2.016 Hydrodynamics
Reading #5
2.016 Hydrodynamics
Prof. A.H. Techet
Fluid Forces on Bodies
1. Steady Flow
In order to design offshore structures, surface vessels and underwater vehicles, an
understanding of the basic fluid forces acting on a body is needed. In the case of steady
viscous flow, these forces are straightforward.
Lift
force, perpendicular to the velocity, and
Drag
force, inline with the flow, can be calculated based on the fluid velocity,
U
, force
coefficients,
C
D
and
C
, the object’s dimensions or area,
A
, and fluid density,
ρ
. For
L
viscous flows the drag and lift on a body are defined as follows
1
2
F
Drag
=
ρ
U
A
C
D
(5.1)
2
1
2
F
Lift
=
ρ
U
A
C
L
(5.2)
2
These equations can also be used in a quiescent (stationary) fluid for a steady translating
body, where
U
is the body velocity instead of the fluid velocity, since
U
is still the
relative velocity of the fluid with respect to the body.
The drag force arises due to viscous rubbing of the fluid.
The fluid may be thought of as
comprised of several “layers” which move relative to one another.
The layer at the surface
of the body “sticks” to the surface due to the
noslip condition
. The next layer of fluid
away from the surface rubs against the layer below, and this rubbing requires a certain
amount of force because of viscosity.
One would expect that in the absence of viscosity,
the force would go to zero.
Jean Le Rond d'Alembert (17171783) performed a series of experiments to measure the
drag on a sphere in a flowing fluid, and on the basis of the potential flow analysis he
expected that the force would approach zero as the viscosity of the fluid approached zero.
However, this was not the case. The net force seemed to converge on a nonzero value as
the viscosity approached zero. Hence, the vanishing of the net force in the potential flow
analysis is known as d'Alembert's Paradox.
version 3.0
updated 8/30/2005
1
©
2005 A. Techet
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2.016 Hydrodynamics
Reading #5
D’Allembert’s Paradox for a fixed sphere in uniform inflow:
Force on a sphere (radius
a
) in an unbounded STEADY moving fluid with velocity
U
is explored in the following
f
discussion.
U
f
z
r
x
θ
The corresponding 3D potential function for a sphere in uniform inflow is simply:
a
3
⎞
φ
(
r
,
θ
)
=
U
⎜
⎛
r
+
2
r
2
⎠
⎟
cos
θ
(5.3)
f
⎝
The hydrodynamic force on the body due to the unsteady motion of the sphere is given as a
surface integral of pressure around the body. Pressure formulation comes from the
unsteady form of Bernoulli. Force in the xdirection is
⎞
⎛
1
2
⎟
F
= −
ρ
⎜
⎜
∂
φ
+
 ∇
φ

⎟
n
dS
(5.4)
x
x
∫∫
⎜
⎝
∂
t
2
⎠
⎟
B
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '05
 AlexandraTechet
 Fluid Dynamics, Force, A. Techet, Prof. A.H. Techet

Click to edit the document details