Journal o f Wind Engineering and Industrial Aerodynamics, 24 (1986) 95--115 95 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

HELMHOLTZ OSCILLATION IN BUILDING MODELS

HENRY LIU and K.H. RHEE

Department of Civil Engineering, University of Missouri-Columbia, Columbia, MO 65211 (U.S.A.)

(Received July 19, 1985; accepted in revised form February 18, 1986)

Summary

The nature of wind-induced Helmholtz oscillation of air pressure in building cavities was studied experimentally by testing an impermeable rigid building model in a wind tunnel. The model had a single room and a single external opening of four different sizes. It was tested in both laminar uniform wind and turbulent boundary-layer wind, with or without a cylinder placed near the building. The RMS value and the spectrum of the fluctuating external and internal pressures were measured and compared with each other. The Helmholtz resonance was found to be present in all the different types of flows studied. The spectral peak showed that the internal pressure generated by different flows all fluctuated at the Helmholtz frequency predicted by using a poly- tropic exponent equal to 1.4 (adiabatic) and an orifice contraction coefficient approxi- mately equal to 0.88. The test results support a recent theory on Helmholtz oscillation based on aerodynamic analysis.

Introduction

The variation of the internal pressure of buildings under the action of wind has become a subject of research only in recent years. In 1970, Eute- neuer [1] analyzed the transient response of the internal pressure caused by the sudden failure of a window. Because he used an approach that ne- glected the inertia effect of the air flow entering the opening, the result does not reveal any periodic oscillation of the internal pressure.

Stathopoulos et al. [2], in 1979, carried out a wind tunnel investigation of the internal pressure of low-rise buildings. The study showed the inten- sity of the internal pressure fluctuations under various degrees of building permeability and with various areas of building openings. It did not report any spectral measurement and did not mention whether the internal pressure oscillated at any predominant frequency.

In 1979, Holmes [3] was the first to report that a building with a single room and a single opening behaves like a Helmholtz resonator in acoustics. The internal pressure that fluctuates at a dominant frequency can be predicted from the following equation derived more than a century ago by Helmholtz:

0167-6105/86/$03.50 © 1986 Elsevier Science Publishers B.V.

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All4 fH ='-~7~ (1)

where fH is the Helmholtz frequency; A is the area of the opening; n is the polytropic exponent (equal to 1.4 for adiabatic air); R is the engineering gas constant for air; T is the absolute temperature of the air and V is the internal volume of the room. Note that eqn. (1) is for any Helmholtz res- onator without a neck at the opening. Should a neek be present, a slightly different form of the equation, described earlier [4] , must be used. It is not diseussed herein because few buildings have necks.

The equation used by Holmes to prediet the variation of internal pres- sure is

pLV . pqV: AnPa Cpi + 2(knAPa)2 ICpilCpi + Cpi = Cpe (2)

where Cpi and Cpe are, respectively, the internal and external pressure coefficients; dot and double dots on Cpi represent, respectively, the first and the second derivatives with respect to time, t; p and Pa are, respectively, the density and the pressure of the ambient air; q is the dynamic pressure (stagnation pressure); k is the orifice coefficient of the opening and L is the length of the air plug at the opening. For a building or Helmholtz oscillator without neck

1//~A L = r ~ (3)

Equation (2) was derived by assuming that a building behaves like a Helm- holtz oscillator which in turn is analogous to a single-degree-freedom me- chanical vibration system. The first, second, third and four th terms in eqn. (2) represent, respectively, the inertia, viscous damping, elastic and forcing-function terms. Without the viscous-damping and the forcing-func- tion terms, solution of eqn. (2) yields eqn. (1).

Liu and Saathoff [5] , in 1981, gave a more rigorous derivation of the internal pressure variation by using a special form of the Bernoulli equation in fluid mechanics. Assuming that the air density variation is small, their result leads to [6]

pLV Cpi + PqYa kAnp a 2(knAPa)~ ICpilCpi + Cpi = Cpe (4)

The only difference between eqns. (4) and (2) is the presence of the orifice contraction coefficient, k, in the first term of eqn. (4). Because eqn. (4) is based on a rigorous derivation, it is believed to be more correct than eqn. (2).

