_{1}

^{*}

A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to the interaction of a longitudinal vortex with a laminar boundary-layer flow. These three-dimensional equations are cast into a Lorenz-format system of equations for the spectral velocity component solutions. Tsallis-form empirical entropic indices are obtained from the solutions of the modified Lorenz equations. These solutions are sensitive to the initial conditions applied to the time-dependent coupled, non-linear differential equations for the spectral velocity components. Eighteen sets of initial conditions for these solutions are examined. The empirical entropic indices yield corresponding intermittency exponents which then yield the entropy generation rates for each set of initial conditions. The flow environment consists of the flow of hydrogen gas with impurities at a given temperature and pressure in the interaction of a longitudinal vortex with a laminar boundary layer flow. Results are presented that indicate a strong correlation of predicted entropy generation rates and the corresponding applied initial conditions. These initial conditions may be ascribed to the turbulence levels in the boundary layer, thus indicating a source for the subsequent entropy generation rates by the interactive instabilities.

Results are reported for an innovative computational procedure applied to a study of the sensitivity to initial conditions in the computation of entropy generation rates generated by the non-linear interaction of a longitudinal vortex with a laminar boundary layer flow. The equations for the fluctuating velocity components in a three-dimensional shear flow have been presented by Townsend [

The nonlinear, time-series solutions for the spectral velocity wave components are obtained from a modified Lorenz-type set of equations that is sensitive to the initial conditions applied to the integration of the equations. The control parameters for these equations are the steady state boundary layer velocity gradients that are determined by the particular value of the kinematic viscosity for the system. Experimental measurements of the unsteady fluctuation levels in laminar boundary layers when subjected to free stream turbulence have been presented by Walsh and Hernon [

However, the initial conditions for the integration of the modified Lorenz equations are the actual turbulent intensity levels applied to the flow system at the specific location within the boundary layer where the computational results are obtained. We have selected eighteen sets of initial conditions for the initial values for the computation of the time development of the spectral stream wise, normal and span wise velocity components.

The boundary-layer environment used in the study reported here is the flow of helium with slight impurities at a temperature of T = 320.0 K and a pressure of p = 1.01325 × 10^{5} N/m^{2} at a normalized boundary-layer vertical location of η = 3.00. The kinematic viscosity for the helium mixture at these conditions is ν = 1.384696 × 10^{−4} m^{2}/s.

The Falkner-Skan transformation, in the form

η = ( u e ν x ) 1 / 2 y (1)

provides the definition of the normalized distance, η from the surface of the boundary layer flow. In this expression, u_{e} is the boundary layer edge velocity, x is the stream wise distance and the edge value for the normalized vertical distance is η_{¥} = 8.0.

This article includes the following sections:

In Section 2, the laminar boundary-layer flow configuration considered in this study is described. In Section 3, eighteen sets of initial conditions for the integration of the time-dependent modified Lorenz equations are presented. Section 4 presents a brief review of the results of the study of the sensitivity of the solutions of the modified Lorenz equations for the entropy generation rates to the initial conditions. In Section 5, the fluctuation equations of Townsend [

The article closes with a discussion of the results and final conclusions.

The solution of the modified Lorenz equations requires the input of various equation control parameters. The steady state boundary-layer velocity gradients and the time-dependent spectral wave component solutions serve as control parameters for the solution of the time-dependent fluctuating spectral velocity equations. The mathematical and computational methods used for the computation of the x-y plane and the z-y plane steady-state laminar boundary-layer velocity gradients are summarized in this section. The boundary layer steady velocity gradients vary with the stream wise distance x, indicating the initiation of instabilities within the boundary layer for several stream wise stations, similar to the development of a young turbulent spot.

