CHAPTER 2

Turbulence

Turbulence is a flow regime characterizing many aspects of our lives, be- cause it si often present both in nature and in engineering applications. As examples for turbulent natural phenomena, one may notice the motion of cumulus clouds in the atmosphere, the flow of the water in a river, or the smoke from a chimney. Flows are characterized by turbulence in almost all industrial applications, such as for fluids in pipes or a flame in a combustion process. In fluid dynamics laminar flows are the exception, not the rule:

small dimensions and high viscosities are needed for laminar flow regimes [9].

Turbulence is a widely studied phenomenon in all branches of Science. Re- garding engineering problems, the understanding of turbulence can lead to the improvement of both safety and efficiency of equipments, hence it is of fundamental importance for all those processes involving fluids, including many aspects of Nuclear Power Plants (NPPs).

Since the purpose of the present thesis work is the assessment of a SEM code for DNS of fluid-structure thermal interactions, turbulence analysis is fundamental to evaluate simulations quality. In fact, in order to fully depict the transfer and transport of Energy in Conjugate Heat Transfer problems, it is important to fully characterize the flow both at largest and smallest scales of motion.

8

2.1 Characteristics of turbulence

In spite of more than one century of researches on turbulence, it is difficult to give a precise and unique definition of it, even though it is possible to describe turbulence through some of its main characteristics.

Observing a turbulent flow, such as the smoke from a chimney or the flame in a fire, it reveals unsteadiness and irregularity, or randomness, like driven by a chaotic and unpredictable motion of particles and eddies. This makes impos- sible a deterministic approach. Anyway, characteristic structures are present in all turbulent flows, and are responsible for the velocity re-distribution, which implies that the velocity decreases in some regions, increasing in oth- ers. As an example, in a turbulent channel the eddies increase the velocity in the wall region, reducing it at the bulk.

As a consequence, this behaviour is not completely random and may be physically depicted by an irregular and significantly varying velocity field in both space and time, which can be decomposed in a chaotic rapid fluctuat- ing part around a mean value, which changes more slowly in time [10], as shown in figure 2.1. Since the velocity field characterizes the turbulent flow behaviour, all other flow parameters show this random fluctuation around a

Figure 2.1: Example of time history of a velocity component in a point. The total trend (left) can be decomposed in a fluctuating and a mean part (right).

main trend.

Because of the chaotic nature of such fluctuations, it is impossible to ex- actly predict the instantaneous behaviour of turbulent flows. Indeed, such instantaneous fields would change even if an experiment is repeated always in the same conditions, while the main trends remain the same. Hence, such instantaneous description of the flow is not of interest for engineering pur- poses. It is preferable to depict the mean trend of all flow variables, adopting statistical descriptors to evaluate fluctuations, as described in section 2.2.

Another important feature of all turbulent flows is the turbulence diffusivity, which causes rapid local mixing and increased rates of momentum, heat and mass transfer [9]. This characteristic is important for engineering applica- tions since it is one of the main parameters affecting turbulent flows and, hence, equipment behaviour. In absence of this turbulent mixing, heat or material quantities would be carried along streamlines of the flow and slowly diffuse by molecular transport, instead of being rapidly dispersed across the flow.

Turbulence diffusivity was depicted by Osborne Reynolds in his famous ex- periment in 1883, in which he also observed the transition from laminar to turbulent regime [11]. With further researches he established that such a transition occurs in presence of perturbations of the laminar flow at high values (from hundreds to thousands) of the non-dimensional quantity

Re = U D

ν = U Dρ

µ (2.1)

being: U the mean velocity, D the hydraulic diameter, ρ the fluid density, ν the kinematic viscosity and µ the dynamic viscosity. This non-dimensional parameter is called as Reynolds number, and it is one of the most famous parameters in fluid dynamics.It represents the ratio between inertial and vis- cous forces in the fluid. For high values the laminar flow becomes unstable, amplifying any generated perturbation and determining the transition to the turbulent regime. Anyway, such perturbations depend on many factors, so that transition to turbulence cannot be identified uniquely by a certain value of the Reynolds number for every condition [9].

