© 2009 International Association of Hydraulic Engineering and Research
Discussion
A turbulent approach to unsteady friction
By IVO POTHOF, Journal of Hydraulic Research, IAHR, 2008, 46(5), 679–690.
Discusser:
HUAN-FENG DUAN, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong. E-mail: ceduan@ust.hk
The Author proposed a new way to express the one-dimensional (1D) unsteady friction model by introducing the two novel con-cepts of history velocity (HV) and transient vena contracta (TVC). Compared to former models, this approach involves a simpler explanation of the turbulence diffusion and the flow reversal processes.
The HV method develops with an exponential rate related to the diffusion time, the TVC method involves a steady orifice during the deceleration stage, during which the shear pulse prop-agates at constant rate from the pipe wall to the core region. However, the radial diffusion as expressed in Eq. (25) approx-imates the initial transients from the tests of He and Jackson, becoming problematic in the final transient stage. As a result, it is important to validate the model from case studies.
Four different systems with eight transient scenarios were used by the Author, with the former two relevant for validating his approach to violent flow reversals. The result for the Perugia sys-tem consisting of visco-elastic pipes is obviously poor in terms of physics. It is known that both unsteady friction and visco-elastic creep may produce pressure head damping, such that the contribution of unsteady friction should be smaller than the total damping. Accordingly, it is unreasonable that the pressure damp-ing from the Author’s model (includdamp-ing only the unsteady friction effect) is larger than the test data.
In the visco-elastic waterhammer models (1D model by Covas
et al. 2004, 2D model by Pezzinga 2002), the contribution of
visco-elasticity to the pressure head damping is comparable to unsteady friction in this system (Fig. D1). The quasi-steady model by Vardy-Brown (2004) is typical for 1D, whereas a typical 2D
k–ε model was presented by Zhao and Ghidaoui (2006). The
quasi-steady model does not capture all the unsteady friction effect for P = 1.8, because this is only a portion of total pressure head damping.
In the Adelaide system, the range of P was 30–90, i.e. P 1. The 1D Vardy-Brown model predicted the measured data well (Ghidaoui and Mansour, 2002). It is not clear which type of quasi-steady model was used in the Author’s comparisons. Therefore, the value of the Author’s proposed model is difficult to assess.
In conclusion, the Author proposed a novel unsteady fric-tion model. His model was not fully validated by a comparative
Figure D1 Contributions of unsteady friction and viscoelasticity to pressure head damping
numerical study. It is suggested that, except for pressure head, the profiles of discharge and turbulent wall shear stress are also important to validate a new model.
References
Covas, D., Stoianov, I., Mano, J., Ramos, H., Graham, N., Maksimovic, C. (2005). The dynamic effect of pipe-wall vis-coelasticity in hydraulic transients 2: Model development, calibration and verification. J. Hydr. Res. 43(R1), 56–70. Ghidaoui, M., Mansour, S. (2002). Efficient treatment of the
Vardy-Brown unsteady shear in pipe transients. J. Hydr. Engng. 128(R1), 238–244.
Pezzinga, G. (2002). Unsteady flow in hydraulic networks with polymeric additional pipe. J. Hydr. Engng. 128(R2), 238–244.
Vardy, A.E., Brown, J.M.B. (2003). Transient turbulent fric-tion in smooth pipe flows. J. Sound and Vibrafric-tion 259(5), 1011–1036.
Zhao, M., Ghidaoui, M. (2006). Investigation of turbulence behavior in pipe transient using a k–ε model. J. Hydr. Res. 44(5), 682–692.
Discusser:
UNO LIIV, Dr. sc., Corson Consulting LCC, Akadeemia tee
21-D201, 12618 Tallinn, Estonia. E-mail: uno@corson.ee
The problems of unsteady liquid flow is relevant in modern tech-nology including mass and heat transfer phenomena or space technology. The key problem relates to friction during a flow transient. Although first investigations originate from Daily et al. (1956), relatively few additions were made within the past half decade. It was mainly found that unsteady friction depends on the flow acceleration causing inertial forces, which are added to the pressure and friction forces as for steady flow.
