Solid body model: industrial steel pipe transporting clean water.

## Abstract

This chapter examines the flow of swirling liquid in a duct. In many cases, circumferential velocity in the cross-section of a cylindrical duct is a remarkably linear function of radius up to the proximity of the duct wall. This is similar to the behaviour of a twisting solid shaft and the analogy leads to a solid body model for swirl flow in ducts. Helically profiled lobate duct walls provide a twisting torque, while wall friction in simple circular ducts causes swirl to decay. The liquid counterpart of the solid body is represented as a first-order system in downstream distance because of the way torque is transmitted by duct walls rather than by shaft stiffness as in the solid case. The effect of the inertia of the rotating and twisting cylinder is unchanged from its solid counterpart, and damping is related to the viscosity of the liquid acting over the annulus between the rotating liquid cylinder and the duct wall. The shear stress in the liquid is shown to be linearly related to the intensity of the swirl. The generation of swirl is briefly described with reference to lobate designs, their development of shape and helix.

### Keywords

- swirl
- solid-liquid pipeflow
- slurry transport
- computational fluid dynamics

## 1. Introduction

Why impart swirling flow to a stream of fluid? Spanner, a much respected naval architect, invented a helical lobate tube which increased the efficiency of heating of water in the boilers of ships [1, 2]. Importantly, his design could be manufactured economically by drawing cylindrical tube through special dies (see Figure 1).

For particle-bearing liquids, swirl puts particles into suspension at lower axial velocities than would be the case for a cylindrical duct. Once in suspension, particles (or debris for downstream collection) remain in full or partial suspension long after the swirl has decayed to negligible proportions.

Lower axial velocity implies much lower pumping power: a strong economic reason to swirl the flow in the first place. The efficacy of swirl generation in pumping particulate liquids, particularly river slurries, has been recognized for many years. The Gordon patent for a duct with internal vanes to encourage swirl was published in 1899 [3].

Economic benefits are not the only reasons for studying swirl. In some instances, enhanced swirl is required irrespective of increased pressure losses and pumping power requirement.

The data in this chapter originate from several sources. Much of it comes from validated computational fluid dynamics (CFD) code using RANS (Reynolds-averaged Navier Stokes equations). The basis of these equations is ‘Reynolds decomposition’, whereby an instantaneous quantity is decomposed into time-averaged quantities and fluctuating quantities. In cylindrical polar co-ordinates (*r*,*θ*,*z*), the time-averaged velocities are *u, v* and *w* and the fluctuating velocities are *u*′, *v*′ and *w*′. In essence, the fluctuating quantities can be assumed to have a temporal mean of zero. This makes for an increase in the number of unknowns because the cross products (

RANS turbulence modelling techniques are often classed by the number of equations used to model the flow field. Early results were obtained using two-equation *k* represents the turbulent kinetic energy, *ω* represents the specific rate of dissipation of turbulent kinetic energy into internal thermal energy. As greater computer power became available, the six-equation Reynolds stress model (RSM) became the method of choice for swirling flows. There are variations in RSM solutions. The CFX RSM*-ω* [5] has been used since it can give more accuracy near the wall.

## 2. Characterizing swirl

Firstly, I should explain what I mean by ‘swirl’ and then define precise ways to assess it. In turbulent pipe flow, there are many eddies and circulations cascading from the large to the small. Kolmorogov [6] showed that most of the kinetic energy in the flow is contained in large-scale structures. Energy ‘cascades’ to smaller scales by an inviscid mechanism until it is small enough for viscous dissipation to take place. I define swirl as large-scale, one-way circulation surrounding the geometric centre of the duct.

With a definition of swirl in place, I now need to explain the mathematical measures of this behaviour. Is it useful, strong or weak, efficient or profligate in its expenditure of pipeline pressure? The first and most obvious measure is the circumferential velocity, sometimes referred to as tangential velocity, *wmax*, taken at its maximum in the cross-section. Useful in itself, this direct measure is important in application to downstream devices—cyclones or pumps for example.

Circumferential velocity as a measure of swirl takes no account of the axial velocity required to generate or maintain it. In contrast, the *swirl angle*, incorporating the axial velocity, indicates the angular deflection of the flow and can be clearly seen in transparent pipe sections (see Figure 2). Tonkin [7] used these images to infer tangential velocity for a series of particle concentrations and axial velocities.

