A parametric study of the hydrodynamic roughness produced by a wall coating layer of heavy oil
A parametric study of the hydrodynamic roughness produced by a wall coating layer of heavy oil
S. Rushd 0
R. S. Sanders 0
0 Department of Chemical and Materials Engineering, University of Alberta , Edmonton, AB , Canada
In water-lubricated pipeline transportation of heavy oil and bitumen, a thin oil film typically coats the pipe wall. A detailed study of the hydrodynamic effects of this fouling layer is critical to the design and operation of oil-water pipelines, as it can increase the pipeline pressure loss (and pumping power requirements) by 15 times or more. In this study, a parametric investigation of the hydrodynamic effects caused by the wall coating of viscous oil was conducted. A custom-built rectangular flow cell was used. A validated CFD-based procedure was used to determine the hydrodynamic roughness from the measured pressure losses. A similar procedure was followed for a set of pipe loop tests. The effects of the thickness of the oil coating layer, the oil viscosity, and water flow rate on the hydrodynamic roughness were evaluated. Oil viscosities from 3 to 21300 Pa s were tested. The results show that the equivalent hydrodynamic roughness produced by a wall coating layer of viscous oil is dependent on the coating thickness but essentially independent of oil viscosity. A new correlation was developed using these data to predict the hydrodynamic roughness for flow conditions in which a viscous oil coating is produced on the pipe wall.
Pipeline transportation; Heavy oil fouling; Lubricated pipe flow; CFD simulation
& R. S. Sanders
List of symbols
Aeff Effective cross-sectional area (m2)
D Internal diameter of the pipeline (m)
Dh Hydraulic diameter (m)
Deff Effective diameter (m)
f Friction factor
H Nominal height of the test cell (m)
heff Effective height (m)
htp Height of the test plate (m)
k Turbulence kinetic energy (m2/s2)
ks Nikuradse sand grain equivalent hydrodynamic
Mass flow rate of water (kg/s)
Static (thermodynamic) pressure (Pa)
Pressure loss (Pa)
Pressure gradient (Pa/m)
Water density (kg/m3)
The reserves of non-conventional oils, such as heavy oil
and bitumen, form a substantial part of the known global
petroleum resources (IEA 2014; CAPP 2014). These oil
reserves are asphaltic, dense, and highly viscous, with
bitumen being more dense and viscous than heavy oil
(Saniere et al. 2004). Densities of these oils are nearly the
same as that of water, whereas their viscosities are higher
than that of water by orders of magnitude (McKibben et al.
2000). Therefore, the production of these non-conventional
oils requires extraordinary techniques that are not needed
to recover traditional petroleum deposits. Various mining
and in situ production technologies are being used to
extract non-conventional oils in Canada. After extraction,
the oil is transported from a production site to an upgrading
facility. Since pipeline transportation is a cost-effective
technology, the non-conventional oil industry is keen to use
this technology for transporting both bitumen and heavy oil
(Nunez et al. 1998; Saniere et al. 2004; Hart 2014).
Water-lubricated pipeline transportation of
non-conventional oils, known as lubricated pipe flow (LPF), is one
option for transporting these viscous fluids. It is more
economical when compared with other technologies, such
as heating, solvent dilution, emulsification, and partial
upgrading (Nunez et al. 1998; Saniere et al. 2004). In LPF,
a thin water annulus prevents continuous contact between
the pipe wall and the viscous oil core, resulting in much
lower energy requirements than would be needed to
transport the viscous oil alone in the pipeline (Arney et al.
1993; Joseph et al. 1999; McKibben et al. 2000; Rodriguez
et al. 2009; de Andrade et al. 2012). The water could be
naturally present in the oil or could be injected for the
purpose of producing LPF. A concern for the application of
LPF is the formation of an oil film on the wall (Nunez et al.
