Manometer tubes for monitoring coastal water levels: New frequency response factors

Transcription

1 Manometer tubes or monitoring coastal water levels: New requency response actors Author Jaari, Alireza, Cartwright, Nick, Nielsen, P. Published 0 Journal Title Coastal Engineering DOI Copyright Statement 0 Elsevier B.V.. This is the author-manuscript version o this paper. Reproduced in accordance with the copyright policy o the publisher. Please reer to the journal's website or access to the deinitive, published version. Downloaded rom Griith Research Online

2 MANOMETER TUBES FOR MONITORING COASTAL WATER LEVELS: NEW FREQUENCY RESPONSE FACTORS A. Jaari, N. Cartwright, P. Nielsen Griith School o Engineering, Gold Coast Campus, Griith University, Southport QLD 45 Australia, a.jaari@griith.edu.au, n.cartwright@griith.edu.au School o Civil Engineering, The University o Queensland, Brisbane St Lucia, QLD 407, Australia, p.nielsen@uq.edu.au ABSTRACT This paper presents indings o new laboratory experiments on the requency response o semi-rigid manometer tubes or use in monitoring coastal hydrodynamic processes, extending and improving upon the previous work o Nielsen et al. (993). An adjustable speed rotary electromotor was used to drive a known water level variation at one end o the tube and the pressure was measured at both ends o the tube, enabling a direct determination o the tube s requency response. Testing was conducted over a range o requencies (0.0067Hz< <Hz) and tube lengths (0m<L<900m). New requency response actors were determined by itting the semi-empirical gain unction equation o Nielsen et al. (993) to the observed gain data. New predictive ormulas or the empirical coeicients are provided which yields substantial improvements over the existing constants deduced by Nielsen et al. (993). Keywords: Manometer tubes, coastal water level monitoring, Gain Function, requency response unction. Introduction The ability to accurately monitor water levels in the coastal environment is essential across the ields o Engineering, Environmental and Geophysical Sciences. Whilst technological advances have addressed the issue o accuracy, the resultant products are costly and, in particular, their use under extreme weather conditions is risky. Nielsen(988) irst introduced the idea o using manometer tubes to monitor coastal water levels as both a cheap, but importantly, robust and accurate system or use in all conditions. Illustrated in Figure, this system consists o an array o semi-rigid nylon manometer tubes extended rom land to the water and weighted down with chain. At the seaward end, the tubes are screened to prevent blockage by sand and at the landward ends the tubes terminate into either a glass riser or manual observation o the mean water surace or into a maniold with pressure transducers attached to capture high requency pressure luctuations induced by waves. Corresponding author. Tel.: +6 (07) ; ax: +6 (07) addresses: a.jaari@griith.edu.au(a. Jaari), Postal Address: Griith School o Engineering, Gold Coast Campus, Griith University, QLD 4, Australia

3 Figure. Schematic illustration o the manometer system o Nielsen(988) Numerous researchers have successully used open-ended manometer tube systems to monitor the hydrodynamic characteristics o coastal and estuarine water bodies (e.g. Nielsen, 988; Schubert and Bokuniewicz, 99; Hanslow and Nielsen, 993; Aagaard et al., 994; Jaari et al., 00; Jaari et al., 0). However, the use o tubes closed by the connection o pressure transducers to monitor waves has had only limited application under ield conditions (e.g. Nielsen et al., 993; Hanslow et al., 995; Jaari et al., 0).. Theoretical background This section outlines the theoretical development o Nielsen et al. (993) (see also Nielsen and Dunn, 998). A water wave-induced pressure wave inside the tube is relected by the transducer at the landward end which sets up a quarter length resonating system. The requency response unction or such a system can be approximated by, F( ) cos 0 id E 0 () where is the requency o the pressure luctuations, 0 is the lowest resonant requency o the tube i.e. 0 =c/4l and D E (/ 0 ) is an energy dissipation unction. This can be used to extract both gain and phase inormation via F and Arg(F) respectively. In this application however, only wave statics were o interest and so only the gain was examined. The gain unction corresponding to equation () is, G( ) F( ) () cos D E 0 0

