# Entropy Generation through Deterministic Spiral Structures in Corner Flows with Sidewall Surface Mass Injection

## Abstract

**:**

## 1. Introduction

## 2. Selection of Heated Air as the Working Substance

^{5}N/m

^{2}

^{−4}m

^{2}/s

## 3. Review of Program Components

#### 3.1. Steady-Flow Boundary-Layer Development: Velocity Gradients

#### 3.2. Modified Lorenz-Form Equations: Spectral Velocity Components

#### 3.3. Synchronization Properties of the Modified Lorenz Equations

#### 3.4. Power Spectral Density within the Deterministic Spectral Velocity Fluctuations

_{j}that is eventually available for dissipation into the internal energy of the final equilibrium thermodynamic state.

#### 3.5. Empirical Entropies from Singular Value Decomposition

_{j}across the component modes, j, for these spectral velocity components. These modes will be denoted simply as empirical modes.

_{j}, defined by the expression (Isaacson [3]):

_{j}is the empirical eigenvalue computed from the singular value decomposition procedure applied to the nonlinear time-series solution. The distribution of the empirical entropy across the decomposition empirical modes has been shown in [2].

#### 3.6. Empirical Entropic Indices for the Ordered Structures

_{j}is introduced to describe the entropy of an ordered structure described by the empirical eigenvalue, λ

_{j}, for the singular value decomposition empirical mode, j. Hence, we simply adopt, in an ad hoc fashion, an expression from which we may extract an empirical entropic index, q

_{j}, from the empirical entropy, Semp

_{j}. This expression may be written as [2]:

_{j}the empirical entropic index or simply entropic index.

#### 3.7. Empirical Intermittency Exponents for the Ordered Structures

_{j}. This intermittency exponent describes the fraction of fluctuating kinetic energy within the deterministic ordered structure that is dissipated into thermodynamic internal energy [2].

#### 3.8. Kinetic Energy Available for Dissipation

^{2}/2, at the normalized vertical distance, η = 3.00 from the horizontal surface, is considered as the source of kinetic energy to be dissipated through the spiral structures. This available kinetic energy is distributed over the stream wise component, the normal component and the span wise component. The fraction of kinetic energy in the stream wise velocity component is denoted as κ

_{x}, the fraction of kinetic energy in the normal velocity component is denoted as κ

_{y}and the fraction in the span wise velocity component denoted as κ

_{z}. The fraction of dissipation kinetic energy within each empirical mode of the power spectral energy distribution is denoted as ξ

_{j}. Then the total rate of dissipation of the available fluctuating kinetic energy for the stream wise, normal and span wise velocity components is the summation, over the empirical modes, j, of the product of the kinetic energy fraction of each mode times the intermittency exponent for that mode, ζ

_{j}[5].

#### 3.9. Entropy Generation Rates through the Ordered Structures

_{j}with the corresponding intermittency exponent for the singular value decomposition mode, ζ

_{j}. This association comes about through the Wiener–Khintchine theorem (pp. 19–21, [6], and references cited therein). As shown in Figure 2, both the power spectral density and the autocorrelation function of the singular value decomposition computation process the same non-linear time series solution of the spectral velocity wave component computations. According to the Weiner–Khintchine theorem, both quantities contain the same spectral information as contained in the original non-linear time series solution. We therefore associate the peaks of the power spectral density with the corresponding intermittency exponents from the singular value decomposition.

_{e}f’, where f’

^{’}is the derivative of the Falkner–Skan stream function f with respect to the normalized distance η. The expression for the entropy generation rate through the deterministic ordered structures may then be written as [1]:

**Figure 2.**This flow chart summarizes the overall computational procedure used to compute the entropy generation through deterministic ordered structures.

_{m}is defined as the “eddy viscosity”, (Cebeci and Bradshaw, pp. 153–159, [8]), having the dimensions of (viscosity)/(density) which relates the turbulent shear stress to the stream wise boundary layer velocity gradient.

## 4. Results

#### Computational Results for the Receiver Stations

_{x3}, for the third receiver station, as a function of the time step number over the total time range of the integration process. Figure 4 shows the corresponding phase diagram for a

_{z3}-a

_{y3}, where a

_{y3}is the normal spectral velocity wave component and a

_{z3}is the span wise spectral velocity wave component, again at the third receiver station.

**Figure 3.**The stream wise spectral velocity wave component, a

_{x3}, is shown as a function of time step number at the third receiver station x = 0.12 with w

_{e}= 0.086375.

**Figure 4.**The phase diagram of the span wise and normal spectral velocity components, a

_{z3}-a

_{y3}, is shown for the third receiver station x = 0.12 with w

_{e}= 0.086375.

_{y3}at the third receiver station at x = 0.12, are presented in Figure 5. Very fine grids are required to extract these results. We have assigned empirical mode numbers to these peaks, starting with mode j = 1 representing the highest peak in the distribution, continuing to mode j = 16, representing the corresponding lowest peak among the sixteen peaks.

**Figure 5.**The power spectral density for the normal spectral velocity component is shown for the third receiver station x = 0.12 with w

_{e}= 0.086375.

