Transient turbulent flow in a pipe

Transcription

1 Tranient turbulent flow in a pipe M. S. Ghidaoui A. A. Kolyhkin Rémi Vaillancourt CRM-3176 January 25 Thi work wa upported in part by the Latvian Council of Science, project , the Natural Science and Engineering Reearch Council of Canada and the Centre de recherche mathématique of the Univerité de Montréal Hong Kong Univerity of Science and Technology, Department of Civil and Structural engineering, Clear Water Bay, Hong Kong; ghidaoui@ut.hk Department of Engineering Mathematic, Riga Technical Univerity, Riga, Latvia LV 148; akolikin@rbi.lv Department of Mathematic and Statitic, Univerity of Ottawa, Ottawa, ON K1N 6N5, Canada; remi@uottawa.ca

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3 Abtract The flow before the rapid deceleration i aumed to be fully developed. Eddy vicoity model i ued to decribe the flow. The olution i obtained by mean of the Laplace tranform and the method of ingular perturbation theory. It i hown that the tructure of the flow hortly after rapid deceleration i not affected by the eddy vicoity and can be atifactorily decribed by a laminar mode Mathematic Subject Claification. Primary: 65L6; Secondary: 65D5, 65D3. Keyword and Phrae. method of matched aymptotic expanion, Laplace tranform, eddy vicoity. To appear in Scientific Proceeding of Riga Technical Univerity, Computer Science Rśumé On modélie le ralentiement rapide d un écoulement pleinement développé au moyen de tourbillon viqueux. On obtient la olution du problème au moyen de la tranformée de Laplace et de méthode de perturbation ingulière. On voit que la vicoité de tourbillon n affecte pa la tructure de l écoulement peu de temp aprè la décélération et que cette tructure rete en mode laminaire.

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5 1 Introduction Recently everal turbulence model for the analyi of water hammer flow were propoed in the literature [1]-[4]. Thee model are ued to calculate velocity and preure ditribution for turbulent flow ubject to rapid deceleration and/or acceleration. Experiment decribed in [5] how trong flow aymmetry with repect to the axi of the pipe hortly after a udden cloure of the pipe. Laminar rapidly decelerated flow were tudied in [6]-[8]. It i found in [6]-[8] that the velocity profile after a udden blockage of the flow contain inflection point and therefore are potentially highly untable [9]. The analyi in [6], [8] i performed under the aumption that the flow before deceleration i laminar. The cae of a turbulent flow i conidered in [7]. Approximate velocity profile after udden blockage are obtained in [7] under the aumption that the velocity ditribution before the deceleration i taken in the form of the ratio of two Beel function. The Beel function are ued in [7] to fit the velocity ditribution for a particular range of Reynold number. It i hown in [8] that the flow tructure after udden blockage of a fully developed laminar flow in a pipe can be conveniently decribed by the method of matched aymptotic expanion. Similar olution were found in [1] and [11] for flow between two parallel plane and in an annulu. In application the flow before deceleration i uually turbulent. An approach baed on an eddy vicoity concept i often ued in turbulent flow modeling. Two-layer and five-layer eddy vicoity model are ued in [4]. It i hown in [12] that in many cae one can approximate the eddy vicoity ditribution in a pipe by a contant in the core region and by a linear function near the wall. Thi eddy vicoity model i ued in the preent paper together with the method of matched aymptotic expanion to contruct an aymptotic olution for the velocity ditribution valid for a hort time after the flow i uddenly decelerated. 2 Aymptotic olution Conider an infinitely long horizontal pipe with radiu R filled with a vicou incompreible fluid. At time t = the flow i intantaneouly decelerated o that the total fluid flux through the croection of the pipe i equal to zero for all t. It i aumed that the flow before deceleration i a fully developed teady turbulent flow which depend on the radial coordinate only. We aume that the velocity vector ha only one non-zero component, Ũ( r, t), depending on the radial coordinate and time. The velocity ditribution in thi cae i given by the following equation Ũ t = 1 p ρ z + 1 r r ( r(ν + ν T ) Ũ r ), (1) where the cylindrical polar coordinate ( r, θ, z) are choen o that the origin of the ytem i on the axi of the pipe, p i the preure, ρ i the denity of the fluid,ν i the vicoity of the fluid and ν T i the turbulent (eddy) vicoity. Let the meaure of length, time, preure, and velocity be R, T, ρu R/T, and U, where T i the characteritic time and U i the characteritic velocity. Equation (1) then can be written in the following dimenionle form: U t = ϕ(t) + ε r [ rν(r) U ], (2) r r where ε = νt, ϕ(t) = p and the function ν(r) i ued to model the ditribution of the eddy R 2 z vicoity acro the pipe. We aume that the eddy vicoity depend only on the radial coordinate, r. 1

