Received 20 June 2014; revised 23 July 2014; accepted 2 August 2014

ABSTRACT

Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with time-dependent structural coefficients are developed to examine kinematic and dynamic problems for a deterministic chaos of N stochastic waves in the two-dimen- sional theory of the Newtonian flows with harmonic velocity. The Dirichlet problems are formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains for stochastic waves vanishing at infinity. Development of the novel method of solving partial differential equations through decomposition in invariant structures is resumed by using experimental and theoretical computation in Maple™. This computational method generalizes the analytical methods of separation of variables and undetermined coefficients. Exact solutions for the deterministic chaos of upper and lower cumulative flows are revealed by experimental computing, proved by theoretical computing, and justified by the system of Navier-Stokes PDEs. Various scenarios of a developed wave chaos are modeled by 3N parameters and 2N boundary functions, which exhibit stochastic behavior.

Keywords:

Stochastic Waves, Invariant Structures, Experimental Computation, Theoretical Computation

1. Introduction

The two-dimensional (2d) Navier-Stokes system of partial differential equations (PDEs) for a Newtonian fluid with a constant density and a constant kinematic viscosity in a gravity field is

(1)

(2)

where is a vector field of the flow velocity, is a vector field of the gravitational acceleration, is a scalar field of the total pressure, and are the gradient and the Laplacian in the 2d Cartesian coordinate system of the three-dimensional (3d) space with unit vectors, respectively, and t is time.

By a flow vorticity of the velocity field

(3)

Equation (1) may be written into the Lamb-Pozrikidis form [1] [2]

(4)

which sets a dynamic balance of inertial, potential, vortical, and viscous forces, respectively.

Using a dynamic pressure per unit mass [3]

(5)

where is a reference pressure, a kinetic energy per unit mass the 2d Helmholtz decomposition [4] of the velocity field

(6)

and the vortex force

(7)

Equation (4) is reduced to the Lamb-Helmholtz PDE [5]

(8)

for a scalar Bernoulli potential and a vector Helmholtz potential, respectively,

(9)

(10)

where and d are scalar potentials, and are vector potentials, and b are pseudovector potentials of and, respectively. The Lamb-Helmholtz PDE (8) means a dynamic balance between potential and vortical forces of the Navier-Stokes PDE (1), which are separated completely. Reduction of (1) to (8) means the potential-vortical duality of the Navier-Stokes PDE since writing equation (8) as

(11)

shows that a virtual force of (1) may be represented both in the potential form and the vortical form.

The exponential Fourier eigenfunctions obtained by the classical method of separation of variables of the 2d Laplace equation in [1] and [4] were primarily used for a linear part of the kinematic problem for free-surface waves of the theory of the ideal fluid with in [6] . This analytical method was recently developed into the computational method of solving PDEs by decomposition into invariant structures. Topological flows away from boundaries were computed by the Boussinesq-Rayleigh-Taylor structures in [3] . Spatiotemporal cascades of exposed and hidden perturbations of the Couette flow were modeled by the trigonometric Taylor structures and the trigonometric-hyperbolic structures, respectively, in [7] . Dual perturbations of the Poiseuille-Hagen flow were treated by the invariant trigonometric, hyperbolic, and elliptic structures in [8] . Exact solutions for the conservative interaction of N internal waves were recently obtained by experimental and theoretical computing with kinematic Fourier (KF) structures with space-dependent structural coefficients and exponential kinematic Fourier (KEF) structures, dynamic exponential Fourier (DEF) structures, and KEF-DEF structures with constant structural coefficients in [5] .

To examine linear and nonlinear parts of kinematic and dynamic problems for 2d stochastic waves in the theory of Newtonian flows with harmonic velocity, the KEF structures, the DEF structures, and the KEF-DEF structures with time-dependent structural coefficients are developed in the current paper. The structure of this paper is as follows. The KEF structures are used to compute theoretical solutions of the kinematic problems for the velocity components and the dual potentials of the velocity field in Section 2. The KEF and KEF-DEF structures are employed for theoretical computation of the dynamic problems for the Helmholtz and Bernoulli potentials, the kinetic energy, and the total pressure in Section 3. Verification of the experimental and theoretical solutions is also provided in Section 3. Various scenarios of a developed wave chaos are treated in Section 4. A summary of main results is given in Section 5.

