Numerical Study of Steady Pipe Flows and Head Loss Coefficients

. The study concerns steady pipe flows within 90 o -bend and T-junctions, the head loss coefficients, and the physical mechanisms responsible for the energy loss. Numerical viscous flow modeling based on SimpleFOAM solver was implemented in the study. Although secondary flow acts as the mechanism of energy loss in both the bend and the branch flow of the T-junction, the head loss coefficient in the former more critically depends on the Reynolds ( Re ) number. This is supported by the comparison of the streamlined fields and the radially varying profiles of the pressures from both cases. The location of flow separation prior to the secondary flow in the bend changes with the Re number, unlike the fixed location at the junction in the branch flow. The study confirms a practical implication related to the dependence of the head loss coefficients for pipes with 90 o -bend and T-junction on flow velocity in a pipe network analysis.


INTRODUCTION
A pipe network is essential to daily human lives because it provides clean water and gasses for daily home and industrial activities.With the increasing number of human populations worldwide, underground pipe networks have become more complex.The network typically requires the installation of pipe junctions and bends to direct the flow to the consumers' destination and valves to dictate the flow discharge.Ensuring that the pressure and velocities have values lower than their safe maximum and achieve their required values is utterly essential.As a result, pressure and head loss associated with friction, valves, pipe junctions, and bends must be accurately predicted when designing a pipe network.
Commercial and open-source software has been employed by many designers to model and predict a pipe network.Such software typically assumes independence of head loss coefficients associated with friction on flow parameters.This may be acceptable when the flow is fully developed and the Reynolds (Re) number is huge; the latter is applied for a Re number larger than 10 8 in the case of a smooth pipe [1].However, flow discharge and pipe diameters utilized for water consumption may be set such that the Re number is much lower than 10 8 .Furthermore, flow separation may occur and disrupt a fully developed pipe flow that enters a bend or a junction.It is well known that flow separation around a circular cylinder is generally influenced by Re number [2,3].Nevertheless, the head loss coefficients for a pipe bend and junction are commonly treated as independent constants of Re number [4,5].As a result, evaluating the potential dependence of these minor head loss coefficients is essential.
The present study aims to investigate the head loss coefficients for cylindrical pipe numerically flows within bends and T-junctions and physically understand the mechanisms that cause their dependency on the Re number.Few studies have employed computational fluid dynamics to evaluate this topic [6].In this study, SimpleFOAM, a viscous flow solver for steady flows available in OpenFOAM (Version 2.3.0)[7], was utilized to model three-dimensional steady pipe numerical flows within the two geometries.This modeling is the most appropriate for pipes with circular cross-sections.Notably, the numerical model provides access to a wealth of data that helps a researcher understand the underlying physics.Since three-dimensional modeling is computationally costly, this study considers bend or Tjunction within a pipe segment bounded by an inlet and outlet.

Test Cases
In the present study, pipes with 90 o -bends and T-junctions are considered.Table 1 presents the dimensions of the two types of pipes.These dimensions were determined from the pipe network data that serves the PDAM Tirtanadi consumers in the Sei Agul District of Medan, Indonesia and are presented in [8].Given the radius of the pipe, rp, relative to the radius of bend, bend, the case of 90 o -bend is classified as a curved pipe [6].