4 Energy dissipation in the tubes is caused by (a) energy dissipation inside the luid and (b) energy absorbed by the deormation o the tube wall. The energy lost due to viscosity over one wave period in luid can be estimated as ollows (Nielsen et al., 993), DE Fluid R0 Eh LT (3) where R 0 is the initial uniorm tube radius, E is the elastic modulus o the tube, h is tube wall thickness, ω is the angular requency, ν is the kinematic viscosity o the luid, L is the tube length and T is the wave period. The energy damping due to lexing o the tube wall is estimated as (Nielsen et al., 993), p R0 p R DE wall L Eh c 3 0 L (4) where, c is wave celerity in tube, ρ luid density, and p is excitation pressure. Combining equations (3) and (4) plus given that input energy at the open end over one wave period corresponds to p + cr 0 T, where p + is seaward wave pressure at open end, Nielsen et al.(993) expressed the relative energy loss as, DE energy loss energy input G p L C 3 c C R 0 (5) where, C and C are dimensionless coeicients and the gain, G is the ratio o the pressures at each end o the tube i.e. p=g p + (see Figure ). Inserting equation (5) into equation () yields the ollowing implicit expression or the tube s gain unction, G cos o G p L C 3 c C R 0 (6) Based on the general orm o equation(6), Nielsen et al.(993) proposed the ollowing semiempirical, explicit gain unction, G cos o B L c B ID (7)

5 where ID is the internal tube diameter, ν is kinematic viscosity (or water ν= m /s) and the empirical coeicients B and B are ound via calibration o equation (7) with laboratory data and c is speed o waves sound in lexible tubes given by, c ( S D ) (8) where S is the compressibility o the luid ( Pa - ) and D is the distensibility o the tube ( Pa - ). Nielsen et al. (993) undertook laboratory calibration on two semi-rigid nylon tubes o lengths 59m and 0m and ound best it values or B and B to be the constants 0.58 and 5.0 respectively. The agreement between equation (7) and their calibration data is shown in Figure. It is noted that although these values provide good agreement or both tube lengths tested, it is expected that this will not be the case or longer tube lengths and so the present study extends the work o Nielsen et al. (993) to cover a much wider range o tube lengths (0m <L< 900m). Figure. Gain unction or 0m and 59m tube length (rom Nielsen et al., 993). The smooth curve is equation (7) and the noisy curve shows the data 3. Approach 3.. Experimental setup The basic concept or the empirical calibration is to induce a dynamic pressure luctuation at the open ( seaward end) ends o the tubes and then to measure the pressures at both ends o the tubes. The ratio o the two signals is the response o the tube system at that particular requency and so repeated tests over a wide range o oscillation requencies (0.0050Hz < <.0Hz in this case) yields data on the system s requency response unction as exempliied by Figure.

6 The presence o air in the system can drastically alter the response o the tube system due to the air s compressibility (Nielsen and Dunn, 998) and so a small amount o dye was added to the water to acilitate a clearer view o the water in the translucent tubing to ensure no air was present. A gravity head o about.8m was used to ill the shorter tubes (L< 00m) and or the longer tubes a suction pump was used to provide additional pressure head. Table summarises the properties o the tubes that have been used. Figure 3, illustrates the experimental setup. The dynamic pressure signal at the seaward end was induced by oscillating a water reservoir with a simple-harmonic rotating arm driven by a variable speed electromotor. The "seaward" end pressure was measured using a secondary tube so that measurement o any relection back rom the landward transducer (PT) was avoided. The range o oscillation provided by the arm was 80cm. Vibration noise rom the electro-motor was iltered rom the signal and unwanted noise rom the movement o the tubes hanging rom the reservoir was minimised using a plastic box illed with water placed at the point that tubes touched the ground. It was ound during the higher requency testing ( > 0.3 Hz), the vibration noise rom the electro-motor increased signiicantly due to the high speed o rotation and could not be adequately iltered and so or these requencies manual excitation o the reservoir was adopted. Figure 3. Sketch o the experimental setup where PT is the pressure transducer or recording the actual input signal representing the "seaward" end o the tube and PT is the sensor recording the resultant landward end signal (G 80 cm) 3.. Data Analysis The raw times series rom the two ends o the tubes were initially analysed using both time domain (zero-crossing) and requency domain (spectral) methods to assess whether the results were sensitive to analysis method. The system gain (G) was then computed as the ratio o p / p +, where p and p + are the magnitudes o the pressure luctuations at the landward (PT) and seaward (PT) ends o the tube respectively (see Figure and Figure 3). An

