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Tensorial Extensions of Tunnel Mathematics for the
Navier-Stokes Equations
Rˆomulo Damasclin Chaves dos Santos
Technological Institute of Aeronautics
romulosantos@ita.br
February 4, 2025
Abstract
This paper extends the tensorial foundation of tunnel mathematics, introducing
a refined framework for incorporating tensor analysis into the study of the Navier-
Stokes equations. By developing a rigorous mathematical formulation, we demon-
strate how spatial complex numbers can be applied to fluid dynamics, particularly in
the context of turbulence modeling. We derive new spatial Cauchy-Riemann con-
ditions and establish their connection with turbulence models, providing a more
precise bridge between tunnel mathematics and classical fluid mechanics. Addi-
tionally, we explore higher-order tensor structures that extend the conventional
spatial complex number framework, offering new insights into flow instability and
turbulence energy cascades. The results highlight the potential of this approach
for advancing our understanding of turbulent flows and improving the accuracy of
turbulence models.
Keywords: Cauchy-Riemann Conditions. Fluid Dynamics. Energy Spectrum.
Navier-Stokes Equations. Tensor Analysis.
Contents
1 Introduction 2
2 Spatial Complex Numbers and Tensor Extensions 2
3 Derivation of Extended Spatial Cauchy-Riemann Conditions 3
4 Application to Navier-Stokes Equations 4
5 New Theoretical Advances in Turbulence Modeling 5
5.1 Turbulence Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 6
5.2 Derivation of the Energy Spectrum . . . . . . . . . . . . . . . . . . . . . 6
5.3 Implications for Turbulence Modeling . . . . . . . . . . . . . . . . . . . . 7
1
6 Results 7
6.1 Extended Spatial Cauchy-Riemann Conditions . . . . . . . . . . . . . . . 7
6.2 Tensorial Formulation of Navier-Stokes Equations . . . . . . . . . . . . . 7
6.3 Turbulence Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Conclusion 8
A Mathematical Background on Spatial Complex Numbers and Tensor
Analysis 10
A.1 Properties of Spatial Complex Numbers . . . . . . . . . . . . . . . . . . . 10
A.2 Tensor Analysis and Spatial Complex Numbers . . . . . . . . . . . . . . 10
A.3 Derivation of Extended Spatial Cauchy-Riemann Conditions . . . . . . . 11
A.4 Fourier Transform and Energy Spectrum . . . . . . . . . . . . . . . . . . 11
1 Introduction
The theory of functions of a complex variable has played a fundamental role in classi-
cal fluid mechanics, particularly in the description of incompressible and laminar flows.
However, its applicability has been largely confined to two-dimensional problems. Tunnel
mathematics seeks to extend these ideas into three-dimensional space using spatial com-
plex numbers, as initially explored by Shvydkyi [5]. This paper deepens the mathematical
foundation of this theory by developing advanced tensorial formulations and proving key
results that further connect tunnel mathematics to the Navier-Stokes equations.
Building upon the foundational work in tensor analysis by McConnell [1], we introduce
new derivations of spatial Cauchy-Riemann conditions that ensure analyticity in three-
dimensional space. These conditions are analogous to those in classical complex analysis,
as discussed by Lavrentiev and Shabat [3], but are extended to handle the additional
complexity of three-dimensional flows. We explore the implications of these conditions
in turbulence modeling, drawing on the seminal work of Kolmogorov [4] on the local
structure of turbulence.
Furthermore, we extend the mathematical structure by incorporating higher-rank
tensors to describe the behavior of velocity fields under perturbations. This extension is
inspired by the comprehensive treatment of fluid mechanics by Landau and Lifshitz [2],
and it allows for a more detailed analysis of flow instability and the transition from laminar
to turbulent motion. The result is a more robust analytical toolset for studying these
phenomena, providing new insights into the energy cascades and dissipation mechanisms
in turbulent flows.
The primary objective of this article is to bridge the gap between the theoretical
framework of tunnel mathematics and its practical application to the Navier-Stokes equa-
tions. By doing so, we aim to offer a more precise and comprehensive understanding of
turbulence, which is a critical area of study in fluid dynamics.
