Full Length Article

Numerical study on flow separation in 90° pipe bend under highReynolds number by k-ε modellingPrasun Dutta *, Sumit Kumar Saha, Nityananda Nandi, Nairit PalDepartment of Aerospace Engineering and Applied Mechanics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India

A R T I C L E I N F O

Article history:Received 7 October 2015Received in revised form15 December 2015Accepted 15 December 2015Available online 25 January 2016

Keywords:90° pipe bendk-ε turbulence modelTurbulent flowFlow separation

A B S T R A C T

The present paper makes an effort to find the flow separation characteristics under high Reynolds numberin pipe bends. Single phase turbulent flow through pipe bends is investigated using k-ε turbulence model.After the validation of present model against existing experimental results, a detailed study has beenperformed to study the influence of Reynolds number on flow separation and reattachment. The sepa-ration region and the velocity field of the primary and the secondary flows in different sections have beenillustrated. Numerical results show that flow separation can be clearly visualized for bend with low cur-vature ratio. Distributions of the velocity vector show the secondary motion clearly induced by themovement of fluid from inner to outer wall of the bend leading to flow separation. This paper providesnumerical results to understand the flow characteristics of fluid flow in 90° bend pipe.

© 2016, Karabuk University. Publishing services by Elsevier B.V.

1. Introduction

Pipe bends are themost important part of any pipeline networksystem as these provide flexibility in routing. Investigations of theflow through bends are of great significance in understanding andimproving their performance andminimizing the losses. It is alreadywell known that the flow of incompressible viscous fluids throughpipe bends is characterized by flow separation, secondary flow andunsteadiness, which are dependent on Reynolds number as well asthe radius of curvature of the bend.Whenever a fluid flows througha bend, there is a radial pressure gradient developed by the cen-trifugal force acting on the fluid. Because of this, a double spiral flowfield and a pair of counter-rotating vortices can also be observedinside the bend i.e. because of the presence of pressure gradient,fluid at the centre of pipe moves towards the outer side and comesback along the wall towards the inner side. Now if the bend cur-vature ratio is very small (Rc/D ≤ 1.5), the adverse pressure gradientnear the inner wall and immediately downstream of the bend maylead to flow separation, giving rise to a large increase in pressurelosses [1–3].However, theflowcharacteristics of incompressibleflowsin pipe bends are not fully clarified yet. Accurate estimation of massflow rate and losses is critical formost incompressible flow systems.The applications of water-flows through pipe bends are found inmany engineering applications. Some of the excellent reviews bear

testimony to this fact. A number of researchers have investigatedturbulent flows in pipe bends by means of theoretical, experimen-tal and numerical methods [4–9]. To perform numerical simulationof fluid flow in curved pipes, on the other hand, the Navier–Stokesequation has to be expressed in curvilinear or body fitted coordi-nate system. A very useful database for direct numerical simulation(DNS) and large eddy simulation (LES) on pipe bend is provided bytwo studies [10,11]. Recently, in the nuclear sector due to the fatigueby the unsteady motion of the vortices, this has also attracted theinterest of the researchers [2,12,13]. Hence, it is interesting to seethe flow separation and reattachment under high Reynolds number.Micro and nano size particle erosion in 90° pipe bend and over thebackward-facing steps were studied numerically [14–18]. Very re-cently, studies on turbulentmixed convection heat transfer [19–25]attracted the interest of investigators;many researchers used LatticeBoltzmannmethods to solvenatural convectionheat transfer problem[26,27].Magneto hydrodynamic flow (MHD) has attractedmuch in-terest of researchers in recent years due to the effect of magneticfield on the boundary layer flow control [28–33]. Different exact andapproximate techniques have also been used to solve the differentproblems in fluid mechanics [34–38]. In this paper, the flow sep-aration in themost common 90° pipe bend is studied by numericalmethods based on computational fluid dynamics. The paper is struc-tured in the following fashion. Section 1 gives a brief idea on theprevious researchworks andmotivation for thepresentwork. Section2 contains the necessary theoretical background. Problem defini-tion with validation is provided in section 3. Section 4 contains thestudy on various parameters affecting the flow pattern and is fol-lowed by summary bulletin of the study under section 5.

