Velocity distribution in non-uniform/unsteady flows and the validity of log law

Transcription

1 University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 3 Velocity distribution in non-uniform/unsteady flows and te validity of log law Israq Alfadli University of Wollongong, ia79@uowmail.edu.au Su-qing Yang University of Wollongong, suqing@uow.edu.au Muttucumaru Sivakumar University of Wollongong, siva@uow.edu.au Publication Details Alfadli, I., Yang, S. & Sivakumar, M. (3). Velocity distribution in non-uniform/unsteady flows and te validity of log law. SGEM 3: 3t International Multidisciplinary Scientific Geoconference (pp ). 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

2 Velocity distribution in non-uniform/unsteady flows and te validity of log law Abstract Tis study investigates te longitudinal velocity profiles in steady and unsteady non-uniform open cannel flows by analyzing te data available in te literature. It was found tat for steady/unsteady flow in te Log law is applicable only in te inner region were y/g Keywords flows, validity, log, law, velocity, distribution, non, uniform, unsteady Disciplines Engineering Science and Tecnology Studies Publication Details Alfadli, I., Yang, S. & Sivakumar, M. (3). Velocity distribution in non-uniform/unsteady flows and te validity of log law. SGEM 3: 3t International Multidisciplinary Scientific Geoconference (pp ). Bulgaria: SGEM. Tis conference paper is available at Researc Online: ttp://ro.uow.edu.au/eispapers/466

3 Section Hydrology and Water Resources VELOCITY DISTRIBUTION IN NON-UNIFORM/UNSTEADY FLOWS AND THE VALIDITY OF LOG LAW Researc Student, Israq Alfadli Assoc. Prof. Su-Qing Yang Assoc. Prof. Muttucumaru Sivakumar Scool of Civil, Mining and Environmental Eng., University of Wollongong, Australia ABSTRACT Tis study investigates te longitudinal velocity profiles in steady and unsteady nonuniform open cannel flows by analyzing te data available in te literature. It was found tat for steady/unsteady flow te Log law is applicable only in te inner region were y /. wen compared wit te measured longitudinal velocity. However, in te main flow, te measured velocity deviates significantly from te Log law s prediction. Te detail assessment sows tat te deviation is negative wen a flow is accelerating, and positive wen decelerating. Te reason for tis deviation is explained by linking tese deviations wit te presence of wall normal velocity. A new equation was developed to express te velocity in non-uniform flow wic is valid for uniform flows (Log law) and accelerating and decelerating steady/unsteady flows, in wic Log law, Cole s Wake law and Dip law ave been combined togeter to reflect te influence of acceleration on te velocity profile. Tis modification as been verified wit experimental data sets available in te literature, and a reasonable agreement between te measured and predicted mean velocity profile is obtained. Keywords: uniform/non-uniform flows; velocity distribution; Log-Wake Dip law; steady/unsteady flows; accelerating/decelerating flows. INTRODUCTION Accurate knowledge of te velocity distribution in an open cannel is of crucial importance to practicing engineers as its elps in te estimation of erosion and sediment transport. Extensive and intensive researc as been conducted over te past century. [] proposed te universal wall function or Log law of te wall as: u yu* ln B () u* k were te integral constant B =5.9; te universal von Kármán constant k =.4; u * is te sear velocity; is te kinematic viscosity; y is te vertical distance measured from te reference level; and u is te point mean orizontal velocity at y. For a roug bed, Equation can be rewritten as follows: u u * () ln k y y k s B were ks is te rougness eigt = d 5 ; y is te reference bed level =. k s and B is te constant of integration. In Equations and, tere are two constant ( k and B ) and tese can be calculated from te experimental results. However, te experimental data from flat boundary layer flows [5], [8] and pipe flows [4] suggested tat te Log-law may be not always be correct in describing te velocity distribution. Tis was also observed in open cannel flows over years ago by river engineers, suc as [6], wo discovered from teir measurement in rivers tat maximum velocity does not appear at te free surface as te Log 45

