IMPROVED METHOD FOR SIMULATING TRANSIENTS OF TURBULENT PIPE FLOW


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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 1, pp , Warsaw 2011 IMPROVED METHOD FOR SIMULATING TRANSIENTS OF TURBULENT PIPE FLOW Zbigniew Zarzycki Sylwester Kudźma Kamil Urbanowicz West Pomeranian University of Technology, Faculty of Mechanical Engineering and Mechatronics, Szczecin, Poland; The paper presents the problem of modelling and simulation of transients during turbulent fluid flow in hydraulic pipes. The instantaneous wallshearstressonapipewallispresentedintheformofintegralconvolution of a weighting function and local acceleration of the liquid. This weighting function depends on the dimensionless time and Reynolds number. Its original, very complicated mathematical structure is approximated to a simpler form which is useful for practical engineering calculations. The paper presents an efficient way to solve the integral convolution based on the method given by Trikha(1975) for laminar flow. AnapplicationofanimprovedmethodwiththeuseoftheMethodof Characteristic for the case of unsteady flow(water hammer) is presented. This method is characterised by high efficiency compared to traditional numerical schemes. Key words: unsteady pipe flow, transients, waterhammer, efficient numerical simulation Notation c 0 [ms 1 ] acousticwavespeed p[kgm 1 s 2 ] pressure s[s 1 ] Laplaceoperator t[s] time v[ms 1 ] instantaneousmeanflowvelocity t=νt/r 2 [ ] dimensionlesstime
2 136 Z. Zarzycki et al. v[ms 1 ] averagevalueofvelocityincrosssectionofpipe v z [ms 1 ] axialvelocity w[ ] weightingfunction z[m] distancealongpipeaxis L[m] pipelength R[m] radiusofpipe Re=2Rv/ν[ ] Reynoldsnumber µ[kgm 1 s 1 ] dynamicviscosity ν[m 2 s 1 ] kinematicviscosity λ[ ] DarcyWeisbach friction coefficient ρ 0 [kgm 3 ] fluiddensity(constant) τ w,τ wq,τ wu [kgm 1 s 2 ] wallshearstress,wallshearstressforquasisteady flow, unsteady wall shear stress, respectively Ω=ωR 2 /ν[ ] dimensionlessfrequency 1. Introduction There are many problems concerning the issue of unsteady flow in pipes, and they include such phenomena as: energy dissipation, fluidstructure interactions, cavitation(for example column separation) and many others. It would beverydifficult,ifnotimpossible,toaccountforallthephenomenainjustone mathematical model. This paper is devoted to the problem of energy dissipation and it concerns unsteady friction modelling of the liquid in pipes during turbulent liquid pipe flow. The commonly used quasisteady, onedimensional(1d) model in which the wall shear stress is approximated by means of DarcyWeisbach formula is correct for slow changes of the liquid velocity field in the pipe crosssection, i.e. either for small accelerations or for low frequencies(if the flow is pulsating). Theformulaisalsocorrectforthewaterhammereffectforitsfirstwavecycle. Thismodeldoesnottakeintoaccountthegradientofspeedchangingintime, whichisdirectedintotheradialdirection,andso,asaresult,thechanging energy dissipation is not taken into account either. Models which represent unsteady friction losses can be divided into two groups.thefirstarethoseinwhichtheshearstressconnectedwiththeunsteady friction term is proportional to the instantaneous local acceleration(daily et al., 1956; Carrtens and Roller, 1959; Sawfat and Polder, 1973). This group
