Measurements in Fluid Mechanics
058:180 (ME:5180)
Time & Location: 2:30P - 3:20P MWF 3315 SC
Office Hours: 4:00P – 5:00P MWF 223B-5 HL
Instructor: Lichuan Gui
lichuan-gui@uiowa.edu
Phone: 319-384-0594 (Lab), 319-400-5985 (Cell)
http://lcgui.net
Lecture 3. Similarity and flow motion patterns
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Similarity and non-dimensionalization
Different ways to perform measurement:
1. Actual system under actual operation condition - expensive and impossible in most cases
2. Actual system under modified condition
- expensive and impossible in many cases
3. Model system under controlled condition - low cost and possible
Similarity - enable application of measured properties with model system and
modified condition to actual system under actual condition.
Requirements of similarity
1. Geometrically similar - same shape, the same ratios of all corresponding dimensions
2. Kinematically similar - same velocity directions and constant ratio of magnitudes
3. Dynamically similar
- same force directions and constant ratio of magnitudes
Non-dimensionalization - convert measured properties into dimensionless numbers
1. Convenience in presentation
2. Independent of unit system
3. Used as guide for selection of optimal geometrical and operating conditions
3
Similarity and non-dimensionalization
Navier Stokes equations - viscous incompressible flows
𝐷𝑈𝑖
1 𝜕𝑝
𝜕 2 𝑈𝑖 𝜕 2 𝑈𝑖 𝜕 2 𝑈𝑖
= 𝑔𝑖 −
+𝜈
+
+
𝐷𝑡
𝜌 𝜕𝑥𝑖
𝜕𝑥1 2 𝜕𝑥2 2 𝜕𝑥3 2
𝑖 = 1,2,3
- Characteristic properties
L – length scale V0 – velocity scale
- Dimensionless variables
𝑥𝑖
𝑝
∗
∗
𝑥𝑖 =
𝑝 =
𝐿
𝑝0
p0 – reference pressure
𝑉0 𝑡
𝑈𝑖
∗
𝑡 =
𝑈𝑖 =
𝐿
𝑉0
𝜕 2 𝑉0 𝑈𝑖 ∗
𝜕 2 𝑉0 𝑈𝑖 ∗
𝜕 2 𝑉0 𝑈𝑖 ∗
+𝜈
+
+
𝜕 𝐿𝑥1 ∗ 2
𝜕 𝐿𝑥2 ∗ 2
𝜕 𝐿𝑥3 ∗ 2
𝑔𝑖 ∗ =
𝐷 𝑉0 𝑈𝑖 ∗
1 𝜕 𝑝∗ 𝑝0
∗
= 𝑔𝑔𝑖 −
𝐷 𝐿𝑡/𝑉0
𝜌 𝜕 𝐿𝑥𝑖 ∗
𝑔𝑖
𝑔
g – gravitational acceleration magnitude
∗
𝑉0 2 𝐷𝑈𝑖 ∗
𝑝0 𝜕𝑝∗ 𝜈𝑉0 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗
∗
= 𝑔𝑔𝑖 −
+
+
+
𝐿 𝐷𝑡 ∗
𝜌𝐿 𝜕𝑥𝑖 ∗ 𝐿2 𝜕𝑥1 ∗ 2 𝜕𝑥2 ∗ 2 𝜕𝑥3 ∗ 2
𝐷𝑈𝑖 ∗ 𝑔𝐿 ∗
𝑝0 𝜕𝑝∗
𝜈 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗
= 2 𝑔𝑖 −
+
+
+
𝐷𝑡 ∗
𝑉0
𝜌𝑉0 2 𝜕𝑥𝑖 ∗ 𝑉0 𝐿 𝜕𝑥1 ∗ 2 𝜕𝑥2 ∗ 2 𝜕𝑥3 ∗ 2
- Non-dimensinlized Navier-Stokes equations
𝐷𝑈𝑖 ∗
1 ∗ 1
𝜕𝑝∗
1 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗ 𝜕 2 𝑈𝑖 ∗
= 2 𝑔𝑖 − 𝐶𝑝
2+
2+
∗+𝑅
∗
∗
𝐷𝑡 ∗
2
𝜕𝑥
𝜕𝑥
𝜕𝑥
𝜕𝑥3 ∗ 2
𝐹𝑟
𝑖
𝑒
1
2
4
Common dimensionless parameters
Reynolds number - ratio of inertia forces to viscous forces
𝑅𝑒 =
𝜌𝑉𝐿 𝑉𝐿
=
𝜇
𝜈
V – characteristic velocity
 – kinematic viscosity
L – characteristic length
 – dynamic viscosity ( = )
Mach number - used to describe effects of compressibility
𝑉
V – flow velocity
c – speed of sound
𝑀=
𝑐
Euler number (pressure coefficient) - ratio of pressure and inertia forces
𝑝 − 𝑝𝑟𝑒𝑓
p – pressure
pref – reference pressure
𝐶𝑝 (≡ 𝐸𝑢) =
1 2
 – density
V – flow velocity
2 𝜌𝑉
Drag coefficient - ratio of drag force and inertia forces
𝐶𝐷 =
𝐹𝐷
1
2
2 𝜌𝐴𝑉
FD – drag force
A – frontal area
 – density
V – flow velocity
Lift coefficient - ratio of lift force and inertia forces
𝐶𝐿 =
𝐹𝐿
1
2
𝜌𝐴𝑉
2
FL – lift force
A – frontal area
 – density
V – flow velocity
5
Common dimensionless parameters
Prandtl number - ratio of rates of diffusion of momentum and heat due to molecular motions
𝜈 𝑐𝑝 𝜇
𝑃𝑟 = =
𝛾
𝑘
 – thermal diffusivity
 – kinematic viscosity
cp – specific heat under contant pressure
