Review for exam3_2007 - Reviews for Exam3 Fall 2007 Chapter...

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Unformatted text preview: Reviews for Exam3 Fall 2007 Chapter 8 Viscous Flow in Pipes 1. Entrance Region and Fully Developed Flow Critical Reynolds number: Re Entrance length 2000 : 0.06Re 4.4 Re for laminar flow (Re ⁄ for turbulent flow (Re Re Re ) ) 2. Shear Stress and Head loss for a Pipe Flow Continuity: where Momentum: average velocity across pipe section where = radius of pipe Energy: where Combining Energy with Momentum, length of pipe Reviews for Exam3 Fall 2007 Friction factor (definition): 4 1 2 Darcy-Weisbach equation: 2 loss due to pipe friction where Need to know 3. Laminar Flow Exact solution to Navier-Stokes equations for a laminar flow in circular pipe: 1 where 2 Flow rate: Average velocity: Wall shear stress: Friction factor: exact solution ; Since Horizontal pipe Δ 0 Non-horizontal pipe Δ Δ 32 Δ sin 32 Δ Δ sin 128 128 Δ Δ 4 8 sin 4 2 Δ 8 2 Δ 4 2 sin sin 4 2 Reviews for Exam3 Fall 2007 4. Turbulent Flow 4.1 Description of Turbulent Flow (1) Randomness and fluctuations: Turbulence is irregular, chaotic, and unpredictable. However, structurally stationary flow, such as steady flows, can be analyzed using Reynolds decomposition. = where, = superimposed random fluctuation, 0 = Reynolds stresses (2) Nonlinearity: Reynolds stresses and 3D vortex stretching are direct results of nonlinear nature of turbulence. (3) Diffusion: Large scale mixing of fluid particles greatly enhances diffusion of momentum (and heat). (4) Vorticity/eddies/energy cascade: Turbulence is characterized by flow visualization as eddies, which vary in size from the largest (width of flow) to the smallest. Largest eddies contain most of energy, which break up into successively smaller eddies with energy transfer to yet smaller until Kolmogorov scale is reached and energy is dissipated by molecular viscosity (i.e. viscous diffusion). (5) Dissipation: = rate of dissipation = energy/time. Dissipation occurs at smallest scales. 4.2 Reynolds-Averaged Navier-Stokes (RANS) equations: , , Reynolds decomposition: Continuity: 0 i.e. 0 and 0 Momentum: or where, 1) equations are for the mean flow 2) differ from laminar equations by Reynolds stress terms = 3) influence of turbulence is to transport momentum from one position to another in a similar manner as viscosity are unknown, the problem is indeterminate: the central problem of turbulent flow 4) since analysis is closure. 4 equations and 4+6=10 unknowns Reviews for Exam3 Fall 2007 4.3 Turbulence Modeling (a) eddy-viscosity theories: In analogy with the laminar viscous stress, i.e., , where mean-flow rate of strain = eddy viscosity Mixing-length theory: based on kinetic theory of gasses ℓ , where ℓ ℓ One and two equation models: , where = constant, = turbulent kinetic energy, = turbulence dissipation rate (b) mean-flow velocity profile correlations 1. laminar sub-layer (viscous shear dominates) Very near the wall: 0 5 law-of-the-wall where, ⁄, ⁄ , and = friction velocity = 2. Outer layer (turbulent shear dominates) velocity defect law 3. overlap layer (viscous and turbulent shear important) 20 10 ln where, 0.41 and log-law 5.5 ⁄ Reviews for Exam3 Fall 2007 4.4 Smooth Pipes Assuming log-law is valid across entire pipe, the velocity profile is 1 ln Average velocity: 2.44 ln 1.34 By using following relations, ⁄ 8 1 2 1 2 ⁄ ⁄ 8 the relation between the friction factor and Reynolds number is (Prandtl, 1935), 1 1.99 log ⁄ ⁄ 1.02 ⁄ 0.8 With slightly adjusted constants to fit friction data better: 1 ⁄ 2.0 log Alternative approximation: Power law (H. Blasius, 1911) 0.316 ⁄ 4000 10 Then, the head loss is 0.316 ⁄ . 4.5 Rough Pipes Non-dimensional roughness 1) 2) 5 3) 5 70 70 hydraulically smooth (no effect of roughness) transitional roughness ( dependence) fully rough (independent ) Reviews for Exam3 Fall 2007 For fully rough flow, 70, the log law is modified as 1 ln 8.5 Then the average velocity becomes 2.44 ln 3.2 or 1 2 log ⁄ ⁄ 3.7 Colebrook (1939) combined the smooth wall and fully rough relations, 1 2.0 log ⁄ ⁄ 3.7 2.51 ⁄ Moody Diagram 4.6 Three Canonical Types of Problems 1. Determine the Head Loss 2 Δ , 2. Determine the Flow Rate 2 ⁄ Guess → → ⁄ ⁄ → , repeat until converged 3. Determine the Pipe Diameter 8 Guess → → , ⁄ , → , repeat until converged ⁄ ⁄ Reviews for Exam3 Fall 2007 4.7 Minor Losses 1. 