Convergence Study of Pipe Flow Within the 90 o -Bend
The first numerical viscous flow modeling aims to investigate the head loss coefficients for the steady flows within the pipe with the 90 o -bend.Three types of boundary surfaces, namely wall, inlet, and outlet, enclose the computational fluid domain within the pipe.The fluid flow enters the domain through the inlet and exits the domain through the outlet.The three boundary conditions for all the boundary surfaces were set as follows.The boundary surface attached to the wall was defined as a no-slip boundary condition.The pressure variable on this surface had a zero gradient boundary condition.On the inlet of the fluid domain, the velocity was set to nonzero while the pressure was to be predicted.As a result, FixedValue was chosen for the boundary condition of the velocity variable, and ZeroGradient was set as the boundary condition of the pressure variable.For the outlet, the vice versa was applied to achieve numerical stability.In other words, ZeroGradient and FixedValue were thus chosen as the boundary conditions for the velocity and the pressure variables on the outlet, respectively.
Before running a numerical simulation, a pre-processing that generates mesh must be undertaken.The study employed SALOME, an open-source mesh-generating software.The first step of the pre-processing in this software was to define a divided disk, also known as a primary object.In order to set up the pipe with the 90 o -bend, five blocks were defined, while the pipe radius, rp, was prescribed.The following step was to make points and a connecting line that formed the pipe.This stage helped create the pipe bend by choosing an option of the fillet.In this stage, the pipe bend radius, rbend, had to be defined.Finally, the complete pipe geometry with the 90 o -bend was created by calling the extrusion along the path command.For this final stage, the pipe length was prescribed in any direction.An x-y plan view of the complete pipe geometry with the 90 o -bend and the locations of the inlet and outlet are presented in Fig. 1.Having created the pipe geometry, a structured mesh consisting of a finite number of hexahedral cells was generated.This mesh is considered to have better accuracy than an unstructured mesh [9,10].In the case of pipe with the bend, the fluid velocity on the inlet, uinl, was set to 0.12 m/s.This was estimated by undertaking an EPANET simulation of the pipe network that serves the PDAM Tirtanadi consumers in the Sei Agul District of Medan, North Sumatra, Indonesia [8].In this case, the radius of the pipe, r p , and the radius of bend, r bend , correspond to Case 1 outlined earlier in Table 1.In respect of the pressure acting on the outlet, the value was set to zero.The pressure predicted on the inlet thus physically means the pressure drop or pressure difference that arises when the water flows from the inlet to the outlet.In the present study, the pressure drop, P, is the average of the pressures computed on all surfaces of the numerical cells adjacent to the inlet.Furthermore, the pipe length in x and y directions, lx and ly, were both set to 5 m; this being approximately equal to 55D, where D the diameter of pipe.The choice of dimension follows the suggestion given in [6] and [11] to ensure that the inlet and outlet boundaries were sufficiently far enough not to be disturbed by local pressure changes due to the bend.
In undertaking a convergence study for this steady flow, various levels of mesh densities were considered.The level of mesh density depends on the radial size of the cell, Dr, defined as D/Nr, with Nr being the number of cells across the diameter of the pipe.Four levels of mesh densities were implemented, referred to as very coarse, coarse, medium, and fine mesh.For each level of mesh density, the Nr value, the number of cells generated in the computational domain, Ncell, the computational time required to run a simulation, Tsim, the pressure drop, P, and the percentage of difference, PD, are presented in Table 2. Figure 2 shows the cross-sectional views of the numerical cells generated under the various levels of mesh densities.It shows that the fine mesh has the most refined cells.Furthermore, the PD value herein quantitatively indicates the discrepancy between the pressure drops, P, generated in one case of level of mesh density and the case of fine mesh.Each numerical simulation was run on a personal computer with an Intel(R) Core (TM) i5-8265U CPU processor @ 1.60GHz 1.80 GHz.Based on the data in Table 2, two critical points can be deduced herein.First, the computational time, tsim, increases with the Nr value and Ncell value.Second, the percentage of difference, PD, indicates the convergence of the present numerical simulations.Indeed, Fig. 3(a) confirms that the pressure drop, P, computed using the various levels of mesh densities, converges to 7.80 Pa.To further confirm the convergence, the head loss coefficient, k, was computed for each level by using this modified Bernoulli equation.
Pinl and Pout, respectively, denote the averaged pressures acting on the inlet and outlet.The difference between these two defines the pressure drop, P-similarly, uinl and uout express the averaged water velocity at the inlet and the outlet.In the present model, the elevations of these two boundaries are equal.Following [11], the averaging was undertaken by implementing a surface integration on each variable's inlet and outlet.Figure 3(b) shows the head loss coefficient, k, for each level of mesh density.The coefficients converge as the number of cells increases.Given the short computational time for the case of medium mesh and its small percentage of difference (see again Table 2), the medium mesh is considered optimal for predicting the head loss coefficient accurately.The head loss coefficient, k, predicted from the present numerical study for the bend case (i.e.k=1.198, see again Fig. 3(b)), lies in the range of coefficient values experimentally measured by [4] (i.e.0.3 <=k<=1.5).The lowest k value in the range was observed in the case of flanged bend with long radius, while the highest one was proven from the opposite cases.The k value obtained from the numerical simulation is comparable to the experimental data.Given this comparison, the present numerical model of the steady flow within the pipe bend using the SimpleFOAM is validated.It should be noted, however, that the comparison is satisfactory.This is at least due to the negligible wall roughness considered in the numerical modeling and the potential dependence of the k value on the fluid velocity (or Re number).

