ANALYSIS Of ONE-DIMENSIONAL UNSTEADY SHEAR STRESS MODELS IN WATERHAMMER 

Mohamed S. Ghidaoui¹ and Sameh G. S. Mansour²

¹Assist. Prof., Dept. of Civ. Engrg., The Hong Kong Uni. of Sci. & Technol.,

Hong Kong SAR, (852) 2358-7174, E-mail: ghidaoui@ust.hk,

(to whom correspondence should be addressed).

²Assist. Prof., Hydr. Resea. Institute, delta Barrage, Egypt. Currently:

Visiting Scholar, Dept. of Civ. Engrg., The Hong Kong Uni. of Sci. & Technol.,

Hong Kong SAR, E-mail: sameh_mansour@hotmail.com. 

Abstract: Practical applications such as the design of pipeline systems, the study of water quality in closed conduits and the application of transients for inexpensive, wide-coverage data collection require efficient mathematical models that are capable of accurately solving water-hammer problems beyond the first wave cycle. In this paper, the two most promising unsteady shear stress models, namely the instantaneous acceleration (IA) model and the Vardy-Brown (VB) convolution integral model, are investigated. The VB model is physically based and does not contain free parameters (i.e., applicable in the absence of data). Computations show that the VB model produces results that are, generally, in good agreement with laboratory data. In the case where good quantitative agreement between the VB model and data could not be achieved, it is found that this lack of agreement is due to the failure of the flow axisymmetry assumption and not the VB model. The IA model lacks physical basis and contains a free parameter that needs to be determined from data fitting. The value of this parameter is highly dependent on the flow conditions. Therefore, in the absence of elaborate and detailed data, the authors recommend that the IA model be abandoned in favor of the VB model.

1    INTRODUCTION

There are numerous applications that warrant accurate prediction of waterhammer pressures and flows over a time scale well in excess of single wave cycle. Examples include, design, analysis and safe operation of complex pressurized pipeline systems, investigation of transient-induced water quality problems and application of transients for inexpensive wide-coverage data collection for leakage detection and calibration and validation of models, to name a few only. The reliable prediction of waterhammer pressures and flows beyond the first wave cycle requires accurate representation of shear stresses and energy dissipation in transient flows.

Traditional one-dimensional waterhammer models, in which the shear stress is approximated by steady-state dissipation formulas (e.g., Darcy-Weisbach equation), fail to correctly predict transient pressures beyond the first wave cycle ([1] and [4]). Therefore, the development of practical yet accurate one-dimensional unsteady friction model capable of predicting transient pressures beyond the first wave cycle has been the subject of numerous works (e.g.,  [7]; [6]; [2]; [8]; [9] and [1]).  The simplest approach, which has appeared in the literature, assumes that the unsteady part of the shear stress is proportional to the instantaneous acceleration of the flow with the coefficient of proportionality being an empirical parameter. In [1] a derivation of an instantaneous acceleration based unsteady friction formulas from extended irreversible thermodynamics was proposed and it was concluded that this class of unsteady friction formulas is limited to transients in which the waterhammer time scale is significantly smaller than the diffusion time scale. The empirical parameter in this model lacks physical basis and its quantitative value is always obtained from data fitting.  In [8], a physically based approach was used to derive an unsteady friction formula for turbulent waterhammer flows. This model is free of data-fitted parameters (i.e., does not require calibration). The objective of this paper is to investigate the above mentioned two unsteady friction models (i.e., namely the instantaneous acceleration based unsteady shear stress model, and the Vardy-Brown unsteady shear stress expression). The laboratory tests in [2] and [5] are crucial for the present investigation especially that these experiments cover a wide range of waterhammer frequencies and flow Reynolds number.