By setting the damping and the force-function terms of eqn. {4) to be zero, the equation yields

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A 1/4 ~/knRT fH = ns/-'-"7 r 2 V (5)

Comparing eqns. (5) and (1) shows that the two equations differ by a factor k in the square-root sign. The difference is due to the presence of k in the coefficient of the first term of eqn. (4) which does not exist in the first term of eqn. (2). The wind-tunnel data presented herein pro- vides a check on the validity of eqn. (5).

Experimental set-up

The tests were carried out in the Civil Engineering Wind Tunnel of the University of Missouri-Columbia. The tunnel had a test section o f 910 mm width X 910 mm height X 3.05 m length. It was an aerodynamic tun- nel. Without using artificial roughness and spires, the wind through the tunnel had a uniform velocity across the test section and a low turbulence intensity of slightly less than 1% of the mean velocity. Such wind will hereafter be referred to as the "laminar uniform wind".

Tests were also conducted in a turbulent boundary-layer wind obtained by using an array of spires combined with artificial roughness on the floor. Where the test model was mounted, the boundary-layer thickness was 508 mm. Pitot tube measurements indicated that the temporal-mean-velo- city profile there can be described by a power law with an exponent of 0.43. The turbulence intensity was measured with a hot-wire anemometer (TSI Model 1054a). At one-third boundary-layer height which corresponds to approximately the height of the windows on the model, the longitudinal component of the turbulence intensity was 10% of the freestream mean velocity. The integral time-scale of the turbulence, found by integrating the auto-correlation function, was approximately 0.002 s. Using the local mean velocity of 18.6 m s -1, this yielded a length-scale of turbulence of 38 mm, approximately. Although this boundary layer does not simulate any particular atmospheric boundary layer correctly, it does provide a means to check the correctness of eqn. (5) in a flow quite different from the laminar-uniform-wind case.

To test. the general validity of eqn. (5), experiments were also run in flows affected by the wake and vortex shedding generated by a cylinder placed upstream of the building model. More about the wind tunnel and the measuring procedure can be found in work by Rhee [7].

The building model tested in the wind tunnel was a single-room, block- type, flat-roof building of internal dimensions of 140 mm width X 140 mm depth X 290 mm height. The model was made of 6 mm-thick plexi- glass. It had two 40 mm X 40 mm square openings {windows), one on the windward wall, and the other on leeward. The area of each opening could be reduced to 30 X 30 ram, 20 X 20 mm, 10 X 10 mm, or 0 X 0 mm {closed). Reduction of area was accomplished by plugging the 40 X 40 mm

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opening of the model with a piece of plexiglass plate having a smaller or no opening. The centers of the openings were always located at the same place: 160 mm above the building base which is slightly higher than one- half of the building height. To prevent air leakage through building joints, the joints were glued together except for one wall which was sealed with a rubber gasket and bolted to the other walls. This side was used for hand- ling the pressure transducers mounted in the model.

In certain tests, a vertical cylinder of 1-in. (25.4 mm) diameter was placed upstream of the building at locations (X, Y), where X is the longi- tudinal distance between the cylinder center and the outer surface of the windward wall, and Y is the transverse distance between the cylinder center and the longitudinal axis of the building passing through the center of the opening. While all the tests were conducted with X=4 in. (102 mm), two different values of Y were used in different tests: Y=0.5 in. (13 mm) for one set of tests with windward openings in un i fo rm winds, and Y=I in. (25.4 mm) for the other tests. It was found that a small change of Y greatly alters the influence of the cylinder on internal pressure fluctuations.

Four pressure transducers were used for measuring external pressure: two on the windward wall at 10 mm above and below the 40 X 40 mm openings, and two on the leeward wall at the same relative positions. They were f lush-mounted on the wall to have high-frequency response. Two additional pressure transducers were mounted inside the model. The use of two transducers on each wall and inside provided a redundancy to check the consistency or lack of consistency of the data collected. The pressure transducers were Sen-Sym Type LX0503A. They were piezoresistive in- tegrated circuits having the advantages of small size (5 mm diameter), high frequency response (10 kHz, and low cost. They were not suitable, however, if the pressure fluctuations were too low. For this reason, all tests in this study were conducted at 29 m s -1 wind speed which is the maximum that could be generated in the wind tunnel. The signal-to-noise- ratio of the pressure measurements conducted at this speed was at least greater than ten.

