**6. Anatomy of the shift using expansions of general results into power series**

#### **6.1. Series expansions of Kelvin functions**

The unsteady part of the solution, which is given by *<sup>u</sup> ' <sup>U</sup>* , Eq. (22), can be written to show the pressure gradient explicitly as shown below.

$$\frac{\boldsymbol{\mu}^{\prime}}{\boldsymbol{\Pi}} = \sum\_{n=1}^{\circ} \mathcal{W} \begin{Bmatrix} \mathbf{r}\_{\prime} \ \mathbf{a} \ \mathbf{a} \end{Bmatrix} \begin{Bmatrix} \boldsymbol{\kappa}\_{\times} \ \mathbf{a} \end{Bmatrix} + \frac{\boldsymbol{\kappa}\_{\times}}{\boldsymbol{\kappa}\_{\times}} \sin(\boldsymbol{\mu} \ \mathbf{r} \ \boldsymbol{\varrho}) + \frac{\boldsymbol{\kappa}\_{\times}}{\boldsymbol{\kappa}\_{\times}} \sin(\boldsymbol{\mu} \ \mathbf{r} \ \boldsymbol{\varrho}) \end{Bmatrix} \tag{26}$$

Where

**Figure 9.** Velocity profiles in pulsating flow at different instants of one period. (a) Pressure gradient variation with time. (b) Duct flow, a/*h* =10, α *h=1*: solid line, *x*/*a* = 0.5; dot, *x*/*a* = 0.1; dashed, *x*/*a* = 0.025; dot-dashed, *x*/*a* = 0.01. (c)

**Figure 10.** Velocity profiles in pulsating flow at different instants of one period. (a) Pressure gradient variation with time. (b) Duct flow, a/*h* =1, α *h=8*: solid line, *x*/*a* = 0.5; dashed, *x*/*a* = 0.25; dot-dashed, *x*/*a* = 0.1. (c) Flow between two

Flow between two parallel plates.

114 Wind Tunnel Designs and Their Diverse Engineering Applications

parallel plates.

$$\mathcal{W}(r,\ a,\ k) = \frac{8B}{(ka)^2} \left[B^2 + (1 - A)^2\right]^{1/2} \tag{27}$$

And *tan*(*φ*(*r*, *<sup>a</sup>*, *<sup>k</sup>*))= 1 - *<sup>A</sup> B*

After a considerable amount of algebra using series expansions for the ber and bei functions, it can be shown that

$$\mathcal{W}(r,\ a,\ k) = 2\left(1 - \frac{r^2}{a^2}\right) \mathcal{D}(r,\ a,\ k) \tag{28}$$

Where D(r, a, k) is a dimensionless factor that is defined as shown below

$$D(r, \ a, \ k) = \left\{ \frac{\stackrel{\equiv}{\sum}\_{n'=0} F\_n(x, \ y)}{\frac{\binom{\text{len}}{\text{len}}\_n + \text{bei}^2 \text{ka}}{\binom{\text{len}}{\text{len}}\_n + \text{bei}^2 \text{ka}}} \right\}^{1/2} \tag{29}$$

Where m = 4n', with n' = 0,1,2,3,…, *<sup>x</sup>* <sup>=</sup>*ka* , y = *<sup>r</sup> <sup>a</sup>* , and *Fm*(*x*, *y*) denotes a family of polyno‐ mials a sample of which is shown below

