Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible flows. (i.e., M=0.3 which for air at standard conditions corresponds to a speed of approximately 100m/s). Internal flows can be classified as either laminar, transitional or turbulent. The particular flow regime is primarily dependent on the Reynolds number, surface roughness, and level of initial disturbances present within the flow. The Reynolds number is defined as; ρ μ 4/16/2018 C6A: Internal Incompressible Viscous Flow 2 1
6.1 Introduction Under normal conditions transition from a laminar to turbulent flow occurs at a critical Re ~ 2,000 in a circular tube or pipe. However, minimizing the disturbances within the flow has, for a smooth circular tube, remained laminar up to Re ~ 100,000. Note: Turbulent flows have random fluctuating velocities which result a in violent mixing action. 4/16/2018 C6A: Internal Incompressible Viscous Flow 3 6.1 Introduction Recall for a laminar flow the shear stress is: τ = μ du/dy Shear stress in turbulent flow is written as: 4/16/2018 C6A: Internal Incompressible Viscous Flow 4 2
6.1 Introduction u z ú z U z average u r ú r U r average p P Time p average 4/16/2018 C6A: Internal Incompressible Viscous Flow 5 6.1 Introduction Laminar flow: τ = μ du/dy Turbulent flow: 4/16/2018 C6A: Internal Incompressible Viscous Flow 6 3
6.1 Introduction Note: To account for any non-uniformity in the velocity profile a kinetic energy coefficient, is defined; where represents the average cross sectional velocity. For fully developed laminar flows the velocity profile is parabolic and the kinetic energy coefficient = 2. However, for most internal turbulent flows the velocity profile is much more uniform and 1.05, so for convenience unity is typically used. 4/16/2018 C6A: Internal Incompressible Viscous Flow 7 6.2.1 Fully Developed Laminar Flow Consider a steady laminar flow of a viscous fluid inside a circular tube. 4/16/2018 C6A: Internal Incompressible Viscous Flow 8 4
6.2.1 Fully Developed Laminar Flow Let the fluid enter the tube with a uniform velocity. As the fluid moves along the tube a shear layer forms. This layer of low speed fluid grows on the tube wall as a result of viscous effects, i.e., the no-slip condition. As the viscous fluid moves down the tube a shear layer on the tube wall continues to grow and meet at the tube centerline. 4/16/2018 C6A: Internal Incompressible Viscous Flow 9 6.2.1 Fully Developed Laminar Flow Pipe Entrance v At this location the velocity profile becomes developed (i.e., self-similar) and no longer changes with downstream distance. In this self-similar state the velocity profile is said to be Fully Developed. The distance between the tube inlet and location where the velocity profile becomes invariant (i.e., fully developed), is referred to as the hydrodynamic entrance length, L e. In many engineering applications the flow is turbulent and the Re is between 10 4 and 10 5, typically producing an Le/D ~ 25. 4/16/2018 C6A: Internal Incompressible Viscous Flow 10 v v 5
6.2.2 Analysis of Flow in a Circular Tube Use of the fully developed flow (FDF) assumption implies that the hydrodynamic state of the fluid remains constant along the length of the tube and that the radial velocity, v r = 0 and / x = 0. For a time independent (i.e., steady) axisymmetric flow in a circular tube the reduced form of the x-momentum eq (in cylindrical coordinates) follows: So if the flow is assumed to be FDF, and since p was shown to vary linearly with x, implying p/ x, we obtain; (6.1) (6.2) 4/16/2018 C6A: Internal Incompressible Viscous Flow 11 6.2.2a Determine the Velocity Profile Integrate Eq 6.2 w.r.t. r and apply the following boundary conditions; So; Applying boundary condition b one obtains (6.3) 4/16/2018 C6A: Internal Incompressible Viscous Flow 12 6
6.2.2a Determine the Velocity Profile Integrating Eq 6.3, Applying boundary condition a, 4/16/2018 C6A: Internal Incompressible Viscous Flow 13 6.2.2a Determine the Velocity Profile The resulting velocity profile, (6.4a) It can be quickly observed that the velocity profile is parabolic, and the maximum velocity occurs at the centerline, r = 0. (6.4b) 4/16/2018 C6A: Internal Incompressible Viscous Flow 14 7
6.2.2b Average Streamwise Velocity, where the cross sectional area of the tube, A x = r 2 and da=2 rdr 4/16/2018 C6A: Internal Incompressible Viscous Flow 15 6.2.2b Average Streamwise Velocity, Rewriting (6.4b) in terms of the mean velocity, (6.4c) 4/16/2018 C6A: Internal Incompressible Viscous Flow 16 8
6.2.2c Volume Flow Rate Computation (6.5) 4/16/2018 C6A: Internal Incompressible Viscous Flow 17 6.2.2d Pressure Drop Determination Approximate the pressure gradient and solve for P in Eq 6.5, Multiply and divide right hand side by 4/16/2018 C6A: Internal Incompressible Viscous Flow 18 9
6.2.2d Pressure Drop Determination (6.6) 4/16/2018 C6A: Internal Incompressible Viscous Flow 19 6.2.2e Evaluation of Shear Stress The x- component of shear stress acting at the tube wall, the only component to survive in the present example is rx since V r = 0 (6.7) Note: Shear stress is maximum at the wall of the tube and linearly decreases to zero at the tube centerline. 4/16/2018 C6A: Internal Incompressible Viscous Flow 20 10
6.2.2e Evaluation of Shear Stress Since the pressure gradient is approximated by p/l then or in terms of the mean velocity (6.8) 4/16/2018 C6A: Internal Incompressible Viscous Flow 21 6.3 Friction Factor (f ) The shear stress can be presented in nondimensional terms by normalizing it by the dynamic pressure; where f is the Fanning friction factor. (6.9) Note: 1- There is a second quantity known as the Darcy friction factor, f D that is often used. 2- The two friction factors are related by; f D 4f For the present example, start with eq 6.9 4/16/2018 C6A: Internal Incompressible Viscous Flow 22 11
6.3 Friction Factor (f ) (6.10) Typically the mean velocity is defined as shown, where A represents the cross sectional area. 4/16/2018 C6A: Internal Incompressible Viscous Flow 23 In many cases involving internal flow a characteristic length, referred to as the Hydraulic Diameter (Dh) is used. 6.3 Friction Factor (f ) The Hydraulic Diameter (Dh) is defined as, (6.11) For a circular tube, Note: 1 - The above relation for f was determined assuming a laminar flow. 2 - A laminar flow can exist if the Re is below a critical value. 3 - The typical value is quoted as, Re crit = 2000, beyond which the flow undergoes transition and becomes turbulent. 4/16/2018 C6A: Internal Incompressible Viscous Flow 24 12
6.3.1 Turbulent Flow Friction Factor (f ) As the flowrate is increased and the Re crit is exceeded the streamlines are no longer straight and become unsteady as the flow transitions from a laminar state. Re UD inertia Viscous forces The primary attribute of turbulence is it variations or fluctuations in both space and time. u z u r p ú z Time U zavg ú r Time U r avg P Time p avg 4/16/2018 C6A: Internal Incompressible Viscous Flow 25 6.3.1 Turbulent Flow Friction Factor (f ) The Moody Diagram Transitional Flow f =16/Re Note: Surface roughness also has an affects the f. 4/16/2018 C6A: Internal Incompressible Viscous Flow 26 13