It has been observed that turbulence is rotational and three dimensional, because it is characterized by high levels of fluctuating vorticity. This plays a fundamental role in flows behaviour and, due to the absence of vorticity- maintenance mechanisms, could not be sustained by two dimensional veloc- ity fluctuations only [9]. Hence, the analysis of turbulent phenomena has to be performed at a three dimensional level. Even in case of symmetries, as for those analyses performed for the present thesis work, simulations have to be performed on three-dimensional computational domains.

All turbulent flows are characterized by a dissipative behaviour due to vis- cous shear stresses which deform flow paths, increasing the internal energy

of the fluid at the expense of the kinetic energy [9]. As a result, turbulence needs a continuous supply of energy to face these viscous losses, otherwise turbulence decays rapidly. Energy dissipation by turbulent eddies increases resistance to flow through pipes and it increases the drag on objects in the flow [12].

It is worth mentioning that turbulence is a continuum phenomenon, i.e.

even the smallest scales of turbulence (see Section 2.4) are larger than any molecular length scale [9]. As a consequence, turbulence is not controlled by molecular properties of the fluid, but is still governed by equations of fluid mechanics (see section 2.3). With other words, at large enough values of Reynolds number, turbulence is a characteristic of the flow and not of the fluid, whether it is a liquid or a gas.

2.2 Statistics

Turbulence is characterized by chaotic fluctuations of parameters around av- erage values. The former have a stochastic nature, with very rapid changes in time. The latter are characterized by much slower variations in time. Be- cause of such chaotic fluctuations, the instantaneous trend is unpredictable and is not of interest for engineering purposes. Anyway, in the long term, fluctuations may strongly influence certain phenomena (e.g. the Thermal Fatigue descried in Chapter 1). Therefore, statistical descriptors are adopted to describe the nature of fluctuations and their distribution.

The evolution in time of a generic parameter c (such as a velocity component) may be expressed as:

c(~r, t) = c(~r, t) + c^{0}(~r, t) (2.2)
being: ~r the position in space, and t the time. From now on, the notation
c and c^{0} will indicate the averaged value of the intensive variable c and
its fluctuating value, respectively. In reference to figure 2.1 at page 9, the
average value of c may be expressed by:

c(~r, t) = 1

∆t
Z _{t+}^{∆t}

2

t−^{∆t}_{2}

c(~r, τ ) dτ (2.3)

being the averaging interval ∆t chosen long enough with respect of the fluctuations’ temporal scale, and short enough with respect of the long term variation of the average quantity [10]. In agreement with such a definition, the fluctuations have a zero time averaged value.

In case of discrete distributions, as for experimental data acquisition or for numerical simulations, the integral operation in equation (2.3) becomes a

summation. Considering all N available values cn as independent events, the mean (or ensemble) average is an arithmetic average expressed by:

C ≡ hc(~r, t)i = lim

N→∞

1 N

XN n=1

cn(~r, t)

where the notation hci is equivalent to c for discrete distributions. Since it is impossible to compute an infinite number N of samples, such arithmetic mean is actually an estimator for the ensemble average [13]. Hereafter in this work, given that N is large enough to let the estimator be close to the ensemble averaging, the following notations will assume the same meaning:

C≡ hci ≡ c

being clear that all notations and equations will be referred to discrete dis- tributions of values, unless explicitly declared.

The nature of the fluctuations is depicted by describing their distribution around the mean value. Such distributions are characterized via statistical descriptors based on moments, i.e. specific quantitative mathematical in- struments to measure the shape of a distribution. A generic m-th moment of the random variable c is defined as [13]

hc^{m}i = lim

N→∞

1 N

XN n=1

c^{m}_{n}

whereas the generic m-th central moment of the same variable is referred to its mean value:

h(c − C)^{m}i = hc^{0m}i = lim

N→∞

1 N

XN n=1

[c_{n}− C]^{m}

By such definitions, the ensemble average results to be the first moment of a variable, while fluctuations may be defined as the first central moment.

