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The mapping of vortex fields around single and dual bluff bodies
J. P. Bentley and A. R. Nichols- The mapping of vortex fields around single and dual bluff bodies
278
The mapping of vortex fields around single and dual bluff bodies J. P. BENTLEY and A. R. N I C H O L S
The development and use of a system to measure important parameters of the vortex field in a coordinate grid around several single bluff bodies and dual bluff body combinations are described. Two hot wire sensors were used: a reference sensor at a fixed position and a moving sensor with position coordinates adjusted by using computer control. The strength, frequency and regularity of the vortex shedding are found from the auto-power spectral density of the moving sensor signal. The phase difference between moving and reference sensor signals is found from the cross-power spectral density function between the signals. The results are presented as maps of vortex parameters plotted as a function of sensor position coordinates. The main features of these maps, including the enhancement of vortex shedding from certain dual bluff body combinations, is then discussed. Keywords: mapping, vortex fields, bluff bodies, hot wire sensors
Nomenclature M
R d D
mm mm
s x y U0
mm mm mm m s -1
Uw
u(t)
ms -~ ms -~
u
ms -~
Uv E
m s-~ V
e(t)
V
f Hz PMM(f) V2 HZ -1 PRR(f)
V2 Hz -1
PuR(f) V 2 Hz -I W
W
Wv
W
WT
W
S/N moving sensor reference sensor width of vortex flowmeter internal diameter of wind tunnel bluff body separation (near edges) sensor x-coordinate sensor y-coordinate free stream velocity at pipe centre mean velocity at sensor fluctuating velocity component at sensor RMS value of total velocity fluctuations RMS value of vortex velocity mean value of sensor circuit output voltage fluctuating component of sensor output frequency auto-power spectral density function for moving sensor auto-power spectral density function for reference sensor cross-power spectral density function for moving-reference total power of sensor signal fluctuation power of vortex component in sensor signal power of turbulence component
Dr J. P. Bentley is with the School of Science and Technology at Teeside Polytechnic, UK, and Dr A. R. Nichols is with the Department of Energy at Aberdeen, UK 0955-5986/90/050278-09
dB
e
V
ev
V
RMR(,8) V 2
deg
vortex signal power to turbulence power ratio RMS value of total fluctuations RMS value of vortex component cross-correlation function between moving and reference sensor signals phase difference between moving and reference sensor signals
Introduction In order to understand the phenomenon of vortex shedding it is important to have quantitative information on the variations in vortex field around the bluff bodies that create the vortices. Moreover it has been shown 1,2 that certain combinations of two bluff bodies in tandem give stronger and more regular shedding than single bluff bodies with the same blockage ratio. This vortex enhancement occurs when the separation of the bluff bodies is small and less than the critical separation ~. Again it is important to have quantitative information on the vortex field around and in between the bluff bodies in order to establish the exact mechanism of this vortex enhancement. In principle, quantitative data on vortex shedding could be obtained from numerical solution of the appropriate Navier-Stokes and continuity equations. However, in practice computation times of several hours are necessary to obtain solutions in simple cases, e.g. a single rectangular body at low Reynolds numbers 4,s. These times would be even greater for two bluff bodies at higher Reynolds numbers, where a satisfactory turbulence model is necessary. Experimental methods are therefore required; flow visualization can provide qualitative information on vortex (~ 1990 Butterworth-Heinemann Ltd
Flow Meas. Instrum.
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Vol I October 1990
shedding. For example, Igarashi 6 has used smoke tunnel visualization techniques to confirm the nature of boundary flows in the narrow gap between two halves of a circular cylinder. However, if quantitative experimental data are required, then the two most useful techniques are laser Doppler anemometry and hot wire anemometry 7. This paper describes a dual hot wire system for quantitative measurement of vortex fields and presents results for both single rectangular bluff bodies and combinations of two rectangular bluff bodies in tandem. The measurement system consists of two hot wire sensors; a reference sensor with fixed position coordinates and a moving sensor with its position adjusted under computer control over a coordinate grid. The strength, regularity and frequency of the vortex shedding are then found from the auto-power spectral density function of the moving sensor signal. The phase difference between moving and reference sensor signals is found from cross-power density function between the signals. The results are presented as maps of vortex parameters plotted as a function of sensor position coordinates.
Wind tunnel and measurement system
Figure 1 shows the wind tunnel and associated system used for vortex measurement. The wind tunnel has an internal diameter D of 300 mm and the free stream velocity U0 (at pipe centre) can be varied, under computer control, between 0.7 and 1 4 m s -1. The control loop consists of an ultrasonic cross-correlation velocity meter (to measure U0), a microcomputer and a stepper motor that adjusts the position of the fan speed-adjustment potentiometer.
