Transient wall shear stress measurements and estimates at high Reynolds numbers

Author’s Accepted Manuscript Transient wall shear stress measurements and estimates at high Reynolds numbers L.R. Joel Sundstrom, Michel J. Cervantes

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To appear in: Flow Measurement and Instrumentation Received date: 5 June 2017 Revised date: 3 October 2017 Accepted date: 5 October 2017 Cite this article as: L.R. Joel Sundstrom and Michel J. Cervantes, Transient wall shear stress measurements and estimates at high Reynolds numbers, Flow Measurement and Instrumentation, https://doi.org/10.1016/j.flowmeasinst.2017.10.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Transient wall shear stress measurements and estimates at high Reynolds numbers L.R. Joel Sundstrom Lule˚ a University of Technology, 971 87 Lule˚ a, Sweden

Michel J. Cervantes Lule˚ a University of Technology, 971 87 Lule˚ a, Sweden Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

Abstract Transient wall shear stress measurements using hot-film anemometry have been performed in a large-scale laboratory setup at high Reynolds numbers. Starting from Reynolds numbers 1.7 × 106 and 0.7 × 106 , the flow was brought to a complete rest by closing a knife gate thus replicating a pressure-time (also know as Gibson) flow rate measurement in a hydropower plant. Ensembleaveraged mean wall shear stresses obtained from 22 repeated runs have been compared with estimates obtained using the pressure-time method. The objective of the work has been to assess the accuracy of the frictional formulation entering the pressure-time integral. It is shown that both the standard method, a quasi-steady approach as well as the recently introduced unsteady method all reproduce the measured wall shear stresses quantitatively during most of the transient. The last phase, following the complete closure of the gate, which is characterized by a slow decay towards zero shear stress at the wall is, however, not captured by the available methods. In general, the unsteady formulation produces the smallest flow rate estimation error, which in turn, implies the best modeling of the frictional losses. Email addresses: [email protected] (L.R. Joel Sundstrom), [email protected] (Michel J. Cervantes)

Preprint submitted to Journal of LATEX Templates

October 13, 2017

Keywords: Wall shear stress, Transients, Pipe flow, Pressure-time method 2010 MSC: 00-01, 99-00

Nomenclature

5

10

15

Q

Flow rate (m3 /s)

A

Cross-sectional area (m2 )

ρ

Fluid density (kg/m3 )

L

Distance between pressure taps (m)

Δp

Differential pressure (Pa)

ξ

Frictional loss (Pa)

t

Time (s)

tu

Upper integration limit (s)

ql

Leakage flow after gate closure (m3 /s)

C

Loss coefficient for standard Gibson method (Pa s2 /m6 )

Re

Reynolds number (-)

ν

Kinematic viscosity (m2 /s)

f

Steady-state friction factor (-)

fb

Brunone model friction factor (-)

k

Coefficient in Brunone model (-)

ΔT

Gate closing time (s)

τ

Wall shear stress (Pa)

1. Introduction The pressure-time method, also known as Gibson’s method in memorial of the pioneering work by N.R. Gibson [1], is a cost-efficient method to measure the flow rate in closed conduits. The principle of the method is derived by considering a control volume bounded by the pipe periphery and two axial cross sections. The pressure and frictional forces acting on the control volume are proportional to the rate of change of momentum inside the control volume.

2

Thus, if an initially steady flow is decelerated to a complete rest, the discharge Q0 before the commencement of the transient is given by  tu A (Δp + ξ)dt + ql . Q0 = ρL 0

(1)

Where A, ρ, L, Δp, ξ, ql and tu are the cross-sectional area, the fluid density, the measuring length, the differential pressure, the frictional losses, the leakage flow, and the upper integration limit, respectively. The differential pressure is, in many instants, straightforward to measure with high accuracy when performing a pressure-time measurement in a full-scale hydropower plant. The frictional losses do, however, require modeling. In the ‘standard Gibson’ method described in IEC41 [2], ξ is calculated by ξ = CQ(t)2 .