Singer [_{e} = 0.08. This is the value we use for the span wise velocity. Schmid and Henningson (pp. 429-436 [

The computer source code listings that we have used to compute the steady laminar boundary layer velocity profiles for both the x-y plane and the z-y plane were developed by Cebeci and Bradshaw [

steady boundary layer velocity gradient control parameters for the solution of the modified Lorenz equations. The working gas for these studies is a mixture of helium with several impurities, as described in [

The boundary layer profiles are determined at six stream-wise stations, with the first station designated as the transmitter station. The following five stations are designated as receiver stations, with the first receiver station designated as station 1. The results obtained for the entropy generation rates at the receiver station 3, at x = 0.120, as a function of the initial conditions, are presented in depth in this article. A computational flow chart is presented in

An essential aspect of the computational procedure discussed in this article is the inclusion of the time-dependent, coupled, nonlinear modified Lorenz equations for the prediction of the development of ordered regions within the nonlinear time series solutions. For the studies reported in [

Application of the computational procedure outlined in

Initial conditions, set number | Initial spectral velocity component, a_{x} [ | Initial spectral velocity component, a_{y} [ | Initial spectral velocity component, a_{z} [ | Equivalent turbulence level, percent |
---|---|---|---|---|

1 | 0.00500 | 0.000200 | 0.000200 | 0.332 |

2 | 0.00560 | 0.002240 | 0.002240 | 0.371 |

3 | 0.01000 | 0.004000 | 0.004000 | 0.663 |

4 | 0.01500 | 0.006000 | 0.006000 | 0.995 |

5 | 0.02000 | 0.008000 | 0.008000 | 1.327 |

6 | 0.03000 | 0.012000 | 0.012000 | 1.990 |

7 | 0.04000 | 0.016000 | 0.016000 | 2.653 |

8 | 0.05000 | 0.020000 | 0.020000 | 3.317 |

9 | 0.06000 | 0.024000 | 0.024000 | 3.980 |

10 | 0.07000 | 0.028000 | 0.028000 | 4.643 |

11 | 0.08000 | 0.032000 | 0.032000 | 5.307 |

12 | 0.09000 | 0.036000 | 0.036000 | 5.970 |

13 | 0.10000 | 0.040000 | 0.040000 | 6.633 |

14 | 0.12000 | 0.048000 | 0.048000 | 7.960 |

15 | 0.14000 | 0.056000 | 0.056000 | 9.287 |

16 | 0.16000 | 0.064000 | 0.064000 | 10.613 |

17 | 0.18000 | 0.072000 | 0.072000 | 11.940 |

18 | 0.20000 | 0.080000 | 0.080000 | 13.266 |

generation rate for each set of initial conditions shown in ^{−4} m^{2}/s.

These results indicate a strong correlation of the predicted entropy generation rates with the corresponding initial conditions applied to the equations for the time dependent solutions. If the assumption is made that these initial conditions are provided by the attenuated free stream turbulence, these computational methods may then provide a path to understanding bypass transition. A detailed explanation of the procedures used in the calculation of these results and a discussion of the possible sources for the indicated entropy generation rates are presented in the next sections.

For the flow of a wall shear layer with velocity fluctuations, the computational procedure may be separated into the evaluation of the steady state velocity profiles and a set of equations for the fluctuating velocity field (Townsend [

The solutions of the modified Lorenz equations yield the spectral velocity components within the nonlinear time series solutions. Statistical analysis of these spectral time-series solutions yields the power spectral densities and the empirical entropies over a range of sixteen empirical modes. The correspondence of the peaks of the spectral power spectrum and the empirical modes of the singular value decomposition analysis is achieved by invoking the Weiner-Khintchine theorem (Attard (pp. 354-355 [

The equations for the velocity fluctuations within a wall shear layer may be written as (Townsend (pp. 46-49 [

∂ u i ∂ t + U j ∂ u i ∂ x j + u j ∂ U i ∂ x j + u j ∂ u i ∂ x j = − 1 ϱ ∂ p ∂ x i + ν ∂ 2 u i ∂ x j ∂ x j . (2)