The vortex measurement system consists of two hot wire sensors, a reference sensor and a moving sensor. The velocity at each sensor consists of a mean or d.c. component Uw and a fluctuating component u(t); the fluctuating component in general consists of a sinusoidal vortex component and a random turbulence component. It is important that the fluctuating component of the reference signal contains as little turbulence as possible, i.e. is as close as possible to the sinusoidal vortex signal. For this reason the reference sensor is a 100mm long nickel wire so that random turbulence tends to average out over the length of the wire. The (x, ,v) position coordinates of the reference sensor are adjusted manually until a point is reached where the most regular vortex signal is obtained and then fixed at that position. The moving sensor is a 10 mm nickel wire and is moved automatically under computer control in a predetermined grid around the bluff bodies in the x - y plane. The positioning of the sensor is controlled from the microcomputer by using two stepper motors: one for x movement and one for y. The search grid takes account of the coordinates of the bluff bodies and the reference sensor so that any possible collision is avoided. Both sensors are incorporated into constant temperature circuits, in each case the relation between mean velocity Uw and mean circuit output voltage E is nonlinear and can be accurately represented by the quartic equation Uw = f(E) where (1)
f(E) = ao + a l E + a2E 2 + a3E 3 + a4E 4
In addition there is a fluctuating component e(t) in the circuit output voltage corresponding to the velocity fluctuations u(t) due to turbulence and vortices. Figure 2 shows the vortex measurement system in further detail. The Kent cross-correlation flowmeter
Kent
I
~-
Correlator
Vortex
measurement
~r-Yersing Ultrasonic receiver/ transmitter
v
.e,eren0e seo
>
r
1T thin
> > 0 Q Foam
300 mm Ten section
rubber seal Fan speed controller
< Figure I Wind tunnel arrangement
5m
>
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
280
eM (t)
UM
CTA
Moving sensor
/.JR Reference sensor
circuit
iill
Solartron spectrum
Frequency counter
analyser
CTA circuit
IEEEbus
l
Voltage scalingand smoothing
IEEE interface
Monitor
ER t EM Kent correlator
l
Voltage scaling
I
BBC masterturbo microcomputer
Disk drive
Printer
Figure 2 Schemat~'cdiagram of the measurement system gives a O - 1 0 V output signal proportional to free stream velocity U0 (at pipe centre) in the range 0 to 20 ms -~. This voltage is then scaled and applied to the analogue input port of a BBC microcomputer. The moving and reference sensor circuit output voltages are filtered to remove the fluctuating components, while the remaining mean values EM and ER are scaled and applied to a second analogue input port. The fluctuating voltages eu(t) and eR(t) consisting of vortex and turbulence components are connected to a Solartron 1201 FFT spectrum analyser and a Phillips 6673 frequency counter. The analyser calculates the auto-power spectral density functions PMM(f) for eM(t), PRR(f) for eR(t) and the cross-power density function PMR(f) between eM(t) and eR(t). These functions are specified by 501 points and are transferred to the computer via an IEEE bus. The counter measures the period for each of 300 cycles of the vortex component in the moving sensor signal eM(t) and these 300 period measurements are again transferred to the computer via the IEEE bus.
variables specified below. The calculations use the input measurement data specified in section 3. Free stream velocity This (U0) was obtained by taking 300 samples of the scaled input voltage from the Kent correlator, computing the average and multiplying by an appropriate conversion factor. Mean velocity at sensor This (Uw) wa~ calculated for both moving and reference sensors. Again 300 samples of the scaled mean value E of the sensor circuit output voltage were taken and averaged. Uw is computed from this average value by using U = 0.5If(E+ e) + f ( E - e)]
(2)
where f(E) is defined by equation (1) and e by equation (5).
Vortex signal power to turbulence power ratio Calculation of vortex variables The system incorporates a BBC Master Microcomputer with monitor, 80OK disk drive and printer. This was used to calculate, store, display and print the vortex
This (S/N) is calculated for both the moving and reference sensor by using the auto-power spectral density functions PMM(f) and PaR(f). The computer calculates the total area under each function to give
Flow Meas. Instrum.
Vol I October 1990
281
the total power W of the sensor signal fluctuations. It also calculates the area under the vortex signal peak to give the power Wv of the vortex component in e(t). The power W of the turbulence component in e(t) is
We assume that the reference sensor signal is a pure sinusoidal vortex signal of frequency f, i.e.
(3)
and the moving sensor signal consists of a vortex component at the same frequency f, but with a phase difference 9, together with a random turbulence component n(t), i.e.
