TRANSSIENT ANALYSIS OF FLUID STRUCTURE INTERACTION IN STRAIGHT PIPE BADREDDIN GIUMA S.K ELGHARIANI

Transcription

1 TRANSSIENT ANALYSIS OF FLUID STRUCTURE INTERACTION IN STRAIGHT PIPE BADREDDIN GIUMA S.K ELGHARIANI A project report submitted in partial fulfillment of the Requirements for the award of the degree of Master of Engineering (Mechanical) Faculty of Mechanical Engineering Universiti Teknologi Malaysia NOVEMBER 2007

2 To my father, mother, brothers and sisters ii

3 iii ACKNOWLEDGEMENT I would like to express my deepest gratitude to Dr. Kahar Osman my supervisor for his continued support during my study and the encouragement, guidance and dedication he provided for this project. Without him this project would have not been possible I wish to express my gratitude also to all who teach me during my study in the Universiti Teknologi Malaysia My deep gratitude goes to my whole family, especially to my father and mother for their encourage and support through the years

4 iv ABSTRACT Water hammer phenomenon is a common problem for flows in pipes. Water hammer usually occurs when transfer of fluid is quickly started, stopped or is forced to make a rapid change in direction. The aim of this study is to use method of characteristics to study water hammer phenomenon. In this study, computational method is used to investigate the transient water hammer problem in a straight pipe. Method of characteristics is applied to constant density flow in a simple reservoir-pipeline-valve system. The water hammer effect is produced via suddenly closing the valve located at the upstream and downstream ends, respectively. Quasi steady shear stress is assumed for the flow. This study also considers steady and unsteady friction. Fluid structure interaction will also be analyzed. The results obtained show slightly higher pressure than that of published experimental data. This could be due to the Quasi steady shear stress assumption. Final results show that when fluid structure interaction is considered, more accurate answers were determined.

5 v Abstrak Fenomena Water Hammer adalah satu masalah yang biasa dalam aliran dalam paip. Water Hammer biasanya berlaku apabila perpindahan bendalir berlaku dengan cepat, berhenti atau dipaksa untuk melakukan perubahan arah dengan tiba-tiba. Tujuan kajian ini adalah untuk menggunakan Method of characteristic untuk mengkaji fenomena Water Hammer. Di dalam kajian ini, kaedah berkomputer digunakan untuk mengkaji masalah aliran peralihan water hammer di dalam paip. Method of characteristic di aplikasikan kepada ketumpatan malar di dalam takungan-paip-injap mudah. Kesan Water hammer di hasilkan melalui penutupan injap yang berada di atas dan di bawah takungan secara tiba-tiba. Tegasan ricih di anggap tidak berubah dengan masa dalam kajian ini. Geseran tidak bergantung pada masa dan bergantung pada masa juga digunakan dalam kajian ini. Struktur interaksi bendalir juga akan di analisis. Hasil kajian menunjukan tekanan sedikit tinggi jika dibandingkan dengan kajian melalui eksperimen yang sudah di publikasikan. Ini mungkin kerana anggapan tegasan ricih tidak bergantung kepada masa. Keputusan akhir menunjukan apabila struktur interasi bendalir diambil kira dalam kajian akan menghasilkan keputusan yang lebih jitu.

6 vi CONTENTS CHAPTER SUBJECT PAGE ABSTRACT LIST OF FIGURES iv viii CHAPTER INTRODUCTION. Introduction.2 Objective 3.3 Scope 3 CHAPTER 2 LITERATURE REVIEW 5 2. history of water hammer analysis unsteady friction fluid structure interaction basic equations 2.4. Classical water hammer theory Brunone unsteady friction model Fluid structure interaction Initial and boundary conditions Initial conditions 5

7 vii The boundary conditions 5 CHAPTER 3 Methodology 7 3. Numerical solution of the classical water 7 hammer with quasi steady shear stress 3.2 Numerical solution of the classical water 25 with unsteady steady shear stress 3.3 Numerical solution of fluid structure 27 interaction with quasi steady shear stress 3.4 Numerical solution of fluid structure 32 with unsteady shear stress CHAPTER 4 Result and Discussions Comparison of numerical and experimental result effect of time closure and initial velocity 4 CHAPTER 5 Conclusions 43 REFERENCES 44 APPENDIX A 46 APPENDIX B 72

8 viii LIST OF FIGURES FIGURE TOPIC PAGE 3. interpolations of H and V values on the Δx Δt flow chart of the methodology 38 A. the flow chart of the programs that is used to 46 solve water hammer and FSI with steady and unsteady friction A.2 Computer program for solving the classical 47 water hammer by using MOC A.3 Computer program for solving the water hammer 50 with Brunone Unsteady Friction Model by using MOC A.4 Computer program for solving the fluid structure 55 interaction by using MOC A.5 Computer program for solving the fluid structure 62 interaction with Brunone Unsteady Friction Model by using MOC B. Variation of piezometric head with time at: (a) the 72 downstream end; and (b) the mid-point for V 0 =0. m/s. Experiment (black line) classical water hammer (green line), water hammer with unsteady friction (red line)

9 ix B.2 Variation of piezometric head with time at: (a) the 73 downstream end; and (b) the mid-point for V 0 =0. m/s. Experiment (black line) FSI with steady state friction (blue line) FSI with unsteady friction (yellow line) B.3 Variation of piezometric head with time at: (a) the 74 downstream end; and (b) the mid-point for V 0 =0.2 m/s. Experiment (black line) classical water hammer (green line), water hammer with unsteady friction (red line) B.4 Variation of piezometric head with time at: (a) the 75 downstream end; and (b) the mid-point for V 0 =0.2 m/s. Experiment (black line) FSI with steady state friction (blue line) FSI with unsteady friction (yellow line) B.5 Variation of piezometric head with time at: (a) the 76 downstream end; and (b) the mid-point for V 0 =0.3 m/s. Experiment (black line) classical water hammer (green line), water hammer with unsteady friction (red line) B.6 Variation of piezometric head with time at: (a) the 77 downstream end; and (b) the mid-point for V 0 =0.3 m/s. Experiment (black line) FSI with steady state friction (blue line) FSI with unsteady friction (yellow line) B.7 Variation of piezometric head with time at: (a) the 78 downstream end; and (b) the mid-point for L=43.7 m. Experiment (black line) classical water hammer (green line), water hammer with unsteady friction (red line)

