**4. Pulsating flow through pipes**

### **4.1. Basic equations**

**2. Richardson's annular effect in a wind tunnel**

100 Wind Tunnel Designs and Their Diverse Engineering Applications

tunnel (Njock Libii, 2011).

and graphical illustrations.

**3. Stokes' second problem**

into the fluid, is given by *<sup>δ</sup>* =( <sup>2</sup>*<sup>v</sup>*

Unsteady pulsating flows occur in many situations that have a practical engineering im‐ portance. These include high- speed pulsating flows in reciprocating piston-driven flows, rotor blade aerodynamics and turbomachinery. They also arise in wind-tunnel flows. When the velocity distribution is measured across the test section of a subsonic wind tunnel that is driven by a high speed fan, it has been observed experimentally that, in addition to the effect of the boundary layer that is expected near the wall, Richardson's annular effect can be demonstrated as well. Indeed, published experimental results from our laboratory have demonstrated that Richardson's annular effect can occur in a wind

The purpose of the remainder of this chapter is to summarize the theoretical basis of the Ri‐ chardson's annular effect in pipes of circular sections and in rectangular tubes, illustrate its

First Stokes' second problem is reviewed briefly. The theory of pulsating flows in pipes and ducts is summarized. The anatomy of the shift in the location of the maximum ve‐ locity from the center to points near the wall is presented using series approximations

Fundamental studies of fully-developed and periodic pipe and duct flows with pressure gradients that vary sinusoidally have been done (Sexl, 1930). From such studies, we know that, when an incompressible and viscous fluid is forced to move under a pulsating pressure difference in a pipe or a duct, some characteristic features are always observed. Some of these features are similar to those that are observed to occur in the boundary layer adjacent to a body that is performing reciprocating harmonic oscillations. These features are related to the results of a classic problem solved by Stokes, known as Stokes' second problem, which gives details of the behavior of the boundary layer in a viscous fluid of kinematic vis‐ cosity, *v* , that is bounded by an infinite plane surface that moves back and forth in its own

Stokes solution shows that, for this type of flow, 1) transverse waves propagate away from the oscillating surface and into the viscous fluid; 2) the direction of the velocity of these waves is perpendicular to the direction of propagation; 3) the oscillating fluid layer so gen‐ erated has a phase lag, *φ* , with respect to the motion of the wall; and 4) that phase lag,

length scale introduced by Stokes; that length, called the depth of penetration of the wave

the penetration depth, *δ* . The value of the constant of proportionality varies with the point

*<sup>ω</sup>* )1/2 . The thickness of the boundary layer is proportional to

*<sup>δ</sup>* , where *δ* represents a

results graphically, and relate them to what happens in a wind tunnel.

plane with a simple harmonic oscillation that has a circular frequency, *ω* .

which varies with y, the distance from the wall, is given by, *<sup>φ</sup>* <sup>=</sup> *<sup>y</sup>*

The flow of a viscous fluid in a straight pipe of circular cross-section due to a periodic pres‐ sure gradient was examined experimentally and theoretically by Richardson and Tyler (1929) and theoretically by Sexl (1930). If the pipe is sufficiently long, variations of flow pa‐ rameters along its axis may be neglected and the only component of flow is that along the axis of the pipe. The Navier-Stokes equations become

$$\frac{\partial \ln u}{\partial t} = -\frac{1}{\rho} \frac{\partial \ln p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{1}{r} \frac{\partial u}{\partial r} \right) \tag{4}$$

*u*(*r* =*a*, *t*)=0 ; *and u*( *r* =0, *t*)= *finite*

$$-\frac{1}{\rho} \frac{\partial \rho}{\partial \mathbf{x}} = a \begin{pmatrix} \text{function of time} \\ \end{pmatrix} \tag{5}$$

Where u = u(r, t) is the component of velocity in the axial direction x, <sup>∂</sup> *<sup>p</sup>* <sup>∂</sup> *<sup>x</sup>* is the pressure gra‐ dient in the axial direction, t is the time, *v* is the kinematic viscosity of the fluid, r is the radi‐ al distance measured from the axis of the pipe, and a is the inside radius of the pipe. For a given pressure gradient, one seeks solutions that are finite at r = 0 and satisfy the no-slip condition u = 0 on the wall of the pipe at all times. We present two cases: First, the case of a sinusoidal pressure gradient that was first solved by Sexl (1930) and then that of a general periodic pressure gradient that was first solved by Uchida (1956).