Pothof reviewed the existing unsteady friction models, and concluded that the unsteady wall shear stress equals τtot = τs+
τuf, or τtot= τqs+τuf, with subscript s referring to a steady-state
preceding a transient and subscript qs to the quasi-steady state derived from instantaneous velocity. In the following, the existing models for τuf will be discussed relative to three aspects:
• Effect of initial Reynolds number, • Linearity in dv/dt, and
• Symmetry of unsteady shear stress during acceleration. The start-up of unsteady flow as a special case of transient flow was considered at the Laboratory of Hydrodynamics, Tallinn University of Technology, since 1975, resulting in a first paper by Koppel and Liiv (1977) on the unsteady friction factor and the laminar-turbulent transient delay under constant accelera-tion. Later, based on 2D LDA velocity tests, velocity profiles and turbulence at 16 different points of the pipe radius over a cross section and simultaneous shear detection using flush-mounted CTA tests on the wall for unsteady flow starting from the rest were made in a 19.5 m long stainless steel pipe of diam-eter 0.061 m (Liiv 2003). Figure D2 shows an example of the ensemble-averaged measurements and the predicted quantities (Liiv 2004).
Figure D2 Measured quantities for typical start-up flow including instantaneous mean velocity U, wall shear stress τo, shear intensity
(τ2
o)1/2and pressure p. Calculated quasi-steady quantities of shear stress
and intensity are τo,qsand (τ 2 o)1/2
The results of Fig. D2 relating to the ensemble-averaged test data indicate that during the initial phase τo τo,qs. The
first bursts of turbulent intensities (τo2)1/2arise at the near wall
region, whereas the wall shear stress increases rapidly at time
t ∼= 0.4 s during maximum acceleration. The local
instanta-neous axial velocity data and the flow visualization demonstrate that first turbulent bursts and small radius vortices appear at the dimensionless pipe radius r/R ∼= 0.96 to 0.98. The cal-culated transition Reynolds number from laminar to turbulent flow was R = Ud/ν = 1.89 × 105. At time t = 1.4 s, as
U becomes constant, both τo and (τo2)1/2 are much larger than
under quasi-steady state. Accordingly, a later turbulence energy redistribution must exist demanding for much more time than the valve regulating time generating the transient flow. Therefore it seems that the friction factor of the described case is non-linearly related to dU/dt. The results of Lefebvre and White confirm these phenomena.
References
Daily, J.W., Hankey, W.L., Olive, R.W., Jordaan, J.M. (1956). Resistance coefficients for accelerated and decelerated flows through smooth tubes and orifices. Trans. ASME 78, 1071– 1077.
Koppel, T.A., Liiv, U.R. (1977). Experimental investigation of the development of liquid motion in conduits. Fluid Dynamics 12(6), 881–887.
Lefebvre, P.J., White, F.M. (1989). Experiments of transition to turbulence in a constant-acceleration pipe flow. J. Fluids
Engng. ASME 111, 428–432.
Liiv, U. (2003). Using CTA and LDA techniques in unsteady pipe flow investigations. Proc. 30th IAHR Congress Thessaloniki, Greece, Theme D, 441–448.
Liiv, U. (2004). Unsteady pipe flow transition to turbulence under constant acceleration. Proc. 10th Asian Congress of Fluid
Mech. Peradeniya, Sri Lanka, CD ROM, 6 p.
Discussers:
BRUNO BRUNONE, Department of Civil and
Environmen-tal Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy, E-mail: brunone@unipg.it (author for correspondence)
SILVIA MENICONI, Department of Civil and Environmental
Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy, E-mail: silvia.meniconi@unipg.it
MARCO FERRANTE, Department of Civil and Environmental
Engineering, University of Perugia, via G. Duranti 93, 06125 Perugia, Italy, E-mail: ferrante@unipg.it
The Discussers appreciate the novelty of the Author’s approach to turbulent unsteady friction modelling based on the simili-tude between transient energy dissipation and steady-state minor losses due to the vena contracta. However, its larger number of parameters to be calibrated with respect to standard unsteady
friction models as well as the restriction to turbulent flow are to be considered.
