Application of relative entropy theory to streamwise velocity profile in open-channel flow: effect of prior probability distributions

Abstract

Applying the concept of relative or cross-entropy and the principle of minimum cross-entropy, this study derives the velocity distribution in a wide open channel. Previous studies have employed Shannon entropy and the principle of maximum entropy to derive distributions of various flow variables, including velocity. Relative entropy is a generalized form of entropy and can specialize into Shannon entropy if the prior probability distribution is taken to be a uniform distribution. The prior distribution is often formulated, based on intuition, experience, or thought experiment. When deriving the velocity distribution in wide open channels, this study assumes four prior probability distributions and analyzes the effect of these assumed priors. It is found that a normal-type and a gamma-type prior can significantly influence the velocity profile, especially near the channel bed, and their prediction accuracies are superior to the previously obtained velocity distribution based on Shannon entropy. Furthermore, closed-form explicit analytical solutions are obtained for the nonlinear differential equations that arise while incorporating these priors. Experimental and field data are used to verify the derived velocity distributions, and an error analysis is carried out to evaluate their performance.

Appendix A: Convergence theorems for series solution Eq. (45)

Theorem 1

If the homotopy series $\sum \nolimits _{m=0}^\infty {w_{m}(\hat{y})}$ and $\sum \nolimits _{m=0}^\infty {{w'}_{m}(\hat{y})}$ converge, then $R_{m}\left( \vec {w}_{m-1} \right)$ given by Eq. (43) satisfies the relation $\sum \nolimits _{m=1}^\infty {R_{m}\left( \vec {w}_{m-1} \right) } =0.$

Proof

The auxiliary linear operator is defined as follows: $\square$

$$\begin{aligned} \mathcal {L}\left[ w \right] =\frac{13}{12}\frac{\hbox {d}w}{d\hat{y}}-A_{1}\lambda _{1}w \end{aligned}$$

(54)

According to Eq. (36), we obtain

$$\begin{aligned} \mathcal {L}\left[ w_{1} \right]= & {} \hslash R_{1}\left( \vec {w}_{0} \right) \end{aligned}$$

(55)

$$\begin{aligned} \mathcal {L}\left[ {w_{2}-w}_{1} \right]= & {} \hslash R_{2}\left( \vec {w}_{1} \right) \end{aligned}$$

(56)

$$\begin{aligned} \mathcal {L}\left[ {w_{3}-w}_{2} \right]= & {} \hslash R_{3}\left( \vec {w}_{2} \right) \end{aligned}$$

(57)

$$\begin{aligned} \mathcal {L}\left[ {w_{m}-w}_{m-1} \right]= & {} \hslash R_{m}\left( \vec {w}_{m-1} \right) \end{aligned}$$

(58)

Adding all of the above terms, we get

$$\begin{aligned} \mathcal {L}\left[ w_{m} \right] =\hslash \sum \limits _{k=1}^m {R_{k}\left( \vec {w}_{k-1} \right) } \end{aligned}$$

(59)

As the series $\sum \nolimits _{m=0}^\infty {w_{m}(\hat{y})}$ and $\sum \nolimits _{m=0}^\infty {{w'}_{m}(\hat{y})}$ are convergent, $\hbox {lim}_{m{\rightarrow }{\infty }}w_{m}\left( \hat{y} \right) =0$ and $\hbox {lim}_{m{\rightarrow }{\infty }}{w'}_{m}\left( \hat{y} \right) =0$. Now, recalling the above summand and taking the limit, the required result follows as

$$\begin{aligned} \hslash \sum \limits _{k=1}^\infty {R_{k}\left( \vec {w}_{k-1} \right) } =\lim _{m{\rightarrow }{\infty }}{\hslash \sum \limits _{k=1}^m {R_{k}\left( \vec {w}_{k-1} \right) } =\lim _{m{\rightarrow }{\infty }}{\mathcal {L}\left[ w_{m} \right] }=\lim _{m{\rightarrow }{\infty }}\, \left[ \frac{13}{12}{w'}_{m}-A_{1}\lambda _{1}w_{m} \right] }=0 \end{aligned}$$

(60)

Theorem 2

If $\hslash$ is so properly chosen that the series $\sum \nolimits _{m=0}^\infty {w_{m}(\hat{y})}$ and $\sum \nolimits _{m=0}^\infty {{w'}_{m}(\hat{y})}$ converge absolutely to $w(\hat{y})$ and $w'(\hat{y})$, respectively, then the homotopy series $\sum \nolimits _{m=0}^\infty {w_{m}(\hat{y})}$ satisfies the original governing Eq. (40).