Another statistical descriptor is the variance, defined as the second central moment. Using the definition of Reynolds operator (see appendix A), it becomes

var[c] = c^{02}=hc^{02}i = h(c − C)^{2}i = hc^{2}i − C^{2}

The square root of the variance is a widely used statistical descriptor called root mean square (rms) or standard deviation. In a turbulent flow it is used to describe the turbulent intensity of a variable, such as a velocity compo- nent. More in detail, the following quantity is called turbulence intensity

I = q

u^{02}+ v^{02}+ w^{02}

being u,v and w the three velocity components. It is strictly related to the turbulent kinetic energy per unit mass k, defined by

k = 1

2(u^{02}+ v^{02}+ w^{02}) (2.4)
and described in Section 2.6.

Higher moments give more detailed indication of the temporal distribution around the mean value, i.e. they are descriptors of the shape of the probabil- ity density function (p.d.f.) associated to such distribution. The third and fourth central moments are used to define the dimensionless factors called respectively skewness (S ) and kurtosis (or flatness, K ) [13]

S(c^{0}) = hc^{03}i

hc^{02}i^{3}^{2} = hc^{3}i − 3hc^{2}iC + 2C^{3}
(hc^{2}i − C^{2})^{3}^{2}
K(c^{0}) = hc^{04}i

hc^{02}i^{2} = hc^{4}i − 4hc^{3}iC + 6hc^{2}iC^{2}− 3C^{4}
(hc^{2}i − C^{2})^{3}^{2}

These two central moments are scaled by the variance, and their expressions are derived in appendix A.

The skewness is a measure of the asymmetry of the p.d.f. about its mean value. As shown in figure 2.2, a negative value indicates a pronounced tail of the p.d.f. for the lowest values of the distribution, while in case of positive skewness the tail is on the right side. The closer to zero is the skewness, the more symmetric is the distribution; as an example, distributions c and d of figure 2.2 are symmetric, hence they have zero skewness.

The kurtosis is a measure of how all values are concentrated around the mean value. As shown in figure 2.2, the lower the value of the kurtosis, the

Figure 2.2: Skewness and kurtosis of a generic distribution: positive skewness (a);

negative skewness(b); low kurtosis (c); high kurtosis (d ). The dotted line represents the mean value of the distribution.

flatter the distribution is. On the other hand, an high value of the kurtosis indicates a relatively low probability of excursions from the mean value.

In complex phenomena like turbulence, where a large number of variables
is involved, it is important to analyse also the mutual influence among
such variables. The statistical descriptor for such influence is called cross-
correlation (or covariance), which defined by hcic_{j}i, being ci and c_{j} two
variables. A positive value indicates that the two variables are increasing
or decreasing together, while a negative one indicates that one variable is
increasing meanwhile the other is decreasing (or viceversa). A zero value
indicates that the two variables are not correlated, but it does not mean
that the two variables are statistically independent [13].

2.3 Governing equations

The motion of viscous fluid flows is physically described by continuity equa- tions of mass and momentum, which for an incompressible fluid may be written as, respectively

∇ · ~u = 0 (2.5)

ρD~u

Dt = µ∇^{2}~u− ∇p + ~F (2.6)

being ~u the flow velocity, D~u/Dt its material derivative, ρ the fluid density, µ the dynamic viscosity, p the pressure and ~F an external force per unit volume which, considering the gravity (~g) only, is expressed by ρ~g. In the present thesis work the contribution of gravity is not taken into account. The equations for the momentum are better known as Navier-Stokes equations.

Their solution is the velocity field, i.e. ~u. Once it is known, pressure may be found.

An additional equations has to be taken into account if the heat transfer has to be considered. This is the energy equation, which may be expressed in terms of temperature T :

D ~T

Dt = α∇^{2}T + q~ (2.7)

being α the thermal diffusivity and q the specific heat source.

In the present thesis work the temperature will be considered as a passive scalar, i.e. fluid properties will not vary with the temperature, therefore resulting in constant values. It implies that the velocity field is affecting the temperature distribution, while the former is not affected by the latter.

Furthermore, such an assumption allows to compute more thermal/passive scalar fields within the same simulation, being the velocity field the same for all of them.

Direct Numerical Simulations provide a transient numerical solution to the mentioned equations as they are, without involving additional simplifying models to reduce the large computational power required.

2.4 Turbulence scales

One of the main characteristic of turbulence is its continuous range of scales.

In fact, turbulent flows are constituted by eddies of various sizes, and are associated to phenomena operating on different time and length scales.

