TP2 - Fluid Mechanics Flashcards
1
Q
Whatโs kinetic and dynamic viscosity?
A
Dynamic Viscosity (ฮผ): Measure of internal resistance
Kinematic viscosity:
Ratio of dynamic viscosity to density
v = ๐ / ๐
2
Q
Whatโs an open channel flow?
A
Fluid flow which experiences a liquid-gas boundary layer
3
Q
How can open channel flow be described?
A
Flow can be uniform or varied (non-uniform)
UF - uniform flow
GVF - gradually varied flow
RVF - rapidly varied flow
4
Q
How does the velocity profile change along the length of a channel with steady, uniform flow?
A
It does not change.
5
Q
What is a prismatic channel?
A
A channel with:
- a constant cross-sectional area
- constant surface roughness
- a constant clope
6
Q
What are the (2) forces balanced in an open channel?
A
Gravitational = Drag
Forces acting to speed the fluid = forces acting to slow the fluid.
7
Q
Involved with open channel equations, what do the following symbols represent?
W ฮฑ ๐โ L HL y A P Rโ C f
A
W - Weight of water (N)
ฮฑ - Angle of slope (ยฐ)
ฯโ - Shear stress (Pa)
L - Length of bed (m)
HL - change in elevation across length m(m)
y - depth of liquid
A - CSA
P - wetted perimeter
Rโ - hydraulic radius
C - Chezy coefficient
f - fanning friction factor
8
Q
What is the force balance for an open channel?
A
Gravitational = Drag
๐ sin ๐ผ = ๐โ ร ๐ค๐๐ก๐ก๐๐ ๐๐๐๐
๐ sin ๐ผ = ๐โ ร ๐ฟ๐
(since sin ๐ผ โ ๐ = ๐ป๐ฟ/๐ฟ)
๐จ๐๐ ๐ฏ๐ณ/๐ณ=๐โ ๐ท
9
Q
How is hydraulic radius calculated?
A
Rh = cross-sectional area / wetted perimeter
= A / P
(Rh for a half-full pipe = D/4)
10
Q
What is the Chezy equation?
A
๐=โ((๐/๐)*๐น_๐ฏ ๐๐บ)
Where:
f - fanning friction factor
Rh - hydraulic radius
S - slope
It describes the mean flow velocity of turbulent open channel flow.
11
Q
What is the empirical Manning equation?
A
C = (1/n)*Rh^1/6
Where:
C - Chezy coefficient
n - manning coefficient
Rh - hydraulic radius
12
Q
What is the equation combining the Chezy and Manning equations?
A
u = (1/n)Rh^(2/3)S^(1/2)
which calculates velocity.
Since flowrate, Q = Au,
Q = (1/n)A*Rh^(2/3)S^(1/2)
13
Q
At what Reynolds number does an open channel system flow change from laminar to turbulent?
A
When Re is approximately 600.
14
Q
Whatโs a compressible fluid?
A
A fluid whose volume is dependent on its temperature and pressure.
Itโs density can change.
15
Q
What does isentropic mean?
A
Constant entropy
16
Q
Whatโs choked flow?
A
Choked flow is a limiting condition where the mass flow will not increase with a further decrease in the downstream pressure environment for a fixed upstream pressure and temperature.
Choked flow is a compressible flow effect. The parameter that becomes โchokedโ or โlimitedโ is the fluid velocity.
17
Q
What is the simplified energy balance equation?
A
udu + VdP = 0
Assumptions:
- Fully turbulent flow โ Uniform Velocity Profile (ฮฑ = 1)
- Horizontal Flow (dz = 0)
- Ideal Gas
- No external work or heat transfer to the surroundings
- Frictional losses are neglected
18
Q
What is sonic velocity?
A
The velocity of sound in that fluid
19
Q
Are velocity and mass flowrate for isothermal and adiabatic/isentropic flow systems calculated in the same way?
A
No.
