## 1. Introduction

## 2. Warm Dense Plasma Environment

^{+}, and electrons, e

^{−}, in chemical equilibrium. The equilibrium composition and the thermodynamic properties are computed using the statistical thermodynamic methods outlined by Sonntag and Van Wylen [7]. The statistical parameters required in these calculations are obtained from the NIST-JANAF Thermochemical Tables [8]. The transport properties of the mixture are computed using the expressions given by Spitzer [9].

^{5}K and a static pressure of 4.6 × 10

^{9}N/m

^{2}. For the upstream Mach number of 2.80, a wedge deflection angle of 18°, and a ratio of specific heats of 1.54, the equations of gas dynamics [6] yield a strong oblique shock wave at a shock wave angle of approximately 81°. The subsonic, deflected flow downstream of this strong oblique shock wave is now considered as our warm dense plasma environment. Table 1 presents the thermodynamic and transport properties used throughout the computational procedures.

## 3. Boundary-Layer Development

_{m}, having the dimensions of (viscosity)/(density), by:

## 4. Equations of Lorenz Form for the Spectral Velocity Wave Components

_{i}represent the mean velocity components with i = 1, 2, 3 indicating the x, y, and z components, and x

_{j}, with j = 1, 2, 3, designate the x, y and z directions. The pressure term is eliminated by taking the divergence of Equation (13) and invoking incompressibility, yielding:

_{i}(k), are then given as:

_{i}are obtained from the general equations for the balance of transferable properties:

_{i}, normal to the direction of the corresponding wave number component, k

_{i}. A model equation for this expression in the form:

**= 8.00, with the boundary layer instability observed within the boundary layer at a normalized distance of η = 3.00. The initial values for the wave number components are k**

_{e}_{x}[1] = 0.04, k

_{y}[1] = 0.02 and k

_{z}[1] = 0.02, while the initial conditions for the velocity wave components are a

_{x}[1] = 0.20, a

_{y}[1] = 0.01 and a

_{z}[1] = 0.001. For the thermodynamic and transport conditions given in Table 1, the weighting factor that yields a flow instability has been found to be K = 0.005.

## 5. Power Spectral Density Within the Deterministic Structure

_{x}, for each segment in the time series.

## 6. Singular Value Decomposition and Empirical Entropy

_{j}across the empirical modes, j, for the flow conditions listed in Table 1. The empirical entropy, Semp

_{j}, is defined from these eigenvalues by the expression (Rissanen [20,21]):

_{j}across the empirical modes, j for the given segment [9]. The particular empirical entropy of a particular mode j provides an indication of the nature of the directed kinetic energy that exists within that particular mode.

_{j}is that they represent twice the kinetic energy within each eigenmode distributed across j values [23]. These eigenmodes are obtained for the streamwise velocity wave components across a specified set of time series data, taken as a total ensemble of data values, and not as a sequence of values. Thus, these empirical entropy values exist as a collection of values within the nonlinear time series, and not as a cascade from low entropy modes to high entropy modes. The computation of the empirical entropy has thus provided additional insight into the thermodynamic nature of our nonlinear time series solutions. We have, thus far, demonstrated the production of instabilities at a particular vertical station within a three-dimensional laminar boundary layer as a result of nonlinear interactions within the boundary layer. The empirical entropy further characterizes the nonlinear time series into regions of low empirical entropy, a transition region, and an extensive region of high empirical entropy, coexisting simultaneously within the time series solutions.

## 7. Empirical Entropic Index for Deterministic Structures

_{i}is the probability of the subsystem to be in the state i, and W is the total number of microscopic possibilities of the system. The Tsallis entropic index, q, would be found from this expression for an ensemble of accessible microscopic subsystems.

_{j}is that this is the entropy of an ordered region described by the empirical eigenvalue, λj, 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 index, q

_{j}, from the empirical entropy. This expression may be written as:

_{j}, from the empirical entropy for streamwise stations x = 0.08. Figure 8 shows the empirical entropic indices for the streamwise velocity wave components at this station as a function of the empirical mode. The empirical entropic indices for empirical modes one through four indicate a zero value. Mariz [25], indicates that for an entropic index of zero value, dSemp

_{j}/dt = 0. These modes thus contain a high fraction of directed kinetic energy, flowing in the streamwise direction in a reversible and adiabatic process.

_{j}/dt < 0. When the Tsallis entropic index is negative, Mariz [25] found that the empirical entropy change is also negative, dSemp

_{j}/dt < 0. The results presented in Figure 8 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 from empirical modes five to sixteen at the streamwise station x = 0.08 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 may thus provide a representation of the nonlinear, non-equilibrium structures in a significant way.

