Distribution of reynolds shear stress in steady and unsteady flows

University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 13 Distribution of reynolds sear stress in steady and unsteady flows Israq Alfadli University of Wollongong, ia179@uowmail.edu.au Su-Qing Yang University of Wollongong, suqing@uow.edu.au Muttucumaru Sivaumar University of Wollongong, siva@uow.edu.au Publication Details Alfadli, I., Yang, S. & Sivaumar, M. (13). Distribution of reynolds sear stress in steady and unsteady flows. SGEM 13: 13t International Multidisciplinary Scientific Geoconference (pp. 19-116). Bulgaria: SGEM. Researc Online is te open access institutional repository for te University of Wollongong. For furter information contact te UOW Library: researc-pubs@uow.edu.au

Distribution of reynolds sear stress in steady and unsteady flows Abstract Tis study investigates te Reynolds sear stress distribution in steady and unsteady non-uniform flows. Specifically, it deals wit ow to express te deviation of tis turbulence caracteristic from tat of uniform flow line; it is found tat flow acceleration can well represent te deviation of Reynolds sear stress from its standard linear distribution. By connecting te flow acceleration wit Reynolds sear stress, te study demonstrates empirically tat te linear distrubiton of Reynolds sear stress can be observed wen te flow acceleration is zero; te concave distribution of Reynolds sear stress can be observed wen te flow acceleration is negative or wen te flow velocity is decreased along te cannel; te convex distribution of Reynolds sear stress can be observed wen te flow acceleration is positive or te flow velocity is increased along te cannel. Tese empirical results ave been verified using publised experimental data and good agreement between te predicted and observed profiles as been acieved. Keywords steady, distribution, flows, reynolds, unsteady, sear, stress Disciplines Engineering Science and Tecnology Studies Publication Details Alfadli, I., Yang, S. & Sivaumar, M. (13). Distribution of reynolds sear stress in steady and unsteady flows. SGEM 13: 13t International Multidisciplinary Scientific Geoconference (pp. 19-116). Bulgaria: SGEM. Tis conference paper is available at Researc Online: ttp://ro.uow.edu.au/eispapers/1465

Section Hydrology and Water Resources DISTRIBUTION OF REYNOLDS SHEAR STRESS IN STEADY AND UNSTEADY FLOWS Researc Student, Israq Alfadli Assoc. Prof. Su-Qing Yang Assoc. Prof. Muttucumaru Sivaumar Scool of Civil, Mining and Environmental Eng., University of Wollongong, Australia ABSTRACT Tis study investigates te Reynolds sear stress distribution in steady and unsteady nonuniform flows. Specifically, it deals wit ow to express te deviation of tis turbulence caracteristic from tat of uniform flow line; it is found tat flow acceleration can well represent te deviation of Reynolds sear stress from its standard linear distribution. By connecting te flow acceleration wit Reynolds sear stress, te study demonstrates empirically tat te linear distribution of Reynolds sear stress can be observed wen te flow acceleration is zero; te concave distribution of Reynolds sear stress can be observed wen te flow acceleration is negative or wen te flow velocity is decreased along te cannel; te convex distribution of Reynolds sear stress can be observed wen te flow acceleration is positive or te flow velocity is increased along te cannel. Tese empirical results ave been verified using publised experimental data and good agreement between te predicted and observed profiles as been acieved. Keywords: Uniform/non-uniform flow, unsteady flow, Reynolds sear stress, flow acceleration. INTRODUCTION Te distribution of Reynolds sear stress can determine te distribution of sediment concentration and pollution dispersion. Terefore, its distribution is crucial for predicting sediment transport in river systems were te influence of steadiness or unsteadiness is more significant. Because all river flows are unsteady or non-uniform, uniform flow is very rare to occur, tus it is necessary to investigate te influence of non-uniformity and unsteadiness on Reynolds sear stress distribution, te nowledge of te distribution of Reynolds sear stress in non-uniform open cannel flows is of essential in ydraulic engineering. But its distribution in steady/unsteady flows is not fully understood and tere exist no widely accepted equations available in te literature, tus it requires more investigations. Many researcers ave studied te distribution of Reynolds sear stress in uniform flow [], [3], [4], [6], [7], [9]; owever, few researcers ave studied te influence of non-uniform flow on te distribution of Reynolds sear stress. [5], [8] measured Reynolds sear stress in accelerating and decelerating flows. Tey observed tat an accelerating flow generally ampers te local Reynolds sear stress, vice versa; tese observations clearly demonstrate tat te influence of unsteadiness/non-uniformity on te Reynolds sear stress is not negligible. Unfortunately, te mecanism for tis penomenon as not been well revealed, and te quantitative description for te Reynolds sear stress is not available in te literature. [8] measured Reynolds sear stress in steady/unsteady non-uniform flows using an Acoustic Doppler Velocity Profiler (ADVP). Te measured data sow tat te concave distributions are observed in accelerating flow were te bed slope was negative; wile convex distributions are observed for decelerating flow wit positive bed slope. However, te 19

GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems reason for tis deviation was not discussed in is study. [8] also establised equations to predict te Reynolds sear stress profiles in steady and unsteady flows i.e. ( m) / m ( y) / 1 y / / U d / dx[( m 1) /( m m)]( y / (1) ) 1/ m ( y) / 1 u y / / U / t [ y / 1] y / U / (m) / m [( m 1) /( m m)] y / U / / t / x ( m1)/ m ( m)/ m [1/ m y / ( m1) /( m )( y / ) ] () were Equations 1 and developed for tis prediction in steady and unsteady flows, respectively; and are te pressure gradient parameter in steady and unsteady flows u respectively; is te water dept; m =5.5; is te bed sear stress wic can be determined using te following equation: g [ S {1 ( m 1) /( m m) Fr } d / dx] (3) were Fr is te Froude number. Based on is teoretical distribution, Song investigated tat a good agreement between te measured and predicted Reynolds sear stress can be observed wen te bed slope is not very large, for example, te teoretical distributions matc well wit is measurements wen te bed slope =-.15% and.3%, wic is better tan wen te slope =-.6%. [1] investigated teoretically te deviation of Reynolds sear stress distribution in accelerating/decelerating non-uniform flow. Tey reported tat tis deviation is related to te presence of upward/downward vertical velocity. Based on tis reason, tey developed formula tat can be predicted te full profiles of Reynolds sear stress in uniform and non-uniform flows: u v y y uv (1 ) b (4) u u were u v/ u is te normalized Reynolds sear stress; is te water dept; y is te vertical distance measured from te reference level; u v are te momentum flux caused by te vertical velocity; and b is a parameter and can be expressed as follows: u b ( ) d / dx (4a) u It can be seen from Equation 4a tatb is always positive in an accelerating flow were d/ dx and becomes negative in a decelerating flow were d / dx. Wile it becomes zero wen te flow is uniform and tus te first term on te rigt and side of Equation 4 can be used to estimate te Reynolds sear stress in uniform flow i.e. u v y (5) (1 ) u Based on te review outlined, te Reynolds sear stress in non-uniform flows as little investigation in ydraulic engineering, te conclusions from different autors are different, and tus more researc is needed to clarify wy te measured Reynolds sear in nonuniform flow deviates from tat in uniform flow. From te practical point of view, almost all flows in rivers are unsteady or non-uniform flows, and in te literature tere is no a universal model to express te Reynolds sear stress in te complex flow conditions, tis leads to te present researc aims to develop a universal model to express te Reynolds sear stress profile in uniform/non-uniform and steady/unsteady flows. THE RELATIONSHIP BETWEEN THE FLOW ACCELERATION AND THE DISTRIBUTION OF RYNOLDS SHEAR STEESS 11

