Basic Fluid Mechanics for Geologists
Training Course on Fluid Physics in Geological Environments Jointly Organized
by C-MMACS and JNCASR, BangaloreJanuary 19 - 23, 2004
Raghuraman N. GovardhanMechanical Engineering
Indian Institute of Science, Bangalore
Outline of Lecture
• Fundamental concepts & Fluid Statics
- Fluid definition, Continuum, description and classification of fluid motions, viscosity and other basics, Fluid statics in incompressible and compressible fluids
• Governing equations for fluid flow & Applications
- Integral & differential form of the governing equations, Pipe flow, friction losses, flow measurement & Rainfall-run-off modelling
Fundamental concepts
- Definition of a fluid- Continuum- Velocity field (streamlines)- Thermodynamic properties (p, T, ρ)- Viscosity- Reynolds number- Non-Newtonian fluids
Fluid Statics
What is a fluid ?
Definitions of fluid on the Web:
• Any substance that FLOWs, such as a liquid or gas.
• A substance that is either a liquid or a gas
• Fluids differ from solids in that they cannot resist changes in their shape when acted upon by a force.
• Anything that flows, either liquid or gas. Some solids can also exhibit fluid behavior over time.
• any substance that cannot maintain its own shape
Not directly relevant:
• in cash or easily convertible to cash; "liquid (or fluid) assets"
Fluid – Solid : Distinction
Reaction to an applied shear
SOLID
FLUID
F
F
θθθθ(t)
F
F
θθθθ
flow
Staticdeformation
Fluid Definition
A fluid cannot resist a shear stress by a static deformation.
Fluid includes Liquids and Gases –
Distinction between the two comes from the effect of cohesive molecular forces.
Fluid as a Continuum
Before defining Fluid property like density, pressure at a “point” :
Note:- Fluids are aggregations of molecules- Moving freely relative to each other (unlike a solid)
Fluid density : mass / unit volume depends on elemental volume
= → V
mVV δ
δρ δδ *lim
*Vδ
*Vδ
Density at a “point”
*Vδ
*Vδ3910* mmV −≈δ
ρ
Microscopicuncertainty
Macroscopicuncertainty
Vm
δδ
= → V
mVV δ
δρ δδ *lim
Density field
Most problems are concerned with physical dimensions much larger than this limiting volume
So density is essentially a point function and can be thought of as a continuum
),,,( tzyxρρ =
Velocity field
Perhaps the most important property in a flow is the velocity vector field:
),,,( tzyxVV =
wkvjui ˆˆ ++=)
u, v, w are f(x,y,z,t)
(taken from www.amtec.com)
Eulerian representation
Velocity field
Lagrangian representation
Flow quantities are here defined as functions of time and the choice of a material element of fluid
),( taVV =
awhere = location of fluid particle at t=0
Lagrangian specification describes the dynamical history of the selected fluid element
Material derivative
4ρ
Density variation following a fluid particle
3ρ1ρ 2ρ
Convectivederivative
Localderivative
StreamlineVisual representation of a velocity or flow field: Streamlines
Streamlines are lines drawn in the flow field so that at a given instant they are tangent to the velocity vector at every point in the flow.
Local velocity vector
Thermodynamic properties
- Pressure (p)- Density (ρ)- Temperature (T)
When work, heat and energy balances are treated
- Internal energy (e)
- Enthalpy (h = u + p/ρ)- Entropy (s)- Specfic heats (Cp & Cv)
Transport properties
- Viscosity (µ)- Thermal conductivity (k)
All these are functions of (x,y,z,t)
Viscosity
Shear stress causes continuous shear deformation in a fluid.
Newtonian fluidshow a linear relation between applied shear (ττττ) and resulting
strain rate (dθ/dt)
τ ∝∝∝∝ (dθ/dt)
τ = µ µ µ µ (dθ/dt) τ = µ µ µ µ (du/dy)
µ = viscosity coefficient
ττττ
ττττ
θθθθ(t)u
u + du
dy
Newtonian fluid
Viscosity coefficient (µ)
Kinematic viscosity (ν) = µ/ρ
µ kg/(m s) ν kg/(m s)Air: 1.8 x 10-5 1.5 x 10-5
Water: 1.0 x 10-3 1.0 x 10-6
Newtonian fluid (µ) depends on (T, P)• Generally variation with pressure is weak
less than 10% for 50 times increase in P for air
• Temperature has a strong effect
Factors affecting viscosity
LIQUIDS
• decreases with increasing temperature, since the interatomic forces weaken
• increases under very high pressures.