2 Spatial Complex Numbers and Tensor Extensions
We begin by recalling the definition of a spatial complex number. Let Lbe a spatial
complex number defined as:
2
L= Re(L) + ixImx(L) + iyImy(L) + izImz(L),(1)
where Re(L) is the real part, and Imx(L), Imy(L), Imz(L) are the imaginary components
associated with the unit spatial imaginary numbers ix,iy, and izrespectively. These unit
spatial imaginary numbers satisfy the following multiplication rules:
i2
x=i2
y=i2
z=−1, ixiy=iz, iyiz=ix, izix=iy.(2)
To extend this framework, we introduce a second-rank tensor operator Tdefined as:
T=Tij iiij(3)
where Tij is a second-rank tensor that represents the transformation properties of a spatial
complex number under coordinate rotations. The indices iand jrange over {x, y, z}, and
the Einstein summation convention is implied.
The tensor Tij can be expressed in matrix form as:
T=
Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tz z
,(4)
where each element Tij describes how the components of the spatial complex number
transform under rotations.
This tensorial structure allows us to model velocity fields as tensor-valued functions.
Specifically, if vis a velocity field, we can express it in terms of spatial complex numbers
and the tensor operator Tas:
v=T · u,(5)
where uis a vector field represented in the spatial complex number framework. This
formulation captures the full three-dimensional nature of turbulence by incorporating
the interactions between different spatial components.
Furthermore, the tensor Tcan be used to derive higher-order tensor structures that
describe more complex interactions in the velocity field. For example, the third-rank
tensor Tijk can be defined as:
Tijk =Tij Tj kiiijik,(6)
where the indices i, j, k range over {x, y, z}. This higher-order tensor structure provides a
more detailed description of the velocity field under perturbations, allowing for a deeper
analysis of flow instability and turbulence.
3 Derivation of Extended Spatial Cauchy-Riemann
Conditions
In classical complex analysis, the Cauchy-Riemann conditions ensure the analyticity of a
complex function. Specifically, for a function f(z) = u(x, y) + iv(x, y) to be analytic, the
following conditions must hold:
∂u
∂x =∂v
∂y ,∂v
∂x =−∂u
∂y .(7)
3
In the context of spatial complex numbers, we extend these conditions to ensure
analyticity in three-dimensional space. Let L=u+ixv+iyw+iztbe a spatial complex
number, where u, v, w, and tare real-valued functions of the spatial coordinates x, y,
and z. The extended spatial Cauchy-Riemann conditions are derived by requiring that
the spatial complex function Lis analytic. This leads to the following set of partial
differential equations:
∂u
∂x =∂v
∂y =∂w
∂z ,(8)
∂v
∂x =−∂u
∂y ,(9)
∂w
∂x =∂v
∂z ,(10)
∂w
∂y =−∂u
∂z .(11)
These conditions ensure that the spatial complex function Lis differentiable in the
sense of spatial complex analysis. To derive these conditions, consider the spatial complex
derivative of L:
∂L
∂x +ix
∂L
∂y +iy
∂L
∂z .(12)
For Lto be analytic, this derivative must be independent of the direction in which it
is taken. This independence leads to the extended spatial Cauchy-Riemann conditions.
Furthermore, we can introduce a more general formulation involving higher-order
derivatives and tensorial structures. Consider the second-order derivatives of L:
∂2L
∂x2+∂2L
∂y2+∂2L
∂z2= 0 .(13)
This Laplace-like equation ensures that the spatial complex function Lsatisfies a
harmonic condition in three-dimensional space. By incorporating the tensor operator T,
we can express this condition in a more compact form:
T:∇2L= 0 ,(14)
where ∇2is the Laplacian operator and Tis the second-rank tensor operator defined
earlier. This formulation provides a deeper insight into the analyticity and harmonic
properties of spatial complex functions, extending the classical Cauchy-Riemann condi-
tions to three-dimensional space.
4 Application to Navier-Stokes Equations
The incompressible Navier-Stokes equations describe the motion of fluid substances and
are given by:
∂v
∂t + (v· ∇)v=−1
ρ∇p+ν∇2v,∇ · v= 0 ,(15)
4
where vis the velocity field, pis the pressure, ρis the fluid density, and νis the kinematic
viscosity. The continuity equation ∇ · v= 0 ensures the incompressibility of the fluid.
To incorporate the framework of spatial complex numbers and tensor extensions, we
express the velocity field vin terms of spatial complex numbers. Let v=u+ixvx+
iyvy+izvz, where u,vx,vy,vzare the real and imaginary components of the velocity
field in the spatial complex number framework.
Using the tensor operator Tdefined earlier, we reformulate the Navier-Stokes equa-
tions as a tensor differential equation:
T · ∂v
∂t +T · (v· ∇)v=−1
ρT · ∇p+νT · ∇2v.(16)
This formulation allows for a more natural incorporation of vorticity and turbulence
effects. The tensor operator Tcaptures the interactions between different spatial com-
ponents of the velocity field, providing a more comprehensive description of the fluid
motion.