The *swirl angle*,

where *um* is the mean axial pipe velocity.

Measurement transducers can be corrupted by swirling flow, and International Standard ISO 5167 specifies a maximum swirl-angle limit of 2° at or near transducer stations.

The swirl angle does not take account of the angular momentum given to the flowing liquid. The ratio of angular momentum flux to the product of pipe radius and axial momentum flux is known as the *swirl intensity* or *swirl number*, *Ω*. (Note that a simple ratio of angular to axial momentum would not be dimensionless.) There are several versions of this definition and the version defined in Eq. (2) allows for variations in the axial velocity with radial displacement.

where *R* is the pipe bore radius (of the cylindrical delivery pipe), *u* is the axial velocity at radius *r*, and *w* is the circumferential velocity at radius *r*.

Swirl number gives a simple way to classify swirl for computational calculation methods. If

Swirl intensity and swirl angle are closely related measures and in many cases an almost linear relation exists between them.

Pressure loss is an inevitable consequence of swirl generation and it is important to use that pressure effectively. Ganeshalingam [8] developed a dimensionless group, *swirl effectiveness*, *S*, given by

where *ΔP* is the pressure loss over a length of duct and *ρ* is the fluid density.

This measure has proved invaluable in optimizing Spanner-type duct designs.

Another pressure-related metric for use when a Spanner-type duct generates swirl is the pressure loss for an equivalent length of smooth circular tube. The well-known Darcy-Weisbach equation can be used to calculate this:

where *f* is the friction factor for a smooth duct, *L* is the duct length and *D* is the duct diameter. A simple ratio can be used to gauge the magnitude of the pressure loss suffered as a result of the use of the duct:

## 3. The solid body model

In many cases of developed swirling flow, the swirl angle, *θ,* is found to be almost constant over most of the cross-section. Figure 3 shows a tangential velocity profile for a nominal axial velocity of 2 m/s. The near linearity over approximately 84% of the bore also indicates that angular velocity, *ω*, is similarly constant in this range. Constant angular velocity is a characteristic of a solid rotating shaft and this concept suggests a simple mechanical analogy of *the solid body model* which can be used to describe swirling flow.

Damping friction (directly proportional to tangential velocity) clearly has little effect in the central 84% of Figure 3. The peripheral 16% of the velocity profile indicates gathering damping friction as the radius increases. At the outer radial extremity, the circumferential velocity falls to zero in accordance with the *no-slip* principle of Newtonian mechanics. This outer damping annulus is characterized by a dimensionless distance from the wall, with

where *τw* is the wall shear stress, *y* is the distance to the wall and *ν* is the kinematic viscosity.

In turbulent pipe flow, close to the wall, is a *laminar sub-layer* of width *buffer layer* gradually develops the laminar sub-layer into fully turbulent flow. This is much smaller than the outer 16% of Figure 3 and later results will show that the solid body starts at approximate distance

The simplified system dynamics of the analogy of a solid-liquid cylinder are described by three elements: the *inertia* of the shaft and any rotating components such as a flywheel, the *stiffness* of the shaft transmitting torque down the shaft and the *damping* of the speed of rotation. Straight away the stiffness of the shaft can be eliminated from the model. By definition, liquids do not have stiffness and they adapt to the shape of the containment without coercion. We are left with a shaft, length one pitch (for one 360° rotation), subjected to a torque *M* rotating at temporal rate *dθ/dt*.

where *J* is the polar second moment of mass of the cylinder and *c* represents a damping coefficient dependent upon the area of the shearing surfaces. The torque moment *M* can be positive for a Spanner-type pipe or can approach zero for a frictionless cylindrical tube.

In the model, the coefficient of damping, *c*, is provided by the viscosity of the liquid. Consider fully developed swirling flow in the core of a cylindrical duct (Figure 4). In the example above (mean axial velocity = 2 m/s), there is a zone of approximate width *y+ =* 72 in which all the damping occurs. Note that if the swirl is generated by a Spanner-type duct, there will be other shearing surfaces in the core flow where the rotating annuli meet the as yet non-rotating central portion. This is sometimes referred to as a *wall-jet* profile (Steenbergen and Voskamp [9]).

In order to quantify the damping coefficient, *c*, I concentrate on the viscous part of Eq. (7). Put another way, I assert

Newton’s law of viscosity gives

Torque is applied at the outer radius as wall friction or reaction from the pipe profile, so

Comparing (10) with (8), we obtain the coefficient of damping per unit length.

where *y* is the distance from the wall of the duct.

