**4.5. Case of a general periodic pressure gradient: Graphical illustrations of Uchida's results**

Uchida presented graphical illustrations of these results for four different values of the pa‐ rameter ka: 1, 3, 5, and 10.

At each value of the parameter ka and using the angle, nt, as the variable, he plotted twelve different snapshots of the velocity profiles of the unsteady component of velocity for the fol‐ lowing angles:

*nt* =0<sup>0</sup> , 30<sup>0</sup> , 60<sup>0</sup> , 90<sup>0</sup> , 120<sup>0</sup> , 150<sup>0</sup> , 180<sup>0</sup> , 210<sup>0</sup> , 240<sup>0</sup> , 270<sup>0</sup> , 300<sup>0</sup> , 330<sup>0</sup> .

His plots showed that, as the value of ka was increased, the location of maximum velocities shifted progressively away from the axis of the pipe and moved towards the wall. At ka = 1, all maximums of velocity distributions occurred on the axis of the pipe. At ka = 3, two maxi‐ mums of velocity distributions had shifted away from the axis and moved toward the wall of the pipe. These occurred at nt = 0o and nt = 180o . At ka = 5, half the maximums of velocity distributions had shifted away from the axis and moved toward the wall of the pipe. These occurred at nt = 0o , 30 o, 60 o, 180 o, 210 o and 240 o. At ka = 10, all of the maximums of velocity


occurred at nt = 0o and nt = 180o. At ka = 5, half the maximums of velocity distributions had shifted away from the axis and moved pect ratios (a/h =1 and a/h = 10). They presented results for low frequencies ( *αh* =1 ) and

moderate frequencies ( *αh* =8 ). They indicated that results for frequencies higher, *αh* ≥10,

were very similar to those for moderate frequencies. The other conclusions that they came

up with are summarized below.

distributions had shifted away from the axis and occurred by the wall of the pipe. These re‐ sults are summarized in Table 1 and Uchida's (1956) plots are reproduced in enlarged for‐

0.2 0.4 0.6 0.8 1.0

Nondimensional radius �r�R�

**Figure 4.** As the frequency of pressure pulsations increases, the point of maximum velocity shifts progressively away

ka Total maximums Maximums on the axis of the pipe Maximums away from the axis of the pipe

Yakhot, Arad, and Ben-dor conducted numerical studies of pulsating flows in very long rec‐ tangular ducts, where a and h were the horizontal and the vertical dimensions, respectively,

low and high frequency regimes ( 1≤ *αh* ≤20 ) in rectangular ducts using two different as‐

<sup>2</sup>*<sup>v</sup>* )1/2

, they performed calculations for

from the axis of the pipe and moves towards its wall (plots of Eq. (2), for increasing values of n).

1 12 12 0 3 12 10 2 5 12 6 6 10 12 0 12

**Table 1.** Data extracted from Uchida's papers ( his Figures 1, 2, 3, and 4 are shown below).

**5. Pulsating flow through rectangular ducts**

of the cross-section of the duct, Fig. 7. Letting *<sup>α</sup>* =( *<sup>ω</sup>*

**5.1. Summary of the results of analysis**

profiles of the unsteady component of velocity for the following angles:

INLINE FORMULA �� � 0�, 30�, 60�, 90�, 120�, 150�, 180�, 210�, 240�, 270�, 300�, 330�.

mats in Figures 5(a), 5(b), 6(a), and 6(b), as shown below.

106 Wind Tunnel Designs and Their Diverse Engineering Applications

Then, the velocity takes the form

�� ��� � ��� ��� �

Consider very fast pulsations of the pressure gradients. If �� � ��

Near the center of the pipe, ��� � ���������� � 0, one gets

� �

cn 2 2 n 1 0

  � ��� ��� � �

�

u r <sup>8</sup> a k(a r) k(a r) 2 1 sin(nt) exp sin nt U r a ka 2 2

 

<sup>8</sup> a k(a r) k(a r) cos(nt) exp cos nt . r ka 2 2

��� ��∑ �����

� �� ���.�(23)

periodic function of time and is always in phase with the driving pressure gradient.

pipe is different from that near the wall of the pipe. So, they are discussed separately.

� ��� ��� � �

of 90o relative the driving pressure gradient and its amplitude decreases as the frequency of pulsation increases.

� � ��� .(24)

Near the wall of the pipe, ��� � ��� � �, and one uses asymptotic expansions of Bessel functions to get

�

**4.5 Case of a general periodic pressure gradient: graphical illustrations of Uchida's results** 

At each value of the parameter ka and using the angle, nt, as the variable, he plotted twelve different snapshots of the velocity

ka = 3, two maximums of velocity distributions had shifted away from the axis and moved toward the wall of the pipe. These

toward the wall of the pipe. These occurred at nt = 0o, 30 o, 60 o, 180 o, 210 o and 240 o. At ka = 10, all of the maximums of velocity distributions had shifted away from the axis and occurred by the wall of the pipe. These results are summarized in Table 1 and

Uchida's (1956) plots are reproduced in enlarged formats in Figures 5(a), 5(b), 6(a), and 6(b), as shown in the appendix.

Uchida presented graphical illustrations of these results for four different values of the parameter ka: 1, 3, 5, and 10.