1998; Saniere et al. 2004). This oil layer is usually
identified as ‘‘wall fouling.’’ The probable flow regime in LPF
is schematically presented in Fig. 1. In this figure, a fouling
oil layer is shown to surround a thin water annulus that
lubricates the oil-rich core. The mechanism of wall fouling
has not previously been studied in any detail. Previous
Wall fouling oil-layer
Lubricating water annulus
Lubricated oil core
Fig. 1 Illustration of the flow regime in a LPF pipeline
experimental works suggest the fouling layer is a natural
consequence of the lubrication process (Joseph et al. 1999;
Vuong et al. 2009). Frictional pressure losses in a fouled
pipe are much higher (say, 8–15 times) than those
measured in an unfouled pipe (Arney et al. 1996), but still
orders of magnitude lower than those would be expected
for transporting only heavy oil or bitumen. It has been
found in repeated tests that the formation of this wall
coating is practically unavoidable in the industrial
application of LPF technology (McKibben et al. 2000, 2016).
Different degrees of wall fouling occur depending on the
conditions of LPF, e.g., water cut, oil viscosity, and
superficial velocity (Joseph et al. 1999; Schaan et al. 2002;
Rodriguez et al. 2009; Vuong et al. 2009).
The wall fouling layer in a water-lubricated pipeline can
be considered as a stationary coating film of viscous oil
adhered on the pipe wall. This is because the relative
velocity of this layer is negligible compared to the average
mixture velocity (Joseph et al. 1999; McKibben et al.
2000, 2016; Shook et al. 2002; Schaan et al. 2002; Vuong
et al. 2009). This wall coating layer can produce a large
equivalent hydrodynamic roughness value: The typical
equivalent roughness of a commercial steel pipe is about
0.045 mm, while the hydrodynamic roughness (inferred
from pressure loss measurements) of a pipeline with a
viscous oil layer on the pipe wall can be greater than 1 mm
(Brauer 1963; Shook et al. 2002). The roughness is
produced primarily through contact between the viscous oil
coating and the turbulent water layer that flows over the
film while lubricating the oil core. The result is a rippled/
rough wall that is associated with very large hydrodynamic
roughness values (Brauer 1963; Picologlou et al. 1980;
Shook et al. 2002). While the presence of the coating
reduces somewhat the cross-sectional area available for
flow, which also causes an increase in pressure loss for a
given throughput, the increased hydrodynamic roughness
plays a much more important role in this increase.
The conventional method for describing the
hydrodynamic roughness produced by a rough solid wall is the
Nikuradse sand grain equivalent (Perry and Green 1997;
Whyte 1999). This definition of equivalent roughness is
extensively used for commercial metal pipes or channels
and has also been used to describe the hydrodynamic
roughness caused by a biofilm on a solid wall (Picologlou
et al. 1980). Much like an oil fouling layer, the biofilm is
conformable and can substantially increase the
hydrodynamic roughness, in turn increasing power requirements for
pumping water through bio-fouled pipes and channels
(Barton et al. 2008; Andrewartha et al. 2008).
Previous studies of equivalent hydrodynamic roughness
involved either a rectangular flow cell or a pipe for
experiments (Barton et al. 2008; Andrewartha et al. 2008). In
rectangular flow cells, one wall is typically ‘‘roughened’’
(e.g., through the formation of a biofilm) while the other
three walls are kept hydrodynamically smooth. The
timeaveraged velocity profile perpendicular to the rough wall is
then measured to determine the hydrodynamic roughness on
the basis of correlations, such as the law of the wall
(Andrewartha et al. 2008). The reliability of the measurement
was subject to the type of instrumentation selected for the
measurements and also the size of the flow cell. Typically, a
large channel was used to ensure that the measured velocity
profile would not be affected by the presence of the walls.
Pressure loss measurements have been typically used to
determine the hydrodynamic roughness for the pipeline
tests using some basic equations of fluid dynamics, such as
the Darcy–Weisbach equation or the Colebrook correlation
(e.g., Barton et al. 2008). A basic analytical approach such
is appropriate when the hydrodynamic roughness can be
represented by a single, constant value. In other words, this
approach is not applicable for the flow cells with
asymmetric wall roughness.
