Process and Fluid Dynamics Simulation - Appunti, domande ed esercizi per l'esame

RANDOM FIELDS

In a turbulent flow, velocity is a time dependent random vector field. The one-point

(, )

one-time JCDF and JPDF are: { }

( ; ; ) = ( ; ) < ℎ = 1, 2, 3

3

∂

( ; ; ) = ( ; ; )

∂ ∂ ∂

1 2 3 15

The mean velocity field is

The fluctuating velocity field is

The Reynolds stress is defined (assuming constant density) as and it’s

− ρ ' ( ; ) ' ( ; )

often abbreviated as − ρ ' '

( )

The N-point N-time joint PDF is defined as: , , ,.., , ,

1 1 1

REYNOLDS STRESS

Because of turbulent motion, an element of fluid jumps across a unit area of the plane at

constant y. Considering a 2D flow where

●​ there is no general movement in direction y → = 0

●​ the fluid moves in direction x → ≠ 0

●​ there are fluctuations both in direction x and y → and

' ≠ 0 ' ≠ 0

the instantaneous rate of transfer of x momentum in the y direction across a surface

( )

orthogonal to y is ρ ' + '

Considering all vortices crossing the plane in both direction, the average rate of transfer of x

( )

momentum in the y direction (per unit area) is → the

ρ ' + ' = ρ ' + ρ ' '

( )

average of the mean becomes 0 → → the Reynolds stress is

ρ ' + ' = 0 + ρ ' '

therefore τ ' =− ρ ' '

TURBULENT INTENSITY and TURBULENT

KINETIC ENERGY

The relative turbulent intensity (also simply called turbulent intensity) is

1 1 1

( )

2 2

( ) 2 2

⎡ ⎤ ⎡ ⎤

' 2

2

1 1 where .

⎢ ⎥ ⎢ ⎥

(, ) = ∑ = ∑ = ∑

⎢ ⎥

⎢ ⎥

2

3 3

⎣ ⎦

⎣ ⎦

The maximum observed T is 0,19.

i

The mean turbulent kinetic energy is 2

( )

1

(, ) = ∑ '

2 16

HOMOGENEOUS and ISOTROPIC TURBULENCE

A random field is statistically homogeneous if all statistics are invariants under

() () () ()

[ ] [ ]

a shift in position. is therefore unchanged if is replaced by

() , + ,

for all N points → is uniform.

There is an homogeneous turbulence if the fluctuating velocity is statistically

∂

homogeneous ( isn’t zero but uniform).

∂

If a statistically homogeneous field is invariant under rotations and reflections of

the coordinate system, then it is statistically isotropic, meaning that there is no

dependence on directionality for the turbulent field. In isotropic turbulence we have that:

2 2 2

( ) ( ) ( ) 2

' = ' = ' =

3

1 2 3

ADDITIONAL STATISTICS

The simplest statistics containing some information on the spatial structure of the random

field is the two-point one-time autocovariance or “two-point correlation”:

The two-point correlation can be used to define various integral lengthscales, for example

+∞ ( )

1

Λ

(

,

) = ∫ , ,

(

0,,

) 11 1

11 0

The integral length scale is the average size of turbulent phenomena, such as turbulent

vortices. 17

TURBULENT FLOWS

ENERGY CASCADE

Richardson suggested that turbulence can be described as an energy cascade.

Energy is transferred from the higher scales and is then “cascaded” down in an inviscid way

through even smaller scales until it is dissipated by viscous phenomena. Turbulent motion

is characterised by a large number of rotational flow structures, called eddies (vortices).

Larger eddies are unstable and break down into progressively smaller eddies.

Largest eddies have the largest energy per eddy; however, since there are many more of the

smaller eddies, the total rate of mechanical energy passed down the chain is nearly constant

→ steady state.

The energy of larger eddies has to be maintained because they are continuously losing energy

to smaller eddies. Largest eddies extract energy from the mean flow (experimental evidence

suggests that the most effective eddies at extracting energy from the mean flow are the

vortices whose principal axis is roughly aligned with that of the mean strain [deformation]

rate).

The energy transfer mechanism is believed to be associated with vortex stretching (see

figure): eddies are strained by shear [sforzo di taglio], thus reducing their length scale; the

conservation of angular momentum tends to maintain the good correlation between

velocities (as required in the Reynolds stress definition) and amplifies the vorticity

component w. At some point, eddies are stretched so much that they break down into

eddies that are further stretched by larger turbulent eddies. 18

Let us assume that the Reynolds number ( is the dynamic viscosity) is large.