The pressure transducer signals were connected to a set of amplifiers/ filters (ITHACO 4210). The ITHACO could be used either as a band-pass filter, a band-reject filter, or a DC-coupled low-pass filter. For the pressure f luctuat ion intensity (RMS) and spectral measurements, the low-pass filter was set at 1 kHz and the high-pass filter was set at 0.01 Hz. These settings were justified because the signal (Helmholtz oscillation) was expected to be in the 50--200 Hz range. For the measurement of the temporal mean pressure, the instrument was used as a DC-coupled low-pass filter which only passed frequencies lower than 0.01 Hz.

The outputs of the amplifiers/filters were connected to a seven-channel magnetic tape recorder (Ampex SP 300 FM/Direct). The signal on the tape was anlyzed on a mini-computer (TEKTRONIX 4052 Graphic Sys- tem). A program was used to control the A/D converter and the ROM pack.

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The ROM pack computed the RMS, the spectra and the correlation func- tions. Results of the computat ion were both displayed on a terminal screen and plotted by the computer. All the spectra shown herein are plotted by the computer.

Test results

Spectra in laminar uniform wind (1% freestream turbulence) Spectral analyses were done to determine the frequency distribution

of the wind-induced fluctuating pressures -- both the internal and the external pressures. The Helmholtz frequency indicated by the spectral peak of each case was compared to the Helmholtz frequency calculated from eqn. (5) using n=l.4 (adiabatic air), k=0.88, R=287 J kg -1 K -1, T=293 K and V=5.68 X 10-3 m 3. As will be shown later, k=0.88 is an average value that yields best agreement between theory and experiment for the model used in this study.

Figures la--d give the power spectra of the internal pressure fluctuations for the four window areas tested. The openings were on the windward wall, and laminar (1% turbulence intensity) uniform wind was used. Figure l a shows that for the 10 X 10 mm opening, the theoretical Helmholtz frequency was 72 Hz, whereas the spectral peak was 68 Hz. This small difference is at least partially due to the influence of a 60-Hz noise present in the measurement. Figures lb- -d show that the measured Helmholtz frequency for 20 X 20 mm, 30 X 30 mm, and 40 X 40 mm openings was, respectively, 95, 120 and 144 Hz. They are within 93% of the predicted values. Note that for the laminar uniform wind cases, the signal level was low, and hence the signals were affected to a certain degree by the 60-Hz noise and its harmonics. Efforts were made to minimize the 60-Hz noise by proper grounding, using shielded cables, and making measurements at night when the background noise was low. No efforts were made to filter out the 60-Hz noise for fear that a portion of the signal, near 60-Hz, would be lost as a result. This explains why some 60-Hz noise and their harmonics appear in the spectra in Figs. 1 and 2, but not in the other figures.

Spectra of the external pressure measured on the windward wall are shown in Figs. 2a--d. They show that the external pressure on the wind- ward face near the window where the pressure transducers were located was affected by the Helmholtz oscillation. This is due to the reciprocating motion of the air through the window produced by the Helmholtz oscilla- tion.

The spectra of the internal pressure fluctuations caused by a leeward window opening of different sizes are shown in Figs. 3a--d. Much larger fluctuations of the internal pressure was encountered in this case than in the case of windward openings. The reason for this large fluctuation is that although the laminar-uniform-wind condition upstream contained little turbulence and hence little pressure fluctuation, the downwind region

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Spectra in turbulent boundary-layer wind (10% freestream turbulence) Figures 4a--d give the spectra of the internal pressure fluctuations caused

by a windward opening of different sizes in a turbulent boundary-layer wind having 10% freestream turbulence intensity. As can be seen from the spectra, large openings produce large Helmholtz peaks. The measured peaks again agreed closely with those predicted from eqn. (5).

The corresponding spectra of the external pressure fluctuations on wind- ward wall near the window where the windward pressure transducer was located are given in Figs. 5a--d. Note that even for this external pres- sure, a Helmholtz peak is seen when the opening is large.

When the opening was on the leeward wall, the spectra of the internal measured were given in Figs. 6a--d. The Helmholtz peak was evident for all the openings greater than 10 X 10 mm.

