The prediction of turbulence intensities in unsteady flow

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University of Wollongong Research Online Faclty of Engineering and Information Sciences - Papers: Part A Faclty of Engineering and Information Sciences 24 The prediction of trblence

ed.a Pblication Details, I., Yang, S. & Sivakmar, M. (24). The prediction of trblence intensities in nsteady flow. 35 th Hydrology and Water Resorces Symposim (HWRS 23) (pp. 86-866).

1 University of Wollongong Research Online Faclty of Engineering and Information Sciences - Papers: Part A Faclty of Engineering and Information Sciences 24 The prediction of trblence intensities in nsteady flow Ishraq University of Wollongong, ia79@owmail.ed.a Sh-qing Yang University of Wollongong, shqing@ow.ed.a Mttcmar Sivakmar University of Wollongong, siva@ow.ed.a Pblication Details, I., Yang, S. & Sivakmar, M. (24). The prediction of trblence intensities in nsteady flow. 35 th Hydrology and Water Resorces Symposim (HWRS 23) (pp ). Astralia: Engineers Astralia. Research Online is the open access instittional repository for the University of Wollongong. For frther information contact the UOW Library: research-pbs@ow.ed.a

2 Abstract This stdy investigates the distribtion of trblence intensities in nsteady non-niform flows. Yang & Chows (28) work was extended to express this distribtion based on the relationship between Reynolds shear stress and trblence intensities in nsteady flow. It was fond a self-similarity relationship between Reynolds shear stress and trblence intensities in nsteady flow. This relationship has been developed as empirical eqations based on experimental data available in the literatre. By applying the self-similarity relationship, good agreements between the measred and predicted trblence intensities have been achieved. Keywords prediction, flow, nsteady, intensities, trblence Disciplines Engineering Science and Technology Stdies Pblication Details, I., Yang, S. & Sivakmar, M. (24). The prediction of trblence intensities in nsteady flow. 35 th Hydrology and Water Resorces Symposim (HWRS 23) (pp ). Astralia: Engineers Astralia. This conference paper is available at Research Online:

3 Ishraq Research Stdent, School of Civil, Mining and Environmental Eng., University of Wollongong, NSW 2522, Astralia, Sh-Qing Yang Assoc. /Prof., School of Civil, Mining and Environmental Eng., University of Wollongong, NSW 2522, Astralia, Mttcmar Sivakmar Assoc. /Prof., School of Civil, Mining and Environmental Eng., University of Wollongong, NSW 2522, Astralia, Abstract: This stdy investigates the distribtion of trblence intensities in nsteady non-niform flows. Yang & Chow s (28) work was extended to express this distribtion based on the relationship between Reynolds shear stress and trblence intensities in nsteady flow. It was fond a self-similarity relationship between Reynolds shear stress and trblence intensities in nsteady flow. This relationship has been developed as empirical eqations based on experimental data available in the literatre. By applying the self-similarity relationship, good agreements between the measred and predicted trblence intensities have been achieved.. INTRODUCTION Unsteady flow is a recrrent phenomenon in natre and its trblence characteristics are difficlt to nderstand de to its dependence of time and space. It is important to nderstand nsteady flow becase its trblence characteristics are crcial for predicting sediment transport and polltion dispersion in river systems, lakes and coastal waters where the inflence of nsteadiness is more significant. The nsteady flow is complicated becase in practice it is difficlt to predict it de to its variation with time and space and no widely accepted theory has been established and therefore, in this paper, the fll profile of trblence intensities in nsteady flow will be estimated. Many researchers have stdied the distribtion of trblence intensities in niform flow (Grass, 97, Lafer, 974, Eckelmann, 974, Nakagawa et al., 975, Do, 98, Steffler et al., 985, Kironoto & Graf, 994, Nez & Azma, 24 etc.). While Song (994), Kironoto & Graf (995) and Song & Chiew (2) measred trblence intensities in accelerating and decelerating non-niform flows and they demonstrated that the distribtion of trblence intensities in non-niform flow deviates from those in niform flow. In accelerating flow, this distribtion is decreased more than those in niform flow whereas in decelerating flow their distribtion is higher. This deviation has been explained by Yang & Chow (28) sing the Reynolds eqation and they conclded that the vertical velocity (v ) generated from non-niform flow is responsible for the deviation of trblence intensities in non-niform flows from those in niform flow. They obtained that this vertical velocity can be indced by non-niform flows. Accelerating flows prodce downward velocity whereas decelerating flows generate pward velocity. In order to prove this investigation, Yang & Chow (28) sed experimental data sets from Song (994) and they fond that the measrement of trblence intensities in decelerating flows is higher than those in niform flow if v, whereas its measrement in accelerating flow is lower from niform flow if v. Bt in practice, the vertical velocity is generally too small to measre. So, they obtained their distribtion empirically based on the relationship between Reynolds shear stress and trblence intensities in non-niform steady flow. In nsteady flow, few researchers have stdied the distribtion of trblence intensities and none of them restrict their research to flows with a free srface. Over a gravel bed and sing an Acostic Doppler velocity profile (ADVP), Song (994) measred horizontal and vertical trblence intensities in an entire water colmn throgh negative and positive bed slope. He sed niform eqations obtained from Nez & Rodi (986) to compare with the measred data sets in nsteady flow. From this comparison, he obtained that the rns with small nsteadiness, there is no difference between the

