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 (
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
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,
Circumferential velocity as a measure of swirl takes no account of the axial velocity required to generate or maintain it. In contrast, the
The
where
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
where
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,
where
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
3. The solid body model
In many cases of developed swirling flow, the swirl angle,
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
where
In turbulent pipe flow, close to the wall, is a
The simplified system dynamics of the analogy of a solid-liquid cylinder are described by three elements: the
where
In the model, the coefficient of damping,
In order to quantify the damping coefficient,
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
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
Dividing throughout by
Note that the group of variables at the left-hand side of Eq. (3)
where the time constant
The
The solution to Eq. (14) has another (steady-state) part, the
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
When
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
Differentiating (19) gives
when
Equating exponents in Eqs. (18) and (21), we have a first estimate of the time constant
The solid body model gives us
From this, and the time constant
It is not possible to specify the total extinction of swirl. For some purposes, the point of 95% reduction in swirl angle (
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
The half-life distances
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
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
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,
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 (
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,
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
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
The constant
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
The length of the wake (the point at which swirl has decayed by 95%) is 3
Note the apparent anomaly between the calculated constant
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
Originally, a
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
Eq. (16) gives the complementary function as in the previous cases
The
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
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.
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