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- 1. SIMULATION OF LAMINAR AND TURBULENT FLOW INSIDE A PIPE BY: ANDI FIRDAUS SUDARMA (432107963) LECTURER: DR. JAMEL ALI ORFI SIMULATION PROJECT NUMERICAL METHODS IN THERMOFLUIDS (ME 578) MAGISTER PROGRAM OF MECHANICAL ENGINEERING COLLEGE OF ENGINEERING KING SAUD UNIVERSITY RIYADH - KSA FIRST SEMESTER 1433/1434 H
- 2. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) ABSTRACTThe steady-state three-dimensional water flows inside a pipe are investigated by thenumerical simulation using Fluent. Both problems, laminar and turbulent, are simulatedunder the same model. The solutions are compared with experimental results. The results areillustrated in form of velocity profile and maximum velocity along the pipe. 2
- 3. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) TABLE OF CONTENTAbstract ...................................................................................................................................... 2Table of Content ........................................................................................................................ 3Nomenclature ............................................................................................................................. 4I. Introduction ........................................................................................................................... 5II. Theoritical Analysis ............................................................................................................. 6III. Numerical Simulation ....................................................................................................... 11IV. Result and Discussion ....................................................................................................... 17V. Conclusions. ....................................................................................................................... 21References ................................................................................................................................ 22 3
- 4. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) NOMENCLATURE A Area ………………………………………………….… m2 D Diameter of tube ……………………………………… m L Channel length ……………………………………….. m P Pressure ………………………………………………. Pa R Radius of tube ………………………………………… m r Radial coordinate Re Reynolds number, ⁄ u Axial temperature m/s V Volume m3 x Axial coordinate ……………………………………... mGreek symbols Dynamic viscosity …………………………………… Kg / ms Kinematic viscosity ………………………………….. m2 / s Density ………………………………………………. Kg / m3 4
- 5. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)I. INTRODUCTIONFlows completely bounded by solid surfaces are called internal ﬂows. Thus internal ﬂowsinclude many important and practical ﬂows such as those through pipes, ducts, nozzles,diffusers, sudden contractions and expansions, valves, and ﬁttings. The pipe networks arecommon in any engineering industry. It is important to know the development of a flow at thepipe entrance and pressure drop taking place along the pipe length. The flow of fluids in apipe is widely studied fluid mechanics problem. The correlations for entry length andpressure drop are available in terms of flow Reynolds number.Internal ﬂows may be laminar or turbulent. Some laminar ﬂow cases may be solvedanalytically. In the case of turbulent ﬂow, analytical solutions are not possible, and we mustrely heavily on semi-empirical theories and on experimental data. For internal ﬂows, the ﬂowregime (laminar or turbulent) is primarily a function of the Reynolds number. In this projectwe will only consider incompressible ﬂows; hence we will study the ﬂow of water inside asmooth surface pipe.1.1. Problem DescriptionThe purpose of this project is to illustrate the setup and solution of a 3D turbulent and laminarfluid flow in a pipe using Fluent. This project will consider the flow inside a pipe of diameter1 m and a length of 20 m (Figure 1). The geometry is symmetric therefore this project willmodel only half portion of the pipe. Water enters from the inlet boundary with a variousvelocity (depend on Reynolds number). The flow Reynolds number is 8500 and 300 toillustrate the turbulent and laminar flow respectively. Inlet Outlet 𝐹𝑙𝑜𝑤 Pipe Figure 1. Problem descriptionThe objectives of this study are examining the results, such as velocity profile and entrancelength, compare them with experimental data and visualize the flow using animation tool. 5