They are quite different and adiabatic/isentropic flow consider the ratio of Cp to Cv too (gamma)
20
Q
Whatโs a Laval nozzle?
A
A converging-diverging nozzle used to expand gases when P drop is large, with minimal energy loss (reversible process).
(Typically non-isothermal processes)
21
Q
What is transonic flow?
A
where air flows above, at, and below the speed of sound at the same time at different points on an object.
For example, the air on a wing flows faster, so that air could be supersonic while the air flowing over the body of an airplane could be subsonic
22
Q
How is sonic velocity calculated for an ideal gas under adiabatic conditions?
A
u.w = (ฮณRT/M)ยน/ยฒ
When Wc = Pโ/Pโ, the velocity of the fluid is known as the sonic velocity.
23
Q
Whatโs the Mach number, Ma?
A
The ratio of the fluid velocity, u, to the sonic velocity, uw.
Ma = u / uw
24
Q
What is critical flow?
A
When fluid velocity, u, is equal to sonic velocity, uw, and so the Mach number is 1.
25
What is critical, supersonic and subsonic flow?
Critical:
velocity = sonic velocity (u = uw) therefore Ma = 1
Supersonic:
u > uw therefore Ma > 1
Subsonic:
u < uw therefore Ma < 1
26
What do de Laval nozzles do?
They can accelerate hot, pressurised gas to a supersonic speed and, upon expansion, shape the exhaust flow so that the heat energy propelling the fluid flow is 'maximally' converted into directed kinetic energy.
27
What are the key concepts and assumptions made for when using Laval nozzles?
1. Non-Isothermal. High flow rates means no time for temperature equilibration (a temperature gradient exists along the nozzle)
2. Isentropic. No shock waves should be propagating within the nozzle
3. Fully turbulent flow โ Uniform Velocity Profile (ฮฑ = 1)
4. Horizontal Flow (dz = 0)
5. Ideal Gas
6. No external work or heat transfer to the surroundings
7. Frictional losses are neglected
28
How is minimum throat area for a Laval nozzle calculated?
(Gยฒ*(Vโ/Pโ)*((ฮณ-1)/2ฮณ)*(Wc^(-2/ฮณ)/(1-Wc^(ฮณ-1/ฮณ)))^ยน/ยฒ
29
What's choked flow?
When a flowing fluid at a given pressure and temperature passes through a constriction (such as the throat of a convergent-divergent nozzle or a valve in a pipe) into a lower pressure environment the fluid velocity increases.
At initially subsonic upstream conditions, the conservation of mass principle requires the fluid velocity to increase as it flows through the smaller cross-sectional area of the constriction.
At the same time, the Venturi effect causes the static pressure, and therefore the density, to decrease at the constriction.
Choked flow is a limiting condition where the mass flow will not increase with a further decrease in the downstream pressure environment for a fixed upstream pressure and temperature.
30
Whatโs the simplified energy balance when designing pipe systems?
udu + VdP + dF = 0
```
Assumptions
1. Fully turbulent flow โ Uniform Velocity Profile (ฮฑ = 1)
2. Horizontal Flow (dz = 0)
3. Ideal Gas
4. No external work or heat transfer to
the surroundings
```
31
What are the limiting factors when designing piping systems?
Erosion (e.g. entrained dust/liquid) and stress on pipe wall โ maximum safe velocity
Compressors are expensive โ Maximum allowable pressure drop
Maximum velocity (and flowrate) (critical flow at exit)
32
How can the critical pressure ratio be solved / how must the Pw equation be solved?
This cannot be solved analytically โ it needs to be solved one of two ways:
1. Iteratively (not preferable)
2. Using a graph someone else has made solving this equation already
33
What is the general energy balance for compressible flow?
1/a * udu + gdz + vdP + ๐ฟWs + ๐ฟF = 0
34
What is transition length, xt?
The length required for fluid flow to become fully developed.
Boundary layer develops over this length.