4 normalised trblence intensities distribtion in nsteady and niform flows; while the rns with large negative bed slope or with high nsteadiness, the horizontal and vertical trblence intensities distribtion are generally different from those of niform flow. Nez et al. (997) stdied the trblence characteristics in nsteady open channel flow over a smooth bed. They obtained that the vales of trblence intensities may not be affected by the nsteadiness of flow when they compared their measrements with the determined trblence intensities sing the niform formlas. The reason for this relates to low flow conditions to generate weakly nsteady flow as compared with Song (994) s flow conditions. Based on the review otlined before, trblence intensities in nsteady flow have little investigations in hydralic engineering as almost flows in rivers are nsteady or non-niform flows, and in the literatre there is no a niversal model to express the distribtion of these characteristics in the complex flow conditions, ths more research is needed to predict the distribtion of trblence intensities in nsteady flow. Therefore, the aim of this present stdy is to extend the work of Yang & Chow (28) in nsteady flow. The primary objectives for this paper are: () Establish a new relationship between Reynolds shear stress and trblence intensities in nsteady flow; and (2) Verify the developed empirical eqations sing available experimental data. 2. THE DISTRIBUTION OF TURBULENCE INTENSITIES In steady non-niform flow, the empirical estimations of horizontal and vertical trblence intensities have been developed by Yang & Chow (28) who introdced a new formla to estimate the horizontal and vertical trblence intensities depending on the measred Reynolds shear stress. Yang & Chow s formal is proposed based on the mixing length theorem which states that the horizontal and vertical v trblence intensities are proportional to the prodct of mixing length. In this stdy this relationship between trblence intensities and Reynolds shear stress will be extended to nsteady flow as Song (994) defined that there is no inflence of the nsteadiness on the mixing length. This means that the mixing length distribtion in nsteady flow is the same as in steady flow. Therefore, in this stdy, we follow a similar way that has been sed by Yang & Chow (28) to express new formla for the estimation of horizontal/vertical trblence intensities in nsteady flow. Song s (994) experimental data sets in nsteady flow have been plotted in Figre (a and b). In each legend, there are different variables which has been described as Song s definition, for example, S is the bed slope, NO refers to the nmber of hydrograph in nsteady flow and (t) refers to the time compared with nsteady flow. The measred data sets of horizontal/ vertical trblence intensities and Reynolds shear stress, i.e. (,v and v ), respectively, selected from Song (994) are presented in the form of normalised trblence intensities verss the normalised Reynolds shear stress. This normalization is made with respect to the calclation of trblent intensities and Reynolds shear stress in niform flow. HWRS 24,Yang,Sivakmar 2 of 7

