Fluids Flashcards
(127 cards)
1
Q
Which forces are supported in fluids?
A
- tension is not supported
- compression is supported and results in a small elastic deformation
- shear is supported but results in flow
2
Q
du/dx + dv/dy + dw/dz = 0
A
all flux in all three directions must equal to zero
mass must be concerved
3
Q
In which direction of pressure change is fluid driven?
A
From high to low pressure
4
Q
τ = F/A
A
Shear stress= force/area
5
Q
τ= μ δu/δy
A
Shear stress = dynamic viscosity * (velocity/ depth)
6
Q
What is the shear stress in viscous flow?
A
No longer zero
7
Q
Kinematic viscosity?
A
ν=μ/ρ (m^2/ s)
8
Q
Dynamic viscosity?
A
μ (kg/ms)
9
Q
What navier stokes simplification can you make by assuming the flow to be steady?
A
All time derivatives will be zero (d/dt)=0
10
Q
What navier stokes simplification can you make by assuming the flow to be fully developed?
A
The velocity component in the x-direction will remain constant (du/dx = 0)
11
Q
What navier stokes simplifications can you make when the sides are incredibly long. (Flow in one direction, boundaries in another)?
A
No flow in the the third direction and no effect by the boundaries.
W=0
d/dz=0
12
Q
What simplification to navier stokes equations by assuming its laminar flow?
A
The gradient of v is zero and v is zero at the boundary, v must be zero everywhere. (V is in the y direction)
13
Q
Re= ρUL/μ
A
Inertial forces/ viscous forces
Reynolds number= densityvelocity scale length scale/ dynamic viscosity
14
Q
What are Reynolds numbers for laminar flow?
A
Re < Recrit
15
Q
What Reynold numbers are turbulent flow?
A
Re> Recrit
16
Q
Q= π/4 D^2 U
A
Volume flux= π/4 * diameter^2 * velocity
(Only for uniform flow)
17
Q
What is Q along a pipe?
A
Constant
18
Q
What is M along a pipe
A
Momentum is constant along a pipe
19
Q
M= π/4 D^2 U^2 = QU
A
Momentum = π/4 * diameter ^2 * velocity ^2
20
Q
Hydraulic grade line?
A
Represents piezometric head (h)
21
Q
Energy grade line?
A
Represents total head line (H)
22
Q
Piezometric head?
A
Pressure head + gravitational head combined
23
Q
p1/ρg + z1 = p2/ρg + z2 + τ0PL/ ρgA
A
p= pressure
ρ= density
g= gravitational acceleration
z= height
τ0= shear stress
P= wetted perimeter
L= length
A= area
24
Q
τ0= f/8 ρ U^2
where
f=f(Re,ks/D)
A
Shear stress = function of Reynolds and relative roughness * density * velocity^2/8
25
In what conditions is f= 64/Re
For laminar flow/ Re< 2300
26
In what condition is 1/f^(1/2) = -2log10[k/D/3.71 + 2.51/Re*f^(1/2)]
For turbulent flow/ for Re> 4000
27
When is 1/f^(1/2)= 2log10[Ref^(1/2)] -0.8
For turbulent flow in smooth pipes/ for Re>4000
28
hL = εU^2/2*g
minor head loss= loss coefficient* velocity^2/2*gravitational acceleration
29
ε= (1-Α1/A2)^2
Loss coefficient for sudden pipe enlargement = (1-small area before/ small area after)^2
30
RH= A/P
Hydraulic radius= flow area/ wetted perimeter
31
hf= f* L/4RH * U^2/2*g
head loss= function * length/4*hydraulic radius * velocity^2/2*gravitational acceleration
32
Entry length?
Region where boundary layer develops
33
Does laminar or turbulent flow boundary layers take longer to develop?
It takes longer for laminar pipe flow boundary layers to develop with longer entry lengths
34
Fully developed flow?
When boundary layers converges to pipe centreline
35
How does shear stress vary across a pipe?
Linearly with being 0 at the centreline
36
What effect of shear force have a long a plate?
The effect of shear force increases along a plate, increasing the thickness of fluid layer that is affected. Reynolds number also increases along a plate leading to turbulent flow conditions.