(2)

C is a constant loss-coefficient that is extracted from the pre-transient pressure loss, Q(t) is the instantaneous value of the time-varying flow rate. Adamkowski and Janicki [3] argued that it is more physical to take into consideration the sign of Q(t) when specifying ξ, i.e. ξ = CQ|Q|, 20

(3)

since, in the case of flow reversal, the frictional term changes sign (| · | denotes the absolute value). In here, Equation (3) will be used since [3] showed that this formulation of ξ produces more accurate flow rate estimates than Equation (2). Although Equation (3) is not the formulation of ξ specified in the IEC41 standard, the results obtained using Equation (3) will be termed ‘standard Gibson’. Jonsson, Ramdal and Cervantes [4] studied the effect of varying L and Q0 on the accuracy of the flow rate calculated using the Gibson method. They found a systematic error of the calculated flow rate that was dependent on the initial Reynolds number (Re = U D/ν with U , D and ν denoting the bulk velocity, the pipe diameter and the kinematic viscosity, respectively). When the initial Reynolds number was increased from 0.6 × 106 to 1.7 × 106 , the error decreased from positive to negative. In a later work, Jonsson, Ramdal and Cervantes [5] 3

suggested that the discrepancy originated from an inaccurate formulation of the frictional losses (Equation 2), and to overcome this deficiency, the authors introduced an ‘unsteady’ Gibson method. In the unsteady formulation, the instantaneous value of ξ is estimated by the friction factor that would prevail in a steady-state flow at the instantaneous Reynolds number, plus a friction factor fb , due to the unsteadiness of the flow. The unsteadiness was modeled by the time dependent part of the Brunone friction factor (see, Ghidaoui et al. [6]) fb = 25

kD dU , U |U | dt

(4)

where the convective part of the formulation has been neglected. dU/dt denotes the instantaneous bulk flow acceleration. The Reynolds-number-dependent coefficient k is determined through an empirical relation (see Jonsson et al. [5], for details on the determination of k and the justification of neglecting convective effects). With the inclusion of unsteady friction, the systematic error decreased

30

and the method outperformed the standard Gibson approach. In the present paper, selected findings from wall shear stress measurements performed in a laboratory during a Gibson flow rate measurement are presented. The ensemble-averaged mean wall shear stress obtained from many repeated runs was measured using hot-film anemometry. The measured wall shear stress

35

is compared with estimates from the standard, a quasi-steady and the unsteady Gibson methods. The objective of the work is to assess how the current means of modeling frictional losses in the Gibson method compare with measurements of the wall shear stress. Other aspects of the pressure-time method such as the effect of varying L and Q0 , choosing the upper integration limit correctly,

40

how to determine the leakage flow, or which evaluation procedure to incorporate have been described elsewhere (Jonsson et al. [4], Adamkowski and Janicki [7], Bortoni [8], Dunca et al. [9]) and will thus not be reconsidered in here. 2. Experimental setup and measuring methods To investigate the Gibson method, measurements have been taken in a 26.67

45

m long 300 mm internal diameter pipe as depicted schematically in Figure 1. 4

:DWHU WDQNV .QLIHJDWH 'RXEOHEHQG

+ P

+RWILOP VHQVRUV /SLSH P

H

3UHVVXUHWDSV

&HQWULIXJDO SXPS

6XPS

Figure 1: Schematic illustration of the experimental setup. Table 1: Summary of the investigated cases.

Re0

1.7 × 106

1.7 × 106

0.72 × 106

0.69 × 106

Case

C1

C2

C3

C4

ΔT (s)

4

9

4.5

8.5

Water was supplied to the test section from a 9.75 m high constant-head tank. The flow rate retardation was realized by closing a computer-controlled, hydraulically driven knife gate. Four cases, as summarized in Table 1, were investigated. Measurements of absolute pressure and wall shear stress were performed. 50

The reference flow rate before the commencement of the Gibson measurement was extracted using a KROHNE optiflux electromagnetic flow meter. The accuracy of the flow meter at steady state is ±0.3 %.