## 8. Intermittency Exponents for Deterministic Structures

_{j}, the intermittency exponent, ζ

_{j}for the mode, j, is extracted from this expression by the use of Brent’s method [29] (pp. 397–405). The intermittency exponent is shown in Figure 9 as a function of empirical mode, j. Arimitsu and Arimitsu [27] derive the intermittency exponents for turbulent eddies that dissipate turbulent kinetic energy into thermal energy within the flow.

## 9. Entropy Generation Rate through the Deterministic Boundary-Layer Structures

^{2}/2 while the kinetic energy dissipation rate finally available is that found from the summation of the fraction of kinetic energy dissipation rate in each empirical mode, ξ

_{j}times the intermittency exponent for that mode, ζ

_{j}[27]. We consider the dissipation of the ordered structures into background thermal energy as a relaxation process of the streamwise velocity in the initial state to the final equilibrium state of the streamwise velocity over the internal parameter x. The expression for the entropy generation rate may then be written as:

_{e}f′ and the streamwise Mach number is given by:

## 10. Discussion

## 11. Conclusions

## Nomenclature

a_{i} | Fluctuating i-th component of velocity wave vector |

b | Coefficient in Equation (11) |

b | Fluctuating Fourier component of the static pressure |

b_{1} | Coefficient in modified Townsend equations defined by Equation (33) |

E | Power spectral density for a particular frequency, f |

E_{avail} | Available kinetic energy dissipation rate for a given mode |

f | Frequency for power spectral density distribution |

f_{1} | Initial frequency for integration of power spectral density, Equation (34) |

f_{2} | Final frequency for integration of power spectral density, Equation (34) |

F | Time-dependent feedback factor |

f_{r} | Power spectral density of the r-th time series segment |

j | Mode number empirical eigenvalue |

j | Spectral entropy segment number |

J | Net source of kinetic energy dissipation rate, Equation (39) |

k | Time-dependent wave number magnitude |

k | Dimensional constant, Equation (36) |

k_{i} | Fluctuating i-th wave number of Fourier expansion |

K | Adjustable weighting factor |

m | Pressure gradient parameter, Equation (8) |

M_{1,2} | Flow Mach number upstream and downstream of oblique shock wave |

M_mol | Molecular weight of plasma mixture |

n | Time step number |

p | Local static pressure |

p_{2} | Static pressure in the boundary layer |

p_{i} | Probability of being in a state i, Equation (36) |

p_{1,2} | Static pressure upstream and downstream of oblique shockwave |

q | Tsallis nonextensive entropic index |

q_{j} | Empirical entropic index for the empirical entropy of mode, j |

r_{1} | Coefficient in modified Townsend equations defined by Equation (35) |

R_{u} | Universal gas constant |

s | Entropy per unit mass |

s_{1} | Coefficient in modified Townsend equations defined by Equation (32) |

Semp_{j} | Empirical entropy for empirical mode, j |

S_{q} | Tsallis entropy, Equation (36) |

${\stackrel{\u2022}{S}}_{gen}$ | Entropy generation rate through kinetic energy dissipation |

${\stackrel{\u2022}{S}}_{turb}$ | Entropy generation rate in a turbulent boundary layer |