Smallest eddies behaviour is governed by two main quantities, the kinematic viscosity ν, and the turbulent kinetic energy dissipation in time, defined as

ε =−dk dt

Smallest eddies scales are determined by the combination of such quantities:

η_{K} =

ν^{3}
ε

^{1}_{4}

(2.8)

τ_{K} =ν
ε

^{1}_{2}

(2.9)
υ_{K} = (νε)^{1}^{4}

referred as, respectively, the Kolmogorov length, time and velocity scales (or microscales) for the motion of smallest eddies [10].

The length scales of largest eddies (l) are in the order of the characteristic length of those bodies which generate them. Therefore, they are orders of magnitude larger than those of smallest eddies. Small and large length scales are related by the following expression [10]:

l

η_{K} ≈ (ReT)^{3}^{4} (2.10)

being Re_{T} a Reynolds number expressed via the turbulent kinetic energy k:

Re_{T} = k^{1}^{2} l
ν

From the above equation (2.10) it is noticeable that higher turbulence levels (i.e. Reynolds number) corresponds to smaller Kolmogorov length scales for a fixed geometry (i.e. fixing l). A similar relation exist for time and velocity scales. This is of primary importance for Direct Numerical Simu- lation (DNS) because, given a certain computational domain, the solution

of a case characterized by high turbulence levels requires finer mesh and smaller time step with respect to a low turbulence levels case. The former may result even not affordable because of the high computational power required. In fact, in order to consider all involved turbulent phenomena, DNS computations require a mesh grid characterized by distances between points approximately equal to the local length microscales. Given a fixed integral scale l, equation (2.10) applies to all spatial directions, so that for a 3D computational domain the total number of points is [9]

NK ∝ Re^{9}^{4} (2.11)

For those turbulent scenarios involving heat transfer or the transport of chemical species, Kolmogorov microscales might be not small enough to describe all involved processes. In fact, because of diffusion mechanism characteristic of heat or chemical species, there might be exchange processes operating at scales smaller than Kolmogorov ones. The diffusion mechanism is similar for the transfer of both heat and chemical species, so that one may refer to a generic molecular diffusion of a scalar quantity.

If diffusivity of momentum prevails the diffusivity of the scalar, the scalar transport at smallest length scales is mainly driven by flow motion, i.e.

smallest eddies. In the opposite case, diffusion spawns exchanges of scalar
quantities at length scales smaller than the characteristic lengths of smallest
eddies [9]. These are called Batchelor length scales (η_{B}), and are related to
the Kolmogorov length scales via the Schmidt number Sc:

Sc = ν
α_{m}
η_{B}

η_{K} = Sc^{−}^{1}^{2} for Sc > 1
ηB

η_{K} = Sc^{−}^{3}^{4} for Sc < 1

(2.12)

being ν the ratio of momentum diffusivity (kinematic viscosity), and αm the molecular diffusivity.

In case heat transfer is considered, molecular diffusivity corresponds to the thermal diffusivity. The Schmidt number may be replaced for the Prandtl number (P r), and the Batchelor length scales may be expressed as follows:

α = λ
c_{p}ρ
P r = ν

α
η_{B} =

α^{2}ν
ε

^{1}_{4}

for P r > 1

η_{B} =

α^{3}
ε

^{1}_{4}

for P r < 1

(2.13)

being α the thermal diffusivity, λ the thermal conductivity and cp the spe- cific heat capacity at constant pressure. It is worth mentioning that length scales are estimated on the basis of the strain rate, which is defined as a function of momentum diffusivity. However, when the molecular/thermal diffusivity is much bigger than the momentum diffusivity (i.e. for very low Prandtl numbers, like for liquid metals), Batchelor length scales become independent from ν, and the strain rate can be defined as a function of the thermal/molecular diffusivity [9]. Therefore, a double relationship is needed for equations (2.12) and (2.13), one for P r > 1 and the other one for P r << 1. However, for P r∼ 1 it is α ∼ ν, so that the second relationships of the (2.12) and of the (2.13) are approximately true for every Prandtl number value lower than one.

