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Introduction

An understanding of uid ow through pipes and ducts is a

fundamental requisite for the design and optimization of uid–ow

systems. In particular, the relationship between ow rate and pressure

variation is critical and is among the most studied problems in uid

mechanics. Great efforts have been made to optimize uid systems

and these optimization efforts generally involve the maximization of

uid ow and/or the minimization of pressure loss. Pressure variations

within piping systems are a result of so–called “friction pressure

losses” and “minor pressure losses”. Friction pressure losses refer to

pressure drop caused by friction between the uid and the bounding

wall. Minor losses, on the other hand, are associated with pressure drop

that occurs when the ow passes through some obstacles in the piping

system, such as a bend, a valve, an orice plate, a change in diameter,

or other. It should be noted that “minor” losses are often signicant

components of pressure loss within a system. There is a design

balance between minimization of pressure loss and maximization

of ow that is often sought by a uid engineer. Reasonable efforts

to reduce pressure losses yet increase ow rates are of signicant

value for both the scientic and the engineering elds. In this regard,

the present study aims at assessing the impact of dimpling on pipe

walls to improve the optimal piping performance. That is, dimpled

surfaces have been shown to increase hydrodynamic performance (for

instance, dimpled surfaces reduce drag). Often these improvements

are related to ow over external bodies and the dimples modify the

boundary layer to reduce ow separation. For internal ows (i.e., ow

through pipes/tubes/ducts), on the contrary, separation is not germane

however the dimples still impact the ow. In fact, it has been noticed

that dimpling can affect the heat/mass transfer processes between the

uid and the wall1 and it often leads to an increase in the thermal or

mass transport behavior.

The history of similar studies is rich and includes fully developed

friction in coiled pipes2 as well as the impact of dimples on external

ow over at surfaces.3 Generally, it was found that the presence

of dimples increases both the rate of heat transfer as well as the

frictional pressure losses. These ndings are conrmed in multiple

independent studies4–10 and are evident for a wide range of geometries

and Reynolds numbers.8 These ndings are also relevant for

combined force/natural convection internal ows,11 for ows that

are transitional12 and for ows within narrow channels.13 With the

impact of dimples on the ow and heat transfer characteristics well

established, some works have also turned to design optimization14–21

and to the localized inuences of the dimples on the ow patterns

(both adjacent to and nearby the dimples). The ultimate goal of these

studies is to nd a strategy to optimize both the size and the density

of dimples, to dene their distribution on the tube/duct wall, and to

characterize the hydrodynamic performance of dimpled tubes/ducts

compared to undimpled ones.22–29 Here, a systematic study of the

pressure/ow relationship is made using numerical simulation. The

study encompasses a wide range of dimple shapes. Comparisons of

pressure/ow results with pre–existing solutions are made in order to

establish the veracity of the solutions.

The model

For the present investigation, a circular tube with uniform

dimpling is considered. The dimpling may be fabricated by a number

of manufacturing processes but it is envisioned as uniform both

along the length of the duct as well as around the tube perimeter. The

dimples are semicircles with radius R that are formed on the tube/

duct wall. Dimples are arranged uniformly around the perimeter of

the duct and are not staggered in the ow direction. For the thus–

described situation, the ow behavior in a short segment of duct will

be repeated along the duct length. Consequently, ow periodicity can

be enforced, i.e., the ow patterns through one row of dimples can be

repeated through subsequent rows. The solutions are obtained using

computational uid dynamics wherein the uid region is subdivided

into a multitude of small elements over which conservation equations

of mass and momentum are solved. The results presented here

correspond to a laminar ow with Reynolds numbers of approximately

1150 (halfway to the onset of transitional ow and turbulence). The

qualitative behavior and interpretation of the results is the same for

other Reynolds numbers, provided the ow is laminar. The relevant

equations are the steady state, incompressible laminar Navier Stokes

MOJ Civil Eng. 2018;4(3):150‒154. 150

©2018 Abraham. This is an open access article distributed under the terms of the Creative Commons Attribution License, which

permits unrestricted use, distribution, and build upon your work non-commercially.

Hydrodynamics of laminar ow through dimpled

pipes

Volume 4 Issue 3 - 2018

John Abraham, Ryan Maki

School of Engineering, University of St. Thomas, USA

Correspondence: John Abraham, School of Engineering,

University of St. Thomas, 2115 Summit Ave, St. Paul, MN 55105–

1079, USA, Tel 6129-6321-69, Email jpabraham@stthomas.edu

Received: June 07, 2018 | Published: June 15, 2018

Abstract

Laminar ow through a dimpled pipe has been investigated with a focus to determine the

relationship between hydrodynamic resistance and dimple geometry. Using a periodic

solution domain, in–line dimples were simulated with diameters that varied from 0.125mm

to 2mm (with a pipe diameter of 6 mm). It was found that for small dimples, the ow does

not enter into the dimple to forming a coherent recirculation zone. On the contrary, as

the dimple diameter increases, a recirculation zone forms and moves in the downstream

direction for increasing dimple size. The ow patterns within the dimple are connected to

changes of hydrodynamic resistance. For dimples whose sizes are small enough to prevent

recirculation, the ow resistance is constant with dimple size. Once the dimples become

sufciently large to allow recirculation, there is an inverse relationship between ow rate

and pressure drop. These ndings allow optimization of hydrodynamic performance when

a balance between pressure drop, ow rate, and wall mass transfer is required.