t | Time |

T_{1,2} | Static temperature upstream and downstream of oblique shockwave |

u | Mean streamwise velocity in the streamwise direction in Equation (1) |

u′ | Fluctuating streamwise velocity in Equation (1) |

u_{e} | Streamwise velocity at the outer edge of the x-y plane boundary layer |

u_{i} | The i-th component of the fluctuating velocity |

U_{i} | Mean velocity in the i-th direction in the modified Townsend equations |

v | Mean normal velocity in Equation (1) |

v′ | Fluctuating normal velocity in Equation (1) |

V_{2} | Flow velocity downstream of oblique shock wave |

w_{e} | Spanwise velocity at the outer edge of the z-y plane boundary layer |

W | Total number of microscopic states in a system, Equation (36) |

x | Streamwise distance |

x_{i} | i-th direction |

x_{j} | j-th direction |

y | Normal distance |

z | Spanwise distance |

Greek Letters | |
---|---|

γ | Ratio of specific heats |

δ | Surface deflection angle |

δ | Boundary layer thickness |

δ_{lm} | Kronecker delta |

ε | Oblique shock wave angle |

ε_{m} | Eddy viscosity |

${\epsilon}_{m}^{+}$ | Normalized eddy viscosity |

ζ_{j} | Intermittency exponent for the j-th mode in Equation (38) |

η | Transformed normal distance parameter |

λ_{j} | Eigenvalue for the empirical mode, j |

μ | Mechanical potential in Equation (39) |

v_{1} | Kinematic viscosity of the gas mixture |

ξ_{j} | Kinetic energy dissipation rate in the j-th empirical mode |

ρ | Density |

σ_{y} | Coefficient in modified Townsend equations defined by Equation (29) |

σ_{x} | Coefficient in modified Townsend equations defined by Equation (30) |

Subscripts | |
---|---|

e | Outer edge of the x-y plane boundary layer |

i, j, l, m | Tensor indices |

x | Component in the x-direction |

y | Component in the y-direction |

z | Component in the z-direction |

## Conflicts of Interest

## References

- Atzeni, S.; Meyer-ter-Vehn, J. The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter; Oxford University Press: New York, NY, USA, 2004. [Google Scholar]
- Drake, R.P. High-Energy-Density Physics: Fundamentals, Inertial Fusion, and Experimental Astrophysics; Springer: Dordrecht, The Netherlands, 2006. [Google Scholar]
- Harding, E.C. Observations of Shear Flows in High-Energy-Density Plasmas; ProQuest LLC: Ann Arbor, MI, USA, 2010. [Google Scholar]
- Cebeci, T.; Bradshaw, P. Momentum Transfer in Boundary Layers; Hemisphere: Washington, DC, USA, 1977. [Google Scholar]
- Cebeci, T.; Cousteix, J. Modeling and Computation of Boundary-Layer Flows; Horizons Publishing: Long Beach, CA, USA, 2005. [Google Scholar]
- Zucrow, M.J.; Hoffman, J.D. Gas Dynamics; Wiley: New York, NY, USA, 1975; Volume 1, pp. 356–372. [Google Scholar]
- Sonntag, R.E.; van Wylen, G.J. Fundamentals of Statistical Thermodynamics; Robert E. Krieger Publishing Company: Malabar, FL, USA, 1985. [Google Scholar]
- Chase, M.W., Jr. NIST-JANAF Thermochemical Tables, 4th ed.; American Institute of Physics: Woodbury, NY, USA, 1998. [Google Scholar]
- Spitzer, L., Jr. Physics of Fully Ionized Gas; Interscience Publishers: New York, NY, USA, 1956. [Google Scholar]
- Cambel, A.B. Plasma Physics and Magnetofluidmechanics; McGraw-Hill Book Company: New York, NY, USA, 1963. [Google Scholar]
- Isaacson, L.K.; Spectral, Entropy. Empirical Entropy and Empirical Exergy for Deterministic Boundary-Layer Structures. Entropy
**2013**, 15, 4134–4158. [Google Scholar] - Isaacson, L.K. Spectral Entropyin a Boundary Layer Flow. Entropy
**2011**, 13, 1555–1583. [Google Scholar] - Hansen, A.G. Similarity Analyses of Boundary Value Problems in Engineering; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964. [Google Scholar]
- Townsend, A.A. The Structure of Turbulent Shear Flow, 2nd ed.; Cambridge University Press: Cambridge, UK, 1976. [Google Scholar]
- Hellberg, C.S.; Orszag, S.A. Chaotic behavior of interacting elliptical instability modes. Phys. Fluids
**1988**, 31, 6–8. [Google Scholar] - Isaacson, L.K.; Ordered, Regions. within a Nonlinear Time Series Solution of a Lorenz Form of the Townsend Equations for a Boundary-Layer Flow. Entropy
**2013**, 15, 53–79. [Google Scholar] - Manneville, P. Dissipative Structures and Weak Turbulence; Academic Press: San Diego, CA, USA, 1990. [Google Scholar]
- Chen, C.H. Digital Waveform Processing and Recognition; CRC Press: Boca Raton, FL, USA, 1982; pp. 131–158. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; Wiley: Hoboken, NJ, USA, 2006. [Google Scholar]
- Rissanen, J. Complexity and Information in Data. In Entropy; Greven, A., Keller, G., Warnecke, G., Eds.; Princeton University Press: Princeton, NJ, USA, 2003; pp. 299–327. [Google Scholar]
- Rissanen, J. Information and Complexity in Statistical Modeling; Springer: New York, NY, USA, 2007. [Google Scholar]
- Li, M.; Vitanyi, P.M.B. An Introduction to Kolmogorov Complexity and Its Applications, 3rd ed.; Springer: New York, NY, USA, 2008. [Google Scholar]
- Holmes, P.; Lumley, J.L.; Berkooz, G.; Rowley, C.W. Turbulence, Coherent Structures, Dynamical Stations and Symmetry, 2nd ed.; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: New York, NY, USA, 2009; pp. 37–43. [Google Scholar]
- Mariz, A.M. On the irreversible nature of the Tsallis and Renyi entropies. Phys. Lett. A
**1992**, 165, 409–411. [Google Scholar] - Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability and Fluctuations; Wiley: London, UK, 1971. [Google Scholar]
- Arimitsu, T.; Arimitsu, N. Analysis of fully developed turbulence in terms of Tsallis statistics. Phys. Rev. E
**2000**. [Google Scholar] - Mathieu, J.; Scott, J. An Introduction to Turbulent Flow; Cambridge University Press: New York, NY, USA, 2000; pp. 251–261. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes in C: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, UK, 1992. [Google Scholar]
- Thomas, J.W. Numerical Partial Differential Equations: Finite Difference Methods; Springer: New York, NY, USA, 1995. [Google Scholar]
- Isaacson, L.K. Transitional intermittency exponents through deterministic boundary-layer structures and empirical entropic indices. Entropy
**2014**, 16, 2729–2755. [Google Scholar] - Ghasemi, E.; McEligot, D.M.; Nolan, K.P.; Crepeau, J.; Tokuhiro, A.; Budwig, R.S. Entropy generation in a transitional boundary layer region under the influence of freestream turbulence using transitional RANS models and DNS. Int. Comm. Heat. Mass Transfer
**2013**, 41, 10–16. [Google Scholar] - Ghasemi, E.; McEligot, D.M.; Nolan, K.P.; Crepeau, J.; Siahpush, A.; Budwig, R.S.; Tokuhiro, A. Effects of adverse and favorable pressure gradients on entropy generation in a transitional boundary layer region under the influence of freestream turbulence. Int. J. Heat. Mass. Transfer
**2014**, 77, 475–488. [Google Scholar] - De Groot, S.R.; Mazur, P. Non-Equilibrium Thermodynamics; North-Holland Publishing: Amsterdam, The Netherlands, 1962. [Google Scholar]
- Truitt, R.W. Fundamentals of Aerodynamic Heating; The Ronald Press Company: New York, NY, USA, 1960. [Google Scholar]
- Bejan, A. Entropy generation minimization; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Fung, J.C.H.; Vassilicos, J.C. Fractal dimensions of lines in chaotic advection. Phys. Fluids A
**1991**, 3, 2725–2733. [Google Scholar] - Hurst, D.; Vassilicos, J.C. Scalings and decay of fractal-generated turbulence. Phys. Fluids
**2007**, 19, 035103. [Google Scholar] - Seoud, R.E.; Vassilicos, J.C. Dissipation and decay of fractal-generated turbulence. Phys. Fluids
**2007**, 19, 105108. [Google Scholar] - Mazellier, N.; Vassilicos, J.C. Turbulence without Richardson-Kolmogorov cascade. Phys. Fluids
**2010**, 22, 075101. [Google Scholar] - Valente, P.C.; Vassilicos, J.C. The decay of turbulence generated by a class of multiscale grids. J. Fluid Mech.
**2011**, 687, 300–340. [Google Scholar]