5

2.1. Measurement methods Measurements of the absolute pressure were performed using two UNIK 5000 55

sensors located 39D and 52D away from the inlet, respectively. The corresponding distances away from the gate are 50D and 37D. The measuring range of the sensors are 0-5 bar, with an accuracy of 0.04% of full-scale reading. The differential pressure required in the Gibson calculation (Equation 1), was extracted by subtracting the pressures measured by the two sensors. An example of a

60

measured differential pressure time-history is shown in Figure 2(a), and the corresponding response of the wall shear stress is plotted in Figure 2(b). Wall shear stresses were measured at one axial location 55D from the inlet (34D from the gate) at three circumferential positions using 55R46 hot-film sensors from Dantec Dynamics. The hot-film sensors measures the wall shear stress

65

over a surface area being 0.2 mm × 0.75 mm. At the highest Reynolds number, the longer side corresponds to 160 viscous units. It is commonly recommended to keep sensor dimension below 20 viscous units in order to accurately resolve turbulence fluctuations (see [10], e.g.). Time-averaged mean values, on the other hand, are largely insensitive to the dimensions of the measuring element [11].

70

In here, only the mean wall shear stress is of interest, hence, the influence of spatial averaging should be negligible. Both the pressure and the wall shear stress are considered to be fully developed since Marusic et al. [12] argued that the mean velocity, and hence the mean wall shear stress, requires approximately 50D to become fully developed.

75

To obtain reliable results using hot-films, frequent updating of the sensor’s calibration curves are important. To assure a proper functioning of the hotfilms, the sensors were calibrated before and after each set of measurement. The calibration procedure was as such: the output voltages of the hot-film sensors and the flow rate were recorded at 12-15 steady-state Reynolds numbers. The

80

wall shear stresses were subsequently extracted using the friction factor f = Δp/(0.5ρU 2 L/D), the bulk velocity U , and the fluid’s density ρ, through τ = ρf U 2 /8. The friction factor was determined for the present setup by measuring the differential pressure as a function of the flow rate in the interval 0.1 m3 /s 6

< Q < 0.4 m3 /s. An implicit logarithmic expression similar to Prandtl’s friction 85

law (see Zagarola and Smits [13]) was fitted to the data. For Q < 0.1 m3 /s, however, the differential pressure was too small to obtain an accurate value of the friction factor, therefore, the formula presented by Zagarola and Smits [13] was used in this interval. This calibration approach is approximate, in contrast to considering a force balance between the pressure drop over a pipe section L

90

and the wall shear stress; τ = ΔpD/(4L). However, it provides a consistent way to calibrate the sensors and was therefore utilized. For Re > 5 × 105 the hot-film voltages, E, aptly fitted the estimated wall shear stresses by a logarithmic expression, log10 (τ ) = A E + B  , where A and B  are calibration constants. For Re < 5 × 105 , the ‘standard’ relation τ 1/3 =

95

A E 2 + B  was used. A threshold value of 4 Pa was used, i.e, for τ > 4 Pa, the logarithmic calibration expression was used whereas for τ < 4 Pa, use was made of the power-law relation. This calibration procedure is essentially similar to that of He, Ariyaratne and Vardy [14] who used hot-film sensors to measure the response of the wall shear stress in an accelerating pipe flow.

100

2.2. Data reduction Transient flows require large ensembles of data to enable the calculation of reliable statistics. Ideally, to achieve converged ensemble averages, thousands of repetitions of nominally equivalent runs have to be performed. Performing this many repetitions is not feasible. In here, 50 realizations of nominally similar

105

runs were performed; i.e., the same signal was sent to the system controlling the knife gate. In conjunction to measurements of instantaneous flow quantities, the movement of the gate was logged. It turned out that the control system could not produce highly repeatable conditions. Therefore, a considerable amount of the performed realizations had to be discarded. Typically, only 22 realizations

110

of each case were of good enough quality; judged by the authors to be the closing times that were within ±0.1 s. A way to increase the available data for computing statistics is to perform spatial averaging in addition to time averaging. Since the flow studied in here constitutes a fully developed pipe flow, it is 7

reasonable to assume circumferential symmetry to increase the number of data. 115

In Figure 3 (a) we have tested the adequacy of the assumption of circumferential symmetry by plotting the difference between the wall shear stresses measured by the sensors exhibiting the largest difference for case C1. An error bound corresponding to ±40% of the mean value measured by one of the sensors have been included. As depicted by Figure 3(a), the difference between the shear

120

stresses measured by the sensors fall within the error bound at all times. The justification of using the above mentioned error bound is that the root mean square turbulent fluctuating wall shear stress in steady-state wall bounded flows is approximately 40% of the mean wall shear stress (see Alfredsson et al. [10]). Hence, any difference smaller than 40% of the mean value can be attributed

125

to turbulent fluctuations rather than to a circumferential dependence of τ . In addition to the assumption of symmetry, the fluctuations were further dampen by using ‘window-averaging’. This method was introduced by He and Jackson [15] in their study of linearly accelerating/decelerating flows. The essence of the method is to subdivide the signal into smaller windows (also called bins),

130

each window containing a bundle of samples. The mean value in each window is subsequently used as a single time step. The method is thus acting like a filter. In here, 20 samples per window was used. Figure 3 (b) shows that windowaveraging reduces the fluctuations while the overall time-development overlaps the original signal.