Spectra in flows affected by vortex shedding Figure 7 gives the spectra of internal pressure fluctuations for wind-

ward opening in uniform wind with 1% freestream turbulence but with a 25.4 mm diameter cylinder at X=102 mm and Y=13 mm. For the two smaner openings there is only one spectral peak in each graph, corresponding to the vortex shedding frequency 84 Hz which is constant for all four cases. For the two larger openings, there are two peaks in each case, one corresponding to the vortex-shedding frequency, and the other to the Helmholtz frequency.

Due to the proximity of the cylinder to the building model and the asymmetric nature of the flow, it was found during the experiments that a small adjustment of the lateral position of the cylinder can cause a sig- nificant shift in the vortex-shedding frequency. This can be seen from the spectra in Fig. 8 which were taken under the same conditions as those in Fig. 7 except that Y=25 mm. The vortex shedding frequency in this case is 102 Hz. Since this is very close to the Helmholtz frequency for the 2 cm X 2 cm opening (fH = 102 Hz), the two spectral peaks for the 2 cm × 2 cm opening merged into a strong peak in Fig. 8b. This is a double- resonance effect predicted in 1982 by Liu and Saathoff [4].

Many additional spectral data were taken in this study. However, they are not reported herein due to page limitation; they are documented in detail in work by Rhee [7].

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Coefficients k, n, Cp and Cp, Using the spectral graph of each case that showed a Helmholtz peak,

the value of k for the case was determined from eqn. (5) by using the mea- sured Helmholtz frequency and by assuming n=l.4. This was done for a large number of runs studied by Rhee [7]. It was found that the range of k was between 0.75 and 0.99, with the average being 0.88. Thus, the average value k=0.88 and the constant value n=1.4 were used herein for the prediction of the Helmholtz peak shown on each spectral graph. The results are summarized in Tables 1--4.

Instead of using eqn. (5), Rhee [7] also used eqn. (1) (the Helmholtz equation) to fit his data. He found that in this case n=1.23 gave the best fit. Note that Holmes [3] also found that n=1.2 gave the best fit when eqn. (1) is used.

From the foregoing results, it appears that both Helmholtz's equation, eqn. {1), and Liu's revision of the equation, eqn. (5), gave equally satis- factory results. However, to use eqn. (1) one must use an n value in the neighborhood of 1.2, in addition to using an empirical value for k. On the other hand, if eqn. (5) is used, n remains 1.4 {adiabatic) while a con- traction coefficient, k, the same as for that of a steady orifice flow, can be used. Since the Helmholtz oscillation involves rapid variation of pressure of relatively small amplitude, the process is expected to be adiabatic. There- fore, it appears that eqn. {5) is more appropriate than eqn. (1).

Tables 1--4 also list the values of the mean pressure coefficient, Cp, and the pressure-fluctuation-intensity coefficient, Cp,, for each run. Table 1 shows that with a windward opening in a laminar uniform wind with 1% freestream turbulence, the values of Cp of the internal pressure were approximately the same as that of the external pressure at the opening. They are in the range 0.95--0.98. The values of Cp, were approximately 0.02, 0.01 and 0.14, respectively for the internal, windward external and leeward external. Two things about these values are noteworthy: (1) the fluctuation of the external pressure on leeward was more than ten times stronger than on windward -- a phenomenon caused by the low freestream turbulence upstream and the large fluctuations downstream generated by building wake; (2) the fluctuation of the internal pressure was approxi- mately twice as strong as the fluctuation of the external pressure at the opening (i.e., the windward wall), a result of the Helmholtz resonance. Table 1 gives much higher values of Cp, for both internal and windward external pressure fluctuations when a cylinder was present to disturb the flow.

Table 2 is the same as Table 1 except that the opening was on the lee- ward wall, and Tables 3 and 4 are the counterparts of the Tables 1 and 2 for the model tested in a turbulent boundary-layer wind of 10% free- stream turbulence.

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Conclusions and discussions

(1) Equation (5) derived by Liu using fluid mechanics is identical to eqn. (1) derived by Helmholtz using an analogy between acoustic oscillation and single-degree-freedom mechanical vibration, except that an orifice contraction coefficient, k, appears in eqn. (5).