#### **4.2. Case of a sinusoidal pressure gradient: Sexl's method (1930)**

If the pressure gradient is sinusoidal and given the form

$$\frac{\partial \rho}{\partial x} = \rho C \cos(\omega t),\tag{6}$$

then, the solution is given by the real part of

$$\mu(r\_{\prime},t) = -i\frac{C}{\omega} \left| 1 - \frac{J\_{\rho}\left(\left.\frac{1}{\left(\cdot,ix\right)^{\frac{1}{\frac{1}{\omega}}}}\right|\_{\frac{1}{\omega}}\right)}{J\_{\rho}\left(\left.\frac{1}{\left(\cdot,ix\right)^{\frac{1}{\omega}}}\right|\_{\frac{1}{\omega}}\right)}\right|e^{i\omega t}\tag{7}$$

Where Jo is the Bessel function of the first kind and of zero order (Watson,1944) and, here, x is defined as shown below:

$$
\infty = \frac{\omega u^2}{v}.\tag{8}
$$

For small values of the parameter x, the real part of the velocity u can be written as

$$u(r,t) = \frac{\mathbb{C}}{4v} \{a^2 \text{ - } r^2\} \cos(\omega t) \tag{9}$$

and for large values of the parameter x and of ( *<sup>r</sup> a* )2 , the velocity can be represented by

$$\mu(r,t) = \frac{c}{\omega} \left| \sin(\omega t) \cdot \left(\frac{a}{r}\right)^{1/2} \exp(-\alpha) \sin[\omega t \cdot \alpha] \right|;\tag{10}$$

where

$$
\alpha = \left(\frac{x}{2}\right)^{1/2} \left(1 - \frac{r}{a}\right). \tag{11}
$$

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

$$
\begin{aligned}
\text{mean velocity squared computed with respect to time is found to be} \\
\overline{\ln^2(r)} &= \frac{c^{\frac{\alpha}{2}}}{2\omega^3} \Big[ \mathbf{1} - 2 \left( \frac{a}{r} \right)^{\frac{1}{2}} \exp(-\alpha) \cos(\alpha) + \left( \frac{a}{r} \right) \exp(-2\alpha) \Big].
\end{aligned}
\tag{12}
$$

These well-known results indicate that the representation of the velocity changes radically as one varies the parameter x from very small to very large values. For example, the maxi‐ mum velocity reaches its maximum amplitude on the axis of the pipe when x is very small. However, when the frequency of fluctuations becomes large, the location of the maximum velocity shifts away from the axis of the pipe and moves closer and closer to the wall of the pipe as the parameter increases, Fig. 4. Indeed, in the latter case, the expression for the loca‐ tion of maximum velocity is given by

$$r = a \left(1 - 3.22x^{-1/2}\right). \tag{13}$$

#### **4.3. Case of a general periodic pressure gradient: Uchida's general theory**

The case of a general periodic pressure gradient was solved by Uchida (1956), whose solu‐ tion is summarized below.

Consider a general periodic function that can be expressed using a Fourier series as follows:

$$-\frac{1}{\varrho}\frac{\partial \mathbf{p}}{\partial \mathbf{x}} = \varkappa\_0 + \sum\_{\mathbf{n}=1}^{\bullet} \varkappa\_{\mathbf{cn}} \cos \mathbf{n} \mathbf{t} + \sum\_{\mathbf{n}=1}^{\bullet} \varkappa\_{\mathbf{sn}} \sin \mathbf{n} \mathbf{t} \,, \tag{14}$$

Where n is the frequency of oscillation and *ϰcn* and *ϰsn* are Fourier coefficients.

In complex form, the solution to Eq. (4) is given by

$$\dot{\lambda}\_M = \frac{\varkappa\_o}{4\upsilon} \left( \mu^2 - r^2 \right) - \sum\_{n=1}^{\infty} \frac{i\varkappa\_n}{n} \left[ 1 - \frac{\int\_{\upsilon} \left( \frac{\upsilon}{k\upsilon} \right)^2}{\int\_{\upsilon} \left( \frac{\upsilon}{k\upsilon} \right)^2} \right] e^{i nt \mathbf{f}} \tag{15}$$

Where

*u*(*r*, *t*)= - *i*

102 Wind Tunnel Designs and Their Diverse Engineering Applications

is defined as shown below:

where

*C <sup>ω</sup>* {1 - *<sup>J</sup> <sup>o</sup>* ((-*ix*) 1 <sup>2</sup> *<sup>r</sup> a* )

<sup>2</sup> ) }*eiω<sup>t</sup>* (7)

*<sup>v</sup>* . (8)

