# Spectral Entropy in a Boundary-Layer Flow

## Abstract

**:**

## 1. Introduction

## 2. Boundary-Layer Flow Environment

#### 2.1. Computational Model for the Boundary-Layer Flow

**Figure 1.**The three-dimensional flow model and the coordinate system for the boundary layer flow environment. Note the location for the laminar flow velocity profile computations.

_{∞}) = 1

_{e}, and m as functions of the axial direction x; GRID generates the computational grid across the boundary layer; IVPL generates the initial velocity profile for the laminar portion of the flow; the GROWTH subroutine provides for the growth of the boundary layer; EDDY contains the expressions for the inner and outer eddy viscosity formulations used in the transitional and turbulent portions of the flow; CMOM computes the coefficients of the differenced form of the transformed momentum equation; SOLV3 calculates the recursion formulas that occur in the block elimination of the Keller-Cebeci box method; and OUTPUT provides the output of the computations for the boundary-layer parameters and profiles. The computational data output is stored on external data files for access by the remaining computational procedures in this project.

_{0}= 307.0 K and a stagnation pressure P

_{0}= 10132.5 N/m

^{2}. These values yield a kinematic viscosity of 1.64 × 10

^{−4}m

^{2}/s. To simplify the approach to the computational process, we consider this value of the kinematic viscosity as applicable to boundary layers of interest and thus use this value throughout the boundary layer calculations. The density is calculated from the ideal gas equation of state and the dynamic viscosity for air is calculated from the expression given by Zucrow and Hoffman [10]. These values then yield the kinematic viscosity value close to that used in the example program of [1]. We have chosen this approach to allow a straightforward application of the methods of [1] to the numerical solution of the equations describing the basic flow environment.

_{e}is the dimensionless edge velocity and x is the dimensionless axial distance. The corresponding pressure gradient parameter, m, is given by

_{e}= 0.01, with a pressure gradient of zero in the z direction. The computations are made at a span wise station of nz = 4 (z = 0.08), at the axial station of nx = 4 (x = 0.08), with the span wise results held constant as the computations proceed in the axial direction. This procedure allows us to estimate the required velocity gradients at the initially laminar region in a three-dimensional configuration, as indicated in Figure 1.

#### 2.2. Entropy Production within the Boundary-Layer Flow

**Figure 2.**The dimensionless rate of entropy production in the laminar boundary layer at the axial station nx = 4 (x = 0.08) and the span wise station nz = 4 (z = 0.08).

**Figure 3.**The dimensionless rate of entropy production in the transitional boundary layer at the axial station, nx = 10 (x = 0.20) and the span wise station nz = 4 (z = 0.08).

**Figure 4.**The dimensionless rate of entropy production in the transitional boundary layer at the axial station, nx = 16 (x = 0.32) and the span wise station nz = 4 (z = 0.08).

**Figure 5.**The dimensionless rate of entropy production in the transitional boundary layer at the axial station, nx = 36 (x = 0.72) and the span wise station, nz = 4 (z = 0.08).

## 3. Mathematical Model of the Flow Instability

#### 3.1. Transformation of the Townsend Equations

_{i}, with i = 1,2,3 representing the x, y, and z components, while x

_{j}, with j = 1,2,3, denote the x, y, and z distances. The three mean velocity components and the nine gradients in the mean velocities are obtained from the solutions for the boundary-layer flow as outlined in previous sections.

_{0}t), where ω

_{0}is an amplification factor for the frequency of the periodic disturbances. This set of equations forms a set of external control parameters, as discussed by Klimontovich [37]. A second set of internal control parameters is provided by the values of the mean velocity gradients determined in the solution of the boundary-layer flow as discussed in previous sections.