10 x B.8 Variation of piezometric head with time at: (a) the 79 downstream end; and (b) the mid-point for L=43.7 m. Experiment (black line) FSI with steady state friction (blue line) FSI with unsteady friction (yellow line) B.9 Variation of piezometric head with time at: (a) the 80 downstream end; and (b) the mid-point for L=77.8 m. Experiment (black line) classical water hammer (green line), water hammer with unsteady friction (red line) B.0 Variation of piezometric head with time at: (a) the 8 downstream end; and (b) the mid-point for L=77.8 m. Experiment (black line) FSI with steady state friction (blue line) FSI with unsteady friction (yellow line) B. Variation of piezometric head with time at (a, c) the 82 downstream end and (b, d) the mid-point for V 0 =0. m /s. classical water hammer (green line), water hammer with unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady friction (yellow). B.2 Variation of piezometric head with velocity at (a) the 83 downstream end and (b) the mid-point for T c = 09 sec. Classical water hammer (green line), water hammer with unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady friction (yellow). B.3 Effect of time of close in the maximum pressure 84 with different initial velocity (classical water hammer

11 xi B.4 Effect of time of close in the maximum pressure with 85 different initial velocity (classical water hammer with unsteady friction) B.5 Effect of time of close in the maximum pressure with 86 different initial velocity (FSI with unsteady friction) B.6a Effect of time of close in the maximum pressure with 87 different initial velocity (classical water hammer, water hammer with unsteady friction and FSI) B.6b Effect of time of close in the maximum pressure with 88 different initial velocity (classical water hammer, water hammer with unsteady friction and FSI)

12 CHAPTER INTRODUCTION. Introduction Pipes installed in water supply systems, irrigation networks, hydropower stations, nuclear power stations and industrial plants are required to convey liquid reliably, safely and economically. Modern hydraulic systems operate over a broad range of operating regimes. Any change of flow velocity in the system induces a change in pressure. The sudden shut-down of a pump or closure of a valve causes a pressure wave develops which is transmitted in the pipe at a certain velocity that is determined by fluid properties and the pipe wall material. This phenomenon, called water hammer, can cause pipe and fittings rupture. The intermediate stage flow, when the flow conditions are changed from one steady state condition to another steady state, is called transient state flow or transient flow; water hammer is a transient condition caused by sudden changes in flow velocity or pressure. The classical theory of water hammer [, 2] describes the propagation of pressure waves in fully liquid filled pipe system. The theory correctly predicts extreme pressures and wave periods, but it usually fails in accurately calculating damping and dispersion [3] of wave fronts. In particular, field measurements usually show much more damping and dispersion than the corresponding standard water-hammer calculations. The

13 2 reason is that a number of effects are not taken into account in the standard theory for example: Generally friction losses in the simulation of transient pipe flow are estimated by using formulae derived for steady state flow conditions, this is known as the quasi-steady approximation. This assumption is satisfactory for slow transients where the wall shear stress has a quasi-steady behaviour. Experimental validation of steady friction models for rapid transients [4, 5, 6, 7] previously has shown significant discrepancies in attenuation and phase shift of pressure traces when the computational results are compared to the results of measurements. The discrepancies are introduced by a difference in velocity profile, turbulence and the transition from laminar to turbulent flow. The magnitude of the discrepancies is governed by flow conditions (fast or slow transients, laminar or turbulent flow) and liquid properties (viscosity) [7]. Also the waves have an acoustic pressure that acts against the surface of the pipe. Consequently, the fluid flow and the solid surface are coupled through the forces exerted on the wall by the fluid flow. The fluid forces cause the structure to deform, and as the structure deforms it then produces changes in the flow. As a result, feedback between the structure and flow occurs: action-reaction. This phenomenon what is call fluid structure interaction that can be attributed to three coupling mechanisms [8] Friction coupling is due to shear stresses resisting relative axial motion between the fluid and the pipe wall. These stresses act at the interface between the fluid and the pipe wall. Poisson coupling is due to normal stresses acting at this same interface. For example, an increase in fluid pressure causes an increase in pipe hoop stress and hence a change in axial wall stress.[8] The third coupling mechanism is junction coupling, which results from the reactions set up by unbalanced pressure forces and by changes in liquid momentum at discrete locations in the piping such as bends, tees, valves, and orifices. These include unsteady friction and fluid structure interaction which are taken into account in this study. In addition the discrepancies between the computed and measured water hammer waves may originate from some other assumptions in standard water hammer, i.e. the

14 3 flow in the pipe is considered to be one-dimensional (cross-sectional averaged velocity and pressure distributions), the pressure is greater than the liquid vapour pressure, the pipe wall and liquid behave linearly elastically, and the amount of free gas in the liquid is negligible. Also from discretization error in the numerical model, approximate description of boundary conditions and uncertainties in measurement and input data. In this study unsteady friction and fluid structure interaction are taken into account. Because of the interaction between the fluid flow and the solid surface the equations of motions describing the dynamics are coupled. This makes the problem more challenging, and even worse when the flow is turbulent. In addition, this means that the Navier-Stokes equation and the structure equation for the solid surface must be solved simultaneously with their corresponding boundary conditions [9]. In this project Method of characteristics is used to solving classical water hammer with unsteady friction and fluid structure interaction which solved one-dimensional, four-coupled first- order, nonlinear hyperbolic partial differential equation (PDE) model, which governs axial motion and includes Poisson, junction and friction coupling..2 Objective The objective of this project is to investigate the unsteady friction and fluidstructure interaction that may affect water hammer wave attenuation, shape and timing for single phase fluid in a simple reservoir-pipeline-valve system by using the method of characteristics which compared with experimental result [3, 0].3 Scope We consider cylindrical pipes of circular cross-section with thin linearly-elastic walls and filled with incompressible liquid, the flow velocities are small, the absolute pressures are