3 Improved method for simulating transients contains a popular model developed by Brunone et al.(1994, 1995) in which theunsteadyfrictiontermisasumoftwoterms:thefirstofwhichisproportional to the instantaneous local acceleration and the other is proportional to the instantaneous convective acceleration. Vitkovsky et al.(2000) improved Brunone s model by introducing a sign to the convective acceleration. Inthesecondgroupofmodels,theunsteadywallshearstressdependson the past flow accelerations. These models are based on a twodimensional(2d) equationofmotionandtheytakeintoaccountthevelocityfieldwhichcanbe changeable in time in the crosssection of the pipe. Zielke(1968) proposed forthefirsttimealaminarflowmodelinwhichtheinstantaneousshearstress depends on the instantaneous pipe flow acceleration and the weighting function ofthepastvelocitychanges.inthecaseofunsteadyturbulentflow,modelsof friction losses are mainly based on the eddy viscosity distribution in the crosssection of the pipe determined for a steady flow. The following models should be listed: two and three region models(vardy, 1980; Vardy and Brown, 1995, 1996, 2003, 2004, 2007; Vardy et al. 1993) and as well as Zarzycki s fourlayer model(zarzycki, 1993, 1994, 2000). Lately, Vardy and Brown(2004) used an idealised viscosity distribution model for a rough pipe flow. Unsteady friction models are used for simulation of transients in pipes. The traditional numerical method of unsteady friction calculation proposed byzielke(1968),requiresalargeamountofcomputermemoryandistime consuming. Therefore, Trikha(1975) and then Suzuki(1991) and Schohl(1993) improved Zielke s method making it more effective. In the case of unsteady turbulent flow, Rahl and Barlamond(1996) and then Ghidaoui and Mansour(2002) used Vardy s model to simulate the waterhammer effect. These methods provide only approximated results. They are not very effective and cannot be considered to be satisfactory. Theaimofthepresentpaperistodevelopaneffectivemethodofsimulating shear stress for unsteady turbulent pipe flow, a method that would be similar to that proposed by Trikha and Schoohl for laminar flow. That also involves developing an approximated model of the weighting function, which can be useful for effective simulation of transients. 2. Equations of unsteady turbulent flow Unsteady, axisymmetric turbulent flow of a Newtonian fluid in a long pipe with a constant internal radius and rigid walls is considered. In addition, the following assumptions are made:
4 138 Z. Zarzycki et al. constant distribution of pressure in the pipe crosssection, body forces and thermal effects are negligible, mean velocity in the pipe crosssection is considerably smaller than the speedofsoundinthefluid. The following approximate equation, presented in the previous papers by Ohmi et al.( , 1985), Zarzycki(1993), which omits small terms in the fundamental equation for unsteady flow and in which the Reynolds stress termisdescribedbyusingeddyviscosityν t,isused v z t = 1 p ρ 0 z +ν 2 v ( z Σ r 2 + r ν ) Σ r +ν Σ vz r r (2.1) wherev z,p averagedintime,respectively:velocitycomponentintheaxial directionandpressure, ρ 0 densityoftheliquid(constant), t time, z distancealongthepipeaxis,r radialdistancefromthepipeaxis. Themodifiededdyviscositycoefficient ν Σ isthesumofthekinematic viscositycoefficientνandtheeddyviscositycoefficientν t,thatis ν Σ =ν+ν t (2.2) Itisassumedthatchangesofthecoefficientν t resultingfromtheflowchange intimearenegligible,whichmeansthat ν t isonlyafunctionoftheradial distance r. In practice, this limits the analysis to relatively short periods of time Turbulent kinematics viscosity distributions Inordertodescribethedistributionoftheturbulentviscositycoefficientν t throughout the pipe crosssection, the flow region is divided into several layers and,foreachofthem,thefunctionν t (r)hastobedetermined.byanalogy to steady pipe flow, the following regions of flow can be distinguished: VS (Viscous Sublayer), BL(Buffer Layer), DTL(Developed Turbulent Layer) and TC(Turbulent Core)(Hussain and Reynolds(1975)). This is shown in Fig.1.ThedistributionofeddyviscositypresentedinFig.1wascreatedon thebasisofexperimentaldataofthecoefficient ν t providedbylaufer(for Re= )andnunner(forre= )andpublishedbyhinze(1953)as wellasdataaboutaregionclosetothewall,whichwasprovidedbyschubauer and published by Hussain and Reynolds(1975). That was presented in depth by Zarzycki(1993, 1994).
