different velocities and are, in a manner of speaking, the eddies of a tur-
bulent universe. Turbulence is also produced in the Earth’s outer magne-
tosphere, due to the development of instabilities caused by the interaction
of the solar wind with the magnetosphere. Numerous other examples of
turbulent flows arise in aeronautics, hydraulics, nuclear and chemical engi-
neering, oceanography, meteorology, astrophysics, and internal geophysics
(Lesieur [2008]).
Figure 2.1: Eddies in Jumpiter observed by JunoCam imager on NASA’s Juno
spacecraft in December 2018 (Nasa [2018])
2.2 FieldEquationsforNewtonianFluidsFlows
The necessity to study the spectacular behavior of fluid and eddies motion
led to conceive an elegant formulation made by equations of Motion, Mass
and energy balance. This concept can be expressed in a simple formulation
suggested from the study of General Relativity that assumes the following
formalism (Landau and Lifshitz [2013]),
∂T
k
i
/∂x
k
=0
T
k
i
represents the stress tensor energy and∂/∂x
k
express the partial deriva-
tivein4-space, whicharethethreeCartesiancoordinatesandthetime. But
omitting the relativistic approach in classic formalism the equations of mo-
tion for a general Newtonian fluid will now be established. In doing so the
fluid will be considered to be a continuum. In a continuum, the smallest
volume element considered dV is still homogeneous, i.e. the dimensions
6
of dV are still very large compared to the average distance between the
molecules in the fluid. In three–dimensional motion the flow field is given
by the velocity vector
⃗ v =⃗ e
x
u+⃗ e
y
v+⃗ e
z
w
with the three components u,v,w in a Cartesian coordinate system with
unit vectors⃗ e
x
,⃗ e
y
,⃗ e
z
and also by the pressurep and the temperatureT. To
determinethesefivequantities, therearefiveequationsavailable: continuity
equation, three momentum equations and energy balance (Schlichting and
Kestin [1961]).
2.2.1 The continuity equation
The continuity equation is a statement about the conservation of mass. It
expresses the fact that, per unit volume, the sum of all mass flowing in and
out per unit of time must be equal to the change in mass due to a change in
density per unit of time. For unsteady flows of a general fluid, this yields:
Dϱ
Dt
+ϱ∇⃗ v =0 (2.1)
HereDϱ/Dt is the total or substantial derivative of the density with respect
to time, i.e:
Dϱ
Dt
=
∂ϱ
∂t
+⃗ v·∇ϱ (2.2)
Itfollowsimmediatelyfromthecontinuityequation2.1thatincompress-
ible flows, i.e. flows of incompressible fluids, are source free. We have:
Dϱ
Dt
=0 ∇⃗ v =0 (2.3)
2.2.2 The momentum equation
The momentum equation is the basic law of mechanics which states that
mass times acceleration is equal to the sum of the forces. Both body forces
and surface forces (pressure and friction forces) act. If
⃗
f is the body force
per unit volume (e.g.
⃗
f = ϱ⃗ g where ⃗ g is the vector of gravitational accel-
eration) and
⃗
P the surface force per unit volume, the momentum equation
leads to the following vector notation:
ϱ
D⃗ v
Dt
=
⃗
f +
⃗
P (2.4)
where the material derivative D⃗ v/Dt can be expressed as
D⃗ v
Dt
=
∂⃗ v
∂t
+⃗ v∇⃗ v
7
The total surface force
⃗
P can be expressed in the following form,
⃗
P =
∂p
x
∂x
+
∂p
y
∂y
+
∂p
z
∂z
Here p
x
, p
y
and p
z
are vectors that can be decomposed into compo-
nents. This decomposition will be carried out by denoting the components
perpendicular to each surface element, that is the normal stresses, by σ ,
and by giving the direction of the normal stresses as an index. The state of
stress is therefore determined by nine scalar quantities and these form the
stress tensor. The nine components of the stress tensor are also called the
stress matrix:
σ =
σ x
τ xy
τ xz
τ yx
σ y
τ yz
τ zx
τ zy
σ z
The stress tensor and its matrix are symmetric: thus two tangential
forces whose indices only differ in their order are equal. This will be shown
by considering the equation of motion of a fluid element. In general, this
motion can be decomposed into a translation and a rotation. The first
invariant of the stress tensor will “initially” be denoted as the pressure p:
p=− 1
3
σ x
+σ y
+σ z
It is useful to separate the pressure from the normal stresses:
τ xx
=σ x
+p, τ yy
=σ y
+p, τ zz
=σ z
+p
Here the stresses have been decomposed additively into a part with the
normal stress− p that is the same in all directions, and a part that deviates
from this (deviator stresses).
