# Derivation of 2D Power-Law Velocity Distribution Using Entropy Theory

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of 2D Power Law Velocity Distribution

_{max}is the maximum velocity. Two constraints must been defined to derive the 1D power law velocity distribution:

_{1}and λ

_{2}are the Lagrange multipliers, calculated according to Equations (2) and (3).

_{2}+1 = n, Equations (6) can be rewritten as:

_{1}and λ

_{2}, which can be calculated using Equations (2) and (3). Inserting Equation (4) into Equation (2) and integrating, one obtains:

_{max}, n, and the 2D CDF. The derived equation formally coincides with the equation obtained by Singh [24] for 1D domain, although in this case F is a function of x and y.

_{av}:

## 3. Comparison with Entropy-Based Logarithmic 2D Velocity Distribution

_{max}, an entropic parameter (here called G), and the 2D CDF. Parameter G can be calculated using the following equation, depending on the mean of velocity distribution and the maximum velocity [3]:

#### 3.1. 1D Velocity Distribution and Maximum Velocity on the Water Level

_{max}= 1 m/s; u

_{av}= ū = 0.8 m/s, resulting in G = 4.8 from Equation (20) and 1/n = 0.25 from Equation (22). The velocity profiles obtained from the power law velocity distribution [Equation (17)] and the logarithmic velocity distribution [Equation (18)] were almost the same (negligible differences), as shown in Figure 1.

#### 3.2. 1D Velocity Distribution and Maximum Velocity below the Water Level

_{0}= 0.8 m from the bed channel. Also in this case, the CDF is well-known [5]:

**Figure 2.**Velocity profiles calculated using Equations (17) and (20) for 1D domain, y

_{0}< H and F(u) proposed by Chiu [5].

**Figure 3.**Velocity profiles calculated using Equations (17) and (20) for 1D domain, y

_{0}< H and F(u) proposed by Marini et al. [19].

#### 3.3. 2D Velocity Distribution where Maximum Velocity Occurs below Water Level

_{0}= 0.8 m; u

_{max}= 1 m/s; u

_{av}= 0.8 m/s, we obtain 1/n = 0.64 from Equation (19) and G = 1.22 from Equation (30). The velocity profiles at different abscissas were plotted in Figure 4, resulting in an excellent agreement.

