## 3 Fine-Structures Model  
The concept of the energy cascade by [PERSON] [29] is that the turbulence can be considered to be composed of eddies of different sizes. The large eddies are unstable and break up, which transfers their energy to somewhat smaller eddies. These smaller eddies undergo a similar break-up process, and transfer their energy to even smaller eddies. This energy cascade, in which energy is transferred to successively smaller and smaller eddies, continues until the Reynolds number is sufficiently small enough (the unity Reynolds number based on the Kolmogorov scales) that the eddy motion is stable and molecular viscosity is effective in dissipating the kinetic energy. [PERSON] [1; 2] presented an energy cascade model with a characteristic frequency or strain rate. This model is used to simulate the transfer of mechanical energy from the mean flow to heat by turbulent effects, as illustrated in Figure 2, which includes infinity levels. \(w^{\prime}\) represents the feed of mechanical energy from the mean flow to the turbulence. For steady-state turbulence, this is the production of turbulent kinetic energy \(k\). The sum of \(q^{\prime}+q^{\prime\prime}+\ldots\,+q_{n}+\ldots\,+q^{*}\) is the dissipation rate of turbulent kinetic energy \(\varepsilon\). When turbulent quantities, such as \(k\), are found from transport equations, \(w^{\prime}\) represents the total supply of turbulent kinetic energy [5]. Figure 2: Energy cascade model for the transfer of mechanical energy from the mean flow, through turbulent energy, to heat. Reprinted with permission from Ref. [27]. 2022, Elsevier Ltd. The first level in the turbulent structure is the large, energy-rich eddies that are characterized by the velocity scale \(u^{\prime}\) and length scale \(L^{\prime}\). The velocity scale \(u^{\prime}\) is related to the turbulent energy \(k\) by \(k=\frac{3}{2}u^{\prime 2}\). A frequency or strain rate is expressed as [1; 2; 5]  
\[\omega^{\prime}=\frac{u^{\prime}}{L^{\prime}} \tag{12}\]  
Likewise, the \(n\)th level was characterized by \(u_{n}\) and \(L_{n}\). In the smallest eddies (fine structures), \(u^{*}\) and \(L^{*}\) are of the same order of magnitude as the Kolmogorov scales. For high and moderate Reynolds numbers, the dissipation is small at the upper levels. This means that for a small \(n\), \(q_{n}\) is negligible compared to \(w_{n}\), and \(w_{n}\) is approximately equal to \(w_{n+1}\). Therefore, for \(n=2\), \(u^{\prime 2}\) can be approximately regarded as \(\frac{1}{2}u^{\prime 2}\). Similarly, there are [5]  
\[\left\{ \begin{aligned} u^{\prime 2}&\approx\frac{1}{2}u^{ \prime 2}\\ w^{\prime\prime}&=C_{\text{DI}}\omega^{\prime} \frac{3}{2}u^{\prime 2}&=C_{\text{DI}}\omega^{\prime}k=\frac{3}{2}C_{ \text{DI}}\frac{u^{\prime 3}}{L^{\prime}}\end{aligned} \right. \tag{13}\]  
When the Reynolds number is high, the transfer from the first level to the second level, \(w^{\prime\prime}\), is much larger than the direct dissipation from the first level, i.e., \(q^{\prime}\). This indicates that the dissipation \(\varepsilon\) is approximately equal to the energy transfer \(w^{\prime\prime}\), i.e.,  
\[\varepsilon=w^{\prime}=w^{\prime\prime}+q^{\prime}\approx w^{\prime\prime}= \frac{3}{2}C_{\text{DI}}\frac{u^{\prime 3}}{L^{\prime}} \tag{14}\]  
At the last level, the fine structures are [5]  
\[w^{*}=\frac{3}{4}\varepsilon=\frac{2}{3}C_{\text{DI}}\frac{u^{\prime 3}}{L^{*}} \tag{15}\]  
and  
\[q^{*}=\frac{3}{4}\varepsilon=C_{\text{DI}}v\frac{u^{*2}}{L^{*2}} \tag{16}\]  
The two aforementioned equations give the characteristic scales for the fine structures, the length of which can be expressed as:  
\[\left\{ \begin{aligned} L^{*}&=\frac{2}{3}\left( \frac{3C_{\text{DI}}^{3}}{C_{\text{DI}}^{3}}\right)^{1/4}\eta\\ &\eta&=\left(\frac{\
u^{2}}{\varepsilon}\right)^{1/4 }\end{aligned} \right. \tag{17}\]  
and the velocity scale can be expressed as  
\[\left\{ \begin{aligned} u^{*}&=\left( \frac{C_{\text{DI}}}{3C_{\text{DI}}^{3}}\right)^{1/4}u_{\eta}\\ & u_{\eta}=\left(\
u\varepsilon\right)^{1/4}\end{aligned}