GASES
• increases with increasing temperature, since the rate of interatomic collisions increases and
• is typically independent of pressure and density.
( ) ( )200
0
ln TTcTTba ++=
µµ
7.0
00
=
TT
µµ
Effect of temperature on viscosity
Non-Newtonian fluids
Do not follow linear relationship between applied shear (ττττ) and resulting strain rate (dθ/dt)
ττττ
(dθ/dt)
Newtonian
Pseudoplastic
Dilatant
Bingham Plastic
Yieldstress
Plastic
Rheology
Magma Viscosity
Magmatic liquid viscosity depends on:
composition (especially Si), temperature, time and pressure, each of which effect the melt structure.
It is possible to estimate the viscosity of a magmaticliquid at temperatures well above liquidustemperatures (that is, temperatures at which only liquid is present) from chemical compositions and empirical extrapolation of experimental data. The range of temperatures of naturally flowing magmas, however, is near or within the crystallization interval, where stress-strain relationships are not linear (that is, they are crystal-liquid mixtures and show Bingham behavior). Under such conditions, the only way to predict viscosities is by analogy with similar compositions investigated experimentally.
Information source and for further reading:http://www.geo.ua.edu/volcanology/lecture_notes_files/controls_on_magma_viscos.html
Magma viscositySilica compositionThe strong dependence of viscosity of molten silicateson Si content can be illustrated by those of various Na-Si-O compounds:
0.24:1:4
1.52:1:3
281:1:2.5
10100:1:2
(poise)Na:Si:O
The decrease in viscosity can be attributed to a reduction in the proportion of framework silica tetrahedral, and therefore strong Si-O bonds in the magma.
TemperatureTemperatures of erupting magmas normally fallbetween 700° and 1200°C; lower values, observed in partly crystallized lavas, probably correspond to the limiting conditions under which magmas flow. Magmas do not crystallize instantaneously, but over an interval of temperature.
Temperature has a strong influence on viscosity: as temperature increases viscosity decreases, an effect particularly evident in the behavior of lava flows. As lavas flow away from their source or vent, they lose heat by radiation and conduction, so that their viscosity steadily increases.For example:a) measured viscosity of a Mauna Loa flowincreased 2-fold over a 12-mile distance from vent;b) measured viscosity of a small flow fromMt. Etna increased 375-fold in a distance of about 1500 feet.
The decrease in viscosity can be attributed toan increase in distance between cations and anions, and therefore, a decreasein Si-O bond strength.
Magma viscosity
Time
At temperatures below the beginning of crystallization viscosity also increases with time. If magma is undisturbed at a constant temperature, its viscosity may continue to increase for many hours before it reaches a steady value. The viscosity increases with time results partly an increasing proportion of crystals (which raise the effective magma viscosityby their interference in melt flow), and partly from increasing orderingandpolymerizing (linking) of the framework tetrahedra.
PressureThe effect of pressure is relatively unknown, but viscosity appears to decrease with increasing pressure at least at temperatures above the liquidus. As pressure increases at constant temperature, the rate at which viscosity decreases is less in basaltic magma than that in andesitic magma. The viscosity decrease may be related to a change in the coordination number of Al from 4 to 6 in the melt, thereby reducing the number of framework-forming tetrahedra.
• Bubble content
• Crystal content
Fluid density
Liquids
Water ~ 1000 kg/m3
- Density in liquids is nearly constant - water density increases by 1% if the pressure is increased by a factor of 220 !- for a temperature increase of 100 K, density decreases by 5%
- Magma - Magma densities range from about 2200 kg/m3 to 2800 kg/m3, illustrating a close density-melt composition relationship. Magma density decreases with increasing temperature and gas content. These densities increase a few percent between liquid and crystalline states.
