History Fluid

Introduction to Fluid Mechanics*

CFDEFDAFD

57:020 Fluid Mechanics 1

2

01

Re i jD p u uDt

UU U

Fluid Mechanics

• Fluids essential to life• Human body 65% water• Earth’s surface is 2/3 water• Atmosphere extends 17km above the earth’s surface

• History shaped by fluid mechanics• Geomorphology• Human migration and civilization• Modern scientific and mathematical theories and methods• Warfare

• Affects every part of our lives


• Fluids essential to life• Human body 65% water• Earth’s surface is 2/3 water• Atmosphere extends 17km above the earth’s surface

• History shaped by fluid mechanics• Geomorphology• Human migration and civilization• Modern scientific and mathematical theories and methods• Warfare

• Affects every part of our lives

HistoryFaces of Fluid Mechanics


Archimedes(C. 287-212 BC)

Newton(1642-1727)

Leibniz(1646-1716)

Euler(1707-1783)

Navier(1785-1836) Stokes

(1819-1903)Reynolds(1842-1912)

Prandtl(1875-1953)

Bernoulli(1667-1748)

Taylor(1886-1975)

Significance

• Fluids omnipresent• Weather & climate• Vehicles: automobiles, trains, ships, and

planes, etc.• Environment• Physiology and medicine• Sports & recreation• Many other examples!


• Fluids omnipresent• Weather & climate• Vehicles: automobiles, trains, ships, and

planes, etc.• Environment• Physiology and medicine• Sports & recreation• Many other examples!

Weather & Climate

Tornadoes Thunderstorm


HurricanesGlobal Climate

Vehicles

Aircraft Surface ships


SubmarinesHigh-speed rail

Environment

Air pollution River hydraulics


Physiology and Medicine

Blood pump Ventricular assist device


Sports & Recreation

Water sports Offshore racingCycling


Auto racing Surfing

Fluids Engineering

Reality

Fluids Engineering System Components Idealized


22 PBUD

2 2s SM SNU U U

EFD, Mathematical Physics Problem Formulation

AFD, CFD,mU

Analytical Fluid Dynamics

• The theory of mathematical physicsproblem formulation

• Control volume & differential analysis• Exact solutions only exist for simple

geometry and conditions• Approximate solutions for practical

applications• Linear• Empirical relations using EFD data


• The theory of mathematical physicsproblem formulation

• Control volume & differential analysis• Exact solutions only exist for simple

geometry and conditions• Approximate solutions for practical

applications• Linear• Empirical relations using EFD data


• Lecture Part of Fluid Class• Definition and fluids properties• Fluid statics• Fluids in motion• Continuity, momentum, and energy principles• Dimensional analysis and similitude• Surface resistance• Flow in conduits• Drag and lift


• Lecture Part of Fluid Class• Definition and fluids properties• Fluid statics• Fluids in motion• Continuity, momentum, and energy principles• Dimensional analysis and similitude• Surface resistance• Flow in conduits• Drag and lift


Schematic

• Example: laminar pipe flowAssumptions: Fully developed, LowApproach: Simplify momentum equation,integrate, apply boundary conditions todetermine integration constants and useenergy equation to calculate head loss

UD 2000Re

00

0


Exact solution :2 21( ) ( )( )4

pu r R rx

Friction factor:88 64

Re2 2w

dudywf

V V

xgyu

xu

xp

DtDu

2

2

2

2

Head loss:1 2

1 2 fp pz z h

2

2

322fL V LVh fD g D

00

0

Analytical Fluid Dynamics• Example: turbulent flow in smooth pipe( )

0 5y

1 lnu y B

u y

Re 3000

*y yu *u u u *

wu Three layer concept (using dimensional analysis)

1. Laminar sub-layer (viscous shear dominates)

2. Overlap layer (viscous and turbulent shear important)

3. Outer layer (turbulent shear dominates)


1 lnu y B

520 10y

*0

1U u rfu r

510y

*0

*

1 lnu r r r u

Bu

Three layer concept (using dimensional analysis)

1. Laminar sub-layer (viscous shear dominates)

2. Overlap layer (viscous and turbulent shear important)

3. Outer layer (turbulent shear dominates)

Assume log-law is valid across entire pipe:

Integration for average velocity and using EFD data to adjust constants:

1 21 2log Re .8ff

( =0.41, B=5.5)

