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Analytical Formulation of the Correction Factor Applied in Einstein and Barbarossa Equation (1952)

Analytical Formulation of the Correction Factor Applied in Einstein and Barbarossa Equation (1952) J. Hydrol. Hydromech., 57, 2009, 1, 40­44 DOI: 10.2478/v10098-009-0004-2 ACHANTA RAMAKRISHNA RAO, BIMLESH KUMAR Department of Civil Engineering, IISc, Bangalore-India; mailto: ark@civil.iisc.ernet.in; bimlesh.iisc@gmail.com Einstein-Barbarossa velocity or resistance equation (1952) is widely used to find resistance to flow in alluvial channel. In order to validate the equation in all ranges (smooth to rough); they introduced a correction factor based on the Nikuradse measurement. This correction factor is determined from the graphical method, which can be erroneous. Present work reanalyzes the Nikuradse measurements and gives an analytical formulation for the correction factor. KEY WORDS: Alluvial Channel, Einstein-Barbarossa Equation, Logarithmic Velocity Profile, Hydraulic Rough Boundary, Hydraulic Smooth Boundary, Sediment Transport. Achanta Ramakrishna Rao, Bimlesh Kumar: ANALYTICKÁ FORMULÁCIA KOREKCNÉHO FAKTORA APLIKOVANÉHO V EINSTEINOVEJ­BARBAROSSOVEJ ROVNICI (1952). J. Hydrol. Hydromech., 57, 2008, 1; 8 lit., 2 obr. Einsteinova­Barbarossova rovnica (1952) sa casto pouzíva na urcenie odporu voci prúdeniu v kanáloch. Autori do nej zaviedli korekcný faktor, zalozený na meraniach Nikuradzeho, aby overili platnos rovnice v celom rozsahu drsností (od hladkých stien po drsné). Tento korekcný faktor sa urcuje grafickou metódou, ktorá môze vies k chybným výsledkom. V tejto práci sa znova analyzujú výsledky Nikuradzeho meraní a je navrhnutá analytická formulácia na výpocet korekcného faktora. KÚCOVÉ SLOVÁ: aluviálny kanál, Einsteinova-Barbarossova rovnica, logaritmický profil rýchlosti, hydraulicky drsné hranice, hydraulicky hladké hranice, transport sedimentov. Introduction One of the most important aspects in open channel flow computations is the estimation of hydraulic flow resistance. Knowledge about the hydraulic resistance is important for the understanding and handling of engineering and environmental problems involving rivers and streams. Its estimation has direct or indirect consequences in the planning, design, and operation of water resources projects including flood control, erosion control and channel stabilization. In an alluvial stream, the mobile bed formed by cohesionless alluvium is seldom flat; rather, it is covered by periodic bed deformations, known as bed forms. These bed forms change in type and size depending on the flow conditions. They constitute an important obstacle to the flow, and thus, the resistance of alluvial channels changes as bed forms change. 40 Einstein and Barbarossa (EB) in 1952 provided a semi-analytical method for the computation of flow resistance in alluvial channels. Although the technique is very old, it is still probably the most widely quoted of any existing techniques. They suggested that the resistance of an alluvial stream consists of bed resistance and bank resistance. Furthermore, the bed resistance consists of grain friction and bed form resistance. According to EB (1952), the shear stress or drag force acting along an alluvial bed can be divided into two parts, i.e, ' " = '+ " = S ( Rb + Rb ) , (1) where ­ the total drag force acting along an alluvial bed, ' and '' ­ the drag force due to grain roughness and form roughness, respectively, ­ the specific weight of water, S ­ the energy or channel " slope, and Rb and Rb ­ the hydraulic radii due to grain roughness and form roughness, respectively. The grain friction denotes the resistance to a two- dimensional flow, which is not affected by side banks, with a plane bed. The grain friction can be described by the following equation (EB 1952): R' = 5.75log 12.2 b , ks u* u (2) where ­ the average velocity, u* ­ shear velocity due to grain roughness = (gR'b S)0.5, ks ­ a representative roughness, which is taken as D65, the particle size of bed material of which 65 per cent by weight is finer and ­ a function of ks/, where is the thickness of laminar sublayer (= 11.6/u*). The relationship between and ks/ (= R*/11.6, R*= u*ks/ and is called particle Reynolds number) has been presented through a graph based on the measurement of Nikuradse's experimental data on sand roughened pipes. Although the EB (1952) equation was intended to be universal, embracing all sediment sizes and depth of flow, in practice it has on occasions given results which have been clearly very considerably in error (Smith, 1970). It may be or may not be, but the authors feel that can be attributed to the graphical determination of the parameter . Although Smith (1970) has presented the modified bed-form resistance diagram, he kept the same correction factor as devised by the EB (1952). Brownlie (1981) has analyzed the Nikuradse measurements and presented the three different and distinct equations in order to measure the analytically or explicitly. As said earlier, the graphical relationship between and ks/ has been derived based on the Nikuradse measurement, by carefully analyzing the Nikuradse measurement one distinct and unique semi-empirical relationship can be obtained, which can hopefully replace the graphical determination of . Analytical approach The wide acceptance of the log law of velocity distribution could be due to the fact that it can be justified with certain theoretical arguments, for example, Prandtl's mixing length assumption, von Karman's dimensional reasoning or Millikan's asymptotic analysis (Kundu et al., 2004). However, these arguments can be considered theoretically correct only in a limited region of flow, although the log law may apply practically beyond the region. Generally, it is believed that wall-bounded turbulent flows are characterized by two kinds of length scales. In the inner region near a smooth boundary, fluid viscosity is important, and thus the acceptable length scale is the viscous length scale. In the outer region that is sufficiently far from the boundary, flow inertia is significant. In addition, it is assumed that the shear velocity is a global velocity scale applicable both for the inner and outer region (Nezu and Nakagawa, 1993). The established laws of velocity distribution for turbulent flows can be expressed as: u u yu* for smooth pipes and a' y ks for rough pipes, b' (3) (4) where A ­ the inverse of Von-Karman's constant (= 2.45), a' and b' are constants, u ­ the velocity at a distance y measured from the pipe wall, u* ­ the friction velocity, ks ­ the Nikuradse's sand roughness height and is the kinematic viscosity of the fluid. As seen from the Eqs. (3) and (4), the characteristic length l for non-dimensionalizing the depth y is /u* for smooth turbulent flows and ks for rough turbulent flows. So it is proposed that l is actually a linear combination of both (/u* and ks) with a correction function, covering the all ranges i.e., smooth, transition and rough regimes of turbulent flows. Thus l = (a ' + b ' k s ) (R * ) , (5) where R* ­ equal to ks u* / and the correction function is assumed to be a function of R*. At R*0, pipe is said to be in smooth condition and for rough pipe R*. For large values of /u* the term a'/u* dominates making the second term b' ks negligible in comparison with it. So also for small values of /u*, the second term becomes important allowing the neglect of the first term. Thus the velocity laws covering all the three regimes can be summarized as, A. R. Rao, B. Kumar * a ' * + b ' ks (R ) u y ks A ln . a' * * + b ' (R ) R (6) Now, if a condition that (R* ) = 1 for both when R*0 and is imposed due to established physical conditions of hydraulically smooth and rough regions, Eq. (6) reduces to Eqs. (3) and (4) respectively. By assuming Eq. (6) valid for the entire pipe radius (r), an expression for /u* can be obtained by integrating Eq. (6). Resistance equation for free surface flows can be obtained by Eq. (7) by suitably adjusting the terms in it depending upon the geometry of flow region between pipes and free surface flows. If logarithmic law of velocity distribution is assumed to be valid throughout the radius, r, in case of pipes and the flow depth, D, in case of free surface flows; then r is to be replaced by D and a multiplying factor e-0.5 is to be introduced to B* in Eq. (7) as shown in Fig. 1. r k s B* u u (7) where a + bR* * B* = (R ) . * R (8) Fig. 1. Velocity profile in channel and pipe. Obr. 1. Profily rýchlostí v kanáli a v potrubí. Integrating the velocity profile over flow depth, it follows that u isequal to the valueof u corresponding todimensionallevel and y = 1/ e for freesurfaceflows D y = 1/ e1.5 for pipe flows ( Raudkivi,1967). r Point of application of average velocity in free surface flows: u* D y 1D u ( y ) dy = A ln dy (where C issomeconstant) D0 D0 C u u ( D ) - 1 = ( D ) + ln(0.37) * u = A[ ln 0.37 D ] point of application of u is y = 0.37 D u* y = 0.37 1/ e D Same principlecan be appliedin the turbulent pipeflow. However Pao (1961) hasgiven an empricalformula to calculate y/r , which u= isequal to0.216 1/ e1.5 y ks 1.5 y ks u u For Pipe e and e (in caseof channels). * * * B u B u Now comparing, B = e-0.5 B* e e1.5 = B B* (9) For free surface flows, it can be written as: D ks . e-0.5 B* (10) Substituting the values of a, b and (R*) in Eq. (13), can now be analytically determined from the following equation: Or the resistance equation for free surface flows can be written as: R* D ks . B (11) It is of interest to express Eq. (11) in a form given by EB (1952) as: 2 R* -0.33ln 6.5 * 0.444 + 0.135R 1 - 0.55e (15) u u 12.2 D , ks (12) where ­ a correction factor roughness introduced by EB (1952) and it is a function of ks/ or R*/11.6. Thus the expression for can be expressed as: In order to compute , EB (1952) have given a curve relating and 11.6R*. However, one can now use Eq. (15) for the determination of instead of their curve. The validity of the expression for in Eq. (15) is shown in Fig. 2 by using the Nikuradse's experimental data. Conclusions 1 1 R* . = 12.2 B 7.4 a + bR * (R * ) (13) Now re-analyzing Nikuradse experimental data (1937) on pressure drop measurements in sand roughened pipes, the following values of a = 0.444 and b = 0.135 are obtained and an expression for (R*) is given by R* -0.33ln 6.5 * (R ) =1 - 0.55e 1. In sediment transport, the curve given by EB (1952) can now be replaced by the analytical expression of Eq. (14). 2. Eqs. (7) and (11) are valid for all the three roughness regions, can now be used as a unique equation to find the resistance characteristics of smooth as well as sand roughened pipes and open channels respectively. (14) Nikuradse's Experimental data r / k =507 r / k =252 r / k =126 r / k =60 r / k =30.6 r / k =15 Rough Flow Sm oo th Fl ow Eq. 15 Fig. 2. Validity of . Obr. 2. Platnos . A. R. Rao, B. Kumar List of symbols A ­ inverse of Von-Karman's constant, a' ­ constant, a ­ constant, b' ­ constant, b ­ constant, B* ­ function of R*, D ­ flow depth in open channel [m], D65 ­ the particle size of bed material of which 65 per cent by weight is finer [m], ks ­ representative roughness scale = D65 [m], r ­ pipe radius [m], R ­ Reynolds number, R* ­ particle Reynolds number, R'b ­ the hydraulic radius due to grain roughness [m], R"b ­ the hydraulic radius due to form roughness [m], S ­ the energy or channel slope, ­ the average velocity [m s-1], u* ­ shear velocity due to grain roughness = (g R'b S) 0.5 [m s-1], ­ correction factor = a function of ks/, ­ 11.6/u*, the thickness of laminar sub layer, ­ kinematic viscosity [m2 s-1], ­ the total drag force acting along an alluvial bed [N], ' ­ the drag force due to grain roughness [N], '' ­ the drag force due to form roughness [N], ­ specific weight of water [N m-3]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Analytical Formulation of the Correction Factor Applied in Einstein and Barbarossa Equation (1952)

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de Gruyter
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Copyright © 2009 by the
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0042-790X
DOI
10.2478/v10098-009-0004-2
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Abstract

J. Hydrol. Hydromech., 57, 2009, 1, 40­44 DOI: 10.2478/v10098-009-0004-2 ACHANTA RAMAKRISHNA RAO, BIMLESH KUMAR Department of Civil Engineering, IISc, Bangalore-India; mailto: ark@civil.iisc.ernet.in; bimlesh.iisc@gmail.com Einstein-Barbarossa velocity or resistance equation (1952) is widely used to find resistance to flow in alluvial channel. In order to validate the equation in all ranges (smooth to rough); they introduced a correction factor based on the Nikuradse measurement. This correction factor is determined from the graphical method, which can be erroneous. Present work reanalyzes the Nikuradse measurements and gives an analytical formulation for the correction factor. KEY WORDS: Alluvial Channel, Einstein-Barbarossa Equation, Logarithmic Velocity Profile, Hydraulic Rough Boundary, Hydraulic Smooth Boundary, Sediment Transport. Achanta Ramakrishna Rao, Bimlesh Kumar: ANALYTICKÁ FORMULÁCIA KOREKCNÉHO FAKTORA APLIKOVANÉHO V EINSTEINOVEJ­BARBAROSSOVEJ ROVNICI (1952). J. Hydrol. Hydromech., 57, 2008, 1; 8 lit., 2 obr. Einsteinova­Barbarossova rovnica (1952) sa casto pouzíva na urcenie odporu voci prúdeniu v kanáloch. Autori do nej zaviedli korekcný faktor, zalozený na meraniach Nikuradzeho, aby overili platnos rovnice v celom rozsahu drsností (od hladkých stien po drsné). Tento korekcný faktor sa urcuje grafickou metódou, ktorá môze vies k chybným výsledkom. V tejto práci sa znova analyzujú výsledky Nikuradzeho meraní a je navrhnutá analytická formulácia na výpocet korekcného faktora. KÚCOVÉ SLOVÁ: aluviálny kanál, Einsteinova-Barbarossova rovnica, logaritmický profil rýchlosti, hydraulicky drsné hranice, hydraulicky hladké hranice, transport sedimentov. Introduction One of the most important aspects in open channel flow computations is the estimation of hydraulic flow resistance. Knowledge about the hydraulic resistance is important for the understanding and handling of engineering and environmental problems involving rivers and streams. Its estimation has direct or indirect consequences in the planning, design, and operation of water resources projects including flood control, erosion control and channel stabilization. In an alluvial stream, the mobile bed formed by cohesionless alluvium is seldom flat; rather, it is covered by periodic bed deformations, known as bed forms. These bed forms change in type and size depending on the flow conditions. They constitute an important obstacle to the flow, and thus, the resistance of alluvial channels changes as bed forms change. 40 Einstein and Barbarossa (EB) in 1952 provided a semi-analytical method for the computation of flow resistance in alluvial channels. Although the technique is very old, it is still probably the most widely quoted of any existing techniques. They suggested that the resistance of an alluvial stream consists of bed resistance and bank resistance. Furthermore, the bed resistance consists of grain friction and bed form resistance. According to EB (1952), the shear stress or drag force acting along an alluvial bed can be divided into two parts, i.e, ' " = '+ " = S ( Rb + Rb ) , (1) where ­ the total drag force acting along an alluvial bed, ' and '' ­ the drag force due to grain roughness and form roughness, respectively, ­ the specific weight of water, S ­ the energy or channel " slope, and Rb and Rb ­ the hydraulic radii due to grain roughness and form roughness, respectively. The grain friction denotes the resistance to a two- dimensional flow, which is not affected by side banks, with a plane bed. The grain friction can be described by the following equation (EB 1952): R' = 5.75log 12.2 b , ks u* u (2) where ­ the average velocity, u* ­ shear velocity due to grain roughness = (gR'b S)0.5, ks ­ a representative roughness, which is taken as D65, the particle size of bed material of which 65 per cent by weight is finer and ­ a function of ks/, where is the thickness of laminar sublayer (= 11.6/u*). The relationship between and ks/ (= R*/11.6, R*= u*ks/ and is called particle Reynolds number) has been presented through a graph based on the measurement of Nikuradse's experimental data on sand roughened pipes. Although the EB (1952) equation was intended to be universal, embracing all sediment sizes and depth of flow, in practice it has on occasions given results which have been clearly very considerably in error (Smith, 1970). It may be or may not be, but the authors feel that can be attributed to the graphical determination of the parameter . Although Smith (1970) has presented the modified bed-form resistance diagram, he kept the same correction factor as devised by the EB (1952). Brownlie (1981) has analyzed the Nikuradse measurements and presented the three different and distinct equations in order to measure the analytically or explicitly. As said earlier, the graphical relationship between and ks/ has been derived based on the Nikuradse measurement, by carefully analyzing the Nikuradse measurement one distinct and unique semi-empirical relationship can be obtained, which can hopefully replace the graphical determination of . Analytical approach The wide acceptance of the log law of velocity distribution could be due to the fact that it can be justified with certain theoretical arguments, for example, Prandtl's mixing length assumption, von Karman's dimensional reasoning or Millikan's asymptotic analysis (Kundu et al., 2004). However, these arguments can be considered theoretically correct only in a limited region of flow, although the log law may apply practically beyond the region. Generally, it is believed that wall-bounded turbulent flows are characterized by two kinds of length scales. In the inner region near a smooth boundary, fluid viscosity is important, and thus the acceptable length scale is the viscous length scale. In the outer region that is sufficiently far from the boundary, flow inertia is significant. In addition, it is assumed that the shear velocity is a global velocity scale applicable both for the inner and outer region (Nezu and Nakagawa, 1993). The established laws of velocity distribution for turbulent flows can be expressed as: u u yu* for smooth pipes and a' y ks for rough pipes, b' (3) (4) where A ­ the inverse of Von-Karman's constant (= 2.45), a' and b' are constants, u ­ the velocity at a distance y measured from the pipe wall, u* ­ the friction velocity, ks ­ the Nikuradse's sand roughness height and is the kinematic viscosity of the fluid. As seen from the Eqs. (3) and (4), the characteristic length l for non-dimensionalizing the depth y is /u* for smooth turbulent flows and ks for rough turbulent flows. So it is proposed that l is actually a linear combination of both (/u* and ks) with a correction function, covering the all ranges i.e., smooth, transition and rough regimes of turbulent flows. Thus l = (a ' + b ' k s ) (R * ) , (5) where R* ­ equal to ks u* / and the correction function is assumed to be a function of R*. At R*0, pipe is said to be in smooth condition and for rough pipe R*. For large values of /u* the term a'/u* dominates making the second term b' ks negligible in comparison with it. So also for small values of /u*, the second term becomes important allowing the neglect of the first term. Thus the velocity laws covering all the three regimes can be summarized as, A. R. Rao, B. Kumar * a ' * + b ' ks (R ) u y ks A ln . a' * * + b ' (R ) R (6) Now, if a condition that (R* ) = 1 for both when R*0 and is imposed due to established physical conditions of hydraulically smooth and rough regions, Eq. (6) reduces to Eqs. (3) and (4) respectively. By assuming Eq. (6) valid for the entire pipe radius (r), an expression for /u* can be obtained by integrating Eq. (6). Resistance equation for free surface flows can be obtained by Eq. (7) by suitably adjusting the terms in it depending upon the geometry of flow region between pipes and free surface flows. If logarithmic law of velocity distribution is assumed to be valid throughout the radius, r, in case of pipes and the flow depth, D, in case of free surface flows; then r is to be replaced by D and a multiplying factor e-0.5 is to be introduced to B* in Eq. (7) as shown in Fig. 1. r k s B* u u (7) where a + bR* * B* = (R ) . * R (8) Fig. 1. Velocity profile in channel and pipe. Obr. 1. Profily rýchlostí v kanáli a v potrubí. Integrating the velocity profile over flow depth, it follows that u isequal to the valueof u corresponding todimensionallevel and y = 1/ e for freesurfaceflows D y = 1/ e1.5 for pipe flows ( Raudkivi,1967). r Point of application of average velocity in free surface flows: u* D y 1D u ( y ) dy = A ln dy (where C issomeconstant) D0 D0 C u u ( D ) - 1 = ( D ) + ln(0.37) * u = A[ ln 0.37 D ] point of application of u is y = 0.37 D u* y = 0.37 1/ e D Same principlecan be appliedin the turbulent pipeflow. However Pao (1961) hasgiven an empricalformula to calculate y/r , which u= isequal to0.216 1/ e1.5 y ks 1.5 y ks u u For Pipe e and e (in caseof channels). * * * B u B u Now comparing, B = e-0.5 B* e e1.5 = B B* (9) For free surface flows, it can be written as: D ks . e-0.5 B* (10) Substituting the values of a, b and (R*) in Eq. (13), can now be analytically determined from the following equation: Or the resistance equation for free surface flows can be written as: R* D ks . B (11) It is of interest to express Eq. (11) in a form given by EB (1952) as: 2 R* -0.33ln 6.5 * 0.444 + 0.135R 1 - 0.55e (15) u u 12.2 D , ks (12) where ­ a correction factor roughness introduced by EB (1952) and it is a function of ks/ or R*/11.6. Thus the expression for can be expressed as: In order to compute , EB (1952) have given a curve relating and 11.6R*. However, one can now use Eq. (15) for the determination of instead of their curve. The validity of the expression for in Eq. (15) is shown in Fig. 2 by using the Nikuradse's experimental data. Conclusions 1 1 R* . = 12.2 B 7.4 a + bR * (R * ) (13) Now re-analyzing Nikuradse experimental data (1937) on pressure drop measurements in sand roughened pipes, the following values of a = 0.444 and b = 0.135 are obtained and an expression for (R*) is given by R* -0.33ln 6.5 * (R ) =1 - 0.55e 1. In sediment transport, the curve given by EB (1952) can now be replaced by the analytical expression of Eq. (14). 2. Eqs. (7) and (11) are valid for all the three roughness regions, can now be used as a unique equation to find the resistance characteristics of smooth as well as sand roughened pipes and open channels respectively. (14) Nikuradse's Experimental data r / k =507 r / k =252 r / k =126 r / k =60 r / k =30.6 r / k =15 Rough Flow Sm oo th Fl ow Eq. 15 Fig. 2. Validity of . Obr. 2. Platnos . A. R. Rao, B. Kumar List of symbols A ­ inverse of Von-Karman's constant, a' ­ constant, a ­ constant, b' ­ constant, b ­ constant, B* ­ function of R*, D ­ flow depth in open channel [m], D65 ­ the particle size of bed material of which 65 per cent by weight is finer [m], ks ­ representative roughness scale = D65 [m], r ­ pipe radius [m], R ­ Reynolds number, R* ­ particle Reynolds number, R'b ­ the hydraulic radius due to grain roughness [m], R"b ­ the hydraulic radius due to form roughness [m], S ­ the energy or channel slope, ­ the average velocity [m s-1], u* ­ shear velocity due to grain roughness = (g R'b S) 0.5 [m s-1], ­ correction factor = a function of ks/, ­ 11.6/u*, the thickness of laminar sub layer, ­ kinematic viscosity [m2 s-1], ­ the total drag force acting along an alluvial bed [N], ' ­ the drag force due to grain roughness [N], '' ­ the drag force due to form roughness [N], ­ specific weight of water [N m-3].

Journal

Journal of Hydrology and Hydromechanics – de Gruyter

Published: Mar 1, 2009

References

  • Re-examination of Nikuradse Roughness Data
    W. BROWNLIE
  • River Channel Roughness
    H. EINSTEIN