2. Kinematic Problems for Internal Waves

The following solutions and admissible boundary conditions for the kinematic problems of Section 2 in the KEF and DEF structures with time-dependent coefficients were primarily computed via experimental programming techniques, which use lists of equations and expressions of Maple™ in the virtual environment of a global variable Eqs with 25 procedures of 600 code lines in total.

2.1. Formulation of Theoretical Kinematic Problems for the Velocity Field

Theoretical kinematic problems for harmonic velocity components and of a cumulative flow of a Newtonian fluid are given by vanishing the y-component of the vorticity Equation (3) and the continuity Equation (2), respectively,

(12)

To consider a deterministic chaos of N internal, stochastic waves, the cumulative flow is decomposed into a superposition of local flows

(13)

such that the local vorticity and continuity equations are

(14)

where

An upper cumulative flow is specified by the Dirichlet condition in the KF structure on a lower boundary of an upper domain and (see Figure 1).

(15)

and a vanishing condition as

(16)

A lower cumulative flow is identified by the Dirichlet condition on an upper boundary of a lower domain and (see Figure 1).

(17)

and a vanishing condition as

(18)

Thus, an effect of surface waves on the internal waves is described by the Dirichlet conditions (15) and (17). Here, a structural notation

(19)

is used for kinematic structural functions and, where and are time-dependent boundary functions, is an argument of the kinematic and dynamic structural functions, is a propagation variable, is a wave number, is a celerity, and is an initial coordinate for all n. Similar to [5] , boundary conditions for are then redundant since boundary parameters of depend on boundary parameters of for the upper and lower flows, respectively, as

(20)

(21)

Similarly to w, u vanishes as

(22)

for the upper and lower cumulative flows, respectively.

2.2. Theoretical Solutions for the Velocity Field

Theoretical solutions of kinematic problems (12)-(18) are constructed in the KEF structure of two spatial variables x, u and time t with a general term, which in the structural notation may be written as

(23)

where signs “−” and “+” of the exponential term refer to the upper and lower flows, respectively, first letters f and g of structural coefficients and refer to the kinematic structural functions, and a second letter to the expanded variable p. General terms of the velocity components of the upper and lower flows become, respectively,

(24)

(25)

It may be shown that spatial derivatives of are

(26)

(27)

Application of (26) (27) to (24) (25), substitution in (14), collection of the structural functions, and vanishing their coefficients reduce two vorticity and continuity PDEs to the following system of four vorticity and continuity algebraic equations (AEs) with respect to for the upper flows

(28)

(29)

the lower flows

(30)

(31)

and all

Solving AEs (28) and (30) yields for the upper and lower flows, respectively,

(32)

(33)

Substitution of solutions (32) in (29) and (33) in (31) reduces them to identities, showing that vorticity and continuity AEs (28)-(31) are compatible. Finally, substitution of (32) (33) in the KEF structures (24) (25) and solving the Dirichlet boundary conditions (15) and (17) with respect to and produces velocity components for the upper and lower cumulative flow, respectively,

(34)

while vanishing boundary conditions (16) and (18) are obviously satisfied.

2.3. The DEF Structure and Theoretical Jacobian Determinants of the Velocity Field

Define two KEF structures and with general terms and by using the generalized Einstein notation for summation that is extended for exponents in [5]

(35)

Following [5] , define structural functions, and of the DEF structure

(36)

where capital letters C and S stand for dynamic structural functions cosine and sine, letter a for arguments, , letters s and d for sum and difference of arguments and.

Computation of a general term by summation of diagonal terms yields

(37)

A general term computed by rectangular summation of non-diagonal terms becomes

(38)

By triangular summation, is reduced to

(39)

By (37) and (39), summation formula for the product of the KEF structures may be written as the DEF structure with time-dependent structural coefficients

(40)

with the following structural coefficients:

(41)

where first two letters, and of structural coefficients, and stand for dynamic structural functions, and, respectively, and a third letter for variable p.

Computation of local JDs for the velocity components of the upper and lower flow, respectively, yields

(42)

Thus, velocity components and are independent for non-trivial structural coefficients and since the local JDs vanish when.