2    FUNDAMENTAL WATERHAMMER EQUATIONS

Unsteady closed conduit flow is often represented by a set of one-dimensional hyperbolic partial differential equations.  In their simplified hydraulic grade line form, the continuity and momentum equations are: ([3] and [10])

                              (1)

                             (2)

in which t = time; x = distance along the pipe centerline; H = H(x,t) =  piezometric head; v = v(x,t) = instantaneous average fluid velocity; g = acceleration due to gravity; D = inside diameter of pipe; t = shear stress at pipe wall; r = fluid density; and a = acoustic wave speed. 

In transient flow, the wall shear stress t is the sum of a quasi-steady state shear component t_s and an unsteady state shear component t_u (i.e., t = t_{s +}t_u). 

3    INSTANTANEOUS ACCELERATION SHEAR STRESS MODEL (IA)

In this model, the unsteady shear stress is a function of the instantaneous flow acceleration, the unsteady shear stress term t_u is evaluated as follows [1]:

                                  (3)

where T is a numerical parameter, which must be determined from laboratory or field data.

4    VARDY-BROWN MODEL (VB)

In [9], a physically based derivation of the wall shear stress in transient flows was suggested:

 such that  and    (4)

where f= Darcy-Weisbach friction factor;m= dynamic viscosity of the fluid and  weighting function; ; ;  kinematic viscosity;  Reynolds number; and . It must be noted that all parameters in (4) are determined from the properties of the flow and the fluid and does not need waterhammer data for their evaluation. That is, the Vardy-Brown model does not require calibration and, thus, applicable in the absence of data.

5    NUMERICAL SOLUTION

System (1), (2) and (3) and system (1), (2) and (4) form a closed system of nonlinear hyperbolic partial differential equations.  Therefore, these two system of equations are amenable to a solution by the method of characteristics (MOC). In this paper, the MOC solution presented in [1] and [4] is adopted.

6    TESTS AND RESULTS

Waterhammer experiments where conducted and reported in [2] and  [5]. This published laboratory data is used in this paper in order to investigate the IA and VB models.  To this end, Figs. 1, 2, 3 and 4 depict the measured and the computed head trace results. For reference, the head trace results produced by the Darcy-Weisbach steady state shear stress model (SS) model are included in all figures.

Figs. 1, 2 and 3 show that the head trace results of the VB model and the data are in excellent agreement. However, Fig. 4 shows that the head trace results of the VB model and the data are in good qualitative but not quantitative agreement. The lack of good agreement between the VB model and the data in [2] is due to flow asymmetry in the experiments. To explain, the experiments in [2] showed that, a short time after the wave passage, the flow became asymmetric. However, the VB model assumes that the flow is axisymmetric. Therefore, the difference between model and data provides a measure of the effects of flow asymmetry on energy dissipation. Note that, at present, there is no waterhammer model that incorporates the effects of flow asymmetry on energy dissipation. It must be emphasized that the VB model has no free parameter and can thus be applied in the absence of laboratory or field data.

Figs. 1, 2 and 3 show that good agreement between the IA and the data can be achieved if the free parameter T is 0.039. The parameter value of 0.039 is determined from data fitting and not from the physics of the problem.  Fig. 4 shows that the IA and data are in poor agreement when the parameter value of 0.039 is used. However, a good agreement between the IA and data is achieved if the parameter T is 0.2 (see Fig. 4). It is interesting to note that this agreement between model and data is obtained despite the fact that the IA model incorporates neither the flow asymmetry reported in [2] nor the details of the instantaneous velocity profiles. Therefore, the apparent good agreement between model and data for does not prove the accuracy of the IA model; instead, it proves that it is possible to select a parameter value by force-fitting the model results to the experimental data. In fact, the four test cases presented in this paper and the analysis in other papers (e.g., [1], [2] and [9]) show that an empirical coefficient of the IA model can always be determined by fitting the model results to the measured data.

In terms of computational efficiency, the IA model is three times faster to execute than the VB model. In addition, the VB model requires more memory storage than the IA model. However, with today’s computers, the additional computational requirement of the VB model is trivial. Therefore, given the ample speed and memory of today’s computers and the fact that the IA model lacks physical foundation, requires data for the evaluation of its free parameter and since this parameter is highly sensitive to flow conditions suggest that the IA model be abandoned in favor of the VB model.    

