TurbulentFlowinPipes.pdf

Birla Institute of Technology and Science, Pilani Hyderabad Campus Topic: Turbulent Flow in pipes Course No. CE F312 Course Title: Hydraulic Engineering Instructor in Charge : Jagadeesh Anmala 1 Introduction:...

Birla Institute of Technology and Science, Pilani Hyderabad Campus Topic: Turbulent Flow in pipes Course No. CE F312 Course Title: Hydraulic Engineering Instructor in Charge : Jagadeesh Anmala 1 Introduction: Flow through pipes are of two types: 1. Laminar Flow: Fluid particles moves in straight path line in laminar or layers. 2. Turbulent flow: Irregular movement of fluid particles in random direction. 2 Reynold’s Number: Reynold’s Number : A parameter to predict whether a flow is laminar or turbulent 𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒 𝐹 𝑅𝑒 = 𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒 = 𝐹𝑖 𝑣 𝜌𝑉𝐿 𝑅𝑒 = 𝜇 Where, 𝜌= mass density 𝜇= viscosity of the flowing fluid 3 4 5 Laws of fluid friction for turbulent flow: 1. Proportional to the (velocity)n. 2. Independent of the pressure. 3. Proportional to the density of the flowing fluid. 4. Slightly affected by the variation of the temperature of flowing fluid. 5. Proportional to the area of surface in contact. 6. Dependent on the nature of the surface in contact. 6 Characteristics of Turbulent Flow: 1. Re > 4000 2. Fluid particles are in extreme disorder, haphazard movement and eddies are developed 3. Irregular velocity and pressure fluctuations of high frequency superimposed on main flow 4. Velocity distribution is relatively uniform. 5. Velocity profile is flatter to the corresponding laminar flow. 7 Shear Stresses in Turbulent Flow: Additional shear stresses of high magnitude developed due to transfer of momentum between adjacent layers Expression for turbulent shear stress by J.Boussinesq 𝑑𝑣 𝑑𝑣 𝜏=η = 𝜌𝜀 𝑑𝑦 𝑑𝑦 Where, 𝑣 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑦 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝜂 = 𝑒𝑑𝑑𝑦 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 𝜂 𝜀 = =′ 𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 ′ 𝑜𝑟 ′𝑣𝑖𝑟𝑡𝑢𝑎𝑙 ′ 𝑜𝑟 ′𝑒𝑑𝑑𝑦 ′ 𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦. 