1999 iahr-1999.pdf

Entropy Velocity Distribution in a River M. Greco Department of Environmental Engineering and Physics Basilicata University – Via della Tecnica, 3 – 85100 Potenza, Italy ABSTRACT In the paper the results of the extension of the entropy velocity distribution to a data set collected in field and relative to a natural rivers are presented. The analysis represent the first attempt to use the entropy model to describe the natural flow assessment. Further, the velocity distribution along the vertical has been explained employing the entropy model at the local scale and calibrating the entropy parameter through the average and maximum velocity observed per each vertical. That is, a scaling approach of the entropy profile is proposed and the comparison between the classical formulation and the scaled one is reported in terms of average absolute and global errors observed. 121 1 INTRODUCTION A relevant challenge for hydraulic engineers has been and still is to give a suitable description of flow field in terms of velocity distribution inside the body current. Sediment transport processes as well as pollutant diffusion are classic hydraulic examples in which the knowledge of cinematic assessment of the flow plays a fundamental role for prediction and design purposes. The real needed is to handle an easy velocity distribution which gives suitable results by the use of few parameters simple to measure or derive. Recent studies outline the opportunity to relate the local energy budget to the informational content hold into the point velocity measurements through a velocity distribution profile derived by an entropy-probabilistic approach (Chiu, 1987; 1989; 1991). These analyses enable an easy employment in the field of the Applied Hydraulic Engineering due to the use of simple derivable parameters instead of very difficult and hardly measurable ones. The use of a velocity profile, defined through a few data set collected in situ or obtained via the knowledge of the mean cross velocity (Chiu et al., 1995; Xia, 1997), plays an important role in the study of open channel flow. That is a clever contribute to obtain more suitable information about the flow field and relative processes ongoing with respect to the application limits and suitability of the results. The entropy model has been tested in several regular flows and in different flow regions (De Araújo et al., 1998) emphasising a good performance also for delicate portion of the flow field. Exception has been observed whenever the entropy model has been employed to represent the flow field in the boundary layer region where further boundary conditions "mechanically based" are required to calibrate and efficaciously correct the velocity profile (Pulci Doria, 1992). Nevertheless in the study of natural current with high relative submergence (water depth over roughness dimension) the boundary effects are constrained in a bounded zone, small with respect to the flow field generally, so that the motion of the fluid particles can be though as a turbulent kernel at whole. Through this assumption the entropy model can take place giving a satisfactory description of the velocity field. In the paper velocity data sets referred to natural flows are described by the use of entropy model. Further, stressing the representative scheme of the flow, a scaling approach for the entropy model is proposed obtaining a better description of the velocity field features with respect to the geometry assessment and changes of the flow field. Computed velocity profiles are compared to the measured ones for different river cross section at different water discharge stages enabling an evaluation of the entropy model consistence. 2 DATA COLLECTION The data sets used in the present paper are obtained through direct measurement in field in several cross sections of the Basento River in Southern Italy. For each data collection were placed 7-12 verticals along the measured cross section and, per each 122 vertical, several point measurements were obtained using a one-dimensional currentmeter. These data were coupled to other sets of data referred to experimental field measurement obtained in the 1983 by the U.S. Geological Survey for selected streams in Colorado (Marchand et al., 1984). Based on the point measurements the average vertical velocity and the maximum one were obtained through computations and data comparison, while the average cross section velocity were derived as ratio between the water discharge and the cross section area. Further, data are collected at the same cross section at different time corresponding to different discharge stages. Precisely, two main scenarios were observed: a very low discharge stage (data collected on the Basento River) and medium discharge stage allowing a different flow condition and, thus, general applicability of the data analysis. 