5 Improved method for simulating transients Fig.1.Schematicrepresentationofthefourregionmodelofν t Theradialdistancesr 1,r 2 andr 3 fromthepipecentrelineareasfollows r j =R y(y j + ) j=1,2,3 y 1 + =0.2R+ y 2 + =35 y+ 3 =5 (2.3) where: y + =yv /νisthedimensionlessdistance y, y distancefromthe pipewall,v = τ ws /ρ 0 dynamicvelocity,τ ws wallshearstressforsteady flow,r + =Rv /ν dimensionlessradiusrofthepipe. Thecoefficientofturbulentviscosity ν t isexpressedforparticularlayers (except the buffer layer) as follows ν t =α i r+β i i=1,3,4 α 1 =0 α 3 = ν t1 ν t2 r 2 r 1 α 4 =0 (2.4) β 1 =0 β 2 =ν t2 α 3 r 2 β 4 =ν t1 Forthebufferlayer(r 2 r<r 3 ),ν t isexpressedasfollows ν t =0.01(y + ) 2 ν (2.5) Expressions,whichdefinethequantitiesν t1,ν t2,r 1,r 2 andr 3 havethefollowing forms ν t1 =0.016 v 2 ν ν t2 =12.25ν (2.6) and r 1 =0.8 r 2 = ( ) v R r 3 = (1 20 2) v R (2.7)
6 140 Z. Zarzycki et al. Thequantitiesν t1,r 2 andr 3,whichdefinethedistributionofthecoefficientν t, dependonadimensionlessdynamicvelocity v,whichisdeterminedinthe following way v = 4 2R v = λ s Re s (2.8) ν where:λ s isthefrictioncoefficientforsteadyflow,re s =2Rv s /ν Reynolds numberforsteadyflow,v s meanfluidvelocityinthepipecrosssection. Sincewehave λ s =λ s (Re s )then,forsmoothpipes,theprofileofthe coefficientν t inthepipecrosssectiondependsonlyonthere s number. However,if insteadofthere s weassumetheinstantaneousreynolds number Re = 2Rv/ν (v instantaneous, averaged in the pipe crosssection fluidvelocity)and insteadofλ s weassumeaquasisteadyfrictioncoefficientλ q =λ q (Re),thenthepresenteddistributionofthecoefficientν t will becomeaquasisteadyone.inthispaperλ q isdeterminedfromtheprandtl formula 1 =0.869ln(Re λ q ) 0.8 (2.9) λq 2.2. Momentum equations for particular layers For the fourlayer model of the flow region, equation(2.1) breaks up into four equations. Introducing the differential operator D = / t into equation (2.1), we obtain: for the viscous sublayer Dv z1 = 1 ρ 0 p z +ν ( 2 v z1 r r v ) z1 r (2.10) fortheremaininglayersoftheflowregion Dv zi = 1 p ρ 0 z +ν 2 v ( zi Σ r 2 + νσ ) r +ν Σ vzi r r (2.11) where:i=2,3,4 (i=2forthebl, i=3forthedtland i=4forthe TC),v zi istheaxialcomponentofthevelocityforparticularlayersoftheflow region,ν Σ =ν+ν t effectivecoefficientofturbulentviscosity. Equations(2.10)and(2.11)(fori=3,4)canberesolvedusingrelationship (2.4) into the modified Bessel equations. Equation(2.11)(for i = 2), taking into account the fact that
7 Improved method for simulating transients r R r 1 2 for r 2 r<r 3 (2.12) assumes the form of a differential equation of Euler type. The solutions to the momentum equations for particular layers are shown in papers(zarzycki, 1993, 1994, 2000). 3. Wall shear stress and weighting function Integratingequation(2.1)overthepipecrosssection(fromr=0tor=R) andconsideringthat / t=d,weobtain ρ 0 Dv+ p z +2 R τ w=0 (3.1) Bydeterminingthewallshearstressatthepipewallτ w asfollows τ w = ρ 0 ν v z1 r we obtain a relationship of the following type (3.2) r=r τ w =f(d,re) p z (3.3) Equations(3.1)and(3.3)enableonetodeterminetransferfunctionĜτvthat describes hydraulic resistance, relating the pipe wall shear stress and the mean velocity Ĝ τv ( D,Re)= τ w v (3.4) where v=(r/ν)v, τ w =[R 2 /(ρ 0 ν 2 )]τ w, p=[r 2 /(ρ 0 ν 2 )]pdenotedimensionless velocity, wall shear stress, and pressure, respectively. AprecisemathematicalformofthetransferfunctionĜτvisverycomplex and was presented by Zarzycki(1994). Thequantity D=(R 2 /ν) / tpresentsthedimensionlessdifferentialoperator. Takingintoaccounttransferfunction(3.4)for D=ŝ(whereŝ=sR 2 /ν isthedimensionlesslaplaceoperator)onecanexpressthestress τ w asa sumofquasisteadywallshearstressτ wq andintegralconvolutionofthefluid acceleration v/ t and the weighting function w(t), i.e. as a convolution of