This discussion is divided into two parts: (1) Valuable labora-tory results, yet known only to a few, relating to fully turbulent flow expansion downstream of an orifice are presented; and (2) Comments are offered about laboratory tests carried out at the Water Engineering Laboratory (WEL), University of Perugia, Italy, used by the Author to validate his model along with other experimental data from the University of Adelaide and WL|Delft Hydraulics, respectively.
Flow expansion in pressurised pipes was investigated by the eminent hydraulicians including Kalinske (1944) or Rouse and Jezdinsky (1966). From their analyses, a synthesis of which may be found in Idel’cik (1986), possible improvements in modelling transients at a vena contracta may be derived. In contrast, the paper of Travaglini (1964) is forgotten by now since it was pub-lished in Italian, as an internal report of the current Department of Agricultural Engineering and Agronomy, University of Napoli Federico II. In the Discussers’ opinion, such a work merits to be cited for its relevance and reliability of laboratory investiga-tions carried out decades ago when neither automatic data system acquisition nor electronic probes were available. To present prac-tical design guidelines of an energy dissipation device, which may be particularly applied if a pressurised irrigation system has to switch to open channel flow (Fig. D3), Travaglini considered dif-ferent flow configurations by changing the diameters doand Ddw
of both the orifice (subscript o) and the downstream (subscript
dw) pipe branch.
For each value of a = do/Ddw < 1, the behaviour of the
contracted core region (Fig. D4a), in Italian corrente viva (Russo Spena 1951), and the decay of the flow power E∗(Fig. D4b) were investigated in terms of distance s from the orifice, by means of simultaneous pressure and total head measurements. In Fig. D4,
r∗= thickness of vena contracta and E0= approach flow power.
The results of Citrini (1946, 1947) for the limit case a = 0 are also included. Figure D4 confirms the crucial role played by a for both the geometry of the vena contracta and the dissipation phenomena downstream of the orifice.
Two of the tests considered by the Author to validate the pro-posed unsteady friction model were taken from the EU project on transients in pipe systems (BHR Group et al. 1997, Brunone 1999). From the Author’s remarks on the tests conducted at
Figure D3 Experimental set-up (modified from Travaglini 1964)
Figure D4 Flow behaviour downstream of orifice (a) contracted core region, (b) relative flow power (modified from Travaglini 1964)
Perugia University, selected aspects are examined below. These include the characteristics of the manoeuvres.
As indicated in the reports, transients of the Perugia system were generated by a hand-operated closure of a ball valve placed at the downstream end of the HDPE pipe at WEL. For ball valves, the discharge-throttling curve, i.e. the behaviour with time t of the product CvAvwith Cv= discharge coefficient and Av= area
of valve opening, defines the hydraulic valve behaviour during a manoeuvre. It can be considered linear only approximately, as stated in the reports. As a consequence, if used to validate a refined model like that proposed by the Author, it would be bet-ter to debet-termine the discharge-throttling curve within an Inverse Transient Analysis (ITA) by assuming the measured time-history of the head H upstream the valve, as a known boundary condition, according to Greco et al. (1984) and Brunone and Morelli (1999). As shown in Fig. D5a relating to the dimensionless valve open-ing F = CvAv/(CvAv)s, where subscript s refers to steady-state
conditions, the behaviour of the so obtained discharge-throttling curves differs noticeably from the linear model. If such ITA curves were used for direct simulations, a better prediction of transient pressure traces would result. Figure D5b shows, as an example, the simulation of the transient with a manoeuvre dura-tion of T = 1.66 s for a steady-state discharge Qs = 8.2 l/s
obtained by the Method of Characteristics (MOC).A quasi-steady approach was adopted to evaluate friction and pipe-wall visco-elasticity was neglected. Note that during the first test phase the shape of the pressure rise is much better captured yet without any change in the extreme of H. No difference between the numerical and experimental extremes of H during the first pipe period is observed for the other test carried out with the Perugia system, for which T = 0.43 s and Qs = 3.6 l/s. This phenomenon is however
evident in Test 2 described by Brunone et al. (2000). Discrep-ancies in the elastic and visco-elastic behaviour during the first characteristic time were noticed and carefully analysed by Covas
et al. (2004, 2005). Even if further research is needed to reliably
model the visco-elastic behaviour during the first characteristic time, the role, to be further on verified, of the dimensionless
Figure D5 Features of Perugia system using ITA (a) dimensionless valve openings F versus relative time t/T , with straight line included as reference, (b) H(t) for T = 1.66 s.