15 4 above vapour pressure and the pipe is thin walled and linear, homogeneous and isotropic elastic. The method of characteristics (MOC) is used to solve classical water hammer with quasi-steady shear stress, and with unsteady shear stress. To solving FSI, we used single procedure which treats the whole fluid structure domain as a single entity and describes its behaviour by a single set of equations. these are solved using a single numerical method (MOC-MOC) The main focus will be in compare between water hammer with and without unsteady friction and FSI at different initial velocity and time closure and compare both with experimental results [3, 0]

16 5 CHAPTER 2 Literature review In this section we apply some previous works done by other researchers which has used as reference but most of them are briefly mentioned us, however we have divided this section into fourth parts History of water hammer analysis, some previous unsteady shear stress and fluid structure interaction research and the fourth part is Basic equations. 2. History of water hammer analysis. During the second half of the 9th century and the first quarter of the 20th century, the majority of the publications on water hammer came from Europe. The conception of the theory of surges can, amongst others, be traced to Ménabréa (858, 862), Michaud (878), Von Kries (883), Frizell (898), Joukowsky (900) and Allievi (902, 93) [, 2, 3, 4, 5]. Joukowsky performed classic experiments in Moscow in 897/898 and proposed the law for instantaneous water hammer in a simple pipe system. This law states that the (piezometric) head rise H resulting from a fast (T c < 2L/a) closure of a valve, is given by: [] (.)

17 6 In which, a = pressure wave speed, V 0 = initial flow velocity, g = gravitational acceleration, L = pipe length and T c = valve closure time. The period of pipe, 2L/a, is defined as the return time for a water hammer wave to travel from a valve at one end of the pipeline to a reservoir at the other end, and back to the valve. The theoretical analyses performed independently by Joukowsky and Allievi formed the basis for classical water-hammer theory. Joukowsky s work was translated by Simin in 904. Allievi's work was not known generally outside Europe until Halmos made an English translation in 925. Gibson (908) presented one of the first important water hammer contributions in English. He considered the pressures in penstocks resulting from the gradual closure of turbine gates [2]. In the 930s, friction was included in the analysis of water hammer problem and first symposium of water hammer was held in Chicageo in 933. Topics covered included high-head penstocks, compound pipes, surge tanks, centrifugal pump installations with air chambers, and surge relief valves []. In 937, the second water hammer symposium was held in New York with presentations by both American and European engineers. The leaders in the field were in attendance as paper were presented on air chambers, surge valves, water hammer in centrifugal pump lines, and effects of friction on turbine governing []. During these period graphical techniques of analysis thrived under the work of Allievi, Angus, Bergeron, Schnyder, Wood, Knapp, Paynter, and Rich. In later years moves were made to more accurately incorporate frictional effects into the equations. Also more sophisticated boundary conditions were employed and more general forms of basic equations were used in analysis [, 2].

18 7 2.2 Unsteady friction Several researchers have proposed the unsteady friction models for transient pipe flow, the early model developed by Daily, Hankey, Olive, and Jordaan [7] in which the unsteady friction is dependent on instantaneous mean flow velocity and instantaneous local acceleration. In (973) Safwat and Polder developed unsteady loss models by adding correction terms that were proportional to fluid acceleration for laminar pipe flow [6]. Brunone, Golia, and Greco, [7] deduced an improved version in which the convective acceleration is added to Golia s version of the basic Daily model. The Brunone model is relatively simple and gives a good match between the computed and measured results using an empirically predicted (by trial and error) Brunone friction coefficient k. Zielke [7] found a frequency dependent model for laminar transient flows. He applied the Laplace transform to the momentum equation for parallel axisymmetric flow of an incompressible fluid and derived the equation which relates the wall shear stress to the instantaneous mean velocity and to the weighted past velocity changes. The advantage of this approach is that there is no need for empirical coefficients that are calibrated for certain flow conditions. The Zielke model has been modified by several researchers to improve computational efficiency and to develop weights for transient turbulent flow. In (993) Vardy, Hwang, and Brown extended Zeilke s model to moderately turbulent, unsteady flow [7] and. In (995, 996) Vardy and Brown to turbulent flow at high Reynolds numbers. The approach was to split the flow into an outer viscosity-dominated region and an inner turbulent region characterized by uniform velocity. Other researchers have proposed different models [6], in 989 Suo and Wylie proposed a frequency dependent friction factor model for application with the impedance

19 8 method, in 989 Jelev argued that it was reasonable to assume that energy dissipation is proportional to internal forces in the liquid and at the pipe wall, but with a quarter period phase shift. Motivated by experimental results, and in 99 Brunone assumed unsteady shear was proportional to the mean local and convective accelerations of the fluid. From previous research can classify the unsteady friction into six groups: [7] - The friction term is dependent on instantaneous mean flow velocity V (Hino et al., Brekke, Cocchi ) 2- The friction term is dependent on instantaneous mean flow velocity V and instantaneous local acceleration / (Daily et al., Carstens androller, Safwat and van der Polder, Kurokawa and Morikawa, Shuy and Apelt, Golia, Kompare, 3- The friction term is dependent on instantaneous mean flow velocity V, instantaneous local acceleration / and instantaneous convective acceleration / (Brunone et al., Bughazem and Anderson ) 4- The friction term is dependent on instantaneous mean flow velocity V and diffusion (Vennatrø, Svingen ) 5- The friction term is dependent on instantaneous mean flow velocity V and weights for past velocity changes W(τ) (Zielke, Trikha, Achard and Lespinard, Arlt, Kagawa et al., Brown, Yigang and Jing-Chao, Suzuki et al., Schohl, Vardy, Vardy et al., Vardy and Brown, Shuy, Zarzycki. 6- The friction term is basedon cross-sectional distribution of instantaneous flow velocity (Wood and Funk, Ohmi et al. Bratland, Vardy and Hwang, Eichinger and Lein Vennatrø, Silva-Araya and Chaudhry, Pezzinga). 2.3 Fluid structure interaction