W T =
W-
W v
and the corresponding power ratio, i.e. signal-to-noise ratio is S N
10 log
dB
(4)
eR(t) = asin2#ft
(10)
e(t) = bsin(27rft + 4) + n(t)
(11)
This gives
Root m e a n square total v e l o c i t y
This (u) is again calculated for both moving and reference sensor signals. The RMS value of the total signal e(t)is e = W 1/2 (5)
RMR(,8) = ab/2 cos(2~f/~ + q~)
(12)
i.e. u is equal to one half of the difference between f(E + e) and f ( E - e).
i.e. the cross-correlation function is a cosine function at the vortex frequency f, with a phase difference of and no turbulence correlation present. Thus the phase difference ~ between the moving and sensor vortex signals, in the presence of random turbulence, can be found from the cross-correlation function. Here we measure ~ by using the cross-power spectral density function P(f), which is the Fourier transform of R(/~); this is a peak at frequency f, with magnitude ab/2 and phase 4.
Root m e a n s q u a r e v o r t e x v e l o c i t y
Experimental
By using similar arguments, the RMS value of the vortex component in e(t) is
Figure 3 specifies the single bluff bodies and dual bluff
The corresponding RMS value u of the total velocity fluctuations u(t) is found using u = 0.5If(E+ e)-
f(E-
e)]
ev = Wlv/2
(6)
(7)
Bluff combination
(8)
.I II
and the RMS vortex velocity is Uv = 0.5[f(E + ev) - f(E - ev)] for both moving and reference sensors.
programme
V o r t e x f r e q u e n c i e s f r o m s p e c t r u m analyser
The reference sensor auto-power spectral density function PRR(f) is first searched. The vortex frequency peak is very well defined, being very much higher than the rest. The frequency is measured from a centre of area calculation on this peak; this is the frequency that divides the area into two equal parts. The next highest peak corresponds to twice the vortex frequency; this frequency is found in a similar way. The moving auto-power spectral density function PMM(f) is then searched by using the above reference frequencies as target frequencies. This is because the moving sensor signal may contain more turbulence than the reference sensor signal, so that the vortex frequency peaks may be less well defined. Vortex frequency from counter
The reciprocal of each of 300 periods of the moving sensor signal from the counter is calculated to give 300 frequency values and the mean frequency then found. This is a check on the above spectrum analyser values. Phase d i f f e r e n c e b e t w e e n sensor a n d r e f e r e n c e v o r t e x signals
The cross-correlation function between moving and reference sensor signals eM(t) and eR(t) is defined by 7"
RMR(~ = limit I_'° eR(t -- /~)eM(t)dt To--,= T 1O Ju
(9)
---II
Bluff number
Turbulence level
Bluff separation (mm)
10
low low
50
1
low
10
3
low low medium
15 10 10
low
20
low low low
35 35 25
low low medium
15 60 15
medium
10 (1)
--II ---- II
II II II
8 13
square
low medium
plate
low
(2)
Note (1). The movement of air in the gap between the bluff bodies was blocked using a thin sheet of plastic to provide an air-tight seal. Note (2}. The free stream velocity was reduced to 2.5 m s-1 to initiate a weak vortex shedding regime.
Figure3 List of the 17 experiments undertaken for vortex mapping
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
282
body combinations for which vortex mapping was done. These combinations were chosen as a result of earlier work 1'2, which showed that certain dual combinations, with certain separations, give stronger and more regular shedding than the equivalent single bluff bodies. Other combinations, however, give weaker and less regular shedding than the equivalent single bluff bodies. The width d of each combination was 25 mm and since the internal diameter D of the wind tunnel was 300 mm, the blockage ratio d i d was 0.083. All combinations were tested at a free stream velocity U0 = 6.5 ms -1 and at one or two levels of turbulence intensity: low (0.5%) and medium (1.5%). The moving sensor search grid will be different for each combination. Figure 4 shows the search grid for dual combination 1.
y-axis
5 mm --~ ~---
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)
x
x
,
=
: : : :
:
x-axis
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Figure 4 Search grid around bluff body combination 1
Discussion of results Four maps are presented for each geometry. These are: [ ] Contour plot of RMS vortex velocity Uv (expressed as a percentage of free stream velocity) versus moving sensor (x,y) coordinates [ ] Contour plot of 5/N versus (x, y) [ ] Contour plot of mean velocity at sensor Uw (expressed as a percentage of free stream) versus (x, y) [ ] Phasor plot of phase difference q~ between moving and reference sensor signals versus (x, y) Results for single bluff bodies and dual combinations, which show enhanced shedding relative to single bluff bodies are shown. Figure 5 shows the maps for a single square 25 mm x 25 mm at low turbulence intensity. Maximum vortex strength of 20% is obtained at x = 40 mm, y~- 23 mm. We see from the phase map that a strong, periodic correlation between phases is established within a few millimetres of the rear edge of the bluff body. Figure 6 shows the maps for a single plate 25 mm x 2.5 mm at low turbulence. Maximum vortex strength of 14% is obtained at x = 1 1 5 m m , y = 24 mm, i.e. much further behind the bluff body than for the square. Correspondingly, a strong periodic correlation between phases only begins to become established approximately 35 mm behind the bluff body.