5 Eqation v v.5.5 (a) v v S=-.6, NO.935, t=82 &86 sec S=-.6, NO.936, t=6 & 63 sec S=-.25, NO.93 t=74&78 sec S=-.45, NO.933,t=55 & 57 sec S=-.45, NO.933 t=49&5 sec S=-.45, NO.935, t=2&6 sec S=-.45, NO.935, t=78&82sec predicted line Figre : Relationship between horizontal and vertical trblence intensities and Reynolds shear stress in nsteady non-niform flow based on selected data sets from Song s (994) where solid lines refer to the predicted Eqations () and (2), S is the bed slope, NO refers to the nmber of hydrograph in nsteady flow and (t) refers to the time. (b) v v S=-.6, NO.935,t=82&86 sec S=-.6, NO.936, t=6 & 63 sec S=-.25, NO.93, t=74 &78 sec S=-.45, NO.933, t=55&57 sec S=-.45, NO.933, t=49&5 sec S=-.45, NO.935, t=2&6 sec S=-.45, NO.935, t=78&82 sec predicted line In Figre (), the relationship between the trblence intensities and Reynolds shear stress in nsteady flow has been drawn based on the some profiles selected from Song (994) and the best fit for these data sets can be approximated by Eqations () and (2) with most data between the 5% error band, which are similar to the relationship has been fond in steady flow by Yang & Chow (28). v v v * () v v * (2) v Eqations () and (2) describe the relative horizontal ( / ) and vertical ( v / v ) trblence intensities in nsteady flow with respect to these trblences in niform flow, where the sbscript refers to these trblence intensities in niform flow and ns. is the predicted trblence intensities in nsteady flow. These two eqations demonstrate the vales of these trblence intensities depending on the relative Reynolds shear stress in nsteady flow with respect to niform flow, i.e. v / v. This means that if Reynolds shear stress deviates from the linear distribtion, then the distribtion of trblence intensities will be different from that in niform flows. If the vale of Reynolds shear stress in nsteady flow is less than that in niform flow and then similar observation for trblence intensities distribtion can be expected, vice versa. In order to check the validity of Eqations () and (2), the remaining datasets from Song s (994) experiments except those shown in Figre () are plotted in Figre (2), where (UN) refers to nsteady flow, (S ) refers to the bed slope, (NO) refers to the nmber of each hydrograph and (t) is time. HWRS 24,Yang,Sivakmar 3 of 7

6 UN S-6%, NO.934, t=54 s UN S-45%, NO.935, t=3 s Eqation /* v/* Eqation /* v/*.2.2 (a) / *, v/ * (b) / *, v/ * UN S-6%, NO.936, t=45 s UN S-6%, NO.936, t=67 s Eqation /* v/* Eqation /* v/* (c) / *, v/ * UN S-6%, NO.936, t=6 s Eqation /* v/* (d) / *, v/ * UN S-25%, NO.93, t=54 s Eqation /* v/* / *, v/ (e) * UN S-6%, NO.936, t=35 s Eqation /* v/* (f) / *, v/ * UN S-45%, NO.934, t=82 s Eqation /* v/*.2 (g) / *, v/ * (h) / *, v/ * Figre 2: Comparison of measred and predicted trblence intensities in nsteady flow based on Song s (994) experimental data, where UN refers to nsteady flow, S is the bed slope, NO. the nmber of hydrograph and t is the time. HWRS 24,Yang,Sivakmar 4 of 7

7 Figre (2 from a to h) shows the comparisons of measred and predicted trblence intensities in nsteady flow based on Song s (994) experimental data, where the open circles represent the measred horizontal trblence intensity i.e.( / * ), the open sqare are the measred vertical trblence intensity i.e. ( v / * ), the solid lines are the calclated vales of / * sing Eqation () and the dashed lines are the predicted v / * sing Eqation (2). In each diagram, the measred and predicted vales are plotted against the relative water depth i.e. ( y / h ) for different bed slope, hydrograph and times. In order to apply Eqations () and (2), the fll profiles across the water depth for Reynolds shear stress in nsteady flow shold be available which is already measred by Song (994) and therefore, Song s experimental data is considered to be one of the best available in the literatre. While the fll profiles of Reynolds shear stress and horizontal/vertical trblence intensities in niform flow have been determined based on Yang s (29) and Kironoto & Graf s (994) eqations, respectively, which are selected for rogh beds becase Song s data was condcted on a 2 2 gravel bed with d 5 =2.3 mm. After knowing the vales of / *, v / *, v / * and v / *, where * is the shear velocity, the vales of / * and v / * are predicted sing Eqations () and (2), respectively. Based on the above comparison shown in Figre (2), it is clearly seen that the agreement between the measred and predicted vales are acceptable. 3. COMPARISON WITH NEZU ET AL. S (997) EXPERIMENTAL DATA: Nez et al. (997) measred the fll profiles of Reynolds shear stress and horizontal and vertical trblence intensities in nsteady flow in an open channel m long,.4m wide and.5m deep sing a Laser Doppler Anemometer (LDA). They condcted their experiments over a smooth channel. They measred horizontal mean velocity, Reynolds shear stress and horizontal and vertical trblence intensities sing a Laser Doppler Anemometer (LDA). This experimental data will be sed to verify Eqations () and (2). Their measred data of / * and v / * have been plotted against y / h as open circles and sqares, respectively, while the solid and dashed lines are the predicted vales of / * and v / * sing Eqations () and (2), respectively. In each figre, there is a label which refers to the series name (SC3T) and (t) refers to the rising time (i.e. 36 and 48 s) and the falling time (i.e. 84 and 96 s). Based on Eqations () and (2), the measred Reynolds shear stress is the main factor to predict other trblence characteristics, sch as ( and v ). Therefore, the predicted vales of these trblence intensities shown in Figre (3a to 3d) do not appear as smooth lines. Overall, the agreements between the calclated vales and experimental measrements are very reasonable. HWRS 24,Yang,Sivakmar 5 of 7

