The analysis of temperature field, flow streamlines, and various entropy generation rates will be discussed with spatial-temporal evolution in this section, respectively.

#### 3.1. Analysis of Flow and Temperature Field

Figure 2 describes the isotherms’ temperature distributions with time evolution at

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. Here, τ (

$\tau =\sqrt{H/\beta g\mathsf{\Delta}\theta}$) is the characteristic time of the computing system. As described in

Figure 2, it can be seen that a few thermal plumes ascend in the region of dimensionless bottom boundary (0.5), two big thermal plumes descend in the region of dimensionless top boundary (1.5) at time

t/τ = 8, and the large-scale thermal plumes descend in the region of dimensionless top boundary (0.5) and ascend in the region of dimensionless bottom boundary (1.5) at time

t/τ = 16. According to the development phenomenon of thermal plumes at times

t/τ = 8 and

t/τ = 16, the thermal convective motion of the whole field is still in the initial stage of turbulent development. It was seen that with time evolution, a large-scale thermal plume ascends, strikes on the top plate, and in-volutes several thermal plumes to both sides in the left half of the system, and two large-scale thermal plumes descend at

t/τ = 32. These thermal plumes interact and strike on the top and bottom plates with time evolution, a number of small-scale thermal plumes appear at

t/τ = 32, which demonstrates that the physical system of thermal convection gradually evolves from the large-scale to small-scale thermal plumes with time evolution [

3]. In the process of energy cascade of turbulent thermal convection, the energy of the first large vortex comes from the thermal buoyancy of the outside world, which produces the second small vortex. After the small vortex loses its stability, it produces a smaller vortex process [

5]. At

t/τ = 64, some smallest plumes can be coagulated into the big plumes, several big plumes reappear with time evolution again. The above phenomenon of temperature distributions with time evolution is consistent with the previous studies [

11].

To further demonstrate the above thermal convection flow phenomenon of the whole field, the streamlines of the thermal convection flow at four same time evolution steps are shown in

Figure 3. As illustrated in

Figure 3, one can clearly see that two large vortexes occur in the central region of the whole field due to the injection of energy at the early characteristic time; two small vortexes appear at dimensionless bottom boundary (0.5) and dimensionless top boundary (1.5) at time

t/τ = 8 respectively. In addition, two small vortexes ascend in the region of the dimensionless bottom boundary (0.5), two small vortexes descend in the region of dimensionless top boundary (1.5). At time

t/τ = 16, two large vortexes in central region of the whole field become unstable, and more small vortexes appear in the dimensionless top and bottom boundaries (0.5 and 1.5). It was seen that at time

t/τ = 32, the early large vortexes evolve into a large number of small scale vortexes, and a large number of small scale vortexes generate due to energy transfer process in the whole field. However, many small vortexes disappear in main flow field, and two relatively big vortexes reappear

t/τ = 64. The above phenomenon demonstrates that several large vortexes interact and develop to a large number of small vortexes and a few small vortexes dissipate and big vortexes reappear with temporal evolution, which qualitatively depicts that the state of flow gradually develops from laminar flow to full turbulent thermal convection motion, and further evolve from full turbulent thermal convection to dissipation flow in the process of turbulent energy transfer.

#### 3.2. Analysis of Entropy Generation Rate

The isotherms temperature distributions and streamlines with time evolution are represented in the above section and several analyses of the entropy generation rate will be discussed in the following section.

Figure 4 describes the viscous entropy generation rate at times

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. As shown in

Figure 4, it is clearly seen that the high viscous entropy generation rate mainly appears in the intersectional region between the main flow and the top and bottom boundaries and in the intersectional region between big vortexes at times

t/τ = 8 and

t/τ = 16. It was seen that the high viscous entropy generation rate mainly appears in the high shear region between main flow region and vortex, the low viscous entropy generation rate occurs near the central region of various vortex, which indicates that the viscous flow loss mainly occurs in the high shear region. Meanwhile, at time step

t/τ = 32, the viscous entropy generation rate evidently increases with temporal evolution. Nevertheless, the viscous entropy generation rate evidently decreases at time

t/τ = 64 compared to that of

t/τ = 32, which indicates that the whole mainstream field has already entered the state of turbulent dissipation.

Figure 5 illustrates the thermal entropy generation rate at times

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. Plotted in

Figure 5, it is obviously observed that at times

t/τ = 8 and

t/τ = 16, the high distribution value of thermal entropy generation rate mainly dominates in the high gradient fields of temperature, especially near the top and bottom boundaries compared with the corresponding temperature fields in

Figure 2. The low distribution value of thermal entropy generation rate mainly occurs in the homogenetic temperature fields. It is seen that with spatial-temporal evolution, the high distribution value of thermal entropy generation rate gradually increases due to the interaction and strike of these thermal plumes at time

t/τ = 32, which indicates that the order degree of thermal movement gradually tends to be disordered in the whole closed system. However, the plume scale of thermal entropy generation rate gradually decreases at time

t/τ = 64 compared to that of

t/τ = 32, which further demonstrates that a great deal of large scale turbulent structures interact and develop into a large number of small scale turbulent structures; the thermal dissipation also appears with time evolution in the closed system.

