Friction Factor for Corrugated Pipe_JOMAE

Ftite inortentan

onvouentrou

e imexph

essaluFinipee f14

pipe

1

tcvery devices to main distribution ducts such as transport of liquidnatural gas from ships to the mainland distribution network.

Flexible ducts are often comprised of fabric wrapped over asvmslssio

ssTflatpfTfis

2 Models and Boundary Conditions for TurbulentFlows

psl

J

Downloapiral metal framework. Due to this construction, they respondery well to bending, are cheaper, and much easier to install thanetal pipes. Figure 1 shows a typical flexible pipe. It is con-

tructed from a neoprene impregnated polyester fabric encapsu-ating a helix spring of steel wire. This tube has an excellenttrength/weight ratio and is able to withstand severe flexing. Theteel spiral wire gives strength to the pipe, while the use of fabricnstead of a hard material such as metal allows for a high degreef flexibility.

Because of the specific construction, the pipe walls are nottraightthey are corrugated. Moreover, the fabric that covers theteel spiral is much rougher than the wall of a smooth metal pipe.his requires more energy higher pressure difference to drive theow. Therefore, an important factor in flexible pipe design is tottain minimum pressure loss throughout the distribution line andhus minimize the transportation costs. The pressure loss along aipe is caused by the friction at the wall. In stationary flow, theriction is proportional to the pressure loss per unit distance 1.herefore, in this investigation, we are interested in estimating the

riction factor for turbulent flow at Reynolds numbers around 106n a flexible pipe with a specific configuration. In particular, wetudy the influence of the roughness of the fabric on the friction

There are basically three ways to simulate turbulent flow: directnumerical simulation DNS, large-eddy simulation LES, andReynolds averaged NavierStokes RANS models. Due to theirrandomness, turbulent flows are difficult to simulate. The moredetails we want to obtain from a simulation, the higher is thecomputational cost. DNS and LES offer a high degree of details,but require prohibitively large times for the flow simulations,which are of interest to us. Given our purposes and the availablecomputational power, we will use RANS models specifically k and, to a lesser extent, k for our simulations. The RANSequations are time- or ensemble-averaged equations of motion forfluid flow. The averaging process brings new unknown terms intothe NavierStokes equation. Therefore, additional closure equa-tions are needed to be able to solve the system. These equationsare derived by taking higher-order moments of the averagedNavierStokes equation and making additional assumptions basedon the knowledge of the properties of the turbulent flow. Thisprocess results in a modified set of equations that is computation-ally less expensive to solve. Below, we give the equations, whichdefine the k model and the k model 2.

2.1 Mean Flow EquationsMass conservation

Ujxj

= 0 1

Momentum conservation

1Corresponding author.Contributed by the Ocean Offshore and Arctic Engineering Division of ASME for

ublication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manu-cript received September 7, 2009; final manuscript received January 20, 2010; pub-ished online November 3, 2010. Assoc. Editor: Sergio Sphaier.

ournal of Offshore Mechanics and Arctic Engineering FEBRUARY 2011, Vol. 133 / 011101-1Maxim Pisarenco1e-mail: [email protected]

Bas van der Linden

Arris Tijsseling

Eindhoven University of Technology,P.O. Box 513,

5600 MB Eindhoven, The Netherlands

Emmanuel OrySingle Buoy Moorings Inc.,

P.O. Box 199,MC 98007 Monaco Cedex, Monaco

Jacques DamStork Inoteq,P.O. Box 379,

1000 AJ Amsterdam, The Netherlands

FrictionTurbulenPipes wThe motivation of thused for LNG transpfactor for the turbullence models (kturbulent flow in a cthe chosen models, bcondition is implemmodel the effects ofnolds numbers). Thexperimental data. Nconsidered. New flowpointed out and thefactor for different vset of simulations.flexible corrugated pnot by the type of thDOI: 10.1115/1.400

Keywords: flexiblelaw of the wall

IntroductionNonmetallic flexible pipe products have found wide usage in

he industry. Areas of application include heating, ventilation, air-onditioning, and, most importantly, connecting terminal or deli-Copyright 20

ded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMactor Estimation forFlows in Corrugatedh Rough Wallsvestigation is the critical pressure loss in cryogenic flexible hosesin offshore installations. Our main goal is to estimate the frictionflow in this type of pipes. For this purpose, two-equation turbu-

d k) are used in the computations. First, the fully developedentional pipe is considered. Simulations are performed to validatendary conditions, and computational grids. Then a new boundaryed based on the combined law of the wall. It enables us toghness (and maintain the right flow behavior for moderate Rey-plemented boundary condition is validated by comparison with

t, the turbulent flow in periodically corrugated (flexible) pipes isenomena (such as flow separation) caused by the corrugation are

ence of periodically fully developed flow is explained. The frictiones of relative roughness of the fabric is estimated by performing aally, the main conclusion is presented: The friction factor in ais mostly determined by the shape and size of the steel spiral, and

abric, which is wrapped around the spiral.39

, friction factor, roughness modeling, corrugated pipe, modified

factor. The performance of different two-equation turbulencemodels, boundary conditions, and computational grids is investi-gated.11 by ASME