_{j}is given in Equation (5). We have used this expression to extract the empirical entropic index from the empirical entropy for the third receiver station at x = 0.12. Figure 6 shows the entropic index for the normal spectral velocity wave component at this station as a function of the empirical mode j, for a mass injection velocity of w

_{e}= 0.08375. The first six modes indicate a fairly smooth region with entropic indices slightly below zero, indicating that these structures are only of a slightly dissipative nature. Modes seven through eleven indicate a significant transition region into non-equilibrium ordered structures. Modes twelve through sixteen indicate the possible further transition into further ordered states. We should note at this point that these indices are not in a sequence in time, but exist as a collection of modes within the fluctuating time series solution (Attard, pp. 409–414, [6]).

**Figure 6.**The empirical entropic index for the normal spectral velocity component, a

_{y3}, is shown as a function of the empirical mode, j for the third receiver station at x = 0.12 with the surface mass injection velocity w

_{e}= 0.08375.

_{j}/dt < 0. When the Tsallis entropic index is negative, Mariz [28] found that the empirical entropy change is also negative, dSemp

_{j}/dt < 0. The results presented in Figure 6 indicate that significant deterministic structures exist within the specified time frame of the nonlinear time series solution. These regions may therefore be classified as ordered, dissipative structures. Therefore, the significant negative nature for the extracted empirical entropic indices at the third receiver station at x = 0.12 is in agreement with both the Prigogine criterion and the Mariz results for the Tsallis entropic index. The ad hoc introduction of an empirical entropy index thus provides a representation of the nonlinear, non-equilibrium ordered structures in a significant way.

_{j}, the intermittency exponent, ζ

_{j}for the mode, j, is extracted from the expression given by Arimitsu and Arimitsu [22] by the use of Brent’s method (pp. 397–405, [18]). The intermittency exponent for the normal spectral velocity component, a

_{y3}, at the third receiver station at x = 0.12 for the surface injection velocity w

_{e}= 0.086375 is shown in Figure 7 as a function of the empirical mode, j.

**Figure 7.**The intermittency exponent for the normal spectral velocity component, a

_{y3}, is shown for the third receiver station at x = 0.12 with w

_{e}= 0.086375 as a function of empirical mode, j.

_{e}over the distance x. At the final equilibrium state, the stream wise velocity of the dissipated structure vanishes. The entropy generation rates through the deterministic ordered structures may then be determined from Equation (7) [1].

_{e}/x.

**Figure 8.**The entropy generation rates for the ordered structures are shown for various mass injection velocities at several stream wise stations.

_{e}= 0.08675, and not for the highest injection velocity. This result implies that there is an optimum surface injection velocity for the creation of fluctuations within the deterministic spiral structures. The entropy generation rates then decline over the remaining stream wise receiver stations.

**Figure 9.**The entropy generation rates for the deterministic ordered structures at x = 0.12 for various injection velocities are compared with the entropy generation rates across turbulent boundary layers with and without an applied adverse pressure gradient as a function of the normalized distance from the horizontal surface.

## 5. On the Transition of Non-Equilibrium Ordered Structures into Equilibrium Thermodynamic States

_{c}, Huang (pp. 370–373, [32]) presents an approximate expression for the specific heat of the material as a function of the normalized temperature, kT, as the temperature is varied through the critical point. Here, k is Boltzmann’s constant and T is the absolute temperature. The behavior of the Ising model specific heat Cv/k as a function of the temperature is shown in Figure 10.

**Figure 10.**The specific heat of the two-dimensional Ising model is shown as a function of the Ising temperature, kT.

**Figure 11.**The entropy of the two-dimensional Ising model obtained from the specific heat values for the model is shown as a function of the Ising temperature, kT.

**Figure 12.**The specific heat of a one-dimensional Ising model using a Monte Carlo simulation is shown as a function of the Ising 1d temperature, kT.

**Figure 13.**Comparison of a one-dimensional Monte Carlo simulation and a two-dimensional approximate solution for the entropy of the Ising model is shown as a function of the Ising temperature, kT.

## 6. Discussion

_{e}= 0.0860, 0.086375, 0.08750, 0.87125, at six stream wise stations along the x-axis. These values are well above the threshold value for the initiation of instabilities. However, it is surprising that significant variation in the magnitude of the ordered structures occurs over a very narrow range of injection velocities.

## 7. Conclusions

## Conflicts of Interest

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Isaacson, L.K.
Entropy Generation through Deterministic Spiral Structures in Corner Flows with Sidewall Surface Mass Injection. *Entropy* **2016**, *18*, 47.
https://doi.org/10.3390/e18020047

**AMA Style**

Isaacson LK.
Entropy Generation through Deterministic Spiral Structures in Corner Flows with Sidewall Surface Mass Injection. *Entropy*. 2016; 18(2):47.
https://doi.org/10.3390/e18020047

**Chicago/Turabian Style**

Isaacson, LaVar King.
2016. "Entropy Generation through Deterministic Spiral Structures in Corner Flows with Sidewall Surface Mass Injection" *Entropy* 18, no. 2: 47.
https://doi.org/10.3390/e18020047