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Measurements in Fluid Mechanics 058:180:001 (ME:5180:0001) Time & Location: 2:30P - 3:20P MWF 218 MLH Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui

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2 Lecture 3. Similarity and flow motion patterns

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3 Similarity and non-dimensionalization Different ways to perform measurement: - expensive and impossible in most cases1. Actual system under actual operation condition 1. Geometrically similar 2. Actual system under modified condition 3. Model system under controlled condition - expensive and impossible in many cases - low cost and possible - enable application of measured properties with model system and modified condition to actual system under actual condition. Similarity Requirements of similarity 2. Kinematically similar 3. Dynamically similar - same shape, the same ratios of all corresponding dimensions - same velocity directions and constant ratio of magnitudes - same force directions and constant ratio of magnitudes - convert measured properties into dimensionless numbers Non-dimensionalization 1. Convenience in presentation 2. Independent of unit system 3. Used as guide for selection of optimal geometrical and operating conditions

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4 Similarity and non-dimensionalization - Non-dimensinlized Navier-Stokes equations Navier Stokes equations - viscous incompressible flows L – length scale V 0 – velocity scale p 0 – reference pressure g – gravitational acceleration magnitude - Characteristic properties - Dimensionless variables

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5 Common dimensionless parameters Reynolds number – kinematic viscosity V – characteristic velocity L – characteristic length – dynamic viscosity ( = ) - ratio of inertia forces to viscous forces Mach number- used to describe effects of compressibility V – flow velocityc – speed of sound Euler number (pressure coefficient) - ratio of pressure and inertia forces p – pressure – density p ref – reference pressure V – flow velocity Drag coefficient- ratio of drag force and inertia forces F D – drag force – density A – frontal area V – flow velocity Lift coefficient F L – lift force – density A – frontal area V – flow velocity - ratio of lift force and inertia forces

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6 Common dimensionless parameters Prandtl number- ratio of rates of diffusion of momentum and heat due to molecular motions Schmidt number c – molecular diffusivity of a fluid mixture of species in a fluid mixture Froude number - square represents ratio of inertia to gravitational forces (free surface flows) Weber number- ratio of inertia to surface-tension forces Capillary number – kinematic viscosity – thermal diffusivity c p – specific heat under contant pressure – dynamic viscosity k – thermal conductivity - ratio of rates of diffusion of momentum and mass in fluid V – flow velocityL – characteristic length g – gravitational acceleration magnitude – surface tension – density V – flow velocity L – characteristic length V – flow velocity – surface tension – dynamic viscosity - ratio of viscos forces to surface-tension forces

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7 Common dimensionless parameters Cavitation number Nusselt number Biot number Peclet number- ratio of heat convection and heat conduction Grashof number - ratio of total and conductive heat transfer rates in a fluid T – temperature difference – thermal expansion coefficient - ratio of buoyancy forces and viscous forces P v – vapour pressure h – overall heat transfer coefficient k – thermal conductivity of fluid - ratio of heat transfer rates to surrounding fluid and solid interior h – overall heat transfer coefficient k – thermal conductivity of solid V – flow velocity L – characteristic length – thermal diffusivity

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8 Common dimensionless parameters Rayleigh number Richardson number- for density-stratified flows Marangoni number- for convection induced by surface-tension gradients - ratio of potential energy associate with gravity and kinetic energy. - for free thermal convection - for concentration gradients - for temperature gradients

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9 Common dimensionless parameters Taylor number - for rotation flows – rotation rate Rossby number - for rotation flows - ratio of inertia and Coriolis forces Strouhal number - for periodic vortex shedding from bluff objects f – frequency of vortex shedding Knudsen number - for gas – mean free path

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10 Patterns of fluid motion Pathlines - trajectories of individual fluid particles - may be visualized with multiple exposed particle images Stroboscopic illumination of oil drop in laminar pipe flow

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11 Patterns of fluid motion Timelines - each formed by a set of fluid particles at a previous instant in time, and displaced in time as the particles move Consecutive rows of hydrogen bubbles indicating Velocity profiles a flat plat boundary layer - may be visualized with Hydrogen-bubble technique

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12 Patterns of fluid motion Streaklines Vortex flow behind a yawed cylinder visualized with mixture of ink, milk and alcohol - may be visualized with dye lines in water flow - each formed by locus of all fluid particles passing through a fixed position

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13 Patterns of fluid motion Streamlines - instantaneous curves tangent to the velocity vector of flow, i.e. Smoke lines around an airfoil model in a wind-tunnel - may be visualized with smoke lines in stead air flow Trailing vortices behind an inclined delta-wing visualized by a tuft screen - may be visualized with tuft screen method

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14 Patterns of fluid motion In steady flows, pathlines, streaklines and streamlines coincide. In unsteady flows, they may be vastly different. Red – Pathline Blue – Streakline Dash – Sreamline

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15 Homework - Questions and Problems: 10, 14 on page 17 and 18 - Read textbook on page Send MS Word or PDF file to - Due on Monday, 08/29

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