Continuous formulation for bottom friction in free surface flows modelling

Transcription

1 River Basin Management V 81 Continuous formulation for bottom friction in free surface flows modelling O. Maciels 1, S. Erpicum 1, B. J. Dewals 1, 2, P. Arcambeau 1 & M. Pirotton 1 1 HACH Unit, Department ArGEnCo, University of Liege, Belgium 2 Belgian Fund for Scientific Researc F.R.S-FNRS, Belgium Abstract Bottom friction modelling is an important step in river flows computation wit 1D or 2D solvers. It is usually performed using empirical laws establised for uniform flow conditions or a modern approac based on turbulence analysis. Following te definition of te flow validity field of te main friction laws proposed in te literature, an original continuous formulation as been developed. It is suited to model river flows wit a wide range of properties (water dept, discarge, rougness ). Te efficiency of tis new formulation, teoretically establised and numerically adjusted, is demonstrated troug various practical applications. Keywords: sallow water, bottom friction, empirical laws, modern laws. 1 Introduction Sallow Water equations are commonly used to model numerically river flows. Indeed, teir main assumption states tat te flow velocity component normal to te main flow plane is smaller tan te flow velocity components in tis plane. Tis is te case for te majority of river flows were vertical velocity component is negligible regarding te orizontal ones, except in te vicinity of singularities suc as weirs for example. Tis paper focuses on bottom friction modelling in suc matematical models. Tis effect is indeed of very ig importance for real flow computation despite it is generally evaluated from empirical formulations experimentally determined for uniform flow conditions. doi: /rm090081

2 82 River Basin Management V Te bottom friction term sould represent te wole of te energy losses induced by te friction of te fluid on te roug river bed. It is tus related to te bed caracteristics (sape, rougness), to te fluid caracteristics (viscosity) and to te flow features (water eigt, velocity). Te concept of friction slope [1] as been early used to caracterize bottom friction. It is a non-dimensional number corresponding to te slope of a uniform cannel were a uniform flow establises for a given discarge. Tis concept is at te basis of te first friction laws proposed in te second part of te 18 t century by te researcers of te so called empirical scool. Autors suc as Cézy [2] and Manning [3] proposed laws based on experimental results consisting in measuring te friction slope for a number of idealized flows in a laboratory flume. A second approac appeared later following wors of Prandtl [4]. It provided laws issued from analysis of te pysics of te sear layer penomena, referred to as te modern scool. Today, bot approaces are used by free surface flow modellers, and tese laws are sometimes applied to flow conditions far from tose on te basis of wic tey ave been developed. It is tus important to eep in mind te validity ranges of eac of tese laws and to underline te lac of a single formulation able to describe te bottom friction penomena for largely variable flow conditions. Te aims of te developments presented in tis paper were to define and validate an original continuous formulation for bottom friction, contributing to filling tis lac. 2 Main friction laws 2.1 Empirical laws Te laws of te so called empirical scool ave all been developed on te basis of experimental tests. Tese tests consisted in measuring te slope of a uniform cannel were a uniform flow can be obtained [1]. In tese flow conditions, te effects of bottom friction are exactly counterbalanced by te gravitational forces. Tus te friction slope is equal to te bottom slope of te cannel and simple formulae can be set up to lin te cannel rougness, te flow variables and te bottom slope. Replacing te bottom slope by te friction slope, te friction effects can be computed for oter flow conditions tan te uniform ones. Te general form of empirical friction formulations writes: 1 2 U J R (1) It relates te friction slope J to ydraulic and geometric parameters affecting te bottom friction suc as U te mean flow velocity, α a friction coefficient and R te ydraulic radius. Tis last parameter reflects te effect of te cross section sape. Te differences between te different friction formulations of te empirical scool are in te χ exponent value and in te α coefficient form.

3 River Basin Management V 83 It is important to note tat tese formulations do not explicitly tae into account te turbulence regime of te flow, despite it is well nown tat tis flow state is of great influence on te friction losses. Cézy [2] and Manning [3] formulations are te most widely used empirical friction laws because of te extensive nowledge of te friction coefficient α available in te literature for bot of tem. Oters exist, suc as te laws of Gaucler, Forceimer, Cristen, Hagen and Tillman [1]. Tey differ only by te exponent of te ydraulic radius in te general formulation (1) as sown in table 1. Table 1: Value of exponent of te ydraulic radius for different empirical friction formulations. Autor χ Cézy 0.5 Manning Gaucler 0.4 Forceimer 0.7 Cristen Hagen Tillman Modern scool In contrast wit empirical laws, formulations from te modern scool rely on a sound teoretical bacground on te pysics of friction penomena [1]. Te developments of te modern scool appear one century later tan te first empirical developments. Under te leadersip of Prandtl, researcers from te University of Göttingen (Germany) developed formulations of a friction coefficient, λ, function of te turbulence of te flow troug te Reynolds number Re, and te size of te rougness elements of a pipe,. Modern formulations ave been initially developed for pressurized flows to determine ead losses in pipes. Te Darcy-Weisbac relation [5] lins te friction slope J to te friction coefficient λ: 2 U J 4 R g (2) 2 were 4 R is te equivalent diameter of a cannel and g is de gravity acceleration. In 2D flow modelling, te ydraulic radius R is equivalent to te water dept. In te rest of tis paper, bot expressions will be used similarly. Te friction coefficient λ evaluation depends on te flow turbulence regime, and tus te Reynolds. Te main modern laws are sown in table 2 and classified depending on te flow turbulence regime.

4 84 River Basin Management V Developments on macro-rougness are more recent and are directly related to te development of ydrological modelling. Tey are associated to te modern scool because of te similar form of te matematical formulations. Table 2: Flow turbulence regime Laminar Main modern laws for friction coefficient depending on te flow turbulence regime. Autor Law Teoretical validity range Poiseuille 64 [6] Re (3) Re < 5000 Smoot turbulent Prandtl [4] log (4) Re / negligible Transitional Roug turbulent Macrorougness Colebroo [7] 2 log (5) / < Re 2240/Re Niuradse 1 2 log [8] Baturst 1 [9] log 5.15 (6) / > 2240/Re (7) / > 0.25 Te implicit caracter of some modern equations gives tem an uneasily use. Tat s te reason wy different autors developed explicit equivalent formulations, suc as Barr [10] wo provides an explicit form of te Colebroo formulation (5), wit less tan 1% error on te friction coefficient values: Re log 1 7 2log Re Re (8)

5 3 Validity fields of te mean friction laws River Basin Management V 85 On one and, te empirical laws ave not been developed regarding te variation in turbulence regime of real water flows. On te oter and, modern laws tae better into account te turbulence regime of te flow but none of tem can be applied to te wole flow conditions of real river flows. It is tus important to eep in mind te validity fields of te different friction laws, as defined in [11] function of teir practical use. Tese validity fields are sown in terms of relative rougness in te table 3. Table 3: Validity fields of te principal friction laws in terms of relative rougness [11]. Autor Practical validity range (/) Cézy Near 0 Cristen [0 ; 0.032] Manning [0.007 ; 0.1] Tillman [0.023 ; 0.29] Hagen [0.034 ; 0.38] Gaucler No validity Poiseuille Only laminar Prandtl Near 0 Colebroo [0 ; 0.1] Barr [0 ; 0.1] Niuradse [0 ; 0.1] Baturst [0.1 ; 5.15] Today, te empirical laws of Manning and Cézy are te most widely used formulations for friction modelling. Tis success comes from teir simplicity of use and te large existing literature on teir parameters values. However, te modern laws are more representative of te pysics of bottom friction. Furtermore, explicit forms of te modern laws exist suc as te one of Barr (8). Te modern laws ave tus an important interest for flow modelling. In practice, oter losses, suc as tose due to te turbulence, are included in te friction term used by most flow solvers. In tis case, it is ten important to eep in mind tat te friction slope J terms in te matematical model not only represents te bottom friction penomenon. 4 Continuous friction formulation As sown in table 3, no single formulation is suitable to compute friction effects on te wole range of relative rougness encountered in real river flow, were goes from 0 on te bans to several meters in te cannel centre wit an essentially constant rougness eigt. However, te / validity ranges of several laws are contiguous suc as for example for Colebroo or Barr and Baturst.

6 86 River Basin Management V Terefore, an original approac as been developed on te basis of te tree following statements: Barr formula applies for turbulent flows wit relative rougness / lower tan 0.1. Baturst formula applies to compute friction effect on macro-rougness, i.e. for / iger tan 0.1. But tese two formulations are not equal for a relative rougness / in teir respective validity fields. Developments ave been performed to lin continuously tese two formulations close to relative rougness / equal to 0.1 [11]. Tey finally lead to te following expressions wic can be used to compute continuously te bottom friction effects in rivers or cannels watever te value of /: For : Barr formula For : (9) For 5 Validations 5.1 2D approac : Baturst formula Two Belgian river reaces were considered to validate te continuous approac: te Ourte near te town of Hamoir and te Semois near Membre, fig. 1. Tese two reaces, respectively 2.6 m and 1.6 m long, were selected because of te presence of two gauging stations on bot of tem. Te downstream one combined wit te discarge measurement provides te necessary boundary conditions. Bot river reaces ave been modelled using te 2D-orizontal finite volumes flow solver WOLF2D, developed at te University of Liege [12], using different friction laws suc as Manning, Barr, Baturst and continuous formulations, wit a regular 2 x 2 m mes. Te comparison between te upstream water depts computed using te laws of Barr, Manning, Baturst and te continuous formulation, and te water dept measurements at te upstream gauging station for different discarges as been used to sow te interest of te continuous formulation, fig. 2 and 3, and table 4. It must be noted tat te computation time remains similar watever te friction law. Te Manning s coefficient n value, wic is te inverse of te α coefficient of equation (1), was fitted regarding te real data for te igest discarge. It is

7 River Basin Management V 87 equal to s/m 1/3 in te Ourte and to s/m 1/3 in te Semois. Te constant value for Barr, Baturst and continuous formulations was set up to get close to te measurements for bot te lowest discarge wit Baturst formulation and te igest one wit Barr equation. Its value is 0.09 m in te Ourte and 0.3 m in te Semois. Te water dept is not omogeneous along te river reaces. Te / ratio indicated in table 4 is tus te ratio value at te upstream limit of te river reaces, at te centre of te cross section. Tat s te reason wy te results provided by te continuous formulation are not exactly equivalent to tose provided by te Barr and te Baturst formulations, respectively for / < 0.05 and / > Tabreux (Water dept, discarge) Membre upstream (Water dept) Hamoir (Water dept) Membre (Water dept, discarge) Figure 1: Numerical representation of te topograpy of te Ourte River near Hamoir and te Semois River near Membre and location of te gauging stations. On te Ourte River, for / ratios lower tan 0.05, te water depts computed using Manning, Barr and continuous formulations are relatively close to te measurements (less tan 2.5%), wereas te Baturst results are less satisfactory (more tan 10% error). Tis expresses well te validity of Barr and continuous formulations for 2D free surface flow wit low relative rougness

8 88 River Basin Management V / < 0.05 H (m) < / < / > 0.15 Manning Baturst Barr Continuous formulation Measured Q (m³/s) Figure 2: Computed and measured rating curves in Hamoir on te Ourte River / < 0.15 H(m) Manning / > 0.15 Baturst Barr Continuous formulation Measured Q (m³/s) Figure 3: Computed and measured rating curves in Membre on te Semois River. modelling. Tis also expresses te efficiency of te Manning s formulation for flow conditions close te validation ones. Finally, tis confirms tat Baturst formula does not apply for low relative rougness, as sown in table 3.

9 River Basin Management V 89 Table 4: Mean relative error between measured and computed water depts (%). Situation Modelling law / <0.05 Hamoir Ourte Membre - Semois 0.05 < / < 0.15 / > 0.15 Manning Barr Baturst Continuous formulation Manning Barr Baturst Continuous formulation For / ratios iger tan 0.15, te water depts, provided using Baturst and continuous formulations, are te closest to te measurements. Te important value of te relative error is partially due to te important effect of measurement uncertainty for low water depts. Indeed, water dept measurements accuracy is close to 5 cm in tese cases. However, te results sow te interest of Baturst and continuous formulations, compared to Manning and Barr ones, for flow modelling on ig relative rougness. Tey also sow te limitations of te Barr formulation for ig relative rougness and of te Manning s one wen flow conditions differ from te validation conditions. For intermediary / ratios, Baturst formulation remains attractive wen te water depts ave a low variability on te river reac suc as on te Ourte River. However, wen te water depts are more variable, suc as on te Semois River, te continuous formulation becomes more accurate D approac Based on water dept and discarge measurements and considering a uniform flow, Martiny [13] determined te α coefficient of Manning equation (1), best nown as Stricler coefficient K, for a number of Belgian rivers. Te uniform flow ypotesis is inaccurate for low water dept. From Martiny s results, tree cases, corresponding to te largest rivers studied, ave tus been considered: te Warce River in Tioux as well as te Ourte River in Wyompont and in Amberloup. For tese tree places, Martiny calculated K values depending on te water dept. To compare Martiny s results wit tose provided by te continuous formulation (9), Stricler coefficient was expressed as a function of te friction coefficient λ. Tis was obtained by insertion of Manning equation (1) in Darcy- Weisbac relation (2):

10 90 River Basin Management V U KJ 2 8g 1 6 U K H (10) J 4 2g Considering te λ value provided by continuous formulation (9), equation (10) gives te value of te Stricler coefficient to use for an exact modelling of te continuous friction formulation. Fig. 4, 5 and 6 sow te comparison between Martiny s results and tose provided by equation (10) considering a particular bottom rougness for eac river reac K H (m) Martiny Continuous formulation ( = 0,2) Figure 4: Stricler coefficient value in Tioux on te Warce River. K H (m) Martiny Continuous formulation (=0.16) Figure 5: Stricler coefficient value in Wyompont on te Ourte River.

11 River Basin Management V 91 In te tree cases, te mean relative error on te K value provided by te continuous formulation remains lower tan 5%. Tis comparison igligts tus te ability of te continuous formulation to describe te friction penomenon for 1D flow modelling wit a single rougness value wile te Manning formulation wit a single K value is only suited to model te friction for a particular flow K H (m) Martiny Continuous formulation ( = 0,1) Figure 6: Stricler coefficient value in Amberloup on te Ourte River. 6 Conclusions Friction is a complex penomenon wit a non negligible influence on river flows caracteristics. It is tus necessary to tae it into account for a correct flow modelling. Many autors ave developed friction formulations. But tese laws are not always suited to describe te friction penomenon in te wole range of real varied flow conditions. In tis study, an original friction law as been developed to fill te lac of a continuous law able to describe te friction penomenon for te igly variable flow conditions often met in river flows. Tis law as been validated for 2D modelling by comparison of water dept values on two different river reaces in Belgium and for 1D modelling by comparison of experimental investigations on tree Belgian river reaces. References [1] Carlier, M., Hydraulique générale et appliquée, Eyrolles, Paris, [2] Cézy, A., Formule pour trouver la vitesse de l eau conduite dans une rigole donnée, Dossier 847 (MS 1915) of te manuscript collection of te École National des Ponts et Caussées, Paris, [3] Manning, R., On te Flow of Water in Open Cannels and Pipes, Institute of Civil Engineers of Ireland, 1890.

12 92 River Basin Management V [4] Prandtl, L., Über Flussigeitsbewegung bei ser leiner Reibung, Ver. III Intl. Mat. Kongr., Heidelberg, [5] Weisbac, J., Lerbuc der Ingenieur- und Mascinen-Mecani, Braunscwieg, [6] Poiseuille, J.L.M., Sur le Mouvement des Liquides dans le Tube de Très Petit Diamètre, Comptes Rendues de l'académie des Sciences de Paris, Vol. 9, [7] Colebroo, C.F., Turbulent Flow in Pipes wit particular reference to te Transition Region between te Smoot and Roug Pipe Laws, J. Inst. Civ. Engr, No. 4, [8] Niuradse, J., Strömungsgesetze in rauen Roren, VDI-Forscungseft, No. 361, [9] Baturst, J.C., Flow resistance estimation in mountain rivers, J. Hydraul. Eng. - ASCE, 111(4), [10] Barr, D.I.H., Solution of te Colebroo Wite functions for resistance to uniform turbulent flows, Proc. Inst. Civ. Eng., [11] Maciels, O., Erpicum, S., Arcambeau, P., Dewals, B.J. & Pirotton, M., Bottom friction formulations for free surface flow modelling, proc. 8 t National Congress on Teoretical and Applied Mecanics, Brussels, [12] Erpicum, S., Arcambeau, P., Detrembleur, S., Dewals, B.J. & Pirotton, M., A 2D finite volume multibloc flow solver applied to flood extension forecasting, In: P. Garcia-Navarro, E. Playan (Eds), Numerical modelling of ydrodynamics for water resources, Taylor & Francis, London, [13] Martiny, A., Contribution à la détermination du coefficient de rugosité des rivières, Ministère de la région wallonne, Direction des cours d eau non navigables, Namur, 1995.