20 9 Three coupling mechanisms determine fluid structure interaction in pipelines. Friction coupling is due to shear stresses resisting relative axial motion between the fluid and the pipe wall. These stresses act at the interface between the fluid and the pipe wall. Poisson coupling is due to normal stresses acting at this same interface. For example, an increase in fluid pressure causes an increase in pipe hoop stress and hence a change in axial wall stress. Junction coupling takes place at pipe boundaries that can move, either in response to changes in fluid pressure or because of external excitation. The literature review concerning these follows. In liquid-filled pipes, Poisson coupling results from the transformation of the circumferential strain, caused by internal pressure, to axial strain and is proportional to Poisson ' s ratio. Skalak [] was among the first to extend Joukowski ' s method to include Poisson coupling. Williams [8] conducted a similar study. He found that structural damping caused by longitudinal and flexural motion of the pipe was greater than the viscous damping in the liquid. These researchers did not include the radial inertia of the liquid or the pipe wall. Lin and Morgan [8] included the pipe inertia term and the transverse shear in their equations of motion. Their study was restricted to waves which have axial symmetry and purely sinusoidal variation along axis. Walker and Phillips [8, 2] extended the study by Lin and Morgan to include both the radial inertia of the pipe wall in the fluid and the axial equations of motion. Their interest in short duration, transient events produced a one-dimensional, axisymmetric system of six equations. Vardy and Fan [8, 3, and 4] conducted experiments on a straight pipe, generating a pressure wave by dropping the pipe onto a massive base. Their results showed good agreement with the analytical model by Wilkinson and Curtis. Wiggert and Otwell [8] neglected the radial acceleration in their studies using the six equation model of Walker and Phillips. This simplification reduced the mathematical model to four equations. Budny [8, 3] also reduced the six-equation model, but he included viscous damping and a fluid shear stress term to account for the structural and liquid energy dissipation. Experimental tests verified that the model satisfactorily predicts the wave speeds, fluid pressure, and

21 0 structural velocity of a straight pipeline for several fluid periods after a transient has excited the fluid. The junction coupling mechanism is generated by the pressure resultants at elbows, reducers, tees, and orifices act as localized forces on the pipe. For pipes with only a few bends, a continuous representation of the piping was devised by Blade, Lewis, and Goodykoontz [8]. Experimental tests were conducted to analyze the response of an L- shaped pipe to harmonic loading. The experimental setup included a restricting orifice plate at the downstream end of the pipe. They concluded that an uncoupled analysis does not produce accurate estimates of natural frequencies, and that the elbow, which provides coupling between the pipe motion and liquid motion, causes no appreciable reflection, attenuation, or phase shift in the fluid waves. Wiggert, Hatfield, Lesmez, and Wiggert, Hatfield and Stuckenbruck[8, 2, 3, and 5] used a one dimensional wave formulation in both the liquid reaches and the piping structure resulting in five wave components and fourteen variables. The method of characteristics was used to solve for the fourteen variables and to find the expressions for the wave speeds. Joung and Shin [8] developed a model that takes into account the shear and flexural waves of an elastic axisymmetric tube. The method of characteristics was used in the solution for four families of propagating waves. Their results compared closely to Walker and Phillips results for relatively small pipe deformations. Wood [8] studied a pipe structure loaded with a harmonic excitation. He found that the natural frequencies of liquid were shifted, especially when the frequency of the harmonic load is near one of the natural frequencies of the supporting structure. Ellis [8] reduced a piping structure to equivalent springs and masses by selectively lumping mass and stiffness at fittings and releasing specific force components at bends, valves and tees. His formulation of axial response was a modification of the method of characteristics and included pipe stresses and velocities. The finite element method is used to model the structure, treating each pipe element as a beam. Schwirian and Karabin [8] generalized this approach by using a finite element representation of the liquid and the piping. Their

22 studies imposed coupling at fittings only. The effect of the supports and piping stiffness was shown to be significant. 2.4 Basic equations 2.4. Classical water hammer theory The classical theory of liquid transient flow in pipelines (water hammer) usually draws on the following basic assumptions [, 2] -The liquid flow is one-dimensional with cross-sectional averaged velocity and pressure distributions 2- Unsteady friction losses are approximated as quasi-steady state losses. 3- The pipe is full and remains full during the transient. 4- There is no column separation during the transient event, i.e. the pressure is greater than the liquid vapour pressure. 5- Free gas content of the liquid is small such that the wave speed can be regarded as a constant. 6- The pipe wall and the liquid behave linearly elastically. 7- Structure-induced pressure changes are small compared to the water hammer pressure wave in the liquid. Water hammer equations include the continuity equation and the equation of motion. [, 2] (2.2) (2.3)

23 2 Where, P=pressure, V=averaged fluid velocity, =the pipe inclination angle, =acceleration due to gravity, =density of fluid, =velocity of pressure wave is defined by (2.4) Where k is a parameter depending on the constraint conditions [, 2] : is the shear stress between the pipe and fluid and expressed as the quasi steady part (2.5) Where f =friction factor Brunone Unsteady Friction Model The shear stress explicitly used in equations (2.2) and (2.3) is expressed as the sum of the quasi-steady part and the unsteady part (called the full friction coupling model hereinafter). The computation of the quasi-steady shear stress is straightforward, whereas the unsteady shear stress is related to the instantaneous local (temporal) acceleration and instantaneous convective (spatial) acceleration [3], i.e. (2.6) In which sign (V) = {+ for V 0 and - for V < 0} the Brunone friction coefficient can be predicted either empirically by the trial and error method or analytically using Vary, s shear decay coefficient C