Analytical Fluid Dynamics• Example: turbulent flow in rough pipe

u u y k

1 ln yuk

Both laminar sublayer and overlap layerare affected by roughnessInner layer:

Outer layer: unaffected

Overlap layer: constant


1 ln yuk

1 2log3.7k D

f 1 ln 8.5 Reyu f

k

Three regimes of flow depending on k+

1. K+<5, hydraulically smooth (no effect of roughness)2. 5 < K+< 70, transitional roughness (Re dependent)3. K+> 70, fully rough (independent Re)

Inner layer:

Outer layer: unaffected

Overlap layer:

Friction factor:

For 3, using EFD data to adjust constants:

constant

Analytical Fluid Dynamics• Example: Moody diagram for turbulent pipe flow

Composite Log-Law for smooth and rough pipes is given by the Moody diagram:


1 1 22

1 2.512log3.7 Rek D

ff

Experimental Fluid Dynamics (EFD)

Definition:Use of experimental methodology and procedures for solving fluidsengineering systems, including full and model scales, large and tabletop facilities, measurement systems (instrumentation, data acquisitionand data reduction), uncertainty analysis, and dimensional analysis andsimilarity.

EFD philosophy:• Decisions on conducting experiments are governed by the ability of the

expected test outcome, to achieve the test objectives within allowableuncertainties.

• Integration of UA into all test phases should be a key part of entireexperimental program• test design• determination of error sources• estimation of uncertainty• documentation of the results


Definition:Use of experimental methodology and procedures for solving fluidsengineering systems, including full and model scales, large and tabletop facilities, measurement systems (instrumentation, data acquisitionand data reduction), uncertainty analysis, and dimensional analysis andsimilarity.

EFD philosophy:• Decisions on conducting experiments are governed by the ability of the

expected test outcome, to achieve the test objectives within allowableuncertainties.

• Integration of UA into all test phases should be a key part of entireexperimental program• test design• determination of error sources• estimation of uncertainty• documentation of the results

Purpose

• Science & Technology: understand and investigate aphenomenon/process, substantiate and validate a theory(hypothesis)

• Research & Development: document a process/system,provide benchmark data (standard procedures,validations), calibrate instruments, equipment, andfacilities

• Industry: design optimization and analysis, provide datafor direct use, product liability, and acceptance

• Teaching: instruction/demonstration


• Science & Technology: understand and investigate aphenomenon/process, substantiate and validate a theory(hypothesis)

• Research & Development: document a process/system,provide benchmark data (standard procedures,validations), calibrate instruments, equipment, andfacilities

• Industry: design optimization and analysis, provide datafor direct use, product liability, and acceptance

• Teaching: instruction/demonstration

Applications of EFD


Application in research & development

Tropic Wind Tunnel has the ability to createtemperatures ranging from 0 to 165 degreesFahrenheit and simulate rain

Application in science & technology

Picture of Karman vortex shedding

Applications of EFD (cont’d)


Example of industrial application

NASA's cryogenic wind tunnel simulates flightconditions for scale models--a critical tool indesigning airplanes.

Application in teaching

Fluid dynamics laboratory

Full and model scale


• Scales: model, and full-scale

• Selection of the model scale: governed by dimensional analysis and similarity

Measurement systems

• Instrumentation• Load cell to measure forces and moments• Pressure transducers• Pitot tubes• Hotwire anemometry• PIV, LDV

• Data acquisition• Serial port devices• Desktop PC’s• Plug-in data acquisition boards• Data Acquisition software - Labview

• Data analysis and data reduction• Data reduction equations• Spectral analysis


• Instrumentation• Load cell to measure forces and moments• Pressure transducers• Pitot tubes• Hotwire anemometry• PIV, LDV

• Data acquisition• Serial port devices• Desktop PC’s• Plug-in data acquisition boards• Data Acquisition software - Labview

• Data analysis and data reduction• Data reduction equations• Spectral analysis

Instrumentation

Load cell

Pitot tube


Load cell

Hotwire 3D - PIV

Data acquisition system

Hardware

Software - Labview


Data reduction methods

EXPERIMENTALRESULTS

EXPERIMENTAL ERROR SOURCES

INDIVIDUALMEASUREMENT

SYSTEMS

MEASUREMENTOF INDIVIDUAL

VARIABLES

DATA REDUCTIONEQUATIONS

TEMPERATUREWATER

TEMPERATUREAIR

fB , P

VENTURIPRESSURE

PIPEPRESSURE

f = F( , , z , Q = )a a

wg D8LQ

Q = F( z )

w

w

T

TB T, Pz

zB , P

f f

SM

SMww

DM

SM

2

2

5

aT

TB T, Paa zSM

z

zB , PDM

DM zDM

= F(T )