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Figure 5 (a) Vortex strength (Uv/Uo(%), contour step 2%); (b) S//N ratio (dB, contour step 2 dB); (c) average velocity Uw (% Uo, contour step 20%) and (d) phase difference ~# for the square bluff body (25 mm x 25 mm) at low turbulence. Reference coordinates are (75, 25)
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284
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
Figure 7 shows the maps for dual combination 1 comprising the thin plate 10mm upstream of the square, at low turbulence. Maximum vortex strength of 32% is obtained over a wide area x~-5 to 25 mm and y = 14 to 22 mm. The strength of vortex shedding is therefore considerably enhanced relative to the two individual bluff bodies. From the phase map, we see that the phases of the boundary layers in the gap are not only strongly correlated with each other, but also with the phases of oscillations of boundary layers moving around the bluff bodies. Figure 8 shows the maps for dual combination 1, but with the gap between the bluff bodies sealed by a thin sheet of plastic. This means that boundary layers of fluid cannot move through the gap. The strength and regularity of vortex shedding are now significantly reduced. The phases of oscillations of the boundary layers in the map are not only not correlated with each other, but are also not correlated with the oscillations of the boundary layers moving around the bluff bodies. Figure 9 shows the maps for dual combination 3 made up of the thin plate 25 mm x 2.5 mm, 15 mm upstream of a rectangle 25 mm x 12.5 mm, at low turbulence intensity. Maximum vortex strength of 32% and signal to noise ratio of +10 dB is obtained just above the downstream corner of the second bluff body. Again the strength and regularity of vortex shedding are significantly enhanced relative to the two individual single bluff bodies. Figure 10 shows the maps for dual combination 5
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Conclusions The results confirm that the system can provide quantitative maps of the vortex field around single bluff bodies and dual bluff body combinations. The results for the single rectangular bluff body show the effect of changing the depth of the bluff body on the vortex field. With a thin plate, the vortex field is much further downstream than for a square. For dual bluff body combinations, the results confirm that the movement of boundary layers through the gap is essential to vortex shedding enhancement. Moreover this enhancement only takes place if the phases of the boundary oscillations in the gap are closely related to the phases of oscillating layers moving around the bluff bodies. The results therefore support the mechanism proposed by Igarashi ° for vortex enhancement. Figure 11 shows the enhancement mechanism for two separate half cycles. During
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again made up from the above thin plate and rectangle, but this time the rectangle is 20 mm upstream from the thin plate. The test was again performed at low turbulence. There are two positions where vortex strength and signal to noise ratio have maximum values: around 30% and 12 dB respectively; one just above the upstream bluff body, the other just above the downstream bluff body. This dual combination also outperforms the individual single bluff bodies.
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Flow Meas. Instrum.
Vol 1 October 1990
285
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J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
286
High velocity low pressure
a
Low velocity high pressure
b
Figure I I Movement of shear layers on halves of the vortex cycle
separate
the first half-cycle, boundary layers move up through the gap to join with layers moving above the bluff bodies to give enhanced vortex formation behind the top surface of the downstream bluff body (Figure 1 la). During the next half-cycle, boundary layers move down through the gap to join with layers moving underneath the bluff bodies to give enhanced vortex formation behind the bottom surface of the downstream bluff body (Figure 1 lb).
