# Influence of Right Triangular Prism Rough Beds on Hydraulic Jumps
## 2 Mathematical Method
### Entropy Production by Dissipation  
The specific entropy production rate can be defined as Equation (7) [18].  
\[\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D}=\frac{\dot{Q}}{T} \tag{7}\]  
where \(\dot{Q}\) represents the energy transfer rate. In a steady state, \(\dot{Q}\) is equal to the dissipation energy; and \(T\) is the temperature.  
For turbulent flows, after time-averaging of the entropy production by dissipation, two groups of terms appear, one caused by time-averaged movement and another caused by velocity fluctuations. Then, \(\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D}\) can be expressed as [19; 20]:  
\[\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D}=\dot{S}^{{}^{\prime\prime\prime}}_{ PRO,\overline{D}}+\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D^{\prime}} \tag{8}\]  
where \(\dot{S}^{{}^{\prime\prime\prime}}_{PRO,\overline{D}}\) and \(\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D^{\prime}}\) represent the particular entropy production rate due to the time-averaged movement and the specific entropy production rate due to velocity fluctuations, which can be calculated as follows:  
\[\begin{array}{l}\dot{S}^{{}^{\prime\prime\prime}}_{PRO,\overline{D}}=2\frac{ \mu_{eff}}{T}\bigg{[}\bigg{(}\frac{\partial\overline{u}}{\partial x}\bigg{)}^ {2}+\bigg{(}\frac{\partial\overline{u}}{\partial y}\bigg{)}^{2}+\bigg{(}\frac{ \partial\overline{u}}{\partial z}\bigg{)}^{2}\bigg{]}\\ +\frac{\mu_{eff}}{T}\bigg{[}\bigg{(}\bigg{(}\frac{\partial\overline{u}}{ \partial y}+\frac{\partial\overline{u}}{\partial\overline{x}}\bigg{)}^{2}+ \bigg{(}\frac{\partial\overline{u}}{\partial\overline{x}}+\frac{\partial \overline{u}}{\partial\overline{x}}\bigg{)}^{2}+\bigg{(}\frac{\partial\overline {u}}{\partial\overline{x}}+\frac{\partial\overline{u}}{\partial\overline{y}} \bigg{)}^{2}\bigg{]}\end{array} \tag{9}\]  
\[\begin{array}{l}\dot{S}^{{}^{\prime\prime\prime}}_{PRO,D^{\prime}}=2\frac{ \mu_{eff}}{T}\bigg{[}\bigg{(}\frac{\partial\overline{u}^{\prime}}{\partial x} \bigg{)}^{2}+\overline{\bigg{(}\frac{\partial\overline{u}}{\partial y}\bigg{)} ^{2}}+\overline{\bigg{(}\frac{\partial\overline{u}^{\prime}}{\partial\overline {x}}\bigg{)}^{2}}\bigg{]}\\ +\frac{\mu_{eff}}{T}\bigg{[}\bigg{(}\frac{\partial\overline{u}^{\prime}}{ \partial y}+\frac{\partial\overline{v}^{\prime}}{\partial\overline{x}}\bigg{)} ^{2}+\bigg{(}\frac{\partial\overline{u}^{\prime}}{\partial\overline{x}}+ \frac{\partial\overline{u}^{\prime\prime}}{\partial\overline{x}}\bigg{)}^{2}+ \bigg{(}\frac{\partial\overline{v}^{\prime}}{\partial\overline{x}}+\frac{ \partial\overline{u}^{\prime\prime}}{\partial\overline{y}}\bigg{)}^{2}\bigg{]} \end{array} \tag{10}\]  
where \(\mu\) denotes the dynamic viscosity; \(\mu_{t}\) stands for the turbulent viscosity; \(\overline{u}\), \(\overline{v}\), and \(\overline{w}\) are time-averaged velocity components; \(u^{\prime}\), \(v^{\prime}\), and \(w^{\prime}\) are velocity fluctuation components; and \(\mu_{eff}\) represents the effective viscosity (\(\mu_{eff}=\mu+\mu_{t}\)).  
Equation (9) represents the entropy production by dissipation in the mean flow field, referred to as direct dissipation. Meanwhile, Equation (10) is the so-called indirect or turbulent dissipation.