135

All data were logged at 2 kHz using National Instruments NI-9239 modules mounted in a National Instruments compactDAQ chassi. 2.3. Gibson calculation of wall shear stress The relation between the frictional loss, ξ, entering Equation (1) and the wall shear stress, τ , is ξ = (2L/R)τ . Wall shear stress estimates were obtained using

140

the standard Gibson method, a quasi-steady Gibson method and the unsteady Gibson method. For the standard Gibson method, τ was estimated using Equation (3). For the quasi-steady approach, the friction factor, f , that would prevail in a statistically steady flow at the instantaneous Reynolds number was used 8

4

2.5

x 10

70 2

60 1.5

50

τ (P a)

Δp (P a)

1 0.5

40 30

0

20 −0.5

10 −1

(a)

(b)

0 −1.5 0

1

2

3

t(s)

4

5

0

6

1

2

3

t(s)

4

5

6

Figure 2: Examples of time-histories from a single realization of case C1 showing (a) differential pressure and (b) wall shear stress.

60

20

Mean of 3 sensors Window−averaged

Difference Error bound

15

50

10

τ (P a)

τs1 − τs3 (P a)

40 5 0 −5

30

20

−10

−20 0

10

(a)

−15

1

2

3

t(s)

4

5

6

0 0

(b)

1

2

3

t(s)

4

5

6

Figure 3: Data reduction procedures illustrating; (a) the adequacy of assuming circumferential symmetry by showing the difference in the wall shear stress measured by sensor 1 and sensor 3; (b) the reduction in the fluctuations using window-averaging.

9

for estimating the wall shear stress, i.e., τ = ρf U |U |/8 (similarly as in Equation 145

(3), the sign of U (t) has been taken into account). For the unsteady Gibson method, estimates of the wall shear stress were obtained using the Brunone friction (Equation 4) in addition to the quasi-steady friction. Note that the calculation procedure for the different Gibson methods is iterative. To initiate the calculation, U (t) is assumed to decrease from an initial guess U0 till zero in

150

a linear fashion. This assumed distribution of U (t) is subsequently used in the various frictional formulations to calculate an updated flow rate using Equation (1). The calculations are repeated until convergence. The estimates of the wall shear stresses presented in Figures 4-5 have been computed from the ensembles of ξ that were available from the 22 repetitions of each case.

155

3. Results 3.1. Wall shear stresses Figure 4 shows the predicted and measured responses of the ensembleaveraged mean wall shear stresses for cases C1 and C2 (Re0 ≈ 1.7 × 106 ). The data is the average of 22 repetitions, both for the measured wall shear stress and

160

for the estimates. The figures in the left and in the right columns correspond to C1 and C2, respectively. Initially, τ decreases slowly because there is hardly any change in the flow rate owing to the characteristics of the knife gate. When the rate of change of the flow velocity starts to decrease appreciably, τ also decreases quickly and does so until right before the gate is completely closed (Figure 4

165

(a) at t ≈ 3.9 s, e.g.). Following this point in time, the measured τ exhibits a small increase, and subsequently decays to zero. This small increase in τ is, however, believed to be an artifact owing to deficiencies of the hot-film sensors. For, a hot-film sensor measures the magnitude of the shear stress only, and not the sign. Instead of the experimentally observed increase, it is believed that the

170

wall shear stress changes sign at this point in time. Through a numerical study of the pressure-time method, Saemi et al. [16] found that τ changes sign right before the gate is completely closed. They showed that the mean velocity next