In the present study, a customized rectangular flow cell
was used to perform a parametric investigation of the
equivalent hydrodynamic roughness produced by a wall
coating of viscous oil. A relatively small test cell was
chosen because the goal was to test a number of different
oils under a wide range of flow conditions. As a result, it
was very difficult to make accurate measurements of the
velocity profile of the flow above the coated surface and
instead the pressure loss (under fully developed flow
conditions) was measured for the different flow conditions
tested. The asymmetry of wall roughness in the flow cell
(one rough wall and three smooth walls), however, meant
that a simple analytical approach to relate pressure loss to
hydrodynamic roughness could not be used. Therefore, the
flow conditions in the experimental cell were modeled
using a commercially available CFD package (ANSYS
CFX 13.0) and simulations were conducted to determine
the hydrodynamic roughness that would give a predicted
pressure loss equal to that measured during an experiment.
The validated CFD-based procedure has been described in
detail elsewhere (Rushd et al. 2016). Based on the results
presented here, a new correlation is proposed for the
equivalent hydrodynamic roughness produced by a viscous
layer of wall coating in terms of the coating thickness. This
correlation can be used to estimate the hydrodynamic
roughness from a measured or a known value of the
physical wall coating thickness.
2 Experimental setup and procedure
A 2.5-m-long rectangular flow cell was designed and
fabricated for the present study. The flow cell consisted of a
channel whose lower surface was comprised of segmented
steel plates. These plates were coated with a measured,
constant thickness (tc) of oil prior to the start of each flow
experiment. The effective cross section of the flow channel
without a wall coating was 15.9 mm 9 25.4 mm. Its
entrance length was 1.5 m, which was more than 60Dh;
where Dh = 19.6 mm is the hydraulic diameter defined as
4A/P, where A is the cross-sectional area and P is the
wetted perimeter of the cross-sectional area. The flow cell
had two Plexiglas windows so that it was possible to
observe the shape of oil–water interface. This custom-built
cell was placed in a 25.4-mm pipe loop as shown in
Fig. 2a. A cross-sectional view of the flow cell is presented
as Fig. 2b. A photograph of the cell under actual flow
conditions, after the rough/rippled topology was developed
on the wall coating layer, is given in Fig. 2c.
The steady-state pressure loss across the flow cell was
measured with a differential pressure transducer (Validyne
P61). The experiments were conducted for a range of water
flow rates, coating thickness, and oil viscosities. Typical
pressure gradient measurements (30 s averages) are
presented for a specific flow condition in Fig. 3. These results
demonstrate that the change in pressure loss for a given
water flow rate is negligible when the thickness of the wall
coating layer is constant.
For experiments, each of the segmented plates
comprising the bottom wall of the flow visualizing section was
coated separately with a specific thickness of the viscous
oil and placed in the flow cell to form a coating layer of
uniform thickness. The average thickness of the coating
layer (tc) was determined by weighing the test plates
without and with coating oil. It should be noted that the
coated plates were also weighed before and after each
experiment. The difference between the measured weights
was negligible, i.e., tc was unaffected by the flow rate and
thus was taken as a controlled parameter.
Please refer to Rushd (2016) for more detailed
descriptions of the experimental apparatus, procedures, and
3 Experimental parameters
The rectangular flow cell was used to study the
hydrodynamic effect of different viscous wall coatings. The
measured variable was the pressure loss (DP). The controlled
parameters are listed in Table 1. The most important of
these parameters are the average thickness (tc) and the
viscosity (lo) of the coating oil. Recall that only the bottom
wall of the rectangular flow cell was coated with oil. The
experimental value of tc for an oil was selected depending
on oil viscosity (lo) and flow rate of water (mw). The
thickness (tc) that could be maintained under the highest
flow rate for the lower viscosity oils (lo * 65 and
Fig. 2 Illustration of the experimental facility. a Schematic presentation of the flow loop. b Cross-sectional view (section A–A’) of the flow cell.