Section Hydrology and Water Resources In te literature, it as already been discussed tat te consistency of flow velocity and water dept in open cannel flows from upstream to downstream generate te uniform flow. However, in steady/unsteady non-uniform flows, te flow velocity and water dept are different upstream to downstream. Tese differences relate to flow acceleration. Generally, te flow acceleration i.e. a means tat tere is a difference in velocities in two adjacent measuring stations or different time at te same location. Based on tis definition, te flow acceleration is equal to zero wen te flow is uniform wile in non-uniform flows, te flow acceleration is different. Wen te flow velocity increases along te open cannel te flow acceleration is increased or as a positive value and tis type of flow is accelerating steady/unsteady non-uniform flows. In contrast, te flow acceleration is less tan zero or as a negative value wen te flow is decelerating steady/unsteady non-uniform flows due to te decrease of flow velocity along te cannel. According to tis, flow acceleration is te most important parameter to distinguis te flows, tus it is possible to develop empirical formulas to predict te mean orizontal velocity using te flow acceleration. Tis explanation can be seen from experimental runs of [5] as sown in Figure 1, in wic te measured data in uniform and accelerating/decelerating non-uniform flows are included for comparison legends. It is clearly seen from Figure 1 tat te measured data points of Reynolds sear stress in non-uniform flow deviate from tat in uniform flow (Equation 5). Wen te flow is accelerating or positive flow acceleration te value of Reynolds sear stress is lower tan te uniform flow line and vice versa in decelerating flow. Terefore, Equation 5 needs to be modified to extend its applicability in bot uniform and non-uniform flows, it is reasonable to express te deviation wit te flow acceleration. But te difficulty is tat te flow acceleration is a variable in spatial and temporal domain, it is needed to develop a proper way to express te flow acceleration, and ten investigate te relationsip between te deviations sown in Figure 1 and te caracteristic acceleration. Figure 1: Te measured Reynolds sear stress in uniform and non-uniform flows. DETERMINATION OF FLOW ACCELERATION FOR BOTH STEADY AND UNSTEADY FLOWS Tis paper maes an attempt to express te Reynolds sear stress in steady non-uniform flow or unsteady flow, to acieve tis, appropriate expression of flow acceleration is needed; te point and instantaneous acceleration can be written as follows: u u a t x uv y were a is te flow acceleration in eac point in a flow field, te first term on te rigt and side of Equation 6 becomes zero in a steady flow. In tis study, we use te dept averaged flow acceleration as te caracteristic acceleration for bot steady/unsteady non-uniform flow, it is defined as: 1 a ady (7) (6) 111

GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems were a is te dept averaged flow acceleration in steady and unsteady flows. Ten, inserting Equations 6 into Equation 7 one as: a u 1 dy t u 1 dy x 1 (8) uv dy y Using te Leibnitz teorem to integrate Equations 8 wit respect to water dept cannel bed ( y ) to te free water surface ( y ), one as: from te du u d du u d 1 a ( ) ( ) ( u v ) (9) dt dt dx dx were U = dept average velocity; = water dept; v = wall-normal velocity or vertical velocity; te subscript denotes te free surface were y ; is te momentum flux correction factor tat taes into te non-uniform of flow velocity across te inlet and outlet. Te value of ranges between 1.1 and 1.4 [1] and in tis study, te value of is assumed to be 1.3. In Equation 9, one needs to determine te wall-normal velocity at te free surface; it is a difficult tas for measurement in practice. Te vertical velocity at te free surface can be determined from te continuity equation, i.e. u v (1) x y Te wall-normal velocity at te free surface can be expressed as follows: u d d v dy udy u x dx (11) dx Equation 11 can be alternatively expressed as: d( U) d (1) v dx u dx Equation 1 relates te wall-normal velocity wit streamwise velocity and water dept only. Tus, using Equation 9 and 1 one is able to express te flow acceleration witout te wallnormal velocity tat is difficult for an ordinary engineer to determine. Te first term of Equation 1 on te rigt-and-side is zero in a steady flow, but it becomes non-zero in an unsteady flows. Te dept averaged flow acceleration can be obtained witout te wall normal velocity, tus it avoids te sortcomings of [1] s metod as only tese parameters are required: longitudinal velocity at te water surface, u ; vertical velocity at te water surface, v ; te variation of water dept wit time, d / dt ; te variation of water dept along te longitudinal direction, d / dx ; water dept, ; te variation of dept averaged longitudinal velocity wit time, du / dt ;and te variation of dept averaged longitudinal velocity squared along te cannel, du / dx. THE INFLUENCE OF FLOW ACCELERATION ON REYNOLDS SHEAR STESS IN STEADY AND UNSTEADY FLOWS In order to verify tat flow acceleration is responsible for te deviation of te measured Reynolds sear stress in non-uniform from tat in uniform flow, [8]experimental data is used as it may be one of te most compreensive dataset in te literatures, all parameters needed for Equation 9 were measured and documented. Te full profiles for ( u v / u ) are plotted in Figure for varying flow conditions using dimensionless velocity wit respect to te sear velocity against y /. In Figure, te open and solid symbols represent te measured Reynolds sear stress in non-uniform flow i.e. u v / u wile te straigt solid line is te predicted Reynolds sear stress in uniform flow using Equation 5. It can be seen 11