24 3 (2.7) The Vardy, s shear decay coefficient C from[7] is: - Laminar flow C= Turbulent flow In which Re=Reynolds number (Re=VD/v) Fluid-structure interaction Normally when it is desired to obtain the fluid velocity in a pipe, equations are applied with the assumption of no wall deformation. If the walls deform, the deformation will affect fluid thus creating a fluid structure interaction. The following extended water hammer equations are obtained [9, 0] (2.8) (2.9) Where v : is the Poisson, s ratio is the axial pipe velocity

25 4 E: is Young, s modulus K f: is bulk modulus In the pipe domain, the equations of motion for the pipe in the axial direction is given by Wiggert et al [9, 0] (2.0) (2.) Where : is the density of the pipe : is the axial stress wave (precursor wave) speed i.e. : is the shear stress between the pipe wall and fluid, and expressed as the sum of the quasi-steady part and unsteady part the computation of the quasi-steady shear stress is (2.2) And the unsteady shear stress is (2.3) The last terms on the left-hand side of Equations (2.8) and (2.9) represent shear stress coupling whereas the last terms in the left-hand side of Equations (2.7) and (2.0) denote Poisson coupling.

26 Initial and boundary conditions All these equations are solved subject to boundary conditions at the upstream and downstream ends of the pipeline and initial conditions Initial conditions The boundary conditions For a pipeline connected to a reservoir and a valve at the upstream and downstream ends, respectively. While the pipeline with a fixed valve and a reservoir at the upstream and downstream ends, respectively,

27 6

28 7 CHAPTER 3 Methodology The basic equations presented in the previous chapter can be numerically solved in many ways. For fluid-structure interaction problems the fluid equations are often solved by the method of characteristics (MOC) and the structural equations by the finite method (FEM) but it is advantageous to use one method for all equations. In this study we have used method of characteristics (MOC) to solve both the fluid domain and pipe domain. This chapter is divided into four sections to show how all these equations can be solved by method of characteristics which the partial differential equations transform into ordinary differential equations along characteristics line. First the classical water hammer is described and then water hammer with unsteady friction, and fluid structure interaction. The fourth one is fluid structure interaction with unsteady friction. 3. Numerical solution of the classical water hammer with quasi-steady shear stress From equation of motion and equation of continuity (2.2, 2.3), there are two independent variables, x and t, and two dependent variables, P and V. Other variables are characteristics of the conduit system and are time invariant but may the function of x. Laboratories test have shown that wave velocity, a f, is significantly reduced by reduction

29 8 of pressure even when it remains above the vapor pressure. The friction factor, f, varies with the Reynold number, but f is considered constant because the effects of such a variation on the transient state conditions are negligible. With a nonlinear resistance terms for friction and other effects, no general solution to these equations is known, but they are solved by the method of characteristics for a convenient finite-difference solution with the digital computer due to its large storage capacity and its ability to operate at very high computing rates. In the method of characteristics, the partial differential equations are first converted into ordinary differential equations, which are then solved by an explicit finitedifference technique. Since each boundary condition and each conduit section are analyzed separately during a time step, The simplified equations of motion and continuity are identified as L and L 2 from equations 2.2 and 2.3 as (3.4) (3.5) Where (3.6) These equations are combined linearly using an unknown multiplier λ Regrouping terms, in the equation in a particular manner

30 9 (3.7) Both variables V and P are functions of x and t, and the independent variable x is a functions of t, from calculus (3.8) (3.9) Note that if Is to be replaced by Then And if

31 20 Is to replaced by Then Rewriting the restriction equations for dx/dt, And Equating the two expressions for dx/dt and solving we get Our characteristic equations are in this case And The result is that we have replaced two partial differential equations with two ordinary differential equations provided we follow certain rules which relate the independent variables x and t in each case. If, in addition, we replace P with then we can visualize better the propagation of the pressure waves because H is the height of the EL-HGL above the datum. This substitution gives only if (3.20) only if (3.2)

32 2 The numerical solution procedure first assumes that we can approximate the characteristic curve as straight lines over each time interval. This assumption appears to be promising because a>>v, however, it should be carefully noted that the slope of each characteristic is generally slightly different than that of any other. The problem this creates in the finite difference approximation to the differential equations can be seen on figure 3.. The procedure is to find the value of V and H at new point (P) the curved characteristics intersecting at P are approximated by straight lines. The slope of straight lines is determined by the known value of velocity at the earlier time. It is important to note that the characteristics passing through P do not pass through the grid points Le and Ri, but pass through the t= constant line at points L and R somewhere in between. Figure 3. interpolations of H and V values on the Δx Δt In this case the finite difference approximation to equations (2.20) and (3.2) becomes (3.22) (3.23) The values V L, H L, V L, and H L are not known. However the values of V Le, H Le, V Ri, H Ri, V c and H c are known. The unknown values of H and V at points L and R can be

33 22 estimated by interpolation in this case we will use linear interpolation and the sketch below illustrates the relationships Considering the C + characteristic Where Solving for V L and H L gives And Substituting the value of gives

34 23 (3.24) And (3.25) A similar analysis for the C - characteristic gives (3.26) (3.27) Because is of order which is very small compared to one it is good approximation to neglect the second terms in the denominators of equations (3.24) and (3.26). The result is (3.28) (3.29) The simultaneous solution of equations (3.22) and (3.23) for V P and H P gives (3.30) (3.3)

35 24 The value of is still determined by the number of sections into which we have chosen to divide the pipe. Because our interpolation procedure implies that the points R and L are between points Ri and Le, we must choose so small as to guarantee this always occurs. The equations suggest that [2] (3.32) Where is the maximum expected absolute value of the sum of the wave speed and flow velocity. If locations of points R and L fall outside the grid points Le and Ri, numerical stability problems and accuracy problems begin to develop. The computer program which is used to solve the water hammer equations by using method of characteristics is presented in Figure (A.2) In summary the program reads in the basic information, generates steady state H and V values at the grid intersection points (nodes) along the pipe and then begins the unsteady flow calculations. The interior grid intersection points are first calculated using equations (3.30) and (3.3). The upstream and downstream boundary conditions are used to get values of H P and V P at each end of pipe to simulate transient problems with other boundary conditions, it is necessary only to change those parts of the program listed under upstream and downstream boundary conditions. The whole process begins again using the just- computed values of H P and V P as the known values. The process continues to loop until the time has reached Tmax. Before execution is terminated values of H and V are printed in the matrices H_OUT and V_OUT respectively for each node. The output of this program is the variation of piezometric head with time at the downstream end and mid point for different time closure and initial velocity.