c Photograph showing a test with a rough wall coating of viscous oil (lo = 21300 Pa s)
Fig. 3 Illustration of pressure gradients (DP/L) measured over time
(t) for different water flow rates (mw) (tc = 0.2 mm; lo = 21300 Pa s)
320 Pa s) in the flow cell was 0.2 mm. Similarly, the
maximum tc for the higher-viscosity oils (lo * 2620 and
21300 Pa s) was 1.0 mm. Coating thickness values tested
Table 1 Controlled parameters for the rectangular flow cell
Thickness of wall coating tc, mm
Viscosity of coating oil lo, Pa s
Mass flow rate of water mw, kg/s
Water Reynolds numbers Rew
Flow temperature T, C
for these oils were 0.2, 0.5, and 1.0 mm. The overall
uncertainty associated with the measurement of tc in the
flow cell was 10%. Thus, the coating thickness (tc) for the
first phase of experiments was selected so that it would not
change significantly with water flow rate. The purpose of
these tests was to evaluate the effects of the flow rate and
the viscosity on the hydrodynamic roughness while
keeping the coating thickness constant.
4 CFD simulations
As mentioned previously, the CFD simulations were used
to determine the unknown equivalent sand grain roughness
(ks) of the oil-covered bottom wall of the flow cell. This
was done by modeling the water flow through the cell over
the viscous coating. The CFD software package, ANSYS
CFX 13.0, was used for simulation. The software solves the
governing differential equations that include
Reynoldsaveraged Navier–Stokes (RANS) continuity and
momentum equations. The Reynolds stress term in RANS was
modeled using an omega-based Reynolds stress model,
xRSM. Full details of the governing equations are given in
The geometry of the 3D computational domain used for
the simulation was identical to the rectangular flow cell;
however, two different flow cell lengths (l = 1.0 m;
l = 2.0 m) were used for computations even though the
actual flow cell was 1.0 m in length. This was done to
ensure the length independence of the simulations. The
width (w) was equal to that of the flow cell (25.4 mm). The
height (h = 15.9-tc mm) was varied depending on the
average thickness (tc) of oil coating on the bottom wall.
The values of tc tested during the present study are shown
in Table 1.
The flow geometry was created and meshed with
ANSYS ICEM CFD. The software was used to discretize
the flow domain into structured grids, one for the bulk of
the flow and one for the near-wall region. Coarse,
intermediate, and fine mesh grids were tested. The mesh
resolution was based on the number of nodes, n, in each mesh.
In the present study, the mesh resolution is classified as
follows: coarse (n \ 50000), intermediate (50000 \
n \ 500000), and fine (n [ 500000). The total number of
nodes found to be sufficient for grid independence was
n = 670200. An example of the fine mesh used here is
Fig. 4 Two-dimensional illustration of the fine mesh used for
shown in Fig. 4. The number of nodes in the near-wall
region was selected so that y? [ 11.06. At y? = 11.06,
ANSYS CFX 13.0 uses the logarithmic law of the wall
(i.e., the wall function). For these simulations, typically,
y? = 25. All computations were performed to obtain
steady-state solutions. Double precision was used in the
computations, and solutions were considered converged
when the normalized sum of the absolute dimensionless
residuals of the discretized equations was less than 10-6.
The typical computational time required for the
convergence of a simulation was 45 min.
At the inlet of the flow domain, the experimental mass
flow rate of water and a turbulent intensity of 5% were
prescribed as the boundary condition. A zero pressure
condition was specified at the outlet. The no-slip condition
was used at the boundaries representing walls. The two side
walls and the upper wall in the rectangular domain were
taken as hydrodynamically smooth (ks = 0) based on the
results of simulations conducted for clean walls (Rushd
et al. 2016). Flow conditions where the bottom wall was
coated with oil required one to specify the ks value for this
wall. However, the values of ks were unknown for the
oilcoated bottom wall of the flow cell for any given flow
condition. A trial and error procedure was adopted to
determine the appropriate ks value. Starting from a low
value, ks was increased in steps and the simulation was
repeated until a reasonable agreement between the
measured and predicted pressure loss (maximum 5%
difference) was observed. The final value of ks at which this
condition was met was considered to be the equivalent
hydrodynamic roughness of the corresponding rough wall.