The next major challenge to the solid body model is the transmission of torque. In a solid shaft, the torque is transmitted by its stiffness, but stiffness has been discounted as a factor in liquids. In the case of a profiled swirl tube, the torque comes from the interaction of the axial flow with the walls of the tube, an interaction the author describes as the *driving function*. This becomes clear when Eq. (7) is rewritten so that the dependent variable becomes axial distance along the cylinder (*z*). Putting *G* = twist gradient *dθ/dz*

Dividing throughout by *cu*

Note that the group of variables at the left-hand side of Eq. (3) *G* and is the *driving function*, i.e.

where the time constant

The *complementary function*, or transient

The solution to Eq. (14) has another (steady-state) part, the *particular integral* (PI), *Gpi(z)*, which depends on the driving function *GD*(*z*). The total response is a combination of these two components

There are a series of driving functions of interest and I shall start with the simplest: the decay of swirl angle downstream of swirling flow from, for example, a pump output or double elbow. The driving function *GD*(*z*) in this case is a negative step change from the initial swirl angle to zero.

When *z =* 0, *G*(*z*) *= Go*, the initial swirl gradient. Eq. (16) gives

Halsey [10] studied the swirl in clean water following a double elbow. His work was aimed at measurement devices for which swirling flow is disruptive. ISO 5167 specifies a 2° swirl-angle limit for measurement purposes and Halsey came up with an empirical law for its decay as follows

where *θo* is the swirl angle at commencement, *θ* is the swirl angle at a downstream distance *z*, *f* is the friction factor and *D* is the diameter of the bore. Steenbergen and Voskamp [9] arrived at an almost identical equation in terms of *swirl intensity, Ω,* instead of swirl angle. Ganeshalingam’s work [8] achieved close agreement with these models and a simulation exercise (below) confirms the relationships.

Differentiating (19) gives

when

Equating exponents in Eqs. (18) and (21), we have a first estimate of the time constant *Τ* for decay of swirl and the length of the swirling wake *Τu*

The solid body model gives us *J*, of a solid cylinder, density *ρ*, per unit length (*L*) is given by

From this, and the time constant *T*, a value of *c/L* can be deduced which can be used with Eq. (11) to estimate the distance, *y*, of the solid body from the wall of the duct.

It is not possible to specify the total extinction of swirl. For some purposes, the point of 95% reduction in swirl angle (*L95*) after a downstream distance of 3*Τu* should be a useful approximation. If swirl is a desirable property (to keep solids in suspension for example), the *half-life* distance (

u | Re | f | T | L50 | c/L | y | y+ |
---|---|---|---|---|---|---|---|

m/s | — | — | s | m | Nms/m | m | — |

1 | 50,000 | 0.024 | 1.37 | 0.95 | 0.00045 | 0.0065 | 82.3 |

1.5 | 75,000 | 0.023 | 0.97 | 1.00 | 0.00064 | 0.0050 | 76.8 |

2 | 100,000 | 0.022 | 0.75 | 1.04 | 0.00082 | 0.0040 | 71.8 |

2.5 | 125,000 | 0.022 | 0.61 | 1.06 | 0.00101 | 0.0034 | 67.5 |

3 | 150,000 | 0.022 | 0.52 | 1.07 | 0.00119 | 0.0029 | 63.9 |

3.5 | 175,000 | 0.021 | 0.45 | 1.08 | 0.00137 | 0.0026 | 60.8 |

4 | 200,000 | 0.021 | 0.39 | 1.09 | 0.00156 | 0.0023 | 58.0 |

It is generally accepted that a *y*^{+} value of about 35 indicates the edge of the buffer layer next to the wall. The range of values of *y*^{+} significantly greater than this value suggests that there is an annulus of turbulent flow between the buffer layer and the solid body.

The half-life distances *L50* are approximately constant for axial pipe velocities in the range [1.0, 4 m/s]. This is an important observation for designers of pipe systems in which the axial velocity might vary.

### 3.1. Wall shear stress in the solid body model

Starting from an analysis by Kitoh [4], the tangential momentum equation for axi-symmetric flow gives an equation for circumferential shear stress at the wall in the decay of swirl in a circular pipe.

Now

where *τW* is the circumferential wall shear stress.

