**1.2. Flows driven by a sinusoidal pressure gradient through a pipe of circular cross section**

Things become more complicated if the pressure gradient varies with time. When, for exam‐ ple, the pressure gradient fluctuates with time in such a way that that gradient can be ex‐

> © 2013 Libii; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Libii; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

pressed as a simple sinusoidal function, the velocity profile remains parabolic only at very low frequencies of fluctuation. At very high frequencies, the location of the maximum veloc‐ ity moves away from the axis of the pipe and towards the wall. The higher the frequency of oscillations of the pressure gradient, the farther away the point of maximum velocity moves from the axis of the pipe. Sample plots of velocity profiles that were generated at high fre‐ quencies of fluctuations are shown in the literature by Uchida (1956). Here, Figure 2 is one such example, where five snapshots of velocity profiles at different times are displayed, from left to right, within one complete cycle: at the beginning, one-quarter, half-way, threequarters of the way, and at the very end of the cycle. The values of the parameters that were used to generate these plots are summarized below:

$$k - \frac{1}{\rho K} \frac{\partial \cdot p}{\partial \ x} = \cos(\nu t); \quad k = \sqrt{\frac{n}{v}} R = \mathbf{5}; \mathbf{c} = \frac{\boldsymbol{\kappa} \, k^{\; \; \;}}{8n} = \mathbf{3}.125 \frac{\boldsymbol{\kappa}}{n}$$

Where n is the circular frequency, p the pressure, *ρ* the mass density of the fluid, t the time, x the axial coordinate, R the inside radius of the pipe, u the axial speed of the fluid, *v* the coeffi‐ cient of kinematic viscosity, k a dimensionless ratio used by Schlichting to denote the magni‐ tude of the frequency of oscillation, and K is a constant that indicates the size of the pressure gradient.

**Figure 2.** Sample velocity profiles for flow driven by a sinusoidal pressure gradient in a circular pipe [Uchida]

#### **1.3. The mean velocity squared and Richardson's annular effect**

The higher the frequency of oscillations of the pressure gradient, the farther away the point of maximum velocity moves from the axis of the pipe. The phenomenon in which the point of maximum velocity moves away from the axis of the pipe and shifts towards its wall is known as Richardson's annular effect. It was demonstrated experimentally by Richardson (1929), proved analytically by Sexl (1930), and demonstrated to hold for any pressure gradi‐ ent that is periodic with time by Uchida (1956).

When the sinusoidal pressure gradient that drives the flow in a circular pipe has fast oscilla‐ tions, the mean velocity squared computed with respect to time is found to be

A Method of Evaluating the Presence of Fan-Blade-Rotation Induced Unsteadiness in Wind Tunnel Experiments http://dx.doi.org/10.5772/54144 99

$$\begin{aligned} \text{In dimensions of drawing the } n \text{-axis are:}\\ \text{In } \overline{\text{L}^2(r)} = \frac{\overline{\text{L}^2}}{2n} \Big[ 1 - 2\sqrt{\frac{R}{r}} \exp\left[ -\sqrt{\frac{n}{2v}}(R-r) \right] \cos\left[ \sqrt{\frac{n}{2v}}(R-r) \right] + \frac{R}{r} \exp\left[ -2\sqrt{\frac{n}{2v}}(R-r) \right] \Big] \end{aligned} \tag{11.57}$$

Where r is the radial distance from the axis of the pipe; and letting *y* =(*R* - *r*) be a new varia‐ ble that represents the distance from the wall of that pipe, a dimensionless distance from that wall can be defined as η = y *<sup>n</sup>* <sup>2</sup>*v* . Using this distance, one can nondimensionalize the mean velocity squared as shown below :

pressed as a simple sinusoidal function, the velocity profile remains parabolic only at very low frequencies of fluctuation. At very high frequencies, the location of the maximum veloc‐ ity moves away from the axis of the pipe and towards the wall. The higher the frequency of oscillations of the pressure gradient, the farther away the point of maximum velocity moves from the axis of the pipe. Sample plots of velocity profiles that were generated at high fre‐ quencies of fluctuations are shown in the literature by Uchida (1956). Here, Figure 2 is one such example, where five snapshots of velocity profiles at different times are displayed, from left to right, within one complete cycle: at the beginning, one-quarter, half-way, threequarters of the way, and at the very end of the cycle. The values of the parameters that were

used to generate these plots are summarized below:

98 Wind Tunnel Designs and Their Diverse Engineering Applications

*<sup>v</sup> <sup>R</sup>* =5; *<sup>c</sup>* <sup>=</sup> *K k* <sup>2</sup>

<sup>8</sup>*<sup>n</sup>* =3.125 *<sup>K</sup>*

**Figure 2.** Sample velocity profiles for flow driven by a sinusoidal pressure gradient in a circular pipe [Uchida]

The higher the frequency of oscillations of the pressure gradient, the farther away the point of maximum velocity moves from the axis of the pipe. The phenomenon in which the point of maximum velocity moves away from the axis of the pipe and shifts towards its wall is known as Richardson's annular effect. It was demonstrated experimentally by Richardson (1929), proved analytically by Sexl (1930), and demonstrated to hold for any pressure gradi‐

When the sinusoidal pressure gradient that drives the flow in a circular pipe has fast oscilla‐

tions, the mean velocity squared computed with respect to time is found to be

**1.3. The mean velocity squared and Richardson's annular effect**

ent that is periodic with time by Uchida (1956).

*n* Where n is the circular frequency, p the pressure, *ρ* the mass density of the fluid, t the time, x the axial coordinate, R the inside radius of the pipe, u the axial speed of the fluid, *v* the coeffi‐ cient of kinematic viscosity, k a dimensionless ratio used by Schlichting to denote the magni‐ tude of the frequency of oscillation, and K is a constant that indicates the size of the pressure

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup>*cos*(*nt*);*<sup>k</sup>* <sup>=</sup> *<sup>n</sup>*


gradient.

$$\begin{aligned} \text{Mictivity squared as shown below} &: \\ \frac{\sqrt{\frac{n^2(r)}{2}}}{\left(\frac{\sqrt{\frac{n}{2v}}{2v}\right)^2}} & \left[1 - 2\sqrt{\frac{R}{r}} \exp\left[-\sqrt{\frac{n}{2v}}(R-r)\right] \cos\left[\sqrt{\frac{n}{2v}}(R-r)\right] + \frac{R}{r} \exp\left[-2\sqrt{\frac{n}{2v}}(R-r)\right]\right] \end{aligned} \tag{2}$$

When one is very close to the wall of the pipe, r and R are very close in magnitude and *R <sup>r</sup>* ≈1 . This causes the expression in Eq. (2) to become

precession in Eq. (2) to become

$$
\frac{\sqrt{\frac{\mu}{\epsilon\_{2}}}}{\left(\frac{\mu}{2^{n}}\right)^{2}} = 1 - 2\exp(-\eta)\cos\eta + \exp(-2\eta).\tag{3}
$$

When the variation of the expression of the mean velocity squared in Eq. (3) is plotted against the dimensionless distance η, as shown in Figure 3, one can see that the location of the maximum velocity is not on the axis of the pipe as is the case in steady flow and at very low oscillations of the pressure gradient. Instead, it occurs near the wall of the pipe at a di‐ mensionless distance η = y *<sup>n</sup>* <sup>2</sup>*v* = 2.28. This value has been shown to agree with experimen‐ tal data collected by Richardson (1929).

**Figure 3.** Variation of the mean with respect to time of the velocity squared for periodic pipe flows that are very fast

In this Figure 3, y is the distance from the wall of the pipe and *u<sup>∞</sup>* <sup>2</sup> <sup>=</sup> *<sup>K</sup>* <sup>2</sup> <sup>2</sup>*<sup>n</sup>* <sup>2</sup> *represents* the mean with respect to time of the velocity squared at a large distance from the wall.