At the end the Navier Stokes equation can be written as:
ϱ
D⃗ v
Dt
=
⃗
f−∇ p+∇τ (2.5)
whereτ represents the tensor friction forces.
τ =µ ˙ ϵ − 2
3
δ ij
div⃗ v
with δ ij
being the unit tensor of Dirac, µ the dynamic viscosity and ˙ ϵ the deformation tensor function of the field velocity (Landau and Lifshitz
[2013]). An explicit form of τ can be expressed as
τ ij
=
µ 2
∂u
i
∂x
j
+
∂u
j
∂x
i
− δ ik
2
3
∂u
k
∂x
k
2
8
There is no analytical solution for Navier-Stokes with the exception of basic
examples. Inothersituations, anumericalorexperimentalsolutionmustbe
used. However, it must be taken into account that the great mathematical
difficulty of these equations means that only very few solutions are known
where the convective terms interact quite generally with the friction terms.
However, known particular solutions, such as laminar pipe flow well agree
with the experimental results that there is hardly any doubt about the
general validity of the Navier–Stokes equations. As a consequence of the
Navier–Stokes equations, an equation for the mechanical energy can be
derived. If the Navier–Stokes equation in the x direction is multiplied by
u, the one in the y direction by v and the one in the z direction by w,
and they are summed up, the energy equation for the mechanical energy is
found (Schlichting and Kestin [1961]).
2.2.3 Energy Equation
Inordertosetuptheequationfortheenergybalanceinaflow, weconsidera
fluid particle of massdM =ϱdV and volumedV =dx· dy· dz in a Cartesian
coordinate system and follow it on its path in the flow. According to the
first law of thermodynamics, the gain in total energy DE
t
(the index t
stands for total energy) in unit time Dt is equal to the heat supplied to the
mass element
˙
QDt and the work done on the element
˙
W Dt. Therefore we
have:
DE
t
Dt
=
˙
Q+
˙
W (2.6)
Where DE
t
/Dt is the energy variation,
˙
Q the heat flux and
˙
W the power
made by mechanical energy. The
˙
Q term can be expressed in the following
way,
˙
Q=− dV div⃗ q (2.7)
where⃗ q representsaheatfluxvector. ThetotalenergyE
t
generallyconsists
of three parts: the internal energy e, the kinetic energy 1/2dM· v
2
and the
potential energy ψ . The following statement holds:
dE
t
=dV ϱ
e+
1
2
⃗ v
2
+ψ (2.8)
Thus the substantial change of the total energy follows as
DE
t
Dt
=dV ϱ
D
e+
1
2
⃗ v
2
+ψ Dt
(2.9)
The total rate of work done on the mass element of volume dV is:
9
˙
W =dV ∇(σ ⃗ v) (2.10)
if we combine Eq. 2.6, 2.7, 2.8 and 2.9 we can obtain:
ϱ
D
e+
1
2
⃗ v
2
+ψ Dt
=−∇ ⃗ q+∇
σ ⃗ v
(2.11)
Last we can write the heat flux ⃗ q as
⃗ q =− λ ∇T (2.12)
T corresponds to the scalar temperature field, λ to the heat conduction
coefficient and σ represents the stress tensor.
2.3 Turbulent Flows: introduction and basic
concepts
As before mentioned, most flows which occur in practical applications are
turbulent. This term denotes a motion in which an irregular fluctuation
(mixing, or eddying motion) is superimposed on the mainstream. The
details of the fluctuating motion superimposed on the main motion are so
hopelessly complex that even describing them theoretically seems futile.
However, the resulting mixing motion is of great importance for the course
of the flow and the balance of the forces. Turbulence is generated by more
factors, such as high Reynolds Number R
e
, viscosity ν , variation of density
ρ , and Temperature T gradient. On the other hand, the dissipation of the
energy mainly takes place in the small eddies, as it will be described in
section 2.3.2 with the Kolmogorian Cascade. The turbulent mixing motion
is important for the great drag of turbulent flows in pipes, for the friction
drag encountered by ships and airplanes, and for the losses in turbines and
turbocompressors. On the other hand, it is only turbulence that makes a
large pressure increase in diffusers or along airplane wings and compressor
blades (Lesieur [1987]).
