fluid mechanics slides chapter 1

Universität Stuttgart Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierunglehre/VL-HM/E-HYDRO-LECTURE-NOTES/HYDROFOLIEN/einf_F.tex Introduction to hydromechanics

Solid bodies, fluids

Solid bodies can resist to shear stress,fluids are set in motion immediately by the impact of shear stress.

Fluids: liquids and gasesCaused by gravity, liquids form free surfaces,gases expand themselves.

��

. – p.1/17


Fluids as continua

Definition of the fluid density

= lim∆V →∆V0

∆m

∆V(1)

If the characteristic length of ∆Vis in the order of magnitude of thedistance between the molecules:→

∆m∆V

discontinuous,→ density cannot be determineduniquely

=ρ i

∆m

/ ∆

0

∆

V

V0

fluid as continuum

homogeneousfluid

inhomogeneousfluid

volume

Hydromechanics:consideration of the fluids as continua

. – p.2/17


Primary and secondary dimensions

primary dimensions

mass (M ) kglength (L) mtime (T ) stemperature (Θ) K

All other dimensions can bederived thereof.

e.g., 2. Newtonian Axiom:

force =mass × acceleration

(F = MLT−2)

secondary dimensions

force (MLT−2) Narea (L2) m2

volume (L3) m3

velocity (LT−1) m/sacceleration (LT−2) m/s2

pressure, stress (ML−1T−2) N/m2=Paangular velocity (T−1) 1/senergy, heat, work (ML2T−2) J=N/mpower (ML2T−3) W = J/sdensity (ML−3) kg/m3

dynamic viscosity (ML−1T−1) kg/(m s)kinematic viscosity (L2T−1) m2/s

. – p.3/17


Eulerian/Lagrangian description

. – p.4/17


Eulerian/Lagrangian description

Lagrange:

describing the path of a fluidparticle as a function of time

application predominantly insolid mechanics

velocity change of a fluid particle:∂v∂t

Euler:

describing the flow field in a fixedcontrol volume

field properties at a fixed point inspacevelocity field v(x,y,z)pressure field p(x,y,z)

velocity change in a controlvolume:dvx

dt= ∂vx

∂t+ vx

dvdx

. – p.5/17


Fluid properties and equations of state

velocity field:

v =

0

B

@

vx

vy

vz

1

C

Aor

0

B

@

u

v

w

1

C

A

strongly coupled with thethermodynamic dimensions pressure p,temperature T and density (orspecific volume v)

coupled by equations of state

pressure: unit: Pa = Nm−2, bar, atmThe pressure gradient is the drivingforce of fluid flow.Usually, absolute values of thepressure are of no consideration(exception: cavitation).

temperature: unit: ◦C, KConversion Celsius scale in absolutetemperature: T [K] = T [◦C] + 273.15

density: unit: kg/m3

Liquids are by approximationincompressible ( ≈ const.).Density of gases is strongly dependenton pressure and temperature.

. – p.6/17


Equations of state

ideal gases:

p =n Ru T

V= R T

p [Pa] pressuren [mol] amount of substanceV [m3] volumeRu gas const. (8.314 J/(mol K))T [K] absolute temperatureR individual gas const.

R = Ru/Mz.B. Rair ≈ 287 J/(kg K)

M [ kgmol ] molecular weight

cp − cv = R

cp [ Jkg K ] spec. heat capacity

at constant pressure

cv [ Jkg K ] spec. heat capacity

at constant volume

There is no perfect liquid law!For most applications inhydromechanics, it is valid, e.g.

≈ const. cp ≈ cv ≈ const.

. – p.7/17


Viscosity

��

τ xy

τ xy

τ xyτ xy

τ xy

αddt

α

= f( )

x

y

x

y

velocityprofile

αu(y)

at the wallno−slip condition

viscous fluidsolid

αd=

dydu

shear stress, viscosity, velocity gradient: τ = µ dudy

No-slip condition:flow velocity of real fluids at fixed walls equals zero→ finite values for the velocity gradient and the shear stress

. – p.8/17


Viscosity

Newtonian fluids:

τ = τf + µ“

dudy

”n

Newtonian fluids (n=1)

plastic fluids (n<1)

dilatant fluids (n>1)(e.g., w

ater, air)

(e.g., toothpaste)

ideal Bingham plastic fluids (n=1)

(e.g., gelatine)