Figure 6 describes the total entropy generation rate at times

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. As described in

Figure 6, it can be seen that the high distribution value of total entropy generation rate mainly dominates in the largest temperature velocity gradient compared with the corresponding temperature fields in

Figure 2. The low total entropy generation rate mainly clusters in the region of the homogenetic temperature fields. The distribution size trend of total entropy generation rate is well consistent with that of thermal entropy generation rate in the corresponding time step. In the spatial evolution, the shape of high entropy generation rate congeals into a large number of varied plumes, which indicates that the role of thermal entropy generation rate gradually improves with time evolution in the heat transfer irreversibility. It can be clearly seen that with time evolution, a great deal of large scale plumes interact and develop to a large number of small scale plumes in the closed system, and the value of total entropy generation rate increases, which indicates that the order degree of energy dissipation in the whole closed system gradually tends to be disordered and increase. The viscous, thermal and total entropy generation rates with evolution can promote the idea that the type of mixed bottom boundary condition and thermal configuration can be extensively applied in a wide variety of practical engineering applications, such as the solar thermal absorber plate or the electronic existing plates.

#### 3.3. Quantitative Analysis of Entropy Generation Rate with Time Evolution

The probability density function (PDF) is used to reveal the distribution aggregation situation of physics variable. Wei et al. [

24] argued that the PDFs of

S_{u},

S_{θ} and

S with increase of Prandtl number, the tails of high entropy generation rates can fit well into the curve of the log-normal coordinate and the departure and the distribution of log-normality, gradually becoming more robust with the decrease of Prandtl number. In this paper, an exponential expression is implemented for PDF. Its exponential expression is as follows [

24]:

in which

m,

α and

C represent the fitted parameters, and

Y =

X −

X_{mp} with

X =

S_{u}/(S_{u}) _{rms},

S_{θ}/(S_{θ}) _{rms},

S/(S) _{rms} and

X_{mp} being the abscissa of the most probable amplitude. The best fit of Equation (21) to the data yields

m = 0.86 and

α = 0.72 for

S_{u},

m = 1.15 and

α = 0.69 for

S_{θ} and

m = 1.06 and

α = 0.72 for S.

To highlight the distribution aggregation differences of

$\stackrel{\xb7}{{S}_{u}}$,

$\stackrel{\xb7}{{S}_{\theta}}$ and

$\stackrel{.}{S}$ with time evolution, the PDFs of

$\stackrel{\xb7}{{S}_{u}}$,

$\stackrel{\xb7}{{S}_{\theta}}$ and

$\stackrel{\xb7}{{S}_{0}}$ are plotted, respectively, where

$\stackrel{\xb7}{{S}_{u}}$,

$\stackrel{\xb7}{{S}_{\theta}}$ and

$\stackrel{\xb7}{{S}_{0}}$ represent the value distributions of

S_{u},

S_{θ} and

S in the whole region.

Figure 7 describes the PDFs’ distributions of

$\stackrel{\xb7}{{S}_{u}}$ at four times

t/τ = 8,

t/τ = 16,

t/τ = 32 and

t/τ = 64. As described in

Figure 7, we can see that the high value of

$\stackrel{\xb7}{{S}_{u}}$ decreases in a range of

$\stackrel{\xb7}{{S}_{u}}>10$ with time evolution. This is mainly due to the fact that the flow characteristic velocity of the large-scale flow in early characteristic time at

t/τ= 8 is relatively large, the large-scale flow is broken into more small-scale flows, the viscosity entropy rate decreases in high value with time evolution, and the viscosity entropy generation rate of the small-scale flow is smaller than that of the large-scale flow.

Figure 8 shows the PDF distributions of thermal entropy generation rate at four times

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. Plotted in

Figure 8, it is clearly obtained that the high values of

$\stackrel{\xb7}{{S}_{\theta}}$ keeps almost the same in a wide range of

$\stackrel{\xb7}{{S}_{u}}>100$ with time evolution, the low and middle values of

$\stackrel{\xb7}{{S}_{\theta}}$ keep light difference in a wide range of

$\stackrel{\xb7}{{S}_{u}}<100$ with time evolution.

Figure 9 illustrates the PDF distributions of total entropy generation rate

$\stackrel{\xb7}{{S}_{0}}$ at four times

t/τ = 8,

t/τ = 16,

t/τ = 32, and

t/τ = 64. As illustrated in

Figure 9, it can be seen that the high values of

$\stackrel{\xb7}{{S}_{0}}$ keep almost the same with

$\stackrel{\xb7}{{S}_{\theta}}$, which indicates that the thermal entropy generation rate has a dominant position in the total entropy generation rate with time evolution.