parameter ys= gJsL/(cVs) is significant (Pezzinga 2000), with
Js= steady-state friction slope, L = pipe length, c = wave speed,
and V = mean flow velocity. Specifically, the larger ys, the larger
the effects of unsteady friction and viscoelasticity also in the first pipe characteristic time.
An analysis of Fig. D5b allows to discuss two other points. By assuming a constant value c, no valuable phase shift is noted between the experimental and numerical pressure traces. The assumed value of |c| = 324.18 m/s is slightly different from 320 m/s indicated in project reports. It was determined by ana-lyzing the H time-history by means of the wavelet transform, a powerful tool in the edge detection with respect to both time domain and spectral analysis (Ferrante et al. 2009).
As shown in Fig. D6, the period of these edges, i.e. the extreme pressure values, is almost constant such that the experimental
Figure D6 Distance Lversus time t of Perugia system for 1.66 s closure
data follow a straight, whose slope is|c|. In Fig. D4, L= dis-tance covered by pressure waves generated by the manoeuvre and reflected by the reservoir for a completely closed ball valve.
References
BHR Group, WL|Delft Hydraulics, University of Lisbon, Flow-master International LTD, University of Newcastle-upon-Tyne, University of Perugia (1997). Transient pressures in
pressurized conduits for municipal water and sewage water transport. European Union project (contract no.
SMT4-CT97-2188).
Brunone, B. (1999). European standards for pipelines and pressure transients. J. Hydraulic Engng. 125(3), 221–222. Brunone, B., Morelli, L. (1999). Automatic control valve induced
transients in an operative pipe system. J. Hydraulic Engng. 125(5), 534–542.
Citrini, D. (1946). Ricerca sperimentale sulla diffusione di una vena liquida effluente in un campo di liquido in quiete.
L’Energia Elettrica 23(8), 302–315 [in Italian].
Citrini, D. (1947). Nuove ricerche sulla diffusione di una vena liquida in un campo di liquido in quiete. L’Energia Elettrica 24(6), 177–192 [in Italian].
Covas, D., Stoianov, I., Mano, J.F., Ramos, H., Graham, N., Maksimovic, C. (2004). The dynamic effect of pipe-wall viscoelasticity in hydraulic transients 1: Experimental analysis and creep characteristics. J. Hydraulic Res. 42(5), 516–530.
Covas, D., Stoianov, I., Mano, J.F., Ramos, H., Graham, N., Maksimovic, C. (2005). The dynamic effect of pipe-wall vis-coelasticity in hydraulic transients 2: Model development, calibration and verification. J. Hydraulic Res. 43(R1), 56–70. Ferrante, M., Brunone, B., Meniconi, S. (2009). Leak-edge
detection. J. Hydraulic Res. 47(R2), 233–241.
Greco, M., Brunone, B., Golia, U.M. (1984). Water-hammer in long aqueducts: Mathematical models and laboratory data. Proc. Intl. Conf. Hydrosoft ’84 5, 17-29, C.A. Brebbia, C. Maksimovic, M. Radojkovic, eds. Elsevier, Portoroz SL. Idel’cik, I.E. (1986). Memento des pertes de charge. Eyrolles,
Paris [in French].