(e.g., certain varnishes)

τf

τ

d ud y

relation between shear stress andviscosity for Newtonian fluids:

τij = µ“

∂vi

∂xj+

∂vj

∂xi

”

ideal fluids:

viscosity effects are neglected(assumption: µ = 0).

only inertial, pressure, andgravitational forces

no-slip condition is not valid

infinite velocity gradient at fixedwalls: du

dy= ∞

→ simplified relations

. – p.9/17


Reynolds number

Dimensionless number to characterize the viscous behavior of flowing Newtonian flu-ids:

Re =vL

ν:

inertia effectsviscosity effects

v and L: characteristic velocity and characteristic length scale of a flow (for example:diameter of a duct)

Re very small:→ creeping flow with negligible inertia effects (e.g., groundwater flow)

Re moderately large:→ laminar flow

Re very large:→ turbulent flow

. – p.10/17


Vapor pressure and cavitation

saturated vapor pressure curve ofwater

T [ C]

p [bar]

0

0.5

1

1.5

2

0 50 100 150

vapor pressure is dependent ontemperature(at 100 ◦C: pv = patm)

Forming of steam bubbles due toa pressure lowering down to thevapor pressure level:→ cavitation

Abrupt collapsing of the steambubbles has strong materialdestructive effect.

Cavitation is to be avoided byappropriate measures of thewater engineer.Endangered are buildings withlocally very high flow velocitiesinside or around, e.g. ductcontractions, turbine buckets etc.

. – p.11/17


Surface tension and capillarity

p

p

θ

2 r

p -

hrcos θ2 γ

∆p =2γ cos θ

r

γ [ Jm2 = N

m] surface tension

θ [◦] contact angle

θ < 90◦: wetting liquide.g. water

θ > 90◦: non-wetting liquide.g. mercury

capillary height:

h =2γ cos θ

gr

. – p.12/17


Compressibility

Hooke’s Law:

σ = Eε

analogous:volume reduction due to pressureincrease in a fluid:

∆V

V0

= −

∆p

E

elastic modulus of water:≈ 2 · 109 N/m2

→ compressibility negligible for manyapplications

For gases with small volume changes itis valid:

pV = const. (Boyle-Mariotte)

(p0 + ∆p)(V0 + ∆V ) = p0V0

or p0∆V + ∆pV0 + ∆p∆V = 0

neglecting ∆p∆V yields:

∆V

V0

≈ −

∆p

p0

Elastic modulus of a gas correspondsto its pressure p0. Estimation of theinfluence of compressibility by meansof the Mach number Ma = v/a.

. – p.13/17


Speed of sound

Definition of the speed of sound a (propagation velocity of small pressure disturbances- sound waves):

a2 =cp

cv

„

∂p

∂

«

T

ideal gases:

a2

id.Gas =cp

cv

RT

example: air (20 ◦C = 293.15 K)

aa =q

1005

718· 287 · 293.15 = 343 m/s

water:

a2

w =Ew

w

→ aw ≈ 1420 m/s

. – p.14/17


Forms of energy

total energy of a system:

E = m(u +1

2v2 + gz)

mu: thermal energy, internalenergy of a system (total energyof the molecules)

1

2mv2: kinetic energy, work by

the inertial force, accelerationwork

mgz: potential energy, work to liftup a mass in a gravitational field

energy transfer by

work

heat

. – p.15/17


Summary

continuum approach: fluids are considered as continua

primary dimensions: mass, length, time, temperatureall other dimensions can be derived thereof

descriptions: Eulerian (control volumes) and Lagrangian (particle motion)

properties of the fluids and equations of state:functional relations for pressure, temperature, density etc.viscosity, Newtonian fluids, Reynolds number

vapor pressure: cavitation, danger for water engineering constructions

surface tension and capillarity

compressibility: water mostly considered as incompressible, gas flow dependon their Mach number

forms of energy: internal energy, kinetic energy, and potential energy

. – p.16/17


Balance equations and methods in hydromechanics

balances

mass(continuity)

momentum(2. Newtonian Axiom)

energy(1. fundamental theorem ofthermodynamics)

equations of state

boundary conditions

methods for the description andanalysis of fluid flows:

integral description in a controlvolume

differential description of a fluidelement (infinitesimal system)

experimental investigation anddimensional analysis(not topic of this course)

. – p.17/17

fluid mechanics slides chapter 1

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