𝜌 𝜌 = 𝑚𝑎𝑠𝑠 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑 Kinematic eddy viscosity (ε) is: ▫considered to be independent of the properties of the fluid ▫depends on the characteristics of the flow 8  Total Shear Stress for turbulent flow is given by: 𝑑𝑣 𝑑𝑣 𝑑𝑣 𝜏=𝜇 + 𝜂 = 𝜇+𝜂 𝑑𝑦 𝑑𝑦 𝑑𝑦 The total shear stress at any point is sum of viscous shear stress and turbulent shear stress 𝑑𝑣 𝑑𝑣 𝜏= 𝜇 + 𝜌𝑙 2 ( )2 𝑑𝑦 𝑑𝑦 9 Prandtl – mixing length hypothesis 10 Prandtl – mixing length hypothesis 11 Formation of Boundary Layer in pipes: Fluid is retarded close to the pipe wall resulting in formation of boundary layer Boundary layer in a pipe attains a maximum thickness equal to the pipe radius The velocity in the central core increases For laminar flow , the Reynolds number is 𝜌𝑉𝐷 𝑅𝑒 = 𝜇 Where, 𝐷= Diameter of pipe 𝑉= Mean velocity of the pipe 𝜌= mass density 𝜇= Viscosity of flowing fluid 12 13 Hydrodynamically Rough and Smooth surfaces: Hydrodynamically Rough Boundary: Irregularities of large average height k Hydrodynamically Smooth Boundary: Irregularities of smaller values of k Nikuradse’s Experiment: If 𝜅 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑖𝑟𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑖𝑡𝑖𝑒𝑠 And 𝛿 ′ = 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝑠𝑢𝑏𝑙𝑎𝑦𝑒𝑟, Then , 𝑘 Hydrodynamically Smooth Boundary: 𝛿′ < 0.25 ( due to greater thickness of laminar sublayer, the eddies from Outer layer cannot reach the surface of the irregularities). 𝑘 Hydrodynamically Rough Boundary 𝛿′ > 6.0 (Reynolds number increases Thickness of laminar sublayer decreases. Eddies touch the surface irregularities and causes energy loss) 𝑘 Boundary in transition: 0.25 < 𝛿′ < 6.0 14 Velocity Distribution for Turbulent Flow: Assumption by Prandtl for turbulent flow in circular pipes: Mixing length l is linear function of distance y from the pipe wall. 𝑙 = κ𝑦 , where,κ = Karman universal constant According to Nikuradse’s experiment: 𝑦2 𝑙 = 0.4𝑦 − 0.44 𝑅 Turbulent shear stress varies linearly with the Radius of the pipe : 𝑟 𝑅−𝑦 𝑦 𝝉 = 𝜏0 = 𝜏0 = 𝜏0 1 − 𝑅 𝑅 𝑅 𝜏0 = turbulent shear stress at the pipe boundary The following equation shows that the velocity distribution in turbulent flow is logarithmic in nature. 𝑉∗ 𝑣 = 𝑙𝑜𝑔𝑒 𝑦 + 𝐶 𝑘 𝜏 Where, 𝑉∗ = ( 𝜌0)1Τ2 15 The constant C is determined : At boundary 𝑣 = 𝑣𝑚𝑎𝑥 at 𝑦 = 𝑅 𝑉 Hence, 𝐶 = 𝑣𝑚𝑎𝑥 − κ∗ 𝑙𝑜𝑔𝑒 𝑅 Substituting value of C and κ = 0.4 𝑦 𝜈 = 𝜈𝑚𝑎𝑥 + 2.5𝑉∗ 𝑙𝑜𝑔𝑒 𝑅 The above equation is Prandtl’s universal velocity Distribution equation for turbulent flow in pipes dividing both sides by V* and rearranging: 𝜈𝑚𝑎𝑥 − 𝜈 𝑅 = 2.5 𝑙𝑜𝑔𝑒 𝑉∗ 𝑦 Changing to base 10 𝜈𝑚𝑎𝑥 − 𝜈 𝑅 = 5.75 𝑙𝑜𝑔10 𝑉∗ 𝑦 Fritsch experiments -> velocity distribution identical (smooth and rough) except for narrow region in the vicinity of boundary Prandtl’s equation for the central region of the pipe: 𝑣𝑚𝑎𝑥 − 𝜈 𝑅 = 2.5 𝑙𝑜𝑔𝑒 𝑉∗ 𝑦 16 Velocity Distribution Equation for Turbulent flow in Hydrodynamically Rough and smooth surface: 𝑉∗ 𝑣 = 𝑙𝑜𝑔𝑒 𝑦 + 𝐶 κ For y= 0, v= ∞ And for y= y’ (a certain distance above the boundary), v= 0 𝑉 The constant C becomes : 𝐶 = − κ∗ 𝑙𝑜𝑔𝑒 𝑦 ′ 𝑉∗ 𝑦 Substituting the value of C in above equation: 𝑣= 𝑙𝑜𝑔𝑒 𝑦′ κ 17 Karman-Prandtl equation for Velocity Distribution near hydrodynamically smooth boundaries: The velocity distribution curve for laminar flow is parabolic in nature in laminar sublayer 𝑑𝑣 In laminar sublayer shear stress 𝜏 = 𝜇 𝑑𝑦 For linear distribution : 𝑣 𝜏 = 𝜏0 = 𝜇 𝑦 Dividing both sides by ρ and replacing 𝜏0 / ρ by V*2 the equation can be written as 𝑣 𝜌𝑉∗ 𝑦 = 𝑉∗ 𝜇 𝑣 𝑉∗ 𝑦 = 𝑉∗ 𝜈 is the velocity distribution in laminar sublayer, from y = 0 to y = 𝛿 ′ 18 𝑣 𝑉∗ 𝑦 𝜇 = since =ν 𝑉∗ 𝜈 𝜌 From Nikuradse’s experiment, 𝑉∗ 𝑦 𝑣 =11.6 , for y=δ’ 𝑉∗ 𝑦′ So, δ’ = 11.6 ν /V* and = 0.108 ν 0.108ν 𝛿′ Or, y ′ = = 𝑉∗ 107 Substituting the values of y’ the expression becomes 𝑉∗ 𝑉∗ 𝑦 𝑣 = 𝑙𝑜𝑔𝑒 κ 0.108ν Dividing both sides by 𝑉∗ 𝑣 𝑉∗ 𝑦 = 5.75 𝑙𝑜𝑔10 + 5.5 𝑉∗ ν The above equation is Prandtl’s universal velocity Distribution equation for turbulent flow in near the smooth boundaries 19 The velocity distribution for turbulent flow in smooth pipes -> also by exponential equation instead of logarithmic equation 𝑣 𝑉∗ 𝑦 1Τ7 = 8.74 𝑉∗ 𝜈 𝑣 = 𝑣𝑚𝑎𝑥 𝑎𝑡 𝑦 = 𝑅 𝑣𝑚𝑎𝑥 𝑉∗ 𝑅 1Τ7 = 8.74 𝑉∗ 𝜈 𝑣 𝑦 1Τ7 𝑟 1Τ7 = = 1− 𝑣𝑚𝑎𝑥 𝑅 𝑅 General form of the velocity distribution for turbulent flow in smooth pipes: 1ൗ 1ൗ 𝑣 𝑦 𝑛 𝑟 𝑛 = = 1− 𝑣𝑚𝑎𝑥 𝑅 𝑅 20 2. Velocity distribution near hydrodynamically rough boundaries: 𝑉∗ 𝑦 Equation for turbulent flow in hydrodynamically rough boundaries: 𝑣 = 𝑙𝑜𝑔𝑒 κ 𝑦′ From Nikuradse’s experiment, it has been found that y’=(k/30) Substituting the value of y’ and using κ = 0.4 , we get the following equation 𝑉∗ 30𝑦 𝑣= 𝑙𝑜𝑔𝑒 0.4 𝑘 Converting to log base ten and dividing both sides by 𝑉∗ , we get the following equation 𝑣 = 5.75 𝑙𝑜𝑔10 𝑦Τ𝑘 + 8.5 𝑉∗ The above equation is Prandtl’s universal velocity Distribution equation for turbulent flow in the hydrodynamically rough boundaries 21 Criterion for Smooth and Rough pipes Hydrodynamic smoothness or roughness Depends on the : i. The relative magnitude of average height of surface protrusions k ii. The thickness of laminar sublayer δ’ 𝑘 𝑉∗ 𝑘 1 ′ = 𝛿 ν 11.6 Based on Nikuradse’s experiments: 𝑉∗ 𝑘 𝑘  For hydrodynamically smooth: ≤ 3 or ≤ 0.25 ν 𝛿′ 𝑉∗ 𝑘 𝑘  For transition: 3`< < 70 ; or 0.25 < < 6.0 ν 𝛿′ 