To demonstrate the connection to turbulence modeling, we apply the extended spa-
tial Cauchy-Riemann conditions derived earlier. These conditions ensure the analyticity
of the spatial complex velocity field and provide a framework for studying turbulence.
Specifically, we can show that the vorticity ω=∇ × vsatisfies the following equation:
∂ω
∂t + (v· ∇)ω= (ω· ∇)v+ν∇2ω . (17)
By expressing the vorticity in terms of spatial complex numbers and applying the
extended spatial Cauchy-Riemann conditions, we obtain:
T · ∂ω
∂t +T · (v· ∇)ω=T · (ω· ∇)v+νT · ∇2ω . (18)
This tensorial formulation of the vorticity equation provides a deeper insight into
the dynamics of turbulent flows. Furthermore, by analyzing the energy spectrum of
the velocity field, we can establish an explicit connection to the Kolmogorov-Obukhov
turbulence law. The energy spectrum E(k) in the inertial range of turbulence is given
by:
E(k)∼k−5
3,(19)
where kis the wavenumber. By incorporating the spatial complex number framework
and the tensor operator T, we can derive a more detailed energy spectrum that accounts
for the anisotropy and spatial correlations in the turbulent flow:
E(k)∼k−5
3· F(ix, iy, iz),(20)
where Fis a spatial function characterizing the energy dissipation anisotropy.
5 New Theoretical Advances in Turbulence Model-
ing
In this section, we introduce a novel approach to turbulence modeling based on the
decomposition of the turbulence energy spectrum using spatial complex numbers. This
5
approach provides a more detailed understanding of the energy dissipation mechanisms
in turbulent flows, particularly in the inertial range.
5.1 Turbulence Energy Spectrum
The classical Kolmogorov-Obukhov theory predicts that the energy spectrum E(k) in the
inertial range of turbulence follows a power law:
E(k)∼k−5
3,(21)
where kis the wavenumber. This power law describes the distribution of kinetic energy
across different scales in a turbulent flow.
To incorporate the spatial complex number framework, we decompose the energy
spectrum into components associated with the unit spatial imaginary numbers ix, iy,
and iz. Let F(ix, iy, iz) be a spatial function that characterizes the energy dissipation
anisotropy. This function accounts for the directional dependence of energy dissipation
in the turbulent flow.
The novel turbulence energy spectrum based on spatial complex number decomposi-
tion is given by:
E(k)∼k−5
3· F(ix, iy, iz).(22)
5.2 Derivation of the Energy Spectrum
To derive this energy spectrum, we start by considering the Fourier transform of the
velocity field v(x, t):
ˆ
v(k, t) = Zv(x, t)e−ik·xdx,(23)
where kis the wavevector and xis the position vector. The energy spectrum E(k) is
defined as the average of the squared magnitude of the Fourier transform of the velocity
field over a spherical shell in wavenumber space:
E(k) = 1
2Z|k|=k
|ˆ
v(k, t)|2dS(k),(24)
where dS(k) is the surface element on the spherical shell of radius k.
In the spatial complex number framework, the velocity field vcan be expressed as:
v=u+ixvx+iyvy+izvz.(25)
where u,vx,vy,vzare the real and imaginary components of the velocity field.
The Fourier transform of the velocity field in the spatial complex number framework
is:
ˆ
v(k, t) = ˆ
u(k, t) + ixˆ
vx(k, t) + iyˆ
vy(k, t) + izˆ
vz(k, t).(26)
The energy spectrum E(k) can then be written as:
E(k) = 1
2Z|k|=k|ˆ
u(k, t)|2+|ˆ
vx(k, t)|2+|ˆ
vy(k, t)|2+|ˆ
vz(k, t)|2dS(k).(27)
6
By incorporating the spatial function F(ix, iy, iz) that characterizes the energy dissi-
pation anisotropy, we obtain:
E(k)∼k−5
3· F(ix, iy, iz).(28)
This novel energy spectrum provides a more detailed description of the energy dissipa-
tion mechanisms in turbulent flows, accounting for the directional dependence of energy
dissipation.
5.3 Implications for Turbulence Modeling
The introduction of the spatial function F(ix, iy, iz) allows for a more comprehensive
analysis of turbulent flows. By characterizing the energy dissipation anisotropy, we can
gain insights into the directional dependence of energy transfer and dissipation in turbu-
lent flows. This approach provides a more accurate representation of the energy cascade
process, which is crucial for understanding the dynamics of turbulence.