_{0}is an externally applied frequency factor, held constant at ω

_{0}= 4.0 throughout the computations. The initial values for the wave numbers are:

_{x}(t = 0.0) = 0.80; k

_{y}(t = 0.0) = 0.1; k

_{z}(t = 0.0) = 0.1

_{0}= 4.0 yielded consistent results for the computations. However, for deterministic equations of the Lorenz-type, the results are very sensitive to the values of the external control parameters. We are therefore exploring the behavior of the computational results for the deterministic equations for various values of input amplitude of the cosine functions and for various values of the frequency factor. These results will be reported in a future paper.

_{1}will be examined in the on-going numerical study of the effects of the external control parameters on the solutions of the deterministic equations describing the flow instabilities.

_{y}, versus the axial velocity wave vector, a

_{x}, for the transformed vertical location of j = 2 (η = 0.200). These results indicate that the effects of the external periodic disturbance are transmitted through the entire boundary layer structure. As indicated in both Figure 6 and Figure 7, the magnitudes of the a

_{x}and a

_{y}wave vectors are significant, thus representing a largely vertical burst of fluctuating fluid elements.

**Figure 6.**The axial velocity wave vector, a

_{x}, as a function of time step at the transformed vertical station of j = 2 (η = 0.200).

**Figure 7.**The vertical component of the fluctuating wave vector, a

_{y}, versus the axial component, a

_{x}, for the vertical location of j = 2 (η = 0.200).

_{y}, versus the axial wave vector component, a

_{x}, for the vertical location of j = 10 (η = 1.804). These results indicate an oscillatory behavior in the vertical direction still much larger than the extent of the axial fluctuation. Again, these are results obtained within the laminar velocity profile at nx = 4 (x = 0.08).

**Figure 8.**The vertical component of the fluctuating velocity wave vector, a

_{y}, as a function of the axial fluctuating wave vector, a

_{x}, at the vertical location of j = 10 (η = 1.804).

**Figure 9.**The phase plane representation of the vertical fluctuating velocity vector, ${a}_{y}$ against the horizontal fluctuating velocity vector, a

_{x}, at the vertical location of j = 14 (η = 2.608).

_{x}, at the vertical station j = 18 (η = 3.414) and indicates the development of a weak instability in the wave vector results. These results are considerably reduced in value, thus indicating that at this further distance from the surface, the instabilities have decreased significantly.

_{y}versus the axial velocity wave vector, a

_{x}for j = 18 (η=3.414), again demonstrating the reduced level of the instability.

**Figure 10.**The axial velocity wave vector, a

_{x}, as a function of time step at the transformed vertical station of j = 18 (η= 3.414).

**Figure 11.**The phase plane representation of the vertical velocity a

_{y}, versus the horizontal velocity a

_{x}, at the transformed vertical station of j = 18 (η= 3.414).

_{x}at the vertical station j = 22 (η = 4.221) indicating that at this station, the axial velocity wave vector changes direction from positive x to a negative x direction.

**Figure 12.**The axial velocity wave vector, a

_{x}, as a function of time step at the transformed vertical station of j = 22 (η= 4.221).

#### 3.2. The Prediction of Spectral Entropy from the Deterministic Results

_{r}, for each particular segment. The probability values of each set of particular spectral densities for each segment is then computed from ${P}_{r}={f}_{r}/{\displaystyle \sum _{r}{f}_{r}}$. The methods of [43,44,45] are then applied to the probability distributions for each segment to develop the spectral entropy for the given segment. The spectral entropy (dimensionless) is defined as:

**Figure 13.**The spectral entropy of a

_{x}

^{2}+ a

_{y}

^{2}over 2048 time steps from the time step of 8192 to a time step of 10240 for the vertical location of j = 16 (η = 3.011) at nx = 4 (x = 0.08).

**Figure 14.**The spectral entropy of a

_{x}

^{2}+ a

_{y}

^{2}over 2048 time steps from the time step of 8192 to a time step of 10240 for the vertical location of j = 20 (η = 3.817) at nx = 4 (x = 0.08).