4 GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems law predicts, but occurs below te free surface. Tis effect, also called dip-penomenon, remains an open question for researcers wo still debate is mecanism. Consequently, an article in Science [3] commented tat: te law of te wall was viewed as one of te few certainties in te difficult field of turbulence, and now it sould be detroned. Generations of engineers wo learned te law will ave to abandon it. In open cannel flows, te same penomenon as been known for a long time, but te mecanism is still unclear. [5] introduced an additional term (i.e. Coles Wake term) to express te deviation of measured velocity from te prediction of Log law, wic as te following formula: u y y ln( ) B sin ( ) (3) u* k y k were is te wake strengt parameter. Different values ave been found for tis parameter based on te type of flow and bed configuration [], [4], [7], [9], []. Tus, te value of is not universal. From te brief review, te Log law is applicable only in te inner region wit ( y /.). Terefore, te Log law cannot express te velocity distribution accurately in te outer region. Wilst significant advances ave been made by using Cole s Wake law, te mecanism of Cole s Wake law and te associated wake strengt parameter are not fully understood. [] discussed te dip penomenon, in wic te maximum longitudinal velocity occurs below te water surface. Tey suggested tat te Cole s Wake law is not able to describe te entire velocity profile wen te dippenomenon exists. Terefore, [] modified Log law by adding a term to express te dip penomenon instead of te Cole s Wake law based on Reynolds equations: u u y y ln( ) ln( ) k y k * (4) were is te dip correction factor and can be determined by: z.3exp( ) (5) were z is te distance from te sidewall in z direction. It is clearly seen from Equation 4 tat a dip model consists of two logaritmic distances, one from te bed (i.e. Log law) and te oter from te free surface i.e. ln y/. Similarly, te [] model is unable to fit te cases were te measured local velocity is iger tan te prediction by Log law. In te literature owever, tere is no universal model to express te velocity in te complex flow conditions, tus more researc is needed to clarify wy te Log law cannot predict te measured longitudinal velocity well in non-uniform flows. Tis leads to te present researc aims to develop a universal model to express te velocity profile in uniform and nonuniform as well as steady and unsteady open cannel flows. THE RELATIONSHIP BETWEEN THE FLOW ACCELERATION AND VELOCITY DISTRIBUTION 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 and 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 steady and unsteady nonuniform 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 46

5 Section Hydrology and Water Resources 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 nonuniform 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. Te distinction of velocity profiles in a uniform flow wit tose of accelerating and decelerating flows is sown in Figures, were te velocity profiles were measured by [7], in te form of u / u* against y y / ks, in wic y. *ks is te reference bed level; and k s d5. Figure sows tat te measured velocity profiles matc well in te inner region wit te Log law as no influence for te flow acceleration ( a = ). Wile te measured data points bend down from te straigt line of Log law prediction as te flow acceleration ( a >) is positive; and tey bend up over te Log law prediction wen te flow acceleration ( a <) is negative. Figure : Comparison between measured velocity profile wit Log law s prediction based on [7], were a is te dept averaged flow acceleration. Te reason for tis deviation was explained by [3] based on te Reynolds equation, and tey concluded tat te vertical velocity or wall-normal velocity ( v ) is responsible for te invalidity of Log law, tey found tat te Log law is valid if v = (uniform flow); and te [5] model becomes valid only wen v > (decelerating flows), and te maximum velocity is submerged below te water surface wen and only wen v < (accelerating flows). But in practice, te formulae proposed by [3] are very difficult to use because te wall-normal velocity is generally too small in quantity to measure. Terefore, to elp river engineers to solve teir practical problems easily, it is necessary to develop a formula tat is valid for all flow conditions and only te mean streamwise velocity, not te wall-normal velocity is used. To simplify suc as expression, te Log law togeter wit Cole s Wake law and Dip law are combined to express te longitudinal velocity across te wole water dept from te bed to te water surface for bot steady and unsteady flows. u y y y y (6) u ln( k ) k sin ( ) k ln( * ks ) were kand k are coefficients to be determined empirically. Te first term in Equation 6 refers to te Log law wile te second and tird terms are Cole s Wake law and Dip law respectively. Obviously, kand k are a function of flow acceleration, and tey become zero if te acceleration is zero (uniform flow), terms and 3 on te rigt and side of Equation 6 are negligible in te inner region, but tey are noticeable in te main flow region. Due to ln( y / ) as y /, terefore, Equation 6 is invalid in a very tin layer near te free surface just like te classical Log law tat becomes invalid at te layer closer to te boundary, i.e. y. In Equation 6, kand k are empirical coefficients and te relationsips 47