10

to the wall changes direction just prior to the complete closure of the gate, and remains negative until the flow has come to a complete rest. The case inves175

tigated by [16] is largely a numerical replicate of case C3 investigated herein. Except for the sign difference, the development of τ between the experimental and the numerical data is similar (the absolute value of the data of [16] have been included in Figure 6b). Figure 4 shows that the estimates of τ are in excellent qualitative agreements

180

with the measured values until τ exhibits the above mentioned increase. It can therefore be concluded that the time-development of τ , until the near-wall flow (presumably) changes direction, to the leading-order is proportional to the square of the bulk velocity also during the transient realization of a Gibson measurement. Quantitatively, however, there are disagreements throughout the

185

measurement period, the reasons which can be attributed to: i) the accuracy of the measured differential pressure, ii) the accuracy of the measured τ , iii) the simplifications involved when estimating τ . The wall shear stresses estimated using the standard and quasi-steady Gibson methods are characterized by shear stresses being nearly zero following the

190

complete closure of the gate. For the unsteady Gibson method, on the other hand, the estimated wall shear stress oscillates around zero. Recall that the results presented in Figure 4 are ensemble averages of 22 repetitions. In each realization the estimated τ exhibit oscillations (so small for the standard and quasi-steady Gibson methods that they cannot be seen in the figure, though)

195

but these are smeared out in the averaging procedure. Figure 6(a) shows the estimates of τ from a single realization of case C3. The figure presents a timewindow covering 0.4 s before and 1.1 s after the complete closure of the gate. Following the gate closure (t > 0.4 s), there is hardly any oscillations in the wall shear stresses estimated by the standard and quasi-steady Gibson methods

200

whereas for the unsteady Gibson method, τ oscillates around zero with a peakto-peak amplitude being approximately 66% of the initial wall shear stress (≈ 9 Pa). The large oscillations in the wall shear stress estimated by the Brunone model are expected since the model has been developed for instantaneously de11

celerating flows. This type of flow exhibit large wall shear stress oscillations 205

after a valve closure [17]. For the two other methods, the small oscillations are also expected because τ ∼ U |U | which, following the complete closure of the gate, is nearly zero owing to the small magnitude of the bulk velocity. The origin of time in Figures 4-5 was chosen as the commencement of the closing of the gate. However, as commented in section 2.2, the gate closing

210

time varied slightly between each realization. These variations may smear out details in the time-development of τ during the late stages of the deceleration. Therefore, to investigate the behavior of the wall shear stress during the phase just prior to and just after the complete gate closure, ensemble averaging was performed using a different origin of time. The modified origin of time was cho-

215

sen to be when the gate was 1% open. Results from case C3 using this modified approach is shown in Figure 6(b). Following the gate closure, oscillations in the wall shear stress are more readily observed compared to for the ‘orignial’ approach presented in Figure 5(a,c,e). The frequency of the wall shear stress oscillations match those of the differential pressure and the oscillations of the

220

wall shear stress estimated using the Brunone model; the amplitude, though, being much smaller than from the estimate. Following the gate closure, there is (as previously) nothing indicating that τ changes sign. Instead, τ decays to zero while oscillating. Figure 4 and 5 show that there is, in general, good agreement between the measured and estimated wall shear stresses before the gate

225

has closed completely. Figure 6, on the other hand, shows that the estimates do not agree with the measured τ during the phase following the complete closure. Therefore, in order to improve the frictional modeling of the Gibson method, it seems reasonable to develop a model that accounts for the slow decaying towards a zero shear stress following the gate closure. The development of such

230

model is, however, beyond the scope of the present paper. For cases C3 and C4 (Re0 ≈ 0.7 × 106 ), the results are similar as those just presented. Close qualitative agreement between measured and calculated τ is observed until the point in time when τ exhibits the aforementioned increase, which in this case, is relatively more pronounced. 12

50

40

40

35

35

30

30

25

25

20

15

10

10 5

(a)

0 1

2

t(s)

3

50

−5 0

5

40

40 35

30

30

25

25

20 15

3

4

t(s)

5

6

7

8

9

Measured Quasi−steady Gibson

20 15

10

10 5

(c)

0

(d)

0

−5 0

1

2

t(s)

3

50

4

−5 0

5

1

2

3

4

t(s)

5

6

50

Measured Unsteady Gibson

45

40 35

30

30

25

25

τ (P a)

40

20

15

10

10 5

(e)