𝑉∗ 𝑘 𝑘  For hydrodynamically rough : ≥ 70 ; or ≥ 6.0 ν 𝛿′ 22 Velocity distribution equation for turbulent flow in terms of mean velocity Considering a elementary ring in which R = radius of the pipe dr = thickness of the ring at a radial distance r from the centre of the pipe y = distance from the centre of pipe boundary Thus y= (R-r) 𝑄 1 𝑉 = 𝜋𝑅2 = 𝜋𝑅2 ‫𝐴𝑑 𝑣 𝐴׬‬ 1 𝑅 𝑉 = 𝜋𝑅2 ‫׬‬0 𝑣 2𝜋𝑟 𝑑𝑟 1 𝑅 𝑉 (𝑅−𝑟) 𝑉 = 𝜋𝑅2 ‫׬‬0 2𝜋𝑟𝑉∗ 5.75 𝑙𝑜𝑔10 ∗ ν + 5.5 𝑑𝑟 Solving the above equation 𝑉 𝑉∗ 𝑅 = 5.75 𝑙𝑜𝑔10 + 1.75 𝑉∗ ν The above equation is the mean velocity V for turbulent flow in smooth pipes 23 Similarly for rough pipes : 1 𝑅 (𝑅−𝑟) 𝑉 = 𝜋𝑅2 ‫׬‬0 2𝜋𝑟𝑉∗ 5.75 𝑙𝑜𝑔10 𝑘 + 8.5 𝑑𝑟 Integrating and simplifying the relation for mean velocity V for turbulent flow in rough pipe is given as: 𝑉 𝑅 = 5.75 𝑙𝑜𝑔10 + 4.75 𝑉∗ 𝑘 karman-Prandtl equation for velocity distribution in smooth and rough pipes obtained from above two equations 𝑣−𝑉 𝑦 = 5.75 𝑙𝑜𝑔10 + 3.75 𝑉∗ 𝑅 𝑉𝑚𝑎𝑥 −𝑉 At y=R, 𝑣 = 𝑣𝑚𝑎𝑥 , so the equation obtained is : = 3.75 𝑉∗ 24 Resistance to the Flow of fluid in smooth and rough surfaces: Friction factor (f) depends upon Reynold’s number (𝑉𝐷 Τν) The ratio (𝑘Τ𝐷) 𝑉𝐷 𝑘 𝑓=𝜙 , 𝜈 𝐷 𝑉𝐷 Laminar & turbulent flow with smooth boundary ---- 𝑓= 𝜙1 𝜈 𝑘 Turbulent flow with rough boundary ------ 𝑓 = 𝜙2 𝐷 25 a. Variation of Friction factor f for Laminar flow: 64 𝑓= 𝑅𝑒 For 𝑅𝑒 < 2000 26 b. Variation of friction factor f for turbulent flow: 𝑅𝑒 > 4000 is turbulent 2000 < 𝑅𝑒 < 4000 is in transition zone (from laminar to turbulent) Transition zone -- No specific relation between f and Re Turbulent Zone ---- f depends on Re or (k/D) or both --------- depending on whether the boundary is smooth or rough or in transition 27 Variation of friction factor for Turbulent flow in smooth pipes: Blasius’s equation for turbulent flow with smooth boundaries 0.316 𝑓= 1ൗ (𝑅𝑒) 4 So, f varies inversely with the one-fourth power of Reynolds number 𝑅𝑒. f is valid for 4000 < Re < 105 28 For Re > 105 For turbulent flow in smooth pipes 𝑉 𝑉∗ 𝑅 = 5.75 𝑙𝑜𝑔10 + 1.75 𝑉∗ ν 𝑓 Since 𝑉∗ = V so the equation becomes, 8 𝑓 1Τ2 𝑉 𝑉( ) 𝑅 = 5.75 𝑙𝑜𝑔10 8 + 1.75 𝑓 1Τ2 ν 𝑉( ) 8 Rearranging the terms 1 = 2.03𝑙𝑜𝑔10 𝑅𝑒 𝑓 − 0.91 𝑓 But from Nikuradse’s experimental results the relationship obtained is : 1 = 2.0𝑙𝑜𝑔10 𝑅𝑒 𝑓 − 0.8 “Karman-Prandtl resistance equation for 𝑓 turbulent flow in smooth pipes” applicable 5 X 104 < Re < 4 X 107 29 “Karman-Prandtl resistance equation for turbulent flow in smooth pipes” 1 = 2.0𝑙𝑜𝑔10 𝑅𝑒 𝑓 − 0.8 Trial and error 𝑓 