35
What are the 4 boundary conditions when considering fluid flow?
1) No slip at the wall.
When y = 0, ux = 0
2) Stream velocity (us) at edge of boundary layer.
When y = ๐ฟ, ux = us
3) No shear stress at edge of boundary layer.
When y = ๐ฟ, du/dy = 0
๐
= ๐ = 0 ๐๐ก ๐ฆ = ๐ฟ
4) Shear stress is maximum at the wall (i.e. ฮผ, u0 and the surface roughness are constant).
When y = 0, d๐/dy = d2u/dy2 = 0
36
What is the cubic velocity equation, used to derive velocity profiles?
ux = u0 + ay + by^2 + cy^3
37
What is the velocity profile in a laminar boundary layer?
ux / us = 3y/2๐ฟ - y^3/2๐ฟ^3
38
What are the 3 different regions within a turbulent boundary layer?
Laminar sub-layer: ๐ฟb and ub
Transition layer: ๐ฟt and ut
Overall boundary layer: ๐ฟ and ux
Note: us is stream velocity and um is mean velocity at the entrance
! The laminar and transition layers are usually ignored !
39
What is the Blasius Equation (linking R, us and y)?
R/ฯ*us^2 = 0.0228*(ฮผ /us*๐ฟฯ)^0.25
Where R is shear stress (same as ๐)
40
What is the power law relating us, ux, y and ๐ฟ?
ux / us = (y/๐ฟ)^f
Where f is equal to 1/7 (Prandtl one-seventh power law).
41
How is boundary thickness calculated for a laminar and turbulent boundary layer?
Laminar:
๐ฟ = 4.64*Rex^-0.5 *x
Turbulent:
๐ฟ = 0.376*Rex^-0.2 *x
42
How is rate of boundary layer thickening calculated for a laminar and turbulent boundary layer?
Laminar:
d๐ฟ/dx = 2.32*Rex^-0.5 *x
Turbulent:
๐ฟ = 0.301*Rex^-0.2 *x
43
How is Re x calculated?
Re x = us*x*ฯ / ฮผ
Transition to turbulent occurs when Re x ~10^5 to 10^6
44
How does the boundary layer grow within pipes?
Flow develops along the length until the transition length is reached.
Initially there is plug flow. All velocity is the same and us = um (= mean entrance velocity).
Viscosity is not important.
As flow develops, the inviscid core decreases and, around the core, viscosity becomes more important.
Beyond xt, flow is fully developed.
45
How can transition length, xt, be predicted?
If laminar flow (Re < 2000):
xt/d = 0.0575*Re
If transition flow (Re ~ 2500):
xt = 100*d
If turbulent flow (Re > 3000):
xt/d = 4.4*Re^1/6
46
How is particle Reynolds number, Re p, calculated?
Re p = ฯ*uD*dp/ ฮผ
```
Where:
ฯ - fluid density
uD - relative velocity between fluid and immersed body
dp - particle diameter
ฮผ - fluid viscosity
```
47
How is particle diameter, dp, calculated for irregular (non spherical) objects?
Using the equivalent sphere diameter ds:
ds = (6V/pi)^1/3
48
What is stokes flow (aka creeping flow)?
When the particle Reynolds number, Re p, is below 0.1.
Often occurs when the is an object within the flow.
Only viscous drag (aka Stokes drag) is acting on the surface of the body.
Re p < 0.1
49
What is flow separation?
When the particle Reynolds number, Re p, is greater than 0.1.
Streamlines become detached from the surface of the object within the flow.
Eddies are formed in the wake, resulting in low pressures downstream.
This results in 'form drag'
50
What is the difference between stokes drag and flow drag?
Stokes drag is due to skin friction.
Form drag occurs when the fluid flow separates.
51
How are stream velocity, us, and mean entrance velocity, um, related? (Flow through a pipe)
If flow is laminar:
us = 2*um
If flow is turbulent:
us = um / 0.817
For transition flow, there is no particular approximation. Therefore, max and min can be found with the laminar and turbulent assumptions.