Leibnitz’s rule for the differentiation of integrals allows the change of order of integration and differentiation in Eq. (25):

For a constant Reynolds number, the axial velocity

This allows the simplification of swirl intensity to

and substituting for

i.e.

Note that since *τW* must be given a negative sign.

Returning to the analogy of a solid body for the flow, one might reasonably expect a linear relationship between circumferential stress and circumferential strain (swirl intensity or swirl angle) for a given Reynolds number. This can be tested with a straightforward simulation experiment.

The simulation experiment (below) gives

The imposition of pipe roughness considerably increases the friction factor, *f*. For an axial velocity of 1.64 m/s in a commercial steel pipe (*ε* = 0.000046 m), friction factor is sharply increased to 0.0228. If the stress-strain assumption still holds, substituting this increased friction factor in Eq. (31) yields

SIMULATION EXPERIMENT: swirl decay in a cylindrical tube

Figure 5 shows the results of a simple RANS simulation for the flow of clean water through a 50-mm diameter smooth circular tube using the Reynolds stress model (*−ω* version) [5]. The entry plane is furnished with a mean axial velocity of 1.64 m/s and an initial circumferential velocity of 0.72 m/s at the wall and zero at the centre of the tube. The boundary conditions including the law of the wall are allowed to develop over an axial length of 10.0 m. The solid body model implies that wall friction will have reached 95% of its effect on the flow after 3*Tu* m and 99% after 5*Tu* m.

Reynolds number

Friction factor (Blasius equation):

Time constant (Eq. (22))

Effective range downstream:

For the solid body model this is

Measured mean friction factor over range: 0.0197

In this range, the regression law applied to the CFD data is very precise (

where

Note the significant difference in time constant for smooth pipe when compared to commercial industrial steel pipe because of the increased value of the multiplier

Since the swirl angle is linearly related to swirl intensity in most cases, it follows that Halsey’s correlation [10] also fits the data.

## 4. Generating swirl

Previously, we have seen that a solid body model can be applied to the simple case of swirl decaying downstream. In these cases, the driving function is simply a step to zero:

The contours of duct walls should impart torque to the flow while minimizing pressure loss. By designing using this criterion, pressure costs are used in an effective way. Here, Ganeshalingham’s dimensionless group, *swirl effectiveness*, *S*, can be used to evaluate the effectiveness of swirl generation.

### 4.1. Lobate designs

The boiler tube patented by Spanner and illustrated in Figure 1 has only three lobes. Raylor [11] idealized the lobe profiles to form semicircular shapes for his CFD modelling to test the design for the transportation of particle-bearing liquids. The computer modelling was underpinned by experimental work on an extant boiler tube. Later work by Ganeshalingam showed that a four-lobe duct (or a 2-lobe duct) was more efficient when compared on the basis of swirl effectiveness.

In Figure 6, after Ariyaratne [13], it can be seen that the contours of tangential velocity adopt a more circular pattern in the four-lobe variant and that an efficient circulating core flow is produced in consequence.

Simply equating the area of the four-lobe duct to *R*_{4}, to the upstream radius can be derived:

*R*_{4} and sub-multiples have been used in other speculative swirl-duct designs so that lobe sizes can be compared across designs.

### 4.2. Response of the solid body model to a constant-pitch swirl duct

I first consider a four-lobe swirl duct with constant pitch:diameter ratio of 8:1 simply connected in line after a cylindrical duct. The driving function for this is a positive step or Heaviside function in swirl gradient *G*_{0}, this can be tried as the particular integral in Eq. (17):

The constant *B* can be easily obtained by considering the boundary condition *G*(0) which yields

Figure 7 illustrates the response of a system comprising a four-lobe Spanner-type duct with cross-sectional area equal to a cylindrical upstream main of diameter 50 mm carrying clean water at an axial velocity of 2 m/s. The ordinates are tangential velocities at a radius of 0.7*R*. From these ordinates, the response of *G*(*z*) can be deduced given that the twist gradient of the duct is 15.71 radians/m. Fitting Eq. (36) to these data yields *Tu* = 0.1 m. In the first-order system, 63.2% of the final value is reached after *Tu* metres.

The length of the wake (the point at which swirl has decayed by 95%) is 3*Tu*. Hence the standard length of this four-lobe Spanner-type duct with axial velocity 2 m/s is 300 mm.