Section Hydrology and Water Resources tat te measured Reynolds sear stress in non-uniform flow becomes zero as y / approaces to te water surface, and its value becomes 1 as y / approaces to te bed surface ( y ), tese indicate tat at y and y, te measured Reynolds sear stress in non-uniform flows becomes identical as a uniform flow. However, between tese two extremes, te measurement of data points in accelerating and decelerating steady/unsteady flows locate on bot sides of te solid line or Equation 5. From tis comparison, te Reynolds sear stress profiles in unsteady flow are similar to tose in steady flow, tus it is possible to establis a unify formula to express te Reynolds sear stress distribution of steady/unsteady flows based on te effect of flow acceleration on te measured Reynolds sear stress. Figure : Te influence of dimensionless flow acceleration on te deviation of measured Reynolds sear stress in accelerating and decelerating steady and unsteady flows from uniform flow based on [8] experimental data sets. DISTRIBUTION OF REYNOLDS SHEAR STRESS After demonstrating te impact of flow acceleration on te deviation of Reynolds sear stress in steady/unsteady flows from tat in te uniform flow, one may conclude tat te difference of Reynolds sear stress in uniform form/unsteady or non-uniform flows is proportional to ( 1 y / ) and y / as te difference between tese two types of flow must become zero at tese boundary conditions, were y /. Tus te uv proportionality sould depend on a dimensionless flow acceleration i.e. a /( u / ). Terefore, tese empirical formulas are proposed as follows: u v ( u u v 1.5 ) nonuf. ( ) uf. (1 y / ) y / u u v were ( u v / u ) is te normalized Reynolds sear stress, and te subscript uf. and nonuf. refer to te Reynolds sear stress in uniform and non-uniform flows, respectively. In Equation 13, te value of uv as two signs, one positive and te oter negative. Te positive one can be used wen te flow is decelerating wile te negative sign will be used wen te flow is accelerating for bot steady and unsteady flows. To yield te best agreement between Equation 13 and te measured Reynolds sear stress, one can determine uv from experimental data in steady and unsteady flows. For example, te value of uv can be evaluated from Reynolds sear stress defect between te measured Reynolds (13) 113

GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems sear stress in uniform and non-uniform flows i.e. ( u v / u ) ( u v nonuf. u v unf. ) / u. By fitting te Reynolds sear stress difference, one may obtain te expressions of uv. Te values of uv are obtained from [8] s experimental data as sown in Figure 3, tus, te empirical expressions of are obtained: ln{1 5.8[ a /( u / )]} (Accelerating flow) (14) u v u v exp{.[ a /( u / )]} 1 From Equations 14 and 15, te observed value of (Decelerating flow) (15) uv is negative wen te flow is accelerating because te distribution of Reynolds sear stress in accelerating non-uniform flow is lower tan tose in uniform flow (see Figure 3a). Wile tis empirical value of is positive (see Figure 3b) wen te flow is decelerating because te data points for uv Reynolds sear stress in tis flow are iger tan tose in uniform flow. Eq. 14 Eq. 15 (a) (b) re 3: Relationsip between dimensionless flow acceleration and te value of uv Figu for Reynolds sear stress ( u v / u ) in (a) accelerating and (b) decelerating steady and unsteady flow based on selected data sets from [8] experimental data. Te solid symbols in Figure 3 denote te data obtained from unsteady flows, and te open symbols represent te same but in steady flow. In Figure 3b, tere is no decelerating unsteady data available to compare wit decelerating steady flows because [8] reported tat te flow in is experiments was accelerating in unsteady flows. A clear dependence of te coefficient of on te dimensionless acceleration can be observed and solid lines can uv be drawn based on te data points and te condition tat if a =, ten uv = in bot accelerating and decelerating flows, Equations 14 and 15 were obtained. In Figure 3, significant similarity is observed between dimensionless flow acceleration in steady and unsteady flows and tese similar values of flow acceleration give similar values for. uv Tus, tis is te reason wy te acceleration symbol ( a ) is used in Equations 14 and 15 witout any subscript relating to te steady or unsteady flow. Wen te flow is accelerating for bot steady/unsteady cases, a positive a gives a negative value of and te vice versa for decelerating flow. Wile if a =, ten uv uv = and as a result te second term of Equation 13 is negligible and tis is uniform flow. Consequently, if te flow acceleration in steady or unsteady flows is nown and ten te value of can be estimated based on uv Equations 14 and 15 for bot accelerating and decelerating flows. In order to cec te validity of Equation 13 wit tese empirical values for, te remaining datasets in [8] uv 114