* Corresponding author. Tel.: +91 33 2668–4561 to 63 (Extn.: 277).E-mail address: pd.iiest@gmail.com (P. Dutta).Peer review under responsibility of Karabuk University.

http://dx.doi.org/10.1016/j.jestch.2015.12.0052215-0986/© 2016, Karabuk University. Publishing services by Elsevier B.V.

Engineering Science and Technology, an International Journal 19 (2016) 904–910

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2. Governing equations and numerical methodology

Three dimensional Reynolds Averaged Navier–Stokes (RANS)equations are solved using the segregated implicit solver. The rightchoice of a turbulence model is critical when an industrial turbu-lent flow problem is faced, especially when this problem involvesthree dimensional flow phenomena, which needs an accurate mod-elling. The second order scheme is used for the U-RANS equationscalculations, with a pressure velocity coupling achieved using SIMPLEalgorithm. The time step size used in the present study is 0.001 swith 1000 time steps. The default under relaxation factors were usedto aid convergence for all models.

The governing equations for incompressible fluid flowwith con-stant properties are

∂∂

=ux

i

i

0 (1)

∂∂

+ ∂∂

= − ∂∂

+ ∂∂ ∂

ut

uux

fpx

ux x

ij

i

ji

i

i

j j

1 2

ρυ (2)

Equations (1) and (2) are conservations of mass and momen-tum, respectively; fi is a vector representing external forces, υ is thekinematic viscosity.

2.1. Turbulence model

It is well known that turbulent flows are basically designated bythe fluctuations of the velocity fields. Different transported quan-tities such as momentum, energy, etc. also fluctuate for thisfluctuation of velocity field and these fluctuations can be of veryhigh frequency and small scale; they are very difficult andcomputationally crucial to analyse directly in industrial engineer-ing calculations. The turbulence model needs to be selected basedon some considerations, e.g., the physics of the flow, the insight intothe capabilities and limitations of turbulence models, the attemptfor the specific problem by other researchers, the accuracy needed,the available computational resources, and time.

The k-ε turbulence model is adopted for the present study as k-εturbulence model performs better for both single-phase and two-phase flows in pipe bend [3,39–43]. In this model, the turbulencekinetic energy (k) and the turbulence dissipation rate (ε) are solvedto determine the coefficient of turbulent viscosity (μt).

Transport equation for k-epsilon

∂( )∂

+ ∂( )∂

= ∂∂

∂∂

⎡

⎣⎢

⎤

⎦⎥ + −pk

tpku

x xkx

E Ei

i j

t

k jt ij ij

μσ

μ ρε2 (3)

∂( )∂

+ ∂( )∂

= ∂∂

∂∂

⎡

⎣⎢

⎤

⎦⎥ + −p

tp ux x

kx

Ck

E E Ci

i j

t

jt ij ij

ε ε μσ

ε μ ρε

ε ε1 22εε2

k(4)

where ui represents velocity component in corresponding direc-tion, Eij represents component of rate of deformation, and μt

represents eddy viscosity.Equations (3) and (4) also consist of some adjustable constants

[44]; these are as follows

C C Cμ κ κ ε ε εσ σ σ= = = = = =0 09 1 00 1 00 1 30 1 44 1 921 2. . . . . .

3. Problem definition

The problem that is considered here is the fluid flow through90° pipe bends having inner diameter of 0.01mwith curvature ratio(Rc/D) = 1 for different Reynolds numbers ranging from 1 × 105 to10 × 105. The inlet length of straight pipe in the calculations was

set up 50D for all cases to save computational time. The fluidmediumwas air having density (ρ) of 1.2647 kg/m3 and dynamic viscosity(μ) of 1.983 × 10−5 kg/m-s for validation purpose and water havingdensity (ρ) of 990.2 kg/m3 and dynamic viscosity (μ) of 0.0006 kg/m-sfor the present study with working temperature of 300 K in bothcases. Three dimensional structuredmeshwas used containing hexa-hedron elements, which was optimized via a grid-independencestudy. The bend geometry and mesh are shown in Fig. 1a and b re-spectively. It is defined that the axial direction downstream the bendis x-coordinate, the direction from inner core to outer core of thebend is y-coordinate and the perpendicular direction to x and y isz-coordinate.

