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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.
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
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