8 142 Z. Zarzycki et al. the fluid acceleration v/ t and the step response g(t)(zielke, 1968; Zarzycki, 1993, 2000), i.e. τ w (t)=τ wq + 2µ R whereτ wq isdescribedbythefollowingexpression t 0 v (u)w(t u)du (3.5) t τ wq = 1 8 λ qρ 0 v v (3.6) The graph of weighting function w(t) for different Reynolds numbers and for thelaminarflowispresentedinfig.2. Fig. 2. The weighting functions for turbulent and laminar flow The weighting function approaches zero when time tends to infinity. This tendency becomes stronger for higher Reynolds numbers. 4. Approximated weighting function model Theweightingfunctionhasacomplexmathematicalformanditisnotvery useful to calculate and simulate unsteady turbulent flow(zarzycki, 1994, 2000). Zarzycki(1994) approximated it to the following form
9 Improved method for simulating transients w apr =C 1 t Ren (4.1) wherec= ,n= However, numerical calculations that include this function lead to some complications(dividing by zero) as time and the Reynolds number are to be found in the denominator. Hence, in order to improve the effectiveness of calculations, the function will be approximated with a simpler form. Additionally, it is suggested that the approximated function should meet the following assumptions: 1) its form should be similar to Trikha s(trikha, 1975) or Schohl s(schohl, 1993) models(methods of simulating shear stress based on these models are very effective), 2) its mathematical form should allow an easy way of carrying out Laplace transformation. This is important for simulating oscillating or pulsating flow by the matrix method(wylie and Streeter, 1993). In order to meet the above presented criteria, a number of possible forms of the function were investigated. The form which was eventually selected can begivenby n w N ( t)=(c 1 Re c2 +c 3 ) A i e b i t (4.2) wherei=1,2,...,n. Determining the coefficients of the weighting function was carried out in two stages in the following way: a)thefirststage.intherangeofthereynoldsnumbersre ,10 7, a several equally distributed points were chosen, and for each of them weighting functions were estimated using Statistica programme. Estimationwasconductedfortherangeofdimensionaltime t 10 6,10. b) The second stage. Twodimensional weighting function w( t, Re) was estimated using Matlab s module for optimization. For the optimization procedure, the following assumption was formulated: todeterminethedecisionvariablesvector x=[x 1,x 2,...,x 19 ], where: x 1 =c 1,x 2 =c 2,x 3 =c 3,x 4 =A 1,x 5 =b 1,x 6 =A 2, x 7 =b 2,...,x 18 =A 8,x 19 =b 8 (arecoefficientsof(4.2)) i=1 which minimizes the vector objective function minf(x)=[f(x)] (4.3)
10 144 Z. Zarzycki et al. where r ( w N(i) ( t,re) w i ( t,re) ) 2 f(x)= (4.4) i=1 w i ( t,re) and satisfies the following limitation g(x)= 10 7 [ ] w N ( t,re)d t dre [ ] w( t,re)d t dre= (4.5) Inrelation(4.4)thearguments tandrewereselectedsothatthedetermined points(pairs of numbers) on the plane t Re would uniformly cover the interval t 10 5,10 1 andre= ,10 7,andthattheywouldbedifferentfrom thepointsselectedatthefirststage.atotalofr=1000pointswereusedin the calculations. The values of coefficients in Eq.