# 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

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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