( ()) - Po _ j(()) _ 1 _ 4 . 2 ()
1
2
-
-
sm.
2PwuO
(2.59)
Relationship (2.59) is presented in Fig. 2.22b. Distribution of the normalized
excess pressure is symmetrical with respect to both cross- and along-flow axis
of cylinder. Along the flow axis, normalized pressure is equal to +1, while at
lateral points it is equal to -3. The symmetrical distribution of pressure results
in a zero value of force on cylinder.
It is clear that the assumption of an ideal, non-viscous fluid does not reflect
the everyday situation. Therefore, let us take the viscosity into account when
a boundary layer develops along the cylinder. At the upstream centre, the flow
is much the same as that for an ideal fluid (Fig. 2.22c); the velocity is low and
pressure is equal to total head, i.e. Po + Pwu6/2. However, when proceeding
further along the cylinder, the boundary layer increases in thickness. Fluid is
moving from high to low pressure between points A and B, which causes an
increase in speed (Fig. 2.22c).
However, the increase in speed is not as high as it would be in the absence of
viscosity. Since a certain amount of kinetic energy is lost by the friction within
the thin boundary layer, the remaining energy is too small to overcome the
increasing pressure, which forms an a~rse pressure gradient. The fluid can
not travel into this adverse gradient and so separates from the surface at point
S. The location of separation point, S, depends on the level of turbulence (or
on Reynolds number).
Behind the separation point, eddies are formed and the pressure remains close
to its value at the separation point, which is always less than the pressure at the
forward stagnation point. For example, in Fig. 2.22d, the schematic pressure
distribution.for a real fluid when the Reynolds number 40 < Re = uoa/v < 10 5
is shown. Because of predominantly negative dynamic pressure, except on the
frontal area, a net force in the flow direction is created.
Behind the cylinder, a wake of vortices is formed, with each vortex rotating in
a direction opposite to that of its predecessor further downstream (Fig. 2.22e).
This pattern of vortices is known as a 'von Karman's trail' (Batchelor, 1967;
Vogel, 1994; White, 1994). This periodic pattern of vortices exists for flows
up to a Reynolds number of about 2 x 10 5 . At higher Re values, turbulence
reaches the area very near the surface of the cylinder which provides additional
mo~ntum to the fluid. As a result, the flow follows the cylinder surface for
longer before separation, and an abrupt reduction in the wake width and drag
force appears. This reduction depends on the roughness of the cylinder and on
the turbulence level in the free stream. A detailed description of the separation
of boundary layer from a bluff body is given by Batchelor (1967).
What determines the drag force imposed on a stationary cylinder by real fluid
flow? The starting point is to examine the pressure distribution around the
2 Water at Rest and in Motion
we can say that the function j(()), represents an excess of pressure due to
the cylinder presence, normalized by (1/2)Pwu6. In the case of an ideal fluid,
function (2.58) becomes (for derivation see Appendix C.4):
p( ()) - Po _ j(()) _ 1 _ 4 . 2 ()
1
2
-
-
sm.
2PwuO
(2.59)
Relationship (2.59) is presented in Fig. 2.22b. Distribution of the normalized
excess pressure is symmetrical with respect to both cross- and along-flow axis
of cylinder. Along the flow axis, normalized pressure is equal to +1, while at
lateral points it is equal to -3. The symmetrical distribution of pressure results
in a zero value of force on cylinder.
It is clear that the assumption of an ideal, non-viscous fluid does not reflect
the everyday situation. Therefore, let us take the viscosity into account when
a boundary layer develops along the cylinder. At the upstream centre, the flow
is much the same as that for an ideal fluid (Fig. 2.22c); the velocity is low and
pressure is equal to total head, i.e. Po + Pwu6/2. However, when proceeding
further along the cylinder, the boundary layer increases in thickness. Fluid is
moving from high to low pressure between points A and B, which causes an
increase in speed (Fig. 2.22c).
However, the increase in speed is not as high as it would be in the absence of
viscosity. Since a certain amount of kinetic energy is lost by the friction within
the thin boundary layer, the remaining energy is too small to overcome the
increasing pressure, which forms an a~rse pressure gradient. The fluid can
not travel into this adverse gradient and so separates from the surface at point
S. The location of separation point, S, depends on the level of turbulence (or
on Reynolds number).
Behind the separation point, eddies are formed and the pressure remains close
to its value at the separation point, which is always less than the pressure at the
forward stagnation point. For example, in Fig. 2.22d, the schematic pressure
distribution.for a real fluid when the Reynolds number 40 < Re = uoa/v < 10 5
is shown. Because of predominantly negative dynamic pressure, except on the
frontal area, a net force in the flow direction is created.
Behind the cylinder, a wake of vortices is formed, with each vortex rotating in
a direction opposite to that of its predecessor further downstream (Fig. 2.22e).
This pattern of vortices is known as a 'von Karman's trail' (Batchelor, 1967;
Vogel, 1994; White, 1994). This periodic pattern of vortices exists for flows
up to a Reynolds number of about 2 x 10 5 . At higher Re values, turbulence
reaches the area very near the surface of the cylinder which provides additional
mo~ntum to the fluid. As a result, the flow follows the cylinder surface for
longer before separation, and an abrupt reduction in the wake width and drag
force appears. This reduction depends on the roughness of the cylinder and on
the turbulence level in the free stream. A detailed description of the separation
of boundary layer from a bluff body is given by Batchelor (1967).
What determines the drag force imposed on a stationary cylinder by real fluid
flow? The starting point is to examine the pressure distribution around the