E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

TT

wt

T

S

w

wpt

gsl

0

Downloa Uit

+ UjUixj = P

xi+

xj + T Uixj + Ujxi 2

2.2 Transport Equations for Standard k Modelurbulence energy equation

kt

+ Ujkxj

= ijUixj

+

xj + T

k kxj 3

urbulence dissipation equation

t+ Uj

xj= C1

kij

Uixj

C22

k+

xj + T

xj4

ith C1=1.44, C2=1.92, C=0.09, k=1.0, =1.3, andurbulent viscosity T=Ck2 /.

2.3 Transport Equations for Standard k Modelurbulence energy equation

kt

+ Ujkxj

= ijUixj

k +

xj + T kxj 5

pecific dissipation rate equation the -equation

t+ Uj

xj=

kij

Uixj

2 +

xj + Txj

6ith = 59 , =

340,

=

9100, =

12 ,

=

12 , and T=k /.

The following notation was used in Eqs. 16:

Uicomponents of the velocity vector, xicomponents of the position vector, ttime, fluid density, Ppressure, viscosity, Tturbulent viscosity, kturbulence kinetic energy, turbulence dissipation, rate of dissipation per unit turbulence kinetic energy, ijReynolds stress tensor, ij =TUi /xj +Uj /xi.

These models are given, as defined by Wilcox in Ref. 2. It isorth noting here that an updated and improved k model wasresented in 2006 3. However, the earlier standard version ofhe k model was the only option in the used software package.

2.4 The Law of the Wall. Because of the large velocityradients arising in the region near the wall, this area requirespecial treatment. Moreover, the flow in the near-wall region is noonger turbulent at least not everywhere so that the assumptions

Fig. 1 Typical flexible pipe

11101-2 / Vol. 133, FEBRUARY 2011ded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMmade while deriving the turbulence models are not valid.Traditionally, there are two approaches to modeling the flow in

the near-wall region. In one approach, the turbulence models aremodified to enable the viscosity-affected region to be resolvedwith a mesh, all the way to the wall, including the viscous sub-layer. In another approach, the viscosity-affected inner regionviscous sublayer and buffer layer is not resolved. Instead, semi-empirical wall functions are used to bridge the viscosity-affectedregion between the wall and the fully turbulent region. The use ofwall functions obviates the need to modify the turbulence modelsto account for the presence of the wall. These two approaches aredepicted schematically in Fig. 2.

In most high-Reynolds-number flows, the wall function ap-proach substantially saves computational resources, because theviscosity-affected near-wall region, in which the solution variableschange most rapidly, does not need to be resolved. The wall func-tion approach is popular because it is economical, robust, andreasonably accurate. It is a practical option for the near-wall treat-ment in industrial flow simulations.

The boundary conditions derived from the wall functions lawof the wall are applied at a location y=yp in the log-law region yis the direction normal to the wall. We use the subscript p toindicate quantities evaluated at yp, such as Up, kp, p, and Tp.The law of the smooth wall is given by the relation

Upu

=

1

lnuyp + B 7

n kp = 0, p =Ckp

2

uyp, p =

kpuyp

8

where Up is the tangential velocity, u=C1/4kp

1/2 is the shear ve-locity, =0.41, B=5.0, . . . ,5.5, and n is the unit vector normal tothe wall. The value of yp is chosen, such that yp

+=uyp / is

between 30 and 100 i.e., in the range of the log-layer.If the law of the wall would describe the velocity profile ex-

actly, then, assuming a perfect turbulence model, the choice of yp+

in the range between 30 and 100 would not influence the solu-tion. However, the law of the wall is a semi-empirical relation,and the k /k models are based on assumptions, which do notalways hold. Therefore, the solution does depend on the thicknessof the near-wall region. We hope though, that this dependence isnot too strong again, for yp

+ in the range between 30 and 100.Simulations have been performed to observe the influence of

the thickness of the near-wall region on the solution. The value yp+

spans from 6.25 up to 3200 in a geometric progression. To mea-sure the difference between solutions, a certain norm could beused. However, since we are ultimately interested in friction factorestimation, we will use this as a physically relevant indicator.