3 ENTROPY VELOCITY LAW As well known the velocity entropy profile derived by Chiu (1987) is based on the probabilistic approach and represent a very useful tool for researchers and engineers because founded on few synthetic parameters relatively easy to derive. The aim followed during the derivation of the entropy velocity profile as suggested by Chiu (1987) assume the role of well describing tool of the local energy budget as well as the direct instrument for the evaluation of the average and maximum cross velocity. The maximum cross velocity plays a primary part into the entropy description of the velocity profile through the relation (Chiu, 1987): u   0  p (u )du  max   0 0 (1) where u is the local velocity measured in the flow field along a vertical line,  is a dimensionless variable depending on the reference system employed for the local representation of the flow field, 0 and max are the value of the dimensionless variable at which correspond the minimum (u=0) and the maximum (u=umax) of the velocity respectively while p(u) is the density probability function derived by maximising the Shannon's entropy: H (u )    p(u ) ln p(u )du u max (2) 0 and assumed as follows p(u )  e a1  a2u (3) with a1 and a 2 distribution parameters, going through simple math and integrating the (1) the velocity distribution became: 123 u u max    0  1  ln 1  (e M  1)  M   max   0  (4) in which M is the dimensionless entropy parameter needed together to the maximum velocity, umax, for the analytical problem closure. The distribution parameters can be obtained by alternative methods allowing three equations into the three unknown M, umax and a1 as suggested by Chiu (1987) but the most useful one is: u u max  e M (e M  1) 1  1 M (5) where the mean cross velocity, u , is related to the M parameter through the value of the maximum velocity. Recent studies have shown the possibility to describe the relation between mean and maximum velocities through the linear equation (Chiu, 1995; Xia, 1997): u  b  umax (6) where the coefficient b depend on the flow regime, flow field type (pipe or open channel) (Streeter et al., 1979), flow morphology (straight or meandering) (Xia, 1997) and, thus, on the flow discharge. The equation (4) can be reprocessed as b  e M (e M  1) 1  1 M (7) and the solution of the (7) is related to the evaluation of the b parameter thus on the peculiarity of the flow field. In the case study analyses have been performed on the data collected finding a b values ranging in between 0.4 - 0.8 above all investigated flow condition according to the results found in literature and relative to river flows (Xia, 1997). 4 SCALING APPLICATION OF THE VELOCITY PROFILE Entropy velocity profile has been discussed for several regular flow conditions like pipes, wide rectangular open channel or narrow flumes (Chiu, 1991; 1993; Chiu et al., 1995) only recently the field of application has been enlarged to natural flow condition with good performance (Greco, 1998). Transverse flow distribution can be derived by the use of the entropy velocity profile coupled to the equation (7) whose allows to determine the value of the entropy parameter, M, through the knowledge of the ratio between the average cross velocity and the maximum cross velocity. Of course, even if the approximation is nearly good with a small global error in between 13 - 19% for regular flows (De Araújo et al., 1998), the evaluation of the point velocity became much more approximated passing to an irregular flow geometry condition like that observed in natural river. As mention above, first attempt to describe a natural flow through the entropy law gives appreciable results with a general overestimation of the local flow energy budget. 124 Figure 1 shows the observed velocity, measured in field in a cross section of the Basento river at low discharge stage, and the predicted velocity obtained through the entropy law versus dimensionless depth (y/hmax). The plot outlines the gap existing in between the effective velocity and the reproduced one due to the geometry variation characteristic of the flow field, probably. 1.00 0.90 0.80 0.70 1.70 2.70 3.70 4.70 5.70 6.70 7.70 8.70 9.70 10.70 11.70 Chiu 0.60 y/hmax 0.70 0.50 0.40 0.30 0.20 0.10 0.