52
What are the main dimensionless numbers used in fluid flow (TP2)?
Nusselt - Nu: for heat transfer in fluid
Reynolds - Re: for fluid flow involving viscous and inertial forces
Prandtl - Pr: for heat transfer in flowing fluid
Stanton - St: for heat transfer in flowing fluid
Schmidt - Sc: for mass transfer in a flowing fluid
53
How is the Nusselt number calculated?
Nu = hd/k
Where:
h - heat transfer coefficient
d - diameter
k - thermal conductivity
54
How is the Prandtl number calculated?
Pr = Cp*mu/k
55
Whatโs the Stanton number used for?
How is it calculated?
For heat transfer in fluid fluid
St = h / rho*u*Cp
56
Whatโs the Schmidt number used for?
How is it calculated?
For mass transfer in a flowing fluid.
Sc = u / rho*D
Where D is the diffusion coefficient (molecular diffusivity)
57
What are Reynoldsโ assumptions?
The boundary layer is the limiting step (i.e. we ignore the inviscid core)
Heat, mass & momentum transfer is entirely by eddy motion.
Eddies will propagate all the way to the surface
- No laminar regions (i.e. ignore laminar sub-layer and buffer region)
Reynolds Analogy assumes that both the laminar sub-layer and transition
layer are negligible
58
What is the heat balance, considering heat transfer to surface per unit area per second, for fluid flow?
-Q0 = m*Cp*(Ts-Tw) / At
59
Whatโs the mass balance for fluid flow?
-N A0 = m*(C(as) - C(aw))/ A*t*rho
60
Whatโs the momentum balance for fluid flow?
-R0 = m*us/A*t
61
What is the equation for the Reynolds analogy for heat transfer?
R / rho*us^2 = h / rho*us*Cp = St (Stanton number)
62
What is the equation for the Reynolds analogy for mass transfer?
R / rho*us^2 = hD/us
Where hD / us = St D (Stanton number for mass transfer where hD is the mass transfer coefficient)
63
What conditions are needed for creeping/stokes flow and separation flow to occur?
Creeping: Re p < 0.1
Separation: Re p > 0.1
64
How do velocity and pressure vary as a fluid flows past an object?
(Considering point 1 to be the stagnation point, 2 to be on the equator line and point 3 to 4 to be the wake region)
Between point 1 & 2:
- Area between fluid and object decreases
- Fluid velocity increases
- Pressure decreases
At point 2 (max distance from centre of object):
- Area (of contact) is at its minimum
- Fluid velocity is at its maximum
- Pressure will be at its minimum
```
Between point 2 & 3:
- Contact area will increase
- Hence, fluid velocity will decrease
- Pressure will increase
Here, there will be an adverse pressure gradient
```
65
How can you produce a smaller wake (after fluid passes an object)?
Reduce the adverse pressure gradient, dP/dx
This can be done by increasing the length of the object.
Increase Ek in boundary layer
Reduce shear stress
Thinner boundary layer
66
How is drag force calculated for when a fluid passes an immersed body?
FD = CD*rho*uD^2*Ap / 2
Where:
CD is drag coefficient. This combines form and viscous drag and depends on shape and surface roughness.
uD is relative velocity
Ap is the projected surface area
67
Standard golf balls are very similar in diameter to ping pong balls but have a โdimpledโ surface (i.e. it is marked with small indentations) rather than a smooth surface.
Under the same conditions and velocity, explain whether you expect the golf ball to exhibit more or less drag than a ping pong ball.
A dimpled surface will promote turbulence in the boundary layer.
This can delay separation by reducing the likelihood of stopped or reversed flow near the surface. Thus, a golf ball will exhibit less drag than a ping pong ball under the same flow conditions.