 – dynamic viscosity
k – thermal conductivity
Schmidt number - ratio of rates of diffusion of momentum and mass in fluid
𝜈
𝑆𝑐 =
c – molecular diffusivity of a fluid mixture of species in a fluid mixture
𝛾𝑐
Froude number - square represents ratio of inertia to gravitational forces (free surface flows)
𝑉
V – flow velocity
L – characteristic length
𝐹𝑟 =
𝑔𝐿
g – gravitational acceleration magnitude
Weber number
𝜌𝑉 2 𝐿
𝑊𝑒 =
𝜎
- ratio of inertia to surface-tension forces
 – surface tension  – density
V – flow velocity L – characteristic length
Capillary number - ratio of viscos forces to surface-tension forces
𝜇𝑉
𝐶𝑎 =
𝜎
 – dynamic viscosity V – flow velocity
 – surface tension
6
Common dimensionless parameters
Cavitation number
𝑝 − 𝑝𝑣
Pv – vapour pressure
𝜎𝑐 =
1 2
2 𝜌𝑉
Nusselt number - ratio of total and conductive heat transfer rates in a fluid
ℎ𝐿
𝑁𝑢 =
𝑘
Biot number
𝐵𝑖 =
h – overall heat transfer coefficient
k – thermal conductivity of fluid
- ratio of heat transfer rates to surrounding fluid and solid interior
ℎ𝐿
𝑘
Peclet number
h – overall heat transfer coefficient
k – thermal conductivity of solid
- ratio of heat convection and heat conduction
𝑉𝐿
𝑃𝑒 =
= 𝑅𝑒 𝑃𝑟
𝛾
Grashof number
𝛼𝑔𝐿3 ∆𝑇
𝐺𝑟 =
𝜈2
V – flow velocity L – characteristic length
 – thermal diffusivity
- ratio of buoyancy forces and viscous forces
 – thermal expansion coefficient
T – temperature difference
7
Common dimensionless parameters
Rayleigh number - for free thermal convection
𝛼𝑔𝐿3 ∆𝑇
𝑅𝑎 =
= 𝐺𝑟 𝑃𝑟
𝜈𝛾
Marangoni number - for convection induced by surface-tension gradients
𝜕𝜎 𝜕𝑐 2
𝐿
𝜕𝑐
𝜕𝑥
𝑀𝑎 =
𝜇𝛾𝑐
𝜕𝜎 𝜕𝑇 2
𝐿
𝜕𝑇
𝜕𝑥
𝑀𝑎 =
𝜇𝛾
Richardson number
𝑔𝑝
𝑅𝑖 = − 𝑧
𝑉
𝜌 𝑧
2
- for concentration gradients
- for temperature gradients
- for density-stratified flows
1
𝑔𝑧
−2
= − 2 = −𝐹𝑟
𝑉
- ratio of potential energy associate with gravity and kinetic energy.
8
Common dimensionless parameters
- for rotation flows
Taylor number
Ω2 𝐿4
𝑇𝑎 = 2
𝜈
 – rotation rate
- for rotation flows
Rossby number
𝑅𝑜 =
𝑉
𝐿
- ratio of inertia and Coriolis forces
Strouhal number - for periodic vortex shedding from bluff objects
𝑆=
𝑓𝐿
𝑉
f – frequency of vortex shedding
Knudsen number - for gas
𝜆
𝐾𝑛 =
𝐿
 – mean free path
9
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
10
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
- may be visualized with Hydrogen-bubble technique
Consecutive rows of hydrogen bubbles indicating
Velocity profiles a flat plat boundary layer
11
Patterns of fluid motion
Streaklines - each formed by locus of all fluid particles passing through a fixed position
- may be visualized with dye lines in water flow
Vortex flow behind a yawed cylinder visualized
with mixture of ink, milk and alcohol
12
Patterns of fluid motion
Streamlines
- instantaneous curves tangent to the velocity vector of flow, i.e.
- may be visualized with smoke lines in stead air flow
𝑑𝑥1 𝑑𝑥2 𝑑𝑥3
=
=
𝑈1
𝑈2
𝑈3
- may be visualized with tuft screen method
Smoke lines around an airfoil model
in a wind-tunnel
Trailing vortices behind an inclined
delta-wing visualized by a tuft screen
13
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
14
Homework
- Read textbook 1.6-1.7 on page 11-17
- Questions and Problems: one of 10, 14 on page 17 and 18
- Due on Wednesday, 08/29
- Send MS Word or PDF file to lichuan-gui@uiowa.edu
15