**Figure 15.**Comparison of wall shear layer maximum entropy production rates at various axial stations with values of spectral entropy of a

_{x}

^{2}+ a

_{y}

^{2}for the vertical location of j = 22 (η = 4.221).

## 4. Spectral Entropy Transfer Process

**Figure 16.**Downward sweep of vortex motion induced by vortex instabilities in the upstream laminar boundary-layer velocity profile.

**Figure 17.**(a) Photograph of a spiral vortex formed in the vertical shear layer above a cumulus cloud. The atmospheric flow is from left to right. (b) The induced down sweep of the vortex motion. (c) The upper surface of the induced down sweep moves deepest toward the origin. (d) Dissolution of the down sweep structure.

## 5. Conclusions

## Nomenclature

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

b | Eddy viscosity function in transformed boundary layer equation |

f | Dimensionless stream function |

f’ | First derivative of f with respect to η |

f’’ | Second derivative of f with respect to η |

f_{r} | Sum of the squares of the fluctuating axial and vertical velocity wave vectors |

F_{1} | Time-dependent perturbation factor |

j | Vertical station number in the boundary layer computations |

j | Spectral entropy segment number |

k | Time-dependent wave number vector |

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

K_{1} | Adjustable weight factor |

m | Pressure gradient parameter |

M | Molecular weight |

nx | Axial station number in the boundary layer computations |

p | Hydrostatic pressure |

P_{r} | Power spectral density of the r-th spectral segment |

P_{0} | Boundary layer edge pressure |

R | Appropriate gas constant |

s_{j}_spent | Spectral entropy for the j-th time series data segment |

${\stackrel{\u2022}{S}}_{shear}$ | Rate of dimensionless entropy production (1/s) in the shear layer |

${\stackrel{\u2022}{{S}^{\ast}}}_{shear}$ | Rate of volumetric entropy production in the shear layer |

${\stackrel{\u2022}{S}}_{turb}$ | Rate of dimensionless entropy production (1/s) in the turbulent boundary layer |

t | Time |

T | Static temperature |

T_{0} | Boundary layer edge temperature |

u | Axial boundary layer velocity |

u’ | Axial boundary layer fluctuation velocity |

u_{e} | Axial velocity at the outer edge of the boundary layer |

u_{i} | Fluctuating i-component of velocity instability |

U_{i} | Mean velocity in the i-direction |

v | Vertical boundary layer velocity |

v’ | Vertical boundary layer fluctuation velocity |

V_{y} | Mean vertical velocity in the x-y plane |

V_{z} | Mean vertical velocity in the y-z plane |

w | Span wise boundary layer velocity |

W | Mean velocity in the span wise direction |

x | Axial direction |

x_{i} | i-th direction |

x_{j} | j-th direction |

y | Vertical direction |

z | Span wise direction |

## Greek Letters

δ | Boundary layer thickness |

ε_{m} | Eddy viscosity |

ε_{m}^{+} | Dimensionless eddy viscosity |

η | Normalized vertical distance |

μ | Dynamic viscosity |

ρ | Density |

$\nu $ | Kinematic viscosity |

ω | External control parameter, frequency factor |

Ψ | Transformed stream function |

## Subscripts

i, j, l, m | Tensor indices |

r | The r-th index in the j-th time series data segment |

0 | Stagnation state, reference value |

x | Component in the x-direction |

y | Component in the y-direction |

z | Component in the z-direction |

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Isaacson, L.K.
Spectral Entropy in a Boundary-Layer Flow. *Entropy* **2011**, *13*, 1555-1583.
https://doi.org/10.3390/e13091555

**AMA Style**

Isaacson LK.
Spectral Entropy in a Boundary-Layer Flow. *Entropy*. 2011; 13(9):1555-1583.
https://doi.org/10.3390/e13091555

**Chicago/Turabian Style**

Isaacson, LaVar King.
2011. "Spectral Entropy in a Boundary-Layer Flow" *Entropy* 13, no. 9: 1555-1583.
https://doi.org/10.3390/e13091555