Figure 3 shows the dependence of the friction factor on thethickness of the near-wall region at a Reynolds number of order106. Although some preliminary simulations showed a strong de-pendence of f on yp+ 4, after individually adjusting the mesh by

Fig. 2 Schematic representation of the mesh for a wall func-tion and a near-wall model approach

Transactions of the ASMEE license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

uhittcf

iotlBatdRbntrpe

3g

RSoflacdad

astai

cbA

0.012

0.014

Fol

J

Downloasing an adaptive solver for each value of yp+, much better results

ave been obtained, as now shown in the plot. The friction factors almost constant in the range of yp

+ between 50 and 300 betweenhe two vertical lines in the plot, where the variation of f is lesshan 1.5%. Thus, the range of validity of the law of the wall inombination with the k /k model proves to be in practicerom 50 to 300 instead of 30 to 100.

It must be mentioned that the logarithmic law of the wall is notndisputable. After all, Prandtls assumption used in the derivationf the law is based only on dimensional grounds. The scaling inhe inertial sublayer also referred to as overlap region of turbu-ent wall-bounded flows has long been the source of controversy.arenblatt et al. 5 developed theories showing that power lawsre more suitable for describing velocity profiles in wall-boundedurbulent flows. Until recently, this controversy could not be ad-ressed because measurements did not span a sufficient range ofeynolds numbers. However, in 1997 new experiments conductedy Zaragola et al. 6 showed that at sufficiently high-Reynoldsumbers, the mean velocity profile in the overlap region is foundo be better represented by a log-law than a power law. Theseesults suggest a theory of complete similarity instead of incom-lete similarity, contradicting the theories developed by Barenblattt al.

Flow Simulations for Smooth and Rough Noncorru-ated PipesIn this section, we evaluate the performance of two-equation

ANS models, as implemented in the finite element package COM-OL MULTIPHYSICS 7, and validate the boundary conditions basedn the law of the wall, which will be used later for more complexows. To do this, we will simulate the turbulent flow in a pipe andssess the validity of the results by comparing the friction factoromputed from the simulation to the one given by the Moodyiagram. Models used in the simulations are k Eqs. 14nd k Eqs. 1, 2, 5, and 6, written in cylindrical coor-inates.

3.1 Computational Domain Geometry. The fully developednd time-averaged turbulent flow in a smooth pipe is one- dimen-ional and axisymmetric in its nature; the only dimension beingaken in the radial direction. In the following computations, a 2Dxisymmetric model with periodic boundary conditions couplingnflow and outflow will be used; see Fig. 4.

3.2 Boundary Conditions. The axial symmetry boundaryondition is prescribed at the centerline of the pipe. The otheroundary parallel to the flow coincides with the wall of the pipe.long this boundary, the law of the wall Eq. 7 is used to

100

101

102

103

104

0

0.002

0.004

0.006

0.008

0.01

yp+

f

ig. 3 Dependence of the friction factor f on the thickness yp+f the near-wall region at Reynolds numbers of order 106. Solid

ine computed values, dashed line experimental valuesfrom the Moody diagram. The k model is used.

ournal of Offshore Mechanics and Arctic Engineeringded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMprescribe the axial velocity at a certain distance from the wall.An effective method of simulating a fully developed flow on a

small computational domain is the use of periodic boundary con-ditions. Their use is explained by the fact that the fully developedflow has a constant velocity profile, which means that

U1r,0 = U1r,L

U2r,0 = U2r,L

kr,0 = kr,L

r,0 = r,L

r,0 = r,L 9where L is the length of the computational domain in the axialdirection set as L=0.02 m.

At inflow and outflow boundaries, we will prescribe constantpressures Pr ,0= Pin and Pr ,L= Pout, where Pout is taken to bezero. We are entitled to do this because in a fully developed flowthrough a duct with a constant shape cross section, the transversevelocity components vanish and it can be easily proven from themomentum conservation 2 that

Pr

= 0 Pr = constant 10

3.3 Meshing and Solution Procedure. Although our compu-tational domain is very regular, which encourages the use of struc-tured meshes, it was decided to use unstructured grids based onthe Delaunay triangulation for the discretization step because wewant to use similar grids for both corrugated and noncorrugatedpipes, and only unstructured grids can be used for the latter.

The solution procedure consists of the following three steps.

Solve the model on a coarse mesh. Refine the mesh by subdivision to obtain the one shown in

Fig. 4. Solve the model on the refined mesh, using the coarse mesh

solution as an initial guess.