To further reveal the distribution differences of

S_{u},

S_{θ} and

S with time evolution, the average value of

$\stackrel{\xb7}{{S}_{u}}$,

$\stackrel{\xb7}{{S}_{\theta}}$ and

$\stackrel{\xb7}{{S}_{0}}$ are plotted in the whole region, respectively.

$\stackrel{\_}{\stackrel{\xb7}{{S}_{u}}}$,

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$, and

$\stackrel{\_}{\stackrel{\xb7}{{S}_{0}}}$ denote the average value of

$\stackrel{\xb7}{{S}_{u}}$,

$\stackrel{\xb7}{{S}_{\theta}}$ and

$\stackrel{\xb7}{{S}_{0}}$ in the whole region.

Figure 10 shows the time evolution of average viscous entropy generation rate from

t/τ = 0 to

t/τ = 100 in the whole field. Plotted in

Figure 10, it is clearly observed that the average value of

$\stackrel{\_}{\stackrel{\xb7}{{S}_{u}}}$ alternately increases, three peaks successively appear from the time step of

t/τ = 0 to

t/τ = 32 with time evolution. One strong peak appears at the time step of

t/τ = 32, however, the average value of

$\stackrel{\_}{\stackrel{\xb7}{{S}_{u}}}$ gradually decreases from the time step of

t/τ = 32 to

t/τ = 64, and the average value of

$\stackrel{\_}{\stackrel{\xb7}{{S}_{u}}}$ gradually increases in a range of

t/τ > 64. This is mainly due to the fact that the largest length-scales eddy is produced owing to the injection of energy at an early characteristic time; the decrease of flow eddies and the geometric eddy size is associated with the characteristic time scales (

t/τ < 32). However, with time evolution, the large-scale flow is broken into more small-scale flows in a range of

t/τ from 32 to 64; the viscosity entropy rate decreases in high value with time evolution, and the viscosity entropy generation rate of the small-scale flow is smaller than that of the large-scale flow. In a range of

t/τ > 64, some of the smallest eddies can be distorted in this distortion process, which further indicates that the kinetic energy may be dissipated from the dissipation of the smallest eddies owing to the effect of viscous flow.

Figure 11 illustrates the time evolution of average thermal entropy generation rate from

t/τ = 0 to

t/τ = 100 in the whole field. As illustrated in

Figure 11, one can clearly see that at first

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$ is very large due to the extremely thin boundary layer. As time goes by, the boundary layer thickness rapidly increases to the normal level, and

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$ decreases rapidly. After the initial period, the average value of the temperature generation rate

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$ alternately increases, several peaks periodically appear from the time step of

t/τ = 0 to

t/τ = 32 with time evolution. One strong peak appears at the time step of

t/τ = 32, however, the average value of

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$ periodically decreases from the time step of

t/τ = 32 to

t/τ = 64. The average value of

$\stackrel{\_}{\stackrel{\xb7}{{S}_{\theta}}}$ periodically and lightly increases in a range of

t/τ > 64. This is mainly due to the fact that the large scale plumes produce owing to the injection of energy at the early characteristic time, the decrease of thermal plumes size is associated with the characteristic time-scales (

t/τ < 32). Nevertheless, with time evolution, the large-scale plumes are broken into more small-scale plumes in a range of

t/τ from 32 to 64, the thermal entropy rate decreases in high value with time evolution, and the thermal entropy generation rate of the small-scale plumes is smaller than that of the large-scale plumes. In a range of

t/τ > 64, some of the smallest plumes can be coagulated into the big plumes.

Figure 12 shows the time evolution of average total entropy generation rate from

t/τ = 0 to

t/τ = 100 in the whole field. As shown in

Figure 12, one can clearly see that at first

$\stackrel{\_}{\stackrel{\xb7}{{S}_{0}}}$ is very large due to the extremely thin boundary layer. With time evolution, it is clearly seen that the boundary layer thickness rapidly increases to the normal level, and

$\stackrel{\_}{\stackrel{\xb7}{{S}_{0}}}$ decreases rapidly. Plotted in

Figure 12, it can be seen that the high values of

$\stackrel{\xb7}{{S}_{0}}$ remain almost the same with

$\stackrel{\xb7}{{S}_{\theta}}$ with time evolution, which indicates that the thermal entropy generation rate plays a dominated role in the heat transfer irreversibility—the viscous entropy generation can be neglected time evolution. The above phenomenon is well consistent with the importance of heat transfer irreversibility in the previous studies [

23,

24,

25]. Wei et al. [

24] studied the effect of changing the Prandtl number on the entropy generation rate in two-dimensional RB convection, and argued that the thermal entropy generation rate has a dominant role in the heat transfer irreversibility—the viscous entropy generation can be neglected with the increasing Prandtl number. Mohamed et al. [

45,

46,

47] studied a new analytical solution of a longitudinal fin with variable heat generation and thermal conductivity in the mixed convection Falkner-Skan flow of nanofluids with variable thermal conductivity.