8

9

20

15

0

7

Measured Unsteady Gibson

45

35

−5 0

2

45

35

5

1

50

τ (P a)

τ (P a)

4

Measured Quasi−steady Gibson

45

5

(b)

0

−5 0

τ (P a)

20

15

5

Measured Standard Gibson

45

τ (P a)

τ (P a)

50

Measured Standard Gibson

45

(f)

0 1

2

t(s)

3

4

5

−5 0

1

2

3

4

t(s)

5

6

7

8

9

Figure 4: Time-developments of measured and estimated wall shear stresses for Re0 ≈ 1.7 × 106 . Left figures, C1; right figures, C2

13

12

Measured Standard Gibson

10

10

8

8

6

6

τ (P a)

τ (P a)

12

4

4

2

2

(a)

(b)

0

0

0

1

2

t(s)

3

4

5

0

10

10

8

8

6

6

4

2

3

4

t(s)

5

6

7

8

9

Measured Quasi−steady Gibson

4

2

2

(c)

(d)

0

0

0

1

2

t(s)

3

12

4

5

0

1

2

3

4

t(s)

5

6

12

Measured Unsteady Gibson

10

10

8

8

6

6

τ (P a)

τ (P a)

1

12

Measured Quasi−steady Gibson

τ (P a)

τ (P a)

12

4

7

8

9

Measured Unsteady Gibson

4

2

2

(e)

(f)

0 0

Measured Standard Gibson

0 1

2

t(s)

3

4

5

0

1

2

3

4

t(s)

5

6

7

8

9

Figure 5: Time-developments of measured and estimated wall shear stresses for Re0 ≈ 0.7 × 106 . Left figures, C3; right figures, C4

14

4

1.5

Unsteady Gibson Quasi−steady Gibson Standard Gibson

3

C3 (experimental) Saemi et al. (numerical)

2 1

τ (P a)

τ (P a)

1 0 −1

0.5

−2

(a)

−3 −4 0

(b)

0.5

t(s)

1

1.5

0 0

0.5

t(s)

1

1.5

Figure 6: The wall shear stress approximately 0.4 s before and 1.1 s after the gate closure. (a) Predicted using the Gibson method from a single realization. (b) The ensemble-averaged mean wall shear stress measured using hot-film sensor number one and the absolute value of the data of Saemi et al. [16]. The origin of time is chosen to be when the position of the gate is 1% open.

235

3.2. Accuracy of different approaches Figure 7 shows the flow estimation error, i.e., the difference between the flow rate calculated by each method and the reference flow rate obtained from the electromagnetic flow meter. The Gibson calculated flow rates were obtained using an updated version of the code developed by Jonsson et al. [5], and is the

240

average value obtained by integrating over 1, 2, 3, 4 and 5 peaks of the differential pressure, using the final point of integration as described in the IEC41 standard [2]. The bars denote 1.96 standard deviations, i.e., a 95% confidence interval of the calculated flow rate based on the 22 repetitions. Starting with cases C1 and C3 (ΔT ≈ 4 s; Re0 = 1.7 × 106 and 0.7 × 106 , respectively), it is

245

seen that the error for all approaches are smaller at the higher initial Reynolds number. The unsteady Gibson method performs about equally well for both cases having an error of approximately 0.5%. The error for the standard Gibson is nearly three times larger for C3 compared to C1 (0.6% and 0.2%). The quasi-steady method performs better for case C1 than C3 (0.65% and 0.85%).

250

The better performance of the quasi-steady approach for case C1 is likely a result of the smaller turbulence time scale prevailing at the higher Re0 , which 15

thus causes a smaller effect of the deceleration, and hence, less deviation from a quasi-steady flow development. For case C1, the error of the flow rate calculated from the measured wall shear stress is smaller than those of the estimates, 255

and for C3, the error is about equally large as for the estimates. The confidence interval, however, is nearly twice as large compared to those of the other methods. The larger confidence interval can likely be related to the variations in the gate closing time (±0.1 s). For, the flow rate estimates calculated using the standard, quasi-steady and unsteady Gibson methods utilizes a frictional

260

formulation based on the differential pressure entering Equation (1). The measured τ , on the other hand, is the average of 22 realization. Therefore, when the same τ is used for the 22 (slightly) different repetitions, this is a likely source of spread in the data. Note, however, that the flow rate calculated using an instantaneous measurement of τ produces less accurate results than using the