36 Numerical solution of the classical water hammer with unsteady-steady shear stress In this case the shear stress is expressed as the sum of the quasi-steady part equation (2.5) and unsteady part equation (2.6) and the friction coefficient for unsteady shear stress has to determined as demonstrated by Brunone [7], so the equation of motion and equation of continuity are (3.33) (3.34) Where k s is the sign of i.e. for whereas for Again the multiplier λ is used to combine the partial differential equations. Multiplying λ by equation (3.34) and adding the result to equation (3.33) gives To carry forth the same procedure as was used previously, we must break and down into component parts. The result is As before,

37 26 If And If Rewriting the restriction equations for And Equating the two expressions for and solving, we get So our characteristic equations are in this case (3.35) (3.36)

38 27 The final set of equations which compare with equations (3.20) and (3.2) after replacing P with, are (3.37) (3.38) The computer program for solution of the water hammer with unsteady friction is similar in most respects to the program used with the classical water hammer the main differences occur in the requirements to include the Brunone friction coefficient. The computer program is shown in Figure (A.3) 3.3 Numerical solution of Fluid-structure interaction with quasi-steady shear stress The four-coupled first-order, non-linear hyperbolic partial differential equation (2.8), (2.9), (2.0), and (2.) which the friction coupling is mainly based on the assumption where shear has a quasi steady behaviour can be transform into ordinary differential equations to determine V, P,, and as following (3.39) (3.40) (3.4)

39 28 (3.42) Where Both variables V, P,, and σ are functions of x and t, and the independent variable x is a functions of t, from calculus (3.43) (3.44) (3.45) (3.46) Equations (3.39) through (3.46) can be expressed in matrix form as

40 29 dt dx dt dx dt dx dt dx a he vd V v a a V P P P f f f 2 f 2 f ρ ρ ρ ρ ρ x t x u t u x V t V x P t P σ σ = dt d dt du dt dv dt dp C σ ω (3.47) Or [ ] Q x t x u t u x V t V x P t P σ σ = dt d dt du dt dv dt dp C σ ω (3.48)

41 30 Where This system will have a unique solution if the determinant of the matrix [Q] is nonzero. On the other hand, the system will have an infinite number of solutions, if the determinant of the matrix [Q] is zero. vd 2hE V ρ f dx dt ρ a f V dx dt 2 f 2ρ a f dx dt 2 f v ρ a P 2 P ρ P = (3.49) dx dt A characteristic is defined as a curve along which the determinant of the matrix [Q] is zero. Thus the direction of the characteristics can be found from equation (3.49). The slopes of characteristics are defined by (3.50)

42 3 The four roots of this equation (3.50) are denoted as and. The curve along which the slope equal to, -, or - is termed as the characteristic. Thus, there will be four families of characteristic curves in the domain (x, t-plane) of the problem When equation (3.48) holds, Cramer s rule implies that a solution of equation (3.47) cannot be obtained unless the determinants of the matrices obtained by substituting the right hand column of equation (3.47) into the first, second, third, or fourth column of the matrix [Q] are equal to zero ω C dp dt dv dt du dt dσ dt V ρ f dx dt ρ a f V dx dt 2 f 2ρ a f dx dt 2 f v ρ a P 2 P ρ P = (3.5) dx dt The equation (3.5) yield after replacing P with (3.52)

43 32 (3.53) Where and It can see that the hyperbolic partial differential equations (3.39) to (3.42) are now replaced by the ordinary differential equations (3.52) and (3.53). The method of characteristics involves the determination of the characteristic curves as x=x(t) by integrating equation (3.50) and the solution the solution of equations (3.52) and (3.53) by integration along the characteristic curves. The computer program for solving these equations is illustrated in the Figure (A.4) 3.4 Numerical solution of Fluid-structure interaction with unsteady shear stress In this case we consider both the unsteady friction and fluid structure interaction which the continuity equations and motions equations for the fluid and structural are (3.54) (3.55)

44 33 (3.56) (3.57) Both variables V, P, variable x is a functions of t, from calculus, and σ are functions of x and t, and the independent (3.58) (3.59) (3.60) (3.6) Equations (3.54) through (3.6) can be expressed in matrix form as

45 34 Δ dt dx dt dx dt dx dt dx a v a a V P P P f f 2 2 f 2 f 2 f ρ θ ρ ψ φδ ψ φδ ζ δ β ρ ρ ρ x t x u t u x V t V x P t P σ σ = dt d dt du dt dv dt dp C σ ω (3.62) Where

46 35 The direction of the characteristic curve can be obtained by setting the determinant of the coefficient matrix in equation (3.62) equal to zero θ V ρ f dx dt Δ φδ ρ a f β ψ dx dt 2 f δ φδ 2ρ a f ζ ψ dx dt 2 f v ρ a P 2 P ρ P dx dt = (3.63) Expanding the determinant we obtain (3.64) The four roots of this equation (3.64) are denoted as and. The curve along which the slope equal to,,

47 36, or is termed as the characteristic. Thus, there will be four families of characteristic curves in the domain (x, t-plane) of the problem. As in the case of previous section, if the determinant of the coefficient matrix is zero in equation (3.62), the right-hand side must be compatible with this in order to have a solution to equation (3.62). This implies that when the right-hand side is resulting matrix must be zero. For example, when the fourth column of the matrix on the left side is replaced by right-hand-side column of equation (3.62) and the determinant is set to zero, we obtain ω C dp dt dv dt du dt dσ dt V ρ f dx dt Δ φδ ρ a f β ψ dx dt 2 f δ φδ 2ρ a f ζ ψ dx dt 2 f v ρ a P 2 P ρ P = (3.65) dx dt Which yields, upon expansions

