CIVE40008 Fluid Mechanics Flashcards
Describe a fluid?
A substance that deforms continuously under the application of a shear stress
continuum hypothesis?
You can consider the average effects of the molecules in a given volume.
Relies on no. molecules being very large & over a large scale.
A continuum prevails if the number of molecules in a given volume is sufficiently great that the average effects are constant or change smoothly with time.
Specific Gravity = ?
S=ρfluid / ρwater
Specific gravity= density of the fluid/ density of water
Specific weight = ?
γ= ρ*g
Specific weight = density * gravitational acceleration
describe compressibility in fluids
All fluids are compressible
exceptions :
- air incompressible at velocities < speed of sound
- water mostly treated as incompressible, hence constant density
describe normal stresses
Forces acting normally to the surface of the fluid particles.
- Tend to compress/ expand the fluid particle without changing its shape.
describe shear stresses
Forces that shear the particle & deform its shape without changing volume
what stresses act on stationary and moving fluids ?
normal stresses - Both stationary and moving
shear - ONLY moving fluids
viscosity?
A measure of how much resistance a fluid has to shear
kinematic viscosity = ?
ν=μ/ρ
Kinematic viscosity ( m^2/s) = dynamic viscosity (Pa s) / density
used when dealing with motion
describe pressure ?
normal force
p = F / A
due to molecules exerting an equal and opposite force onto another molecules
units of pressure ?
Pascals (1 Pa= 1N/m^2)
why do we assume pressure to be perpendicular to a surface ?
On a molecular scale the surface is never flat. By averaging over billions of collisions ( continuum approach), the resulting force will act perpendicular to the surface. Therefore we assume pressure produces a force perpendicular to the surface.
describe pressure transmission
fast but not instantaneous, dependant on speed of sound in medium & shape of container
absolute pressure ?
Pressure with respect to a vacuum
gauge pressure ?
Pressure measured relative to local atmospheric pressure
relationship between gauge, atmospheric and absolute pressure ?
p gauge = p abs - p atm
when is pressure constant
for hydrostatic pressure distribution
in horizontal planes
pressure equation ?
for hydrostatic pressure distribution
p = p0 + ∫ ρ g dz*
when are hydrostatic pressure changes ignored
in gases
what is a manometer ?
- pressure difference between two locations in a flow
- liquid filled U-tubes
- Δ pressure causes liquid to sit at different levels either side of U - tube
hence height ∝ pressure differenc
equation for Δpressure in manometer ?
Δpressure = density of liquid x gravitational acceleration x difference in height of sides of mono meter
Δp = p1 - p2 = ρw * g * Δh
equation for fluid weight
for hydrostatic pressure distribution
Fv = pgV = (ρgh)(wb) = ρgh*A
density x g x height x width x depth
equation for net force on a surface
for hydrostatic pressure distribution
F h = 1/2 * ρgb*h^2
acts at z = 2h / 3
centre of pressure ?
centre of pressure is the point at which the pressure may be considered to act
-location of where pressure force acts to exert the same moment as the pressure distribution
how do you approach pressure calculation for submerged angled surfaces ?
for hydrostatic pressure distribution
consider a constant and linearly distributed load
- eg a rectangular UDL & triangular LDL
how do you approach pressure calculation for complex geometries ?
projection method : projecting horizonal & vertical planes
since locally (at infintissimaly small strips) same forces as a vertical & horizontal wall
horizontal force ~= force on equivalent vertical surface
vertical force = weight of fluid above up to free surface
LIMITATIONS : no information on centre of pressure, only relevant for forces not moments
buoyancy force ?
net upward buoyancy force = weight of displaced fluid
Fb = ρgVbody
One - dimensional flow ?
Variations only along one spatial coordinate
Two - dimensional flow?
Variations along the direction of the flow and across the flow, two spatial dimensions
Three - dimensional flow?
Flow varies in all three spatial directions
Steady flow?