References I Bentley, J. P. 'Advantages of and developments in dual bluff body vortex flowmeters' In Advances in Ultrasonic and Vortex Flowmeters, Inst. Meas. Control seminar, London, February 1989 2 Bentley, J. P. and Nichols A. R. 'Experimental measurement of vortex shedding from different arrangements of two rectangular bluff bodies in tandem' In FLUCOME '88, Sheffield, September 1988 3 Zdrakovich, M. M. 'Flow induced oscillations of two interfering circular cylinders' In Int. Conf. on Flow Induced Vibrations in Fluid Engineering, Reading, September 1982 4 Davis, R. W. and Moore, E. F. A numerical study of vortex shedding from rectangles. ]. Fluid Mech. 116 (1982) 475-506 5 Davis, R. W., Moore, E. F. and Purtell, L. P. A numericalexperimental study of confined around rectangular cylinders. Phys. Fluids 27(1) (1984) 46-59 6 Igarashi, T. 'Fluid flow around a bluff body used for a Karman vortex flowmeter' In FLUCOME '85, Tokyo, 1985 7 Larsen, P. S. and Buchave, P. 'Flow measurements: why, what and how? Part 1' Dantec Information No. 01 (June 1985)
278
The mapping of vortex fields around single and dual bluff bodies J. P. BENTLEY and A. R. N I C H O L S
The development and use of a system to measure important parameters of the vortex field in a coordinate grid around several single bluff bodies and dual bluff body combinations are described. Two hot wire sensors were used: a reference sensor at a fixed position and a moving sensor with position coordinates adjusted by using computer control. The strength, frequency and regularity of the vortex shedding are found from the auto-power spectral density of the moving sensor signal. The phase difference between moving and reference sensor signals is found from the cross-power spectral density function between the signals. The results are presented as maps of vortex parameters plotted as a function of sensor position coordinates. The main features of these maps, including the enhancement of vortex shedding from certain dual bluff body combinations, is then discussed. Keywords: mapping, vortex fields, bluff bodies, hot wire sensors
Nomenclature M
R d D
mm mm
s x y U0
mm mm mm m s -1
Uw
u(t)
ms -~ ms -~
u
ms -~
Uv E
m s-~ V
e(t)
V
f Hz PMM(f) V2 HZ -1 PRR(f)
V2 Hz -1
PuR(f) V 2 Hz -I W
W
Wv
W
WT
W
S/N moving sensor reference sensor width of vortex flowmeter internal diameter of wind tunnel bluff body separation (near edges) sensor x-coordinate sensor y-coordinate free stream velocity at pipe centre mean velocity at sensor fluctuating velocity component at sensor RMS value of total velocity fluctuations RMS value of vortex velocity mean value of sensor circuit output voltage fluctuating component of sensor output frequency auto-power spectral density function for moving sensor auto-power spectral density function for reference sensor cross-power spectral density function for moving-reference total power of sensor signal fluctuation power of vortex component in sensor signal power of turbulence component
Dr J. P. Bentley is with the School of Science and Technology at Teeside Polytechnic, UK, and Dr A. R. Nichols is with the Department of Energy at Aberdeen, UK 0955-5986/90/050278-09
dB
e
V
ev
V
RMR(,8) V 2
deg
vortex signal power to turbulence power ratio RMS value of total fluctuations RMS value of vortex component cross-correlation function between moving and reference sensor signals phase difference between moving and reference sensor signals
Introduction In order to understand the phenomenon of vortex shedding it is important to have quantitative information on the variations in vortex field around the bluff bodies that create the vortices. Moreover it has been shown 1,2 that certain combinations of two bluff bodies in tandem give stronger and more regular shedding than single bluff bodies with the same blockage ratio. This vortex enhancement occurs when the separation of the bluff bodies is small and less than the critical separation ~. Again it is important to have quantitative information on the vortex field around and in between the bluff bodies in order to establish the exact mechanism of this vortex enhancement. In principle, quantitative data on vortex shedding could be obtained from numerical solution of the appropriate Navier-Stokes and continuity equations. However, in practice computation times of several hours are necessary to obtain solutions in simple cases, e.g. a single rectangular body at low Reynolds numbers 4,s. These times would be even greater for two bluff bodies at higher Reynolds numbers, where a satisfactory turbulence model is necessary. Experimental methods are therefore required; flow visualization can provide qualitative information on vortex (~ 1990 Butterworth-Heinemann Ltd
Flow Meas. Instrum.
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Vol I October 1990
shedding. For example, Igarashi 6 has used smoke tunnel visualization techniques to confirm the nature of boundary flows in the narrow gap between two halves of a circular cylinder. However, if quantitative experimental data are required, then the two most useful techniques are laser Doppler anemometry and hot wire anemometry 7. This paper describes a dual hot wire system for quantitative measurement of vortex fields and presents results for both single rectangular bluff bodies and combinations of two rectangular bluff bodies in tandem. The measurement system consists of two hot wire sensors; a reference sensor with fixed position coordinates and a moving sensor with its position adjusted under computer control over a coordinate grid. The strength, regularity and frequency of the vortex shedding are then found from the auto-power spectral density function of the moving sensor signal. The phase difference between moving and reference sensor signals is found from cross-power density function between the signals. The results are presented as maps of vortex parameters plotted as a function of sensor position coordinates.
Wind tunnel and measurement system
Figure 1 shows the wind tunnel and associated system used for vortex measurement. The wind tunnel has an internal diameter D of 300 mm and the free stream velocity U0 (at pipe centre) can be varied, under computer control, between 0.7 and 1 4 m s -1. The control loop consists of an ultrasonic cross-correlation velocity meter (to measure U0), a microcomputer and a stepper motor that adjusts the position of the fan speed-adjustment potentiometer.