6 GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems will be evaluated wit te flow acceleration instead of te wall-normal velocity. As mentioned before, te wall-normal velocity is equivalent to predict te longitudinal velocity but te flow acceleration is more direct and simpler in te matematical treatment. For tis purpose a simple expression of flow acceleration sould be establised. DETERMINATION OF FLOW ACCELERATION FOR BOTH STEADY AND UNSTEADY FLOW Te flow acceleration in D flows 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 7 becomes zero in a steady flow. Te dept averaged flow acceleration ( a ) for bot steady and unsteady non-uniform flow is defined as: a (8) ady Ten, insert Equations 7 into Equation 8 to obtain: a (9) u dy t u dy x uv dy y Using te Leibnitz teorem, Equations 9 can be integrated wit respect to water dept from te cannel bed ( y ) to te free water surface ( y ), one as: du u d du u d a ( ) ( ) ( u v ) () dt dt dx dx were U = dept average longitudinal 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 takes into te non-uniform of flow velocity across te inlet and outlet. Te value of ranges between. and.4 [] and in tis study, te value of is assumed to be.3. Te dept averaged flow acceleration can be obtained witout te wall normal velocity, tus it avoids te sortcomings of [3] 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. Te vertical velocity at te free surface can be determined from te continuity equation, i.e. u v () x y Te wall-normal velocity at te free surface can be expressed as follows: u d d v dy udy u x dx () dx Equation can be alternatively expressed as: d( U) d (3) v dx u dx Equation 3 relates te wall-normal velocity wit streamwise velocity and water dept only. Tus, using Equations 3 and, one is able to express te flow acceleration witout te wall-normal velocity tat is difficult for an ordinary engineer to determine. Te first (7) 48

7 Section Hydrology and Water Resources term of Equation 3 on te rigt-and-side is zero in a steady flow, but it becomes non-zero in unsteady flows. THE INFLUENCE OF FLOW ACCELERATION ON VELOCITY DISTRIBUTION IN STEADY AND UNSTEADY FLOW In te literature, a compreensive measurement was conducted by [] wo measured mean orizontal velocity (u) in steady and unsteady non-uniform flows using an Acoustic Doppler Velocity profiler (ADVP). Based on [] experimental datasets, Equation is used to determine te flow acceleration in steady and unsteady flows. In order to prove te influence of flow acceleration on te deviation of tese velocity distribution in non-uniform flow from Log law, te measured velocity profiles are plotted in Figure in te form of dimensionless velocity versus te relative distance i.e., y y / k as ' x ', were te acceleration is normalized by u * /. Te typical velocity profiles in accelerating and decelerating flows are sown in Figure, in wic A means accelerating steady; D denotes decelerating steady; AU refers to accelerating unsteady; S represents te bed slope; Q is flow discarge; t means time; and 93 or 936 is te ydrograp s number in Song s experimental datasets. In Figure, te open and solid symbols are te measured point velocities and te solid line is te Log law from Equation. Te calculated values of a /( u * / ) are sown in legend of Figure. It can be seen clearly tat te measured velocity from te inner region in steady and unsteady flows matces well wit Log law s prediction but in te main flow region ( y /. ), te measured velocity becomes larger tan Log s formula wen te flow is decelerating or it as lower value wen te flow is accelerating, tis is wy in Equation 6 te dip-term and wake function are kept in te expression. Terefore, it seems tat te flow acceleration is a controlling parameter wic can affect te sape of te measured longitudinal velocity profile. s Figure : Te influence of dimensionless flow acceleration on te deviation of measured longitudinal velocity in accelerating and decelerating steady and unsteady flows from Log law based on [] s experimental data sets. To yield te best agreement between Equation 6 and te measured velocity profiles, one can obtain k and k from experimental data in steady and unsteady flows. For example, kand k can be evaluated from te velocity defect between te measured and Log law predicted velocities u (i.e. u ( u measure u Loglaw) / u* ). Tis difference indicates te 49

8 GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems amount of deviation from te measured mean velocity and te Log law. By fitting te velocity difference, one may obtain te expressions of k and k. For accelerating flow, k and k were obtained from [] experimental data as sown in Figure 3, in wic only te data from accelerating flows in bot steady and unsteady flows were selected, and te following empirical expressions are obtained:.7 k.4[ a /( u* / )] (4).7 k [ a /( u* / )] (5) A clear dependence of tese coefficients kand k on te dimensionless acceleration can be observed and solid lines can be drawn based on te data points presented in eac figure. If a =, ten k and k =, Equations 4 and 5 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 kand k. Tus, tis is te reason wy te acceleration symbol ( a ) is used in Equations 5 and 6 witout any subscript relating to te steady or unsteady flow. From tese figures, it is clear tat te values of kand k increase wit te flow acceleration. Figu re 3: Relationsip between k, k and dimensionless flow acceleration a/(u * /) in steady and unsteady flow based on []experimental data sets. For decelerating flows, te empirical equations for kand k can be obtained wit te similar metod as sown in Figure 4, and te kcan be evaluated as an exponential function of negative flow acceleration: k exp{.6*[ a /( u* / )]} (6) It was found by analyzing [7] data tat k is very small and k. Figure 4: Relationsip between k and dimensionless flow acceleration a/(u*/) in decelerating steady flow based on [] experimental data sets. In order to ceck te validity of Equation 6 wit tese empirical values for kand remaining datasets in [] experiments expect tose sown in Figures ¾ are plotted k, te 43