There’s a laminar region, transition region and then turbulent region
37
u/u* = u*y/ ν
u= velocity
u* = wall shear velocity
ν= velocity viscosity
y= depth/ distance from sublayer
This works when yu*/ν is between 0 and 3~5
38
Sublayer thickness = (3-5) ν/ u*
ν= velocity viscosity
u* = wall shear velocity
39
Irrationality equation?
dv/ dx - du/ dy = 0
40
What is the velocity normal to the cylinder on the surface of the cylinder?
Zero
41
Φ = Uo (r * D^2/4*r) cos θ
Φ = potential flow around a cylinder
U0 = velocity
r = distance from centre of cylinder
D= diameter
θ = angle to the point
42
ur = dΦ/ dr
uθ = 1/r * dΦ/dθ
Velocity components in polar coordiantes
43
What are velocity on the surface of the cylinder?
ur = 0
uθ = -2U0 sinθ
44
P = 1/2 ρ U0^2 (1-4sinθ^2)
P = inviscid pressure distribution
45
What are the velocities on the surface of the cylinder in a real fluid?
ur = 0 (no flow in/ out of cylinder)
uθ = 0 (friction between surface and fluid)
No slip conditions
46
τ = μ duθ/dr
Shear stress = velocity gradient * dynamic viscosity
47
Drag force = Σ (τ ) ds
Drag force is the sum of all shear stresses taken round the surface of the cylinder
48
Explain the difference between upstream and downstream normal pressures for larger U0 around a cylinder
For larger U0 the difference between upstream and downstream normal pressures contributes more significantly to the drag force. This pressure difference is caused by flow separation from the surface of the cylinder. In real solution pressure is a lot lower downstream than potential.
49
What causes flow seperation
-
50
What does flow separation depend on?
-
51
Moody diagram?
A plot of multiple curves between f and Reynolds number
52
u(r) = - R^2/4μ * dp/dx (1-r^2/R^2)
u(r) = velocity of fluid at a radial position r
R= radius of the pipe
μ = dynamic viscosity of the fluid
dp/dx = pressure gradient along the pipe
53
du/dr = 1/2μ dp/dx r
du/dr = velocity along the radius of pipe (depth)
μ = dynamic viscosity
dp/dx = pressure gradient along pipe
r = depth into pipe from centreline
54
τ (r) = μ du/dr = 1/2 * dp/dx * r
τ = shear stress
r = depth into pipe from centreline
μ = dynamic viscosity
u = velocity at a certain depth
p = pressure
x = distance along pipe
55
u(r)/umax = ((R-r)/R)^(1/n)
u(r) = time averaged velocity
umax= time averaged velocity on centreline is maximum
R = radius
r = depth into radius from centreline
n = exponent varies with Reynolds number
56
U/umax = 2n^2 / (2n^2 + 3n +1)
U = mean cross sectional average velocity
umax = time averaged velocity on centreline is maximum
n = exponent that varies slightly with Reynolds number
57
How is the velocity profile for larger Re numbers
Flatter
58
du/dy = τ0/ μ
u = velocity
y= distance from wall
τ0 = wall shear stress
μ = dynamic viscosity
59
u(y)/u* = C1 ln y*u*/ ν + C2
u= time averaged velocity
u*= wall shear velocity
y= depth from wall
ν= velocity viscosity
for approximately 10~40 < yu*/ν < 70~200
60
u(y)/u* = 2.5 ln y/Ks + 8.5
u(y) = time averaged velocity at y
u* = wall shear velocity
y= depth from wall
Ks= roughness coefficient
rough pipes
10~40 < yu*/ν < 70~200
61
umax- u(y) / u* = -1/k ln (y/R)
umax = velocity at centreline
u(y) = time averaged velocity for y
u* = wall shear velocity
K= roughness coefficient
y= depth from wall
R= radius
core region
yu*/ ν > 70~200
62
U= umax - 3/2 u*/κ
U= average U=Q/A
umax = velocity at the centreline
u* = wall shear velocity
κ = karman’s constant ~ 0.4
63
u* = sqrt(τ0/ρ)
u* = wall shear velocity
τ0 = shear stress at the wall
ρ = density of water
64
- Reynold’s stress/ τ0 ~ 1-y/R
y= depth from wall
R= radius
τ0 = shear stress at wall
For core region
65
hf + hL = KQ^2
Head loss (minor and friction) = volume flux * K
66
Steady flow?