The vortex measurement system consists of two hot wire sensors, a reference sensor and a moving sensor. The velocity at each sensor consists of a mean or d.c. component Uw and a fluctuating component u(t); the fluctuating component in general consists of a sinusoidal vortex component and a random turbulence component. It is important that the fluctuating component of the reference signal contains as little turbulence as possible, i.e. is as close as possible to the sinusoidal vortex signal. For this reason the reference sensor is a 100mm long nickel wire so that random turbulence tends to average out over the length of the wire. The (x, ,v) position coordinates of the reference sensor are adjusted manually until a point is reached where the most regular vortex signal is obtained and then fixed at that position. The moving sensor is a 10 mm nickel wire and is moved automatically under computer control in a predetermined grid around the bluff bodies in the x - y plane. The positioning of the sensor is controlled from the microcomputer by using two stepper motors: one for x movement and one for y. The search grid takes account of the coordinates of the bluff bodies and the reference sensor so that any possible collision is avoided. Both sensors are incorporated into constant temperature circuits, in each case the relation between mean velocity Uw and mean circuit output voltage E is nonlinear and can be accurately represented by the quartic equation Uw = f(E) where (1)
f(E) = ao + a l E + a2E 2 + a3E 3 + a4E 4
In addition there is a fluctuating component e(t) in the circuit output voltage corresponding to the velocity fluctuations u(t) due to turbulence and vortices. Figure 2 shows the vortex measurement system in further detail. The Kent cross-correlation flowmeter
Kent
I
~-
Correlator
Vortex
measurement
~r-Yersing Ultrasonic receiver/ transmitter
v
.e,eren0e seo
>
r
1T thin
> > 0 Q Foam
300 mm Ten section
rubber seal Fan speed controller
< Figure I Wind tunnel arrangement
5m
>
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
280
eM (t)
UM
CTA
Moving sensor
/.JR Reference sensor
circuit
iill
Solartron spectrum
Frequency counter
analyser
CTA circuit
IEEEbus
l
Voltage scalingand smoothing
IEEE interface
Monitor
ER t EM Kent correlator
l
Voltage scaling
I
BBC masterturbo microcomputer
Disk drive
Printer
Figure 2 Schemat~'cdiagram of the measurement system gives a O - 1 0 V output signal proportional to free stream velocity U0 (at pipe centre) in the range 0 to 20 ms -~. This voltage is then scaled and applied to the analogue input port of a BBC microcomputer. The moving and reference sensor circuit output voltages are filtered to remove the fluctuating components, while the remaining mean values EM and ER are scaled and applied to a second analogue input port. The fluctuating voltages eu(t) and eR(t) consisting of vortex and turbulence components are connected to a Solartron 1201 FFT spectrum analyser and a Phillips 6673 frequency counter. The analyser calculates the auto-power spectral density functions PMM(f) for eM(t), PRR(f) for eR(t) and the cross-power density function PMR(f) between eM(t) and eR(t). These functions are specified by 501 points and are transferred to the computer via an IEEE bus. The counter measures the period for each of 300 cycles of the vortex component in the moving sensor signal eM(t) and these 300 period measurements are again transferred to the computer via the IEEE bus.
variables specified below. The calculations use the input measurement data specified in section 3. Free stream velocity This (U0) was obtained by taking 300 samples of the scaled input voltage from the Kent correlator, computing the average and multiplying by an appropriate conversion factor. Mean velocity at sensor This (Uw) wa~ calculated for both moving and reference sensors. Again 300 samples of the scaled mean value E of the sensor circuit output voltage were taken and averaged. Uw is computed from this average value by using U = 0.5If(E+ e) + f ( E - e)]
(2)
where f(E) is defined by equation (1) and e by equation (5).
Vortex signal power to turbulence power ratio Calculation of vortex variables The system incorporates a BBC Master Microcomputer with monitor, 80OK disk drive and printer. This was used to calculate, store, display and print the vortex
This (S/N) is calculated for both the moving and reference sensor by using the auto-power spectral density functions PMM(f) and PaR(f). The computer calculates the total area under each function to give
Flow Meas. Instrum.
Vol I October 1990
281
the total power W of the sensor signal fluctuations. It also calculates the area under the vortex signal peak to give the power Wv of the vortex component in e(t). The power W of the turbulence component in e(t) is
We assume that the reference sensor signal is a pure sinusoidal vortex signal of frequency f, i.e.
(3)
and the moving sensor signal consists of a vortex component at the same frequency f, but with a phase difference 9, together with a random turbulence component n(t), i.e.