In summary, the novel turbulence energy spectrum based on spatial complex number
decomposition offers a more detailed and rigorous framework for studying turbulence.
This approach provides new insights into the energy dissipation mechanisms in turbulent
flows and enhances our understanding of the energy cascade process.
6 Results
In this section, we present the key results obtained from the application of spatial complex
numbers and tensor extensions to the Navier-Stokes equations.
6.1 Extended Spatial Cauchy-Riemann Conditions
We derived the extended spatial Cauchy-Riemann conditions for a spatial complex func-
tion L=u+ixv+iyw+izt:
∂u
∂x =∂v
∂y =∂w
∂z ,(29)
∂v
∂x =−∂u
∂y ,(30)
∂w
∂x =∂v
∂z ,(31)
∂w
∂y =−∂u
∂z .(32)
These conditions ensure the analyticity of the spatial complex function in three-
dimensional space and provide a framework for studying turbulence.
6.2 Tensorial Formulation of Navier-Stokes Equations
We reformulated the Navier-Stokes equations using the tensor operator T:
T · ∂v
∂t +T · (v· ∇)v=−1
ρT · ∇p+νT · ∇2v.(33)
7
This formulation allows for a more natural incorporation of vorticity and turbulence
effects, providing a comprehensive description of the fluid motion.
6.3 Turbulence Energy Spectrum
We derived a novel turbulence energy spectrum based on spatial complex number decom-
position:
E(k)∼k−5
3· F(ix, iy, iz),(34)
where Fis a spatial function characterizing the energy dissipation anisotropy. This energy
spectrum provides a more detailed description of the energy dissipation mechanisms in
turbulent flows, accounting for the directional dependence of energy dissipation.
7 Conclusion
In this work, we have deepened the mathematical foundation of tunnel mathematics by
extending its tensorial formulation and deriving new spatial Cauchy-Riemann conditions.
Our results provide a more rigorous connection between spatial complex numbers and the
Navier-Stokes equations, with significant implications for turbulence modeling.
The extended spatial Cauchy-Riemann conditions ensure the analyticity of spatial
complex functions in three-dimensional space, providing a framework for studying turbu-
lence. The tensorial formulation of the Navier-Stokes equations allows for a more natural
incorporation of vorticity and turbulence effects, offering a comprehensive description of
the fluid motion.
Furthermore, the novel turbulence energy spectrum based on spatial complex number
decomposition provides a more detailed understanding of the energy dissipation mecha-
nisms in turbulent flows. This approach accounts for the directional dependence of energy
dissipation, enhancing our understanding of the energy cascade process.
Overall, this work demonstrates the potential of spatial complex numbers and tensor
extensions for advancing our understanding of turbulent flows and improving the accuracy
of turbulence models. Future research should focus on further developing these theoretical
advances and applying them to practical problems in fluid dynamics.
8
Symbols and Nomenclature
Symbol Description
LSpatial complex number
Re(L) Real part of the spatial complex number
Imx(L) Imaginary component associated with ix
Imy(L) Imaginary component associated with iy
Imz(L) Imaginary component associated with iz
ix, iy, izUnit spatial imaginary numbers
TSecond-rank tensor operator
Tij Second-rank tensor representing transformation properties
vVelocity field
uVector field represented in the spatial complex number framework
ωVorticity
pPressure
ρFluid density
νKinematic viscosity
E(k) Energy spectrum
kWavenumber
F(ix, iy, iz) Spatial function characterizing energy dissipation anisotropy
∇Gradient operator
∇2Laplacian operator
ˆ
v(k, t) Fourier transform of the velocity field
ˆ
v(k, t) Fourier transform of the velocity field in the spatial complex number framework
kWavevector
xPosition vector
9
A Mathematical Background on Spatial Complex Num-
bers and Tensor Analysis
This appendix provides additional mathematical background and derivations that support
the main results presented in the research. Specifically, we focus on the properties of
spatial complex numbers, their application to tensor analysis, and the derivation of key
results.