Flow that does not change with time
describe a streamline ?
A line that is everywhere tangential to the instantaneous flow velocity (= gradient function)
properties of streamlines ?
- streamlines cannot cross
- fluid cannot cross a streamline
- streamlines can meet at a point
- locally the flow must run parallel to solid boundaries
- two adjacent streamlines may be thought of as a ‘stream tube’
- moving fluids cannot suddenly change direction
- if an obstacle is not streamlined (a bluff body), the fluid will separate from the solid boundary.
- there are separation, reattachment and recirculation zones
describe a vena contracta
stream cross-section is smallest, contraction of container leads to separation, reattachment & recirculation zones
define a flux
rate at which a substance flows through a surface
define volume flux
The volume of fluid passing through a surface per unit time
define mass flux
The mass of the fluid passing through a surface per unit time
define momentum flux
The momentum passing through a surface per unit time
equation for volume flux
Volume flux/Discharge= velocity x area
Q = ∫ u dA
Q = U*A
(where U is mean velocity)
units : m^3/s
equation for mass flux
Φρ = ∫ ρu dA (= ρQ for constant ρ)
Mass flux = density x velocity x area
units : kg/s
equation for momentum flux ?
Φρu = ∫ u(ρu) dA
units : Ns/m^3 or N
approximation : ρQU
define an ideal fluid
- incompressible
- no internal resistance to flow ( viscosity = 0 )
- shear forces ignored (frictional forces)
- only experience normal stresses
- no boundary layers
describe no-slip condition
for real fluids :
At the interface between a solid surface and a fluid, the fluid velocity and solid surface velocity are identical
(hence zero is boundary is stationary)
describe boundary layers
A flow region where large differences in velocity occur, often very close to the wall ( large velocity gradient )
define viscosity
measure of resistance of a fluid to shear
equation for shear stress
τ = μ * du/dz
hence a Newtonian fluid
frictional stress per unit area of contact
where μ is dynamic viscosity ( Ns/m^2)
define a newtonian fluid
viscosity does not change with flow rate
hence viscosity independent of shear rate
define a laminar flow
Re < Recrit , viscosity dominates
τw ∝ U
Organised layered flow
define a turbulent flow
Re > Recrit , viscosity negligible
τw ∝ U^2
Disorganised, random, efficient mixer causes large friction and energy loss
describe difference between lagrangian & eulerian approaches
lagrangian - identifying small elements of fluid within the flow for all time, position at some instant
eulerian - bulk description of flow by defining a control volume
mass conservation ?
mass flux in (Φρ in) = mass flux out (Φρ out)
ρU1A1 = ρU2A2
volume conservation
U1A1 = U2A2
volume flux in (Q1) = volume flux out (Q2)
steady continuity equation
unsteady continuity equation ?
dV/dt = Qin - Qout
conservation equation expressed mathematically ?
d/dt ∫ XdV = Φin -Φout + S
storage = fluxes in - fluxes out + sources - sinks
describe application of continuity equation for a leaking container
unsteady cont. eqn :
Qin is constant
Qout non-constant
water depth increases until Qin = Qout
system tries to equilibrize
rate of change of momentum = ?
Rate of change of the momentum is equal to the sum of forces
d(mu) / dt = Σ F
steady continuity equation for momentum
ρQ (Uout- Uin) = Σ Fx
give examples of conditions for a non-zero resultant force acts on a fluid
- pressure gradient
- gravity
- density differences
- friction / viscosity
- externally applied forces, eg by a solid surface
under what conditions can we apply hydrostatic principles to hydrodynamic situations ?
when the vertical acceleration is negligibly small
- streamlines are nearly horizontal, with no significant curvature
explain the pressure distribution in closed conduits
e.g. pipes, ducts
hydrostatic pressure distribution does not influence the net force and is thus omitted from calculation
pressure is modelled as constant
what is the primary principle of the Bernoulli equation ? what are the assumptions ?
energy is constant along a streamline
assuming:
- steady flow
- constant density
- dissipationless (no friction)
describe the components of a total energy head
H = p / ρg + z + U^2/ 2g
total energy head = pressure head + elevation head + velocity head
what is the piezometric head?
h = p / ρg + z
potential energy of flow, due to pressure & elevation heads
constant in vertical when a hydrostatic distribution
Describe the difference between the energy & hydraulic grade lines ?