In this case, the velocity distribution is a quadratic function of the radial distance from the axis of the pipe ; and the corresponding velocity profile is parabolic. This result is similar to what is obtained in steady flow. However, the magnitude of the velocity is a

Uchida used asymptotic expansions of ber(ka) and bei(ka). In this extreme, the expression for the velocity near the center of the

Comparing this to Eq. (14), one sees that when the pulsations are very rapid, fluid near the axis of the pipe moves with a phase lag

(25)

� �� � �, pulsations of the pressure gradients are very fast. Then,

��� � ��� ��� � � �� ��� � �

�� �a� � ��� � <sup>∑</sup> �����

2

sn

n 1 0

 

0.0

0.2

0.4

Nondimensional

 mean of the squared

0.6

0.8

1.0

1.2

1.4

velocity

2

�

� � � ����

� � 2 �1 � ��

A Method of Evaluating the Presence of Fan-Blade-Rotation Induced Unsteadiness in Wind Tunnel Experiments http://dx.doi.org/10.5772/54144 109

108 Wind Tunnel Designs and Their Diverse Engineering Applications

**Figure 5.** (a). Where maximums of velocity distributions occur when the parameter ka = 1. The angle nt is the parame‐ ter; in these plots, *nt* = 0<sup>0</sup> , 30<sup>0</sup> , 60<sup>0</sup> , 90<sup>0</sup> , 120<sup>0</sup> , 150<sup>0</sup> , 180<sup>0</sup> , 210<sup>0</sup> , 240<sup>0</sup> , 270<sup>0</sup> , 300<sup>0</sup> , 330<sup>0</sup> . (b). Where maximums of velocity distributions occur when the parameter ka = 3. The angle nt is the parameter; in these plots, *nt* = 0<sup>0</sup> , 30<sup>0</sup> , 60<sup>0</sup> , 90<sup>0</sup> , 120<sup>0</sup> , 150<sup>0</sup> , 180<sup>0</sup> , 210<sup>0</sup> , 240<sup>0</sup> , 270<sup>0</sup> , 300<sup>0</sup> , 330<sup>0</sup> .

A Method of Evaluating the Presence of Fan-Blade-Rotation Induced Unsteadiness in Wind Tunnel Experiments http://dx.doi.org/10.5772/54144 111

110 Wind Tunnel Designs and Their Diverse Engineering Applications

**Figure 6.** (a) Where maximums of velocity distributions occur when the parameter ka = 5. The angle nt is the parame‐ ter; *intheseplots*, *nt* = 0<sup>0</sup> , 30<sup>0</sup> , 60<sup>0</sup> , 90<sup>0</sup> , 120<sup>0</sup> , 150<sup>0</sup> , 180<sup>0</sup> , 210<sup>0</sup> , 240<sup>0</sup> , 270<sup>0</sup> , 300<sup>0</sup> , 330<sup>0</sup> . (b). Where maximums of velocity distributions occur when the parameter ka = 10. The angle nt is the parameter; *in these plots*, *nt* = 0<sup>0</sup> , 30<sup>0</sup> , 60<sup>0</sup> , 90<sup>0</sup> , 120<sup>0</sup> , 150<sup>0</sup> , 180<sup>0</sup> , 210<sup>0</sup> , 240<sup>0</sup> , 270<sup>0</sup> , 300<sup>0</sup> , 330<sup>0</sup> .

**Figure 7.** Sketch of the rectangular duct used by Yakhot, Arad and Ben-dor (1999) in their numerical studies.

For low pulsating frequencies, *h* =1 , flow in a duct of square cross-sectional area, the ve‐ locity distribution is in phase, that is in lock step, with the driving pressure gradient. This was true at low and at high aspect ratios. This result is the same as what happens in the case of flow between parallel plates. When one compares the amplitudes of the in‐ duced velocity, one finds that the amplitude of flow between flat plates is larger than that in a square duct. This is due to the fact that, in a duct the fluid experiences friction of four sides, whereas in the case of flow between parallel plates, it experiences flow on‐ ly from two sides. When the aspect ratio is increased to a/h = 10, the velocity in the duct differs only with the velocity between parallel plates near the side walls. This is clearly due to the effects of viscosity.

For moderately pulsating frequencies, *αh* =8 , the velocity distribution of the flow in a duct of square cross- sectional area differs considerably from that obtained at low frequencies. The shapes of the velocity profiles are different; results indicate that, at certain instants of time during a complete cycle, the profiles reach maximum values near the wall of the pipe rather than on its axis of symmetry. This is Richardson's "annular effect". The induced ve‐ locity is no longer in phase, that is in lock step, with the driving pressure gradient. Rather, the velocity is shifted with respect to the driving pressure and the magnitude of the shift de‐ pends on how far away points in the flow space are from the wall. Near the wall, the in‐ duced velocity on the axis of the duct lags behind that in the regions that are near the walls of the duct. On the axis, the phase shift is 90o . This was true at low and at high aspect ratios. This result is the same as what happens in the case of flow between parallel plates. When one compares the amplitudes of the induced velocity, one finds that the amplitude of flow between flat plates is larger than that in a square duct. This is due to the fact that, in a duct the fluid experiences friction of four sides, whereas in the case of flow between parallel plates, it experiences flow only from two sides. When the aspect ratio is increased to a/h = 10, the velocity in the duct differs only with the velocity between parallel plates near the side walls. This is clearly due to the effects of viscosity.