2.3.1 General features of turbulence
We can illustrate this by considering figure 2.2, which shows two-time
records of the velocity, U(t), measured within a turbulent jet at the same
location but at different times. They could have been obtained at a point
somewhere in the middle of the jet, as shown in figure 2.2. Note that the
velocity fluctuates randomly and that these fluctuations occur on a range of
timescales, with frequent relatively sudden excursions randomly intermixed
10
with less rapid variations. The two velocity values records look similar, but
it is clear that they differ in detail at the same specific time. It would
never be possible to reproduce these time traces exactly by repeating the
measurement at other times.
Figure 2.2: Two equal-period records of velocity within a turbulent flow, measured
at the same spatial location but separated in time. Lesieur [2008]
The second major characteristic of turbulent flows is that they contain
features at a wide range of spatial and temporal scales - smaller ones within
the larger ones. The temporal scale could be deduced by careful inspection
of the velocity trace in figure 2.2. We will discover that the range of scales
increases with the flow Reynolds number, R
e
, a typical ratio of inertial to
viscous forces. Flows in the atmosphere generally have an extremely wide
range of scales because the associated R
e
values are very large and the
spatial scales can typically range from millimeters to kilometers.
Thirdly, turbulence is rotational and the vortical motions occur at all
scales, right down to the very smallest. Vorticity in a fluid is defined, in
mathematical terms, by the curl∇∧ of the velocity vector i.e.
⃗ ω =∇∧⃗ v (2.13)
2.3.2 Statistic approach and Energy Cascade
As reported in figure 2.2, the velocity profile can be expressed as a steady-
ergotic stochastic function (the average of velocity fluctuation equal to
zero), with this assumption, the flow can be described through the auto-
correlation R(θ ) and a spectral E(k) function.
Being that turbulence is dependent on the statistic, the spectral energy
is related to auto-correlation function R(θ ) along an axis, calculated in the
following way,
R(θ )=<u
′
(x+θ,t
0
)u
′
(x,t
0
)>− <u
′
(x,t
0
)><u
′
(x,t
0
)> (2.14)
11
<∗ > is indicated as the average in space, θ represent an increment length-
spaceandu
′
istheoscillatingcomponentofthevelocity. Iftheflowisergotic
and stationary, the average square <u
′
(x,t
0
)><u
′
(x,t
0
)> is an identity.
This last is true if it satisfies the intrinsic hypothesis of Kolmogorov. While
the spectral density E(k) is defined as the Fourier transform in x of R(θ ).
E(k)=
1
π ∞
−∞
R(θ )dθ (2.15)
The spectral density can be extended in three-dimensional space and
can be assumed in the following form,
E
ij
(k)=
1
π ∞
−∞
R
ij
(θ )dθ (2.16)
with i and j representing the Cartesian component which is calculated as
the spectral density.
The experimental evidence suggests that the large eddies in a turbulent
flow are anisotropic, in the sense that the fluctuating flow properties have
some directional dependence. The width of the shear flow, such as the
diameter of a pipe or the width of a boundary layer along a wall or over the
top of the ocean, determines the size of the greatest eddies. These eddies
extract kinetic energy from the mean field. The vortexes that are smaller
than these are strained by the velocity field and extract energy from larger
eddies by the same mechanism of vortex stretching. Therefore the smallest
eddies do not interact with either the large eddies or the mean field, so the
kinetic energy is therefore cascaded down from large to small eddies in a
series of small steps. Observations show that the large eddies lose most
of their energy during the time they turn over so that the rate of energy
transferred from large eddies ϵ , is proportional to fluctuation velocity u
′
times the frequency spin u
′
/L, suppose with L the dimension of largest
eddies.
ϵ ∼ u
′3
L
Kolmogorov suggested that the size of dissipating eddies depends on the
rate of dissipation energy ϵ and the viscosity ν . He proposed a microscale
η value in which viscous dissipation takes place.