Keywords: dimple geometry, pipe, recirculation, ow resistance, pressure drop, ow

rate, wall mass

MOJ Civil Engineering

Research Article Open Access

Hydrodynamics of laminar ow through dimpled pipes 151

Copyright:

©2018 Abraham et al.

Citation: Abraham J, Maki R. Hydrodynamics of laminar ow through dimpled pipes. MOJ Civil Eng. 2018;4(3):150‒154. DOI: 10.15406/mojce.2018.04.00113

equations, including both conservation of mass and momentum. They

are given below:

0

i

i

u

x

∂=

∂

(1)

and

1, 2, 3 1, 2 , 3

jj

i

i ji i

uu

p

u ij

x xx x

ρµ

∂∂

∂∂

=−+ = =

∂ ∂∂ ∂

(2)

In these equations, the symbols

ρ

and

µ

are the uid density and

dynamic viscosity, respectively while the ui terms are the velocities

in Cartesian tensor notation, and term p represents the uid pressure.

The tube diameter is 6 mm and its length is 6 mm. The inlet and

outlet of the tube are connected by periodic conditions wherein a pre–

dened pressure drop across the pipe segment is specied to be 1Pa.

The periodicity enforces a matching of the velocity prole at the inlet

and outlet. The uid used in the simulation was water, with a density

and kinematic viscosity of 998kg/m3 and 8.9 x 10–7m2/s, respectively.

The uid–ow equations are solved using ANSYS CFX v18.0 which

is a nite volume based solution algorithm. The rst calculation was

performed for the no–dimple case. In this case, in fact, exact solutions

for fully developed laminar ow exist and allow a validation check

on the numerical results. The analytical solution, with the applied

pressure drop along the pipe of 1Pa, gives a mass ow rate of 5.9g/s.

The numerical results, on the other hand, give a mass ow rate of

5.8g/s, in excellent agreement.

Once the ow rate through a straight, non–dimpled tube is

validated, attention is turned to the dimpled cases. Figure 1 has been

prepared to illustrate the distribution of dimples around the pipe

circumference. It can be noticed that eight dimples are positioned

around the pipe circumference, equidistant from each other. The

diameter of the dimples, indicated in Figure 1 by an annotation, is

changed from 0.125mm to 2mm for different simulations. For the

largest diameters, the dimples are sufciently large to merge together

and form a toroidal shape around the pipe. The follow–on gure

(Figure 2) shows an oblique view of the pipe with dimples. It can

be noticed that the dimples are placed midway between the inlet

and outlet surfaces of the periodic segment. Once the dimensions

and boundary conditions are explained, attention is turned to the

computational mesh. Above–described mesh was subject to a mesh–

independence test wherein the number of elements was increased and

the calculations were repeated. For instance, for simulations involving

a 2mm dimple, the initial mesh encompassed 282,541 elements while

the rened mesh totaled 1,497,549 elements. Results obtained from

the two meshes were compared to ascertain whether there was an

appreciable difference. By way of example, mesh independent tests

for dimples of 1 mm diameter dimples showed that the calculated

ow rates differed by less than 0.3%. A similar test for 2mm diameter

dimples revealed that initial mesh and rened mesh results differed

by only ~1%. These results demonstrated that the initial mesh was

suitable to provide results of high accuracy

More details regarding the computational mesh are provided in two

further images which are shown in Figure 3. The two images focus

on the mesh in a straight pipe and in a dimpled pipe near a dimple,

respectively, so that the mesh deployment can easily be seen. It is

seen that for both geometries, a rectangular–element mesh is deployed

along the walls while tetrahedral elements are used within the core of

the ow space. When a dimple is present, the wall–elements follow

the dimple shape and within the dimple space itself, the elements are

ner than at other locations.

Figure 1 Illustration of the deployment of dimples around the pipe perimeter.

Figure 2 Oblique view showing the positioning on the dimples on the pipe

exterior.

Figure 3 Distribution of elements for a straight (top image) and dimpled pipe

(bottom image).

Hydrodynamics of laminar ow through dimpled pipes 152

Copyright:

©2018 Abraham et al.