9 Section Hydrology and Water Resources in Figures 5-7. In order to demonstrate te performance of Equation 6 and te two empirical values of kand k obtained from teir relationsip wit te flow acceleration, in eac figure te relative error between te measured and predicted longitudinal velocity profile is determined as E u m -u c / u m *, were te subscript m and c are te measured and calculated velocities respectively. As can be seen from Figures 5-7, te average values of E are less tan 4% error band between te measurement and calculations velocity profiles in accelerating unsteady flow. Terefore, it can be seen from tis comparison tat Equation 6 is able to capture te velocity distribution in te entire profile tat includes te inner and outer regions depending on te value of flow acceleration. Figure 5: Comparison of measured and predicted mean orizontal velocity profile in accelerating unsteady flow based on [] experimental data. Figure 6: Comparison of measured and predicted mean orizontal velocity profile in accelerating steady flow based on [] experimental data. Figure 7: Comparison of measured and predicted mean orizontal velocity profile in decelerating steady flow based on [] experimental data. CONCLUTIONS It as been widely reported tat te longitudinal velocity distribution deviates from te classical Log law in steady and unsteady open cannel flows. In accelerating flow, te measured data fall below te Log law wile it as iger value wen te flow is decelerating. In tis paper, we attribute tis deviation to te existence of vertical velocity generated from non-uniform flows. Terefore, a new empirical equation was developed to predict te longitudinal velocity in non-uniform flow wic is valid for uniform flows (Log 43

10 GeoConference on Water Resources. Forest, Marine and Ocean Ecosystems law) and accelerating/ decelerating steady and unsteady flows, in wic Log law, Cole s Wake law and Dip law ave been combined togeter to reflect te influence of acceleration on te velocity profile. Wen te longitudinal velocity increases along te cannel te flow acceleration is positive; and te acceleration becomes negative wen tis velocity decreases. Tis flow acceleration or deceleration can be predicted using Equation wic is valid for bot steady and unsteady flows. Te relationsip between tese tree laws wit te influence of flow acceleration as been proposed, in wic two k factors tat depend on dimensionless acceleration are determined. Te one factor of k is introduced for te Cole s Wake law and te oter for te Dip law. Song s experimental data was used to verify te relationsip between te flow acceleration and te values of k. It is found tat for bot positive and negative flow accelerations, te values of k,k are positive. Comparing Equation 6 wit experimental data by [], a good agreement was obtained between te measured and predicted value of velocity profiles depending on te existence of flow acceleration. REFERENCES [] Cardoso, AH, Graf, WH and Gust, G 99, Uniform flow in a smoot open cannel, Journal of Hydraulic Researc, vol. 7, no.5, pp [] Cengel, Y.A. and Cimbala, J.M., Fluid mecanics: fundamentals and application, nd ed., Mc Graw-Hill education, New York. [3] Cipra, B. 996, A new teory of turbulence causes a stir among experts, Science, vol. 7, no. 564, pp 95. [4] Coleman, N L and Alonso, CV 983, Two dimensional cannel flows over roug surfaces, Journal of Hydraulic Engineering, vol. 9, no., pp [5] Coles, D 956, Te law of te wake in turbulent boundary layer, Journal of Fluid Mecanics, vol., pp 9-6. [6] Francis, J.B. 878, On te cause of te maximum velocity of water flowing in open cannels being below te surface, Trans. ASCE., May. [7] Kironoto, B and Graf, WH 995, Turbulence caracteristics in roug non uniform open cannel Flow, In: Proceedings of te institution Civil Engineering Water, Maritime and Energy, UK, vol., pp [8] Monty, J. P., Hutcins, N., Ng, H. C. H., Marusic, I. and Cong M. S. 9, A comparison of turbulent pipe, cannel and boundary layer flows, Journal of Fluid Mecanics, vol.63, pp [9] Nezu, I, Kadota, A and Nakagawa, H 997, Turbulent structures in unsteady deptvaring open cannel flows, Journal of Hydraulic Engineering, vol. 3, no. 9, pp [] Prandtl, L 95, Uber die ausgebildete turbulenz. ZAMM, vol. 5, no. 36. [] Song, TC994, Velocity and turbulence distribution in non-uniform and unsteady open - cannel flow, Doctoral dissertation, Ecole Polytecnique Federale de Lausanne, Switzerland. [] Yang, SQ, Tan, SK, and Lim, SY 4, Velocity distribution and Dip penomenon in smoot uniform open cannel flows, Journal of Hydraulic Engineering, vol. 3, no., pp [3] Yang, SQ and Lee, JW 7, Reynolds sear stress distributions in a gradually varied flow, Journal of Hydraulic Researc, vol. 45, no. 4, pp [4] Zagarola, M. 996, P.D. tesis (Princeton Univ., Princeton). 43