(4.2), determined at the first stage, were used as the starting points(the values of elements of the decision variables vector) for the optimization. Once the optimization problems were formulated in the way described above, calculations were conducted using Matlab and its function fgoalattain. Details of the whole process of a new weighting function estimation were described by Zarzycki and Kudźma(2004). On the basis of the above assumptions and calculations that were conducted later, the final form of the weighting function was established 8 w N ( t)=(c 1 Re c 2 +c 3 ) A i e b i t i=1 (4.6) c 1 = ;c 2 = ;c 3 = ;A 1 =0.224,A 2 =1.644, A 3 =2.934,A 4 =5.794,A 5 =11.28,A 6 =19.909,A 7 =34.869,A 8 =63.668; b 1 = ,b 2 =8.44,b 3 =88.02,b 4 =480.5,b 5 =2162,b 6 =8425, b 7 =29250,b 8 = The present form of the weighting function contrary to the one put forward in(zarzycki and Kudźma, 2004) contains eight terms. As it turned out, the effective method of calculating wall shear stress presented later in this work is more sensitive to inaccurate fit between the proposed function and the original. Therefore, in order to ensure correct mapping, the number of elements of the new weighting function was increased from six to eight. In addition, a comparison was made of new expression(4.6) course and Zarzycki s model with the weighting function of Vardy and Brown(2003)
11 Improved method for simulating transients w( t)= 1 2 π t exp ( t C ) (4.7) where:c =12.86/Re κ ;κ=log 10 (15.29/Re ). The accuracy of Zarzycki s mapping of the weighting function by the new function is as follow: ForarangeofReynoldsnumbersbetween Re 10 7 and dimensionlesstimebetween10 5 t 10 1,therelativeerrorsatisfies condition w N( t) w( t) 0.05 w( t) Graphical comparison of the newly estimated function and models of weighing function by Zarzycki and VardyBrown are presented in Fig. 3. The relative errorofthenewfunctionandzarzycki smodelispresentedinfig.4.asitis shown,theerrorinsomeareasisupto5%butithastobementionedthatthe run of the new function is between the models of VardyBrown and Zarzycki weighting functions. There is no clear evidence of better consistency with the experimentforawiderangeofreynoldsnumbersuptonow,soitisambiguous which models should be mapped more precisely by the new function. The authors did not want to complicate the form of weighting expression through an increase of the number of exponential components thus left it in form(4.6). But,itisworthtomentionthatitispossibletoincreaseaccuracyofmapping the original weight in a wider range of Reynolds numbers or dimensionless time in the case of new more precise and experimentally tested models. The lowest limit of usage of the new weighting function for dimensionless timeis t=10 5.Forthepresentedmethodoftransientssimulationsitisa satisfactory value from the point of engineering view. It permit simulations of low viscosity fluids in quite large pipe radius(e.g. for fluid which viscosity ν=1mm 2 /sacceptablediameter=0.3m).ithastobealsomentionedthat suchabigradiusofthepipelinehasusuallylonglength,whatmeans,in consequence, low frequency of transients pressure oscillations. It allows one to take a quite big time step in numerical calculations, which is in the acceptable range of dimensionless time. In order to quantitatively assess a mapping of Zarzycki s model by the new expression for selected Reynolds numbers, a relative error is calculated according to the expression relative error = w N( t) w( t) (4.8) w( t)
12 146 Z. Zarzycki et al. Fig. 3. Comparison of VardyBrown s, Zarzycki s and new weighting function runs fordifferentreynoldsnumbers:(a)re=10 4,(b)Re=10 6 Fig. 4. The relative error between the new function and Zarzycki s weighting functionmodelforre=10 4 ResultsofitsusageareshownforthecaseofReynoldsnumberRe=10 4 in Fig. 4. In the case of harmonically pulsating flow, a transfer matrix method is usually used(wylie and Streeter, 1993). For this method, an adequate(laminar or turbulent) weighting function transformed into Laplace domain is needed.