6 The boundary condition are The initial condition i U(1, t) = and U(, t) i bounded for all t. (3) U(r, ) = g(r), (4) where g(r) i the function which decribe a fully developed teady turbulent flow before deceleration. In addition, the total fluid flux through the cro-ection of the pipe i zero for all t, that i, 1 2 Applying the Laplace tranform to (2)-(5) we obtain The boundary condition are Ū g(r) = ϕ + ε r ru(r, t)dr =. (5) d dr [ rν(r) du ]. (6) dr Ū(1, ) = and Ū(, ) i bounded. (7) The zero-flux condition i 1 2 rū(r, )dr =. (8) Here Ū(r, ) i the Laplace tranform of U(r, t) and i the parameter of the Laplace tranform. Experiment [5] and theoretical analyi [7] how that udden cloure of the pipe generate additional vorticity near the wall which tart to diffue in the radial direction. Thu, the tructure of the flow in the core region doe not change for a ufficiently hort time, but a boundary layer tart to develop near the wall. Thi phyical proce ugget that the olution to (6) (8) can be found by the method of matched aymptotic expanion where the outer part of the expanion correpond to the core region and the inner part correpond to the boundary layer near the wall. Thi method wa uccefully ued in [8], [1], [11] for the cae of a laminar flow. Here the method i ued for turbulent flow. We aume that the eddy vicoity can be approximated by a linear function of the radial coordinate near the wall and by a contant in the core region a uggeted in [12]. We ue the following boundary layer variable ξ = 1 r ε. The outer expanion (in the core region of the pipe) i ought in the form Ū(r,, ε) = Ū(r, ) + εū1(r, ) + εū2(r, ) +..., (9) ϕ(, ε) = ϕ () + ε ϕ 1 () + ε ϕ 2 () (1) Subtituting (9)-(1) into (6) and collecting the term that do not contain ε, we obtain Ū g(r) = ϕ, (11) 2

7 1 2 It can eaily be hown from (11) and (12) that Ū (r, ) = g(r) where G i the average velocity of unditurbed flow: rū(r, )dr =. (12) 1 G = 2 The inner expanion (near the wall r = 1) i ought in the form G, ϕ () = G, (13) rg(r)dr. (14) Ū(r,, ε) = ū (ξ, ) + εū 1 (ξ, ) + εū 2 (ξ, ) +..., (15) Subtituting (15) into (6) and collecting the term that do not containε, we obtain The general olution to (16) i d 2 ū dξ 2 ū = G. (16) ū = C 1 exp (, ξ ) + C 2 exp ( ξ ) G. (17) Uing the zero boundary condition at ξ = and the matching condition lim ū(ξ, ) = lim Ū (r, ) ξ r 1 we obtain C 1 = and C 2 = G/. Thu, the function ū i ū = G exp ( ξ ) G (18) and a uniformly valid approximation ( r 1) of order unity i Ū(r,, ε) = g(r) G + G exp ( ξ ) + O ( ε ). (19) It i intereting to note that the olution to the leading order (19) i independent of the eddy vicoity ν(r) and ha the ame tructure a the olution for the laminar cae [8]. Thi mean, in particular, that for a very hort time there i no change in the tructure of the turbulence (that i, the turbulence i frozen ). Thi fact i conitent with the numerical experiment [4] where it i hown that for hort time it doe not matter which eddy vicoity model i ued to decribe the turbulence all model give eentially the ame diipation a a laminar flow model. In order to find the higher term of the aymptotic expanion we ubtitute (19) into (8) and ue (9). Thi give the condition 1 2 rū1dr = 2 G. (2) 3

