CIVE40008 Fluid Mechanics Flashcards
1
Q
Describe a fluid?
A
A substance that deforms continuously under the application of a shear stress
2
Q
continuum hypothesis?
A
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.
3
Q
Specific Gravity = ?
A
S=ρfluid / ρwater
Specific gravity= density of the fluid/ density of water
4
Q
Specific weight = ?
A
γ= ρ*g
Specific weight = density * gravitational acceleration
5
Q
describe compressibility in fluids
A
All fluids are compressible
exceptions :
- air incompressible at velocities < speed of sound
- water mostly treated as incompressible, hence constant density
6
Q
describe normal stresses
A
Forces acting normally to the surface of the fluid particles.
- Tend to compress/ expand the fluid particle without changing its shape.
7
Q
describe shear stresses
A
Forces that shear the particle & deform its shape without changing volume
8
Q
what stresses act on stationary and moving fluids ?
A
normal stresses - Both stationary and moving
shear - ONLY moving fluids
9
Q
viscosity?
A
A measure of how much resistance a fluid has to shear
10
Q
kinematic viscosity = ?
A
ν=μ/ρ
Kinematic viscosity ( m^2/s) = dynamic viscosity (Pa s) / density
used when dealing with motion
11
Q
describe pressure ?
A
normal force
p = F / A
due to molecules exerting an equal and opposite force onto another molecules
12
Q
units of pressure ?
A
Pascals (1 Pa= 1N/m^2)
13
Q
why do we assume pressure to be perpendicular to a surface ?
A
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.
14
Q
describe pressure transmission
A
fast but not instantaneous, dependant on speed of sound in medium & shape of container
15
Q
absolute pressure ?
A
Pressure with respect to a vacuum
16
Q
gauge pressure ?
A
Pressure measured relative to local atmospheric pressure
17
Q
relationship between gauge, atmospheric and absolute pressure ?
A
p gauge = p abs - p atm
18
Q
when is pressure constant
for hydrostatic pressure distribution
A
in horizontal planes
19
Q
pressure equation ?
for hydrostatic pressure distribution
A
p = p0 + ∫ ρ g dz*
20
Q
when are hydrostatic pressure changes ignored
A
in gases
21
Q
what is a manometer ?
A
- 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
22
Q
equation for Δpressure in manometer ?
A
Δpressure = density of liquid x gravitational acceleration x difference in height of sides of mono meter
Δp = p1 - p2 = ρw * g * Δh
23
Q
equation for fluid weight
for hydrostatic pressure distribution
A
Fv = pgV = (ρgh)(wb) = ρgh*A
density x g x height x width x depth
24
Q
equation for net force on a surface
for hydrostatic pressure distribution
A
F h = 1/2 * ρgb*h^2
acts at z = 2h / 3
25
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
26
how do you approach pressure calculation for submerged angled surfaces ?
## Footnote
for hydrostatic pressure distribution
consider a constant and linearly distributed load
- eg a rectangular UDL & triangular LDL
27
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
28
buoyancy force ?
net upward buoyancy force = weight of displaced fluid
Fb = ρ*g*Vbody
28
One - dimensional flow ?
Variations only along one spatial coordinate
29
Two - dimensional flow?
Variations along the direction of the flow and across the flow, two spatial dimensions
30
Three - dimensional flow?
Flow varies in all three spatial directions
31
Steady flow?
Flow that does not change with time
32
describe a streamline ?
A line that is everywhere tangential to the instantaneous flow velocity (= gradient function)
33
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
34
describe a vena contracta
stream cross-section is smallest, contraction of container leads to separation, reattachment & recirculation zones
35
define a flux
rate at which a substance flows through a surface
36
define volume flux
The volume of fluid passing through a surface per unit time
37
define mass flux
The mass of the fluid passing through a surface per unit time
38
define momentum flux
The momentum passing through a surface per unit time
39
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
40
equation for mass flux
Φρ = ∫ ρu dA (= ρQ for constant ρ)
Mass flux = density x velocity x area
units : kg/s
41
equation for momentum flux ?