No variation with time
The surface profile is a function of space only
Q= constant
67
Uniform flow?
No variation with x
68
Rapidly varied flow?
Very steep profile (theoretically a step)
69
Gradually varied flow
h is slowly varying with x
70
dH/dx= -Sf
Change in total energy/ change in x (along channel) = energy (friction) slope
71
U^2/gh = Fr
Froude number^2 = inertia force/ gravitational force = velocity ^2/ gravitational acceleration * water height
72
Uniform and steady open channel flow is characterised by? *
- velocity of liquid does not change with time or space
- no change in cross sectional area
- the surface must be parallel to the channel bed
- nearly horizontal flow
- water surface slope is very small but this small slope drives the flow
- generally the flow is at high Reynolds number where viscous effects are independent of Reynolds number and so hf~ U^2 where the constant of proportionality reflects the channel roughness
- forces due to gravity must be in a balance with drag/ shear forces otherwise there would be an acceleration
73
- dη/ dx = tanθ
difference in distance between horizontal datum to free surface / difference in x along pipe = tan (angle of free surface)
Smallθ = tanθ = sinθ = 0
74
S0= -dZ/ dx
Bed slope = change in height of bed/ diffference of x along pipe
75
What is the water surface slope, bed slope and energy slope for uniform flow?
The same
76
pA1 + ρgA Δx tanθ -τ0PΔx/ cos θ -pA2
p1= pressure upstream
A1= area upstream
ρ = density of liquid
g= gravity acceleration
Δx = distance along horizontal datum between 1 and 2
θ = angle of bed
τ0 = shear stress along bed
P= wetted perimeter
p2 = pressure at 2
A2 = pressure at 2
Small θ = cos θ ~ 1, tanθ~θ
77
τ0= ρgRSf
τ0= boundary shear
ρ = density
g= gravitational acceleration
R= hydraulic radius = A/P
Sf= energy slope (for uniform flow = θ)
78
Sf= f/8 h/R U^2/gh = f/8 h/R Fr^2
Sf= surface slope
f= friction factor
h= height of water
R= hydraulic radius
U= velocity
g= gravitation acceleration
h= height of water
Fr= froude number
79
U= Q/A = (8g/f)^0.5 R^0.5 S^0.5
U= average velocity
Q= volume flux
A= area
g= gravitational acceleration
f= friction factor h
R= hydraulic radius
S= slope
80
U= C R^0.5 S^0.5 Chezy
Chezy model
U= average velocity
C= Chezy constant ~ 50
R= hydraulic radius
S= slope
81
U= 1/n R^(2/3) S^0.5
U= average velocity
n= manning number ~ 0.02
R= hydraulic radius
S= slope
82
f/8 = g/C^2
f= friction factor ~ 0.03
g= gravitational acceleration
C= Chezy number ~ 50
83
f/8 = gn^2 / R^(1/3)
f= friction factor
g= gravitational acceleration
n= manning number
R= hydraulic radius
84
hn normal/ uniform depth?
The characteristic depth of the flow of it is in uniform flow I.e. the depth if the gravity force is in balance with drag/shesr forces otherwise
85
For gradually varied flow what is Q?
Constant
h, A, P are all functions of x
86
-Sf = -S0 + dh/dx + d/dx*(U^2/2g)
Sf= energy slope
S0 = bed slope
dh/dx = change in water height/ change in distance along distance along bed
U= average velocity
g= gravitational acceleration n
87
q= Q/b = Uh
q= volume flow rate through section/ unit width
Q= volume flux
b= width of channel
U= velocity average
h= height of water
For wide rectangular channel
88
Mass flux through section / unit width ?
ρ q
89
E(h) = h + q^2/2gh^2
E= Specific energy
h = height of water
q= volume flow rate through section through section/ unit width
g= gravitational acceleration
90
Critical depth?
The value of h that makes E(h) a minimum is called critical depth
The depth of the flow which results in the minimum energy state for the flow Fr=1
91
dE/dh = 0 = 1-q^2/ghc^3
The exponent of E (specific energy) plotted against h (water height)
q= volume flow rate through section/ unit width
g= gravitational acceleration
hc= critical depth
92
Mild (M) slope?