This strategy proved to be faster than immediately solving themodel on the refined mesh. After the solution was obtained, it wasadditionally checked by uniformly refining the mesh once againand comparing the new solution to the previous one. The differ-ence between them was in all cases less than 2%.

3.4 Smooth Wall Validation. To validate the simulation ofturbulent flow in pipes with smooth walls, a post-processing pro-cedure is performed at the end of each simulation. The averagevelocity Vavg, Reynolds number Re, and DarcyWeisbach frictionfactor f are computed as follows:

Vavg =0

R U12r

R2

dr, Re =2RVavg

, f = 2RP0.5Vavg2 L

11

where P is the prescribed difference in pressure between in- andoutflow, R is the radius of the pipe, and L is the length of the pipesection included in the computational domain.

It is worth mentioning here that, because of the way the law ofthe wall is used as a boundary condition in COMSOL MULTIPHYSICS,the computational domain does not include the whole physicaldomain. Specifically, it does not include the thin layer near the

Fig. 4 Computational domain and the fine mesh used for thelast step of the computations

FEBRUARY 2011, Vol. 133 / 011101-3E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

wsbtb

na

pMt1tFtkbmitBn

tt

0.04

0.045kepsilon, B=5.5kepsilon, B=5.0komega, B=5.5

Fp

Fp

0.07

0.08e/D=0.0005e/D=0.001e/D=0.005

0

Downloaall where the velocity profile is given by the law of the wall thehaded area in Fig. 2. Therefore, in Eq. 11, the radius R shoulde understood as R=R+yp, where R is the radius of the pipe inhe simulations. In the computations, the following values haveeen used: R=0.2 m and L=0.02 m.Simulations have been performed for a wide range of Reynolds

umbers from 104 up to 108 using the two turbulence modelsk and k. For each model, two different values of B 5.5.nd 5.0 were used in the boundary condition given by Eq. 7.

Thus, for each Reynolds number, we end up with four com-uted friction factors plus the friction factor taken from theoody diagram. These are shown in Fig. 5. As can be seen from

he plot, in the range of relatively low Reynolds numbers 104 to05, the computed values are far from the measured friction fac-or. The k model seems to perform better in this range of Re.or higher Reynolds numbers, that is, fully developed turbulence,

he computed values follow closely the measured value; for Re5105, the relative error is less than 0.04. Now, the k and

models perform equally well and the choice of B in theoundary condition becomes important. As seen from Fig. 6showing the plot from Fig. 5 zoomed around Re=106, for Re

7105 simulations with B=5.0 give closer agreement with theeasured friction factor, while for Re7105, the friction factor

s better predicted by the simulations with B=5.5. This could behe reason why different authors give slightly different values for

in the law of the wall; its choice depends on the Reynoldsumber characteristic to the flow.

3.5 Rough Wall Modeling and Validation. The asperities onhe pipe wall become important when their size is comparable tohe thickness of the laminar sublayer. Roughness effects can be

102

104

106

108

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Re

f

komega, B=5.0Moody

ig. 5 Computed and measured friction factors for smoothipes e /D=0

106

0.01

0.0105

0.011

0.0115

0.012

0.0125

0.013

0.0135

0.014

Re

f

kepsilon, B=5.5kepsilon, B=5.0komega, B=5.5komega, B=5.0Moody

ig. 6 Computed and measured friction factors for smoothipes e /D=0 zoomed around Re106

11101-4 / Vol. 133, FEBRUARY 2011ded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMaccounted for by using a law of the wall modified for roughness.If e is the equivalent height of the asperities, the law of therough wall is given by the relation see, e.g., Ref. 2

Uu

=

1

ln ype + 8.5 12

Figure 7 shows the friction factors computed using the lawabove. We observe that this law provides a good prediction of thefriction factor only for Reynolds numbers higher than 106, and istotally wrong for Reynolds numbers lower than 105. To get a goodapproximation for the whole spectrum of Reynolds numbers, weneed to combine the law for smooth walls with the law for roughwalls.

Experiments in roughened pipes and channels indicate that themean velocity distribution near rough walls, when plotted in theusual semilogarithmic scale, has the same slope 1 / but a dif-ferent intercept additive constant B in the log-law. Thus, the lawof the wall for mean velocity modified for roughness has the form8

Uu

=

1

lnuyp + B 13

where B is a roughness function that quantifies the shift of theintercept due to roughness effects.