The trial and error approach is summarized in Fig. 5. This
CFD-based trial and error approach of estimating ks was
validated using data on biofilms taken from the literature
and from flow cell tests using materials of known
roughness (Rushd et al. 2016).
5 Results and discussion
As mentioned earlier, two effects were produced by the
wall coating layer: a slight reduction in the effective flow
area and a drastically increased hydrodynamic roughness
(ks). The reduction in the flow area was taken into account
through the average thickness of the wall coating layer (tc),
which is a physical parameter that can be measured
directly. The hydrodynamic roughness (ks) value
corresponding to each combination of viscous wall coating
thickness (tc), and water Reynolds number was determined
using the CFD-based procedure describe above. The results
were used to develop a correlation between ks and tc.
Specify initial boundary conditions in the rectangular flow domain
Inlet: Known flow rate of water
Outlet: Zero pressure gradient
Three walls: Smooth
Bottom wall: Rough, ks=ki (iterative value)
Solve for steady state
Compare the simulation results
with experimental values
Accept ki as the ks
Fig. 5 Flowchart describing the steps involved in the simulation procedure for computing the equivalent sand grain roughness (ks)
5.1 Rectangular flow cell results
The effect of wall coating thickness (tc) on the measured
pressure gradient, for tests involving a specific oil
(lo = 2620 Pa s), is demonstrated in Fig. 6a. It can be
seen from the figure that operation at higher average
velocities (V = mw/(qwAeff)) causes the pressure gradients
(DP/L) to increase approximately with V2, as would be
expected for the turbulent flow of water through a channel
or pipe. Note, however, that compared to the clean wall
condition, the measured pressure gradients are significantly
higher when the wall is coated with oil (tc [ 0). Clearly,
the primary contributor to the measured pressure loss at
any velocity is the presence of the oil coating in the flow
cell. Although four different oils with viscosities ranging
from 65 to 21300 Pa s were tested (see Table 1), the results
for any given oil were almost identical to those presented in
Fig. 6a (Rushd 2016). In other words, oil viscosity played a
negligible role over the range of viscosities tested here. The
observation that viscosity of the coating layer (lo) had no
appreciable effect on the measured pressure gradients (DP/
L) is demonstrated in Fig. 6b.
As can be observed from Fig. 6a, a small increase in
coating thickness (tc) causes a significant increase in
pressure gradient (DP/L). The cause of this substantial
increase is related primarily to the increase in
hydrodynamic roughness of the oil coating layer produced through
its interaction with the turbulent water flow through the
channel. The coating thickness tc reduces Dh by 0.5%–4%
(depending on the value of tc tested). If the wall coating
layers behaved hydrodynamically as ‘‘smooth’’ surfaces
(ks = 0), the reduced Dh would cause a 4%–20% increment
in DP/L. The range can be calculated on the basis of
Blasius law for a rectangular flow cell (Jones 1976). As
Figs. 6a and 7 show, the measured increase in pressure loss
with increasing tc is in the range of 50%–200%. The
substantial increase in DP/L indicates the importance of the
roughness of the coating layer.