7 example comparison or two tube lengths is shown in Figure 4 and it is seen that there is little dierence between both methods with the maximum dierence being.0%. Table. Tube properties that used to extract gain unction, where L= tube length, OD= outside diameter, ID= inside diameter, t= tube wall thickness, c= speed o sound, and 0 =lowest resonant requency L (m) OD (m) ID (m) t (m) c (m/s) 0 (Hz) Figure 4. Comparison o time domain and spectral analysis results or L = 00m (time domain ( ), requency domain ( )) and L = 900m (time domain ( ), requency domain ( )) tubes. 4. Results and Discussion Figure 5 shows the initial comparison between the experimental data and equation(7) using the Nielsen et al. (993) derived constants B = 0.58 and B = 5.0. For L = 750m and 00m (let and right respectively) and qualitative discrepancies are obvious. In order to improve the ability o equation (7) to reproduce the data a -step regression analysis was undertaken. The irst step entails an adjustment o the quarter-length resonant requency, 0, to account or the act that the sensor diaphragms are not rigid but lex under pressure. In this situation a time lag, t, occurs in the resonance requency which can be accounted or using (Nielsen et al., 993),

8 resonance 4L c 0 c t t L 0 (9) where, Δ is adjusted such that the peak requency according to equation () matches the gain s peak requency in the data. According to equation (9), Δ is a unction o inverse tube length and speed o sound and thereore as the tube length increases makes the relative raction, i.e. equation (9), smaller and ultimately Δ approaches to and so Δ is more important or shorter tube lengths than longer ones. This is urther evidenced by the best it Δ values summarized in Table. Figure 5. Comparison o equation (7) with experimental data ( ) using Nielsen et al. (993) constants B = 0.58 and B = 5.0 (dashed line) and the present best-it variable B, B, and Δ values with modiied 0 (solid line) or 750m (let) and 00m (right) tubes. Table. From let to right: tube length (L), inside diameter (ID), tube wall thickness (t), peak requency modiier (Δ), gain unction empirical actors (B & B ), regression coeicient o determination o applying presented B, B, and Δ in relation with lab data L(m) ID (m) t (m) Δ B B R The second step is then to it equation (7) with the modiied 0 to the data by least-squares regression adjustment o B and B. The results rom two o the tube lengths tested are shown

9 in Figure 5. The best-it values o B and B or all o the tubes tested are provided in Table and it is evident that B and B are no longer constant or all tube lengths. Curve itting to the data in Table was undertaken to establish predictive unctions or Δ, B, and B. Since the energy dissipation in the tube is primarily inluenced by the tube dimensions (i.e. tube length and internal diameter), an appropriate choice o dimensionless independent variable is the ratio o tube length to internal diameter (L/ID). The ollowing equations are the best it expressions to the data in Table as plotted in Figure 6 and Figure 7. L ID B L ID L B ID (0) () () Figure 6 and Figure 7 show the agreement between equations (0), () and () with Δ, B and B values respectively which are presented in Table. As shown in the igures, regression coeicient o determination shows high agreement between the predictive equations and these constant values. Figure 6. Agreement between values o Δ( ) with equation(0) (solid line)