Computation of a global JD by using (40) (41) for velocity components of the upper and lower cumulative flows (34) with slant internal waves gives

(43)

So, is a superposition of a propagation JD with general term proportional to, an interaction JD with proportional to, and an interaction JD with proportional to, which describe interaction between parallel and orthogonal internal waves, respectively.

coincides with local JDs (42). They describe propagation of internal waves and vanish only for vanishing waves with. vanishes for parallel waves with

(44)

Global JD (43) then becomes

(45)

Thus, the global JD does not vanish for parallel waves with non-vanishing.

vanishes for orthogonal waves with

(46)

In this case, global JD (43) is reduced to

(47)

So, the global JD does not vanish also for orthogonal waves with non-vanishing In the general case of slant internal waves (43), both and are non-vanishing. Therefore, both propagating and interacting waves are independent for structural coefficients with for all n.

2.4. Theoretical Solutions for the Kinematic Potentials in the KEF Structures

Theoretical kinematic problems for cumulative pseudovector potential and cumulative scalar potential of are set by the global Helmholtz PDEs (6)

(48)

(49)

since the potential-vortical duality the velocity field admits two presentations: for and for. The cumulative kinematic potentials are decomposed into a superposition of local kinematic potentials

(50)

such that the local Helmholtz PDEs are

(51)

(52)

where.

Construct general terms of the kinematic potentials of the local flows in the KEF structure with time-depen- dent coefficients

(53)

(54)

Application of (26) (27) to (53) (54), substitution in the Helmholtz PDEs (51) (52), collection of the structural functions, and vanishing their coefficients reduce four Helmholtz PDEs to the following system of eight Helmholtz AEs for the upper flows

(55)

(56)

the lower flows

(57)

(58)

and all, and.

Solving AEs (55) and (57) with respect to time-dependent structural coefficients, and gives for the upper and lower flows, respectively,

(59)

Substitution of solutions (59) in AEs (56) and (58) reduces them to identities. Finally, substitution of structural coefficients (59) in the KEF structures (53) (54) and superpositions (50) yields the cumulative kinematic potentials in the KEF structures for the upper and lower cumulative flows, respectively,

(60)

The theoretical solutions in the KEF and DEF structures for the kinematic problems of Section 2 were computed utilizing theoretical programming methods with symbolic general terms in the virtual environment of a global variable Eqt with 21 procedures of 522 Maple code lines in total. The theoretical formulas for velocity components (34), the products of the KEF structures (40) (41), and the kinematic potentials (60) of the upper and lower cumulative flows were justified by the correspondent experimental solutions for N = 1, 3, 10.

3. Dynamic Problems for Internal Waves

The following solutions for the dynamic problems of Section 3 in the KEF, DEF, and KEF-DEF structures were primarily computed by experimental programming with lists of equations and expressions in the virtual environment of the global variable Eqs with 18 procedures of 470 code lines in total.

3.1. Theoretical Solutions for the Dynamic Potentials in the KEF Structures

Theoretical dynamic problems in the KF structures for the Helmholtz and Bernoulli potentials of the cumulative flows are set by the Lamb-Helmholtz PDEs (8)-(10) in the vortical presentation with

(61)

Equations (61) are complemented by the local Lamb-Helmholtz PDEs

(62)

since the cumulative dynamic potentials are again decomposed into the local dynamic potentials

(63)

Construct a general term of the Bernoulli potential of the local flows in the KEF structure with time-depen- dent coefficients

(64)

Computation of the temporal derivative of, application of (26) (27), substitution in (62), collection of the structural functions, and vanishing their coefficients reduce two Lamb-Helmholtz PDEs to the following system of four Lamb-Helmholtz AEs with respect to and for the upper flows

(65)

(66)

and the lower flows

(67)

(68)

and all variables, parameters, and functions, and.

