Review Of Basic Fluid Mechanics
Hydromechanics VVR090
Hydrostatics
Fluids at rest: • the pressure at any point in a fluid is the same in every direction • in a continuous fluid with constant density the pressure increases linearly with depth and the pressure is the same along horizontal planes
dp = −γ = −ρg dz
Constant density:
p − pzzh11=−γ() − =γ
1 Pressure Definitions
p Pressure head: h = γ
(pressure may be expressed in height of a fluid column; e.g., mm Hg, m H2O)
p Static head (piezometric head): + z γ
p Hydrostatic pressure distribution: +=z const. γ
Force on a plane area
Total force: FhA= γ c
Ic Center of pressure: yypc=+ yc A
2 Force on Curved Surfaces
Look at horizontal (Fx) and vertical components (Fz) separately.
Total force: 22 Ftot=+FF x z
Flowing Fluid
Basic equations for conservation of:
• mass (continuity equation) • energy (involves potential and kinetic energy + work) • momentum (involves momentum fluxes + forces)
Analysis through control volumes
3 Classification of flow types
• 1-D, 2-D, and 3-D • real and ideal fluid • incompressible - compressible • steady – unsteady • laminar – turbulent • established – unestablished • uniform – non-uniform • subcritical – supercritical • subsonic - supersonic
Continuity Equation
1-D, steady, compressible: ρ111AVAV=ρ 2 22
1-D, steady, incompressible: A11VAV= 2 2
stream tube
4 Energy Equation
With energy losses (hL) and energy input (hM):
⎛⎞⎛⎞pV11 p 2 V 2 ⎜⎟⎜⎟++zh12 +M = ++ z ++ hL ⎝⎠⎝⎠γγ1222gg
No losses:
⎛⎞pV ⎜⎟++z = const. Bernoulli’s equation ⎝⎠γ 2g
Definitions in Fluid Flow
• Pressure head (p/g) • Elevation head (z) • Static head (piezometric head) (p/g + z) • Velocity head (V2/2g) • Total (energy) head (p/g + z + V2/2g)
• Hydraulic grade line Plane and parallell • Energy line streamlines
p +=z const. γ
5 Energy and Hydraulic Grade Lines
exit loss
flow between reservoirs
Momentum Equations
Newton’s second law (vector relationship)
∑ FQVVx =ρ()21xx −
∑ FQVVyyy=ρ()21 −
∑ FQVVzzz=ρ()21 −
Laminar – turbulent flow Characterized by Reynolds number:
DVρ DV Re == μν
Critical Re-value for transition to turbulence: 2000 (pipe flow)
Head Loss in Pipes
Wall shear stress
L ULU22 hf== f L Dg242 Rg
A R = (hydraulic P radius)
Moody’s diagram
7 Minor Losses in Pipelines
• expansion (e.g., exit) • contraction (e.g., entrance) • bends • fittings (e.g., valves)
V 2 General expression: hK= LL2g
Pipe Configurations
Pipes in parallell:
QQQ=+++123 Q ...
hhLL====123 h L h L...
Pipes in series:
QQ===12 Q ...
hhhLL=+++123 L h L...
8 Pumps and Turbines in the System
Pumps supply the system with energy
Turbines extract energy from the system
Pumping Between Two Reservoirs
Pump and system curve
System curve:
2 HzCQP =Δ + PP
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