265

average of τ . For the longer closing time of the gate (cases C2 and C4), the quasi-steady Gibson method produces flow rate estimation errors of 0.2 % and 0.5 %. These errors are significantly smaller than those for cases C1 and C3. For the unsteady Gibson method, the errors are below 0.2 %. The small errors of the quasi-steady

270

and unsteady Gibson methods for these conditions are not surprising since, as the deceleration rate decreases, the flow tends toward a quasi-equilibrium; i.e., the effects of turbulence dynamics becomes negligible compared to the change in the mean flow characteristics. Hence, the quasi-steady approach produces better results and, in addition, by applying a small correction through the Brunone

275

model, the estimated flow rate using the unsteady approach comes even closer to the reference. The standard Gibson method produces a fairly large negative error for case C2, whereas for C4, the error is practically nil. The poor performance of the standard Gibson method for a high Re0 and a long closing time is also expected since, as the flow tends toward a quasi-equilibrium, the use of a

280

constant friction coefficient becomes increasingly more inaccurate. For a static flow, the friction coefficient increases with a decrease in the Reynolds number, thus causing the constant friction approach to be a poor approximation. The 16

3 2.5

2

Flow rate estimation error (%)

Flow rate estimation error (%)

3 2.5

1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5

U

SG

Q

SG

0 −0.5 −1 −1.5

−2.5

τG

3

3

2.5

2.5

1.5 1 0.5 0 −0.5 −1 −1.5

US

G

QS

G

τG

SG

US

G

QS

G

τG

1.5 1 0.5 0 −0.5 −1 −1.5

−2 −2.5

C2 SG

2 Flow estimation error (%)

2 Flow estimation error (%)

1 0.5

−2

C1 SG

2 1.5

−2

C3 SG

G US

G QS

−2.5

τG

C4

Figure 7: Flow rate estimation error. SG, USG, QSG and τ G denote, respectively, Standard Gibson, Unsteady Gibson, Quasi-steady Gibson and the flow rate obtained using the measured wall shear stress.

error of the flow rate calculated using the measured τ is about the same as for cases C1 and C3. The confidence interval is, again, significantly larger compared 285

to those of the estimates.

4. Conclusions A comparison of ensemble-averaged mean wall shear stresses measured using hot-film anemometry and estimated using the Gibson method have been presented for a transient turbulent pipe flow. Initially steady flows starting at 290

Reynolds numbers 1.7 × 106 and 0.7 × 106 were brought to a complete rest by closing a knife gate over time periods of approximately 4 s and 9 s, respectively. The ensemble-averaged mean wall shear stresses were calculated from 22 repeated runs. It has been shown that the measured wall shear stresses and the wall shear stresses calculated with either the ‘standard Gibson’ approach, 17

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the recently proposed ‘unsteady Gibson’ method, and a quasi-steady Gibson method all produce wall shear stress time developments that are similar until the point in time just prior to the complete closure of the gate. Following the gate closure, the wall shear stress is believed to change sign, however, the measured τ remain positive since hot-film sensors cannot measure the direction of

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τ . The characteristics of the data, when compared with a numerical simulation by Saemi et al. [16], suggest that the near wall velocity reverses. The unsteady approach predicts a negative shear stress, and in general, also produce the most accurate flow rate estimation. The unsteady method is, however, based on the Brunone model which has been empirically tuned for water hammer flows,

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which are characterized by the propagation of pressure waves rather than turbulence dynamics. In a Gibson measurements, on the other hand, the closing time of the gate is relatively long, and hence, turbulence dynamics should be taken into consideration as well. Hence, it is expected that even more accurate flow rate estimations can be achieved by using a physically more suitable model

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than the Brunone model. No effort to develop such model has been presented herein, though. The standard Gibson method, which utilizes a constant friction factor, was shown to perform well with errors falling below ±0.75%. Furthermore, the standard Gibson method produced time-developments of τ being in close qualitative agreement with the measured values. The good performance

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of the standard Gibson method is encouraging because it shows that, despite its simplicity, the method produces fairly accurate results. Avoiding transient modeling of the wall shear stress in the Gibson flow rate evaluation is desirable, because such approach bypasses the need of using (presumably) universal empirical parameters entering such formulations. However, the results presented

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herein are still obtained at Reynolds numbers being at least an order of magnitude smaller than typically encountered in a full scale hydropower plant. The validity of the results for higher Reynolds numbers can, therefore, not be guaranteed. Encouraging, though, as the Reynolds number is increased, the turbulent time scale becomes increasingly smaller. As a consequence, the relative impor-

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tance of the flow unsteadiness is expected to become less important, and hence, 18

the accuracy of the standard Gibson method is expected to increase.