7    CONCLUSIONS

Practical applications such as the design of pipeline systems, the study of water quality in closed conduits and the application of transients for inexpensive, wide-coverage data collection require efficient mathematical models that are capable of accurately solving water-hammer problems beyond the first wave cycle. The reliable prediction of waterhammer pressures and flows beyond the first wave cycle requires accurate representation of shear stresses and energy dissipation in transient flows. Therefore, the formulation of accurate shear stress models in waterhammer has been the subject of numerous research papers. In this paper, the two most promising unsteady shear stress models are investigated. The Vardy-Brown model is physically based and free of data fitting parameters (i.e., does not need to be calibrated). The computations show that the energy dissipation produced by Vardy-Brown model is, generally, in good agreement with laboratory data. In the case where good quantitative agreement between the VB model and data was not achieved, it was found that this lack of agreement is due to flow instability and not the VB model. On the other hand, the instantaneous acceleration based unsteady shear stress model lacks physical basis and contains a free parameter that needs to be determined from data fitting (i.e., requires calibration). The lack of physical basis of the instantaneous acceleration unsteady friction model is manifested by the large degree of variability of its empirical parameter when different sets of laboratory data are used to calibrate this model. The high variability of this model’s parameter to flow conditions makes this model impractical in the absence of field data; a significant limitation given that the majority of water distribution systems around the world are not equipped with data acquisition systems for waterhammer events. In the absence of elaborate and detailed data, the authors recommend that the instantaneous acceleration based unsteady friction model be abandoned in favor of the Vardy-Brown model. 

Acknowledgements

The writers wish to thank the Research Grants Council of Hong Kong for financial support.

References

[1]    Axworthy, D. H., Ghidaoui, M. S., and McInnis, D. A. (2000). “Extended Thermodynamics derivation of energy dissipation in unsteady pipe flow” J. Hydr. Engrg., ASCE, 126(4), 276-287.

[2]   Brunone, B., Karney, B., and Mecarelli, M., and Ferrante, M. (2000). “Velocity Profiles and Unsteady Pipe Friction in Transient Flow” J. Water Resources Planning and Management, July/August 2000, 236-244

[3]    Chaudhry, M. H. (1987). Applied hydraulic transients. Van Nostrand Reinhold, New York.

[4]    Ghidaoui, M.S., and Mansour, S.G.S. “Investigation of the Vardy-Brown unsteady friction model and the formulation of an accurate and efficient approximation to this model”, submitted to J. Hydr. Engrg., ASCE, on the 29^th of Sept. 2000.

[5]    Pezzinga, G., and Scandura, P. (1995). “Unsteady flow in installations with polymeric additional pipe” J. Hydr. Engrg., ASCE, 121(11), 802-811.

[6]    Suo, L., and Wylie, E.B. (1989). “Impulse response method for frequency-dependent pipeline transients” J. Fluids Engrg., 111(4), 478-483

[7]    Trikha, A.K. (1975). “An efficient method for simulating frequency-dependent friction in transient liquid flow” J. Fluids Engrg., 97(1), 97-105.

[8]    Vardy, A. E., Hwang, K. L., and Brown, J. M. B. (1993). “A weighting function model of transient turbulent pipe friction” J. Hydr. Res., Delft, The Netherlands, 31(4), 533-548.

[9]    Vardy, A. E., and Brown, J. (1995). “Transient, turbulent, smooth pipe friction” J. Hydr. Res., Delft, The Netherlands, 33(4), 435-456.

[10]    Wylie, E.B., and Streeter, V. L. (1993). Fluid Transient in systems. Prentice-Hall, Englewood Cliffs, N.J.

Fig. 1    Upstream piezometric head trace (Test 2)

Fig. 2    Upstream piezometric head trace (Test 3)

Fig. 3    Upstream piezometric head trace (Test 4)

Fig. 4    Down stream piezometric head trace (Test 1)