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4.2 Acceleration
Acceleration of a fluid particle as it moves along a pathline, as shown in Fig. 4.9, is the rate of change of the
particle's velocity with time. The local velocity of the fluid particle depends on the distance traveled, s, and time,
t. The local radius of curvature of the pathline is r. The components of the acceleration vector are shown in Fig.
4.9b. The normal component of acceleration an will be present anytime a fluid particle is moving on a curved
path (i.e., centripetal acceleration). The tangential component of acceleration at will be present if the particle is
changing speed.
Figure 4.9 Particle moving on a pathline. (a) Velocity. (b) Acceleration.
Using normal and tangential components, the velocity of a fluid particle on a pathline (Fig. 4.9a) may be written
as
where V(s, t) is the speed of the particle, which can vary with distance along the pathline, s, and time, t. The
direction of the velocity vector is given by a unit vector et.
Using the definition of acceleration,
(4.1)
To evaluate the derivative of speed in Eq. (4.1), the chain rule for a function of two variables can be used.
(4.2)
In a time dt, the fluid particle moves a distance ds, so the derivative ds/dt corresponds to the speed V of the
particle, and Eq. (4.2) becomes
(4.3)
In Eq. (4.1), the derivative of the unit vector det/dt is nonzero because the direction of the unit vector changes
with time as the particle moves along the pathline. The derivative is
(4.4)
where en is the unit vector perpendicular to the pathline and pointing inward toward the center of curvature 1.
Substituting Eqs. (4.3) and (4.4) into Eq. (4.1) gives the acceleration of the fluid particle:
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(4.5)
The interpretation of this equation is as follows. The acceleration on the left side is the value recorded at a point
in the flow field if one were moving with the fluid particle past that point. The terms on the right side represent
another way to evaluate the fluid particle acceleration at the same point by measuring the velocity, the velocity
gradient, and the velocity change with time at that point and reducing the acceleration according the terms in Eq.
(4.5).
Convective, Local, and Centripetal Acceleration
Inspection of Eq. (4.5) reveals that the acceleration component along a pathline depends on two terms. The
variation of velocity with time at a point on the pathline, namely "V / "t, is called the local acceleration. In
steady flow the local acceleration is zero. The other term, V"V / "s, depends on the variation of velocity along
the pathline and is called the convective acceleration. In a uniform flow, the convective acceleration is zero. The
acceleration with magnitude V2 / r, which is normal to the pathline and directed toward the center of rotation, is
the centripetal acceleration.*
The concept of convective acceleration can be illustrated by use of the cartoon in Fig. 4.10. The carriage
represents the fluid particle, and the track, the pathline. It is assumed that the track is stationary. One way to
measure the acceleration is to ride on the carriage and read the acceleration off an accelerometer. This gives a
direct measurement of dV / dt. The other way is to measure the carriage velocity at two locations separated by a
distance "s and calculate the convective acceleration using
Both methods will give the same answer. The centripetal acceleration could also be measured with an
accelerometer attached to the carriage or by calculating V2 / r if the local radius of curvature of the track is
known.
Figure 4.10 Measuring convective acceleration by two different approaches.
Example 4.1 illustrates how to find the fluid acceleration by evaluating the local and convective acceleration.
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EXAMPLE 4.1 EVALUATI G ACCELERATIO I A
OZZLE
A nozzle is designed such that the velocity in the nozzle varies as
where the velocity u0 is the entrance velocity and L is the nozzle length. The entrance velocity is
10 m/s, and the length is 0.5 m. The velocity is uniform across each section. Find the acceleration at
the station halfway through the nozzle (x/L = 0.5).
Problem Definition
Situation: Given velocity distribution in a nozzle.
Find: Acceleration at nozzle midpoint.
Sketch:
Assumptions: Flow field is quasi one-dimensional (negligible velocity normal to nozzle centerline).
Plan
1. Select the pathline along the centerline of the nozzle.
2. Evaluate the convective, local, and centripetal accelerations in Eq. (4.5).
3. Calculate the acceleration.
Solution
The distance along the pathline is x, so s in Eq. 4.5 becomes x and V becomes u. The pathline is
straight, so r ".
1. Evaluation of terms:
· Convective acceleration
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Evaluation at x/L = 0.5:
· Local acceleration
· Centripetal acceleration
2. Acceleration
Review
Since ax is positive, the direction of the acceleration is positive; that is, the velocity increases in the
x-direction, as expected. Even though the flow is steady, the fluid particles still accelerate.
Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
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