Note: Dimpling does increase the surface area slightly, so skin friction may be increased slightly as a result. For most real systems, (i.e. non-creeping flows), form drag is much more important than skin friction and reducing this generally compensates for higher skin friction.
Of course, introducing very deep dimples on the surface may increase the form drag by introducing significant vorticity near the surface โ hydrodynamics and aerodynamics often includes this kind of compromise.
68
Why does flow separation occur more easily in laminar than turbulent boundary layers?
Flow separation is a result of stopped/reversed flow in the boundary layer. In a laminar layer, we
have:
* Lower average velocity than a turbulent layer for the same stream velocity โ less kinetic energy and the flow is easier to stop.
* Slower rates of momentum transfer within the boundary layer, so it is difficult to move kinetic energy (momentum) from the bulk fluid to the surface.
* A thicker boundary layer for the same stream velocity โ momentum must be transferred further than in a turbulent system.
69
What assumptions are made about the HMMT mechanisms and structure of the fluid near the surface in Reynolds analogy?
(HMMT = heat, mass and momentum transfer)
* Turbulent boundary layer in which the laminar sub-layer and transition layer are negligible
* Heat, mass and momentum transfer is entirely via eddy motion
* Eddies propagate all the way to the surface
These assumptions mean that the Reynolds analogy is useful for systems at high Re (i.e. very turbulent flows with lots of eddies) and rough surfaces (to help introduce eddies near the surface).
Experimentally, it can be shown that Reynolds analogy is most appropriate to gas flows at high Re, over rough surfaces for systems in which flow separation does not occur.
70
What are the 3 main HMMT analogies?
Reynolds analogy
Film theory
J-factor analogy
71
What are the core assumptions of film theory? (HMMT analogy)
All resistances are in the laminar sub-layer. (This layer is the most important. It will be the rate determining part of the boundary layer)
Transition layer is negligible
Transfer is by molecular diffusion.
(Moderate Re - 10^3 to 10^5 - and smooth surfaces. It is still for turbulent boundary layers)
72
What is the relationship between R/rho*us2 , the Stanton number and the Prandtl number?
R/rho*us^2 = St*Pr
R/rho*us^2 = (h/(rho*us*Cp)*(mu*Cp/k)
This links the heat, mass and momentum balances
73
What equations are used when considering j-factor correlations?
Dittus-Boelter (heat and/or mass transfer) and Prandtl number
[Equations given]
74
What are characteristics of the j-factor analogy?
Empirical relationships based on lots of experimental data โ can be used for a wide range of systems:
* Gases or liquids
* Turbulent (Re > 104)
* Flow separation
* Limited to tubes and fully developed flow
75
What is the value for Re x (Re for pipe flow where Re = velocity x characteristic dimension / kinematic viscosity) for which flow transitions from laminar to turbulent?
At approx. 10^5 to 10^6 , flow transitions from laminar to turbulent.
Must memorise!
76
When is the Reynolds analogy valid?
Turbulent boundary layer (flow in the transition length
* Rough surfaces
* Very high Re
* Gases
* No flow separation
77
When is film theory valid?
* Smooth surfaces
* Turbulent flow
* Moderate Re
* Gas or liquid
* No flow separation
* Flow in the transition length
78
When is the Chilton and Colburn j-factor theory valid?
* Fully developed flow
* Turbulent flow (Re > 10,000)
* Pipe or tube flow
* Gas or liquid
* Flow separation
79
At what Re does flow transition (L to T) for open channel systems?
Re = 600
80
How do you calculate Re for open channel flow?
Re = rho*u*Rh / mu
Where Rh = hydraulic radius (A/P)
81
What is the transition Re for open, turbulent boundary layers?
Approx 10^5-10^6
82
What is Re for boundary layers in pipe flow?
Laminar: Re < 2000
Transition: Re ~ 2500
Turbulent: Re > 3000
83
What is maximum velocity equal to for fully developed flow? (In pipes)
Stream velocity.
Laminar: us = 2um
Turbulent: us = um/0.817