From the above relationships it is noticeable that, for those fluid character- ized by Prandtl number higher than 1, the numerical mesh has to be based on Batchelor length scales instead of Kolmogorov ones, being the former smaller than the latter. Equations (2.12) express the relationship between Kolmogorov and Batchelor scales, i.e. between the smallest needed distances of mesh grid points. It applies to each direction. As a consequence, the to- tal number of mesh points for those cases characterized by Prandtl number higher than 1 is given by

N_{B}= P r^{3}^{2} N_{K}

A similar relationship for Batchelor and Komlogorov time scales can be derived [9]. Therefore, given a fixed computational domain, the required computational effort is higher for those fluids (e.g. water) characterized by high values of the Prandtl number, than for those characterized by low values (e.g. liquid metals).

2.5 Production and transport of turbulence

The largest size eddies, which characterized the mean flow motion, carry the greatest kinetic energy and generate smaller eddies via non-linear processes, transferring the energy to them through inviscid processes (energy cascade).

The smaller eddies spawn smaller eddies, and so on to smaller and smaller scales, till when the energy of the smallest eddies is dissipated by viscosity.

In fact, small eddies have large velocity gradients, upon which viscosity acts, making the turbulent motion highly dissipative. Such behaviour is described in the universal equilibrium theory, stated by Kolmogorov in 1941. A gen- erally accepted hypothesis is the equilibrium state between large and small eddies, so that the latter receive from the former the same rate of energy that is dissipated because of viscosity [10].

Furthermore, small perturbations extract energy from the mean flow produc- ing irregular (turbulent) fluctuations which are self-sustained, and propagate

Figure 2.3: Ideal energy spectrum for a turbulent flow (adapted from [14]). Log- arithmic scale for both axes.

by further extraction of energy [12]. This process is referred to as production and transport of turbulence.

The energy distribution E at different length scales is generally obtained via
Fourier series decomposition of time depending turbulence quantities. It is
expressed in function of the wave number κ, related to the wavelength λ_{κ}
as follows

κ = 2π
λ_{κ}

The qualitative trend of the energy spectrum is shown in figure 2.3. Three separate regions are highlighted, corresponding to the three different phases of the energy cascade. At lowest wave numbers, i.e. largest wavelengths, turbulence takes energy from the mean flow. At intermediate scales (iner- tial subrange) energy is transferred to smallest eddies via the well known Kolmogorov−5/3 law reported in the figure, being Cκ a proportionality co- efficient. Last region is characterized by smallest wavelengths (i.e. highest wave numbers κ) and the energy associated to each wavelength decreases drastically because of the viscous dissipation.

The energy distribution E is related to the turbulent kinetic energy via the following relationship:

k =
Z _{∞}

0

E(κ)dκ

2.6 Turbulent Kinetic Energy

In Section 2.3 the Navier-Stokes equations for incompressible flows (equation 2.6) are introduced. These equations describe the evolution of the velocity field ~u in space and time. If the time averaging process described in Section 2.2 (mainly equations (2.2) and (2.3)) is applied to all its terms, the so called Reynolds-Averaged Navier-Stokes (RANS) equations are obtained [10]:

∂(ρ~u)

∂t +∇ · (ρ~u~u) = ∇ · (~~τ − p~~I) + ρ~g − ∇ · (ρ~u^{0}u~^{0}) (2.14)
being~~I the identity matrix, ~~τ the viscous stress tensor, and gravity (~g) the
only external force considered.

Last term of the Right-Hand Side (RHS) (−ρ~u^{0}u~^{0}) is a tensor similar to
the viscous stress tensor. The latter accounts for viscous normal and shear
stresses, while the former is called the Reynolds stress tensor (~~τ_{Re}) and
expresses the correlation among the velocity fluctuations only. Therefore, it
can be interpreted as a component of the total stress tensor that accounts
for turbulent fluctuations in fluid momentum [15]. From the point of view
of the statistical analysis, its terms are the covariance of fluctuating velocity
components.