2. 3. 4. Entrance and exit effects Expansions and contractions Bends, elbows, tees, and other fittings Valves (open or partially closed) Energy equation: 1 2 Head loss due to minor loss, 1 2 ∑ : 2 Loss coefficient, : ⁄2 1) Flow in a bend: swirling flow and/or flow separation represent energy losses which must be added to head loss 2) Valves: enormous losses 3) Entrances: depends on rounding of entrance 1 4) Exit (to a large reservoir): 5) Contraction and Expansions - Sudden ⇒ Expansion: 0.42 1 Contraction: 1 ⁄ from experiment - Gradual Expansion (Diffuser): where, ⁄ and 1 ⁄ Reviews for Exam3 Fall 2007 Chapter 9 Flow over Immersed Bodies 1. Fluid flow categories: 1. Internal flow: bounded by walls or fluid interfaces Ex) duct/pipe, turbo machinery, open channel/river 2. External flow: unbounded or partially bounded. Viscous and inviscid flow regions Ex) Flow around vehicles and structures a. Boundary layer flow: high Reynolds number flow around streamlined bodies without flow separation b. Bluff body flow: flow around bluff bodies with flow separation 3. Free shear flow: absence of walls Ex) jets, wakes, mixing layers 2. Basic Considerations Drag: 1 ̂ 1⁄2 ̂ : Lift: 1 1⁄2 ⁄ 1 streamlined body ⁄ 1 ̂ bluff body Streamlining: One way to reduce the drag. Reducing vibration and noise. 3. Boundary Layer Flow-field regions for high flow about slender bodies Reviews for Exam3 Fall 2007 - Boundary layer (BL) theory assumes that viscous effects are confined to a thin layer. There is a dominant flow direction such that and . - Gradients across are very large in order to satisfy the no slip condition; thus, . 0 0 1 laminar low turbulent low 1) 0, i.e. = constant across the boundary layer, where Bernoulli equation: = edge value (inviscid flow) constant, i.e., 2) Continuity equation is unaffected 3) Elliptic NS equations → Parabolic BL equations which can be solved using marching techniques 4) Boundary conditions 0 at 0 at 3.1 Momentum Integral Method 1 2 1 • • 2 displacement thickness 1 momentum thickness • shape parameter For flat plates: 1 o o o o ⁄ ⁄ ⇒ 2 Reviews for Exam3 Fall 2007 3.2 The Flat Plate Boundary Layer 3.2.1 Laminar Flow: Blasius solution (1908) Introducing a dimensionless transverse coordinate and a stream function, Ψ Blasius equation: 2 0 at with B.C.’s, 0 an d 1 at 5 1.7208 0.664 2.5916 0.332 ⁄ 0.332 0.664 1⁄2 1 2 1.328 where 3.2.2 Transition to Turbulent Flow • 5 10 , 1 0 Reviews for Exam3 Fall 2007 3.2.3 Turbulent Flow From the momentum integral theorem for flat plates, 2 By using a power-low approximation (Prandtl), ⁄ 0.02 ⁄ ⁄ 1 ⁄ 7 72 Then, the momentum equation becomes, ⁄ 0.02 2 7 72 or ⁄ 9.72 Separate the variables and integrate, assuming . ⁄ 1 8 7 72 1.3 0.027 ⁄ 0.031 ⁄ 7 6 0 at 0: Reviews for Exam3 Fall 2007 4. Drag 4.1 Flat plates 1) Parallel to the flow 1 1⁄2 ̂ 0 1.33 1 1⁄2 laminar low ̂ ⁄ 0.074 turbulent low ⁄ 2) Normal to the flow 1 1⁄2 ̂ 1 2 0 4.2 Bluff Body Drag 1⁄2 Flow separation: The fluid stream detaches itself from the surface of the body at sufficiently high velocities. Only appeared in viscous flow. • • Inside the separation region: o low pressure, existence of recirculating/backflows o viscous and rotational effects are the most significant Important physics related to flow separation: o ‘Stall’ for airplane, vortex shedding Reviews for Exam3 Fall 2007 Terminal velocity is the maximum velocity attained by a falling body when the drag reaches a magnitude such that the sum of all external forces on the body is zero. 0 ∑ or 1 2 For a sphere, V and V The terminal velocity is: 4⁄3 ⁄ Effect of Compressibility on Drag: • • • increases due to shock waves and wave drag 1: 1: upstream flow is not warned of approaching disturbance which results in the formation of shock waves across which flow properties and streamlines change discountinously (sphere) ∼ 0.6, (slender bodies) ∼ 1 5. Lift Lift ⁄2 1 Lift force: the component of the net force (viscous+pressure) that is perpendicular to the flow direction The minimum flight velocity: Total weight 1 2 of the aircraft be equal to the lift 2 , , Magnus effect: Lift generation by spinning. Breaking the symmetry causes the lift. ...
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This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

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