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8/12/2019 Exercise Questions 2011
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Exercise Questions
8/12/2019 Exercise Questions 2011
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Introduction to Turbulenceand Turbulent Flows
Students should attempt exercise questions as soon as relevant
materials are covered in the lecture.
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Introduction to Turbulenceand Turbulent Flows
Exercise 1 -- Scaling Laws
1. A box of volume L3 is filled with air in turbulent motion. Derive the
expression for the decay of turbulent kinetic energy, )(2
1 2222 wvuq ++=
as a function of time when no energy is fed or produced within.
[Hint] The differential equation for the decay of turbulent kinetic energy is
given by ( ) =2qdt
d, where is the dissipation rate.
2. When the Reynolds number is 2x105 in the above question, estimate the
velocity scale of energy containing eddies and the size of Kolmogorov
microscale, whereL= 2 m and = 1.5x10-5
m2/s.
[Hint] The Reynolds number is defined as
LuR
e= , while the Kolmogorov
microscale is given by 43
=e
RL .
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Introduction to Turbulenceand Turbulent Flows
3. A turbulent boundary layer is being developed over a flat plate in a wind
tunnel. An initial test shows that the wall shear stress at the location of
measurement is 1.31 x 10-2
Pa, where the kinematic viscosity and the
density of air are given by 1.5 x 10-5m2/s and 1.23 kg/m3, respectively.
Obtain the dissipation rate of the turbulent boundary layer at the edge of the
viscous sublayer, assuming that the turbulent velocity scale is represented
by the friction velocity. Explain why the dissipation rate is nearly maximum
at this location in the boundary layer.
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Introduction to Turbulenceand Turbulent Flows
Exercise 2 Energy Cascade
Even when viewed on TV, we can often distinguish between an eruption of
volcano and that of a scaled model by observing the pattern of fire flames and
smoke. Describe the reason why this is possible using appropriate relationships for
turbulent flows.
You should then illustrate your answer numerically by assuming values for typical
turbulence scales.
[Hint] Use the relationship: 43
Rel
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Introduction to Turbulenceand Turbulent Flows
Exercise 3 Energy Spectra
The time series of the velocity signal in a turbulent flow is expressed by
u= sin (t)
where, is the angular velocity and tis the time.
(a) Sketch the velocity signal as a function of time, clearly indicating its
amplitude and period.
(b) Sketch the energy spectrum of the velocity signal as a function of frequency.
(c) Give two examples of the flows where this type of energy spectrum can be
observed.
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Introduction to Turbulenceand Turbulent Flows
Exercise 4 Asymptotic Methods
If we consider turbulent shear flows with large Reynolds number (Rl), there
is an overlappedregion in the energy spectrum that satisfies
l small-scale end of the large-scale spectrum,E=E(,,S) and
0
large-scale end of the Kolmogorov spectrum,E=E(,,)
at the same time. Then, show that the inertial subrangeof energy spectrum can
be given by
E() ~ -5/3
where, is the wave number, l the scale of energy containing eddies, the
Kolmogorov scale, the dissipation rate and S (=u/l) the share rate.
[Hint] The dimensions ofE, ,areE= [L3T-2], = [L-1] and =[L2T-3].
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Introduction to Turbulenceand Turbulent Flows
Exercise 5 Probability Density Function
1. Sketch a random signal u(t) as a function of time t and obtain the
corresponding probability density function (PDF) using the graphical
technique as described in this lecturer.
2. Compare this PDF with that of the Gaussiandistribution, which is given by
( ) 21
2
22
2
)2/exp()(
uuBG
=
3. Repeat the question 1 and 2 above for a sine signal,
)sin()( ttu =
4. Observe major differences between the probability density function of a
random signal and that of a sinusoidal signal.
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Introduction to Turbulenceand Turbulent Flows
Exercise 6 Turbulence Modelling
1. The followings are the two-dimensional, steady Navier-Stokes equation for
boundary layer flows and the Continuity equation, where U0 is the free-
stream velocity.
0=y
V+x
U
y
U+
x
UU=
y
UV+
x
UU
2
20
0
Derive the corresponding Reynolds equation indicating the Reynolds stress
terms in the equation.
2. If the mixing length lmfor pipe flows is given by
R
y-10.06-
R
y-10.08-0.14=
R
l42
m
show that lm= 0.4yfor smally(i.e. y/R
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Introduction to Turbulenceand Turbulent Flows
3. In the -model, show that the turbulent viscosity tis given by
2kCt=
where, w+v+u=k222
21 and Cis a constant.
4. We wish to develop a new first order, two-equation turbulence model,
where the turbulent kinetic energy per unit mass Keand the acceleration of
energy containing eddiesAeare chosen as the two variables to be modelled.
(a) Based on dimensional analysis, derive plausible algebraic
dependencies of the turbulence velocity scale uand length scale l upon Ke
andAe.
(b) How are the dissipation rate and the turbulent kinematic viscosity T
expressed in terms ofKeandAe?
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Introduction to Turbulenceand Turbulent Flows
Exercise 7 Experimental Techniques
1. A turbulent boundary layer with freestream velocity U of 3 m/s is
developed over a flat plate in a wind tunnel under zero pressure gradient
condition, where the velocity profiles are measured with hot-wire
anemometer. At this speed, the friction velocity is approximately 1/25 of the
free-stream velocity. The kinematic viscosity and the density of air are 1.5 x
10-5
m2/s and 1.23 kg/m
3, respectively.
(a) It is known that the thickness of the viscous sublayer represents the smallest
scale of turbulence in the boundary layer. Determine the sensor length of the
hot wire that you should use in order to obtain an accurate energy spectrum
of velocity fluctuations from the measurement.
(b) The RMS (root-mean squared) value of voltage output from the hot-wire
anemometer is 1.8 V at the edge of the viscous sublayer, where the local
mean velocity is 1.3 m/s. Obtain the turbulence intensity (u'/U) when the
amplifier setting of the signal conditioner is 50. The calibration constants
for the hot-wire sensor are A = 1.36 and B = 0.66 when Kings law is used.
(c) Boundary layer profiles are measured using hot-wire anemometer in
constant temperature mode, where ambient temperature went up a fewdegrees during the test. Describe how this will affect the velocity
measurement, assuming that the operating temperature of the hot wire is
fixed throughout the test. Justify your answer by considering the heat
transfer balance of a hot-wire sensor.
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Introduction to Turbulenceand Turbulent Flows
2. Water of 20C is flowing through a 0.10m diameter smooth pipe at a bulk
velocity of 3m/s. Estimate the absolute error in static pressure measurement if
a 5mm diameter square-edged tap is used. The kinematic viscosity and the
density of water are 1.0 x 10-6m2/s and 1.0 x 103kg/m3, respectively.
3. The hydrogen bubble technique will be used in an experimental investigation
of flow around a circular cylinder in a uniform stream, where the Reynolds
number is above the critical value for vortex shedding.
(a) Describe the basic principle of this technique, including an appropriate
experimental set-up for this particular flow situation.
(b) What problems are anticipated in interpreting the results?
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