$$F\_0 = 1\tag{30}$$

*<sup>F</sup>*<sup>4</sup> <sup>=</sup> <sup>4</sup> (4 !)2 ( *<sup>x</sup>* 2 )4 (1 + 10*y* <sup>2</sup> + *y* 4) *<sup>F</sup>*<sup>8</sup> <sup>=</sup> <sup>22</sup> (6 !)2 ( *<sup>x</sup>* 2 )8 (1 - <sup>14</sup> <sup>11</sup> *<sup>y</sup>* <sup>2</sup> <sup>+</sup> 186 <sup>11</sup> *y* 4 - <sup>14</sup> <sup>11</sup> *<sup>y</sup>* <sup>6</sup> <sup>+</sup> *<sup>y</sup>* 8) *<sup>F</sup>*<sup>12</sup> <sup>=</sup> <sup>68</sup> (8 !)2 ( *<sup>x</sup>* <sup>2</sup> )12(1 <sup>+</sup> 66 <sup>17</sup> *<sup>y</sup>* <sup>2</sup> - <sup>277</sup> <sup>17</sup> *y* 4 + 948 <sup>17</sup> *<sup>y</sup>* <sup>6</sup> - <sup>277</sup> <sup>17</sup> *y* 8 + 66 <sup>17</sup> *<sup>y</sup>* 10+ *<sup>y</sup>*12) *<sup>F</sup>*<sup>16</sup> <sup>=</sup> <sup>254</sup> (10 !)2 ( *<sup>x</sup>* <sup>2</sup> )16(1 <sup>+</sup> 154 <sup>127</sup> *<sup>y</sup>* <sup>2</sup> <sup>+</sup> 2206 <sup>127</sup> *y* 4 - <sup>10142</sup> <sup>127</sup> *<sup>y</sup>* <sup>6</sup> <sup>+</sup> 21610 <sup>127</sup> *y* 8 - <sup>10142</sup> <sup>127</sup> *<sup>y</sup>* 10+ 2206 <sup>127</sup> *y* 12 + 154 <sup>127</sup> *y* 14 + *y* 16) *<sup>F</sup>*<sup>20</sup> <sup>=</sup> <sup>922</sup> (12 !)2 ( *<sup>x</sup>* <sup>2</sup> )20(1 <sup>+</sup> <sup>1066</sup> <sup>461</sup> *<sup>y</sup>* <sup>2</sup> - <sup>2685</sup> <sup>461</sup> *y* 4 <sup>+</sup> <sup>41964</sup> <sup>461</sup> *<sup>y</sup>* <sup>6</sup> - <sup>158412</sup> <sup>461</sup> *y* 8 <sup>+</sup> <sup>268476</sup> <sup>461</sup> *<sup>y</sup>* 10- 158412 <sup>461</sup> *y* <sup>12</sup> <sup>+</sup> <sup>41964</sup> <sup>461</sup> *y* <sup>14</sup> - <sup>2685</sup> <sup>461</sup> *<sup>y</sup>* <sup>16</sup> <sup>+</sup> <sup>1066</sup> <sup>461</sup> *<sup>y</sup>* <sup>18</sup> <sup>+</sup> *<sup>y</sup>* 20) *<sup>F</sup>*<sup>24</sup> <sup>=</sup> <sup>3434</sup> (14!)2 ( *<sup>x</sup>* 2 ) 24(1 <sup>+</sup> <sup>3238</sup> <sup>1717</sup> *<sup>y</sup>* <sup>2</sup> <sup>+</sup> <sup>13040</sup> <sup>1717</sup> *<sup>y</sup>* <sup>4</sup> <sup>−</sup> <sup>109654</sup> <sup>1717</sup> *<sup>y</sup>* <sup>6</sup> <sup>+</sup> <sup>769653</sup> <sup>1717</sup> *<sup>y</sup>* <sup>8</sup> <sup>−</sup> <sup>2359044</sup> <sup>1717</sup> *<sup>y</sup>* <sup>10</sup> <sup>+</sup> <sup>3530268</sup> <sup>1717</sup> *<sup>y</sup>* <sup>12</sup> <sup>−</sup> <sup>−</sup> <sup>2359044</sup> <sup>1717</sup> *<sup>y</sup>* <sup>14</sup> <sup>+</sup> <sup>769653</sup> <sup>1717</sup> *<sup>y</sup>* <sup>16</sup> <sup>−</sup> <sup>109654</sup> <sup>1717</sup> *<sup>y</sup>* <sup>18</sup> <sup>+</sup> <sup>13040</sup> <sup>1717</sup> *<sup>y</sup>* <sup>20</sup> <sup>+</sup> <sup>3238</sup> <sup>1717</sup> *<sup>y</sup>* <sup>22</sup> <sup>+</sup> *<sup>y</sup>* <sup>24</sup> ) *<sup>F</sup>*<sup>28</sup> <sup>=</sup> <sup>12868</sup> (16 !)2 ( *<sup>x</sup>* <sup>2</sup> )28(1 <sup>+</sup> <sup>25992</sup> <sup>12868</sup> *<sup>y</sup>* <sup>2</sup> <sup>+</sup> <sup>24716</sup> <sup>12868</sup> *<sup>y</sup>* <sup>4</sup> <sup>+</sup> <sup>337040</sup> <sup>12868</sup> *<sup>y</sup>* <sup>6</sup> <sup>−</sup> <sup>2663036</sup> <sup>12868</sup> *<sup>y</sup>* <sup>8</sup> <sup>+</sup> <sup>13416312</sup> <sup>12868</sup> *<sup>y</sup>* <sup>10</sup> <sup>−</sup> <sup>34632404</sup> <sup>12868</sup> *<sup>y</sup>* <sup>12</sup> <sup>+</sup> <sup>48192480</sup> <sup>12868</sup> *<sup>y</sup>* <sup>14</sup> <sup>−</sup> <sup>−</sup> <sup>34632404</sup> <sup>12868</sup> *<sup>y</sup>* <sup>16</sup> <sup>+</sup> <sup>13416312</sup> <sup>12868</sup> *<sup>y</sup>* <sup>18</sup> <sup>−</sup> <sup>2663036</sup> <sup>12868</sup> *<sup>y</sup>* <sup>20</sup> <sup>+</sup> <sup>337040</sup> <sup>12868</sup> *<sup>y</sup>* <sup>22</sup> <sup>+</sup> <sup>24716</sup> <sup>12868</sup> *<sup>y</sup>* <sup>24</sup> <sup>+</sup> <sup>25992</sup> <sup>12868</sup> *<sup>y</sup>* <sup>26</sup> <sup>+</sup> *<sup>y</sup>* <sup>28</sup> )