Gases
Air ~ 1.2 kg/m3
- Density is highly variable- ideal gas law : p = ρRT (perfect gas law)
- real gas: at low temperatures & high pressure – intermolecular forces become important
Reynolds number
Dimensionless parameter correlating viscous behaviour
forcesViscousforcesInertial=
Low Re:
- Viscous forces dominate- Flow is “Laminar”
- flow structure is characterized by smooth motion in laminae or layers
High Re:
- Viscous forces are very small - Flow is “Turbulent”
- flow structure is characterized by random three-dimensional motions of fluid particles
νµρ VLVL ==Re
Low & High Reynolds number
Low Re
ViscousLaminar
High Re
InertialTurbulent
νµρ VLVL ==Re
Reynolds pipe experiment
Laminar
Transition : Re ~ 2000
Turbulent
Low & High Reynolds number
Low Re
ViscousLaminar
High Re
InertialTurbulent
Classification of flows
Continuum
Fluid Mechanics
Inviscid
µ = 0
TurbulentHigh Re
LaminarLow Re
Viscous
Compressible Incompressible
Fluid staticsFluids by definition cannot resist shear
⇒ in fluids at rest there can be no shear
Only stresses are normal pressure forces
Net force in x-direction
dzdyp
z
x
y
dzdydxxpdFx ∂
∂−=
dzdydxxpp )(
∂∂+
Hydrostatic equation
dzdydxzpk
ypj
xpiFd pressure
∂∂+
∂∂+
∂∂−= ˆˆˆ
( )pdzdydx
Fdfd pressure
pressure ∇−==
gfd gravity ρ=
For equilibrium the pressure gradient force has to be balanced by the body forces (like gravity)
per unit volume
0=+ pressuregravity fdfd
gp ρ=∇
gzp ρ−=
∂∂
0=∂∂
yp
0=∂∂
xp
zgp ∆−=∆ ρif incompressible, ρ=constant, then
AtmosphereFor the purpose of calculating the pressure and density of the atmosphere, we can regard air as a perfect gas obeying the perfect gas law equation. Substituting the perfect gas law into the differential equation of force balance, and integrating, we find an expression for the pressure:
where p0 is the atmospheric pressure at the earth's surface, z=0. The density ρ may be found readily by dividing equation by RT(z).
Note that the atmospheric absolute temperature T(z) must be known as a function of altitude in order to evaluate the integral.
Atmosphere
Fluid statics - atmosphere
Can determine pressure, density as functions of altitude from the “hydrostatic equation”.
International Standard Atmosphere
Created by ICAO (International Civil Aviation Organization)
The ISA is a "model" of the atmosphere, designed to allow for standardized comparison of conditions on a given day.
Based on the International Standard Atmosphere:for dry air (ICAO 1964):
1. At mean sea level pressure=101325 Pa, temp=15 deg C
Atmosphere - pressure
Linearly varying temperature
Constant temperature region
Standard Atmosphere
SECOND SESSION
Outline of Lecture
• Fundamental concepts & Fluid Statics
- Fluid definition, Continuum, description and classification of fluid motions, viscosity and other basics, Fluid statics in incompressible and compressible fluids
• Governing equations for fluid flow & Applications
- Integral & differential form of the governing equations,- Pipe flow, friction losses, flow measurement
& Rainfall-run-off modelling
Approach
Fluid flow analysis:
• Control volume, or large-scale
• Differential, or small-scale
Flow must satisfy the three basic laws of mechanics:
• Conservation of mass (continuity)
• Conservation of Linear momentum (Newton’s second law)
• Conservation of energy (first law of thermodynamics)
System
All the laws of mechanics are written for a system, which is defined as an arbitrary quantity of mass of fixed identity.
Mass:(dmsys/dt) = 0
Momentum:F = m (dV/dt)
Energy:
dQ/dt – dW/dt = dE/dt
Difficult to follow a fluid of fixed identity. Easier to look at a specific region ….