Value of f by Nikuradse’s empirical relationship 0.221 5 X 104 < Re < 4 X 107 𝑓 = 0.0032 + (𝑅𝑒)0.237 30 e. Variation factor for turbulent flow in Rough pipes: Mean velocity in turbulent flow in rough pipes 𝑉 𝑅 = 5.75 𝑙𝑜𝑔10 + 4.75 𝑉∗ 𝑘 Substituting value of 𝑉∗ 𝑉 𝑅 = 5.75 𝑙𝑜𝑔10 + 4.75 𝑓 1Τ2 𝑘 𝑉( ) 8 Or , 1 𝑅 = 2.03𝑙𝑜𝑔10 + 1.68 𝑓 𝑘 By Nikuradse’s experimental data on turbulent flow in rough pipes 1 𝑅 = 2.0𝑙𝑜𝑔10 + 1.74 𝑓 𝑘 “Karman-Prandtl resistance equation for turbulent flow in rough pipes” 31  𝑅 Hence, greater value of parameter 𝑘 , larger is Reynold’s number at which the pipe begins to follow the pattern of rough pipe. Criteria for Hydrodynamically Smooth and Rough boundaries: Nature of boundary depends on the value of (k/δ’) 𝑅𝑒 𝑓 The parameter 𝑅 𝛿 ′ = 11.6ν/𝑉∗ 𝑘 𝑓 𝑉∗ = V ( ) 8 11.6ν 11.6ν( 8) So, 𝛿′ = = 𝑓 1Τ2 𝑉 𝑓 𝑉 8 𝛿′ 65.6 𝑘 𝑅𝑒 𝑓 Or, = - = 𝑅 𝑅𝑒 𝑓 𝛿′ 65.6 𝑅Τ𝑘 32 Criteria for Hydrodynamically Smooth and Rough boundaries: 𝑘 𝑅𝑒 𝑓 = 𝛿 ′ 65.6 𝑅Τ𝑘 𝑅𝑒 𝑓 For hydrodynamically smooth pipe 𝑅 < 17 𝑘 𝑅𝑒 𝑓 For hydrodynamically rough pipes 𝑅 > 400 𝑘 For transition 𝑅𝑒 𝑓 17< 𝑅 < 400 𝑘 33 “Karman-Prandtl resistance equation for turbulent flow in smooth pipes” 1 𝑅𝑒 𝑓 1 − 2.0𝑙𝑜𝑔10 𝑅/𝑘 = 2.0𝑙𝑜𝑔10 𝑅 − 0.8 = 2.0𝑙𝑜𝑔10 𝑅𝑒 𝑓 − 0.8 𝑓 𝑘 𝑓 1 𝑅 “Karman-Prandtl resistance equation for turbulent flow − 2.0𝑙𝑜𝑔10 = 1.74 𝑓 𝑘 in rough pipes” 1 𝑅 From which follows that: = 2.0𝑙𝑜𝑔10 + 1.74 1 𝑅 𝑓 𝑘  When pipe is smooth − 2.0𝑙𝑜𝑔10 is a function 𝑓 𝑘 𝑅𝑒 𝑓 of 𝑅 𝑘 1 𝑅  When pipe is rough − 2.0𝑙𝑜𝑔10 attains a 𝑓 𝑘 constant value of 1.74 34 From Nikuradse's Experimental data and graph of 1 𝑅 𝑅𝑒 𝑓 − 2.0𝑙𝑜𝑔10 vs 𝑅 𝑓 𝑘 𝑘 35  f. Variation of Friction factor for commercial pipes: For commercial pipes the surface roughness of the pipe in terms of uniform sand grain diameter k is evaluated The coordinates of the graph are chosen to be 1 𝑅 𝑅𝑒 𝑓 − 2.0𝑙𝑜𝑔10 and 𝑅 𝑓 𝑘 𝑘 The limiting value of 𝑓 is obtained from Darcy-Weisbach equation. 36  Empirical equation of Colebrook and White for the curve which represents the variation friction factor for commercial pipes is given by 𝑅 1 𝑅 ( ) 𝑘 − 2.0𝑙𝑜𝑔10 = 1.74 − 2.0 𝑙𝑜𝑔10 1 + 18.7 𝑓 𝑘 𝑅𝑒 𝑓 37  𝑅 1 𝑅 (𝑘) Moody Daigram: L.F.Moody plotted 𝑓 − 2.0𝑙𝑜𝑔10 𝑘 = 1.74 − 2.0 𝑙𝑜𝑔10 1 + 18.7 𝑅𝑒 𝑓 38 The values of k for some materials are given below Pipe Material k in mm 1. Glass,Brass ,Copper , Smooth Lead 2. steel, Wrought iron 0.045 3. Asphalted cast iron 0.120 4. Galvanised iron 0.150 5. Cast iron 0.260 6. Concrete 0.30 to 3.0 7. Rivetted steel 0.90 to 9.0 39 40 41 Thank you 42