Proof

We first recall the following definition: $\square$

Cauchy Product of two infinite series Let $\sum \nolimits _{i=0}^\infty a_{i}$ and $\sum \nolimits _{j=0}^\infty b_{j}$ be two infinite series of real/complex terms. Then, the Cauchy product of the above two series is defined by the discrete convolution as follows:

$$\begin{aligned} \left( \sum \limits _{i=0}^\infty a_{i} \right) \left( \sum \limits _{j=0}^\infty b_{j} \right) = \sum \limits _{k=0}^\infty \sum \limits _{l=0}^k {a_{l}b_{k-l}} \end{aligned}$$

(61)

Therefore, using the above rule in relation to Eq. (43), we get

$$\begin{aligned}&\sum \limits _{m=1}^\infty \sum \limits _{j=0}^{m-1} {w_{j}w_{m-1-j}} =\left( \sum \limits _{m=0}^\infty w_{m} \right) \left( \sum \limits _{k=0}^\infty w_{k} \right) =\left( \sum \limits _{m=0}^\infty w_{m} \right) ^{2} \end{aligned}$$

(62)

$$\begin{aligned}&\sum \limits _{m=1}^\infty {\sum \limits _{j=0}^{m-1} {w_{m-1-j}\sum \limits _{k=0}^j w_{k} } w_{j-k}} =\left( \sum \limits _{m=0}^\infty w_{m} \right) ^{3} \end{aligned}$$

(63)

$$\begin{aligned}&\sum \limits _{m=1}^\infty {\sum \limits _{j=0}^{m-1} {w_{m-1-j}\sum \limits _{k=0}^j w_{k} } {w'}_{j-k}} =\left( \sum \limits _{m=0}^\infty w_{m} \right) ^{2}\left( \sum \limits _{m=0}^\infty {w'}_{m} \right) \end{aligned}$$

(64)

$$\begin{aligned}&\sum \limits _{m=1}^\infty {\sum \limits _{j=0}^{m-1} {w_{m-1-j}\sum \limits _{k=0}^j w_{j-k} \sum \limits _{l=0}^k w_{l} } w_{k-l}} =\left( \sum \limits _{m=0}^\infty w_{m} \right) ^{4} \end{aligned}$$

(65)

$$\begin{aligned}&\sum \limits _{m=1}^\infty {\sum \limits _{j=0}^{m-1} {w_{m-1-j}\sum \limits _{k=0}^j w_{j-k} \sum \limits _{l=0}^k w_{k-l} \sum \limits _{n=0}^l w_{n} } w_{l-n}} =\left( \sum \limits _{m=0}^\infty w_{m} \right) ^{5} \end{aligned}$$

(66)

Theorem 1 shows that if $\sum \nolimits _{m=0}^\infty {w_{m}(z)}$ and $\sum \nolimits _{m=0}^\infty {{w'}_{m}(z)}$ converge, then $\sum \nolimits _{m=1}^\infty {R_{m}\left( \vec {w}_{m-1} \right) } =0$.

Therefore, substituting the above expressions in Eq. (43) and simplifying further lead to

$$\begin{aligned}&\frac{13}{12}\sum \limits _{m=0}^\infty w_{m}^{'} -\left( \sum \limits _{m=0}^\infty w_{m}^{'} \right) \left( \sum \limits _{m=0}^\infty w_{m} \right) ^{2}\\&\quad -A_{1}\left[ \sum \limits _{m=0}^\infty {(1-\chi _{m+1})} +\lambda _{1}\sum \limits _{m=0}^\infty w_{m} +\left( \frac{\lambda _{1}^{2}}{2!} \right) \left( \sum \limits _{m=0}^\infty w_{m} \right) ^{2}+\left( \frac{\lambda _{1}^{3}}{3!} \right) \left( \sum \limits _{m=0}^\infty w_{m} \right) ^{3}\right. \\&\quad \left. +\left( \frac{\lambda _{1}^{4}}{4!} \right) \left( \sum \limits _{m=0}^\infty w_{m} \right) ^{4}+\left( \frac{\lambda _{1}^{5}}{5!} \right) \left( \sum \limits _{m=0}^\infty w_{m} \right) ^{5} \right] =0 \end{aligned}$$

which is basically the original governing equation Eq. (40). Furthermore, subject to the initial condition $w_{0}\left( 0 \right) =-1/2$ and the conditions for the higher-order deformation equation ${w}_{m}\left( 0 \right) =0$, for $m\ge 1$, we easily obtain

$$\begin{aligned} \sum \limits _{m=0}^\infty {w_{m}(0)} =-\frac{1}{2} \end{aligned}$$

(67)

Hence, the convergence result follows.