For sand-grain roughness and similar types of uniform rough-ness elements, B has been found to be well-correlated with thenondimensional roughness height, e+=eu /. Analysis of ex-perimental data for uniform roughness shows that the roughnessfunction B is not a single function of e+, but takes different formsdepending on the e+ value. It has been observed that there arethree distinct regimes:

hydrodynamically smooth e+e1+,

transitional e1+e+e2

+, and fully rough e+e2

+,

where e1+3, . . . ,5 and e2

+70, . . . ,90.According to the data, roughness effects are negligible in the

hydrodynamically smooth regime, but become increasingly im-portant in the transitional regime, and take full effect in the fullyrough regime. The formulas proposed by Ioselevich and Pilipenkoin Ref. 8 are adopted to compute the roughness function B foreach regime

103

104

105

106

107

0

0.01

0.02

0.03

0.04

0.05

0.06

Re

f

e/D=0.01Moody

Fig. 7 Computed dashed lines and measured solid linesfriction factors for flow in pipes with rough walls. The kmodel is used.


T

Ra

TppTh

ie

te

dctrwamc

mswMTdpgpoe

Ref. 13. Our computations resemble the data from Nikuradsebecause the latter was used by Ioselevich and Pilipenko 8 to fitthe combined law of the wall Eqs. 13, 14, and 16.

e/D=0.0005e/D=0.001e/D=0.005

Ffo

J

DownloaB = B + 8.5 B 1

ln e+, with B = 5.5 14he case of hydrodynamic smoothness corresponds to =0 e+e1

+, whereas the case of full roughness corresponds to =1e+e2

+. The function =e+ for e1+e+e2

+ is obtained inef. 8 from the analysis of experimental data. The followingpproximation is proposed for

= sin2 lne+/e1+

lne2+/e1

+ 15

he values for e1+ and e2

+ recommended by Ioselevich and Pili-enko are 2.25 although outside of the prescribed interval, inractice, this value gives optimal results and 90, respectively.hese values were used in the present investigation. Thus, weave

= 0, e+ 2.25

sin0.4258ln e+ 0.811 , e+ 2.25,901, e+ 90 16

Given the roughness parameter, the roughness function Be+s evaluated using Eq. 14 and the corresponding formula for Eq. 16. The modified law of the wall in Eq. 13 is then used tovaluate the velocity at the wall.

Figure 8 shows the results of the computations. They were ob-ained by varying two parameters of the flow: pressure differenceP= 0.1,1 ,10,100,1000,10000 Pa and relative roughness/2R= 0.0005,0.001,0.005,0.01,0.05. The pressure variationetermines the variation in the Reynolds number, while thehange in relative roughness generates a set of distinct curves onhe diagram. Solid lines indicate the computed friction factor. Aseference, we used the measurements of Nikuradse 9,10, plottedith dashed lines. An analytical expression suggested by Uri

nd Kompare 11 was used to approximate those and his owneasurements. There is an excellent agreement of measured and

omputed values for Re106. In the transitional regime 104Re105, the friction factor deviates slightly from the measure-ents, but has a similar qualitative behavior. Our computations

eem to have a closer agreement with the data of Nikuradse thanith Moodys diagram. Nikuradses data and the data in theoody diagram have a small difference in the transitional regime.

his difference is caused by the fact that the transition from hy-raulically smooth conditions at small Reynolds numbers to com-lete roughness at large Reynolds numbers occurs much moreradually in commercial rough pipes than in artificially roughenedipes used by Nikuradse. Moodys diagram reproduces the dataf Colebrook and White 12, who used commercial pipes in theirxperiments. The history behind Moodys diagram is discussed in

104

105

106

107

0.01

0.02

0.03

0.04

0.05

Re

f

e/D=0.01Nikuradse

ig. 8 Computed dashed lines and measured solid linesriction factors for flow with rough walls using a combined lawf the wall; Eqs. 13, 14, and 16. The k model is used.

ournal of Offshore Mechanics and Arctic Engineeringded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASM4 Flow Simulations for Corrugated PipesAs it was explained in the Sec. 1, flexible ducts have a specific

structure. To enable the simulation of the flow in 2D, the steelspiral is modeled as an annular corrugation and not as a helicalone. Helical corrugation requires more demanding 3D simulationsand is studied in Ref. 14.

The flow inside corrugated pipes has some important proper-ties, which are not characteristic to flows in conventional pipes.One of the expected effects of corrugation is that the transitionfrom laminar to turbulent flow will occur at lower Reynolds num-bers so, at Re2000. Nishimura et al. 15, Russ and Beer 16,and Yang 17 investigated the transitional flow characteristics incorrugated ducts. Indeed, they reported the laminar-turbulent tran-sition to occur at very low Reynolds numbers compared with con-ventional ducts. Another important characteristic of flow in corru-gated pipes is the possible presence of local adverse pressuregradients and, as a result, the appearance of flow separations.