As a stress tensor, it is symmetric, with six independent components: three normal stresses (diagonal terms), and three independent shear stresses (off- diagonal terms):

~~τ_{Re}=−ρ

u^{02} u^{0}v^{0} u^{0}w^{0}
u^{0}v^{0} v^{02} v^{0}w^{0}
u^{0}w^{0} v^{0}w^{0} w^{02}

A dynamical equation for the components of the Reynolds stress tensor can be derived subtracting equations (2.14) from the Navier-Stokes equations (2.6), in which velocity components are now expressed in terms of mean and fluctuating part. The resulting equations can be averaged again and manipulated, obtaining the so called Reynolds stress transport equations, which are in the form [12]

∂u^{0}_{i}u^{0}_{j}

∂t + u_{k}∂u^{0}_{i}u^{0}_{j}

∂x_{k} =−1
ρ

u^{0}_{j}∂p^{0}

∂xi

+ u^{0}_{i}∂p^{0}

∂xj

− 2ν∂u^{0}_{i}

∂x_{k}

∂u^{0}_{j}

∂x_{k}+

−∂u^{0}_{i}u^{0}_{j}u^{0}_{k}

∂x_{k} −

u^{0}_{j}u^{0}_{k}∂ui

∂x_{k} + u^{0}_{i}u^{0}_{k}∂u_{j}

∂x_{k}

+
+ ν∇^{2}u^{0}_{i}u^{0}_{j}

(2.15)

In order to obtain the six independent components of the Reynolds stress tensor, subscripts i, j, k have to be permuted following the Einstein notation:

ui,uj and u_{k} mean the three velocity components u, v and w; xi,xj and x_{k}
mean the three spatial directions x, y and z.

In reference to the definition of Turbulent Kinetic Energy (TKE) k (equation 2.4), the TKE equation can be obtained as half of the trace of the Reynolds stress tensor, expressing the diagonal components via the Reynolds stress transport equations (2.15). Using properties of Reynolds operators, like averaging is, the redistribution term of equation (2.15), i.e. the first term of its RHS, can be re-written as summation of a pressure diffusion and a pressure strain term, as follows [12]:

−

u^{0}_{j}∂p^{0}

∂x_{i} + u^{0}_{i}∂p^{0}

∂x_{j}

=− ∂u^{0}_{j}p^{0}

∂x_{i} +∂u^{0}_{j}p^{0}

∂x_{j}

!
+ p^{0}

∂u^{0}_{j}

∂x_{i} + ∂u^{0}_{j}

∂x_{j}

Due to continuity, pressure strain term has zero trace and, as a consequence, only the pressure diffusion term compares in the turbulent kinetic energy equation. This is given by:

∂k

∂t + u_{k} ∂k

∂x_{k} =−1
ρ

∂u^{0}_{i}p^{0}

∂xi − ν∂u^{0}_{i}

∂x_{k}

∂u^{0}_{i}

∂x_{k} −1
2

∂u^{0}_{i}u^{0}_{i}u^{0}_{k}

∂x_{k} +

− u^{0}iu^{0}_{k}∂u_{i}

∂x_{k} + ν∇^{2}k

The first term on the Left-Hand Side (LHS) is the local change rate of tur- bulent kinetic energy, while the second one is a convection term representing the turbulent transport of TKE due to the mean fluid motion.

The first term of the RHS is the pressure diffusion term, given by the correla- tion of pressure and velocity fluctuations. It describes not exactly a diffusion process, even if it is called in this way due to its conservative nature. In fact, it does not contribute to the production or destruction of turbulent kinetic energy, but only to its redistribution [12].

The second term expresses the dissipation rate (ε) of TKE, i.e. the rate at which it is converted in thermal energy by smallest eddies due to viscosity processes. It is noticeable that such term is always negative. Hence, it rep- resents the only effective decay term of turbulence.

The third term is called turbulent transport term, and represents the con- tribution of velocity fluctuations to the global kinetic energy transport [10].

It is a redistributive term for TKE in space, without generating or destroy- ing it. It is generally supposed to contribute to make spatially uniform the turbulence distribution [12].

The fourth term is the so called production term because it describes the energy transfer rate from the mean flow. Hence, it doesn’t represent a net generation of turbulent kinetic energy, but a transfer from largest to smallest eddies. In spite of the negative sign, the global term is always positive, with the exception of peculiar cases such as flows subjected to strongly stabilizing

forces (e.g. centrifugal acceleration) [12].

Last term is the molecular diffusion term, representing the diffusive trans- port of TKE due to those mechanisms acting at molecular level.