13 Improved method for simulating transients Next,theLaplacevariableissubstitutetbyŝ=jΩ. Conducting the above mentioned transformations, the new weighting function takes the following form w N (jω)=(c 1 Re c 2 +c 3 ) 8 i=1 A i jω+b i (4.9) where:ω=ωr 2 /νisthedimensionlessangularfrequency. InFig.5,acomparisonofrealandimaginarypartsofnewweightingfunction(4.9) with Zarzycki and VardyBrown models is presented. Fig. 5. Comparison of the new weighting function with models by Zarzycki and VardyBrownforRe=10 4 To sum up, it should be pointed out that new approximating function(5.3) well reflects the weight of the unsteady friction model for turbulent flow for t 10 5 and 2000 Re 10 7.However,asfarasfrequencyisconcerned, thepresentedrelationisrightforadimensionlessfrequencyω 10.These particular ranges are most important as far as engineering applications and precision of numerical simulations are concerned. 5. Improvement of the approximation method for simulations of wall shear stress Numerical calculations of the unsteady wall shear stress(convolution of the local liquid acceleration and the weighting function) in the traditional approach is both timeconsuming and takes a lot of computer memory space(suzuki et al., 1991). In papers by Trikha(1975) and Schohl(1993) another, more efficient method of calculating unsteady shear stress for laminar flow was presented.
14 148 Z. Zarzycki et al. This section presents the implementation of Trikha s method into calculationsofwallshearstressforturbulentflowusingthedevelopedhereformof weighting function(4.6). Weighting function(4.6) can be given by 8 w N (t)= w i (5.1) i=1 wherew i =(c 1 Re c 2 +c 3 )A i e b iνt/r 2. Using the above form of the weighting function, the unsteady term of pipe wallshearstressτ wn canbewrittenas τ wu = 2µ R 8 y i (t) (5.2) i=1 where t y i (t)=(c 1 Re c 2 +c 3 )A i e b i ν R 2t 0 e b i ν R 2u v (u)du (5.3) u TheReynoldsnumberwasusedintheaboveexpressionsasavariabletobe updated at every iterative step. This approach was also suggested by Vardy etal.(1993). Because in the characteristics method calculations are carried out for some fixed time intervals, then y i (t+ t)= t+ t 0 w i (t+ t u) v u (u)du= t =e b i ν R 2 t (c 1 Re c 2 +c 3 )A i e b i ν R 2t e b i ν R 2u v } 0 {{ } y i (t) +(c 1 Re c 2 +c 3 )A i e b i ν t+ t R 2(t+ t) e b i ν R 2u v u (u)du } {{ t } y i (t) While comparing relations(5.4) and(5.3), we obtain u (u)du + (5.4) y i (t+ t)=y i (t)e b i ν R 2 t + y i (t) (5.5)
15 Improved method for simulating transients where y i (t)=(c 1 Re c 2 +c 3 )A i e b i ν t+ t R 2(t+ t) e b i ν R 2u v t (u)du (5.6) u Ifintheabovegivenexpressiondescribing y i (t),weassumethatv(u)is alinearfunction [v(u) =au+b]intheintervalfrom tto t+ t,then its derivative calculated after time given by v(u)/ u can be treated as a constant whose value can be determined from the following expression: [v(t+ t) v(t)]/ t.onthestrengthofthisassumption,wecanwrite y i (t) (c 1 Re c 2 +c 3 )A i e b i ν R 2(t+ t) [v(t+ t) v(t)] t =(c 1 Re c 2 +c 3 )A i e b i ν R 2(t+ t) [v(t+ t) v(t)] t b i ν =(c 1 Re c 2 +c 3 ) A ir 2 tb i ν y i (t+ t) y i (t)e b i ν R 2 t + R 2 t+ t e b i ν R 2u du= t (e b i ν R 2(t+ t) e b i ν R 2t) = (1 e b i ν R 2 t) [v(t+ t) v(t)] (5.7) (5.8) +(c 1 Re c 2 +c 3 ) A ir 2 (1 e b i ν R tb i ν 2 t) [v(t+ t) v(t)] τ wu (t+ t)= 2µ 8 y i (t+ t) (5.9) R τ wu (t+ t) 2µ R i=1 8 i=1 +(c 1 Re c 2 +c 3 ) A ir 2 tb i ν y i (t)e b i ν R 2 t + (1 e b i ν R 2 t) [v(t+ t) v(t)] (5.10) Atthebeginningofsimulations(i.e.atthefirstiterativestep),thevalueofall termsofy i (t)equalszero.ateverysubsequenttimestep,thesevalueschange according to Eq.(5.5).