Acknowledgement The research presented was carried out as a part of ”Swedish Hydropower Centre - SVC”. SVC has been established by the Swedish Energy Agency, 330

Elforsk and Svenska Kraftn¨at together with Lule˚ a University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University (see http://www.svc.nu).

References [1] N. R. Gibson, The Gibson method and apparatus for measuring the flow 335

of water in closed conduits, ASME Power Division 45 (1923) 343–392. [2] International Electrotechnical Commision, Geneva, Switzerland, International standard - field acceptance tests to determine the hydraulic performance of hydraulic turbines, storage pumps and pump-turbines, 3rd Edition (1991).

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[3] A. Adamkowski, W. Janicki, Selected problems in calculation procedures for the Gibson discharge measurement method, 8th International Conference on Hydraulic Efficiency Measurement- IGHEM 2010, Roorkee, India. [4] P. P. Jonsson, J. Ramdal, M. J. Cervantes, Experimental investigation of the Gibson method outside standards, in: Proceedings of the 24th IAHR

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Symposium on Hydraulic Machinery and System, Foz do Iguacu, Brazil, 2008. [5] P. P. Jonsson, J. Ramdal, M. J. Cervantes, Development of the Gibson method : unsteady friction, Flow Meas. Instrum. 23 (2012) 19–25. [6] M. S. Ghidaoui, M. Zhao, D. A. McInnis, D. H. Axworthy, A review of

350

water hammer theory and practice, Appl. Mech. Rev. 58 (2005) 49–76.

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[7] A. Adamkowski, W. Janicki, Elastic Water-Hammer Theory-Based Approach to Discharge Calculation in the Pressure-Time Method, J. Hydraul. Eng. 143 (2017) 06017002–8. [8] E. C. Bortoni, New developments in Gibson’s method for flow measurement 355

in hydro power plants, Flow Meas. Instrum. 19 (2008) 385–390. [9] G. Dunca, R. G. Iovanel, D. M. Bucur, M. J. Cervantes, On the use of the water hammer equations with time dependent friction during a valve closure for discharge estimation, J. Appl. Fluid Mech. 9 (2005) 2427–2434. [10] P. Alfredsson, A. Johansson, J. Haritonidis, H. Eckelman, The fluctuating

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wall shear stress and the velocity field in the viscous sublayer, Phys. Fluids 31 (1988) 1026–1033. [11] N. Hutchins, T. B. Nickels, I. Marusic, M. S. Chong, Hot-wire spatial resolution issues in wall-bounded turbulence, J. Fluid Mech. 635 (2009) 103. [12] I. Marusic, B. J. McKeon, P. A. Monkewitz, H. M. Nagib, A. J. Smits,

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K. R. Sreenivasan, Wall-bounded turbulent flows at high Reynolds numbers:Recent advances and key issues, Phys. Fluids 22 (2010) 065103–24. [13] M. V. Zagarola, A. J. Smits, Mean-flow scaling of turbulent pipe flow, J. Fluid Mech. 373 (1998) 33–79. [14] S. He, C. Ariyaratne, A. E. Vardy, Wall shear stress in accelerating turbu-

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lent pipe flow, J. Fluid Mech. 685 (2011) 440–460. [15] S. He, J. D. Jackson, A study of turbulence under conditions of transient flow in a pipe, J. Fluid Mech. 408 (2000) 1–38. [16] S. Saemi, M. J. Cervantes, M. Raisee, A. Nourbakhsh, Numerical investigation of the pressure-time method, Flow Meas. Instrum. 55 (2017) 44–58.

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[17] B. Brunone, A. Berni, Wall shear stress in transient turbulent pipe flow by local velocity measurement, J. Hydraul. Eng. 136 (10) (2010) 716–726.

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