Convergence Study of Pipe Flow Within a T-junction
In this case, the T-junction has one inlet and two outlets.The types of boundary conditions for walls and for inlet and outlet defined in the earlier case were again implemented in the present numerical model.In making the geometry, SALOME was again employed, and T Shape Fluid was chosen as the primary object.For this stage, the radius of the pipe and the pipe length in any direction were prescribed.Figure 4 shows an x-y plane view of the complete geometry of the pipe with a T-junction.This figure shows that the pipe with a T-junction has one inlet and two outlets; the first and second outlets are referred to as outlet1 and outlet2, respectively.Whether a boundary surface acted as an inlet or outlet was also determined from the EPANET simulation conducted earlier [8].The velocity at the inlet, uinl, was estimated to be 0.27 m/s.In this case, the pipe radius, rp, corresponds to Case 2, outlined earlier in Table 1.Given all the data, the corresponding Re number was thus 24300.Having set up the boundary conditions for all boundary surfaces, the flow domain was solved again using the SimpleFOAM viscous flow solver.In the setup, the pipe length in any direction was chosen so that the inlet and outlet were sufficiently far enough not to be disturbed by local pressure changes in the T-junction.
A convergence study involving various levels of mesh densities was also conducted.Table 3 provides the values of Dr, Ncell, tsim, P, and PD for five different levels of mesh density.With the increasing mesh density and decreasing PD value, Table 3 indicates the convergence of the result.To further evaluate the convergence, the variations of the head loss coefficients arising when the fluid flows into outlet1, k1, and outlet2, k2, are evaluated from these modified Bernoulli equations: where uinl, uout1 and uout2 are the averaged velocities at the inlet, outlet1 and outlet2, respectively.Figure 5(a) shows the variation of the pressure drop with the level of mesh density, while Figs.5(b) and (c) present the corresponding variations of the head loss coefficients arising when the fluid flows into outlet1, k1, and outlet2, k2, respectively.These figures confirm that the present simulations have reached converged results with the increasing Nr value.For the fine mesh, the number of cells, Ncell, was 55296, and the required computational time was 24 minutes.
Furthermore, the value of P is estimated to be 34.70Pa, while the k1 and k2 values are found to be 1.21 and 1.95, respectively.Gerhart et al. [4] previously measured the head loss coefficient for a pipe with a T-junction.This measured coefficient varies in the range of 0.2<k<0.9for a line flow and 1.0<k<2.0for a branch flow.The coefficient's exact value depends on the pipes' connection type.Herein, a line flow refers to the flow going into the first outlet (outlet1), and a branch flow corresponds to the flow entering the second outlet (outlet2).Given the range measured from the earlier laboratory study, the head loss coefficient predicted in the present study is considered reasonable.Figures 5(b) and (c)) also confirm that the head loss coefficient for the line flow (outlet1), k1, is smaller than that for the branch flow (outlet2), k2.This is also in agreement with the earlier study.Moreover, the velocities at the first and second outlet, uout1 and uout2, were predicted to be 0.231 m/s and 0.048 m/s, respectively.These two predictions indicate that the law of mass continuity has been satisfied.

Comparison of Fluid Flows
Figure 6(a) presents the field of velocity magnitude, umag, that exists within the bend, while Fig. 6(b) shows the field of umag within the T-junction, with its close-up view provided in Fig. 6(c).The velocity magnitude, umag, is indicated from the legend, while the arrows on the streamlines present the flow direction.The velocity near the outer wall of the bend has larger magnitude.This is similar to the findings in [9], [12] and [13].In both bend and T-junction cases, flow separations occur.This mainly generates multiple vortices when the water passes through the T-junction (see Figs. 6(b) and (c)).Interestingly, a secondary flow exists when the flow direction has to change.This is confirmed from both the bend case, in which the flow direction changes gradually (Fig. 6(a)), and the T-junction case, in which an abrupt change of flow direction occurs in the branch flow (Fig. 6(b)).
The secondary flow in the bend occurs following a flow separation near the bend entrance and inner wall.Interestingly, in the case of T-junction, the secondary flow only occurs in the branch flow and begins at the junction.The branch flow differs from the line flow.This is consistent with the head loss coefficient associated with the outlet2, k2, which has an unidentical value to that corresponding to the outlet1, (see Figs.