( )

w

= F(T )a

zSM i

- zSM j

w

a

• Data reduction equations

• Spectral analysis


EXPERIMENTALRESULTS

EXPERIMENTAL ERROR SOURCES

INDIVIDUALMEASUREMENT

SYSTEMS

MEASUREMENTOF INDIVIDUAL

VARIABLES

DATA REDUCTIONEQUATIONS

TEMPERATUREWATER

TEMPERATUREAIR

fB , P

VENTURIPRESSURE

PIPEPRESSURE

f = F( , , z , Q = )a a

wg D8LQ

Q = F( z )

w

w

T

TB T, Pz

zB , P

f f

SM

SMww

DM

SM

2

2

5

aT

TB T, Paa zSM

z

zB , PDM

DM zDM

= F(T )

( )

w

= F(T )a

zSM i

- zSM j

w

a

Example of data reduction equations

StatSMStagSMa

w zrzgru2

)(

Spectral analysisFFT: Converts a function from amplitude as function

of time to amplitude as function of frequencyAim: To analyze the naturalunsteadiness of the separated flow,around a surface piercingstrut, using FFT.

Fast Fourier Transform


Surface piercing strut Power spectral densityof wave elevation

Free-surface waveelevation contours

FFT of wave elevation

Time history of waveelevation

Uncertainty analysis

r = r (X , X ,......, X )1 2 J

1 2 J

MEASUREMENTOF INDIVIDUALVARIABLES

INDIVIDUALMEASUREMENTSYSTEMS

ELEMENTALERROR SOURCES

DATA REDUCTIONEQUATION

EXPERIMENTALRESULT

XB , P

1

1 1

XB , P

2

2 2

XB , P

J

J J

rB , P

r r

Rigorous methodology for uncertainty assessmentusing statistical and engineering concepts


r = r (X , X ,......, X )1 2 J

1 2 J

MEASUREMENTOF INDIVIDUALVARIABLES

INDIVIDUALMEASUREMENTSYSTEMS

ELEMENTALERROR SOURCES

DATA REDUCTIONEQUATION

EXPERIMENTALRESULT

XB , P

1

1 1

XB , P

2

2 2

XB , P

J

J J

rB , P

r r

Dimensional analysis• Definition : Dimensional analysis is a process of formulating fluid mechanics problems in

in terms of non-dimensional variables and parameters.• Why is it used :

• Reduction in variables ( If F(A1, A2, … , An) = 0, then f( 1, 2, … r < n) = 0,where, F = functional form, Ai = dimensional variables, j = non-dimensionalparameters, m = number of important dimensions, n = number of dimensional variables, r= n – m ). Thereby the number of experiments required to determine f vs. F is reduced.

• Helps in understanding physics• Useful in data analysis and modeling• Enables scaling of different physical dimensions and fluid properties


• Definition : Dimensional analysis is a process of formulating fluid mechanics problems inin terms of non-dimensional variables and parameters.

• Why is it used :• Reduction in variables ( If F(A1, A2, … , An) = 0, then f( 1, 2, … r < n) = 0,

where, F = functional form, Ai = dimensional variables, j = non-dimensionalparameters, m = number of important dimensions, n = number of dimensional variables, r= n – m ). Thereby the number of experiments required to determine f vs. F is reduced.

• Helps in understanding physics• Useful in data analysis and modeling• Enables scaling of different physical dimensions and fluid properties

Example

Vortex shedding behind cylinder

Drag = f(V, L, r, m, c, t, e, T, etc.)

From dimensional analysis,

Examples of dimensionless quantities : Reynolds number, FroudeNumber, Strouhal number, Euler number, etc.

Similarity and model testing• Definition : Flow conditions for a model test are completely similar if all relevantdimensionless parameters have the same corresponding values for model and prototype.

• i model = i prototype i = 1• Enables extrapolation from model to full scale• However, complete similarity usually not possible. Therefore, often it is necessary to

use Re, or Fr, or Ma scaling, i.e., select most important and accommodate othersas best possible.