A.1 Properties of Spatial Complex Numbers
Spatial complex numbers extend the concept of complex numbers to three-dimensional
space. A spatial complex number Lis defined as:
L= Re(L) + ixImx(L) + iyImy(L) + izImz(L) (35)
where Re(L) is the real part, and Imx(L), Imy(L), Imz(L) are the imaginary com-
ponents associated with the unit spatial imaginary numbers ix,iy, and iz. These unit
spatial imaginary numbers satisfy the following multiplication rules:
i2
x=i2
y=i2
z=−1, ixiy=iz, iyiz=ix, izix=iy(36)
The multiplication of two spatial complex numbers L1and L2can be expressed as:
L1L2= (Re(L1) + ixImx(L1) + iyImy(L1) + izImz(L1)) ·(37)
(Re(L2) + ixImx(L2) + iyImy(L2) + izImz(L2)) (38)
Expanding this product and applying the multiplication rules for the unit spatial
imaginary numbers, we obtain the interaction between the real and imaginary components
of the spatial complex numbers.
A.2 Tensor Analysis and Spatial Complex Numbers
Tensor analysis provides a framework for studying the transformation properties of phys-
ical quantities under coordinate rotations. In the context of spatial complex numbers,
we can use tensor analysis to describe the behavior of velocity fields and other physical
quantities in three-dimensional space.
A second-rank tensor Tij can be defined as:
Tij =
Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tz z
(39)
where each element Tij describes how the components of a spatial complex number
transform under rotations.
The tensor operator Tis defined as:
T=Tij iiij(40)
where the indices iand jrange over {x, y, z}, and the Einstein summation convention
is implied.
10
Using the tensor operator T, we can express the Navier-Stokes equations in a tensorial
form:
T · ∂v
∂t +T · (v· ∇)v=−1
ρT · ∇p+νT · ∇2v(41)
where vis the velocity field, pis the pressure, ρis the fluid density, and νis the
kinematic viscosity.
A.3 Derivation of Extended Spatial Cauchy-Riemann Condi-
tions
The extended spatial Cauchy-Riemann conditions ensure the analyticity of a spatial com-
plex function in three-dimensional space. For a spatial complex function L=u+ixv+
iyw+izt, the extended spatial Cauchy-Riemann conditions are given by:
∂u
∂x =∂v
∂y =∂w
∂z (42)
∂v
∂x =−∂u
∂y (43)
∂w
∂x =∂v
∂z (44)
∂w
∂y =−∂u
∂z (45)
These conditions can be derived by considering the spatial complex derivative of L:
∂L
∂x +ix
∂L
∂y +iy
∂L
∂z (46)
For Lto be analytic, this derivative must be independent of the direction in which it
is taken. This independence leads to the extended spatial Cauchy-Riemann conditions.
A.4 Fourier Transform and Energy Spectrum
The Fourier transform of the velocity field v(x, t) is given by:
ˆ
v(k, t) = Zv(x, t)e−ik·xdx(47)
where kis the wavevector and xis the position vector. The energy spectrum E(k) is
defined as the average of the squared magnitude of the Fourier transform of the velocity
field over a spherical shell in wavenumber space:
E(k) = 1
2Z|k|=k
|ˆ
v(k, t)|2dS(k) (48)
where dS(k) is the surface element on the spherical shell of radius k.
In the spatial complex number framework, the velocity field vcan be expressed as:
v=u+ixvx+iyvy+izvz(49)
11
where u,vx,vy,vzare the real and imaginary components of the velocity field.
The Fourier transform of the velocity field in the spatial complex number framework
is:
ˆ
v(k, t) = ˆ
u(k, t) + ixˆ
vx(k, t) + iyˆ
vy(k, t) + izˆ
vz(k, t) (50)
The energy spectrum E(k) can then be written as:
E(k) = 1
2Z|k|=k|ˆ
u(k, t)|2+|ˆ
vx(k, t)|2+|ˆ
vy(k, t)|2+|ˆ
vz(k, t)|2dS(k) (51)
By incorporating the spatial function F(ix, iy, iz) that characterizes the energy dissi-
pation anisotropy, we obtain:
E(k)∼k−5
3· F(ix, iy, iz) (52)
This novel energy spectrum provides a more detailed description of the energy dissipa-
tion mechanisms in turbulent flows, accounting for the directional dependence of energy
dissipation.
12
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physics. Elsevier, 2013.
[3] Lavrent’Ev, M. A., and B. V. Shabat. ”Methods of the Theory of Functions of a
Complex Variable.” (1973).
[4] Kolmogorov, Andrey Nikolaevich. ”The local structure of turbulence in incompressible
viscous fluid for very large Reynolds.” Numbers. In Dokl. Akad. Nauk SSSR 30 (1941):
301.
[5] Shvydkyi, O. G. ”Application of tunnel mathematics for solving the steady Lam´e and
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