Solving AEs (65) and (67) for structural coefficients and yields for the upper and lower flows, respectively,

(69)

Substitution of solutions (69) in AEs (66) and (68) reduces them to identities. Eventually, substitution of structural coefficients (69) in the KEF structure (64) and superpositions (63) returns the cumulative dynamic potentials in the KEF structures for the upper and lower cumulative flows, respectively,

(70)

(71)

3.2. Theoretical Solutions for the Total Pressure in the KEF-DEF Structures

Theoretical dynamic problems in the KEF-DEF structures for, , and of the cumulative flows are formulated by definition

(72)

the Bernoulli Equation (9) with

(73)

and the hydrostatic Equation (5)

(74)

where is the reference pressure at

Computation of by (40) (41) and (34) returns

(75)

for the upper and lower cumulative flows, respectively. Substitution of (71), (75), and (73) in (74) yields

(76)

for the upper and lower cumulative flows, respectively. So, the kinetic energy is obtained in the DEF structures, the dynamic pressure is expressed in the KEF-DEF structures, and the total pressure is computed in the KEF-DEF and polynomial structures.

3.3. Harmonic Relationships between the Kinematic and Dynamic Variables

Similar to the invariant trigonometric, hyperbolic, and elliptic structures [8] , there are two pairs of independent KEF structures: generating structures with general terms, and complementary structures with general terms, for the upper and lower flows, respectively,

(77)

(78)

Expressing velocity components (34), kinematic potentials (60), and dynamic potentials (70) (71) through the generating and complementary structures (77) (78) and solving for and gives algebraic relationships

(79)

Taking derivatives of (34) and (60) with respect to x, z and solving for and yields differential relationships, which extend algebraic ones (79),

(80)

In fluid dynamics, relationships (79) (80) mean that a harmonic flow, which is non-uniform in or directions, produces a complementary flow in z- or x-directions, respectively.

Computing velocity components (34) and dynamic potentials (70) (71) through the generating and complementary structures (77) (78), taking temporal and spatial derivatives, and solving for and returns

(81)

Differential relationships (81) mean that spatial derivatives of the dynamic potentials generate temporal rates of a harmonic flow.

By the following substitutions:

(82)

both local and global vorticity and continuity Equations (12) and (14), Helmholtz Equations (48)-(49) and (51)- (52), and Lamb-Helmholtz Equations (61) and (62), respectively, are reduced to the Cauchy-Riemann equations

(83)

for conjugate functions and [4] . All conjugate functions have orthogonal isocurves, due to the vanishing scalar product of gradients,

(84)

and vanishing Laplacians,

(85)

Thus, , and are six pairs of harmonic functions with orthogonal isocurves.

3.4. Theoretical Verification by the System of Navier-Stokes PDEs

The system of the Navier-Stokes PDEs (1)-(2) in the scalar notation becomes

(86)

(87)

Computation of spatial derivatives of (34) by (26)-(27) reduces (87) to identity both for the upper and lower cumulative flows.

Temporal derivatives of in the KEF structures for the upper and lower cumulative flows, respectively, are

(88)

(89)

The directional derivatives of (86) computed by (40)-(41) in the DEF structures for the upper and lower cumulative flows, respectively, become

(90)

(91)

By using (26)-(27), components of the gradient of (76) may be written in the KEF-DEF structures for the upper and lower cumulative flows, respectively, as

(92)

(93)

Substitution of Equations (88)-(93) and (85) for u and w in (86) reduces then to identities. Thus, Equations (34) and (76) constitute exact solutions in the KEF, DEF, and KEF-DEF structures with time-dependent structural coefficients for deterministic chaos of N stochastic waves both in the upper and lower domains.

The theoretical solutions in the KEF, DEF, and KEF-DEF structures for the dynamic problems of Section 3 were computed by theoretical programming methods with symbolic general terms in the virtual environment of the global variable Eqt with 14 procedures of 410 code lines in total. The theoretical solutions for Helmholtz and Bernoulli potentials (70)-(71), total pressure (76), temporal derivatives (88)-(89), directional derivatives (90)- (91), and pressure gradient (92)-(93) of the upper and lower cumulative flows were justified by the correspondent experimental solutions for N = 1, 3, 10.

All kinematic solutions of Section 2 and dynamic solutions of Section 3 with time-dependent structural coefficients are reduced to the correspondent solutions of [5] when and.

4. Chaotic Scenarios

Boundary functions and, which are shown in Figure 2 for, are constructed from stochastic solutions of the Navier-Stokes equation in one dimension by the following method. Let and be the stochastic solutions of

(94)

where is the Reynolds number.