• Types of similarity:• Geometric Similarity : all body dimensions in all three coordinates have the same

linear-scale ratios.• Kinematic Similarity : homologous (same relative position) particles lie at homologous

points at homologous times.• Dynamic Similarity : in addition to the requirements for kinematic similarity the model

and prototype forces must be in a constant ratio.


• Definition : Flow conditions for a model test are completely similar if all relevantdimensionless parameters have the same corresponding values for model and prototype.

• i model = i prototype i = 1• Enables extrapolation from model to full scale• However, complete similarity usually not possible. Therefore, often it is necessary to

use Re, or Fr, or Ma scaling, i.e., select most important and accommodate othersas best possible.

• Types of similarity:• Geometric Similarity : all body dimensions in all three coordinates have the same

linear-scale ratios.• Kinematic Similarity : homologous (same relative position) particles lie at homologous

points at homologous times.• Dynamic Similarity : in addition to the requirements for kinematic similarity the model

and prototype forces must be in a constant ratio.

EFD process

• “EFD process” is the steps to set up an experiment andtake data

TestSet-up

Facility &conditions

Install model

Preparemeasurement

systems

DataAcquisition

DataReduction

UncertaintyAnalysis

DataAnalysis

Initialize dataacquisitionsoftware

Run tests &acquire data

Store data

Statisticalanalysis

Estimate biaslimits

Compare resultswith benchmarkdata, CFD, and

/or AFD

Evaluate fluidphysics

Calibration

Prepareexperimentalprocedures

Data reductionequations

Estimateprecision limits

Estimate totaluncertainty

Prepare report


TestSet-up

Facility &conditions

Install model

Preparemeasurement

systems

DataAcquisition

DataReduction

UncertaintyAnalysis

DataAnalysis

Initialize dataacquisitionsoftware

Run tests &acquire data

Store data

Statisticalanalysis

Estimate biaslimits

Compare resultswith benchmarkdata, CFD, and

/or AFD

Evaluate fluidphysics

Calibration

Prepareexperimentalprocedures

Data reductionequations

Estimateprecision limits

Estimate totaluncertainty

Prepare report

EFD – “hands on” experience

Lab1: Measurement of density andkinematic viscosity of a fluid Lab2: Measurement of

flow rate, friction factor andvelocity profiles in smooth andrough pipes.

ToScanivalve

Chord-wisePressure

TapsTygonTubing

Load Cell

Load CellL

D


Lab1: Measurement of density andkinematic viscosity of a fluid Lab2: Measurement of

flow rate, friction factor andvelocity profiles in smooth andrough pipes.

Lab3: Measurement of surface pressureDistribution, lift and drag coefficient for an airfoil

ToScanivalve

Chord-wisePressure

TapsTygonTubing

Load Cell

Load CellL

D

Computational Fluid Dynamics• CFD is use of computational methods for

solving fluid engineering systems, includingmodeling (mathematical & Physics) andnumerical methods (solvers, finite differences,and grid generations, etc.).

• Rapid growth in CFD technology since adventof computer


• CFD is use of computational methods forsolving fluid engineering systems, includingmodeling (mathematical & Physics) andnumerical methods (solvers, finite differences,and grid generations, etc.).

• Rapid growth in CFD technology since adventof computer

ENIAC 1, 1946 IBM WorkStation

Purpose• The objective of CFD is to model the continuous fluids

with Partial Differential Equations (PDEs) anddiscretize PDEs into an algebra problem, solve it,validate it and achieve simulation based designinstead of “build & test”

• Simulation of physical fluid phenomena that aredifficult to be measured by experiments: scalesimulations (full-scale ships, airplanes), hazards(explosions,radiations,pollution), physics (weatherprediction, planetary boundary layer, stellarevolution).


• The objective of CFD is to model the continuous fluidswith Partial Differential Equations (PDEs) anddiscretize PDEs into an algebra problem, solve it,validate it and achieve simulation based designinstead of “build & test”

• Simulation of physical fluid phenomena that aredifficult to be measured by experiments: scalesimulations (full-scale ships, airplanes), hazards(explosions,radiations,pollution), physics (weatherprediction, planetary boundary layer, stellarevolution).

Modeling• Mathematical physics problem formulation of fluid

engineering system• Governing equations: Navier-Stokes equations (momentum),

continuity equation, pressure Poisson equation, energyequation, ideal gas law, combustions (chemical reactionequation), multi-phase flows(e.g. Rayleigh equation), andturbulent models (RANS, LES, DES).