Kalinske, A.A. (1944). Conversion of kinetic to potential energy in flow expansions. Trans. ASCE 111, 335–375.
Pezzinga, G. (2000). Evaluation of unsteady flow resistances by quasi-2D or 1D models. J. Hydraulic Engng. 126(10), 778–785.
Rouse, H., Jezdinsky, V. (1966). Fluctuation of pressure in conduit expansions. J. Hydraulics Div. ASCE 92(HY3), 1–12. Russo Spena, A. (1951). Contributo sperimentale allo studio dell’efflusso da tubi addizionali cilindrici. Proc. Accademia
delle Scienze Fisiche e Matematiche Napoli, Ser. 4, 18, 1–35
[in Italian].
Travaglini, G. (1964). Ricerca sperimentale sulla espansione di
vene liquide a valle di diaframmi. Institute of Agricultural
Hydraulics, University of Napoli Federico II, Napoli, Italy [in Italian].
Reply by the Author IVO POTHOF
Discussers Duan, Liiv and Brunone et al. are acknowledged for their constructive comments. The Author agrees with the Discussers that further validation against existing models and measurements is required. This Reply reflects the Author’s opinion on the Discussers’ comments.
Duan asked which type of quasi-steady model was used. It is based on the quasi-steady friction factor, which adjusts the local friction factor in each computational node to instan-taneous velocity. This friction model is used by default in most commercially-available waterhammer software tools. One of its features is the limited pressure damping compared to measurements, as outlined in the paper.
Discusser Duan noted that the computed unsteady friction contribution to the pressure damping should be smaller than the measured pressure damping in the Perugia test data with visco-elastic pipes. This is correct, which has triggered the Author to re-evaluate the damping parameter K and decay parameter d on the Adelaide experiments and to verify these parameters on the Perugia data (Figs. R1 and R2). The new evaluation has led to a proposed damping parameter of K = 0.02 (instead of 0.05) and a decay parameter of d = 1.2 (instead of 0.8). The value
-20 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 t (s) H (m) measurement UF Quasi-steady (a) 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 t (s) H (m) measurement UF Quasi-steady (b)
Figure R1 Model performance of Adelaide experiment with K = 0.02 and d = 1.2 for initial velocity (a) 0.3 m/s, (b) 0.1 m/s
-20 0 20 40 60 80 0 10 20 30 40 t (s) H (m) measurement UF Quasi steady
Figure R2 Model performance of Perugia experiment for K = 0.02 and d = 1.2
of 1.2 is almost equal to the experimental shear pulse propaga-tion parameter of 1.4 (He and Jackson 2000). This observapropaga-tion allows to conclude that the velocity profile is apparently com-pletely adjusted to the new profile after the shear pulses have traveled from the walls to the opposite pipe side.
If Fig. R1 (a) is compared with Fig. 10, then the new param-eters generate slightly less damping, but a better fit for most turbulent experiments of Adelaide. If Fig. R2 is compared with Fig. 4, then the damping reduction is more pronounced. This dif-ference is caused by the different time scale ratios P. The model is less sensitive to the parameters if P is large, i.e. if the turbulence diffusion time scale is significantly larger than the water hammer time scale. Figure R2 shows that the computed pressure damping is slightly smaller than that measured, due to visco-elastic argu-ments. Note that the other Perugia experiments still show slightly more damping during the second and third pressure oscillations. The Author notes that the new unsteady friction model underes-timates damping of the Adelaide experiments at the lowest initial velocity of 0.1 m/s, i.e. at R = 1,940, because the new model accounts for fully turbulent flow.