In Fig. 7, the measured values are shown in comparison
with the predictions obtained from CFD simulations. The
simulated pressure gradients agree well with the
corresponding measurements when the rectangular flow cell was
clean, i.e., when the bottom wall was not coated with oil
(tc = 0). For these simulations, all four walls of the
tc = 1.0 mm
tc = 0.5 mm
tc = 0.2 mm
mw = 1.78 kg/s
mw = 1.20 kg/s
mw = 0.59 kg/s
Fig. 6 Experimental results for the rectangular flow cell. a Pressure gradient (DP/L) as a function of bulk water velocity (V) and oil coating
thickness (tc) (lo = 2620 Pa s). b Effect of water flow rate (mw) and oil viscosity for a fixed coating thickness (tc = 0.2 mm)
Fig. 7 Comparison of simulation and experimental results for the
rectangular flow cell (lo = 2620 Pa s)
rectangular flow cell were considered ‘‘smooth,’’ i.e.,
ks = 0. The agreement between the experimental and
simulation results indicates the clean walls of the
rectangular flow cell to be hydrodynamically smooth. The
figure also shows that when the bottom wall was coated with
oil, the measured values of DP/L could be accurately
predicted. Another important point to note is that the
hydrodynamic roughness (ks) produced by a constant coating
thickness (tc) was not dependent on velocity, for the range
of velocities tested here.
5.2 Pipe loop results
The CFD-based methodology of determining the
equivalent hydrodynamic roughness that was developed for the
flow cell experiments was then applied to determine the
values of ks for comparable tests carried out with a
recirculating pipe loop. A 103.3-mm (ID) pipe having an
internal wall fouled/coated with two different heavy oils
(lo * 3 and 27 Pa s) was used in experiments. The wall
coatings were developed in the course of testing LPF. After
completing a set of LPF tests, water at 20 C was pumped
through the pipeline, replacing the oil core. The flow
scenario for the pipeline testing is shown schematically in
Fig. 8. Pressure loss and wall coating thickness
measurements were made simultaneously at mean (bulk) water
velocities of V = 0.5, 1.0, 1.5, and 2.0 m/s. A custom-built
double pipe heat exchanger (Schaan et al. 2002) and a hot
film probe were used to obtain wall coating thickness
measurements. The wall coating thickness for the pipeline
tests decreased with increasing velocity, i.e., tc values were
dependent on V because the coating was partially stripped
from the wall as the water velocity was increased. The
pressure measurements always reached a steady-state
condition at each velocity, which allowed for the
calculation a steady-state value of ks. A more detailed description
of the apparatus and test procedure is provided by
McKibben et al. (2016).
As was done for the rectangular flow cell tests, CFD
simulations of the pipe loop tests were conducted to
Deff = D - 2tc
Fig. 8 Schematic cross-sectional view of test section in the pipeline
determine the equivalent hydrodynamic roughness (ks). A
typical comparison of the measured pressure gradients and
those obtained from simulations for the pipeline tests are
shown in Table 2. Only the results for the higher-viscosity
oil (lo * 27 Pa s) are shown, as the trend is similar for the
other oil (lo * 3 Pa s). Because of the way the pipe loop
experiments were conducted, different values of tc were
tested at different water velocities. As the results in Table 2
show, the agreement between the measured pressure
gradients and the values determined using the CFD
methodology (where ks is set by trial and error) is excellent.
The values of ks for the pipeline tests were corroborated
by estimating the same values on the basis of the
The values of ks for these tests were determined using
Eq. (1) and the CFD methodology. These are two
completely different approaches for determining ks. The values
calculated on the basis of the Colebrook formula agree
reasonably well with those obtained with the CFD method.
The results from the two calculation methods are presented
in Table 2 for the higher-viscosity oil (*27 Pa s). Similar
agreement was found when comparing the two calculation
methods for the 3 Pa s oil coating as well.
5.3 Correlation development
A correlation between ks and tc is proposed here, on the
basis of the rectangular flow cell data and the pipe flow
tests described previously:
The correlation is illustrated in Fig. 9. The proportionality
constant of the equation is determined with a regression
analysis for which R2 = 0.96. The average uncertainty
associated with the predictions of this correlation is ±14%.
The data set used for developing the correlation set
presented in Table 3.