Slope less than critical slope
93
Critical (C) slope?
Slope such that uniform flow is critical
NDL & CDL are the same
94
Steep (S) slope?
Slope greater than critical slope
95
Horizontal slope?
Slope is horizontal
96
Adverse (A) slope?
Slope is negative I.e. up hill
97
dh/dx = S0(1-Sf/S0)/(1-Fr^2)
dh/ dx = WS gradients = height of water along x length
S0 = bed slope
Sf= energy slope
Fr= froude number
98
Type 1 flow?
Depth greater than both the normal and the critical depth
99
Type 2 flow?
Depth between normal and critical depth
100
Type 3 flow?
Depth less than both the normal and the critical depth
101
What slope class and flow types are impossible?
- Normal/ uniform flow isn’t possible on adverse or zero slope
- Critical slope h0=hc and C2 cannot occur
102
NDL?
Normal depth line
103
CDL?
Critical depth line
104
hn= (fq^2/8gS0)^(1/3)
Water height for wife rectangular channel
hn= water height
f= friction factor
q= volume flux per unit length
g= gravitational acceleration
S0= water surface slope
105
Frn = sqrt(8S0/f)
Frn= Froude number
S0= water surface slope
f= friction factor q
106
hc= (q^2/g)^1/3
For wide rectangular channel
hc= critical height water
h= height
q= volume flux per metre
g= gravitational acceleration
107
Frc^2 = q^2/ ghc^3
Frc= critical froude number
q= volume flux per unit length
g= gravitational acceleration
hc= critical height of water
108
dh/dx = S0 (1-(hn/h)^3)/(1-(hc/h)^3)= So (1-Fr^2/Frn^2)/(1-Fr^2/Frc^2)
x = along length of channel
h= height of water
S0 = water surface slope Sf
hn= height at n=x
h= height of water
hc= critical height
Fr= froude number
Fr= froude number at n
Frc= critical froude number
109
Eulerian?
Considers the fluid motion at a specific location fixed in space
110
Lagrangian?
Considers the motion of individual fluid particles and seeks to define their trajectory within the flow field
111
du/dx + dw/dz =0
Mass continuity
u= coming in x direction
w = flow in z direction
112
dθ/dt = dw/dx
Angular velocity = change in w/dx
θ + in the anti clockwise direction
dθ/δt = -du/dz is the second formula
113
Ω = dw/dx - du/dz
Definition of vorticity
Sum of two angular velocities
114
Irrotational Ω?
0
115
Φ?
Scaler field defined as the velocity potential
116
u= dΦ/dx w=dΦ/dz
u= flow into x
x= x dimension
Φ= scaler field velocity potential
w= flow in z direction
z= z dimension
117
dΦ^2/dx^2 + dΦ^2/dz^2 = 0 = Δ^2Φ
Laplace’s equation in terms of Φ
Φ = scalar field / velocity potential
x= x direction
z= z direction
Vector notation
118
u= dψ/dz. w=-dψ/dx
Definition of (u, w) in terms of stream function
119
du/dx = -dw/dz
Mass continuity
120
dψ^2/dx^2 + dψ^2/dz^2 = 0 = Δ^2ψ
Laplace’s equation in terms of ψ
121
Φ contour lines?
Potential lines which link points of equal pressure gradient
122
Ψ contour lines?
Streamlines
123
-1/ρ dp/dx + X = ax
-1/ρ dp/dz + Z = az
for most flow X= 0 and Z=-g
a= acceleration
ρ= density
p = pressure
124
du/dt
Unsteady acceleration. Change in time at a point
125
u du/dx + w du/dz
Convective accelerations - changes in space
126
Euler equations?
du/dt + u* du/dx + w* du/dz = -1/ρ dp/dx
dw/dt + udw/dx + wdw/dz = -1/ρ dp/dz -g
127
ρ dΦ/dt + p + ρgz + ρ(u^2 + w^2/2) = C
C= Bernoulli constant
ρ= density
Φ= velocity potential
t= time
p= pressure
g= gravitational acceleration
z= z depth
u= velocity in z direction
w= velocity in x direction