Duan claims that Eq. (R1) becomes problematic in the final stage of a transient event, referring to shear pulse propagation measurements by He and Jackson (2000). The Author disagrees because: (1) Pressure waves hardly affect the radial distribution of turbulence, as shown numerically by Zhao and Ghidaoui (2006) and experimentally in Fig. D1 of Discusser Liiv; (2) The new model does not use the shear pulse propagation itself. The time scale of the shear pulse propagation and turbulence diffusion determine the exponential evolution of the History Velocity (HV) and Transient Vena Contracta (TVC), as shown in Eqs. (R2) and (R3) below. Consequently, the unsteady friction time scale equals the turbulence diffusion time scale in the proposed model as
R Td = D 2 u∗√2 D = u∗ √ 2 (R1) vh(t + t) = v(t) − (v(t) − vh(t)) · e− d·u∗,h·t D (R2) µ(t + t) = min µx(t + t); (1 − µ(t)) · e− d·u∗,h·t D (R3)
Discussers Brunone, Meniconi and Ferrante are acknowl-edged for summarizing key literature on the steady-state flow behaviour through orifices. Brunone et al. state that the Author’s model contains a large number of parameters. I do not understand this comment, because the unsteady friction model includes two parameters only, namely the damping coefficient K and the decay coefficient d. The history velocity and the transient vena contracta variables do not need any other parameters.
A second point raised by Brunone et al. is the restriction to turbulent flow. The Author’s model indeed is based entirely on turbulent flow considerations and completely neglects vis-cosity driven friction effects or possible transitions to laminar flow, as pointed out in section 6.1. The Author’s key point is that the proposed unsteady friction model has a significantly improved damping compared to the quasi-steady friction model. It includes physical phenomena that were not yet included in existing unsteady friction models, which were extended from Zielke’s laminar flow models. The exclusion of laminar flow in the Author’s model stems from the experience that the Reynolds number in many practical applications is large enough to develop a ‘turbulent’ unsteady friction model.
Finally, Brunone et al. recommend to determine the discharge-throttling curve from an Inverse Transient Analysis. I agree with the Discussers that this curve does affect the instantaneous decel-eration and therefore the damping. This point has been addressed in the modeling work as follows. The valve stroke in the Perugia models was not modeled as a linear valve closure, but as a two-stage closure to obtain a reasonable match between the measured and computed shapes of the initial pressure rise. The two-stage closure was based on visual inspection of the measured and com-puted pressure time series without a formal Inverse Transient Analysis.
The measurements of Discusser Liiv confirm that the turbu-lence diffusion advances weakly and indirectly depends on the
pressure wave evolution. Liiv’s waterhammer time scale is about
L/c = 19.5/1300 = 0.015 s, his turbulence diffusion time scale
is about 0.5 s, based on an assumed friction factor of 0.03 and an average velocity during start-up of 2 m/s. Liiv’s time scale ratio is about 35, which is similar to the Adelaide experiments. Liiv’s Fig. D1 shows that the total wall shear stress evolves in accordance with the turbulence diffusion time scale. The wall shear stress is not directly influenced by the pressure waves. This observation supports the claim that unsteady friction phenom-ena should be modeled at the turbulence diffusion time scale. An immediate consequence is that instantaneous acceleration models cannot capture unsteady friction phenomena.
The current simulation time for practical applications is one or two orders of magnitude larger than the turbulence diffusion time D/u∗h ∼= 10 s. Therefore, the turbulence diffusion time
scale is required for practical simulations with unsteady friction. All previous 1D unsteady friction models did not include the tur-bulence diffusion time scale. Therefore, these models are limited to simulation times shorter than the turbulence diffusion time (Zhao and Ghidaoui 2006). The Author suggests that the history velocity concept could “melt” the frozen viscosity approach in the Vardy-Brown model, but this is an issue for further research.
References
He, S., Jackson, J.D. (2000). A study of turbulence under conditions of transient flow in a pipe. J. Fluid Mech. 408, 1–38.
Zhao, M., Ghidaoui, M. (2006). Investigation of turbulence behavior in pipe transient using a k–ε model. J. Hydr. Res. 44(5), 682–692.