Fourier transform of these equations yields the equations for the three time-dependent spectral velocity wave components, a_{i}(k), as (pp. 47-49 [

∂ a i ( k ) ∂ t = − ν k 2 a i ( k ) − ∂ U i ∂ x i a i ( k ) + 2 k i k l k 2 ∂ U l ∂ x m a i ( k ) + i ∑ k ′ + k ″ = k ( k l k i k m k 2 − δ i m k l ) a l ( k ′ ) a m ( k ″ ) (3)

The nonlinear coupling terms in the spectral velocity components in Equations (3) are represented in our series of equations by characterizing the transfer matrix

k l ( δ i m − k i k m k 2 ) (4)

as basic to the formation of patterns in nonlinear time series solutions (Mannevile (pp. 302-312 [

( 1 − K ∗ cos ( k ( t ) ) ) (5)

is introduced to provide the proper weighting of the transfer matrix (Equation (4) in our computational procedure. K is an empirical amplitude factor [

k ( t ) = ( k x 2 ) (6)

Substitution of Equation (5) and with F = K cos ( k ( t ) ) , the equations for the spectral velocity components, Equations (3), may be rearranged into Lorenz format as [

d a x n d t = σ y n a y n − σ x n a x n , (7)

d a y n d t = − ( 1 − F ) a x n a z n + r n a x n − s n a y n , (8)

d a z n d t = ( 1 − F ) a x n a y n − b n a z n . (9)

The expressions for the coefficients σ y n , σ x n , r_{n}, s_{n}, and b_{n} are given in detail in [_{i}, and the steady state values of the velocity gradients in the boundary layer flow, as indicated in Equation (3). These equations are designated as the modified Lorenz equations. These equations are solved at the initial station of x_{0} = 0.06, considered as the transmitter station.

The solutions of these equations at the transmitter station require additional assumptions for the modified Lorenz equations. Manneville (pp. 305-312 [

Incorporating Expression (5) in Equation (3), with the definition of the term F, the modified Lorenz equations take the form of Equations (7-9). These equations are solved at the transmitter station for each set of initial conditions listed in _{0} = 0.060.

The following stations at x_{1} = 0.080, x_{2} = 0.100, x_{3} = 0.120, x_{4} = 0.140, and x_{5} = 0.160 are designated as receiver stations. From thermodynamic considerations (Attard (pp. 329-331 [

We apply the transformation of the pattern matrix (Equation (5)) to the transmitter station at x = 0.060. The time-series solutions for this station indicate the generation of nonlinear instabilities in each of the spectral velocity components. These instabilities are then transferred to the next station, or first receiver station. The modified Lorenz equations have been shown to have the property of synchronization or extraction of ordered signals from a chaotic signal. We will apply this property to each of the receiver stations in the system.

The synchronization properties of systems of Lorenz-type equations have been shown by Pecora and Carroll [

We apply the synchronization properties at each of the receiver stations downstream from the initial transmitter station. The various boundary layer control parameters at each of these stations are computed in the same manner as in the transmitter station. Following the results in [

For each receiver station, n, the system of nonlinear dynamic equations is written as:

d a x n d t = σ y n a y n − σ x n a x n , (10)

d a y n d t = − a r n a z n + r n a x n − s n a y n , (11)

d a z n d t = a r n a y n − b n a z n . (12)

Note that for the initial station, characterized as the transmitter station, a_{x0}_{ ,}_{ }is the time-dependent spectral velocity wave component output from the transmitter station. The input driving term for the next station, the first receiver station at x = 0.08, is then given by a_{x}_{0}, where a_{x}_{0} is the output from the initial or_{ }transmitter station at x = 0.06.