W T =
W-
W v
and the corresponding power ratio, i.e. signal-to-noise ratio is S N
10 log
dB
(4)
eR(t) = asin2#ft
(10)
e(t) = bsin(27rft + 4) + n(t)
(11)
This gives
Root m e a n square total v e l o c i t y
This (u) is again calculated for both moving and reference sensor signals. The RMS value of the total signal e(t)is e = W 1/2 (5)
RMR(,8) = ab/2 cos(2~f/~ + q~)
(12)
i.e. u is equal to one half of the difference between f(E + e) and f ( E - e).
i.e. the cross-correlation function is a cosine function at the vortex frequency f, with a phase difference of and no turbulence correlation present. Thus the phase difference ~ between the moving and sensor vortex signals, in the presence of random turbulence, can be found from the cross-correlation function. Here we measure ~ by using the cross-power spectral density function P(f), which is the Fourier transform of R(/~); this is a peak at frequency f, with magnitude ab/2 and phase 4.
Root m e a n s q u a r e v o r t e x v e l o c i t y
Experimental
By using similar arguments, the RMS value of the vortex component in e(t) is
Figure 3 specifies the single bluff bodies and dual bluff
The corresponding RMS value u of the total velocity fluctuations u(t) is found using u = 0.5If(E+ e)-
f(E-
e)]
ev = Wlv/2
(6)
(7)
Bluff combination
(8)
.I II
and the RMS vortex velocity is Uv = 0.5[f(E + ev) - f(E - ev)] for both moving and reference sensors.
programme
V o r t e x f r e q u e n c i e s f r o m s p e c t r u m analyser
The reference sensor auto-power spectral density function PRR(f) is first searched. The vortex frequency peak is very well defined, being very much higher than the rest. The frequency is measured from a centre of area calculation on this peak; this is the frequency that divides the area into two equal parts. The next highest peak corresponds to twice the vortex frequency; this frequency is found in a similar way. The moving auto-power spectral density function PMM(f) is then searched by using the above reference frequencies as target frequencies. This is because the moving sensor signal may contain more turbulence than the reference sensor signal, so that the vortex frequency peaks may be less well defined. Vortex frequency from counter
The reciprocal of each of 300 periods of the moving sensor signal from the counter is calculated to give 300 frequency values and the mean frequency then found. This is a check on the above spectrum analyser values. Phase d i f f e r e n c e b e t w e e n sensor a n d r e f e r e n c e v o r t e x signals
The cross-correlation function between moving and reference sensor signals eM(t) and eR(t) is defined by 7"
RMR(~ = limit I_'° eR(t -- /~)eM(t)dt To--,= T 1O Ju
(9)
---II
Bluff number
Turbulence level
Bluff separation (mm)
10
low low
50
1
low
10
3
low low medium
15 10 10
low
20
low low low
35 35 25
low low medium
15 60 15
medium
10 (1)
--II ---- II
II II II
8 13
square
low medium
plate
low
(2)
Note (1). The movement of air in the gap between the bluff bodies was blocked using a thin sheet of plastic to provide an air-tight seal. Note (2}. The free stream velocity was reduced to 2.5 m s-1 to initiate a weak vortex shedding regime.
Figure3 List of the 17 experiments undertaken for vortex mapping
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
282
body combinations for which vortex mapping was done. These combinations were chosen as a result of earlier work 1'2, which showed that certain dual combinations, with certain separations, give stronger and more regular shedding than the equivalent single bluff bodies. Other combinations, however, give weaker and less regular shedding than the equivalent single bluff bodies. The width d of each combination was 25 mm and since the internal diameter D of the wind tunnel was 300 mm, the blockage ratio d i d was 0.083. All combinations were tested at a free stream velocity U0 = 6.5 ms -1 and at one or two levels of turbulence intensity: low (0.5%) and medium (1.5%). The moving sensor search grid will be different for each combination. Figure 4 shows the search grid for dual combination 1.
y-axis
5 mm --~ ~---
x x
)
x
x
,
=
: : : :
:
x-axis
Coordinate origin (0, O)
Figure 4 Search grid around bluff body combination 1
Discussion of results Four maps are presented for each geometry. These are: [ ] Contour plot of RMS vortex velocity Uv (expressed as a percentage of free stream velocity) versus moving sensor (x,y) coordinates [ ] Contour plot of 5/N versus (x, y) [ ] Contour plot of mean velocity at sensor Uw (expressed as a percentage of free stream) versus (x, y) [ ] Phasor plot of phase difference q~ between moving and reference sensor signals versus (x, y) Results for single bluff bodies and dual combinations, which show enhanced shedding relative to single bluff bodies are shown. Figure 5 shows the maps for a single square 25 mm x 25 mm at low turbulence intensity. Maximum vortex strength of 20% is obtained at x = 40 mm, y~- 23 mm. We see from the phase map that a strong, periodic correlation between phases is established within a few millimetres of the rear edge of the bluff body. Figure 6 shows the maps for a single plate 25 mm x 2.5 mm at low turbulence. Maximum vortex strength of 14% is obtained at x = 1 1 5 m m , y = 24 mm, i.e. much further behind the bluff body than for the square. Correspondingly, a strong periodic correlation between phases only begins to become established approximately 35 mm behind the bluff body.
