Exercise Questions 2011

8/12/2019 Exercise Questions 2011

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Exercise Questions


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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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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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http://u%20fluctuation.jpg/http://u%20fluctuation.jpg/


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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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http://tbl%20profile.jpg/http://tbl%20profile.jpg/


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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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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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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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http://inertia%20subrange.jpg/http://inertia%20subrange.jpg/


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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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http://gaussian%20distribution.gif/http://gaussian%20distribution.gif/http://gaussian%20distribution.gif/


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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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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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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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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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Exercise Questions 2011