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

## 1. Introduction

### 1.1. Flows driven by a constant pressure gradient through a pipe of circular cross section

When the flow of a Newtonian fluid in a pipe of circular cross section is driven solely by a constant pressure gradient, the resulting velocity distribution is a quadratic function of the radial distance from the axis of the pipe. The velocity profile of such a flow has, therefore, a parabolic distribution in which the maximum velocity occurs on the axis of the pipe. A graphical representation of this type of velocity is shown in Figure 1.

### 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 example, the pressure gradient fluctuates with time in such a way that that gradient can be expressed 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 velocity 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 frequencies 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, three-quarters 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:

Where n is the circular frequency, p the pressure,

### 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 gradient that is periodic with time by Uchida (1956).

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

Where r is the radial distance from the axis of the pipe; and letting

When one is very close to the wall of the pipe, r and R are very close in magnitude and

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 dimensionless distance

In this Figure 3, y is the distance from the wall of the pipe and

## 2. Richardson’s annular effect in a wind tunnel

Unsteady pulsating flows occur in many situations that have a practical engineering importance. 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 tunnel (Njock Libii, 2011).

The purpose of the remainder of this chapter is to summarize the theoretical basis of the Richardson’s annular effect in pipes of circular sections and in rectangular tubes, illustrate its results graphically, and relate them to what happens in a wind tunnel.

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 velocity from the center to points near the wall is presented using series approximations and graphical illustrations.

## 3. Stokes’ second problem

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 viscosity,

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 generated has a phase lag,

## 4. Pulsating flow through pipes

### 4.1. Basic equations

The flow of a viscous fluid in a straight pipe of circular cross-section due to a periodic pressure 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 parameters 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

Where u = u(r, t) is the component of velocity in the axial direction x,

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

If the pressure gradient is sinusoidal and given the form

then, the solution is given by the real part of

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

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

and for large values of the parameter x and of

where

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

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 maximum 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 location of maximum velocity is given by

### 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 solution is summarized below.

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

Where n is the frequency of oscillation and

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

Where

The total mean velocity U is defined as

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

Where

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

with

And

### 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

Consider very slow pulsations of the pressure gradients. If

Then, the velocity takes the form

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

Consider very fast pulsations of the pressure gradients. If

Near the center of the pipe,

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

Near the wall of the pipe,

### 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 parameter 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 following angles:

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 maximums of velocity distributions had shifted away from the axis and moved toward the wall of the pipe. These occurred at nt = 0^{o} and nt = 180^{o}. 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 = 0^{o}, 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 below.

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

1 | 12 | 12 | 0 |

3 | 12 | 10 | 2 |

5 | 12 | 6 | 6 |

10 | 12 | 0 | 12 |

## 5. Pulsating flow through rectangular ducts

### 5.1. Summary of the results of analysis

Yakhot, Arad, and Ben-dor conducted numerical studies of pulsating flows in very long rectangular ducts, where a and h were the horizontal and the vertical dimensions, respectively, of the cross-section of the duct, Fig. 7. Letting

For low pulsating frequencies,

For moderately pulsating frequencies, ^{o}. 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.

### 5.2. Graphical illustration of the results of analysis by Yakhot, Arad and Ben-dor (1999)

The velocity profiles in pulsating flow at selected instants within one complete period are shown below. Flow in a duct is compared to flow between parallel plates for different aspect ratios and frequencies.

## 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

Where

And

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

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

Where m = 4n’, with n’ = 0,1,2,3,…,

Note, from the definition of w(r, a, k), Eq. (28), that each of these polynomials will be multiplied 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 below demonstrates this observation:

Using the expression for

After a minor rearrangement of terms, Eq. (34) becomes

Since D(r, a, k), in Eq. (35), consists of the functions

### 6.2. Graphical illustrations of the shape of the F m x , y polynomials

Variation in the shapes of the functions

m = 4n’, with n’ = 1, 2, 3, 4, 5, … In Figures 12 and 13, twelve functions

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

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 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^{o} 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 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 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 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 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%.

## Nomenclature of the symbols (with units)

^{3});

^{2});

^{2}/s);

^{2});

a: the inside radius of a pipe through which an oscillating flow is moving;

c : the magnitude of a reference speed

G: the total mean value of the mass flow, U the total mean value of the velocity

i : the pure imaginary number; it is defined by i^{2} = -1

k: a dimensionless ratio used by Schlichting to denote the magnitude of the frequency of oscillation

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/m^{2})

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)

u_{s}: the steady part 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)