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 u Figure 1. The entropy velocity profile and the whole of the velocity measurement referred to a cross section of the Basento river at low discharge stage Extending the analysis at the whole of the measures collected and plotting the reproduced velocities versus the observed ones (figure 2) the gap seems to be reduced and the general result gives rise a good performance of the entropy law for natural currents. Moreover, the entropy model gives a good evaluation of the real velocity distribution along the river with a general overestimation comparable to that observed for regular flows (De Araújo et al., 1998). Using as much as possible the observed data, taking into account the variability of the velocity distribution inside the cross section as well as the existence of mean and maximum velocity per each vertical, the entropy law can be "stretched" and improved at the vertical scale. That is, a scaling approach for the entropy model can arise computing per each vertical the entropy parameter in order to derive the corresponding velocity profile. The downward approach, as shown in figure 3, give as suitable representation of the flow field through the best interpretation of the single velocity profile generating a set of M’s values. The plot report the local velocity profile, derived through the downward application of the entropy law (Vvertical), and the original law (Vchiu) compared to the measured data. The figure 3 represent the velocity versus the 125 6.00 5.00 Vcomputed (m/sec) 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Vmeasured (m/sec) 1.20 1.20 1.00 1.00 1.00 0.80 0.80 0.80 0.60 V 0.40 0.00 1.00 2.00 3.00 v (m/sec) 4.00 0.00 5.00 0.60 V 0.40 Vchiu 2.00 4.00 6.00 0.00 1.00 0.80 0.80 0.80 Vvertical 0.20 V 1.00 2.00 3.00 v (m/sec) 4.00 5.00 0.00 0.60 0.60 V Vchiu Vvertical 0.20 0.40 0.40 Vchiu 0.20 Vvertical 0.00 0.00 0.00 0.00 0.60 0.40 Vchiu y/hmax 1.20 1.00 y /hm ax 1.20 1.00 0.40 0.20 v (m/sec) 1.20 V Vvertical 0.00 v (m/sec) 0.60 Vchiu 0.20 Vvertical 0.00 0.00 y /hm ax V 0.20 Vvertical 0.20 0.60 0.40 Vchiu y/hmax 1.20 y /hm ax y /hm ax Figure 2. Measured and computed point velocity by the classic model 1.00 2.00 v (m/sec) 3.00 4.00 0.00 0.20 0.40 0.60 v (m/sec) Figure 3. Comparison between the local profile, Vvertical, the Chiu’s profile, Vchiu, and measured data for different vertical of the same cross section dimensionless depth, as defined above, and shows a general underestimation of the classic entropy model (Vchiu) with respect to the local entropy profile (Vvertical). The 126 comparison between the discharge values obtained employing the two laws outlines a lower value for the Vchiu profile ranging from 7 up to 12% of the value obtained by the local profile. The extension of the computations at the whole of the data set is shown in figure 4 where the entropy model (Vchiu) and the scaling approach (Vvertical) are plotted versus the observed values. The capacity prediction of the two models has been compared basing on the evaluation of average absolute error Ej (Araújo et al., 1998) defined as: i i 1 NP uc  um  Ej  NP i 1 u mi (8) where umi is the ith measured velocity, u ci is the ith computed velocity and NP is the number of points measurements, and thus the global error as a weighted average of the average absolute errors (EG= Ej·NPj/ NPj). Entropy model (Chiu) presents a good performance with a global error (EG) close to 31% while the scaling approach (Vertical) shows the smaller value of 10% . Both report a coefficient of variation (CVE)greater than 50% (table 1). Data set Bear 1 Bear 2 Bear 3 Clear 1 Clear 2 Clear 3 Clear Gold 1 Clear Gold 2 Clear Gold 3 Cache La Poudre 1 Cache La Poudre 1 Cache La Poudre 1 EG(%) CVE(%) Error (%) Chiu Error (%) Vertical 32 59 36 31 26 30 11 25 21 23 61 22 31 61 15 14 6 10 6 7 8 11 10 10 15 6 10 56 Table. 1. Experimental data and prediction error 127 6.00 Vchiu 5.00 Vvertical Vcomputed (m/sec) 4.00 3.00 2.00 1.00 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Vmeasured (m/sec) Figure. 