48 37 V (3.66) V (3.67) When and. Are substituted in equations (3.66) and (3.67), and replace P with, we obtain four ordinary differential equations to determine H, V,, and along the and characteristics. Thus the system of hyperbolic partial differential equations and solved by solving four ordinary differential equations. The computer program for solving these equations is shown in the Figure (A.5) The summary of the this chapter has be illustrated in the Figure (3.2) which we have started with classical water hammer to find the numerical expression by using method of characteristic and used same procedure to find the numerical expressions for

49 38 the water hammer with unsteady shear stress, fluid structure interaction with quasi-steady shear stress and with unsteady shear stress. Brunone Unsteady Friction Model had been used. Figure (A.) show the flow chart of the programs that is used to solve these problems which the main different between these programs are the numerical expressions. Figure (3.2) flow chart of the methodology

50 39 CHAPTER 4 RESULT AND DISCUSSIONS In the first part of this chapter will be presenting the numerical results of water hammer with and without unsteady friction and fluid structure interaction with and without unsteady friction that are compared with experimental result [3, 0]. The effect of time closure and initial velocity are presented in the second part. \4 - comparison of numerical and experimental result. In the first example we considers a straight sloping copper pipeline connected with a tank and a ball valve located at the upstream and downstream ends, respectively, Pressures in the pipeline evolves under the action of valve closure with closure time Tc =09s. The physical and geometric parameters, used in the experimental tests [3, 0], are as follows:

51 40 First we examine the variation of pressure with time for transient laminar flow at initial steady flow velocity with Reynolds number. The computational results obtained by the classical water hammer and water hammer with unsteady friction figures B.(a) and B.(b) agree well with the experimental results for the first and the second pressure head rise. The discrepancies between them and the experimental results are magnified for later time. Also can be seen the water hammer with unsteady friction is better agreement than the classical water hammer in phase shift and there is slight difference between them in amplitude. Figures B.2 (a) and B.2(b) show the comparison of experimental results [3, 0] with theoretical results obtained using the fluid structure interaction with full-friction coupling model and the partial-friction coupling model. It can be seen that there is better agreement between the experiment and the fullfriction coupling model (there is almost no phase shift and slight difference in amplitude). Discrepancies between the experiment and the theoretical results increase with time. Second comparison of numerical and experimental results for transient turbulent flow at initial steady flow velocity (law Reynolds number turbulent flows). Figures B.3 and B.5 show the comparisons of the experimental results [3, 0] with the theoretical results obtained using the classical water hammer and water hammer with unsteady friction for low Reynolds number turbulent flow respectively. The computational results from the FSI with fullfriction coupling model and the partial-friction coupling model for low Reynolds number turbulent flow are compared with results of measurements and are depicted in figures B.4 and B.6 respectively. Similar trends in the variation of pressure with time hold, but discrepancies between the experiment and the partial-friction coupling

52 4 model are magnified compared to the results for laminar flow. And also between FSI with full- friction coupling model and classical water hammer. The second example we consider is a pipeline equipped with a ball valve and a tank located at the upstream and downstream ends, respectively (the opposite of the previous example). Its geometrical and physical parameters, used by Pezzinga and Scandura [0], are:,,,,,,,,,,,,,. The variation of pressure with time for transient turbulent flow at initial steady flow velocity is examined. Figures B.7, B.8, B.9 and B.0 show the comparison between our numerical results and the experimental results of Pezzinga and Scandura [0] for L=43.7 and 77.8 m, respectively. Water hammer event starting with an initial small magnitude of pressure is initiated by rapid valve closure. Similar behaviour, which is illustrated in Figures B.-B.6, is observed. 4-2 effect of time closure and initial velocity The physical and geometric parameters given in the first example are used with different time closure of valve to see the effect of time closure in the pressure traces. Figure B. shows the comparison of pressures obtained using the classical water

53 42 hammer, water hammer with unsteady friction, and FSI with steady friction and unsteady. It can be seen that transient shear stress damps pressure fluctuation and decelerates pressure wave propagation more at a smaller valve closure time. Hence the influence of transient shear stress can be significant and varies considerably, depending on the valve closure time. To obtain the effect of initial velocity we used the physical and geometric parameters given in the first example with different initial velocity [0., 0.2, and 0.3] as shown in the Figure B.2. It can be seen the effect of unsteady friction and fluid structure interaction increase as the initial velocity increase Figures B.3, B.4, and B.5 show the maximum pressure as a function of initial velocity for different closing time. For classical water hammer, water hammer with unsteady friction, and fluid structure interaction with unsteady friction. They can be seen a linear behavior between maximum pressure and the initial velocity. In these Figures are evident that the faster the close time higher is the pressure. Also, there isn t different between closing valve at 0, 05 and 0,009 and the difference between closing the valve at 0.35 and 0.4 second is minimal thus,.009and 0.34 second may be taken as the critical values. As expected the fluid will tend to increase it pressure at higher velocities. Figures B.6, and B.7 compare between the maximum pressure of classical water hammer, water hammer with unsteady friction and fluid structure interaction with unsteady friction at different initial velocity and time closure from these figures we can see the higher maximum pressure occur with classical water hammer than water hammer with unsteady friction and fluid structure interaction.

54 43 CHAPTER 6 CONCLUSIONS The method of characteristics is employed to solve the water hammer with quasisteady friction model and Brunone model and FSI with quasi-steady friction model and Brunone model which compared with published data from a fast valve closure for laminar flow (0. m/s) and low Reynolds number turbulent flows ( 0.2, and 0.3 m/s). The unsteady shear stress part, which are related to the instantaneous relative local acceleration and the instantaneous relative convective acceleration, and the steady shear stress part, act collectively to damp pressure fluctuation and reduce phase shift of pressure traces. FSI with full friction coupling is showed better result than that of FSI with quasi steady friction and its effect improved as initial velocity increase. The influence of transient shear stress, which consists of the steady and unsteady shear stress parts, can be significant and varies considerably, depending on the valve closure time. The agreement between experiment and theory shows that the MOC offers an alternative way to investigate the behaviour of transient flow in pipe system with fluid structure interaction.