Note, from the definition of w(r, a, k), Eq. (28), that each of these polynomials will be multi‐ plied by the steady velocity. Clearly, this shows that all components that are added to the velocity due to unsteadiness are essentially various forms of the same steady velocity after it has been modified by the introduction of time variations. The series of equations shown be‐ low demonstrates this observation:

$$\frac{\mu}{\frac{\mu}{\Delta I}} = \frac{u\_\*}{\frac{\mu}{\Delta I}} + \frac{u^\*}{\frac{\mu}{\Delta I}}.\tag{31}$$

$$\mathbf{1} \xrightarrow[\overline{\underline{\mathbf{U}}}{\overline{\underline{\mathbf{U}}}}]{} \mathbf{2} \left(\mathbf{1} - \frac{r^{\*2}}{a^{\*2}}\right) \tag{32}$$

$$\frac{\mu}{\Pi} = 2\left(1 - \frac{r^2}{a^2}\right) + \sum\_{n=1}^{\infty} W\left(r, \text{ a. } k\right) \Big| \frac{\varkappa\_m}{\varkappa\_0} \cos\{nt \cdot \varphi\} + \frac{\varkappa\_m}{\varkappa\_0} \sin\{nt \cdot \varphi\} \Big|\_{\text{}} \tag{33}$$

$$\frac{du}{d\Omega} = \mathcal{Q}\left(1 - \frac{r^2}{a^2}\right) + \mathcal{Q}\left(1 - \frac{r^2}{a^2}\right) \sum\_{n=1}^{\infty} D\left(r, \text{ a, } k\right) \Big| \frac{\varkappa\_m}{\varkappa\_0} \cos\{nt \cdot \varphi\} + \frac{\varkappa\_m}{\varkappa\_0} \sin\{nt \cdot \varphi\} \Big|\_{\prime} \tag{34}$$

$$\mathbf{x}\_{\overline{\Pi}}^{\underline{u}} = \mathcal{Q} \left( \mathbf{1} - \frac{r^2}{a^2} \right) + \mathcal{Q} \left( \mathbf{1} - \frac{r^2}{a^2} \right) \sum\_{n=1}^{\infty} \frac{D(r, a, k)}{\varkappa\_0} \{ \varkappa\_{cn} \cos(\mathbf{nt} \cdot \boldsymbol{\varphi}) + \varkappa\_{sn} \sin(\mathbf{nt} \cdot \boldsymbol{\varphi}) \}. \tag{35}$$

Fig. 13. Each coordinate frame shows plots of three functions����, �� vs. y: ����, ��, �����, ��, and �����, ��; x is the parameter. Note that larger values of x indicate higher rates of pulsations by the pressure gradient. **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 the parameter. Note that larger values of x indicate higher rates of pulsations by the pressure gradient.

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

7. Compiled summary of results from several investigators and conclusions

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 which the pulsating flow was subjected. The phase lag of the velocity variation from that of the pressure gradient increases from zero in the steady flow to 90<sup>o</sup> in the pulsation of infinite frequency. Integration of the work needed for changing the kinetic energy of fluid over a complete cycle yields zero, however, a 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 (1956) under the assumption that that flow was parallel to the axis of the pipe.

Tyler (1929-1930) confirmed these findings with more experiments with a pure periodic flow generated by the

similar integration of the dissipation of energy by internal friction remains finite and an excess amount caused by the components of periodic motion is added to what is generated by the steady flow alone. 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.

necessary over and above what is required when the flow is steady. Recently, Camacho, Martinez, and Rendon (2012) showed that the location of the characteristic overshoot of the The phase lag of the velocity variation from that of the pressure gradient increases from zero in the steady flow to 90o in the pulsation of infinite frequency.

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 Stokes' second problem. 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 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 added to what is generated by the steady flow alone.

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%. 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 when the flow is steady.

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

All these results were obtained in flows through pipes of circular cross-sections and rectan‐ gular ducts. It is reasonable to expect that they would hold in the flow of air in a wind tun‐ nel. 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 evi‐ dence 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%.