- 6. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)II. THEORITICAL ANALYSISThe problem that will be discussed in this project is a two-dimensional single phase forcedconvection flow in a pipe. To obtain the equations that govern the current problem, thefollowing assumption are made for the analysis; i) Steady flow ii) Constant transport properties of fluid iii) Incompressible fluid flow iv) Newtonian fluid v) Continuum fluid 𝑢 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑟 𝑢 𝐹𝑙𝑜𝑤 𝑥 2𝑅 Figure 2. Schematic diagram.2.1. Laminar Velocity ProfileIn the first place we examine the flow of fluid inside the pipe set in motion. The governingequations of this problem are continuity, momentum and energy equations. To get thevelocity profile inside the pipe, the governing equations, namely continuity, momentum andenergy equations have been derived based on the above-mentioned assumptions. D(Continuity equation) V 0 (2.1) Dt ux 1 P(Momentum eq. in x-direction) V ux g x 2u x (2.2) t xBegin by formulating two dimensional continuity equation (2.1) for conditions mentionedabove which can be written with respect to cylindrical coordinate as; 1 1 r ur u ux 0 (2.3) r r r xSince ρ is constant, we will obtain V 0 . Where ur u 0 and the velocity is notchanging with respect to x, it‟s only a function of r u u (r ) . An important feature ofhydrodynamic conditions in the fully developed region is the gradient of axial velocity 6
- 7. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)component is everywhere zero. And from the assumption, there is no velocity in the r anddirections, i.e, ur u 0 , which gives u 0 (2.4) xThe next step is momentum equation formulation. The flow is in the x-direction u x , sour u 0 . Where g x 0 , ux ux (r ) , ux t 0 (steady).We can write momentum equation (2.2) as; 1 P V ux 2u x (2.5) xExpanding the momentum equation, ux 0 1 ux 0 ux 0 1 P 1 u 1 2u 0 2u 0 ur ux r x 2 x x r r x x r r r r 2 x 2 Using continuity equation (2.5) and assumption (iii), where ux u , we can write aboveequation as follows; 1 u 1 P r (2.6) r r r xThe momentum equation can be solved analytically to be used in the energy equation.Multiplying energy equation (2.7) by r and integrating it twice with respect to r, u r 2 P r c1 (2.7) r 2 x r 2 P u c1 ln r c2 (2.8) 4 xThe integration constants may be determined by invoking the boundary conditions u For 0 will give the result c1 0 (2.9) r r 0 And for ur R us 0 (no slip flow condition), will give the result R 2 P (2.10) c2 4 xSubstituting equation (2.9) and (2.10) into equation (2.8) will gives 7
- 8. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) 1 P 2 u 4 x r R2 (2.11)The initial and boundary conditions for constant wall temperature problem are;Boundary Condition 1 at r 0 , u r 0 (2.12)Boundary Condition 2 at r R , T Ts for all x0 (2.13)Then, we formulate the dimensionless form of velocity. Where um Q A and R R Q u dA u (2 r ) dr (2.14) 0 0Substitute equation (2.11) into equation (2.14). 1 P 2 R Q 4 x r R2 dr (2.15) 0And substitute equation (2.14) into um Q A R 2 P um U (2.16) 4 xSubstitute equation (2.12) and (2.15) to obtain dimensionless variable ⁄ 2 u r 1 (2.17) um REquation (2.17) can be used to obtain laminar velocity profile inside the pipe.2.2. Turbulent Velocity ProfileExcept for ﬂows of very viscous ﬂuids in small diameter ducts, internal ﬂows generally areturbulent. As noted in the relation of shear stress distribution in fully developed pipe ﬂow, inturbulent ﬂow there is no universal relationship between the stress ﬁeld and the mean velocityﬁeld. Thus, for turbulent ﬂows we are forced to rely on experimental data.The velocity proﬁle for turbulent ﬂow through a smooth pipe may also be approximated bythe empirical power-law equation 1 u r n 1 (2.18) U R 8
- 9. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)Where the exponent (n) is varies with the Reynolds number. Data from Hinze suggest that thevariation of power-law exponent n with Reynolds number (based on pipe diameter, D, andcenterline velocity, U) for fully developed ﬂow in smooth pipes is given by, n 1.7 1.8log Reu (2.19)For 2Velocity proﬁles for n = 6 and n = 10 are shown in Figure 3. The parabolic proﬁle for fullydeveloped laminar ﬂow is included for comparison. It is clear that the turbulent proﬁle has amuch steeper slope near the wall. Figure 3. Velocity profiles for fully developed flow.2.1. Reynolds Number CorrelationAs discussed previously in introduction, the pipe ﬂow regime (laminar or turbulent) isdetermined by the Reynolds number, where; UD Re (2.20) At low ﬂow rates (low Reynolds numbers) the ﬂow is laminar and at high rates the flow istransition into or turbulent. Laminar flow in a pipe may be only for Reynolds numbers lessthan 2300. 9
- 10. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) Figure 4. Flow in the entrance region of a pipeThe length of the tube between the start and the point where the fully developed flow beginsis called the Entrance Length, denoted by Le. The entrance length is a function of theReynolds Number Re of the flow. LeLa min ar 0.06 Re D (2.21)Where D is the tube diameter.To restore a turbulent flow to parabolic flow, the entrance length is by approximation: LeTurbulent 4.4D Re1/ 6 (2.22) 10
- 11. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)III. NUMERICAL SIMULATIONThe grid (mesh) that used in this project is already included in Fluent Tutorial-4. Using thesame mesh to generate 2 model, that is; Model A. Laminar flow with Re = 300 Model B. Turbulent flow with Re = 8500Reynolds number approximation is based on expectation that fully developed region will beoccurring before the flow reach pipe outlet. Figure 5. Grid displayThe problem is solved in steady state using pressure based solver. Definition of viscousmodel are shown in figure (6), where the laminar and k-epsilon (2 eqn.) selected for laminarand turbulent problem respectively. 11