8 The equation for the function Ū1 and ϕ 1 in the core region of the flow i It follow from (2) and (21) that Ū1 = ϕ 1. (21) Ū 1 = 2 G, ϕ 1 = 2 G. Following [12] we aume that the eddy vicoity ν(r) i contant in the core region of the flow and i a linear function of r near the wall r = 1, that i, ν(r) = 1 + α(1 r) (22) in the boundary layer, where α i a contant. Subtituting (15) into (6) and (7), uing (22) and collecting the term of order ε, we obtain The boundary condition i d 2 ū 1 dξ 2 ū 1 = ϕ 1 (α 1) dū dξ αξ d2 ū dξ 2. (23) Solving (23) and (24), uing (18) and the matching condition we obtain the function ū 1 (ξ, ) in the form ū 1 (, ) =. (24) lim ū1(ξ, ) = lim Ū 1 (r, ), ξ r 1 ū 1 (ξ, ) = 2G [ ( )] αg exp ξ ξ2 exp ( ξ ) + Thu, the Laplace tranform of the olution up to O(ε) i where ξ = 1 r ε. Uing the formula ( G 2 αg ) ξ exp ( ξ ). (25) 4 Ū(r,, ε) = g(r) G + G exp ( ξ ) + [ 2G ε exp ( ξ ) 2G + αg 4 ξ2 exp ( ξ ) ( G + 2 αg ) ξ exp ( ξ )] + O(ε), (26) 4 L 1 [ exp ( β ) ] ( ) [ ] β exp ( β ) = erfc 2, L 1 = 1 ) exp ( β2 t πt 4t and [ ] ) ( ) exp ( β ) t L 1 = 2 ( π exp β2 β β erfc 4t 2, t where L 1 [F ()] = f(t) i the invere Laplace tranform of the function F (), we obtain the invere Laplace tranform of (26) in the form ( ) { ] } (3r 1)G 1 r εt U(r, t) = g(r) G + erfc 2 2 (1 r)2 + 4G exp [ 1 εt π 4εt + αg ] 4 πεt (1 (1 r)2 r)2 exp [ + O(ε). 4εt 4

9 3 Concluion A tranient rapidly decelerated pipe flow i conidered in the preent paper. The flow before deceleration i aumed to be fully developed. At t = the flow i intantaneouly decelerated o that the total fluid flux through the cro-ection of the pipe i equal to zero. The method of matched aymptotic expanion i ued to contruct the olution. Reference [1] Vardy, A.E. & Hwang, K.L (1991). A characteritic model of tranient friction in pipe. J. of Hydraulic Reearch, 29(5): [2] Pezzinga, G. (1999). Quai-2D model for unteady flow in pipe network. J. of Hydraulic Engineering, 125(7): [3] Silva-Araya, W.F. & Chaudhry, M.H. (1997). Computation of energy diipation in tranient flow. J. of Hydraulic Engineering, 123(2): [4] Ghidaoui, M.S, Manour, G.S., & Zhao, M. (22). Applicability of quai-teady and axiymmetric turbulence model in water hammer. J. of Hydraulic Engineering, 128(1): [5] Brunone, B., Karney, B., Mecarelli, M., & Ferrante, M. (2). Velocity profile and unteady pipe friction in tranient flow. J. of Water Reource Planning and Management, 126(4): [6] Da, D. & Arakeri, J.H. (1998). Tranition of unteady velocity profile with revere flow. J. of Fluid Mechanic, 374: [7] Ghidaoui, M.S. & Kolyhkin, A.A. (21). Stability analyi of velocity profile in waterhammer flow. J. of Hydraulic Engineering, 127(6): [8] Ghidaoui, M.S. & Kolyhkin, A.A. (22). A quai-teady approach to the intability of timedependent flow in pipe. J. of Fluid Mechanic, 465: [9] Drazin, P. & Reid, W. (1981). Hydrodynamic Stability. Cambridge Univerity Pre. [1] Kolyhkin, A.A.& Vaillancourt, R. (21). Aymptotic olution for unteady vicou flow in a plane channel. Latvian J. of Phyic and Technical Science, no. 3: [11] Kolyhkin, A.A. & Volodko, I. (22). Tranient vicou flow in an annulu. Mathematical Modeling and Analyi, 7(2): [12] Vardy, A.E., & Brown, J.M.B. (1995). Tranient, turbulent, mooth pipe friction. J. of Hydraulic Reearch, 33(4):