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b
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284
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
Figure 7 shows the maps for dual combination 1 comprising the thin plate 10mm upstream of the square, at low turbulence. Maximum vortex strength of 32% is obtained over a wide area x~-5 to 25 mm and y = 14 to 22 mm. The strength of vortex shedding is therefore considerably enhanced relative to the two individual bluff bodies. From the phase map, we see that the phases of the boundary layers in the gap are not only strongly correlated with each other, but also with the phases of oscillations of boundary layers moving around the bluff bodies. Figure 8 shows the maps for dual combination 1, but with the gap between the bluff bodies sealed by a thin sheet of plastic. This means that boundary layers of fluid cannot move through the gap. The strength and regularity of vortex shedding are now significantly reduced. The phases of oscillations of the boundary layers in the map are not only not correlated with each other, but are also not correlated with the oscillations of the boundary layers moving around the bluff bodies. Figure 9 shows the maps for dual combination 3 made up of the thin plate 25 mm x 2.5 mm, 15 mm upstream of a rectangle 25 mm x 12.5 mm, at low turbulence intensity. Maximum vortex strength of 32% and signal to noise ratio of +10 dB is obtained just above the downstream corner of the second bluff body. Again the strength and regularity of vortex shedding are significantly enhanced relative to the two individual single bluff bodies. Figure 10 shows the maps for dual combination 5
"=
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,
~"~
Conclusions The results confirm that the system can provide quantitative maps of the vortex field around single bluff bodies and dual bluff body combinations. The results for the single rectangular bluff body show the effect of changing the depth of the bluff body on the vortex field. With a thin plate, the vortex field is much further downstream than for a square. For dual bluff body combinations, the results confirm that the movement of boundary layers through the gap is essential to vortex shedding enhancement. Moreover this enhancement only takes place if the phases of the boundary oscillations in the gap are closely related to the phases of oscillating layers moving around the bluff bodies. The results therefore support the mechanism proposed by Igarashi ° for vortex enhancement. Figure 11 shows the enhancement mechanism for two separate half cycles. During
.... - 2 4
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again made up from the above thin plate and rectangle, but this time the rectangle is 20 mm upstream from the thin plate. The test was again performed at low turbulence. There are two positions where vortex strength and signal to noise ratio have maximum values: around 30% and 12 dB respectively; one just above the upstream bluff body, the other just above the downstream bluff body. This dual combination also outperforms the individual single bluff bodies.
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Flow Meas. Instrum.
Vol 1 October 1990
285
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Figure 10 (a) Vortex strength (Uv/Uo(%), contour step 4%); (b) S/N ratio (dB, contour step 3 dB); (c) average velocity Uw (% Uo, contour step 20%) and (d) phase difference dp for combination 5 in the low turbulence, strong shedding regime (separation of 20 mm). Reference coordinates are (0, 20)
J. P. Bentley and A. R. Nichols - The mapping of vortex fields around single and dual bluff bodies
286
High velocity low pressure
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Low velocity high pressure
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Figure I I Movement of shear layers on halves of the vortex cycle
separate
the first half-cycle, boundary layers move up through the gap to join with layers moving above the bluff bodies to give enhanced vortex formation behind the top surface of the downstream bluff body (Figure 1 la). During the next half-cycle, boundary layers move down through the gap to join with layers moving underneath the bluff bodies to give enhanced vortex formation behind the bottom surface of the downstream bluff body (Figure 1 lb).
References I Bentley, J. P. 'Advantages of and developments in dual bluff body vortex flowmeters' In Advances in Ultrasonic and Vortex Flowmeters, Inst. Meas. Control seminar, London, February 1989 2 Bentley, J. P. and Nichols A. R. 'Experimental measurement of vortex shedding from different arrangements of two rectangular bluff bodies in tandem' In FLUCOME '88, Sheffield, September 1988 3 Zdrakovich, M. M. 'Flow induced oscillations of two interfering circular cylinders' In Int. Conf. on Flow Induced Vibrations in Fluid Engineering, Reading, September 1982 4 Davis, R. W. and Moore, E. F. A numerical study of vortex shedding from rectangles. ]. Fluid Mech. 116 (1982) 475-506 5 Davis, R. W., Moore, E. F. and Purtell, L. P. A numericalexperimental study of confined around rectangular cylinders. Phys. Fluids 27(1) (1984) 46-59 6 Igarashi, T. 'Fluid flow around a bluff body used for a Karman vortex flowmeter' In FLUCOME '85, Tokyo, 1985 7 Larsen, P. S. and Buchave, P. 'Flow measurements: why, what and how? Part 1' Dantec Information No. 01 (June 1985)