10 Figure 7. Agreement between values o B ( ) with equation () (solid line) and B ( ) with equation () (dashed line). Inserting equations (0), () and () in to equation (7) yields, G cos o L ID L ID L 0 c 4 L ID d (3) 5. Veriication o Proposed Constant Equations Equation (3) was then veriied by comparison with the laboratory data. Figure 8 provides an example comparison or the 750m tube length which shows good agreement. Table 3 provides summary o coeicient o determination (R ) or all tube lengths. For reerence, the regression or equation (7) with Nielsen s constant B and B (0.58 and 5 respectively) is also shown. The signiicant improvement o equation (3) over the existing constant values o Nielsen et al. (993) is clearly apparent. The 40m tube length has the poorest R value. This occurs mainly because o relatively low number o observations (N = 9) and that one o the values (at.0hz) there is an anomalous 4% dierence between the predicted value and the recorded data. It is noted that the requency o.0hz is outside o the range o requencies normally expected in a natural sea state and i this anomalous data point is not considered in the regression the R is improved to a value o.00 (see Table 3).

11 Figure 8. Veriying Predictive values against recorded data or 750m tube Table 3. Summary o coeicient o determination (R ), let to right: tube length (L), Number o data points which shows the resolution o experiment(n), R o equation(3), R o equation(7), R o equation (3) in requency band width o sea wave (<0.5 Hz) L(m) N R equation(3) (0.0067Hz < < Hz) R constant B & B (Nielsen et al., 993) R equation(3) (sea wave requencies) Conclusion A series o laboratory experiments were conducted to examine the requency response o semi-rigid manometer tubes or use in monitoring coastal hydrodynamic processes. The data collected was used to establish new empirical actors or the semi-empirical gain unction ormula o Nielsen et al. (993). The results provide marked improvement on the predictions made using constant derived o Nielsen et al. (993) who used only two tube lengths. The wide range o tube lengths tested in the current study acilitated the ormulation o predictive equations or the empirical actors as a unction o the tube geometry (length and diameter). The new predictive ormulas in conjunction with Nielsen et al. (993) gain

12 unction ormula provide an accurate means to monitor coastal/sur zone water level variations under any climatic conditions. Acknowledgements The help and assistance o Mr. Ian Underhill and Mr. Duncan Ward through perorming the laboratory test is appreciated. Funding or the present work and inancial support or the irst author has been provided by Australian Research Council grant number DP Reerences Aagaard, T., Nielsen, N. and Nielsen, J., 994. Cross-shore structure o inragravity standing wave motion and morphological adjustment: An example rom northern zealand, denmark. Journal o coastal research, 0(3): Hanslow, D.J. and Nielsen, P., 993. Wave setup on beaches and in river entrances. ASCE, NEW YORK, NY(USA). : Hanslow, D.J., Nielsen, P. and Hibbert, K., 995. Measuring waves with manometer tubes. p.^pp Jaari, A., Cartwright, N. and Nielsen, P., 00. Extracting empirical actor o wave parameters based on iled data recorded under storm condition. International Conerences on Coasts, Ports, and Marine Structures, Tehran, Iran, p.^pp. Jaari, A., Cartwright, N. and Nielsen, P., 0. Stormy wave analysis based on recorded ield data on south-east coasts o queensland, australia. Journal o coastal Research, SI 64: Nielsen, P., 988. Wave setup: A ield study. Journal o Geophysical Research, 93(No C): Nielsen, P. and Dunn, S.L., 998. Manometer tubes or coastal hydrodynamics investigations. Coastal Engineering, 35: Nielsen, P., Hanslow, D.J. and Apelt, C.J., 993. A new type o nearshore wave gauge. th Aust. Con. Coastal Ocean Eng. Townsville.: Schubert, C.E. and Bokuniewicz, H.J., 99. Inragravity wave motion in a tidal inlet. Coastal sediments 9. ASCE, New York, p.^pp. pp