<sup>4</sup>*<sup>v</sup>* (*<sup>a</sup>* <sup>2</sup> - *<sup>r</sup>* 2) *cos*(*ωt*) (9)

, the velocity can be represented by

*exp*(-*α*) *sin ωt* - *α* }; (10)

*<sup>a</sup>* ). (11)

*r* =*a*(1 - 3.22*x* -1/2). (13)

*<sup>r</sup>* )exp(−2*α*)}. (12)

*J o* ((-*ix*) 1

Where Jo is the Bessel function of the first kind and of zero order (Watson,1944) and, here, x

*a* )2

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

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

exp(−*α*)cos(*α*) <sup>+</sup> ( *<sup>a</sup>*

These well-known results indicate that the representation of the velocity changes radically as one varies the parameter x from very small to very large values. For example, the maxi‐ mum velocity reaches its maximum amplitude on the axis of the pipe when x is very small. However, when the frequency of fluctuations becomes large, the location of the maximum velocity shifts away from the axis of the pipe and moves closer and closer to the wall of the pipe as the parameter increases, Fig. 4. Indeed, in the latter case, the expression for the loca‐

*<sup>x</sup>* <sup>=</sup> *<sup>ω</sup><sup>a</sup>* <sup>2</sup>

For small values of the parameter x, the real part of the velocity u can be written as

*<sup>u</sup>*(*r*, *<sup>t</sup>*)= *<sup>C</sup>*

*<sup>ω</sup>* {*sin*(*ωt*) - ( *<sup>a</sup>*

*<sup>α</sup>* =( *<sup>x</sup>* <sup>2</sup> )1/2 (1 - *<sup>r</sup>*

*r* ) 1 2

and for large values of the parameter x and of ( *<sup>r</sup>*

*<sup>u</sup>*(*r*, *<sup>t</sup>*)= *<sup>C</sup>*

*u* 2 (*r*) ¯

tion of maximum velocity is given by

<sup>=</sup> *<sup>C</sup>* <sup>2</sup> <sup>2</sup>*<sup>ω</sup>* <sup>2</sup> {1−2( *<sup>a</sup>*

$$k = \sqrt{\frac{n}{v}}\tag{16}$$

The total mean velocity U is defined as *<sup>U</sup>* <sup>=</sup> *<sup>G</sup> <sup>π</sup><sup>a</sup>* 2 , where G, the total mean mass flow, is given by

$$G = \frac{1}{2\pi} \int\_0^{2\pi} dt \underbrace{\int\_0^a}\_0^a \pi ur dr = \frac{\pi a^4 \varkappa\_0}{8v} \,. \tag{17}$$

When this expression has been rearranged in order to introduce the mean pressure gradient, one gets

$$\begin{aligned} \text{Como:} & \text{ como} \\ & \text{ em:} \\ & G = \frac{1}{2\pi} \int\_0^a dt \int\_0^a 2\pi urr dr = \frac{\pi a^4 \varkappa\_0}{8\mu} \sqrt{-\frac{\partial \cdot p}{\partial x}} \Big|\_{\text{y}} \end{aligned} \tag{18}$$

Where ( <sup>−</sup> <sup>∂</sup> *<sup>p</sup>* ∂ *x* ¯ ) =*ρϰ*0, is the mean pressure gradient taken over time. Therefore, ¯

$$
\Delta U = \frac{\mu^2}{8\mu} \overline{\left(-\frac{\partial \cdot p}{\partial \cdot x}\right)}.\tag{19}
$$

If one uses U as a velocity scale, the nondimensional expression of the velocity is given by

$$\frac{\frac{\mu}{\Delta T}}{\frac{\Delta T}{\Delta T}} = \frac{\frac{\mu\_s}{\Delta T}}{\frac{\Delta T}{\Delta T}} + \frac{\mu\_s}{\Delta T} \tag{20}$$

with

$$\frac{\mu\_\*}{\Delta I} = 2\left(1 - \frac{r^2}{a^2}\right) \tag{21}$$

And

$$\frac{\frac{1}{\omega\_0}}{\frac{1}{\omega\_0}} = \sum\_{n=1}^{\omega} \frac{\varkappa\_m}{\varkappa\_0} \left\{ \frac{8B}{(\ln r)^2} \cos nt + \frac{8(1-A)}{(\ln r)^2} \sin nt \right\} + \sum\_{n=1}^{\omega} \frac{\varkappa\_m}{\varkappa\_0} \left\{ \frac{8B}{(\ln r)^2} \sin nt - \frac{8(1-A)}{(\ln r)^2} \cos nt \right\} \quad \text{(a)}$$