- 12. Simulation ProjectNumerical Methods in Thermo-Fluids (ME 578) Figure 6. Setting of viscous model 12
- 13. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)The fluid that flow inside the pipe is water. The properties of water are obtained from Fluentdatabase. Figure 7. Material propertiesThe models are made with boundary conditions at inlet (at X = 0 m) and outlet (at X = 20 m)is „Inlet‟ and „Outlet‟ respectively. The boundary condition for surface and axis of the pipe is„Wall‟ and „Axis‟ respectively. „Axis‟ boundary condition acts like „Symmetry‟ boundarycondition but it is used for axisymmetric problem such as flow in a pipe.The velocity inlet is obtained by using equation (2.20). The velocity is 0.0003 and 0.0085 m/sfor model A and model B, respectively. Where Turbulent Intensity can be calculated as; T .I . 0.16 Re1/8 (2.23) 13
- 14. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)The CFD calculation is carried out using the SIMPLE algorithm for pressure-velocitycoupling and the second order upwind differencing scheme for momentum equation andturbulent term. These settings are shown in solution controls window figure (8). Model B Model AFigure 8. Settings of algorithm for pressure-velocity coupling and spatial discretizationThe convergence data are plotted to represent the fully developed velocity profile at outletand maximum velocity at centerline. 14
- 15. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)Model A 0.5 Model A 0.4 0.3 0.2 0.1 Radius (m) 0.0 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 -0.1 -0.2 -0.3 -0.4 -0.5 Velocity (m/s) Figure 9. Fully developed velocity profile at outlet for laminar 15
- 16. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)Model B 0.5 Model B 0.4 0.3 0.2 0.1 Radius (m) 0.0 0 0.002 0.004 0.006 0.008 0.01 0.012 -0.1 -0.2 -0.3 -0.4 -0.5 Velocity (m/s) Figure 10. Fully developed velocity profile at outlet for turbulent 16
- 17. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)IV. RESULT AND DISCUSSIONThe maximum velocities at centerline are presented in the chart below. Where fullydeveloped region will occur after the flow reaching entrance length (Le). 0.0006 0.00055 Maximum Velocity (m/s) 0.0005 0.00045 0.0004 0.00035 0.0003 0 2 4 6 8 10 12 14 16 18 20 Pipe Length (m) Figure 11. Maximum velocity of laminar flowThe entrance length of laminar flow can be calculated using equation (2.21). For Re=300, theentrance length may as long as 18 m. Comparing with the result obtained from simulation(figure 11), at length of the pipe above 18 m there velocity is still developing with margin ofincrement 0.052 percent. 17
- 18. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) 0.0105 0.01 Maximum Velocity (m/s) 0.0095 0.009 0.0085 0.008 0 2 4 6 8 10 12 14 16 18 20 Pipe Length (m) Figure 12. Maximum velocity of turbulent flowFor turbulent flow, entrance length can be approximated using equation (2.22). Where atRe=8500, the flow approximated will be fully developed at 18 m length of pipe. Comparingwith the result obtained from simulation (figure 12), at length of the pipe above 19.8 m therevelocity is still developing with margin of increment 0.0384 percent.Using dimensionless form of velocity profile, we comparing experimental data from equation(2.17) for laminar and equation (2.18) for turbulent and data that obtained from thesimulation. The results are illustrated in the figure (13) for laminar and (14) for turbulent. 18
- 19. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) 1.0 0.9 0.8 0.7 0.6rR 0.5 Numerical Experimental 0.4 0.3 0.2 0.1 0.0 0 0.2 0.4 0.6 0.8 1 u/U Figure 13. Velocity profile of laminar flow 1.0 0.9 0.8 0.7 0.6rR 0.5 Numerical Experimental 0.4 0.3 0.2 0.1 0.0 0 0.2 0.4 0.6 0.8 1 u/U Figure 14. Velocity profile of turbulent flow 19
- 20. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)The velocity profile comparison for laminar flow shows that the velocities obtained from thesimulation are similar with the theoretical data. But the turbulent flow chart shows that thereis unmatched data between experimental and simulations. This result happen when theproblem not simulated correctly. After evaluating the turbulent model, we found that theturbulent intensity value was 4.8%, where it should be 0.052% base on equation (2.23). 20
- 21. Simulation Project Numerical Methods in Thermo-Fluids (ME 578)V. CONCLUSIONS.In general, for the above three-dimensional with two boundary conditions stated, Reynoldsnumber affects the velocity profile. When Reynolds number is increasing, the entrance lengthwill also increased. This situation is valid for both cases, laminar and turbulent.The velocity profile of laminar flow is similar with parabolic curve, and at turbulent flowthere is extreme different between internal flow with the flow near the wall. 21
- 22. Simulation Project Numerical Methods in Thermo-Fluids (ME 578) REFERENCES[1] Fluent Inc., “Tutorial 4. Simulation of Flow Development in a Pipe”, 2006[2] Fox, R. W., McDonald, A. T., Pritchard, P. J., “Introduction to Fluid Mechanics”, 6 th ed., John Wiley & Sons, New York, 2003. 22

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