Effects of Reynolds Number Pipe Flows Within Bends
The head loss coefficients for the pipe with bend over varying Re number are investigated.The aim of this investigation is to confirm any dependence of the coefficient on the Re number.The Re number was varied by changing the velocity at the inlet (or inlet velocity), uinl.This velocity was determined using EPANET simulations that considered the fluctuation of water consumption in one day.The time-varying discharge of water refers to the data of water-consumption percentage issued by the General Directorate Cipta Karya Air Bersih [14].
Various Re numbers considered herein lie in the range of 7200 ≤ Re ≤ 20700 in the cases of pipe with bend.By implementing Equation ( 1), the head loss coefficient, k, was predicted.The variation of the head loss coefficient with the Re number is presented on Fig. 7(a).It shows that the head loss coefficient significantly decreases with the Re number.Increasing almost three times the Re number reduces the k value by half of the initial k value (i.e., k = 2.11).This is unexpected given that the head loss coefficient for the flow within a pipe with a bend is traditionally believed to be independent of the Re number [4,5].There might have been the contribution of wall friction (or significant loss) along the pipe length before and after the bend to this dependence.The head loss coefficient was re-computed to verify this potential by considering a new inlet and an existing outlet in the present numerical simulations.The former was chosen at a cross-section where the location was very close to the bend, and the secondary flow had not occurred.Thecated at the belocatedentrance, while the new outlet was set to the exit.By computing the averaged velocities and the averaged pressures at these two cross-sectional areas and implementing Equation (1) again, the head loss coefficient associated with the bend, kbend, was estimated.The variation of kbend with the Re number is presented in Fig. 7(b).It shows that the kbend value indeed varies with the Re number.Increasing almost three times the Re number reduces the kbend value by more than 60% of the initial kbend value (i.e.kbend = 0.49).This indicates that the energy loss arising in the bend due to the change in flow direction is not a function of the square of the fluid velocity.It is a higherorder function of the velocity.Ito's formulae [15] predict that the bend is 0.36 when the Re number is 10500, and the ratio of bend/rp is 11.11.This kbend value is very close to the present prediction (see Fig. 7(b)).In order to further confirm the dependence of the Re number, the radially varying profiles of velocity and pressure on an x-y plane with z = 0 m require an investigation.The velocity magnitudes, umag, and the pressures, p, across the radius of the pipe relative to its diameter, r/D, at certain locations, l', relative to the length of the bend, lbend, are presented.Herein, lbend is defined as the perimeter of the bend and quantified as 0.5πrbend, with rbend being the radius of bend.Figures 8(a)-(d) show the radially varying profiles of the velocity magnitudes, umag, normalised by the corresponding inlet velocities, uinl, respectively at l' = 0.25lbend, l' = 0.50lbend, l' = 0.75lbend and l' = 1.0lbend for three different Re numbers of 7200, 14400 and 20700.The positive r/D values in Fig. 8 indicate the radial locations near the inner wall, while the negative r/D values are closer to the outer wall.The velocity profile at l' = 0.75lbend near the inner wall (positive r/D values), where the secondary flow occurs, appears to vary with the Re number (see Fig. 8(c)).Overall, the velocity profiles at various locations slightly vary with the Re number.The slight variation of the velocity profile with the Re number contrasts with the variations of the pressure profile.Figure 9 confirms the significant variation of the pressure profile with the Re number; herein, the pressure, p, is Re, k bend = 20700, 0.15) Re, k bend = 7200, 0.49) k bend normalized to the square of the velocity at the inlet, u 2 inl.The importance of Re number is most evident at lower range of Re numbers (i.e.7200 ≤ Re ≤ 14400) and becomes less pronounced with the increasing Re number (i.e.14400 ≤ Re ≤ 20700).This suggests the negligible variation of head loss coefficient at huge Re numbers.Nevertheless, this study confirms important practical implications because the present range of Re numbers is chosen from the actual data of an existing pipe network.As such, predicting pressure at the nodes of a pipe network involving pipes with 90 o -bends should consider the dependence of the head loss coefficient on Re number.This contradicts the common practice in an analysis of pipe flow that assumes a fixed head loss coefficient regardless of the flow discharge.The fields of velocity magnitude, umag, at the Re numbers of 7200 and 20700 are presented on Figure 10.It shows that with the increasing Re number, the location of flow separation moves further upstream, changing the extent of the secondary flow.This is consistent with the results presented in [9].Notably, the field further confirms the dependence of the head loss coefficient for pipe with a 90 o -bend on the Re number.