$$\text{where}$$

$$A = \frac{h \tau (k \dot{x}) \dot{\nu} \epsilon (kr) + h \dot{\nu} (\ln r) \dot{\nu} \epsilon (kr)}{\dot{\nu} \epsilon \tau \left(\frac{1}{\dot{\nu}}\right) + h \dot{\nu} \dot{\nu} \left(\frac{1}{\dot{\nu}}\right)}, \quad B = \frac{h \tau (k \dot{x}) \dot{\nu} \epsilon (kr) - \dot{\nu} \epsilon (ka)}{\dot{\nu} \epsilon \tau \left(\frac{1}{\dot{\nu}}\right)} \quad \text{(a)}$$

$$\text{And}$$

$$\int\_{\omega} \left(\frac{1}{\omega\_0} \frac{\dot{\nu}}{\dot{\tau}}\right) = \text{ber} \left(kr \right) + i \dot{\nu} \epsilon \dot{\nu} \left(kr \right) \text{ (b)}\tag{22}$$

In which *ber* and *bei* are Kelvin functions defined using infinite series as shown below:

$$\begin{aligned} ber(z) &= \sum\_{k=0}^{\omega} \frac{(-1)^k \left(\frac{z}{2}\right)^{4k}}{((2k)!)^{2}} \text{(c)}\\ &\text{and} \end{aligned}$$

$$\operatorname{bei}(z) = \sum\_{k=0}^{\cdots} \frac{(-1)^k \left(\frac{z}{2}\right)^{4k+2}}{\left((2k+1)!\right)^2} \dots \quad \text{(d)}$$

#### **4.4. Asymptotic expressions of the velocity distribution**

Two extreme cases were considered by Uchida: the case of very slow pulsations and that of very fast pulsations, depending on the magnitude of the dimensionless parameter *ka* <sup>=</sup> *<sup>n</sup> v a*

Consider very slow pulsations of the pressure gradients. If = *<sup>n</sup> <sup>v</sup> a* ≪1 , pulsations of the pressure gradients are very slow. Then, under these conditions and from the behavior of Kelvin functions, it is reasonable to expect that

*berka* →1 *and beika* →0.

Then, the velocity takes the form

$$\frac{\mu}{\Omega} = 2\left(1 - \frac{r^2}{a^2}\right)\frac{1}{\varkappa\_0} \left[ -\frac{1}{\rho} \frac{\partial \, p}{\partial \, x} \right] = \frac{1}{4\mu} \left(a^2 - r^2\right) \left[ -\frac{1}{\rho} \frac{\partial \, p}{\partial \, x} \right].\tag{23}$$

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 periodic func‐ tion of time and is always in phase with the driving pressure gradient.

Consider very fast pulsations of the pressure gradients. If *ka* <sup>=</sup> *<sup>n</sup> <sup>v</sup> a* →*∞*, pulsations of the pressure gradients are very fast. Then, Uchida used asymptotic expansions of ber(ka) and bei(ka). In this extreme, the expression for the velocity near the center of the pipe is different from that near the wall of the pipe. So, they are discussed separately.

Near the center of the pipe, *ka* →*∞ and kr* →0 , one gets

*u <sup>U</sup>* <sup>=</sup> *us*

*us <sup>U</sup>* =2(1 - *<sup>r</sup>* <sup>2</sup>

*n*=1

(*ka*) + *bei* <sup>2</sup>

In which *ber* and *bei* are Kelvin functions defined using infinite series as shown below:

Two extreme cases were considered by Uchida: the case of very slow pulsations and that of very fast pulsations, depending on the magnitude of the dimensionless parameter *ka* <sup>=</sup> *<sup>n</sup>*

pressure gradients are very slow. Then, under these conditions and from the behavior of

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

<sup>4</sup>*<sup>μ</sup>* (*<sup>a</sup>* <sup>2</sup> - *<sup>r</sup>* 2) - <sup>1</sup>

*ρ* ∂ *p*

(*kr*) , *<sup>B</sup>* <sup>=</sup> *bei*(*ka*)*ber*(*kr*) <sup>−</sup> *ber*(*ka*) *bei*(*kr*) *ber* <sup>2</sup>

(*ka*)2 *sinnt* <sup>−</sup> 8(1 <sup>−</sup> *<sup>A</sup>*)

(*kr*) (a)