(2) Although both eqns. (1) and (5) yield satisfactory results, the latter appears to be more appropriate because it always uses n=l .4 , the adiabatic exponent of air. Furthermore, the effect of the orifice geometry can be taken into account by adjusting the value of k, the contraction coefficient, rather than an arbitrary adjustment of the value of n. The average k value for the model tested was 0.88.

(3) The general validity of eqn. (5) has been verified in this study by using different flows with different velocity profiles and turbulence char- acteristics. These include a laminar flow with uniform velocity and 1% free-stream turbulence, a boundary-layer flow with 10% free-stream turbu- lence, and flows affected by a cylinder placed upstream of the building opening.

(4) Even in laminar uniform wind with only 1% of freestream turbulence, the internal pressure oscillates when there is a large external opening. With little freestream turbulence, this oscillation of the internal pressure is in- duced by the external pressure fluctuations generated by the turbulent wake of the building. In this case, the internal pressure fluctuations are much stronger when the opening is on the leeward wall than on the wind- ward wall. The oscillation takes place at the Helmholtz frequency.

(5) When there is a large windward opening, any disturbance of the flow as produced by freestream turbulence or vortex shedding from an upstream cylinder increases the internal-pressure fluctuations.

(6) When the flow is affected by vortex shedding and when the shedding frequency approaches the Helmholtz frequency, a double resonance takes place which can cause a much larger oscillation of the internal pressure than that produced by the Helmholtz oscillation generated by random fluctuations of external pressure as produced by turbulence.

(7) The study must be kept in perspective. The experiments were con- ducted with an impermeable rigid model of a single opening which are idealized conditions conducive to Helmholtz oscillations. In real buildings, large permeability and other factors often cause so much damping that the Helmholtz oscillation may not take place at all [8]. Also, the flow conditions used in the study are quite different from those normally en- countered in the field. Therefore, numerical values of pressure coefficients found in the study should not be used for building designs. The experiments were not intended to produce practical data for use in the design of real buildings. Rather, they were intended for enhancing the basic understanding of the Helmholtz oscillation phenomenon in building cavity. The study

115

is also useful for assessing the limitations of using impermeable rigid models to simulate building internal pressure.

(8) Theoretical analyses and wind-tunnel tests are currently underway at the University of Missouri-Columbia to determine how building internal pressure is affected by building permeability, flexibility and other factors. Results of the study will further clarify many of the issues poorly under- stood at present.

Acknowledgments

This study was supported by the Structural Mechanics Program of the National Science Foundation under grant CME 8012679. The computer program used for calculating and plotting the spectra and the correlation coefficients on the TEKTRONIX 4052 computer was developed by Dr. John P. Barton, Assistant Professor, Department of Mechanical and Aero- space Engineering, University of Missouri-Columbia.

References

1 G.A. Euteneuer, Druckansteig im Innern yon Gebauden bei Windeinfall, Bauingenieur, 45 (1970) 214--216.

2 T. Stathopoulos, D. Surry and A.D. Davenport, Internal pressure characteristics of low-rise buildings due to wind action, 5th Int. Conf. on Wind Engineering, Fort Collins, CO, 1973, pp. 451--463.

3 J.D. Holmes, Mean and fluctuating internal pressures induced by wind, Proc. 5th Int. Conf. on Wind Engineering, Fort Collins, CO, 1979, pp. 435--450.

4 H. Liu and P.J. Saathoff, Internal pressure and building safety, J. Struct. Div. ASCE (1982) 2223--2234.

5 H. Liu and P.J. Saathoff, Building internal pressure: sudden change, J. Eng. Mech. Div. ASCE (1981) 309--321.

6 P.J. Saathoff, The inertial effect of wind on the variation of building internal pres- sure, CE400 Rep., Department of Civil Engineering, University of Missouri-Columbia, 1980, 31 pp.

7 K.H. Rhee, Wind tunnel investigation of building internal pressure, M.Sc. Thesis, Department of Civil Engineering, University of Missouri-Columbia, 1984, 163 pp.

8 H. Liu and M. Fartash, Field measurements of building internal pressure, Proc. 5th U.S. Wind Engineering Conf., Lubbock, TX, 1985, 4B51-58.