**Figure 2.**Shown is the streamwise velocity wave vector component, a

_{x}as a function of the time step, n.

**Figure 3.**Shown is the normal velocity wave vector component, a

_{y}as a function of the time step, n.

**Figure 4.**Shown is the power spectral density of the streamwise velocity wave vector component, a

_{x}as a function of the frequency at station x = 0.08.

**Figure 5.**Shown is the percentage distribution of the available kinetic energy dissipation rate for the streamwise velocity wave vector component, a

_{x}.

**Figure 6.**The fractional distribution of kinetic energy is shown as a function of the empirical mode number from the singular value decomposition process.

**Figure 8.**The empirical entropic index, q

_{j}is shown as a function of the empirical mode, j for the streamwise station x = 0.08.

**Figure 9.**The intermittency exponent, ζ

_{j}is shown as a function of the empirical mode number, j for streamwise station x = 0.08.

**Figure 10.**The entropy generation rate is shown as a function of the normalized distance from the surface.

Parameter | Applied value |
---|---|

Upstream Mach number (M_{1}) | 2.80 |

Upstream temperature (T_{1}) | 1.00 × 10^{5} K |

Upstream static pressure (p_{1}) | 4.63 × 10^{9} N/m^{2} |

Upstream gamma (γ_{1}) | 1.54 |

Flow deflection angle (δ) | 18° |

Strong oblique shock wave angle (ε) | 81° |

Downstream Mach number (M_{2}) | 0.54 |

Downstream temperature (T_{2}) | 2.86 × 10^{5} K |

Downstream static pressure (p_{2}) | 4.20 × 10^{10} N/m^{2} |

Downstream gamma (γ_{2}) | 1.65 |

Downstream kinematic viscosity (v_{2}) | 1.46 × 10^{−4} m^{2}/s |

Downstream velocity (V_{2}) | 3.4 × 10^{4} m/s |

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