3.1. Validation

At the very beginning of our study, our model and simulationsetup are first validated against the existing experimental and nu-merical data in References [3,7,45]. For that intension, samegeometrical configuration is adopted. In their experiment, the authorsof the previously mentioned studies used a circular cross sec-tioned 90° bend with a curvature ratio (Rc/D) of 2 and themeasurements of velocities were performed at a Reynolds numberof 6 × 104. For the validation of our present model, the simulationis performed on a computational mesh containing total 2.85 millionhexahedron elements, whichwas optimized via a grid-independencestudy, see Fig. 1a. The value of non-dimensional distance from wall(Y+) is strictly controlled using standard wall treatment function(30 < Y+ < 90 for a near wall cell used for present study). The mean

(a)

(b)

Fig. 1. Schematic diagram of the bend geometry and present model with compu-tational grid.

905P. Dutta et al. / Engineering Science and Technology, an International Journal 19 (2016) 904–910

axial velocity profile normalized with inlet velocity along symme-try line at bend outlet (α = 90°) shows very good agreement withboth experimental and numerical results [3,7,45], see Fig. 2b. In theinner core region of the bend (negative X-axis in Fig.), some dis-crepancies between experimental and numerical data are observed.This discrepancies may arise due to the adverse pressure gradientnear the inner wall of the bend and this is also consistent with arecent study [3]. From the validation part, it has been seen that themodel is in close approximation with the published results; hence,this procedure of mesh generation and simulation set up has beenused for further analysis.

4. Results and discussions

The main objective of the present study is to characterize theeffect of Reynolds number on flow separation of single phase tur-bulent flow in a 90° pipe bend through numerical simulation. Theresults of the mean velocity profile, velocity fluctuation and sepa-ration reattachment points for different Reynolds numbers alongthe different positions of the bend in central symmetry plane arepresented in this section.

Fig. 3 shows the mean velocity profile normalized with inlet ve-locity (Uin) for different Reynolds numbers (Re = 1 × 105 to 10 × 105).The negative r values represent the inner core of the bend and SP,

RP, MP represent positions of separation point, reattachment point,middle point between separation and reattachment points respec-tively. Velocity acceleration is depicted at the outer core of the bendas expected. It is found that with the increase of Reynolds number,the velocity profile at the inner core of the pipe bend tries to recoverits fully developed shape by decelerating and accelerating effectsat the outer and the inner parts of the pipe bend respectively. Hence,it may be speculated for higher values of the Reynolds number thatpipe curvature effects are reducing.

Fig. 4 shows the normalized flow velocity vector fields at thecentral plane and at the bend outlet (α = 90°) for Re = 1 × 105,Rc/D = 1. The mean velocity is shifted upward due to the fastermoving fluid near the outer core as expressed in Reference [7]. Sec-ondary flow pattern can clearly be observed in this section. At theinner core of the bend, a region having very low velocity was foundand the flow was highly unsteady and complex.

Fig. 5 shows the normalized velocity fields for two different Reyn-olds numbers (Re = 1 × 105 & 10 × 105) at the bend outlet. The topof the each figure is the outer core of the bend and the bottom isthe inner core of the bend. The low velocity region near the innercore of the bend was seen clearly. This region is concluded to bethe separated region [46]. Because of the relation between the cen-trifugal force with high velocity and pressure gradient on the flow,two identical counter rotating Dean vortices were also found for bothReynolds numbers.

As mentioned above, a flow separation region was found at theinner core of bend outlet; another study has been made to under-stand the dependency of separation and reattachment points onReynolds number. Fig. 6 shows various starting points of flow sep-aration from the bend inlet and corresponding reattachment pointsfrom the bend outlet. It was found that the starting point of flow sep-aration moves upstream in the bend as Reynolds number increases,while the reattachment point moves downstream. This trend corre-sponds to the dependency of Reynolds number on the total pressureloss [45] and consistentwith experimental results in References [2,13].

The rms velocity fluctuations at bend outlet in the x and y di-rections are depicted in Fig. 7 for bend with Rc/D = 1 at differentReynolds numbers. Non-dimensional fluctuating velocities Urms andVrms, normalized by the mean flow velocity in x and y directions re-spectively. Negative values of r represent the inner core of the bend.It is found that velocity fluctuation is maximum in both x and y di-rections at the separation region; however, the intensity offluctuation is higher in the x direction. The Urms profiles do not showmuch Reynolds number dependency, the peak position is almostsame for all measurements but the peak value is high for higher

(a) (b)

Fig. 2. Comparison of normalized axial velocity profile of present analysis with published experimental and numerical results.