55 44 LIST OF REFERENCES [] M. Hanif Chaudhry (979), applied hydraulic transients. New York_Van Mostrand Reinhond, 997 [2] Gary Z.Watters (984). Analysis and control of unsteady flow in pipelines. 2 nd Boston,Mass_buttbrworths [3] Anton Bergant and Arris Tijsseling.(200). Parameters Affecting Water Hammer Wave Attenuation, Shape and Timing [4] Anton Bergunt, Arris Tijsseling, John Vitkovsky, Didia Covas, Angus Simpson, and Martin Lambert.(2003). Further Investigation of Parameters Affecting Water Hammer Wave Attenuation, Shape and Timing part : Mathematical Tools [5] Anton Bergunt, Arris Tijsseling, John Vitkovsky, Didia Covas, Angus Simpson, and Martin Lambert. (2003). Further Investigation of Parameters Affecting Water Hammer Wave Attenuation, Shape and Timing part 2: case studies [6] Bruno Brunone, Bryan W.Karney, Michele Mecarelli, and Marco Ferrante (2000). Velocity Profiles and Unsteady Pipe Friction in Transient Flow. Journal of water resources planning and management [7] Anton Bergant, Angus Ross Simpson, and John Vttkovsky (2000, 200). Developments in Unsteady Pipe Flow Friction Modeling. Journal of Hydraulic Research, vol.39,200, No, 3

56 45 [8] Arris Tijsseling (996) fluid-structure interaction in liquid-filled pipe systems: a review. Journal of fluids and structures (996) 0,09-46 [9] A.S. Tijsseling. Water hammer with fluid-structure interaction in thick-walled pipes Eindhoven University of Technology, [0] YongLiang Zhang and K. Vairavamoorthy (2005). Analysis of transient flow in pipelines with fluid structure interaction using method of lines, Int. J. Numer. Meth. Engng 2005; 63: [] Arris Tijsseling, Martin Lambert, Angus Simpson, Mark Stephens, john Vitkovsky, and Anton Bergant. wave front dispersion due to fluid-structure interaction in long liquid-filled pipelines [2] B Sreejith, K Jayaraj, N Ganesan, and Padmanabhan (2004). Finite element analysis of fluid-structure interaction in pipeline systems, nuclear engineering and design [3] David C Wiggert and Arris S Tijsseling. (200) fluid transients and fluid-structure interaction in flexible liquid-filled piping. American Society of Mechanical Engineers [4] L. Zhang, A. S. Tijsseling, and A. E. Vardy(999). FSI analysis of liquid-filled pipes journal of sound and vibration [5] A. G. T. J. Heinsbroek(997) fluid-structure interaction in non-rigid pipeline systems nuclear engineering and design [6] C.J. Greenshields and H. G. Weller (2005). A unified formulation for continuum mechanics applied to fluid structure interaction in flexible tubes. Int. J. Numer. Meth. Enging 2005;64: Published online 29 July 2005 in Wiley interscience [7] K. Namkoong, H. G. Choi, and J. Y. Yoo (2005). Computation of dynamic fluid structure interaction in two dimensional laminar flows using combined formulation. Journal of fluids and structures 20(2005)5-59 [8] S.Mttal, and T. E. Tezduyar (995). Parallel finite element simulation of 3D incompressible flows: fluid structure interactions. International Journal for numerical methods in fluids, vol. 2, (995) [9] M. P. Paidoussis (2005). Some unresolved issues in fluid structure interactions. Journal of fluids and structures 20 (2005)

57 46 Figure (A.) the flow chart of the programs that is used to solve water hammer and FSI with steady and unsteady friction

58 47 Figure-B. Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0. m/s. Experiment (black line)[3, 0] classical water hammer (green line), water hammer with unsteady friction (red line)

59 48 Figure-B.2 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0. m/s. Experiment (black line)[3, 0] FSI with steady state friction (blue line),fsi with unsteady friction (yellow line)

60 49 Figure-B.3 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0.2 m/s. Experiment (black line)[3, 0] classical water hammer (green line), water hammer with unsteady friction (red line)

61 50 Figure-B.4 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0.2 m/s. Experiment (black line)[3, 0] FSI with steady state friction (blue line),fsi with unsteady friction (yellow line)

62 5 Figure-B.5 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0.3 m/s. Experiment (black line)[3, 0] classical water hammer (green line), water hammer with unsteady friction (red line)

63 52 Figure-B.6 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for V 0 =0.3 m/s. Experiment (black line)[3, 0] FSI with steady state friction (blue line),fsi with unsteady friction (yellow line)

64 53 Figure-B.7 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for L=43.7 m. Experiment (black line)[0] classical water hammer (green line), water hammer with unsteady friction (red line)

65 54 Figure-B.8 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for L=43.7 m Experiment (black line)[0] FSI with steady state friction (blue line),fsi with unsteady friction (yellow line)

66 55 Figure-B.9 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for L=77.8 m. Experiment (black line) [0] classical water hammer (green line), water hammer with unsteady friction (red line)

67 56 Figure-B.0 Variation of piezometric head with time at: (a) the downstream end; and (b) the mid-point for L=77.8 m Experiment (black line) [0] FSI with steady state friction (blue line), FSI with unsteady friction (yellow line)

68 57 Figure-B. Variation of piezometric head with time at (a, c) the downstream end and (b, d) the mid-point for V 0 =0. m /s. classical water hammer (green line), water hammer with unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady friction (yellow).

69 58 Figure-B.2 Variation of piezometric head with velocity at (a) the downstream end and (b) the mid-point for T c = 09 sec. Classical water hammer (green line), water hammer with unsteady friction (red line) FSI with steady friction (blue) and FSI with unsteady friction (yellow).