EGL - H(x) energy head
HGL - h(x) piezometric head
H(x)-h(x) = U^2/ 2g
difference is the velocity head, hence associated with the kinetic energy
what is the Bernoulli equation for curved streamlines ?
the centripetal force produces a pressure gradient, hence the Bernoulli equation becomes :
Δp / Δr ~ ρ*U^2 / R
(where R is the radius of curvature)
hence, if R is smaller (stronger curvature), Δp is larger
Describe the pressure in a fluid body, using the second Bernoulli equation
- pressure across straight streamlines is constant in horizontal plane, or hydrostatic in vertical plane
- applying gravity : piezometric head across straight streamlines is constant
since straight streamline assumes R ⇒ ∞
what are the conditions for applying the Bernoulli equation ?
- converging streamlines
- accelerating fluid
- low energy losses (not Turbulence)
what is the discharge formula for a draining reservoir ?
Q = A√(2gh)
where h is the distance from the datum (taken as centre of outflow area) to free surface
how would you manipulate the Bernoulli equation for stagnation points ?
at a given point in the flow :
p = p∞, U = U∞, z = 0
at interface to surface :
p = ps , U = 0, z = 0
since energy is conserved :
p∞ + 1/2 ρU∞^2 = ps
where 1/2 ρU∞^2 is the dynamic pressure
what is a pitot-static tube ?
measures stagnation & static pressure to infer the velocity of a fluid
using U = √(2g∆h)
what is a syphon ?
inverted U-shaped tube which causes a liquid to flow uphill, discharging at a lower relative elevation, through a pressure gradient
what is a venturi tube ?
gently converging & diverging tube to determine the volume flux in a pipe
- with a pressure tapping to identify pressure difference
- a Qideal can be found, and a correction factor can be applied to compensate for energy losses in the system
what are major losses in pipe systems ?
friction with the wall
what are minor losses in pipe systems ?
due to pipe entry, exit, bend, contractions
additional energy dissipation due to secondary flows induced, eg curvature or recirculation
how do you deal with minor energy losses ?
apply a loss factor
- energy loss ∝ local kinetic energy of flow
∆Hlocal = ξ U^2/2g
where 0 < ξ < 1
what is the exit loss for a pipe discharging into a reservoir ?
all kinetic energy is lost, hence
∆Hexit = U^2/2g
what is the Darcy-Weisbach Friction equation ?
∆Hfriction = f * L/D * U^2/2g
for non circular pipes :
∆Hfriction = f * L/De * U^2/2g
where De = 4A / P (perimeter)
Give examples of uses for pumps
irrigation, water supply, sewage & water control
Give examples of uses for turbines
power generation, in hydro dams, or wind farms
describe a centrifugal pump
centripetal accelerations induced by spinning rotor blades
- the centrifugal forces lead to a low pressure in the centre & high pressure at the outlet
- this pressure difference determines the output flow rate
give the energy balence for a control volume with a pump or turbine
dEtotal / dt = (ρgHin)Q - (ρgHout)Q + P (energy added by pump per unit time)
equation for power of a pump
P = ρg∆HQ in J/s
open channel flow ?
flow of fluid with a free-surface that is driven along a conduit by gravity
free surface introduces new degree of freedom (flow geometry now require
hydraulic jump ?