*∞ ϰsn ϰ*0 { <sup>8</sup>*<sup>B</sup>*

(*ka*)2 *sinnt*} <sup>+</sup> ∑

**4.4. Asymptotic expressions of the velocity distribution**

Kelvin functions, it is reasonable to expect that

*u <sup>U</sup>* =2(1 - *<sup>r</sup>* <sup>2</sup>

Consider very slow pulsations of the pressure gradients. If = *<sup>n</sup>*

*<sup>a</sup>* <sup>2</sup> ) <sup>1</sup> *<sup>ϰ</sup>*<sup>0</sup> - <sup>1</sup> *ρ* ∂ *p* <sup>∂</sup> *<sup>x</sup>* <sup>=</sup> <sup>1</sup>

with

And

*u* ' *<sup>U</sup>* <sup>=</sup>∑ *n*=1

where

And

*ber*(*z*)= ∑

*bei*(*z*)= ∑

*k*=0

*k*=0

*berka* →1 *and beika* →0.

Then, the velocity takes the form

*∞* ( − 1) *<sup>k</sup>* ( *<sup>z</sup>* <sup>2</sup> )4*<sup>k</sup>* +2 ((2*<sup>k</sup>* <sup>+</sup> 1)!)2 . (d)

*∞* ( − 1) *<sup>k</sup>* ( *<sup>z</sup>* 2 )4*k* ((2*<sup>k</sup>* )!)2 (c)

*J o* (*kri* 3

and

*∞ ϰcn ϰ*0 { <sup>8</sup>*<sup>B</sup>*

*<sup>A</sup>*<sup>=</sup> *ber*(*ka*)*ber*(*kr*) <sup>+</sup> *bei*(*ka*) *bei*(*kr*) *ber* <sup>2</sup>

(*ka*) + *bei* <sup>2</sup>

<sup>2</sup> ) =*ber*(*kr*) + *ibei*(*kr*) (b)

(*ka*)2 *cosnt* <sup>+</sup> 8(1 <sup>−</sup> *<sup>A</sup>*)

104 Wind Tunnel Designs and Their Diverse Engineering Applications

*<sup>U</sup>* <sup>+</sup> *<sup>u</sup> '*

*<sup>U</sup>* (20)

*<sup>a</sup>* <sup>2</sup> ) (21)

(22)

*v a*

*<sup>v</sup> a* ≪1 , pulsations of the

<sup>∂</sup> *<sup>x</sup>* . (23)

(*ka*)2 *cosnt*} ,

$$\frac{\mathbf{u}}{\mathbf{U}} = \frac{\varkappa\_0}{4\mathbf{v}} \left(\mathbf{a}^2 - \mathbf{r}^2\right) + \sum\_{n=1}^{\omega} \frac{\varkappa\_m}{n} \cos\left(\mathbf{nt} \cdot \frac{\pi}{2}\right) + \sum\_{n=1}^{\omega} \frac{\varkappa\_m}{n} \sin\left(\mathbf{nt} \cdot \frac{\pi}{2}\right). \tag{24}$$

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

Near the wall of the pipe, *kr* →*ka* →*∞* , and one uses asymptotic expansions of Bessel func‐ tions to get

$$\begin{split} \frac{\mathbf{u}}{\mathbf{U}} &= 2\left(\mathbf{1} - \frac{\mathbf{r}^{2}}{\mathbf{a}^{2}}\right) + \sum\_{n=1}^{\bullet} \frac{\varkappa\_{m}}{\varkappa\_{0}} \frac{8}{(\mathbf{ka})^{2}} \Big[ \sin(\mathbf{nt}) - \sqrt{\frac{\mathbf{a}}{\mathbf{r}}} \exp\Big(-\frac{\mathbf{k}(\mathbf{a} - \mathbf{r})}{\sqrt{2}}\right) \sin\Big[\mathbf{nt} - \frac{\mathbf{k}(\mathbf{a} - \mathbf{r})}{\sqrt{2}}\Big] \\ &+ \sum\_{n=1}^{\bullet} \frac{\varkappa\_{m}}{\varkappa\_{0}} \frac{8}{(\mathbf{ka})^{2}} \Big[ -\cos(\mathbf{nt}) + \sqrt{\frac{\mathbf{a}}{\mathbf{r}}} \exp\Big(-\frac{\mathbf{k}(\mathbf{a} - \mathbf{r})}{\sqrt{2}}\Big) \cos\Big[\mathbf{nt} - \frac{\mathbf{k}(\mathbf{a} - \mathbf{r})}{\sqrt{2}}\Big] .\end{split} \tag{25}$$