Citation: Abraham J, Maki R. Hydrodynamics of laminar ow through dimpled pipes. MOJ Civil Eng. 2018;4(3):150‒154. DOI: 10.15406/mojce.2018.04.00113

Results and discussion

The presentation of results will rst focus on quantitative results

and then turn toward qualitative ow patterns and their inuence on

the results. The primary issue to be addressed is the impact of dimples

on the hydrodynamic efciency of the pipe. Thus, for a predened

pressure drop, a comparison of ow rates through the various dimpled

pipes will be made. The effect of dimple size on mass ow rate is

provided in Figure 4, where dimple sizes vary from 0.125mm to a

maximum of 2mm. It is found that very small dimples (0.125mm)

result in a lower ow rate. As dimple sizes increase slightly, there is

a more or less constant mass ow rate until the size of the dimples

reaches ~1.0mm. Then, the ow begins to decrease again with dimple

size. In order to explore the behavior of Figure 4, vectors of local

uid velocity are provided (Figure 5) (Figure 6). These vectors are

arranged for sequentially increasing dimple sizes. It is seen that for

smaller dimples, there is no coherent central eddy conned within

the dimple, i.e., the dimple volume is not large enough to support a

coherent eddy. To the contrary, for larger dimples (~1mm), a coherent

eddy begins to form within the dimple (Figure 5). As the dimple size

increases further, the recirculation patterns within the dimple become

more coherent and they move downstream (Figure 6). For instance,

for the largest dimple shown in Figure 6 (2mm), the center of the

recirculation zone is clearly downstream of the dimple center. The

off–center location is associated with a stronger uid impact on the

downstream face of the dimple and a larger stagnation pressure there.

It is believed that the formation, strengthening, and movement of this

recirculating ow are associated with the mass ow rate behavior

displayed in Figure 4.

Figure 4 Effect of dimple size on mass ow rate.

It should be noted that for dimple sizes of 1mm and larger, the

images show a portion of a second dimple intruding into the image.

The protrusion is the semitransparent dimple walls that have become

large enough to enter into the image. A deeper exploration of the

hydrodynamics is provided by evaluating the pressure within the

tube. Figure 7 shows the pressure along a plane that bisects the

tube; the gure corresponds to a dimple size of 2mm. Flow moves

from left to right in the image. It is seen that there is a large pressure

(stagnation location) at the downstream face of the dimple. This

location corresponds to the impact of ow against the downstream

dimple surface.

A corresponding image is shown in Figure 8. There, streamlines

are illustrated and they are colored according to local pressure

variations along the streamline. The gure has two parts: the bottom

image closely focuses on ow patterns in one of the dimples. The

recirculation zone can be easily identied: it is clear that it is not

centrally located within the dimple, but is rather situated downstream.

Also, the pressure within the uid in the downstream portion of the

dimple exceeds that in the upstream portion – leading to the added

hydrodynamic resistance. The observations obtained from Figure 4–8

provide some guidance for the design of uid conveyance systems.

It is seen that relatively large and constant ow rates can be obtained

(for a predened pressure decrease) provided that the dimples are

small enough to prevent coherent eddy formation. For instance, it

is possible to use dimple–walled pipes to promote mixing and mass

transfer in a uid dynamic application thus limiting the consequence

of added pressure losses. Alternatively, for situations where increased

pressure losses are desired, for instance in ow control applications,

the dimple design should be large enough to allow the formation of

the coherent recirculation region within the dimple. The quantitative

results are specic to the parameters considered in the present

calculations (Reynolds number, tube diameter, number of dimples

deployed around the perimeter, etc.) However, the behavior of

coherent recirculation within the dimple and its effect on ow rate can

be generalized to other situations where these parameters differ from

those presented here.

Figure 5 Changes in ow patterns with increasing dimple size (dimple sizes

from 0.375 to 1.0 mm).

Hydrodynamics of laminar ow through dimpled pipes 153

Copyright:

©2018 Abraham et al.

Citation: Abraham J, Maki R. Hydrodynamics of laminar ow through dimpled pipes. MOJ Civil Eng. 2018;4(3):150‒154. DOI: 10.15406/mojce.2018.04.00113

Figure 6 Changes in ow patterns with increasing dimple size (dimple sizes

up to 2.0 mm).

Figure 7 Pressure variation within tube and dimples, 2mm dimple size.

Figure 8 Pressure along streamlines, 2mm dimple size.

Concluding remarks

In this study, a three–dimensional numerical simulation of

laminar ow in a dimpled pipe has been performed. The simulations

accounted for dimple sizes that range from 0 mm (a smooth pipe)

2mm. It was found that the no–dimple calculations agreed very well

with established laminar ow expectations. It was also found that the

presence of dimples always decreased the ow compared to the smooth

wall situation; however, the size of the dimple played a signicant

role in the ow. For very small dimples, there was a relatively large

effect on the ow. On the contrary, for intermediate sized dimples,

a more–or–less stable ow rate was calculated. As dimple size

continued to increase, a new ow regime was established and a

coherent recirculation formed within the dimple. The recirculation

became more coherent as the dimple size increased and also moved

downstream. The strengthening and location of the recirculation zone

resulted in additional pressure losses for the ow and consequently in

a decrease of ow rate.

Acknowledgements

None.

Conict of interest

The author declares there is no conict of interest.

Hydrodynamics of laminar ow through dimpled pipes 154

Copyright:

©2018 Abraham et al.

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