Fig. 3. Normalized velocity profiles at different positions in the bend for different Re.

906 P. Dutta et al. / Engineering Science and Technology, an International Journal 19 (2016) 904–910

(a) (b)

Fig. 4. Normalized velocity vector field for Re = 1 × 105 and Rc/D = 1. (a) Symmetry plane (centre cross section). (b) Cross section at bend outlet (∝ = 90°).

(a) (b)

Fig. 5. Normalized velocity vector field. (a) Re = 1 × 105. (b) Re = 10 × 105.

(a) (b)

Fig. 6. Dependency of separation and reattachment points on Reynolds number. (a) Separation points. (b) Reattachment points.

907P. Dutta et al. / Engineering Science and Technology, an International Journal 19 (2016) 904–910

Reynolds number. For the Vrms profiles, a clear Reynolds number de-pendency can be observed. The peak value becomes larger andshifted from the inner core towards the centre of the bend with in-creasing Reynolds number.

Asmentioned earlier, a very complicated fluid structure is formedat the downstream of bend outlet due to the flow fluctuation inducedby the movement separation region and the acceleration of flow ve-locity. Fig. 8a–d shows the complex flowmotions in the bend. Fig. 8a

(a) (b)

Fig. 7. rms profiles of velocity fluctuations at the bend outlet. (a) rms in x direction. (b) rms in y direction.

(a)

(c)

(b)

(d)

Fig. 8. Complex flow structure in the pipe bend. (a) Cross sectional planes for flow visualization. (b) Flow at symmetry plane. (c) Flow in separated region. (d) Circumfer-ential flow from outer core to inner core.

908 P. Dutta et al. / Engineering Science and Technology, an International Journal 19 (2016) 904–910

describes the three cross sectional planes for the flow visualiza-tions. Three types of flowmotions are observed. Takamura et al. [2]studied these three motions by statistical analysis (frequency anal-ysis) in terms of Strouhal number (St) and explained as vorticesmotion with St ≈ 1.0 (Fig. 8b), separated vortices shedding withSt ≈ 0.5 (Fig. 8c) and circumferential motion with St ≈ 0.5 (Fig. 8c).

5. Conclusions

Turbulent flow of single phase incompressible fluid through 90°pipe bend has been simulated numerically using k-ε turbulent modelin the present study. The validation of 3Dmodels used for the presentstudy with experiments reported in Reference [7] and numericalworks in References [3,45] indicates a good agreement. The fol-lowing conclusions can be made from the present study:

• It is found that with the increase of Reynolds number, the ve-locity profile at the inner core of the pipe bend tries to recoverits fully developed shape by decelerating and accelerating effectsat the outer and the inner parts of the pipe bend respectively.Hence, it may be speculated for higher values of the Reynoldsnumber that pipe curvature effects are reducing. The separa-tion region for all bends was generated and a clear Reynoldsnumber dependency on flow separation was found. As Reyn-olds number increased, the separation point moved towards theupstream, while the reattachment point moved towards down-stream of the bend. A low velocity region was found near theinner core of the bend leading to the flow separation. Two counterrotating Dean vortices were formed at the bend outlet.

• The rms of the velocity fluctuation in x direction of flow has nosignificant Reynolds number dependence, the peak value and theposition of the peak value are nearly the same for different Reyn-olds numbers; however, the peak value was observed for thehighest Reynolds number. For the rms profile in y direction, alittle Reynolds number dependence can be observed. With in-creasing Reynolds number, the peak value becomes larger andsituated further from thewall. Overall velocity fluctuation becamelarger in the separation region near the inner core of the bend;however, the intensity of fluctuation in x direction is lower thanthat of y direction. The flow became very complex, unsteady andcoherent at the downstream of the bend due to the flow sepa-ration showing three flow motions.

The above conclusions refer especially for present study range.To provide a correlation between the separation region, curvatureratio and the Reynolds number, additional studies is required withdifferent Reynolds numbers and curvature ratio and will remain forfuture work.

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