Pipe Flows Within T-junction
The effect of the Re number on the flow within the pipe with T-junction is also investigated.Similarly, the velocity at the inlet varied while the pipe diameter, pipeand wa, andmeters were kept constant.The variation of the inlet velocity was again determined using the data of water-consumption percentage issued by the Public Work Service of the General Directorate Cipta Karya Air Bersih [14].As a result, the Re numbers considered vary in the range of 6300 ≤ Re ≤ 24300.
Figure 11 presents the variations of the head loss coefficients associated with outlet1 and outlet2, k 1 and k 2 , with the Re number in the cases of pipes with T-junction.Again, outlet1 corresponds to the line flow, and outlet2 is the branch flow.It shows that both the k1 and k2 values decrease less than 50% of its initial values when the Re number increases fourfold.This reduction in the k1 and k2 values is less significant than that in the cases of pipes with bend.Nevertheless, the dependence on the Re number is considered necessary since the decrease is more than 40% of the initial k1 value and 20% of the k2 value at Re=6300; the head loss coefficient associated with the outlet1 (line flow) being more critically dependent on Re number.This dependence is also unexpected given that many literature studies confirm the independence of the head loss coefficient for pipes with T-junction on Re number [4].The fields of umag and the streamlines of the flows near the branch in the three cases of Re numbers further confirm the earlier finding.Figure 14 indicates that the vortex formation at the right corner of the branch varies with the Re number.As the inlet velocity increases, the vortex formation becomes less evident.In contrast, the secondary flows within the branch pipe appear to be similar regardless of the Re number.The secondary flow in the branch pipe always begins at the junction.This contrasts with the bend case (see Figs. 6(a) and 10).The secondary flow begins after the flow separates, and this separation's location depends on the Re number.The comparison indicates that the secondary flow more predictable when flow direction abruptly changes.This is analogous to the fixed location of flow separation when the flow separates from a corner or bluff body.Instead, when the flow encounters a streamlined body, the location of flow separation is more dependent on the Re number [3].The more pronounced dependence of the head loss coefficient for the pipe with a bend than that in the branch flow thus has a physically grounded explanation.The energy loss in these two flows similarly occurs during a secondary flow.However, its starting location and the length over which the secondary flow exists may depend on the Re number.Following on, another case of T-junction with two branch flows is considered.The variations of the head loss coefficients, k1, and k2, with Re number, are presented in Figures 15(a) and (b).Both figures confirm that the k1 and k2 coefficients have equal values.This is unsurprising since the flow from the inlet separates symmetrically at the Tjunction.Furthermore, Figure 15 confirms that the decreases of the k1 and k2 coefficients with the increasing Re numbers are approximately 30%, which is less significant than the percentage of decrease in the bent case (see again Figure 11(a)).Therefore, all the evaluations provided in the present study confirm that the dependence on the Re number for the flow within a pipe with a 90 o -bend is more critical than in the case of a pipe with a T-junction.

CONCLUSIONS
Steady pipe flows within bends and T-junctions were numerically modeled.Convergence studies were undertaken to determine the required grid arrangements to predict the head loss coefficients and pressure differences accurately.The mechanisms that cause the energy loss in the T-junction and the bend have been described.The former is associated with a secondary flow in the branch flow and vortices in the line flow.The latter is driven by a secondary flow following a flow separation at a specific location near the entrance of the bend.
The head loss coefficients in the bends and T-junctions were investigated.This indicates that these coefficients depend on the Re number.The radially varying profiles of pressures and the streamline fields evaluated over various Re numbers support this finding.The study thus recommends considering the variation in the coefficient with the Re number in an analysis of a pipe network that typically assumes constant head loss coefficients regardless of the flow velocity.Furthermore, the flow visualizations help explain why the bent case is more critically dependent on the Re number.The location where flow separates after the entrance of the bend depends on this number, changing the extent of the secondary flow.This significantly differs from the branch flow in which the secondary flow always begins at the junction.