• Coordinates: Cartesian, cylindrical and spherical coordinatesresult in different form of governing equations

• Initial conditions(initial guess of the solution) and BoundaryConditions (no-slip wall, free-surface, zero-gradient,symmetry, velocity/pressure inlet/outlet)

• Flow conditions: Geometry approximation, domain, ReynoldsNumber, and Mach Number, etc.


• Mathematical physics problem formulation of fluidengineering system

• Governing equations: Navier-Stokes equations (momentum),continuity equation, pressure Poisson equation, energyequation, ideal gas law, combustions (chemical reactionequation), multi-phase flows(e.g. Rayleigh equation), andturbulent models (RANS, LES, DES).

• Coordinates: Cartesian, cylindrical and spherical coordinatesresult in different form of governing equations

• Initial conditions(initial guess of the solution) and BoundaryConditions (no-slip wall, free-surface, zero-gradient,symmetry, velocity/pressure inlet/outlet)

• Flow conditions: Geometry approximation, domain, ReynoldsNumber, and Mach Number, etc.

Modeling (examples)

Free surface animation for ship inregular waves

Developing flame surface (Bell et al., 2001)


Evolution of a 2D mixing layer laden with particles of StokesNumber 0.3 with respect to the vortex time scale (C.Narayanan)

Modeling (examples, cont’d)

3D vortex shedding behind a circular cylinder(Re=100,DNS,J.Dijkstra)


3D vortex shedding behind a circular cylinder(Re=100,DNS,J.Dijkstra)

DES,Re=105, Iso-surface of Qcriterion (0.4)for turbulentflow aroundNACA12 withangle of attack60 degrees

LES of a turbulent jet. Back wall shows a slice of the dissipation rate and thebottom wall shows a carpet plot of the mixture fraction in a slice through the jetcenterline, Re=21,000 (D. Glaze).

Numerical methods• Finite difference methods:

using numerical scheme toapproximate the exact derivativesin the PDEs

• Finite volume methods• Grid generation: conformal

mapping, algebraic methods anddifferential equation methods

• Grid types: structured,unstructured

• Solvers: direct methods (Cramer’srule, Gauss elimination, LUdecomposition) and iterativemethods (Jacobi, Gauss-Seidel,SOR)

o x

y

i i+1i-1

j+1j

j-1

imax

jmax x

y

21 1

2 2

2i i iP P PPx x

2

1 12 2

2j j jP P PPy y


• Finite difference methods:using numerical scheme toapproximate the exact derivativesin the PDEs

• Finite volume methods• Grid generation: conformal

mapping, algebraic methods anddifferential equation methods

• Grid types: structured,unstructured

• Solvers: direct methods (Cramer’srule, Gauss elimination, LUdecomposition) and iterativemethods (Jacobi, Gauss-Seidel,SOR)

Slice of 3D mesh of a fighter aircraft

CFD process

ConvergentLimit

Contours

Vectors

StreamlinesVerification

Geometry

SelectGeometry

GeometryParameters

Physics Mesh Solve Post-Processing

CompressibleON/OFF

Flowproperties

Unstructured(automatic/

manual)

Steady/Unsteady

Forces Report(lift/drag, shearstress, etc)

XY Plot

Domain Shapeand Size

Heat TransferON/OFF

Structured(automatic/

manual)

Iterations/Steps

Reports


Viscous Model

BoundaryConditions

InitialConditions

ConvergentLimit

Precisions(single/double)

NumericalScheme

Flowproperties

Domain Shapeand Size

Validation

Commercial software• CFD software

1. FLUENT: http://www.fluent.com2. FLOWLAB: http://www.flowlab.fluent.com3. CFDRC: http://www.cfdrc.com4. STAR-CD: http://www.cd-adapco.com5. CFX/AEA: http://www.software.aeat.com/cfx

• Grid Generation software1. Gridgen: http://www.pointwise.com2. GridPro: http://www.gridpro.com

• Visualization software1. Tecplot: http://www.amtec.com2. Fieldview: http://www.ilight.com


• CFD software1. FLUENT: http://www.fluent.com2. FLOWLAB: http://www.flowlab.fluent.com3. CFDRC: http://www.cfdrc.com4. STAR-CD: http://www.cd-adapco.com5. CFX/AEA: http://www.software.aeat.com/cfx