As shown in Fig. 9, eight data points were used to develop
the correlation. Three of these points were obtained from the
Table 2 Comparison of equivalent hydrodynamic roughness for pipeline tests (lo * 27 Pa s)
Water velocity V, m/s
Wall coating thickness tc, mm
Pressure gradient DP/L, kPa/m
Hydrodynamic roughness ks, mm
experiments conducted with the flow cell, and five points
were obtained from pipeline tests. Multiple combinations of
oil viscosity, water flow rate, and coating thickness were
used for the flow cell experiments. Therefore, the three data
points for the rectangular flow cell actually correspond to 24
different flow conditions, meaning the correlation is based on
29 distinct flow conditions.
The relationship between ks and tc proposed in this work
is the first of its kind. To the best of our knowledge, a
similar correlation is not available in the literature. An
example of its application to predict pressure losses in a
fouled/coated pipeline is presented in Appendix 2. Using
the correlation in its current form is subject to the
knowledge of coating thickness (tc), which is possible to measure
for a LPF system (Schaan et al. 2002). Additional efforts
are currently underway to correlate tc with flow parameters
so that its direct measurement will no longer be necessary.
The objective of the present study is to provide detailed
information about the hydrodynamic roughness that a wall
coating of viscous oil produces. The results reported here
can be summarized as follows:
× Rectangular flow cell
ks = 2.76 tc
Fig. 9 Correlation between hydrodynamic roughness (ks) and coating
Table 3 Data used to develop the correlation between ks and tc (Eq. 2)
2 620, 21 300
A film of viscous wall coating substantially increases
the measured pressure loss, primarily as a result of the
rough/rippled structure that forms on the surface of the
coating, which produces a very large value of the
equivalent hydrodynamic roughness, ks.
Experiments were conducted using two different
geometries—a rectangular flow cell and a 100-mm
(diameter) pipeline loop—and using different oils to
produce the wall coating layer. The oil viscosities
spanned four orders of magnitude (3 B lo B 2.1 9 104
Pa s). Water alone (i.e., no oil in the bulk flow) was
circulated through both test cells under highly turbulent
conditions (2.9 9 104 \ Rew \ 2 9 105).
The results obtained from the two test geometries were
in very close agreement and showed that the
hydrodynamic roughness produced by a wall coating of
viscous oil was essentially independent of oil viscosity.
In fact, the thickness of the coating layer directly
determines its hydrodynamic roughness.
A new correlation was proposed to relate the
hydrodynamic roughness produced by a viscous wall to the
thickness of the coating layer. The correlation will be a
critical component of any advance model of lubricated
pipe flow with wall fouling.
Acknowledgements The research was conducted through the support
of the NSERC Industrial Research Chair in Pipeline Transport
Processes (held by RS Sanders). The contributions of Canada’s Natural
Sciences and Engineering Research Council (NSERC) and the
Industrial Sponsors (Canadian Natural Resources Limited, Fort Hills
LLP, Nexen Inc., Saskatchewan Research Council Pipe Flow
nology CentreTM, Shell Canada Energy, Syncrude Canada Ltd., Total
E&P Canada Ltd., Teck Resources Ltd. and Paterson & Cooke
Consulting Engineers Ltd.) are recognized with gratitude. We are also
grateful to Dr. Adane and Dr. Islam for their kind assistance in
preparing the manuscript.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creative
commons.org/licenses/by/4.0/), which permits unrestricted use,
distribution, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
Appendix 1: description of x-RSM
Most important features of the x-RSM are described here
on the basis of Fletcher et al. (2009) and ANSYS
CFXSolver Theory Guide (2010). In this narrative, the
differential equations are presented with index notation1. The
Reynolds-averaged Navier–Stokes (RANS) equations of
continuity and momentum transport for an incompressible
fluid can be presented with following equations.
1 In Cartesian coordinates, for example, Ui represents all three
components (x, y, z) of the vector U. Likewise, sij stands for the six
components (xx, xy, xz, yx, yy, yz, zx, zy, zz) of the tensor s. The
differential operators are denoted similarly. Also, the summation
convention is implied.
In Eq. (4), p is the static (thermodynamic) pressure; Si is
the sum of body forces, and sij is the fluctuating Reynolds
A number of models are available in ANSYS CFX 13.0
for the Reynolds stresses (sij) in the RANS equations.