= ν

ν

Eddies are characterised by characteristic length (size) ℓ and velocity , which determines

ϑ

1

the eddy time scale (turnover time) and its frequency .

τ = = τ

ϑ

Larger eddies have a characteristic size ℓ comparable to the characteristic length scale of

0

the process L . Assuming that characteristic velocity of larger eddies, , is also

ϑ

C 0

comparable to the process characteristic velocity U , then

C

→ larger eddies are inviscid.

≈ = >> 1

0 ϑ

Energy is transferred to smaller and smaller eddies until the eddy Reynolds number = ν

is “small” and kinetic energy is dissipated by viscous action.

If it is assumed that the energy cascade is a steady-state process, then the turbulent energy

2

⎡⎢ ⎤

dissipation rate at the lower scale is defined by the energy transfer rate at the

ε = ⎥

3

⎣ ⎦

largest scale.

3

2

The ratio is often used to represent a characteristic length scale for turbulence.

ε

Similarly, can be used to represent a characteristic time scale.

ε

TURBULENT ENERGY DISSIPATION

RATE

Let’s consider the Navier-Stokes equations for flows at constant density and viscosity with

constant U and considering its fluctuations u’:

After multiplying by u’ and averaging, this expression is obtained:

In the case of homogeneous turbulence, turbulent statistics do not vary in space. Thus

(when the gradient is inside the average it’s not zero): 19

demonstrating that kinetic energy (per unit of mass) k is dissipated by viscous forces. Also

note that in this example we are not considering a steady-state process.

KOLMOGOROV’s HYPOTHESES

LOCAL ISOTROPY

Larger eddies are anisotropic because of the significant interaction with the mean flow,

however the diffusive effect of viscosity tends to decrease the directionality of those eddies,

thus we can write hypothesis I:

If the Reynolds number is large enough, then the small scale motion of turbulent flows

( ) is statistically isotropic.

<< 0

Commonly, it can be said that below scale ( ) information on directionality is lost

0

≈ 6

and turbulence can be described in a universal way: all turbulent flows at large Re can be

described according to a common approach.

This scale interval ( ) is called universal equilibrium range.

<

FIRST SIMILARITY

In the universal equilibrium range, the key phenomena are energy transport and viscous

dissipation, thus the hypothesis II is:

In every turbulent flow at large Reynolds number, the statistics of small scale motion

( ) have a universal form, which is uniquely determined by and .

< ν ε

The important parameters for Kolmogorov are: 1

( )

3 4

ν

●​ the Kolmogorov scale (or length) η [

] = ε

1

●​ the Kolmogorov velocity 4

ζ = ( εν ) 1

( )

ν 2

●​ the Kolmogorov time scale τ = ε

η ηζ

By the definition of these parameters, so the Kolmogorov scale represents

= = 1

ν

η

smaller eddies, where viscous dissipation is effective. 20

EXAMPLE

In an approximate way, may be described as energy dissipation due to friction. As a matter

ε

of example, let us consider water flowing at a velocity of a 2 m/s through a steel pipe with a

diameter of 10 cm: estimate (average) and the Kolmogorov scale .

ε η

We assume that the energy dissipation is represented by the pressure drops.

2

∆

To find pressure drop we use the fanning equation .

=− 4 2

ρ

To find the friction factor we compute at first the Reynolds number

5 −0,24

2·0,6 so .

= = = 2 · 10 > 2000 = 0, 10 ·

−6

ν 10

The turbulent energy dissipation rate is a dissipated power over a mass. From Fanning

∆ ∆·· ∆·

equation it’s possible to manipulate the correlation: →

− − =−

ρ ρ·· ρ·

2 3

∆· 1

.

ε =− = 4 = 4

2 2

ρ· 3 3 2

1 1

The value is therefore ε = 4 = 4 = 0, 855 3

2 2

1

1 ( )

3

( ) 4

3 −6

( )

4 −5

ν 10

The Kolmogorov scale is .

η η = = = 3, 3 · 10

ε 0,855

According to Kolmogorov, in some specific conditions (i.e., small scale and large Re),

turbulence should only depend on and : thus, the turbulent phenomena related to the

ν ε

dime