### A review of one-dimensional unsteady friction models for transient pipe flow

#### Abstract

**Abstract.** This paper reviews a quasi-steady model and four unsteady friction models for transient pipe flow. One of the factors which may affect the accuracy of the one-dimensional models of transition flow is the friction coefficient. This coefficient can be estimated as steady, quasi-steady, and unsteady. In the steady approach, a constant value of the Darcy-Weisbach friction factor is used. In the quasi-steady approximation, friction losses are estimated by using formula derived for steady-state flow conditions. The fundamental assumption in this approximation is that the head loss during transient conditions is equal to the head loss obtained for steady uniform flow with an average velocity equal to the instantaneous transient velocity. During transient conditions the shear stress at the wall is not in phase with the mean velocity. In addition, the velocity profile can be completely different from a uniform flow profile. Therefore friction losses computed by using steady-state relationships are inaccurate in transient laminar and turbulent flow. To cope with this problem, for both laminar and turbulent flows, it is possible to algebraically add unsteady-flow terms to the quasi-steady resistance term of one-dimensional models. Unsteady models are divided into two groups. The first group includes those models which instantaneous wall shear stress is the sum of the quasi-steady value plus a term in which certain weights are given to the past velocity changes. Three models of this group are presented in this paper: Zielke, Vardy & Brown, and Trikha. The second group of models assumes the wall shear stress due to ﬂow unsteadiness is proportional to the variable ﬂow acceleration. Brunone model from this group is presented in this paper. Numerical results from the quasi-steady friction model and the Zielke, Vardy & Brown, Trikha and the Brunone unsteady friction models are compared with results of laboratory measurements for water hammer cases with laminar and low Reynolds number turbulent flows. The computational results clearly show that Zielke model yields better conformance with the experimental data

#### Keywords

#### Full Text:

PDF#### References

Zielke, W. Frequency dependent friction in transient pipe ﬂow. J Basic Eng Trans ASME, 90(1), pp. 109–15 (1968).

Vardy, A.E. and Brown, J.M. On Turbulent Unsteady, Smooth-Pipe Friction. Proc. of the 7th International. Conf. on Pressure Surges-BHR Group, Harrogate, United Kingdom (1996).

Vardy, A.E. and Brown, J.M. Transient turbulent friction in smooth pipe ﬂows. Journal of Sound Vibration, 259(5), pp.1011–1036 (2003).

Zarzycki, Z. Hydraulic Resistance in Unsteady Liquid Flow in Closed Conduits. Research Reports of Tech. Univ. of Szczecin, No.516, Szczecin (in Polish), (1994).

Zarzycki, Z. On Weighing Function for Wall Shear Stress During Unsteady Turbulent Pipe Flow. Proc. of the 8th International Conf. on Pressure Surges-BHR Group, The Hague, The Netherlands (2000).

Daily, W.L.; Hankey, W.L.; Olive, R.W., and Jordan J.M. Resistance CoefŞcients for Accelerated and Decelerated Flows Through Smooth Tubes and OriŞces. Trans. ASME, vol. 78, pp. 1071–1077 (1956).

Cartens, M.R. and Roller, J.E. Boundary-Shear Stress in Unsteady Turbulent Pipe flow. J. Hydr. Div., 85_HY2_, pp. 67–81 (1959).

Brunone, B., Golia, U.M. and Greco M. Modelling of fast transients by numerical methods. International meeting on hydraulic transients with column separation. ninth round table, IAHR, Valencia, Spain (1991).

Wylie, E.B. and Streeter, V.L. Fluid Transients in Systems. PrentinceHall, New Jersey (1993).

Almeida, A. B. and Koelle, E. Fluid Transients in Pipe Networks” CMP Southampton Boston and Elsevier Applied Science. London, New York (1992).

Chaudhry, M.H. applied Hydraulic Transiets. Vancouver, Van Nostrand Reinhold, pp. 1 (1979).

Vardy, A.E. and Hwang, K.L. A characteristics model of transient friction in pipes. J Hydraul Res, vol. 29(5), pp. 669–684 (1991).

Pezzinga, G. Quasi-2D model for unsteady ﬂow in pipe networks. J. Hydr. Engnrg., ASCE, 125(7), 676–685 (1999).

Bergant, A. and Simpson, A.R. Estimating unsteady friction in transient cavitating pipe flow. Water pipeline systems, D.S. Miller, ed., Mechanical Engineering Publications, London, 3- 16 (1994).

Brunone, B.; Golia U.M. and Greco M. “Effects of two-dimensionality on pipe transients modelling”, Journal of Hydraulic Engineering, ASCE, 121(12), 906912 (1995).

Vardy, A.E., and Brown, J.M.B. On turbulent, unsteady, smooth-pipe ﬂow. Proc., Int. Conf. on Pressure Surges and Fluid Transients, BHR Group, Harrogate, England, 289-311 (1996).

Vardy, A.E . and Brown, J.M.B. Turbulent, unsteady, smooth-pipe ﬂow. Journal of Hydraulic Research, IAHR, 33(4), pp. 435-456 (1995).

Trikha, A.K. An EfŞcient Method for Simulating Frequency Dependent Friction in Transient Liquid Flow. ASME Journal of Fluids Engineering, 97(1), pp. 97- 105 (1975).

Ghidaoui, M.S. and Mansour, S. Efficient Treatment of the Vardy- Brown Unsteady Shear in Pipe Transients. J. Hydr. Div., 128_1_, pp. 102–112 (2002).

Suzuki, K.; Taketomi, T. and Sato, S. Improving Zielke’s Method of Simulating Frequency Dependent Friction in Laminar Liquid Pipe Flow. ASME J. Fluids Eng., 113, pp. 569–573 (1991).

Schohl, G.A. Improved Approximate Method for Simulating Frequency-Dependent Friction in Transient Laminar Flow. ASME J. Fluids Eng., 115, pp. 420–424 (1993).

Bergant, A. and R.Simpson, A. Water Hammer and Column Separation Measurements in an Experimental Apparatus. Report No. R128, Department of Civil and Environmental Engineering, The University of Adelaide, Adelaide, Australia (1995).