In Equations (10-12), the input driving signal, a_{rn}_{,} carrying information from the transmitter and the previous receiver stations to the n-th station is given by the sum of the outputs from the transmitter station and the previous n-1 receiver stations:

a r n = ∑ i = 0 i = n − 1 a x i , n = 1 , 2 , 3 , 4 , 5 (13)

The initial conditions for the fluctuating spectral velocity wave vector components for the transmitter station and for each successive receiver station are set equal to the values listed in

The free stream velocity for the stream wise boundary layer flow is taken as unity, u_{e} = 1.00, while the vortex tangential velocity is w_{e} = 0.08 (pp. 101-102 [_{¥} = 8.00, the instabilities occur at 37.5 percent of the boundary layer thickness. The time step for the time-dependent integration process is taken as h = 0.0001 s. Initial values for the spectral wave number equations are taken as k_{x} [_{y} [_{z} [

The solutions of nonlinear, coupled differential equations, such as the modified Lorenz equations (Equations (7-10) and Equations (10-12)), are sensitive to the particular values of the initial conditions applied in the solutions. _{x} [_{y} [_{z} [_{x0}. The following station is denoted as the first receiver station, n = 1, at x = 0.08, with the stream wise spectral velocity component as a_{x1}. The results presented here are the sensitivity to the initial conditions for the entropy generation rates at the receiver station of n = 3, at the stream wise location of x = 0.120. The three spectral velocity components are denoted as a_{x}_{3}, a_{y}_{3} and a_{z}_{3}.

The initial conditions applied to the spectral velocity wave equations must arise from the attenuation of the external free stream turbulence level through the boundary layer flow. Schmid and Henningson (pp. 401-413 11]) and Sengupta (pp. 171-200 [

The solutions of the modified Lorenz equations have been obtained for each set of intial conditions listed in _{y}_{3} − a_{x}_{3}, where a_{y}_{3} is the normal spectral velocity wave

component and a_{z}_{3} is the span wise spectral velocity wave component, again at the station x = 0.120.

_{y}_{3} − a_{z}_{3}, where a_{z}_{3} is the span wise spectral velocity wave component and a_{y}_{3} is the normal spectral velocity wave component, again at the station x = 0.120. These results indicate the formation of an initially strong spiral cone in the stream wise direction, transforming into a strongly oscillating motion in the stream wise, normal and span wise spectral planes of the flow environment.

In

However, for the spike in entropy generation rate for Set 9, the stream wise spectral velocity component maintains a strong value, with significant structure. This pattern is repeated for initial condition Sets 6, 9, 12, 15 and 18, creating the strong patterns indicated in

Burg’s method (Chen [

The first five sets of initial conditions in _{y}_{3} at the third receiver station at x = 0.120, for initial conditions, Set 6. For the power spectral density spectrum, we have assigned empirical mode numbers to the 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.

The power spectral density for the normal spectral velocity component shown in

Additional fundamental characteristics within the nonlinear time series solutions of the modified Lorenz equations may also be found using the singular value decomposition procedure (Holmes, et al. (pp. 130-152 [

[

The singular value decomposition of the nonlinear time series solutions of the modified Lorenz equations yields the distribution of the spectral velocity component eigenvalues λ_{j} across the empirical modes, j, for each set of initial conditions listed in _{j}, are equivalent to the eigenvalues in the physical plane. These eigenvalues therefore represent the distribution of the kinetic energy of the fluctuating velocity components across the empirical modes, j. The fractional eigenvalues have an approximate exponential distribution over the empirical modes, j, as shown in

Therefore, the analysis of Rissanen (pp. 58-60 [_{j}, may be obtained from these eigenvalues by the expression:

S e m p j = − ln ( λ j ) . (14)

In this expression, λ_{j} is the empirical eigenvalue computed from the singular value decomposition procedure applied to the nonlinear time-series solution.

The peaks of the spectral power density analysis and the empirical modes of the singular value decomposition analysis are computed from the same set of nonlinear time series data. Therefore, the Weiner-Khintchine theorem allows us to relate each power spectral density peak with a corresponding empirical eigenvalue. The Weiner-Khintchine theorem relates the power density spectrum to the autocorrelation function as they are operating on the same nonlinear time series data (Attard (pp. 354-355 [

The results for the empirical entropy value for each of the empirical modes allows us to compute a corresponding entropic index for these modes. This process is described in the next section.