The objective in this section is to set up a model in COMSOLMULTIPHYSICS, which will allow us to perform a set of simulationsand compute the friction factor depending on parameters such asroughness of the hose wall. Before we perform the simulations, aproper turbulence model has to be chosen k, k, or both.Because of the curved surfaces present in the geometry, we expectto encounter separated flow. Prediction of separated flows is anAchilles heel for many turbulence models 18, therefore, simu-lations of turbulent flow over a backward facing step a standardtest problem for separated flow have been performed using thek and k turbulence models. Results, presented in Ref. 4showed that, although both models underpredict the reattachmentlength, the values given by the k model are closer to experi-mental data. Therefore, we have chosen the k model, given byEqs. 14, for our computations.

4.1 Computational Domain Geometry. In general, fully de-veloped turbulent flow in a helically corrugated pipe is of three-dimensional nature. For the annular corrugation considered, it canbe reduced to 2D because of axial symmetry. In other words,characteristics of the flow only depend on the distance from thepipes centerline r and the position along the pipe x within a singleperiod.

Figure 9 shows the computational domain for the simulatedpipe. It includes 1 period of the corrugated pipe. At the left and atthe right, it is bounded by the wall and by the symmetry axis,respectively, while the inflow/outflow boundaries are the ones lo-cated at the top and bottom of the domain. The radius of the spiralis 1 cm.

4.2 Boundary Conditions. As can be seen from Fig. 9, ourcomputational domain has five distinct boundaries at whichboundary conditions have to be prescribed. Axial symmetry isprescribed at the centerline of the pipe. The other boundary par-allel to the flow coincides with the wall of the pipe, which consistsof two different materials: a flexible hose made of fabric and asteel spiral. The spiral, made of steel, is considered a smoothsurface. Therefore, the standard law of the wall Eq. 7 is usedthere. The part made of fabric, on the other hand, is considered tobe rough, so the combined law of the wall modified for roughnessEq. 13 is used.

The boundaries normal perpendicular to the flow representinflow and outflow boundaries. Far from the duct entrance, theflow will be periodically fully developed 19 because of the equi-distant positioning of the corrugations. In a periodic fully devel-oped flow, the velocity repeats itself at corresponding axial loca-tions in successive cycles. Now, it becomes clear why it is enoughto confine the computational domain to a single period cycle ofthe corrugation and use again the periodic boundary conditions

FEBRUARY 2011, Vol. 133 / 011101-5E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

dtgcs

w

coi

Bwoa

w

dtetcstd

cbn

es

0

Downloaefined by Eq. 9.Now remains the question of the conditions for the pressure at

he inflow/outflow boundaries. In the periodic fully developed re-ime, the pressure at periodically corresponding locations de-reases linearly in the downstream direction 20. Thus, the pres-ure P can be expressed as

Px,r = x + px,r 17

here is the mean pressure gradient and px ,r is the periodicomponent of the pressure. The term x represents the nonperi-dic pressure drop that takes place in the flow direction. Keepingn mind that pxin ,r= pxout ,r, we have

Pxin,r Pxout,r = xout xin = L 18

ecause is the mean pressure gradient, we have that L=P,ith P being the pressure difference between the inflow andutflow boundaries. Thus, we obtain the desired periodic bound-ry condition for P

Pxin,r = Pxout,r + P 19

ith P given.

4.3 Meshing and Solution Procedure. We expect large gra-ients around the corrugation, which require refined meshes. Ashe exact position of high activity regions depends on the param-ters of the flow, adaptive mesh generation is used. It is charac-erized by the fact that the construction of the mesh and the cal-ulation of the corresponding solution are performedimultaneously. This technique identifies the high activity regionshat require a high resolution by estimating the errors and pro-uces an appropriate mesh.

To perform the simulations, an adaptive solver was used. Aoarse mesh was used as a starting point. It was adaptively refinedy the solver to finally arrive at the mesh shown in Fig. 9. Weotice that the regions around the corrugation and near the wall

Fig. 9 Computational domain with the adapted msolver. Distances are in meters.

11101-6 / Vol. 133, FEBRUARY 2011ded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMrequire much higher resolution than the area near the center of thepipe.

Figure 10 shows the convergence of the solution when usingthe adaptive solver. First, the model is solved on the coarse mesh.Then the errors are estimated and the mesh is refined at the loca-tions with the larger errors. The model is solved again on the newmesh. The process continues until the errors are smaller than acertain threshold, and the maximum number of mesh refinementsis reached this number is 2 in our case.