Control Volume
Write the basic laws for a specific region:
Consider a fixed Control Volume:
Let B =any property (mass, momentum, energy)
β = B per unit mass = dB/dm
dAnudVdtd
dtdB
CSCV
sys )( ⋅+
= ∫∫∫∫∫ ρββρ
Flux out of the CV
Increase within CV
nu
Integral form
Mass:
B = m β = dm/dm =1
From system, (dmsys/dt) = 0
dAnudVdtd
dtdm
CSCV
sys )( ⋅+
= ∫∫∫∫∫ ρρ
0)( =⋅+
∫∫∫∫∫ dAnudV
dtd
CSCV
ρρ
Integral form
Momentum:
From system,
umB =u=β
dAnuudVudtdF
CSCV
)( ⋅+
= ∫∫∫∫∫∑ ρρ
dAnuudVudtd
dtumd
FCSCV
sys )()(
⋅⋅+
⋅== ∫∫∫∫∫∑ ρρ
dAnuudVudtd
dtumd
CSCV
sys )()(
⋅⋅+
⋅= ∫∫∫∫∫ ρρ
Integral form
Energy:
From system,
EB =e=β
e = einternal + ekinetic + epotential + eelectrostatic
dAnudVdtd
dtdW
dtdQ
CSCV
)( ⋅+
=− ∫∫∫∫∫ βρβρ
dAnudVdtd
dtEd
dtdW
dtdQ
CSCV
sys )()(
⋅+
==− ∫∫∫∫∫ βρβρ
dAnudVdtd
dtEd
CSCV
sys )()(
⋅+
= ∫∫∫∫∫ βρβρ
Control Volume Analysis0=⋅nu
0)( =⋅+
∫∫∫∫∫ dAnudV
dtd
CSCV
ρρ
dAnuudVudtdF
CSCV
)( ⋅+
= ∫∫∫∫∫∑ ρρ
Control Volume
0=⋅nu
1u 2u
Consider steady flow of water through a bend,
Mass:
Steady
2211 AuAu =
Momentum
Steady
)( 22
212
1 AuAuFy
ρρ +−=∑ 0=∑ xF
y
x
Differential form
0)( =⋅+
∫∫∫∫∫ dAnudV
dtd
CSCV
ρρ
0)( =
⋅∇+
∂∂
∫∫∫ dVutCV
ρρ
0)( =⋅∇+∂∂ u
tρρ
Can be written in the form:
Mass
Valid for any volume V, possible only if:
0)( =⋅∇+ uDtD ρρ
)( ρρρ ∇⋅+∂∂= u
tDtD
Integral form :
(or)
Differential form
τρρ ⋅∇+∇−=
∇⋅+
∂∂ pguu
tu
gravitationalforce
Pressuregradient
viscousforce
Momentum
If we assume Newtonian fluid
upguutu 2∇+∇−=
∇⋅+
∂∂ µρρ
⋅∇∇+∇+∇−=
∇⋅+
∂∂ )(
312 uupguu
tu µρρ
ρ = constant
Navier-Stokes equation
Differential form
heatconduction
ViscousDissipation
Energy
steady motion of a frictionless non- conducting fluid
B = constant
Bernoulli equation
(for material fluid element)
Bernoulli equation
Commonly used form in pipe flows (in terms of head):
pumpturbinefriction hhhzg
Vg
pzg
Vg
p −++
++=
++ 2
222
1
211
22 ρρ
1
2
Flow measurement
Flow measurement
Fox & McDonald
Pipe flow – Major loss
Major losses: Frictional losses in piping system
P1
P2R: radius, D: diameterL: pipe lengthτw: wall shear stress
Consider a laminar, fully developed circular pipe flow
p P+dp
τw[ ( )]( ) ( ) ,
,
p p dp R R dx
dpR
dx
p p p hg
LD
f LD
Vg
w
w
Lw
− + =
− =
= − = = FHIK= FHIKFHGIKJ
π τ π
τ
γ γτρ
2
1 22
2
2
42
Pressure force balances frictional force
integrate from 1 to 2
where f is defined as frictional factor characterizingpressure loss due to pipe wall shear stress
∆
=
=
gV
DLf
DL
gh w
L 24 2
ρρρρττττ
=
=
gV
DLf
DL
gh w
L 24 2
ρρρρττττ
W hen the pipe flow is lam inar, it can be show n (not here) that
by recognizing that as R eynolds num ber
Therefore, frictional factor is a function of the R eynolds num ber
S im ilarly, for a turbulent flow , f = function of R eynolds num ber also. A nother param eter that influences the friction is the surface
roughness as relativeto the pipe diam eter D
Such that D
P ipe frictional factor is a function of pipe R eynolds
num ber and the relative roughness of pipe.This relation is sketched in the M oody diagram as show n in the follow ing page.The diagram show s f as a function of the R eynolds num ber (R e), w ith a series of
param etric curves related to the relative roughness D
fVD
VD
f
f F
f F
= =
=
=
= FH IK
FHIK
64
64
µρ
ρµ
ε
ε
ε
, R e ,
R e,
(R e)
.
R e, :
.