Note the apparent anomaly between the calculated constant *Tu* for the swirl generation case (0.1 m) and that for the swirl-decaying case from Table 1 (1.5 m), a multiplicative factor of 15. The time constant of the model must be identical whatever driving function is applied so the only way in which this phenomenon can be explained is by studying the changes in the geometry of the rotating core for the generation case. The circumferential velocity distribution for swirl generation is a *wall jet* [9] in which circumferential velocity is concentrated towards the outer perimeter of the core flow. Effectively, the solid body in this case is not a solid cylinder, but more like a cylindrical tube. Thinking of the time constant of the system (*J*, is significantly reduced from that of a solid cylinder while the shearing surfaces are significantly greater than those of a solid cylinder, increasing the damping coefficient *c*.

### 4.3. Cross-section development for lobate ducts

The example of a fixed-pitch duct is useful in that it gives a standard length for swirl pipe designs. However, a lobate swirl duct cannot be added directly to a cylindrical pipe without incurring wasteful pressure losses. A better solution is to allow the shape to develop in a sigmoidal fashion. A family of sigmoidal coefficients is given by

and illustrated in Figure 8.

For a Spanner-type lobate duct, the sigmoidal function can be used to schedule the growth of lobe area, the expansion of the duct or, usually, the development of the radius of the lobe to its final value. In this case, the factors are

The exponent *n* governs the steepness of the sigmoidal curve. If *n >* 0.5, the initial gradient is effectively zero, giving a gradual increase in shape.

Originally, a *Three-Zone* development was proposed by Jones and Ariyaratne [12] whereby the entry section was a beta transition with *n =* 2, followed by a fixed-pitch section, followed again by an exit transition with *n =* 0.5. This was certainly an improvement on the system without transitions, but the fixed-pitch section constrained the angular acceleration of the liquid in the duct. Later designs embodied a continuous development of shape and helix. The *symmetric* development had an increasing beta function to the central point of the duct followed by a decreasing beta function to the outlet. The *asymmetric* development had an increasing beta function to a point two-thirds along the length of the duct and a decreasing beta function for the latter third. The asymmetric development gave slightly better swirl intensity results than the symmetric development.

### 4.4. Helix development

Raylor [11] showed that advantages accrued from the gradual angular acceleration of twist in a profiled tube. In recent work, this has been combined with the asymmetric beta function to create a duct with developing cross-sections and acceleration of twist throughout the tube. Inserting the driving function for this case we obtain

where *λ* is the rate of increase of *G* along the duct, and *G*_{0} is the starting value.

Eq. (16) gives the complementary function as in the previous cases

The *particular integral* (PI) is obtained by trying

Comparing coefficients we have the solution for the PI

Hence, the complete solution (PI + CF) is given by

Applying the boundary condition

So

Eq. (44) specifies the response of the solid body model to the ramped driving function in Eq. (39). A two-lobe design is illustrated in Table 2. The design is a modestly twisting tube but pressure losses are considerably larger than those expected in a smooth straight duct for the same duty (using the Darcy-Weisbach equation for this prediction). Increasing the amount of twist and increasing the number of lobes to four can improve the performance of the tube at the expense of increased pressure loss.

## 5. Conclusion

The purpose of this chapter has been to examine the technical aspects of swirling flows and to facilitate the design of ducts for specific purposes. Swirling flow is a complex, while stunningly beautiful, phenomenon and my work has been guided by the need to reduce its complexity for the designer. The emphasis has been on Spanner-type profiled tubes, but this is by no means the only way to generate swirl. The fascinating medical prospect that small amplitude helically coiled pipes might be used as bypass grafts to prevent occlusion by thrombosis has been the subject of scholarly study [14, 15].

The efficacy of the first-order solid body model was demonstrated by the simulation of flow through a 10.0-m cylindrical tube. The prediction that the downstream data taken after a distance of 3*Tu* would be representative of fully developed flow was amply demonstrated. Swirling flows are a little more difficult to predict than by using a simple exponential decay formula, but the solid body model introduced in this chapter is a simple and useful tool to apply to the design task.

## Acknowledgments

As always, I am indebted to my research students Benjamin Raylor, Jeyakumar Ganeshalingam, Chanchala Ariyaratne and Ruth Tonkin for their tireless experimental and computational work in the early days. I am particularly indebted to Benjamin Raylor for his continued efforts to the present day, his enthusiasm for swirl ducts, his hard work and unfailing support.