16 150 Z. Zarzycki et al. 6. Results of waterhammer simulations and efficiency of the method 6.1. Governingequations The unsteady flow, accompanying the water hammer effect, may be described by a set of the following partial differential equations(wylie and Streeter, 1993): equation of continuity p v t +c2 0ρ 0 =0 (6.1) x equation of motion v t +1 p ρ 0 x + 2τ =0 (6.2) ρ 0 R wherev=v(x,t)istheaveragevelocityofliquidinthepipecrosssection, p=p(x,t) averagepressureinthepipecrosssection, R innerradiusof thetube,τ wallshearsterss,ρ 0 densityoffluid,c 0 velocityofpressure wavepropagation,t time,x axiallocationalongthepipe. A waterhammer simulation is carried out by the method of characteristics (MOC). This method(for small Mach number flow) transform equations(6.1), (6.2) into the following systems of equations(wylie and Streeter, 1993) if ±dp±ρ 0 c 0 dv+ 2τ wa dt=0 (6.3) R dz=±c 0 dt (6.4) Details of the further procedure of calculation may be found in papers by Zarzycki and Kudźma(2004) or Zarzycki et al.(2007) Numericalsimulation The investigated system(tankpipelinevalve) is presented in Fig. 6. For calculations, the following data were assumed(wichowski, 1984): lengthofhydrauliclinel=3600m innerdiameteroftubed=0.62m thicknessoftubewallg=0.016m modulusofelasticityoftubewallmaterial(castiron)e= Pa bulkmodulusofliquide L = Pa
17 Improved method for simulating transients Fig.6.Schemeofthepipelinesystem;1 hydraulicpipeline,2 cutoffvalve, 3 tank,v 0 directionoffluidflow(initialcondition) velocityofpressurewavepropagationc 0 =1072m/s initialvelocityoffluidflowv 0 =1.355m/s densityofwaterρ 0 =1000kg/m 3 kinematicviscosityoffluidν= m 2 /s The critical Reynolds number is calculated according to the following equation(ohmi et al., 1985) Re c =800 Ω (6.5) whereω=ωr 2 /νisthedimensionlessangularfrequency;ω=2π/t angularfrequency; T =4L/c 0 periodofpressurepulsation, L lengthof pipe. ThevalueofReynoldsnumberattheinitialconditionisRe=2Rv 0 /ν= ,thecriticalReynoldsnumberaccordingtoEq.(6.5)Re c = , so(re>re c )andtheflowisregardedasturbulent. Theunsteadystateofthesystemwasachievedbyacompleteclosureof thevalveforatimelongerthanahalfofthewave speriodinthepipe,i.e. t z =90s>2L/c 0 =6.72s,consequentlyitcanberegardedasanexample of a complex waterhammer(wylie and Streeter, 1993). Closure of the valve proceeded according to the characteristics presented in Fig. 7(Wichowski, 1984).ThepresentedcharacteristicsillustratethevariableliquidflowrateQ z through the valve due to its slow shutdown. Two simulation cases were investigated: inthefirstone,theunsteadystressτ wu wascalculatedbymeansofeq. (5.10) by making use of new weighting function(4.6), inthesecondone,theunsteadystress τ wu wascalculatedusingthe traditional method(zielke, 1968). Theresultsofnumericalsimulationatthevalvecarriedoutforagiven numberoftimestepsispresentedinfig.8.asonecansee,afterthevalve
18 152 Z. Zarzycki et al. Fig. 7. Flow characteristics of closing of the water supply valve closure which lasts 90s, there are still evident significant pressure pulsations. Thiseffectiscausedbyverylargeinertiaoftheliquid(highvalueofthe Reynolds number and a substantial length of the liquid line). Therefore, despite the long time of valve closure, disappearing pressure pulsations are still present intheline. Fig. 8. Pressure fluctuation at the valve after its closure The comparison of two simulations, using the old and new method of calculations of unsteady frictional losses presented in Fig. 8 shows a good agreement of the results obtained, and thus confirms that the model of efficient computation satisfactorily replaces the traditional method. However, an important difference that can be observed in the simulations is the time of numerical calculations. Figure 9 presents the dependence of timenecessarytocarryoutcalculationsonthenumberoftimestepsusedfor the simulations. As it can be clearly seen, the time of numerical calculations carried out by means of the traditional method using Zarzycki s weighting function(4.1) is much longer than that used to calculate it using the presented