4.4 Discussion of Solution. A typical solution for the case ofperiodically fully developed turbulent flow in a corrugated pipeat Re106 is displayed in Fig. 11. The colored surface corre-sponds to the value of pressure, arrows indicate the direction andthe magnitude of the velocity field, while the green lines near thewall are the streamlines of the flow. As expected, there is a regionof higher pressure upstream the bump corrugation, followed by aregion of lower pressure downstream. By doing several computa-tions for different Reynolds numbers, it became clear that thelow-pressure region is located behind the bump in flows at lowReynolds numbers, and it moves toward the top of the bump asthe Reynolds number becomes larger. There is an adverse pressuregradient on the top of the bump. However, due to the large enoughvelocity, the flow has sufficient momentum to overcome it andkeep flowing in the mainstream direction. The situation is not thesame for the flow immediately behind the bump. A continuousretardation of flow brings the velocity as well as its gradient andthe wall shear stress near a certain point on the wall to zero. Fromthis point onwards, the flow reverses and a region of recirculatingflow develops. The shear stress becomes negative. We see that theflow no longer follows the contour of the wall. The flow hasseparated. The point where the shear stress is zero is the point ofseparation. Further downstream, the recirculating flow terminatesand the flow becomes reattached to the wall. Thus, a separation ofthe flow occurs and a recirculation zone a vortex is formed.

The cross-sectional pressure profile is shown in Fig. 12. It is no

h generated from an initial mesh by an adaptive


lpce

J

Downloaonger constant as it was the case for a noncorrugated pipe. Theressure is almost constant near the centerline and strongly in-reases toward the corrugated wall, where higher resistance isncountered.

Fig. 11 A typical solution. Colored surface prstreamlines. This computation was performed fo

Fig. 10 Numerical error versus total iteratioand has its local origin at the end of the pre

ournal of Offshore Mechanics and Arctic Engineeringded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASM4.5 Influence of Fabric Roughness on the Friction Factor.One of the goals of this investigation was to answer the questionwhether the type of fabric which is wrapped around the steelspiral has a strong influence on the friction factor for the resulting

ure, arrows velocity field, and green lines P=3104 Pa, =500 kg/m3, and =0.01 Pa s.

umber. Each curve corresponds to a meshus curve.

FEBRUARY 2011, Vol. 133 / 011101-7n nvioessr E license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

pv=

ip

erfwmtcp

5

wvfsan

mt

ng

e

0

Downloaipe. In order to do this, simulations were performed for a set ofalues for relative roughness of the fabric e /D

0,0.0125,0.025,0.05. The computed friction factors see Eq.

11 are shown in Table 1.Two observations can be made here: a The friction factor

ncreases as the relative roughness increases this is what we ex-ect, and b the increase is small.

The difference between the two limiting cases e /D=0 and/D=0.05 is less than 10%. This entitles us to state that theoughness of the fabric has a negligible influence on the frictionactor. There is no point in looking for new materials, whichould keep the properties of the older ones such as noninflam-able, resistive to high pressures, resistive to extreme tempera-

ures, etc., but would have a lower roughness, because the de-rease in the friction factor is predicted to be too small to be ofractical value.

ConclusionsThe performance of two-equation turbulence models RANS

as assessed. Two popular representatives of this class were in-estigated: the k and the k models. They have been testedor smooth and rough pipe flows. The pipe flow simulations havehown that both models perform reasonably well, with a smalldvantage of the k model over the k model at low Reynoldsumbers 104Re105. Also, it has been shown that the experi-ental results are closer reproduced if the constant B in the law of

he wall has a value of 5.0 for Re7105 and 5.5 for Re7105. Tests with flow over a backward facing step have con-

Fig. 12 Pressure profile alo

Table 1 Friction factor versus fabric roughness

/2R 0 0.0125 0.025 0.05f 0.140 0.146 0.148 0.152

11101-8 / Vol. 133, FEBRUARY 2011ded 03 Dec 2010 to 131.155.70.112. Redistribution subject to ASMfirmed the tendency of two-equation models to underpredict thereattachment length 4. However, the results obtained from thek model were closer to the measurements. For this reason, thek model was chosen for the simulation of fully developed tur-bulent flow in corrugated pipes, where flow separation is encoun-tered in between the corrugations.