The power spectral density spectrum shown in

The empirical entropy, Semp_{j} describes the entropy of an ordered region identified by the empirical eigenvalue, λ j , for the empirical mode, j. We have found that an expression from which we may extract an empirical entropic index, q_{j}, from the empirical entropy, Semp_{j}, may be written in a Tsallis entropic format as [

S e m p j = − ln ( λ j ) = ( λ j ) q j − 1 ( 1 − q j ) . (15)

The empirical entropy index, q_{j}, provides a connection between the empirical entropy obtained from the singular value decomposition to the intermittency exponent of the ordered structures within the time series solutions. The intermittency exponent describes the fraction of the available kinetic energy within each empirical mode, j, that is dissipated into thermodynamic internal energy, thus increasing the entropy of the system. The intermittency exponent for each of the empirical modes is discussed in the next section.

The final phase of the dissipation of fluctuating kinetic energy into thermodynamic internal energy occurs through the process of intermittency exponents and a relaxation process into the final thermodynamic entropy state.

The intermittency exponents for the each of the empirical modes, z_{j}, are obtained from the empirical entropic indices of the Tsallis form extracted from the empirical entropies in Equation (15). Arimitsu and Arimitsu [_{j}, may be extracted from the entropic index of Tsallis. The intermittency exponent provides the fraction of fluctuating kinetic energy within the non-equilibrium empirical mode, j, that is dissipated into thermodynamic internal energy [

The absolute value of the empirical entropic index calculated from Equation (15) is used to extract the intermittency exponent from the equation derived by Arimitsu and Arimitsu [

| q j | = 1 − 1 + ζ j − log 2 ( 1 + 1 − 2 − ζ j ) ⋇ log 2 ( 1 − 1 − 2 − ζ j ) log 2 ( 1 + 1 − 2 − ζ j ) − log 2 ( 1 − 1 − 2 − ζ j ) . (16)

The intermittency exponent, z_{j}, found from this expression, represents the fraction of kinetic energy in the empirical mode, j, that is dissipated into background thermal energy. The kinetic energy contained within the spectral mode, j, of the power spectral density is denoted as x_{j}. Thus, the product of the kinetic energy of the mode, j, and the intermittency exponent for that mode, z_{j}, summed over all of the empirical modes, represents the amount of kinetic energy in the given spectral velocity component that is dissipated into increasing the entropy of the reservoir.

The computation of the intermittency exponents yields two significant results. First, the computed value for each empirical mode allows the computation of the entropy generated through the dissipation of that mode. Second, the particular value for each empirical mode provides us with additional insight into the physical processes involved in the generation of entropy through the dissipation of the empirical modes embedded in the nonlinear solutions of the modified Lorenz equations.

Consider the entropy generation rates for initial conditions Sets 8, 9 and 10 in

Figures 9-11 show the intermittency exponents for the stream wise, normal, and span wise spectral velocity wave components a_{x3}, a_{y3} and a_{z3} as functions of the empirical mode, j for the initial conditions Sets 8, 9 and 10, respectively. Note that the stream wise intermittency is relatively low for all three sets of initial conditions. The increase in value for the stream wise intermittency exponents at the high empirical modes do not make a signifiacant contribution

to the entropy generation rates because these modes contain very low fractions of the available stream wise kinetic energy.

However, for the normal and span wise spectral velocity components for initial conditions Sets 8 and 10, the first and third empirical modes make signifant contributions to the values of the intermittency exponents, with the third mode dominating the contributions. For the initial conditions Set 9, the fifth empirical mode comes into play with a significant increase in value. This indicates that as the initial conditions are increased in magnitude, the nonlinear time dependent solution of the modified Lorenz equations for the normal spectral velocity component predicts the spread of ordered kinetic energy over an additional empirical mode for this component. This results in the predition of a higher rate of entropy generation for this particular set of initial conditions. The repeated pattern shown in

The foundation for the Tsallis entropic index [

To compare our computational results with results presented in the literature, we need to obtain average values for the intermittency exponents computed for a selected set of initial conditions from

spectral velocity components to yield the average intermittency exponent for the entire Set 5 initial conditions. The resulting intermittency exponent will be designated as ζ 5 a v e . The value of the average intermittency exponent for initial conditions Set 5 is found to be ζ 5 a v e = 0.349.

Arimitsu and Arimitsu [

The low-temperature quantum superfluid system studied in [

However, when we move to the initial conditions Set 6, the first significant instability is found in the nonlinear time series solutions of the modified Lorenz equations. These instabilities give rise to a higher rate of dissipation of turbulent kinetic energy, thus increasing the irreversibilities in the process. This increase is reflected in the local peak in the entropy generation rate for Set 6, as indicated in

The computation of entropy generation rates reported here are for a three-dimensional laminar boundary layer located at a stream wise distance of 0.120 on a 1 m scale, with a stream wise velocity of unity, and at a normalized vertical distance in the boundary layer, η = 3.0, with the edge of the boundary layer at η ∞ = 8.00, or 37.5 percent of the boundary-layer thickness. This location is approximately the position of the hot-wire measurements reported by Meneneau and Sreenivan [

An overall average intermittency value was determined by averaging over Sets 6 - 10 and 14 - 16 with the results that ζ 5 a v e = 0.241, which agrees with the universally accepted value of approximately 0.24 [

The source of the kinetic energy to be dissipated through the empirical modes is considered as the local steady-flow kinetic energy, u^{2}/2, at the normalized vertical distance, η = 3.0 in the x-y plane boundary layer. This kinetic energy is assumed to be distributed over the three fluctuating velocity components. The fraction of kinetic energy in the x-direction velocity component is denoted as κ x , the fraction of kinetic energy in the y-direction velocity component is denoted as κ y and the fraction in the z-direction velocity component is 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, ξ j , times the intermittency exponent for that mode, ξ φ [

The empirical intermittency exponent for each of the empirical modes has been obtained from the expression (Equation (16)) given by Arimitsu and Arimitsu [

Initial conditions: Set number | Intermittency exponent: Average value | Reference intermittency value | Reference number |
---|---|---|---|

5 | 0.3487 | 0.345 | [ |

6 | 0.2377 | 0.238 | [ |

7 | 0.2313 | 0.235 | [ |

8 | 0.2267 | ||

9 | 0.2502 | 0.259 | [ |

10 | 0.2187 | ||

14 | 0.2636 | ||

15 | 0.2647 | 0.260 | [ |

16 | 0.2380 | 0.240 | [ |

available in each of the empirical modes within the non-equilibrium ordered regions, and the fraction of the energy in each of the empirical modes that dissipates into background thermal energy, thus increasing the thermodynamic entropy. Concepts from non-equilibrium thermodynamics are used to calculate the dissipation process for the ordered regions as a general relaxation process. This is considered in the next section.

de Groot and Mazur (pp. 221-230 [

∂ s ∂ t = − J ( x ) ∂ μ ( x ) ∂ x . (17)

Here, s is the entropy per unit mass, μ is the mechanical potential for the transport of the ordered regions in an external context and J(x) is the flux of kinetic energy through the ordered regions available for dissipation into thermal internal energy. The dissipation of the ordered regions into background thermal energy may be considered as a two-stage process from the transition of the ordered regions into equilibrium thermodynamic states and a turn-over process of the downstream velocity in the initial state to the final equilibrium state of the velocity over the internal distance x. The local boundary layer steady state velocity is written as u = u 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 (in Joules/(m^{3}∙K∙s)) through the non-equilibrium ordered regions is then written as [

( d S d t ) g e n = ρ [ ( 1 u e 2 2 T ) ( f ′ ) 2 { κ x ( ∑ j = 1 16 ξ j ζ j ) x + κ y ( ∑ j = 1 16 ξ j ζ j ) y + κ z ( ∑ j = 1 16 ξ j ζ j ) z } ( u e x ) ] . (18)

In this expression, ρ is the density of the working substance, T is the temperature and u_{e} is the free stream velocity. The dissipation rate for each of the three fluctuating spectral velocity components is included in Equation (18).

The kinetic energy in each spectral mode available for final dissipation into equilibrium internal energy is computed for each of the spectral peaks shown in

There are two significant issues with the computational procedure and the results reported in this article. First, there have been no comparable computational results which would validate the procedures adopted here. Second, experimental validation is sparse and applies only to selected aspects of the computational approach and the results. However, the computational procedure is innovative in that it provides a method for the incorporation of a deterministic set of equations for the development of instabilities within the steady state environment of a three-dimensional laminar boundary layer flow. The results indicate that the entropy generation rates resulting from nonlinear interactions in a three-dimensional laminar boundary-layer flow are significantly affected by the particular initial conditions that are applied to the longitudinal vortex structure and the adjacent laminar boundary layer flow.

The counter clockwise rotating longitudinal vortex structure creates a viscous boundary layer along the z-y plane of the flow configuration. This viscous boundary layer is orthogonal to the laminar boundary layer in the x-y plane in the stream wise direction. It is shown that this nonlinear interaction creates instabilities within the three-dimensional flow configuration.

The computational results reported here for the entropy generation rates for a helium mixture boundary layer flow are obtained at the stream wise location of x = 0.120, in the range of stream wise locations from x = 0.06 to x = 0.18, for the normalized vertical station of η = 3.00. The weighting factor K in Equation (5) has been found to yield the prediction of instabilities for a value of K = 0.05. In an experimental investigation of laminar boundary layer receptivity to surface mass injection, Sengupta (pp. 158-170 [

Free stream turbulence levels provide the turbulent kinetic energy that enters the wall shear layer. However, this level is attenuated within the layer and only a portion is available to serve as the initial conditions for the solution of the modified Lorenz equations describing the development of instabilities within the layer. Schmid and Henningson (pp. 430-436 [

In the study reported here, we have modeled the interaction of a longitudinal vortex structure with a stream wise developing boundary layer through the time dependent modified Lorenz equations and have applied a range of initial conditions to the solutions of these equations.

Fluctuating spectral velocity components are found within the time-series solutions for the modified Lorenz equations. Statistical processing of the solutions indicates the presence of ordered regions embedded within the nonlinear time-series solutions. The dissipation of these ordered regions into equilibrium thermodynamic states yields the entropy generation rates for the three-dimensional interaction flow environment. Significant entropy generation rates are predicted for the specified sets of initial conditions applied to the solutions. The results for these entropy generation rates indicate a strong correlation of the entropy results with the levels of the applied initial conditions and that a pattern emerges as a function of the increasing levels of the initial condition equivalent turbulence intensities.

The sensitivity to initial conditions of the Lorenz format spectral velocity equations may provide a means of connecting the incorporation of these time dependent spectral equations in the computational procedure with the concept of bypass transition of the boundary layer flow due to outside disturbances.

The flow configuration of a longitudinal vortex structure and an adjacent laminar boundary layer for a helium mixture flow provides a three-dimensional nonlinear interaction of laminar boundary layers. In the study reported in this article, the velocity fluctuations produced by the instabilities that occur in such an interaction have been modeled through the transformation of the Townsend equations for the velocity fluctuations into a set of nonlinear deterministic Lorenz equations for the spectral velocity components of the fluctuations in the time frame.

The steady boundary-layer velocity profiles in the longitudinal-normal plane and the span wise-normal plane are computed using well-established numerical methods and serve as control parameters for the solutions of the modified Lorenz equations for the time-dependent spectral velocity components. It is shown that the solutions of the nonlinear deterministic modified Lorenz equations are strongly dependent on the values for the initial conditions applied to the solutions. Eighteen sets of initial conditions are examined for the resulting effects on the final values for the computed entropy generation rates.

Power spectral density analysis of the nonlinear time series indicates the presence of sixteen ordered spectral velocity modes within the time series data. Integration over each of these modes provides the ordered energy available within that mode for dissipation into background thermal energy, or increase of entropy. From the singular value decomposition of the time series data, the fraction of kinetic energy in each of the sixteen modes yields a corresponding value of empirical entropy for that mode. The value of empirical entropy yields the value for the Tsallis empirical entropic index, from which the corresponding intermittency exponent for each mode is obtained. Combining these results yields the expression of the total entropy generation rate for all of the ordered modes.

The entropy generation rates through the dissipation of these ordered regions are computed for eighteen sets of initial conditions for the given helium boundary layer flow environment. Strong correlation is found between the predicted entropy generation rates and the initial conditions applied for the solutions of the modified Lorenz equations, both in amplitudes of the generation rates and the emergent of significant patterns of the entropy generation rates as a function of the intensity levels of the applied initial conditions.

These results offer a deterministic path to the understanding of bypass transition and a foundation for the development of an understanding of the dynamics of turbulent spots in the transition from laminar to turbulent flows.

The author declares no conflict of interest.

Isaacson, L.K. (2018) Entropy Generation through the Interaction of Laminar Boundary-Layer Flows: Sensitivity to Initial Conditions. Journal of Modern Physics, 9, 1660-1689. https://doi.org/10.4236/jmp.2018.98104

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

b_{n}: Coefficient in modified Lorenz equations defined by Equation (9)

F: Time-dependent feedback factor

h: Integration time step (s)

j: Mode number empirical eigenvalue

J: Net source of kinetic energy dissipation rate, Equation (17)

k: Time-dependent wave number magnitude

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

K: Adjustable weighting factor

n: Stream wise station number

P: Local static pressure (N∙m^{−2})

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

r_{n}: Coefficient in modified Lorenz equations defined by Equation (8)

s: Entropy per unit mass (J∙kg^{−1}∙K^{−1}))

s_{n}: Coefficient in modified Lorenz equations defined by Equation (8)

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

S ˙ g e n : Entropy generation rate through kinetic energy dissipation (J∙m^{−3}∙K^{-1}∙s^{−1})

t: Time (s)

u: Mean stream wise velocity in the x-direction in Equation (4)

u’: Fluctuating stream wise velocity in Equation (4)

u_{e}: Stream wise 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 Lorenz equations

w_{e}: Span wise velocity at the outer edge of the z-y plane boundary layer

x: Stream wise distance

x_{i}: i-th direction

x_{j}: j-th direction

y: Normal distance

z: Span wise distance

Greek Letters

δ: Boundary layer thickness (m)

δ_{lm}: Kronecker delta

ζ j : Intermittency exponent for the j-th mode in Equation (18)

η: Transformed normal distance parameter

η_{¥}: Transformed outer edge of the boundary layer

κ x : Fraction of kinetic energy in the stream wise component

κ y : Fraction of kinetic energy in the normal component

κ z : Fraction of kinetic energy in the span wise component

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

μ: Mechanical potential in Equation (17)

v: Kinematic viscosity of the gas mixture (m^{2}∙s^{-1})

ξ j : Kinetic energy in the j-th empirical mode

ρ: Density (kg∙m^{−3})

σ_{y}: Coefficient in modified Lorenz equations defined by Equation (7)

σ_{x}: Coefficient in modified Lorenz equations defined by Equation (7)

τ_{w}: Wall shear stress (N∙m^{−2})

Subscripts

e: Outer edge of the laminar 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