• Grid Generation software1. Gridgen: http://www.pointwise.com2. GridPro: http://www.gridpro.com

• Visualization software1. Tecplot: http://www.amtec.com2. Fieldview: http://www.ilight.com

“Hands-on” experience using CFDEducational Interface (pipe template)


“Hands-on” experience using CFDEducational Interface (airfoil template)


57:020 Fluid Mechanics• Lectures cover basic concepts in fluid statics,

kinematics, and dynamics, control-volume, anddifferential-equation analysis methods. Homeworkassignments, tests, and complementary EFD/CFDlabs

• This class provides an introduction to all three tools:AFD through lecture and CFD and EFD through labs

• ISTUE Teaching Modules(http://www.iihr.uiowa.edu/~istue) (next two slides)


• Lectures cover basic concepts in fluid statics,kinematics, and dynamics, control-volume, anddifferential-equation analysis methods. Homeworkassignments, tests, and complementary EFD/CFDlabs

• This class provides an introduction to all three tools:AFD through lecture and CFD and EFD through labs

• ISTUE Teaching Modules(http://www.iihr.uiowa.edu/~istue) (next two slides)

TM DescriptionsTable 1: ISTUE Teaching Modules for Introductory Level Fluid Mechanics at Iowa

Teaching Modules TM for FluidProperty

TM for Pipe Flow TM for Airfoil Flow

Overall Purpose Hands-on studentexperience with table-topfacility and simple MS forfluid propertymeasurement, includingcomparison manufacturervalues and rigorousimplementation standardEFD UA

Hands-on student experiencewith complementary EFD, CFD,and UA for Introductory PipeFlow, including friction factorand mean velocity measurementsand comparisons benchmarkdata, laminar and turbulent flowCFD simulations, modeling andverification studies, andvalidation using AFD and EFD.

Hands-on student experience withcomplementary EFD, CFD, and UAfor Introductory Airfoil Flow,including lift and drag, surfacepressure, and mean and turbulentwake velocity profile measurementsand comparisons benchmark data,inviscid and turbulent flowsimulations, modeling and verificationstudies, and validation using AFD andEFD.


http://css.engineering.uiowa.edu/~fluids

Hands-on student experience withcomplementary EFD, CFD, and UAfor Introductory Airfoil Flow,including lift and drag, surfacepressure, and mean and turbulentwake velocity profile measurementsand comparisons benchmark data,inviscid and turbulent flowsimulations, modeling and verificationstudies, and validation using AFD andEFD.

Educational Materials FM and EFD lecture; labreport instructions; pre labquestions, and EFDexercise notes.

FM, EFD and CFD lectures; labreport instructions; pre labquestions, and EFD and CFDexercise notes.

FM, EFD and CFD lectures; labreport instructions; pre lab questions,and EFD and CFD exercise notes.

ISTUE ASEE papers Paper 1 Paper2

FM Lecture Introduction to Fluid Mechanics

Lab Report Instructions EFD lab report Instructions CFD lab report Instructions

Continued in next slide…

TM Descriptions, cont’dTeaching Modules TM for Fluid

PropertyTM for Pipe Flow TM for Airfoil Flow

CFD CFD Lecture Introduction to CFD

Exercise Notes None CFD Prelab1PreLab1 QuestionsCFD Lab 1Lab Concepts

EFD Data

CFD Prelab2PreLab 2 QuestionsCFD Lab2Lab Concepts

EFD Data

EFD EFDLecture

EFD and UA

Exercise Notes PreLab1 QuestionsLab1 LectureLab 1 exercise notesLab 1 datareduction sheetLab concepts

PreLab2 QuestionsLab2 LectureLab 2 exercise notesLab2 Data reductionsheet (smooth & rough)Lab concepts

PreLab3 QuestionsLab3 LectureLab 3 exercise notesData Reduction SheetLab concepts


Exercise Notes PreLab1 QuestionsLab1 LectureLab 1 exercise notesLab 1 datareduction sheetLab concepts

PreLab2 QuestionsLab2 LectureLab 2 exercise notesLab2 Data reductionsheet (smooth & rough)Lab concepts

PreLab3 QuestionsLab3 LectureLab 3 exercise notesData Reduction SheetLab concepts

UA(EFD) Lecture EFD UA Report

Exercise Notes Instructions UA

UA(CFD) Lecture

Exercise Notes None

History Fluid

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