19 Improved method for simulating transients Fig. 9. Comparison of numerical computation times method. Moreover, the time necessary to carry out the calculations conducted for the traditional method increases exponentially, along with the increase of the number of time steps. However, for the effective method the time increases slowlyinalinearway. Onthebasisoftheabovepresentedinvestigations,itcanbestatedthatthe presently developed method of simulating pressure changes and flow velocity, while taking into account the unsteady friction loss expressed by(5.10), is an effective method which can replace the tradition method and which offers satisfactory accuracy. 7. Conclusions The paper presented the transfer function relating the pipe wall shear stress and the mean velocity. It is based on a fourregion distribution of eddy viscosityatthecrosssectionofthepipe.thismodelwasusedtodeterminethe unsteady wall shear stress as integral convolution of acceleration and weighting function similarly as it was done for laminar flow(see e.g. Zielke, 1968). However, the weighting function has a complex mathematical form and it is not very convenient to use it for simulating transient flows. Therefore, it was approximated to a simpler form, which was a sum of eight exponential summands. For this form of the weighting function and approximated in this way, an effective method of simulating wall shear stress was developed. The most important conclusions which can be drawn from this study are as follows: Theweightingfunctiondependsonthedimensionlesstime t=νt/r 2 (asinthecaseforlaminarflow)andadditionallyonthereynoldsnumber Re.
20 154 Z. Zarzycki et al. The weighting function approaches zero as the time increases. For a given valueofdimensionlesstimeitisgreaterinthecaseoflaminarflowandit decreasesalongwiththeincreaseofreynoldsnumberreinthecaseof turbulent flow. It means that the hydraulic resistance with an increase inthereynoldsnumberregoesfastertoaquasisteadyvalue,whichis associated with increase of damping. Approximated weighting function(4.6) proves to be a precise expression intherange 10 5 t 10 1 and Re 10 7 (relativeerror smallerthan5%). The present approximated method of calculating wall shear stress(which incidentally is similar to Trikha s method developed for laminar flow) makes it possible to significantly reduce the time necessary to carry out calculations as compared with the traditional method. Theweightingfunctiondevelopedinthisstudyshowsagoodfitwith Vardy s model for short times(high frequencies). However, the discrepancy between the values of weighting function obtained in both methods increase together with the increase of Reynolds number. It should bepointedoutthatbothfunctionswereobtainedonthebasisofdetermined distribution of eddy viscosity. This is justified for high frequencies (Brownetal.,1969).Itseemsthatforthemeanbandoffrequenciesanother model would be more appropriate. A model in which eddy viscosity, except for viscous sublayer, would be determined using turbulent kinetic energymodels(e.g.k ε,k l). Theweightingfunctionhasaformwhichcaneasilyundergothefirst Laplace transformation and then Fourier transform. This is useful for determination a frequency characteristics of a long hydraulic pipeline systems. References 1. Abreu J.W., Almeida A.B., 2000, Pressure transients dissipative effects: a contribution for their computational prediction, 8th International Conference on Pressure Surges, BHR Group, The Hague, Netherlands, Bergant A., Simpson A.R., 1994, Estimating unsteady friction in transient cavitating pipe flow, Proc., 2nd Int. Conf. On Water Pipeline Systems, BHRA Group Conf. Series Publ., 10, UK, Edinburgh, May, 215
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