EGM 6812, F11, University of Florida, M. Sheplak
1/74
Section 9, Ideal Flow
9
“Ideal” Flow (I
3
)
Now we will focus on a certain class of flows termed “ideal” because they possess negligible
compressibility, rotationality, and viscous effects
Incompressible
:
compressibility effects in the flow are negligible with respect to other
effects
1
0
0
D
V
Dt
5 conditions discussed in class
Irrotational
:
there is negligible rotation of the fluid particles,
0
V
.
For
incompressible flow, irrotationality usually requires zero divergence of shear stress (i.e.,
inviscid flow).
We can see that for a Newtonian, incompressible flow, the divergence of
shear stress and vorticity are related by
2
V
V
V
.
Inviscid:
viscous forces are negligible compared to other forces
b
DV
p
f
Dt
inviscid
b
DV
p
f
Dt
which is Euler’s Equation.
Often, this implies that there is zero strain rate:
0
ij
.
Strictly speaking,
0
ij
is NOT
required. There must be an imbalance of viscous stresses for viscous forces to be
important,
0
.
For a Newtonian fluid with constant viscosity, this means
0
,
which is a broader statement than
0
ij
.
Illustrative example:

This ** preview** has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
EGM 6812, F11, University of Florida, M. Sheplak
2/74
Section 9, Ideal Flow
9.1 Helmholtz Decomposition of Vector Field
Before studying ideal flows, it is beneficial to review the Helmholtz decomposition.
Helmholtz’s theorem states that any vector field
F
, which is finite, uniform, continuous, and
vanishes at infinity, can be uniquely expressed as the sum of the gradient of a scalar potential
and the curl a divergence free vector potential
A
, (Morse and Feschbach, 1953, Karamcheti
1966)
;
0
F
A
A
(0.1)
Constraining
A
to be solenoidal is permissible since it is indeterminate to the extent of the
gradient of a scalar of position and time.
This is not a unique decomposition.
We could still add
an arbitrary gradient of a potential
and still get the same answer.
This will be useful later.
In the case of the velocity field vector in fluid mechanics, this decomposition has the following
physical meaning,
V
V
V
(0.2)
where
V
is the rotational and
V
is the irrotational part.
It follows mathematically that
V
V
(0.3)
and
0
V
(0.4)
where
is the vorticity vector.
If
V
is constrained to be solenoidal (incompressible),
0
V
(0.5)