=
DFf εRe,
Dε
Pipe flow
Losses in Pipe Flows
Major Losses: due to friction, significant head loss is associated with the straight portions of pipe flows. This loss can be calculated using the Moody chart.
Minor Losses: Additional components (valves, bends, tees, contractions, etc) inpipe flows also contribute to the total head loss of the system. Their contributionsare generally termed minor losses.
The head losses and pressure drops can be characterized by using the loss coefficient,KL, which is defined as
One of the example of minor losses is the entrance flow loss. A typical flow pattern for flow entering a sharp-edged entrance is shown in the following page. A vena contracta region is formed at the inlet because the fluid can not turn a sharp corner.Flow separation and associated viscous effects will tend to decrease the flow energy;the phenomenon is fairly complicated. To simplify the analysis, a head loss and the associated loss coefficient are used in the extended Bernoulli’s equation to take intoconsideration this effect as described in the next page.
K hV g
p p K VLL
LV= = =2 2
12
2
2 12/
,∆ ∆ρ ρ so that
Minor Loss
V2 V3
V1
ghK
zzgK
VVppp
gVKhz
gVphz
gVp
LL
LLL
+=−
+=≈==
=++=−++
∞ 12)(2(
11,0,
2,
22:Equation sBernoulli' Extended
313131
23
3
233
1
211
γγ
(1/2)ρV22 (1/2)ρV3
2
KL(1/2)ρV32
p→p∞
gzVp ρρ ++2
2
Open channel flow
Rh = A/P
A= cross-sectional area
P =“wetted perimeter”
Hydraulic radius
Open channel
Rh = A/P
P =“wetted perimeter”
Hydraulic radius
Open Channel
Rainfall/Runoff Relationships
Depending on the nature of precipitation, soil type, moisture history, etc., an ever-varying portion of the precipitation becomes runoff, moving via overland flow into stream channels
• these stormflow events are typically recorded as hydrographs of discharge, or stream height (stage) vs. time
• A hydrograph is a plot of discharge vs. time at any point of interest in a watershed, usually its outlet. Hydrographs are the ultimate measure of a watershed's response to precipitation events
• for any storm, the initial precipitation does not contribute directly to flow at the outlet, instead it is stored or absorbed. This is termed the initial abstraction (Fig. 2), precipitation that falls before the storm hydrograph begins.
• direct runoff is that portion of the precipitation that moves directly into the channel, appearing in the hydrograph
• losses represent storage of precipitation upstream from the outlet after the storm hydrograph begins. Often lumped with abstraction.
• excess precipitation runs off, and forms the storm hydrograph
GEOS 4310/5310 Lecture Notes, Fall 2002Dr. T. Brikowski, UTD
Rainfall/Runoff
Idealized model: Hortonian Overland Flow • when precipitation exceeds infiltration capacity of soil,
Hortonian overland flow results • infiltration rate declines exponentially as soil saturates • Horton model (1940) assumes uniform infiltration capacity for
watershed
Overland Flow (OF) actually unimportant in most watersheds (studies performed in 1960's) • often only 10% of a watershed regularly supplies OF during a
storm event • in those areas, often only 10-30% of precip. becomes OF • vegetation also absorbs much precip. • well-vegetated watersheds in humid climate rarely show OF • arid zones (sparse vegetation) during high-intensity rainfall
will show Hortonian OF
Rainfall/RunoffBest model: variable source area
• interflow (subsurface stormflow) is prime contributor to streamflow• OF is important near streams, where slopes become saturated by
interflow • return flow (emergence of interflow) also important near streams
Baseflow Characteristics
storm hydrograph has two contributions
• ``Fast'' response: overland flow, interflow, etc. direct runoff
• Baseflow: discharge of groundwater flow to stream
hydrograph separation helps distinguish these components
Gaining/losing stream
Flash flood prediction
Starting from precipitation …
…. Storm hydrograph
actual discharge volume flow rate (Q) and height (d) in discharge channel
Outline of Lecture
• Fundamental concepts & Fluid Statics- Fluid definition, Continuum, description and classification of fluid
motions, viscosity and other basics, Fluid statics in incompressible and compressible fluids
• Governing equations for fluid flow & Applications- Control Volume analysis using basic laws of Fluid Mechanics,
Pipe flow, friction losses, flow measurement & Rainfall-run-off-modelling