8 SC3T/t=36s SC3T/t=48 s.8.6 Eqation /* v/*.8.6 Eqation /* v/* (a) / *, v/ * (b) / *, v/ * SC3T/ t=84 s SC3T/t=96 s Eqation /* v/* Eqation /* v/*.2.2 (c) / *, v/ * 2 3 (d) / *, v/ * 2 3 Figre 3: Comparison of measred and predicted trblence intensities in nsteady flow based on Nez et al. (997) experimental data, SC3T is the series name and (t=36s and 48 s) is the rising time while (t=84s and 96s) is the falling time. 4. CONCLUSION The distribtions of horizontal/vertical trblence intensities in nsteady flow have been investigated. This prediction depends on the ratio of Reynolds shear stress in non-niform nsteady flow to that in niform flow, as developed in Eqations () and (2). A good agreement is obtained when compared the measred trblence intensities from Song (994) and Nez et al. (997) with the calclated vales sing Eqations () and (2). HWRS 24,Yang,Sivakmar 6 of 7

9 5. REFERENCES Do, GR. (98), Trblent strctre in open channels and pipes, Sci Sinica, 24 (5), Eckelmann, H. (974), The strctre of viscos sblayer and adjacent wall region in a trblent channel, Jornal of flid Mechanics, 65, Grass, AJ. (97), Strctral feathers of trblent flow over smooth and rogh bondaries, Jornal of flid Mechanics, 5, Kironoto, B and Graf, WH (994), Trblence characteristics in rogh niform open-channel flow, In: Proceedings of the instittion Civil Engineering Water, Maritime and Energy, 6, Kironoto, B and Graf, WH (995), Trblence characteristics in rogh non niform open channel Flow, In: Proceedings of the instittion Civil Engineering Water, Maritime and Energy, UK, 2, Lafer, J (954), The strctre of trblence in flly developed pipe flow, NACA, Technical report, 74. Nakagawa, H, Nez, I and Ueda, H (975), Trblence of open channel flow over smooth and rogh beds, Proc Jpn Soc Civil Engineering, 24, Nez, I and Rodi, W (986), Open channel flow measrements with a laser Doppler anemometer, Jornal of Hydralic Engineering, 2 (5), Nez, I, Kadota, A and Nakagawa, H (997), Trblent strctres in nsteady depth-varing open channel flows, Jornal of Hydralic Engineering, 23 (9), Nez I, Azma R (24), Trblence characteristics and interaction between particles and flid in particle-laden open channel flows, Jornal of Hydralic Engineering, ASCE, 3 (), 988. Song, TC (994), Velocity and trblence distribtion in non-niform and nsteady open- channel flow, Doctoral dissertation, Ecole Polytechniqe Federale de Lasanne, Switzerland. Song, TC and Chiew, YM (2), Trblence measrement in non-niform open-channel flow sing Acostic Doppler velocimeter, Jornal of Hydralic Engineering, 27 (3), Steffler PM, Rajaratnam N, Peterson AW (985), LDA measrements in open channel, Jornal of Hydralic Engineering, ASCE,, 9 3. Yang, SQ and Chow, AT (28), Trblence strctres in non-niform flows, Advances in Water Resorces, 3, Yang, SQ (29), Velocity distribtion and wake-law in gradally decelerating flows, Jornal of Hydralic Research, 47 (2), HWRS 24,Yang,Sivakmar 7 of 7

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