4. Measured and computed point velocity by the entropy model and the scaling approach Further information can be derived observing the bottom shear stress distribution calculated by the use of the local and the global entropy laws. The bottom shear stress depends on the velocity profile distribution by the equation:  u     0    0  (9) (with , fluid density and 0 eddy viscosity) and will change along the wetted perimeter outlining different region with different energy content according to Bathurst et al. (1979). Employing the different result derived by the use of both the local and the global velocity prediction models the resulting bottom shear stress distribution enable to recognised the region of the flow field in which the energy rate of dissipation assume different values giving further information about the energetic processes active into the flow. By the way an interesting features of the scaling approach referred to the bottom shear stress is due to the comparison between the transverse distribution obtained by the use of the two velocity laws. Figure 5 shows the dimensionless bottom shear stress computed by the two laws and referred to the average bottom shear stress derived by the classical relationship  0  Ri (where  is the specific weight, R is the hydraulics radius and i is the local slope) relative to different measurement periods. Both the average bottom shear 128 3.00 3.00 Chiu Vertical 2.50 march april 2.00 2.00 1.50 1.50 1.00 1.00 0.50 0.50 0.00 0.00 0.00 Chiu Vertical 2.50 0.25 0.50 0.75 0.00 1.00 0.25 0.50 Chiu Vertical 2.50 Chiu Vertical 2.50 may 2.00 1.50 1.50 1.00 1.00 0.50 0.50 0.00 0.00 1.00 3.00 3.00 2.00 0.75 x/B x/B may 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 x/B 0.75 1.00 x/B Figure 5. Bottom shear stress distribution computed by the classic entropy model (Chiu) and the scaling approach (Vertical) x/b 0.00 0.00 0.25 0.50 0.75 1.00 hmax/Ri 0.50 1.00 1.50 27-feb-97 20-may-97 13-may-97 2.00 Figure 6. Time evolution of the cross section used for the computation of the bottom shear stress. stress derived by the global entropy law and the local stress are always close to the unity giving a good estimation of the cross section energy rate of dissipation. Further, comparing the shear stress distributions to the time evolution of the relative cross section, reported in figure 6, the local approach allows to identify flow regions in which erosive and deposition processes are active enforcing the possibility to use the local velocity distribution for a better description of the flow field. 129 5 CONCLUSIONS The velocity profile derived by Chiu (1987) through the entropy-probabilistic approach has been applied to describe real data collected in natural rivers. A scaling approach of the entropy model, obtained employing the velocity profile per each measured vertical of the cross section and estimating the entropy parameter (M) through the values of the local vertical maximum and average velocity, is proposed. Measured longitudinal velocity data are compared to both velocity distribution prediction models. The results show that the entropy model frequently overestimate the value of the longitudinal velocity than the modified model, as shown by the computed global error equal to 31% and 10% respectively. Finally, the bottom shear stress distribution has been derived using both the velocity prediction models allowing to recognised the region of the flow field in which the energy rate of dissipation assume different values and erosive and deposition processes take place. REFERENCES 1. De Araújo J.C., Chaudhry F.H. (1998) “Experimental evaluation of 2-D entropy model for open channel flow”, J.Hydr.Engrg. ASCE, 124 (10), 1064-1067. 2. Bathurst J.C., (1979) “Distribution of boundary shear stress in rivers”, Kendall/Hunt Publishing Co., Iowa 3. Chiu C.-L. (1987) “Entropy and probability concepts in hydraulics” J.Hydr.Engrg. ASCE, 113 (5), 583-600. 4. Chiu C.-L. (1989) “Velocity distribution in open channel flow” J.Hydr.Engrg. ASCE, 115 (5), 576-594. 5. Chiu C.-L. (1991) “Application of entropy concept in open channel flow” J.Hydr.Engrg. ASCE, 117 (5), 615-628. 6. Chiu C.-L., Said C.A., (1995) “Maximum and mean velocity and entropy in open channel flow” J.Hydr.Engrg. ASCE, 121 (1), 26-35. 7. Greco M., (1998) “Distribuzione entropica di velocità in un corso d’acqua naturale”, XXVI Convegno di Idraulica e Costruzioni Idrauliche, Vol. 1, 103-112. 8. Marchand J.P., Jarret R.D., Jones L.L. (1984) “Velocity profile, water-surface slope and bed-material size for selected streams in Colorado”, U.S. Geological Survey, Open-File report 84-733. 9. Pulci Doria G., (1992) “Statistical quantities distribution in turbulent flows and use of the entropy concepts”, Entropy and Energy Dissipation in Water Resources, Kluwer Academic Publishers, by V.Psingh and M.Fiorentino, 541586. 10. Streeter V.L., Wylie E.B.,(1979) “Fluid mechanics”, McGraw-Hill Inc., N.Y. 11. Xia R. (1997) “Relation between mean and Maximum velocities in a natural river”, J.Hydr. Engrg. 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