- rapid transition between super- and subcritical flow
- Large amount of mixing and energy dissipation
- Sudden step change in flow depth
laminar flow ?
ocf
Re < Recrit
turbulent flow ?
ocf
chaotic motion and complex flow patterns for Re»_space; Recrit
** Virtually all open channel flows are turbulent **
steady flow ?
no variations in time
- for example : velocity, pressure, and density remain constant at any point in space over time
- example : flow of water through a pipe with a constant velocity and pressure
unsteady flow ?
variations in time :
- fluid properties are not constant at any point in space over time
- e.g. acceleration, deceleration
- flow in a river
uniform flow ?
no along-channel variations in water depth, slope,
discharge etc
e.g. constant diameter pipe flow
steady uniform = equilibrated force balance, both in
space and time
non-uniform flow ?
variations in one or more of the variables
e.g. changing geometry, slope
* Slowly varying flows – gradual variations; pressure is usually
hydrostatic and streamlines are straight.
* Rapidly varying flows – rapid variations; pressure is generally not
hydrostatic and streamlines are curved
usually due to hydraulic structures (weirs, dams, etc,)
when can hydrostatic pressure distribution be introduced to OCFs?
if streamlines are approximately straight and parallel (uniform/slowly-varying flows)
underlying assumtions of Bernoulli for OCFs ?
- uniform/slowly-varying flow,
- incompressible fluid,
- steady flow,
- friction negligible (no major energy losses).
small slopes ?
apply small angle approximations :
sinθ ≈ s
cosθ ≈ 1
tanθ ≈ s
h*cosθ ≈ h
gravity body force for OCF with small slope ?
F g,x = mg ≈ ρgALs
wall shear stress for OCF ?
acts over the entire wetted perimeter P and length L
Ff = τwPL
(since stress = F / A)
when force balence applied in x direction with fluid body weight :
τw = ρgRs
sub-critical ?
Fr < 1
upstream flow is affected (changing
h, U) by an obstacle - downstream control
U < √gh
- most rivers
‘ tranquil , slow ‘
super-critical ?
Fr > 1
upstream flow is not affected by an
obstacle
U > √gh
upstream control
‘shooting, fast’
scale - modelling ?
apply scale factor λ = LF /LM
- apply to Froude’s Number equation
- use dimension to identify scaling for other parameters
- e.g. velocity = L / T
- Um / √ghm = Uf / √ gλhm
solution strategy for hydraulic jump ?
continuity & momentum :
ρQ(U2 -U1) = ΣF
apply Bélanger equation
or for points far enough from the control structure : apply Bernoulli
purpose of control structures ?
- Controlling water depth (ships, increasing ground water level)
- Bed stabilization (reduced flow velocity)
- Flood control (water storage, e.g. if located immediately after a
lake) - Air entrainment etc.
what can you obtain from an energy diagram ?
minimum = critical depth = Fr = 1
for each energy head : there are two intersections with the curve denoting the possible super & sub critical states
sub : piezometric head dominates
super : velocity head dominates
assumptions for broad-crested weir w/ free outfall ?
- length of weir ≫ upstream flow depth
- free outfall (no downstream disturbance)
- free outfall ensures discharge becomes critical
- well-rounded/smooth weir (no major energy loss)
- constant energy head incoming - fed from large reservoir
- upstream velocity head is negligible’
- streamlines ≈ parallel above weir
hence we can apply energy conservation between two points over wier
sharp - crested weir assumptions ?
water level directly related to Q over weir
Assumptions :
- Downstream of weir is free jet at atmospheric pressure (broad-crested = hydrostatic)
- No losses across weir
- Upstream pressure hydrostatic
- flow over weir ≈ upstream flow depth
poleni formula ?
solves bernoulli for discharge as a function of heigh & integrate over area of weir
for rectangular :
Q ideal = 2/3 * √(2g) * wh^3/2
In reality discharge will be lower due to losses and the
assumptions made ∴ correction coefficient Cd is applied:
Qreal = Cd * Qideal
