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6 Non-inertial Reference Frame
Thus far we have assumed use of an inertial reference frame (i.e. fixed with respect to the
distant stars in deriving the CV and differential form of the momentum equation).
However, in many cases non-inertial reference frames are useful (e.g. rotational
machinery, vehicle dynamics, geophysical applications, etc).
In a noninertial frame of reference:
Continuity equation is unchanged.
Momentum equation must be modified.
Reference frame
translates at velocity
⁄
and rotates at angular
velocity
with respect to a stationary
reference frame
. The vectors and
may be resolved in either frame.
.
A fluid particle P can be located in the rotating frame
. The velocity of the fluid particle is:
or in the stationary frame , and
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is from the geometric construction of the cross
product for .
In , the rotation of
causes to trace a
small portion of a cone with radius
:
|
The magnitude of the change in is |
| | , so | |⁄
| |, which is equal
to the magnitude of
.
⁄ is perpendicular to
The direction of
and , which is the direction of
.
⁄
Thus, by geometric construction,
⁄
, and
.
The acceleration of a fluid particle at P, take the time derivative of the velocity :
[
{(
]
)
[
]}
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Jianming Yang
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Fall 2012
In fluid mechanics, the acceleration
( )
(
of fluid particles is denoted
⁄
:
)
The momentum equation of an incompressible flow in a noninertial reference frame:
(
)
[
]
[
]
where the primes denote differentiation, velocity, and position in the noninertial frame.
Thermodynamic variables and the net viscous stress are independent of the reference
frame. The primary effect of a noninertial frame is the addition of extra body force terms
due to the relative acceleration:
⁄
⁄
is the acceleration of with respect to . It provides the
apparent force that pushes occupants back into their seats when a vehicle accelerates.
is the Coriolis acceleration, which depends on the velocity, not on position.
is the acceleration caused by angular acceleration of the noninertial frame
is the centripetal acceleration, which depends strongly on the rotation
rate and the distance of the fluid particle from the axis of rotation.
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Fall 2012
In geophysical flows:
⁄
(earth not accelerating relative to distant stars)
⁄
(for earth)
(can be added to )
is the most important!
⁄
Rossby number
, if
, Coriolis term is larger than the
inertia terms and is important. The Coriolis acceleration is responsible for wind
circulation patterns around centers of high and low pressure in the earth’s atmosphere.
Control volume form of Momentum equation for non-inertial reference frame:
∑
∫
∫
∫
where is the fluid velocity in the non-inertial reference frame (without prime),
fluid velocity relative to the control volume in the non-inertial reference frame.
is the
Usually, the control volume is fixed in a non-inertial reference frame:
∑
∫
∫
∫
Both equations in the non-inertial reference frame have the “same” right-hand sides as
those in the inertial reference frame. Only difference is the “apparent” force ∫
on the left-hand side.
058:0160
Jianming Yang
Fall 2012
7 Curvilinear coordinate systems
Position vector:
Velocity vector:
Gradient of a scalar :
Laplacian of a scalar :
Divergence of a vector:
Gradient of a vector:
Curl of a vector, vorticity:
Laplacian of a vector:
Divergence of a tensor:
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7.1Cartesian Coordinates (Kundu et al., Fluid Mechanics)
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7.2Cylindrical Coordinates (Kundu et al., Fluid Mechanics)
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7.3Spherical Coordinates (Kundu et al., Fluid Mechanics)
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7.4General Orthogonal Coordinates (Bird et al., Transport Phenomena)
The relation between Cartesian coordinates
and the curvilinear coordinates
{
is
or
These can be solved for the
to get the inverse relations
. Then the unit
vectors in rectangular coordinates and the
in curvilinear coordinates are related:
{
∑
∑
in which the “scale factors”
(
(
)
)
∑
(
)
∑
(
)
are given by
∑ (
The spatial derivatives of the unit vectors
)
[∑ (
) ]
can then be found to be
∑
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Fall 2012
The -operator is
∑
The divergence of a vector:
∑
(
)
∑
(
)
The Laplacian of a scalar:
The curl of a vector:
|
|
in which the unit vectors are those belonging to the curvilinear coordinate system.
Volume element:
Surface element:
here
is a surface element on a surface of constant , where
and