Case 1
Pipe with 90 o -bend The radius of the pipe, rp = 0.045 m Radius of bend, rbend = 0.5 m Case 2 Pipe with T-junction The radius of the pipe, rp = 0.045m

FIGURE 1 .
FIGURE 1.An x-y plane view of the complete pipe geometry with the 90 o -bend.

TABLE 2 .FIGURE 2 .
FIGURE 2. Cross-sectional views of the numerical cells generated under different levels of mesh densities: (a) very coarse mesh, (b) coarse mesh, (c) medium mesh, and (d) fine mesh.

FIGURE 3 .
FIGURE 3. Variations of (a) the pressure drop, P, and (b) the head loss coefficient, k, with level of mesh density.

FIGURE 4 .
FIGURE 4.An x-y plane view of the complete pipe geometry with a T-junction

FIGURE 5 .
FIGURE 5. (a) The pressure drops, P, and (b) the head loss coefficients arise when the fluid flows into outlet1, k1, and (c) outlet2, k2 over various levels of mesh density.
Figure6(a) presents the field of velocity magnitude, umag, that exists within the bend, while Fig.6(b)  shows the field of umag within the T-junction, with its close-up view provided in Fig.6(c).The velocity magnitude, umag, is indicated from the legend, while the arrows on the streamlines present the flow direction.The velocity near the outer wall of the bend has larger magnitude.This is similar to the findings in[9],[12] and[13].In both bend and T-junction cases, flow separations occur.This mainly generates multiple vortices when the water passes through the T-junction (see Figs.6(b) and (c)).Interestingly, a secondary flow exists when the flow direction has to change.This is confirmed from both the bend case, in which the flow direction changes gradually (Fig.6(a)), and the T-junction case, in which an abrupt change of flow direction occurs in the branch flow (Fig.6(b)).The secondary flow in the bend occurs following a flow separation near the bend entrance and inner wall.Interestingly, in the case of T-junction, the secondary flow only occurs in the branch flow and begins at the junction.The branch flow differs from the line flow.This is consistent with the head loss coefficient associated with the outlet2, k2, which has an unidentical value to that corresponding to the outlet1, (see Figs. 5(b) and (c)).The k2 coefficient is greater than the counterpart.This comparison and the flow visualizations within the T-junction indicate that the energy loss arising in the branch flow is more significant than in the line flow.

FIGURE 6 .
FIGURE 6. Field of velocity magnitude, umag, and streamlines of the flow within the pipe near (a) the bend and (b) the T-junction with (c) its close-up view.

FIGURE 7 .
FIGURE 7. Variation of (a) the head loss coefficient, k, and (b) the head loss coefficient associated with the bend, kbend, with Re number in the case of pipe with a 90 o -bend.

FIGURE 10 .
Velocity fields and streamlines of the flows within the pipe near the bend at Re numbers (a) 7200 and (b) 20700.

FIGURE 11 .
FIGURE 11.Variation of the head loss coefficient associated with (a) outlet1, k1, and (b) outlet2, k2, with Re number in the case of T-junction with one line flow and one branch flow.

FIGURE 14 .
FIGURE 14. Field of velocity magnitude, umag, and streamlines of the flow near the T-junction at various Re numbers: (a) 6300, (b) 12600, and (c) 21600.

FIGURE 15 .
FIGURE 15.Variation of the head loss coefficient associated with (a) outlet1, k1, and (b) outlet2, k2, with Re number in the case of T-junction with two branch flows.

TABLE 1 .
Cases and pipe dimensions are considered in the present study.

TABLE 3 .
Comparisons of Radial Size of Cell, Number of Cells, Computational Time, Pressure Drop, and Percentage of Difference Generated in Various Levels of Mesh Densities