Among the available models, x-RSM is selected as the
most suitable for the current work. In this model, sij is
made to satisfy a transport equation. A separate transport
equation is solved for each of the six Reynolds stress
components of sij. The differential transport equation for
Reynolds stress is as follows:
The Reynolds stress production tensor Pij is given by:
The constitutive relation for the pressure–strain term Uij in
Eq. (5) is expressed as follows:
Uij ¼ b0C1qx sij þ 3 kdij
In Eq. (7), the tensor Dij and the model coefficients are
a^ ¼ ð8 þ C2Þ=11
In addition to the stress equations, the x-RSM uses the
following equations with corresponding coefficients for the
turbulent eddy frequency x and turbulent kinetic energy k.
In the previously mentioned transport equations, the
turbulent viscosity lt is defined as
lt ¼ q x : ð12Þ
Along with the basic differential equations, the flow near a
stationary wall is significant for the turbulent model,
xRSM. Usually a wall is treated with ‘‘no-slip’’ boundary
condition for CFD simulations. Mesh-insensitive automatic
near-wall treatment is available for the x-RSM
implementation in ANSYS CFX 13.0. The treatment is meant to
control the smooth transition from the viscous sub-layer to
the turbulent layer through the logarithmic zone. Important
features of the near-wall treatment for x-RSM are outlined
(1) In case of a hydrodynamically smooth wall, the
viscous sub-layer is connected to the turbulent layer with a
log-law region. Velocity profiles for the near-wall regions
are as follows:Viscous sub-layer:
u ¼ ð1= kÞ lnðy Þ þ B
Here, uþ ¼ Ut=us, yþ ¼ qDyus=l ¼ Dyus=v, and
us ¼ ðsw=qÞ0:5. In the log-law, B and DB are constants.
The value of B is considered as 5.2 and that of DB is
dependent on the wall roughness. For a smooth wall,
DB = 0. The term Dy, in the definition of y?, is calculated
as the distance between the first and the second grid points
off the wall. Special treatment of y? in CFX allows one to
arbitrarily refine the mesh.
(2) For a hydrodynamically rough wall, the roughness is
scaled with Nikuradse sand grain equivalent (ks).
Dimensional roughness ksþ is defined as ksþus=v. A wall is treated
hydrodynamically rough when ksþ is greater than 70. The
value of DB is empirically correlated to ksþ as follows.
DB represents a parallel shift of logarithmic velocity profile
compared to the smooth wall condition.
(3) At the fully rough condition (ksþ [ 70), the viscous
sub-layer is assumed to be destroyed. Effect of viscosity in
the near-wall region is neglected.
(4) The equivalent sand grains are considered to have a
blockage effect on the flow. This effect is taken into account
by virtually shifting the wall by a distance of 0.5 ks.
Appendix 2: sample calculation illustrating
application of the proposed correlation
An example illustrating the application of the proposed
correlation, i.e., Equation (3) is presented here. The
correlation is used to predict the frictional pressure loss for a
specific pipe flow case, and then, the predicted value is
compared with the measured value. The data for this
example are reported by McKibben et al. (2016).
Internal diameter of the pipeline (D): 103.3 mm
Average water velocity (V): 1.0 m/s
Density of water (qw): 997 kg/m3
Viscosity of water (lw): 0.001 Pa s
Average thickness of wall coating/fouling (tc)
(measured): 2.0 mm
Effective diameter (Deff): Deff ¼ D
Effective velocity (Veff): Veff ¼ V DDeff
Reynolds number (Rew): Rew ¼ DefflVweff qw ¼ 1:1
Equivalent hydrodynamic (ks):
ks ¼ 2:76tc ¼ 5:52 mm
Darcy friction factor (f), obtained using the Swamee–Jain
correlation: f ¼ 0:25 log
Predicted pressure gradient
¼ 0:44 kPa/m
Measured pressure gradient
(DP/L): 0.45 kPa/m
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