Φρu = ∫ u(ρu) dA
units : Ns/m^3 or N
approximation : ρQU
42
define an ideal fluid
- incompressible
- no internal resistance to flow ( viscosity = 0 )
- shear forces ignored (frictional forces)
- only experience normal stresses
- no boundary layers
43
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)
44
describe boundary layers
A flow region where large differences in velocity occur, often very close to the wall ( large velocity gradient )
45
define viscosity
measure of resistance of a fluid to shear
46
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)
47
define a newtonian fluid
viscosity does not change with flow rate
hence viscosity independent of shear rate
48
define a laminar flow
Re < Re*crit* , viscosity dominates
τw ∝ U
Organised layered flow
49
define a turbulent flow
Re > Re*crit* , viscosity negligible
τ*w* ∝ U^2
Disorganised, random, efficient mixer causes large friction and energy loss
50
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
51
mass conservation ?
mass flux in (Φρ in) = mass flux out (Φρ out)
ρU1A1 = ρU2A2
52
volume conservation
U1A1 = U2A2
volume flux in (Q1) = volume flux out (Q2)
steady continuity equation
53
unsteady continuity equation ?
dV/dt = Q*in* - Q*out*
54
conservation equation expressed mathematically ?
d/dt ∫ XdV = Φ*in* -Φ*out* + S
storage = fluxes in - fluxes out + sources - sinks
55
describe application of continuity equation for a leaking container
unsteady cont. eqn :
Q*in* is constant
Q*out* non-constant
water depth increases until Q*in* = Q*out*
system tries to equilibrize
56
rate of change of momentum = ?
Rate of change of the momentum is equal to the sum of forces
d(mu) / dt = Σ F
57
steady continuity equation for momentum
ρ*Q *(U*out*- U*in*) = Σ Fx
58
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
59
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
60
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
61
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)
62
describe the components of a total energy head
H = p / ρg + z + U^2/ 2g
total energy head = pressure head + elevation head + velocity head
63
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
64
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
65
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
66
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 ⇒ ∞
67
what are the conditions for applying the Bernoulli equation ?
- converging streamlines
- accelerating fluid
- low energy losses (not Turbulence)
68
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
69
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 = p*s* , U = 0, z = 0
since energy is conserved :
p∞ + 1/2 *ρ*U∞^2 = p*s*
where 1/2 *ρ*U∞^2 is the dynamic pressure
70
what is a pitot-static tube ?
measures stagnation & static pressure to infer the velocity of a fluid
using U = √(2g∆h)
71
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
72
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
73
what are major losses in pipe systems ?
friction with the wall
74
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
75
how do you deal with minor energy losses ?
apply a loss factor
- energy loss ∝ local kinetic energy of flow
∆H*local* = ξ U^2/2g
where 0 < ξ < 1
76
what is the exit loss for a pipe discharging into a reservoir ?
all kinetic energy is lost, hence
∆H*exit* = U^2/2g
77
what is the Darcy-Weisbach Friction equation ?
∆H*friction* = *f* * L/D * U^2/2g
for non circular pipes :
∆H*friction* = *f* * L/De * U^2/2g
where De = 4A / P (perimeter)
78
Give examples of uses for pumps
irrigation, water supply, sewage & water control
79
Give examples of uses for turbines
power generation, in hydro dams, or wind farms
80
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
81
give the energy balence for a control volume with a pump or turbine
dE*total* / dt = (ρgH*in*)Q - (ρgH*out*)Q + P (energy added by pump per unit time)
82
equation for power of a pump
P = ρg∆HQ in J/s
83
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
84
hydraulic jump ?
- rapid transition between super- and subcritical flow
- Large amount of mixing and energy dissipation
- Sudden step change in flow depth
85
laminar flow ?
| ocf
Re < Recrit
86
turbulent flow ?
| ocf
chaotic motion and complex flow patterns for Re >> Recrit
** Virtually all open channel flows are turbulent **
87
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
88
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
89
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
90
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,)
91
when can hydrostatic pressure distribution be introduced to OCFs?
if streamlines are approximately straight and parallel (uniform/slowly-varying flows)
92
underlying assumtions of Bernoulli for OCFs ?
1. uniform/slowly-varying flow,
2. incompressible fluid,
3. steady flow,
4. friction negligible (no major energy losses).
93
small slopes ?
apply small angle approximations :
sinθ ≈ s
cosθ ≈ 1
tanθ ≈ s
h*cosθ ≈ h
94
gravity body force for OCF with small slope ?
F *g,x* = mg ≈ ρgALs
95
wall shear stress for OCF ?
acts over the entire wetted perimeter P and length L
Ff = τ*w*PL
(since stress = F / A)
when force balence applied in x direction with fluid body weight :
τ*w* = ρgRs
96
sub-critical ?
**Fr < 1**
upstream flow is affected (changing
h, U) by an obstacle - **downstream control**
U < √gh
- most rivers
| ' tranquil , slow '
97
super-critical ?
**Fr > 1**
upstream flow is not affected by an
obstacle
U > √gh
**upstream** control
| 'shooting, fast'
98
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
99
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
100
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.
101
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
102
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
103
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
104
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