η =
ν 3
ϵ 1
4
As previously mentioned, the spectral function E(k), which represents the
turbulent kinetic energy, is a function of the wave number k. Somewhat
vaguely, we shall associate a wavenumber k with ad eddy of size l≡ k
− 1
.
12
Figure 2.3: Schematic of the energy spectrum in Kolmogorov’s theory is shown at
the top. The dissipation spectrum as used in Kolmogorov’s theory is shown at the
bottom in the Boundary layer condition (?.
As shown in figure 2.3, the range of wavenumbers k ≫ L
− 1
is usually
called the inflow Energy and represents the turbulent energy related to
the largest eddies. The dissipation band, correlated to the smallest eddies
k∼ η − 1
, forms the high end of the equilibrium range (Kundu et al. [2015]).
The lower end of this range, for which η ≫ l ≫ L, is called the inertial
subrange, as only the transfer of energy by inertial forces (vortex stretch-
ing) takes place in this range. In addition, in the subrange, the spectrum
allows the− 5/3 low (Goto [2012]), which has become one of the most im-
portant results of turbulence theory. Figure 2.3 shows the Turbulent-Model
limit in the spectral plot. The Turbulent approaches investigations such as
13
RANS or LES Method, are not able to investigate the eddy phenomena in
the dissipation band. For this reason, in the numerical approach, a good
way to obtain the spectral plot is the use of the DNS (Direct Numerical
Simulation), which is treated in the following chapter.
2.4 Drag and Lift force
It is known that external forces occur when an object is moving into a fluid.
Thesearethedrag,lift,andtorqueforce(LandauandLifshitz[2013]). They
depend on the velocity U, the density ρ , the relative contact surface S, and
a constant number C, which can be called Drag C
D
, Lift C
L
, or Torque C
ϕ Coefficient. These coefficients can be calculated through the total force in
x or y direction, called respectively like F
D
and F
L
.
F
D
=
γ p· e
x
dγ +
γ τ · e
x
dγ (2.17)
F
L
=
γ p· e
y
dγ +
γ τ · e
y
dγ (2.18)
whilethetorqueforceisfunctionoftheradiusvector⃗ r andcanbeexpressed
as
F
ϕ =
γ ⃗ τ · ⃗ r dγ (2.19)
n
x
and n
y
represent respectively the normal vector in the direction along x
and y while γ is the perimeter of immerse boundary which is calculated by
the Drag Coefficient. It can be a squared, cylinder, or complex geometry.
The C
D
, C
L
are expressed in the following equation,
C
D
=
F
D
ρU
2
S
D
C
L
=
F
L
ρU
2
S
L
withS
D
andS
L
the projection surface of the object along the flow direction
or perpendicular to it. From the experimental results, it is known that the
C
D
varies with Reynolds Number R
e
. In the case of a cylinder, it was ob-
served that with a very small Reynolds R
e
<<1 the drag is proportional to
thelineardimensionofthebodyandtothevelocityitselfF
D
∼ νρU . While
for largeR
e
, the laminar boundary layer becomes unstable and then turbu-
lent. However, the whole boundary layer does not become turbulence, only
some part of it. Figure 2.4 gives experimentally obtained graphs showing
14
the drag coefficient as a function of the Reynolds NumberR
e
=Ud/ν for a
cylinder with diameter d. For very small R
e
the drag coefficient decreases
according to C
D
=24/R
e
(Stokes formula). The decrease in C
D
continues
more slowly as far asR
e
∼ =5
3
, whereC
D
reaches a minimum, beyond which
it increases somewhat. In the range of Reynolds number 2× 10
4
to 2× 10
5
the C
D
is almost constant (Landau and Lifshitz [2013]).
Figure 2.4: Experimental Plot of CD obtained by Wieselbe (Baracu and Boşneagu
[2019]) for varying Reynolds number
It must be borne in mind that, for the high velocities at which the drag
crisis occurs, the compressibility of the fluid may begin to have a noticeable
effect. The parameter which characterizes the extent of this effect is the
Mach NumberM
a
=U/c, wherec is the velocity of sound. IfM
a
<<1, the
fluid may be regarded as incompressible. The experimental data indicate
that the compressibility has in general a stabilizing effect on the flow in
the laminar boundary layer. When M
a
increases, it can be observed an
increment of the critical value of R
e
.
15
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