Section Hydrology and Water Resources experiments except tat sow in Figure 3 are plotted in Figures 4-6. To demonstrate te performance of Equation 13, [8] s equations for te prediction of Reynolds sear stress in steady and unsteady flows are compared in eac figure based on te determination of te relative error ( E ) between te measured and predicted values as E u v m -u v c / u v c, were te subscript m and c are te measured and calculated values. In eac figure, te symbol E is introduced wit subscripts i.e. (Eq.13) and ([ 8] seq.), were te first subscript refers to te relative error between te measured and predicted values based on te developed model in te present study wile te second one is determined based on te predicted value using Songs equation. Figure 4: Comparison of measured and predicted Reynolds sear stress profile in accelerating unsteady flow based on [8] experimental data. Figure 5: Comparison of measured and predicted Reynolds sear stress profile in accelerating steady flow based on [8] experimental data. Figure 6: Comparison of measured and predicted Reynolds sear stress profile in decelerating steady flow based on Song s (1994) experimental data. In Figures 4-6, te calculated Reynolds sear stress profiles using Equation 13 are plotted as a solid line, te dased line is te calculate Reynolds sear stress using [8] s equation and te open circles denote to te measured data points in non-uniform steady/ unsteady. Te average values of determined E in Figures 4/6 using Equation 13 are less tan tat determined using [8] s equations for all types of flow, indicating tat te proposed model 115

GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems agreed te existing datasets in steady/unsteady flows, especially wen te value of flow acceleration is existed. Overall, it can be found tat te predicted formulas can give similar profiles to te measurements of Reynolds sear stress, concave distribution wen te applying negative value of positive. and convex distribution wen te value of is uv uv CONCLUSIONS Te distributions of Reynolds sear stress in non-uniform steady and unsteady flows ave been predicted. As observed from te literature, te distribution of Reynolds sear stress in non-uniform flow deviate from te linear distribution of uniform flow. Song s (1994) experimental results were used to develop te relationsips between te dimensionless flow acceleration and values of for te prediction of Reynolds sear stress. Tese uv empirical formulas were developed dependent on te impact of flow acceleration on te deviation of te Reynolds sear stress in non-uniform steady and unsteady flows from tat in uniform flow. In Reynolds sear stress, te positive acceleration (accelerating flow) generates negative value of wile te negative value of flow acceleration (decelerating uv flow) generates positive value of uv uv te distribution of. Using tese values of Reynolds sear stress across te wole water dept was ten predicted. Te experimental data from [8] support tese predictions. Wen te effect of flow acceleration is considered, te agreement between te measured and estimated Reynolds sear stress profiles in steady and unsteady flows is found to be good. REFERENCES [1]Cengel, Y.A. and Cimbala, J.M. 1, Fluid mecanics: fundamentals and application, nd ed., Mc Graw-Hill education, New Yor. []Dou GR 1981, Turbulent structure in open cannels and pipes, Sci Sinica, vol. 4, no. 5, pp77 37. [3]Ecelmann, H 1974, Te structure of viscous sublayer and adjacent wall region in a turbulent cannel, Journal of fluid Mecanics, vol. 1974, pp 439-59. [4]Grass, AJ 1971, Structural featers of turbulent flow over smoot and roug boundaries, Journal of fluid Mecanics, vol.5, pp 33-55. [5]Kironoto, B and Graf, WH 1995, Turbulence caracteristics in roug non uniform open cannel Flow, In: Proceedings of te institution Civil Engineering Water, Maritime and Energy, UK, vol. 11, pp 316-48. [6]Laufer, J 1954, Te structure of turbulence in fully developed pipe flow, NACA, Tecnical report, pp 1174. [7]Nezu I and Azuma R 4, Turbulence caracteristics and interaction between particles and fluid in particle-laden open cannel flows, Journal of Hydraulic Engineering, ASCE, vol. 13, no.1, pp 988 11. [8]Song, TC1994, Velocity and turbulence distribution in non-uniform and unsteady open - cannel flow, Doctoral dissertation, Ecole Polytecnique Federale delausanne, Switzerland. [9]Steffler PM, Rajaratnam N, Peterson AW 1985, LDA measurements in open cannel, Journal of Hydraulic Engineering, ASCE, vol.111, pp 119 3. [1]Yang, SQ and Lee, JW 7, Reynolds sear stress distributions in a gradually varied flow, Journal of Hydraulic Researc, vol. 45, no. 4, pp 46-71. 116