**Nomenclature of the symbols (with units)**

*α* : a dimensionless ratio that combines the rate of pressure pulsations and the distance from the wall of the pipe;


a) x = 1 b) x= 5

the parameter. Note that larger values of x indicate higher rates of pulsations by the pressure gradient.

**7. Compiled summary of results from several investigators and**

7. Compiled summary of results from several investigators and conclusions

�1.0 �0.5 0.5 1.0

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4

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118 Wind Tunnel Designs and Their Diverse Engineering Applications

which the pulsating flow was subjected.

Stokes' second problem.

when the flow is steady.

in the steady flow to 90o

in the pulsation of infinite frequency.

90<sup>o</sup>

**conclusions**

the parameter. Note that larger values of x indicate higher rates of pulsations by the pressure gradient.

Fig. 13. Each coordinate frame shows plots of three functions����, �� vs. y: ����, ��, �����, ��, and �����, ��; x is

**Figure 13.** Each coordinate frame shows plots of three functions *Fm*(*x*, *y*) vs. y: *F*4(*x*, *y*), *F*12(*x*, *y*), and *F*28(*x*, *y*); x is

�1.0 �0.5 0.5 1.0

10

20

30

40

50

While conducting experiment on sound waves in resonators, Richardson (1928) measured velocities across an orifice of circular cross-section and found that the maximum velocity could occur away from the axis of symmetry and toward the wall. Sexl (1930) proved analytically that what Richardson observed could happen. Richardson and Tyler (1929-1930) confirmed these findings with more experiments with a pure periodic flow generated by the reciprocating motion of a piston. Uchida (1956) studied the case of periodic motions that were superposed upon a steady Poiseuille flow. An exact solution for the pulsating laminar flow that is superposed on the steady motion in a circular pipe was presented by Uchida (1956) under the assumption that that flow was parallel to the axis of the pipe.

The total mean mass of flow in pulsating motion was found to be identical to that given by Hagen-Poiseuille's law when the steady pressure gradient used in the Hagen-Poiseuille's law was equal to the mean pressure gradient to

While conducting experiment on sound waves in resonators, Richardson (1928) measured velocities across an orifice of circular cross-section and found that the maximum velocity could occur away from the axis of symmetry and toward the wall. Sexl (1930) proved analyt‐ ically that what Richardson observed could happen. Richardson and Tyler (1929-1930) con‐ firmed these findings with more experiments with a pure periodic flow generated by the reciprocating motion of a piston. Uchida (1956) studied the case of periodic motions that were superposed upon a steady Poiseuille flow. An exact solution for the pulsating laminar flow that is superposed on the steady motion in a circular pipe was presented by Uchida

The phase lag of the velocity variation from that of the pressure gradient increases from zero in the steady flow to

Integration of the work needed for changing the kinetic energy of fluid over a complete cycle yields zero, however, a similar integration of the dissipation of energy by internal friction remains finite and an excess amount caused by the

It follows that a given rate of mass flow can be attained in pulsating motion by giving the same amount of average gradient of pressure as in steady flow. However, in order to maintain this motion in pulsating flow, extra work is

The total mean mass of flow in pulsating motion was found to be identical to that given by Hagen-Poiseuille's law when the steady pressure gradient used in the Hagen-Poiseuille's law was equal to the mean pressure gradient to which the pulsating flow was subjected.

 Recently, Camacho, Martinez, and Rendon (2012) showed that the location of the characteristic overshoot of the Richardson's annular effect changes with the kinematic Reynolds number in the range of frequencies within the laminar regime. They identified the existence of transverse damped waves that are similar to those observed in

Integration of the work needed for changing the kinetic energy of fluid over a complete cy‐ cle yields zero, however, a similar integration of the dissipation of energy by internal fric‐ tion remains finite and an excess amount caused by the components of periodic motion is

It follows that a given rate of mass flow can be attained in pulsating motion by giving the same amount of average gradient of pressure as in steady flow. However, in order to main‐ tain this motion in pulsating flow, extra work is necessary over and above what is required

The phase lag of the velocity variation from that of the pressure gradient increases from zero

All these results were obtained in flows through pipes of circular cross-sections and rectangular ducts. It is reasonable to expect that they would hold in the flow of air in a wind tunnel. Experimental results indicate that the Richardson's annular effect does occur in the test section of a subsonic wind tunnel. That behavior first appears unusual and, indeed, odd. However, as shown in this chapter, there is considerable experimental and analytical evidence in the literature that indicates that this behavior is due to high-frequency pulsations of the pressure gradient. Accordingly, in the case of a subsonic wind tunnel, it is probably due to the fast rate of rotation of fan blades. Indeed, in our wind tunnel, results from analysis and those from experiments differed only by about 5.7%.

in the pulsation of infinite frequency.

components of periodic motion is added to what is generated by the steady flow alone.

(1956) under the assumption that that flow was parallel to the axis of the pipe.

necessary over and above what is required when the flow is steady.

added to what is generated by the steady flow alone.


*ϰcn* and *ϰsn* : Fourier coefficients of the pressure gradient; *ϰo* is the steady part of the pressure gradient (m/s2 );


k: a dimensionless ratio used by Schlichting to denote the magnitude of the frequency of os‐ cillation

K: a symbol used by Schlichting to indicate the magnitude of the pressure gradient

n: denotes the circular frequency of pressure oscillations (rad/s)

P: the pressure that drives the flow (N/m2 )

∂ *p* <sup>∂</sup> *<sup>x</sup>* : the pressure gradient in the axial direction of an infinitely long pipe

r: the radial distance measured from the axis of the pipe (m)

R: the inside radius of a pipe of circular cross section (m)

t: time elapsed (s)

u: the axial velocity of the flow (m/s)

us: the steady part of the velocity u (m/s)

*u '* : the unsteady part of the velocity u (m/s

*U* : the mean speed (m/s) of the velocity u (m/s)

x: a dimensionless ratio that measures the rate of pulsations of the pressure gradient

y: a dimensional distance from the wall of the pipe (m)