In agreement with [7] , is composed of an invariant structure with even indices of hyperbolic secant-tangent temporal modes and trigonometric sine spatial modes (HETO structure) and an invariant structure of the same temporal modes combined with trigonometric cosine spatial modes (HETE structure)

(95)

is formed of an invariant structure with odd indices of hyperbolic secant-tangent temporal modes and trigonometric sine spatial modes (HOTO structure) and an invariant structure of the same temporal modes composed with trigonometric cosine spatial modes (HOTE structure)

(96)

Here, a structural notation

(97)

is used for structural functions of temporal and spatial modes, where is an index of spatiotemporal components, is an index of temporal modes, is an index of spatial modes, is a frequency, is a delay parameter, , and are spatial amplitudes. Structural coefficients , and are computed by initialization conditions

(98)

and recurrent relations

(99)

In Figure 2, stochastic boundary functions

(100)

were evaluated for, , , temporal delays, a scale of temporal delays, , scales of spatiotemporal components

(101)

and spatial amplitudes

(102)

which insure convergence of the spatial modes. Frequencies of:

and frequencies of:

were computed by iterations from tolerance equations

(103)

These equations guarantee that absolute values of remainders of the structural approximations of (94)

(104)

(105)

do not exceed for all, and, where is an admissible error of computation of an invariant structure. Chaotic behavior of the boundary functions increases with n, K, and, especially, , while various chaotic scenarios are modeled by parameters, , , , , , , and.

The KEF structures and KEF-DEF structures are visualized in Figure 3 by instantaneous 3d surface plots with isocurves at for scalar potential (60) and the dynamic pressure

with p_t from (76), respectively. Local minimums of the KEF-DEF structure for p_d coincides with local maximums of the DEF structure for, in agreement with [1] . The spatial structure of and p_d for stochastic waves coincides with that shown in Figure 2 and Figure 3 of [5] for conservative

waves. The temporal structure of and p_d for stochastic waves, which is visualized by animations, significantly differs from that of and p_d for conservative waves because of the chaotic behavior of boundary functions and.

5. Conclusions

The computational method of solving PDEs by decomposition in invariant structures, which continues the analytical methods of separation of variables and undetermined coefficients, is generalized in the current paper at the KEF, DEF, and KEF-DEF structures with time-dependent coefficients. This computational method is implemented in the kinematic and dynamic problems for internal waves by 43 procedures with 1070 code lines of the experimental computing in total and 35 procedures with 932 code lines of the theoretical computing in total. These structures with time-dependent structural coefficients are invariant with respect to various differential and algebraic operations.

For internal waves vanishing at infinity in the upper and lower domains, the Dirichlet problems are formulated for the kinematic and dynamic systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations. The exact solutions of the Navier-Stokes PDEs for the deterministic chaos of N stochastic waves are revealed experimentally, proved theoretically, and justified by the system of Navier-Stokes PDEs in the class of flows with the harmonic velocity field. The kinematic and dynamic solutions for stochastic waves coincide with the correspondent solutions for conservative waves [5] when stochastic boundary functions are reduced to constants.

Independence of both propagating and interacting internal waves is shown by computation of the Jacobian determinants in the DEF structures. Conditions for existence of parallel and orthogonal waves with time-de- pendent amplitudes are obtained through the Jacobian determinants, as well. The harmonic relationships between six pairs of the harmonic, fluid-dynamic variables, their temporal derivatives, and their spatial derivatives with respect to x and z are derived both for the upper and lower flows.

The stochastic boundary functions are constructed from the stochastic solutions of the one-dimensional Navier-Stokes equation [7] with hyperbolic temporal modes and trigonometric spatial modes in the HETO, HETE, HOTO, and HOTE structures. Various scenarios of a developed wave chaos are modeled by 3N parameters for internal waves and 2N stochastic boundary functions, which depend on parameters, where K is a number of temporal modes.

Acknowledgements

The author thanks S. P. Bhavaraju for the stimulating discussion at the 2013 SIAM Annual Meeting. Support of the College of Mount Saint Vincent and CAAM is gratefully acknowledged.

References