The effects of wall roughness have been modeled by imple-menting a new boundary condition based on a combined law ofthe wall. The flow in a noncorrugated conventional pipe withrough walls, for which experimental data exists, was used as a testproblem. The results of the test reproduce approximately theMoody diagram except for a small range of Re around 105 andare in close agreement with Nikuradses experimental data. Trus-ting our models and boundary conditions, the question whetherthe type of the fabric used in the corrugated pipe has a stronginfluence on the friction factor was investigated. The performedcomputations have shown that, for this specific geometry, theroughness of the fabric has a small influence on the friction factor.This in turn means that no considerable advantage is gained byreplacing a rough fabric by a smooth one. The bumps created bythe steel spiral cause most resistance, and considerable drag re-duction can only be obtained by optimizing their shape.

References1 Herwig, H., Gloss, D., and Wenterodt, T., 2008, A New Approach to Under-

stand and Model the Influence of Wall Roughness on Friction Factors for Pipeand Channel Flows, J. Fluid Mech., 613, pp. 3553.

2 Wilcox, D. C., 1998, Turbulence Modeling for CFD, 2nd ed., DCW Industries,Inc., La Caada, California.

3 Wilcox, D. C., 2006, Turbulence Modeling for CFD, 3rd ed., DCW Industries,Inc., La Caada, California.

4 Pisarenco, M., 2007, Friction Factor Estimation for Turbulent Flows in Cor-rugated Pipes With Rough Walls, MSc thesis, Eindhoven University of Tech-nology, The Netherlands.

5 Barenblatt, G. I., Chorin, A. J., and Prostokishin, V. M., 1997, Scaling Lawsfor Fully Developed Turbulent Flow in Pipes: Discussion of Experimental

the radial direction at x=0


Data, Proc. Natl. Acad. Sci. U.S.A., 943, pp. 773776.6 Zagarola, M., Perry, A., and Smits, A., 1997, Log Laws or Power Laws: The

Scaling in the Overlap Region, Phys. Fluids, 97, pp. 20942100.7 2006, COMSOL MULTIPHYSICS, Chemical Engineering Module, Users Guide,

COMSOL AB, Stockholm.8 Ioselevich, V. A., and Pilipenko, V. N., 1974, Logarithmic Velocity Profiles

for Flow of a Weak Polymer Solution Near a Rough Surface, Sov. Phys.Dokl., 18, pp. 790796.

9 Nikuradse, J., 1933, Strmungsgesetze in rauhen rohren, VDI-Forschungsh.,361.

10 Nikuradse, J., 1937, Laws of Flow in Rough Pipes, NACA Technical PaperNo. 1292.

11 Uri, M., and Kompare, B., 2003, Improvement of the Hydraulic FrictionLosses Equations for Flow Under Pressure in Circular Pipes, Acta hydrotech-nica, 21/34, pp. 5774.

12 Colebrook, C. F., and White, C. M., 1937, Experiments With Fluid Friction inRoughened Pipes, Proc. R. Soc. London, 161906, pp. 367381.

13 Brown, G. O., 2002, The History of the Darcy-Weisbach Equation for PipeFlow Resistance, Proceedings of the 150th Anniversary Conference of ASCE,pp. 3443.

14 van der Linden, B. J., Pisarenco, M., Ory, E., Dam, J., and Tijsseling, A. S.,

2009, Efficient Computation of Three-Dimensional Flow in Helically Corru-gated Hoses Including Swirl, ASME Paper No. PVP2009-77997.

15 Nishimura, T., Ohori, Y., and Kawamura, Y., 1984, Flow Characteristics in aChannel With Symmetric Wavy Wall for Steady Flow, J. Chem. Eng. Jpn.,17, pp. 466471.

16 Russ, G., and Beer, H., 1997, Heat Transfer and Flow Field in a Pipe WithSinusoidal Wavy SurfaceI. Numerical Investigation, Int. J. Heat MassTransfer, 405, pp. 10611070.

17 Yang, L. C., 1997, Numerical Prediction of Transitional Characteristics ofFlow and Heat Transfer in a Corrugated Duct, ASME J. Heat Transfer, 119,pp. 6269.

18 Patel, V. C., 1998, Perspective: Flow at High Reynolds Number and OverRough SurfacesAchilles Heel of CFD, ASME J. Fluids Eng., 120, pp.434444.

19 Patankar, S. V., Liu, C., and Sparrow, E. M., 1977, Fully Developed Flow andHeat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area, ASME J. Heat Transfer, 99, pp. 180186.

20 Ergin, S., Ota, M., and Yamaguchi, H., 2001, Numerical Study of PeriodicTurbulent Flow Through a Corrugated Duct, Numer. Heat Transfer, 40, pp.139156.

J

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