**Figure 4.**Velocity profiles calculated using Equations (17) and (20) for 2D domain (rectangular cross section).

## 4. Conclusions

## References

- Singh, V.P. Entropy Theory and Its Applications in Environmental and Water Engineering; John Wiley: New York, NY, USA, 2013; p. 662. [Google Scholar]
- de Martino, G.; Fontana, N.; Marini, G.; Singh, V.P. Variability and trend in seasonal precipitation in the continental United States. J. Hydrol. Eng.
**2013**, 18. [Google Scholar] [CrossRef] - Chiu, C.L. Entropy and probability concepts in hydraulics. J. Hydraul. Eng.
**1987**, 113, 583–600. [Google Scholar] [CrossRef] - Chiu, C.L. Entropy and 2-D velocity distribution in open channel. J. Hydraul. Eng.
**1988**, 114, 738–756. [Google Scholar] [CrossRef] - Chiu, C.L. Velocity distribution in open channel flow. J. Hydraul. Eng.
**1989**, 115, 576–594. [Google Scholar] [CrossRef] - Chiu, C.L.; Murray, D.W. Variation of velocity distribution along nonuniform open-channel flow. J. Hydraul. Eng.
**1992**, 118, 989–1001. [Google Scholar] - Chiu, C.L.; Said, C.A.A. Maximum and mean velocities and entropy in open channel flow. J. Hydraul. Eng.
**1995**, 121, 26–35. [Google Scholar] [CrossRef] - Chiu, C.L.; Tung, N.C. Maximum velocity and regularities in open channel flow. J. Hydraul. Eng.
**2002**, 128, 390–398. [Google Scholar] [CrossRef] - Chiu, C.L.; Chen, Y.C. An efficient method of discharge estimation based on probability concept. J. Hydraul. Res.
**2003**, 41, 589–596. [Google Scholar] [CrossRef] - Chiu, C.L.; Hsu, S.M. Probabilistic approach to modeling of velocity distributions in fluid flows. J. Hydrol.
**2006**, 316, 28–42. [Google Scholar] [CrossRef] - Barbe, D.E.; Cruise, J.F.; Singh, V.P. Solution of three-constraint entropy-based velocity distribution. J. Hydraul. Eng.
**1991**, 117, 1389–1396. [Google Scholar] [CrossRef] - Xia, R. Relation between mean and maximum velocities in a natural channel. J. Hydraul. Eng.
**1997**, 123, 720–723. [Google Scholar] [CrossRef] - Araujo, J.C.; Chaudhary, F.H. Experimental evaluation of 2-D entropy model for open channel flow. J. Hydraul. Eng.
**1998**, 124, 1064–1067. [Google Scholar] [CrossRef] - Kirkgoz, M.S.; Akoz, M.S.; Oner, A.A. Numerical modeling of flow over a chute spillway. J. Hydraul. Res.
**2009**, 47, 790–797. [Google Scholar] [CrossRef] - Singh, V.P.; Luo, H. Entropy theory for distribution of one-dimensional velocity in open channels. J. Hydrol. Eng.
**2011**, 16, 725–735. [Google Scholar] [CrossRef] - Luo, H.; Singh, V.P. Entropy theory for two-dimensional velocity distribution. J. Hydrol. Eng.
**2011**, 16, 303–315. [Google Scholar] [CrossRef] - Cui, H.; Singh, V.P. Two-dimensional velocity distribution in open channels using the Tsallis entropy. J. Hydrol. Eng.
**2013**, 18, 331–339. [Google Scholar] [CrossRef] - Moramarco, T.; Saltalippi, C.; Singh, V.P. Estimation of mean velocity in natural channels based on Chiu’s velocity distribution equation. J. Hydrol. Eng.
**2004**, 9, 42–50. [Google Scholar] [CrossRef] - Marini, G.; de Martino, G.; Fontana, N.; Fiorentino, M.; Singh, V.P. Entropy approach for 2D velocity distribution in open channels flow. J. Hydraul. Res.
**2011**, 49, 784–790. [Google Scholar] [CrossRef] - Fontana, N.; Marini, G.; de Paola, F. Experimental assessment of a 2-D entropy-based model for velocity distribution in open channel flow. Entropy
**2013**, 15, 988–998. [Google Scholar] [CrossRef][Green Version] - Dingman, S.L. Probability distribution of velocity in natural channel cross sections. Water Res.
**1989**, 25, 509–518. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423, 623–656. [Google Scholar] [CrossRef] - Singh, V.P. Entropy-Based Parameter Estimation in Hydrology; Kluwer Academic Publishers (now Springer): Dordrecht, The Netherlands, 1998; p. 383. [Google Scholar]
- Singh, V.P. Derivation of power law and logarithmic velocity distributions using the Shannon entropy. J. Hydrol. Eng.
**2011**, 16, 478–483. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics, I. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics, II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Jaynes, E.T. On the rationale of maximum entropy methods. Proc. IEEE
**1982**, 70, 939–952. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Singh, V.P.; Marini, G.; Fontana, N.
Derivation of 2D Power-Law Velocity Distribution Using Entropy Theory. *Entropy* **2013**, *15*, 1221-1231.
https://doi.org/10.3390/e15041221

**AMA Style**

Singh VP, Marini G, Fontana N.
Derivation of 2D Power-Law Velocity Distribution Using Entropy Theory. *Entropy*. 2013; 15(4):1221-1231.
https://doi.org/10.3390/e15041221

**Chicago/Turabian Style**

Singh, Vijay P., Gustavo Marini, and Nicola Fontana.
2013. "Derivation of 2D Power-Law Velocity Distribution Using Entropy Theory" *Entropy* 15, no. 4: 1221-1231.
https://doi.org/10.3390/e15041221