Frictional Field Analysis in Sinusoidal Pulsatile Pipe Flow at Onset of Transition

Transcription

1 Frictional Field Analysis in Sinusoidal Pulsatile Pipe Flow at Onset of Transition EMRAH ÖZAHĐ, MELDA ÖZDĐNÇ ÇARPINLIOĞLU Mechanical Engineering Department University of Gaziantep 731 Gaziantep TURKEY Abstract: - This paper reports the results of an experimenl study which has been carried out in order to perform the frictional and flow field analyses of the sinusoidal pulsatile pipe flow in laminar regime and at onset of transition. Herein the frictional field analysis is presented. The flow resisnce in terms of timeaveraged pressure drop and time-averaged friction factor is investigated and some correlation attempts are given in order to highlight the effects of time-dependent parameters such as Womersley number. A new parameter as a normalized pressure drop which gives a novel contribution to the literature is also introduced. Key-Words: - pulsatile pipe flow; onset of transition; time-averaged friction factor; pressure drop 1 Introduction As a result of the literature survey, some studies related to the frictional field analysis in both laminar and transitional pulsatile pipe flows can be found. The substitution method of the experimenl obined insnneous pressure drop per unit length, P ( L and insnneous mean velocity, U m ( into the below mentioned time-dependent momentum integral equation (Eq. 1) is used in order to estimate the frictional field characteristics. du m ( 4τw( ( ρ + = (1) dt D L In early times, the study of Hershey and Im [1] is seen for the comparison of the experimenl and theoretical results in terms of the friction factors in the laminar pulsatile flows. The experimenl friction factor and theoretical Fanning friction factor for the laminar pulsatile flow were calculated from the following equations, respectively; λ ( ) ave R = () ρu L p,exp 16, = (3) Re λ p theo Comparison of the experimenl friction factors with the theoretical ones has shown an excellent agreement between them. The departure from the theory was accepted as the onset of the transition from laminar to turbulence just after time-averaged Reynolds number, Re =1. Ohmi and Iguchi [] used the conventional momentum integral equation for the evaluations of the frictional losses by means of many considerations to improve the response characteristics of the measuring devices. The insnneous friction factor, λ u ( and the timeaveraged friction factor, follows; t λ u, are described as λu ( = 8τ w( ρu m ( ) (4) 8 λ = τ w( U m ( dt 3 ρu T T (5) Ohmi and Iguchi [3, 4] also investigated the relationships between the frictional losses and the dimensionless frequency in an incompressible pulsatile laminar pipe flow in terms of three kinds of friction factors. They expressed that the insnneous friction factor, λ u ( in an oscillating flow was almost equal to the quasi-steady friction factor, λ ql ( in the quasi-steady region ( ω 1. 3 ). Besides they found that in the intermediate region ( 1.3< ω < 8 ) and the inertia dominant ( ω 8 ) region followed by the quasi-steady one, ( was always larger than λ u ISBN:

2 λ ql ( in the accelerating phase and the first part of the decelerating phase while this was reversed in the rest of the decelerating phase. λ ql 64 ( = (6) ( U ( D ν) m Frictional field analysis at the onset of transition to turbulence in sinusoidal pulsatile pipe flow is investigated by means of the conducted experimenl study. The time-averaged friction factor, λ is evaluated by means of the velocity and pressure measurements. The time-averaged friction factor, λ is an impornt parameter for the time dependent pipe flows. Time-averaged value of mean velocity is evaluated using the following finite Fourier series expansion [5]: U m ( os,1 os, 1 = U + U sin( ωt+ U ) (7) where U os, 1 is the oscillating component of cross sectional mean velocity for the fundamenl first wave in the finite Fourier expansion, ω= πf is the angular frequency, f is the oscillation frequency, t is the time and is the phase lag. In pulsatile pipe flow analyses, the most used parameter is time-averaged Reynolds number as follows: U m, D Re = (8) ν where D is the pipe diameter and ν is the fluid kinematic viscosity. The deils for the basic terminology of timedependent pipe flow, stistical methodology used during the analyses of time-dependent flow parameters and the conducted experimenl study can be found in the literature [6-1]. Correlation Study on Frictional Field Analysis The experimenl study is carried out through 8 runs in the laminar regime and 199 runs at the onset of transition to turbulence regime in the ranges of the time-averaged and oscillating Reynolds numbers of 119 Re 4817 and 17 Re 461 [8, 11-13]. The velocity amplitude ratio of.5 A 1.96 os and oscillation frequency of.1 Hz f 14 Hz corresponding to Womersley numbers of.7 ω 3.1 cover the so-called intermediate region of pulsatile flow with the defined parameters as follows: U m, os,1 D Re os = (9) ν U os,1 A1 = U (1) ω = R ω ν (11) Therefore the effective parameters in timedependent flows can be selected as Re and λ to describe the frictional field. Hence the relationships between them are investigated in both laminar and transitional regimes in this manner and some noteworthy results are found. Figure 1 shows the relationship between the critical time-averaged Reynolds number, Re,, crit where the regime in pulsatile flow passages from laminar to turbulent, and λ in both laminar and transitional regimes as a function of ω for all experimenl da. As is seen from the figure, the flow regime is laminar for Re <45. There is, crit no apparent difference in the magnitudes of λ for laminar and transitional regimes since the transitional regime da belong to the onset of transition. The significant effect of ω on Re,crit and λ is seen for both regimes. For transitional regime, two different characteristics of the relationships between Re, crit and λ are seen as is seen in Fig.. There is a significant increase in the magnitude of λ when Re is increased for, crit ω >8.61. For ω 8.61, there is no sensible effect of ω onλ. The magnitude of almost the same independently of λ is ω when Re,crit is increased. The critical value of ω =8.61 is herein verified once more as is deduced in [14]. It is also found that the flow regime can be kept laminar up to Re, crit =4817 at ω =3.85 in the conducted experimenl study as is shown in Fig.. ISBN:

3 λ u, Laminar Case ω =.7 ω =7.7 ω =1.17 ω =17. ω = Re,crit Fig.1 Time-averaged friction factor variation with respect to critical time-averaged Reynolds number in laminar and transitional pulsatile flow in the range of.7 ω λ u, ω =.7 ω =7.7 ω =1.17 ω =17. ω = Re,crit Fig. Time-averaged friction factor variation with respect to critical time-averaged Reynolds number in transitional pulsatile flow in the range of.7 ω 3.1 ISBN:

4 The effect of ω on an introduced parameter named as normalized pressure drop, P is also investigated with respect to Re in this study., crit The proposed normalized pressure drop parameter is given as follows: ρgh P = (1) h f L U = 4λu, (13) D g f where h f time-averaged frictional head loss and L is the disnce between the pressure transmitters as L= 3D. Figure 3 shows the relationship between Re, crit and P in the transitional regime for all values of ω. It is observed that ω has a significant effect on P regardless of Re, crit. At low ω, the magnitude of P is getting higher. Although the value of Re, crit varies, the magnitude of remains almost consnt at each than ω rather ω =.7 in which the pulsatile flow is in quasi-steady regime. For of is getting too low regardless of more effect of for ω >8.61, the magnitude ω on the magnitude of ω. So no is seen ω >8.61. This deduction also verified the critical limit of ω = ω =.7 ω =7.7 ω =1.17 ω =17. ω = Re,crit Fig.3 Normalized pressure drop variation with respect to critical time-averaged Reynolds number in transitional pulsatile flow in the range of.7 ω Conclusion In this paper, the frictional field analysis in sinusoidal pulsatile pipe flow is carried out in terms of ω varying in the range of.7 ω 3.1 for both laminar regime and at onset of transition. Furthermore a new time-dependent parameter named as normalized pressure drop is proposed resulting in an original contribution to the literature. As a result of the conducted experimenl study, some remarkable conclusions are outlined. One of them is that two typical relationships betweenλ and Re, crit at the onset of transition are observed indicating their dependency on ω. As was ISBN:

5 mentioned before, there is a critical value of ω as ω =8.61 where the flow and frictional fields differ before and after this critical value. The magnitude of λ increases dramatically when the magnitude of u, Re,crit varies at ω >8.61. However, it remains almost consnt at ω 8.61 although there is a significant change in the magnitude of Re. The, crit critical value of ω =8.61 is also verified when the relationship between the new proposed parameter, P and Re, crit is investigated. The magnitudes of P differ from each other at they are the almost same at ω However, ω >8.61. On the other hand, the magnitudes of P remains almost consnt at each ω although Re, crit varies, except that at ω =.7 which is known to be in quasi-steady region. Acknowledgment The authors would like to thank the Research Fund of the University of Gaziantep through the project under grant number of RM References: [1] D. Hershey and C.S. I Critical Reynolds number for sinusoidal flow of water in rigid tubes, AIChE J, Vol. 14, 1968, pp [] M. Ohmi and M. Iguchi, Flow pattern and frictional losses in pulsating pipe flow Part 4 General represention of turbulent frictional losses, Bulletin of the JSME, Vol. 4, 1981, pp [3] M. Ohmi and M. Iguchi, Flow pattern and frictional losses in pulsating pipe flow Part 6 Frictional losses in a laminar flow, Bulletin of the JSME, Vol. 4, 1981, pp [4] M. Ohmi and M. Iguchi, Flow pattern and frictional losses in pulsating pipe flow Part, Effect of pulsating frequency on the turbulent frictional losses, Bulletin of the JSME, Vol. 3, 198, pp [5] M. Ohmi, M. Iguchi and I. Uraha, Transition to turbulence in a pulsatile pipe flow Part 1, Wave forms and distribution of pulsatile velocities near transition region, Bulletin of the JSME, Vol.5, 198, pp [6] M. Ö. Çarpınlıoğlu and M. Y. Gündoğdu, A critical review on pulsatile pipe flow studies directing towards future research topics, Flow Measurement and Instrumention, Vol., 1, pp [7] M. Y. Gündoğdu and M. Ö. Çarpınlıoğlu, Present ste of art on pulsatile flow theory Part 1: Laminar and transitional flow regimes, International Journal Series B Fluids and Thermal Engineering, Vol.4, 1999, pp [8] E. Özahi, Analysis of laminar-turbulent transition in time-dependent pipe flows, Ph.D. Thesis, University of Gaziantep, 11. [9] M. Ö. Çarpınlıoğlu and E. Özahi, Laminar flow control via utilization of pipe entrance inserts (a comment on entrance length concep, Flow Measurement and Instrumention, Vol., 11, pp [1] E. Özahi, M. Ö. Çarpınlıoğlu and M. Y. Gündoğdu, Simple methods for low speed calibration of hot-wire anemometers, Flow Measurement and Instrumention, Vol.1, 1, pp [11] E. Özahi and M. Ö. Çarpınlıoğlu, An experimenl research project on sinusoidal pulsatile pipe flows Part 1: Presention of software programming utilized for measurements and da acquisition, Proceedings of 3 rd International Conference on Fluid Mechanics, Heat and Mass Transfer, ISBN , 1, [1] M. Ö. Çarpınlıoğlu and E. Özahi, An experimenl research project on sinusoidal pulsatile pipe flows Part : Influence of oscillation frequency and amplitude on onset of transition to turbulence, Proceedings of 3 rd International Conference on Fluid Mechanics, Heat and Mass Transfer, ISBN , 1, [13] M. Ö. Çarpınlıoğlu, A practice on the signal processing in a time-dependent flow measurement chain through a research focusing on the passage from laminar to turbulent regime, Proceedings of the 1 th International Conference on Signal Processing, Robotics and Automation (ISPRA'13), ISBN , 13, [14] M. Ö. Çarpınlıoğlu and E. Özahi, An updated portrait of transition to turbulence in laminar pipe flows with periodic time dependence (A correlation study), Flow Turbulence and Combustion, Vol. 89, 1, pp ISBN: