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A computational procedure is developed to determine initial instabilities within a three-dimensional laminar boundary layer and to follow these instabilities in the streamwise direction through to the resulting intermittency exponents within a fully developed turbulent flow. The fluctuating velocity wave vector component equations are arranged into a Lorenz-type system of equations. The nonlinear time series solution of these equations at the fifth station downstream of the initial instabilities indicates a sequential outward burst process, while the results for the eleventh station predict a strong sequential inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The nonlinear time series solutions indicate regions of order and disorder within the solutions. Empirical entropies are defined from decomposition modes obtained from singular value decomposition techniques applied to the nonlinear time series solutions. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The intermittency exponents are then obtained from the entropic indices for these streamwise stations that indicate the burst and autogeneration processes.

Considerable progress has been made in understanding the basic physical processes underlying the transition of a laminar boundary layer into the turbulent state. Recently, a new approach has been introduced which uses dynamical systems theory in which the system’s trajectory traverses among mutually repelling flow states. Some of these flow states exist on the boundaries between laminar and turbulent states within the flow environment. These boundaries are assumed to be on the edge of the turbulent state, and are therefore called “

However, this model does not indicate the occurrence of boundary layer flow “

In this article, we present a nonlinear, low-dimensional computational model for an incompressible, laminar, three-dimensional boundary layer that should provide both the inherent unstable flow structures as input for the “edge states” studies and the prediction of both “

This article is organized in the following sections: In Section 2, we discuss the boundary layer flow environment that forms the basis for the evaluation of boundary layer deterministic structures. We also list the thermo-transport properties that are used throughout the computational scenario. In Section 3, we assume that our flow environment is a three-dimensional laminar flow over a flat plate with a constant velocity in the streamwise (along the x-axis) direction, with no pressure gradient, and that the flow is incompressible. We compute the laminar boundary layer development from the starting point of the plate, providing a priori the three-dimensional boundary layer velocity profiles at various stations in the streamwise direction (Cebeci and Bradshaw [

The fundamental three-dimensional laminar boundary layer flow environment that provides the control parameters for the study of the generation of nonlinear deterministic boundary layer structures is shown in

These numerical velocity gradient values then serve as input control parameters for the computation of unstable nonlinear deterministic boundary layer structures for each of the streamwise stations. Our objective here is to gain some understanding of how the instabilities are initially generated, and how the developing laminar boundary layer characteristics affect the streamwise unstable deterministic structures along the boundary layer. We are not aware of previously published work delineating these processes.

The computational results for the laminar boundary-layer profiles and the initiation of nonlinear deterministic structures are dependent on the particular value of the kinematic viscosity chosen for the overall computational scenario. It has been found that the adiabatic combustion process of methane with 300 percent theoretical air at atmospheric pressure yields the composition and temperature for this pressure provides the value of kinematic viscosity that yields a prediction of strong deterministic structures. The process of finding these particular values of kinematic viscosity is trial and error and we have not discovered a systematic method for finding these values.

The three-dimensional flat plate boundary layer is approximated by assuming a spatially developing Blasius boundary layer in the downstream direction, yielding the mean velocity profiles in the x–y plane and an imposed spanwise Blasius boundary layer yielding velocity profiles in the z-y plane. The computer programs developed by Cebeci and Bradshaw [

Hansen [

The computational results from the evaluation of both the spatially developing boundary layer and the spanwise boundary layer provide a three-dimensional field of mean values for the three boundary layer velocity component gradients. These gradients serve as control parameters for the modified Townsend equations which are then used to obtain the time-dependent solutions for the fluctuating velocity wave components at each streamwise station along the spatially developing boundary layer. It may be helpful to visualize the three-dimensional field of mean boundary layer velocity gradients as a computational container in which the fluctuating velocity field is placed. The time-dependent fluctuating velocity field then responds to the control parameters provided by the three-dimensional mean field of boundary layer velocity gradients at each downstream station of the spatially developing boundary layer.

We wish to obtain a set of coupled, nonlinear differential equations that will describe the time development of the fluctuating spectral components of velocity within the laminar boundary layer environment. This set of differential equations should be coupled and should retain the nonlinear terms that we anticipate will lead to the time-dependent development of unstable velocity fluctuations. The equations of motion for the boundary layer flow may be separated into steady plus fluctuating values of the velocity components. The equations for the velocity fluctuations may then be written as [

In these equations,

In these expressions, _{i}_{j}

The fluctuating velocity and pressure fields are expanded in terms of sums of Fourier components [

and

Substituting these expressions into _{i}(k

From the general equation for the balance of a transferable property, the rate of change of the wave numbers, _{i}

These two sets of equations will now be written in forms that incorporate the mean boundary layer velocity gradients from the steady state laminar boundary layer solution. Including the gradients of the mean velocities in the x–y and z–y boundary layers in the set of equations for the time-dependent wave number components yields the following set of equations:

We next include the mean laminar boundary layer velocity gradients in the transformed equations for the fluctuating velocity wave components. In addition, we introduce an internal feedback parameter for the coefficients of the nonlinear velocity wave components in

Mathieu and Scott [

as a projection matrix, with a given velocity wave vector component, _{i}_{i}

Manneville [

In this expression, the value of

Including the expressions for the mean boundary layer velocity gradients in

The mean boundary layer velocity gradients are computed for each streamwise station and reflect the changing nature of the boundary layer in the downstream direction. The nonlinear time series solutions of the modified Townsend equations are then obtained for each downstream station reflecting the effects of the changing boundary-layer flow. The nonlinear time series for a particular streamwise boundary layer station yield the time dependent values of the local wave number components and the fluctuating velocity wave components at that location. Each of the length parameters is non-dimensionalized by the corresponding plate length that is taken as one meter. The spatially developing boundary layer is computed for downstream locations in increments of 0.02 (2 cm), with the initial location at 0.02 (2 cm) downstream of the leading edge in the x-direction. Thirteen downstream stations are considered, beginning at the x location of 0.08, denoted as station one, through station thirteen, at a location of x = 0.32 from the origin of the boundary-layer flow. Station one at the location of 0.08 in the x direction is denoted as the transmitter station.

The nonlinear equations for the time-dependent velocity wave components are written for the initial (transmitter) station as [

These equations are written in a Lorenz format. However, the coefficients are dependent on both the time-dependent wave number components and the steady boundary-layer velocity gradients for the initial transmitter station. The coefficients are written in the following form [

In our development, the nonlinear coupling for the normal and spanwise velocity wave components is modified by the internal feedback parameter (1–F). This factor is applied to the nonlinear terms, rather than to one of the individual variable terms.

The Lorenz system of equations has been shown to possess the characteristic of enhancing meaningful signals from a time series masked by chaotic signals [

The nonlinear equations for the time-dependent velocity wave components are applied at the initial station, denoted as the transmitter station, and to each of the downstream stations. Each of these downstream stations is denoted as a receiver station, following the nomenclature of communications theory. The time-dependent wave number components and the various boundary layer coefficients are computed for each of the receiver stations in the same manner as for the transmitter station. The output for the streamwise velocity wave vector component from the transmitter station is used as input for the streamwise velocity wave component in the nonlinear terms in the equation set at the first downstream receiver station. The input to the nonlinear coupled terms at the next downstream receiver station consists of the sum of the streamwise velocity wave component signal from the transmitter station plus the streamwise velocity wave component signal from the previous receiver station. The sum of the streamwise velocity wave components from each of the previous stations then serves as the input for the nonlinear terms in each of the subsequent station solutions.

For the nth receiver station, the nonlinear equations for the time-dependent velocity wave components are written as:

In these expressions, the input signal carrying the information from the initial transmitter station and the subsequent receiver stations to the

The computational scenario includes the solution of six simultaneous first-order differential equations, solved at each streamwise station distributed along the x-axis. The three equations for the wave number components,

The solution of the equations for the velocity fluctuations yields the fluctuating velocity wave components for the given normal distance from the surface at the each of the streamwise stations along the x-axis. The spanwise station of z = 0.003, is held constant for each streamwise station. The Falkner-Skan transformation for the laminar boundary layer normal distance is defined as [

It has been found that boundary-layer instabilities occur primarily within the boundary layer at a normal distance of _{e} = 8.00. The initial conditions for the wave number components are _{x1}_{y1}_{z1}_{x1}_{y1}_{z1}

Press [

The computational procedure is initiated at the first or transmitter station, and then repeated for each subsequent streamwise or receiver station in the downstream direction. As indicated in

The streamwise velocity wave component produced by the receiver station at streamwise station eight (x = 0.22) is shown in

_{z8}_{y8}

The streamwise velocity wave component at the streamwise receiver station thirteen is shown in

As a concluding figure for the nonlinear time series solutions for the spatially developing boundary layer,

One of the specific objectives for this study has been to begin to understand the fundamental thermodynamic characteristics of the deterministic structures predicted within the solutions of the coupled, nonlinear Lorenz-type spectral equations [

We wish to extract the detailed power spectral density over a specified range of the output series data from the nonlinear time series solutions for the fluctuating velocity wave components. Burg developed the maximum entropy method for extracting information contained within seismic data, with one of the first computer programs presented by Chen [

The computational methods presented by Press _{r}

Powell and Percival [

Further thermodynamic insight is needed concerning the distribution of fluctuation kinetic energy across the ordered structures identified by the spectral entropy rates for each of the streamwise stations as the boundary layer develops in the downstream direction. Holmes

The application of the singular value decomposition procedure to a specified segment of the nonlinear time-series solution for the streamwise velocity wave component yields the distribution of the component eigenvalues _{j}_{j}

The application of the singular value decomposition procedure to the specified segment of the nonlinear time-series solution yields the various eigenvalues _{j}

The computational procedure for the empirical entropy by the singular value decomposition method is made up of two parts, the computation of the autocorrelation matrix and the singular value decomposition of that matrix [

The empirical entropies for each of the empirical modes from one to sixteen for streamwise station five are shown in

The empirical entropy is shown as a function of empirical mode number in

The interpretation we make for the eigenvalues _{j}

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 effects of the downstream developing laminar boundary layer on the characteristics of these instabilities indicate the prediction of the

We wish to now explore some speculative thermodynamic concepts which may allow us to extend our computational procedure through the deterministic structures produced by the instability process to a connection with the final stages of dissipation within a fully turbulent flow environment. These two exploratory concepts are the empirical entropic index and the empirical intermittency exponent.

The empirical entropy for the fluctuating streamwise velocity wave component time series indicates different characteristics for the various deterministic regions within the time series. These results indicate that the majority of the kinetic energy in the deterministic structures is contained within the first four or five empirical modes of the singular value decompositions, with low empirical entropy. These structures have been classified as coherent [

In this expression, k is a positive constant, _{i}

It is tempting to apply this expression for the ordered structures as calculated in the previous section on empirical entropy. However, the Tsallis entropic form is applicable to an ensemble of microscopic subsystems, while we are working with a set of individual macroscopic systems spread over a limited number of empirical modes, _{j}_{j}

We have to keep in mind that this parameter does not have a mathematical basis in non-extensive thermo-statistics but is simply an artifact that allows us to include the effects of the nonlinear, non-equilibrium nature of the deterministic structures that we are following. The expression has the format of entropic index; hence, we simply call it an empirical entropic index or simply an entropic index. We have used this expression to extract the empirical entropic index, _{j}_{j} / dt = 0

According to the general evolution criterion for non-equilibrium dissipative processes (Glandsdorff and Prigogine [_{j} / dt < 0_{j} / dt < 0

In the preceding sections of the paper, we have focused primarily on the extraction of thermo-statistic properties from the nonlinear time series solution for the fluctuating velocity wave components in a developing laminar boundary layer along a flat plate surface in a three-dimensional configuration. In this section, we introduce a speculative method to connect the deterministic results for the entropic indices with the turbulent dissipation processes occurring in the downstream fully developed turbulent flow. We explore this computational connection through the concept of turbulent intermittency.

The concept of intermittency arises in the observation that within a fully developed turbulent flow, regions of the dissipation of kinetic energy are interspersed with regions in which the dissipation rate is very low, with the regions separated by distinct boundary surfaces. This observation led to the characterization of the dissipation of turbulent kinetic energy in the inertial range as fractal in nature (Tsallis [

The deterministic structures discussed in previous sections are of a macroscopic nature embedded within the nonlinear time series solution of the nonlinear equations for the fluctuating velocity field. We have found, through the singular value decomposition method, that the four lowest empirical modes contain nearly 99 per cent of the kinetic energy. We have also found that the empirical modes obtained from the singular value decomposition indicate empirical entropic indices. We will therefore, heuristically, apply a relationship, found by Arimitsu and Arimitsu [

Given the absolute value of the empirical entropic index, _{j}

The objective of this article has been to present the results of a computational scenario applied to the spatial development of the fluctuating velocity wave field embedded in a developing laminar boundary layer. These results indicate the production of deterministic structures within the boundary layer consisting of burst, sweep and autogeneration processes. The computational scenario also yields the spectral entropy rates indicating ordered and disordered regions within the time series solutions, the empirical entropy values indicating different characteristics of the deterministic structures within the time series solutions, and the empirical entropic indices for those empirical entropies, together with the resulting intermittency exponents for the respective entropic indices. These deterministic structure characteristics have been computed for a number of streamwise stations in the spatially developing boundary layer.

The equations for the fluctuating velocity components within the boundary layer have been Fourier transformed into a set of three time dependent simultaneous equations for the wave number components and a set of three coupled, nonlinear time dependent equations for the velocity wave components within the boundary layer. The mean velocity gradients computed for the laminar boundary layer at each streamwise station serve as control parameters for the solution of each set of equations at each corresponding streamwise station. The computational procedure yields deterministic ordered regions within the resulting solutions for each streamwise station. The synchronization properties of the Lorenz-type equations are exploited in the computational procedures as we progress from station to station. The initial station is characterized as a transmitter station, with the streamwise velocity wave component output transmitted to the next station, called a receiver station. The sum of the streamwise velocity wave component signals from the transmitter station and the first receiver station then serve as similar input to the next receiver station. This process is repeated for each of the receiver stations in the streamwise direction.

Resulting solutions for the three velocity wave components are presented for several streamwise stations. The transmitter station shows several large-scale aperiodic oscillating patterns for each of the three velocity wave components. The autogeneration process is indicated at station three with an upward flow following at station five. These upward flows, or “

A central theme of our computational scenario is to clarify certain thermodynamic aspects of the deterministic structures occurring within the fluctuating velocity field. The power spectrum of the final portion of the overall time series solutions has been computed using Burg’s maximum entropy method. The spectral entropy rates are then computed for that portion of the solution. Applying the computational technique to the initial burst process at streamwise station five, it has been found that a considerable portion of the spectral entropy rate indicates a significant region of ordered structures within the burst process. A similar computation for the autogeneration process at station thirteen indicates a continuous production of sharply ordered structures.

Application of the singular value decomposition technique to the latter portion of the nonlinear time series solutions provides values for the distribution of eigenvalues over a series of empirical decomposition modes. The empirical entropy, defined as the negative natural logarithm of the eigenvalue for each empirical mode, yields a distribution of empirical entropies as a function of the distribution of empirical modes. These results show that for the initial mode numbers, the empirical entropies are near zero, indicating closely ordered behavior. These results also indicate that for a significant range of empirical modes over the high mode numbers, the empirical entropy is near the maximum value.

To further clarify the nature of the ordered and disordered structures, the empirical entropic index is computed from the given values of empirical entropy for each empirical mode at streamwise stations five and thirteen. The empirical entropic index has a value of zero for the initial empirical mode for each of the streamwise stations. We interpret a zero value for the entropic index to indicate an ordered flow with a significant level of directed kinetic energy. This type of flow structure is simply a directed kinetic energy corresponding to a mean streamwise flow component, not of a vortex or similar motion. However, the strong negative entropic indices at empirical mode two for station five and empirical mode five at station thirteen indicate the prediction of significantly ordered structures of a dynamic nature. These structures may appear as deterministic dissipative vortices with negative absolute temperature, which eventually decay to background internal energy with high entropy.

To complete our computational scenario, we compute the intermittency exponents associated with the particular empirical entropic indices across the empirical mode numbers for streamwise stations five and thirteen. These stations indicate the first burst process and the autogeneration process following the burst and sweep processes. For both stations, a peak of intermittency exponent is indicated for empirical modes three at both stations five and thirteen. These sharp values of intermittency could be associated with the “

This article has presented the results of a computational scenario for the fluctuating velocity field embedded in a streamwise laminar boundary layer that yields the prediction of deterministic structures developing within the boundary layer flow. The time integration of the set of time dependent spectral wave number components and the set of time dependent spectral velocity wave components at the fifth streamwise station indicates an outward burst process, while the results for the eleventh station predict a strong inward sweep process. The results for the thirteenth station indicate a return to the original instability autogeneration process. The computational procedure has thus evaluated the initiating boundary layer instabilities, and sequentially predicted the bursting process, the sweep process and the return of the flow to the generation of internal boundary layer instabilities. Spectral entropy rates, obtained by the application of Burg’s maximum entropy method, indicate regions of significant ordered structures ahead of the initial bursting process. When the flow returns to the autogeneration region, the spectral entropy rates indicate the continuous generation of low intensity ordered and disordered regions. The singular value decomposition technique applied to specified segments of the solutions yields empirical entropies over decomposition modes. Empirical entropic indices are obtained from the empirical entropies for two streamwise stations. The entropic indices extracted from the empirical entropies indicate zero values for the first empirical modes and then negative values over the remaining modes. These results indicate that the deterministic structures are non-equilibrium and dissipative in nature. The intermittency exponents are obtained from the entropic indices for the streamwise burst and autogeneration stations. These intermittency exponents closely match the corresponding exponents for the dissipation of kinetic energy in the fully developed regions of turbulent flow. We thus have a computational scenario that evaluates the initiation of instabilities through nonlinear interactions within a three-dimensional laminar boundary layer and then follows the process through to the evaluation of the connecting link to the dissipation of kinetic energy within the fully developed turbulent flow.

The author would like to thank the editor and referees for their timely and helpful suggestions. The editorial assistance of the editors in improving the manuscript is very much appreciated. The comments and observations of the reviewers have clarified several technical aspects of this exploratory study for which the author expresses his thanks and appreciation.

The author declares no conflicts of interest.

_{i}

Fluctuating

Fluctuating Fourier component of the static pressure

_{1}

Coefficient in modified Townsend equations defined by

_{n}

Coefficient in modified Townsend equations at station

Time-dependent feedback factor

_{r}

Power spectral density of the

Mode number empirical eigenvalue

Spectral entropy segment number

Time-dependent wave number magnitude

Dimensional constant,

_{i}

Fluctuating

Adjustable weighting factor

Time step number

Local static pressure

_{1}

Static pressure in the boundary layer channel

_{i}

Probability of being in a state

_{r}

Probability value for the power spectral density of the

Tsallis nonextensive entropic index

_{j}

Empirical entropic index for the empirical entropy of mode,

_{1}

Coefficient in modified Townsend equations defined by

_{n}

Coefficient in modified Townsend equations at station

_{1}

Coefficient in modified Townsend equations defined by

_{n}

Coefficient in modified Townsend equations at station

_{j}

Empirical entropy for empirical mode,

_{j}_spent

Spectral entropy rate for the

_{q}

Tsallis entropy,

Time

_{1}

Combustion process input temperature

_{aft}

Adiabatic flame temperature

_{e}

Streamwise velocity at the outer edge of the x-y plane boundary layer

_{i}

The

_{i}

Mean velocity in the

_{x}

Mean normal velocity in the x-y plane in the modified Townsend equations

_{z}

Mean normal velocity in the z-y plane

_{e}

Spanwise velocity at the outer edge of the z-y plane boundary layer

Mean velocity in the spanwise direction

Total number of microscopic states in a system,

Streamwise distance

_{i}

_{j}

Normal distance

Spanwise distance

Intermittency exponent

_{lm}

Kronecker delta

Transformed normal distance parameter

_{j}

Eigenvalue for the empirical mode,

_{1}

Kinematic viscosity of the gas mixture

Density

_{y1}

Coefficient in modified Townsend equations defined by

_{x1}

Coefficient in modified Townsend equations defined by

Outer edge of the x-y plane boundary layer

Tensor indices

Component in the x-direction

Component in the y-direction

Component in the z-direction

The laminar three-dimensional boundary-layer environment for the computation of the burst and sweep process is shown. Computations are made at each of eleven streamwise stations, from 0.08 to 0.32 along the x-axis. Boundary-layer profiles are shown in the x–y and z–y planes.

Shown is the streamwise velocity wave vector component, _{x5}

The phase plane, _{x5}a_{y5}

The phase plane, a_{z5} - a_{y5}, for the output of the fifth station is shown.

Shown is the streamwise velocity wave component predicted at streamwise station eight as a function of time step.

The phase plane, _{x8} - a_{y8}

The phase plane, _{z8}_{y8}

Shown is the streamwise velocity wave component for the output at streamwise receiver station thirteen as a function of the integration time step.

The streamwise velocity wave component versus the normal velocity wave component is shown for streamwise receiver station thirteen.

The spanwise velocity wave component

The spectral entropy rate for the streamwise velocity wave component at downstream receiver station five is presented as a function of the spectral segment number.

The spectral entropy rate for the streamwise velocity wave component at downstream receiver station thirteen is presented as a function of the spectral segment number.

The empirical entropy is shown as a function of the empirical mode number at streamwise station five (x = 0.16), at the initiation of the “

The empirical entropy is shown as a function of the empirical mode number at streamwise station thirteen (x = 0.32), for the return to the “

The empirical entropic index is shown as a function of the mode number at streamwise station five (x = 0.16).

Shown is the empirical entropic index as a function of the mode number at streamwise station thirteen (x = 0.32).

The intermittency exponents are shown at streamwise station five (x = 0.16).

The intermittency exponents are shown for streamwise station thirteen (x = 0.32).

Thermodynamic and transport properties used for the flow environment.

Title | Title |
---|---|

Inlet temperature, _{1} |
298.15 K |

Adiabatic flame temperature, |
1140.0 K |

Static pressure, _{1} |
1.013 × 10^{5} N/m^{2} |

Kinematic viscosity, _{1}: |
1.523 × 10^{−4} m^{2}/s |