Direct Numerical Simulations of Turbulent Lean Premixed Hydrogren Flames with Detailed Chemistry

INTRODUCTION 7
simulations, several parameters have to be carefully calibrated so to min-
imize the gap between numerical results and experiments. This operation
was effectuated through one-dimensional and two-dimensional codes, which
do not require excessive processing capacities and whose results can be easily
comprehended and compared with a wide literature. In addition, they set
up reference values for three-dimensional flames, enhancing the evaluation
of typical turbulent fluctuations and the comprehension of non-negligible
diffusional processes. Nevertheless these considerations are not enough to
characterize properly a turbulent flame, whose description must take into
account a computation of the averaged fields, thus it is essential to perform
a statistical analysis capable of highlighting the relations among thermody-
namics, fluid-dynamics and chemistry. Results showed a strong dependence
of OH radicals on the progress variable2 and the magnitude of its gradient,
called usually “Surface Density Function” (SDF); in addition the dynamics
of the flame results deeply affected by strain effects and curvature, which
help to understand peculiar behaviors as the increase of the local equiva-
lence ratio in some zones or the quenching effects. At last, even the heat
release has a deep influence on combustion, in fact it was observed to be an
excellent quantity to summarize efficiently many phenomena.
2
it is a dimensionless quantity that distinguish burnt and unburnt gases through tem-
perature or hydrogen concentration
Chapter 1
Introduction to turbulence
In this first chapter it is briefly exposed the theory of turbulence. Reynolds
was the first scientist able to describe efficiently its intrinsic chaotic nature,
using a statistical approach. Later Richardson started to introduce math-
ematical formalism, deducting the famous “energy cascade” concept. More
recently, Kolomogorov revisited the entire theory of turbulent flows apply-
ing his knowledge of probability and statistics and deeply contributing to the
major developments in this complex scientific field. In the last part of the
chapter the main computational approaches for turbulent flows will be dis-
cussed, focusing on the pros and cons of every technique.
1.1 Phenomenology
Turbulence is a particular fluid regime which has a high level of complexity
compared with the laminar flows,although it is important to underline that
it is actually a solution of the Navier-Stokes equations; there are many op-
portunities to observe turbulent flows in everyday surroundings, whether it
be smoke from a chimney, water in a river or waterfall, or the buffeting of a
strong wind.
A main characteristic of turbulent flows is the prevalence of the inertial
forces over the viscous ones. This is the reason why turbulent flows are char-
acterized by a chaotic motion and the fluid properties change significantly
and irregularly both in space and time. An important issue to discuss is the
consistency between the random nature of turbulent flows, and the determin-
istic nature of classical mechanics, embodied in the Navier-Stokes equations;
the explanation lies in the acute sensitivity of turbulent phenomena to the
initial and boundary conditions conditions as well as material properties.
These physical quantities are unavoidably subject to perturbations so even
8
CHAPTER 1. INTRODUCTION TO TURBULENCE 9
Figure 1.1: Flow past a sphere at (a) Re = 2 · 104; (b) Re = 2 · 105.
[Photograph by H. Werle of ONERA, courtesy of J. Delery]
Figure 1.2: Two turbulent jets. Note the similar behavior between two flows
upstream the jet and the evident difference on vortex structures generated.
slight alterations, caused by turbulent fluctuations, affect considerably the
fluid motion. In order to deal with this topic, it is important to recall Navier-
Stokes equations for Newtonian incompressible flows.
Let us consider a spatial domain D ∈ R3 and a temporal interval [0, T ] ∈ R,
where the scalar field ρ (x, t) satisfies the continuity equation:
∂ρ
∂t
+∇· (ρu) = 0 (1.1)
Since turbulence is basically an incompressible phenomenon, Eq. (1.1)
can be rewritten as:
∇·u = 0 (1.2)
CHAPTER 1. INTRODUCTION TO TURBULENCE 10
It is important to underline that the result obtained in Eq. (1.2), points
out the solenoidal structure of velocity field. Furthermore, the velocity
u(x, t) satisfies the Cauchy equations for a generical fluid:
∂u
∂t
+ u· ∇u = ∇·T + 1
Fr2
f (1.3)
where f are the resultant of the external forces, while Fr is Froud num-
ber, a dimensionless value defined as :
Fr = U0√
F0 L0
The quantity T is the stress tensor, which, for a Newtonian fluid, assumes
the following form:
T = −pI + 1Re
[
(∇u) + (∇u)T
]
+
(
2
3
1
Re −
1
Re
κ
µ
)
∇·uI (1.4)
where κ is the viscous dissipation due to normal Shear Stress, while Re
is Reynolds number, given by the ratio between inertial forces and viscous
ones:
Re = U0L0 ρ0
µ
Substituting Eq. (1.4) in (1.3), considering (1.2), the following expression
1.5 can be obtained for the momentum conservation:
∂u
∂t
+ u· ∇u = −∇p+ 1Re∇
2u+ 1
Fr2
f (1.5)
where the scalar p (x, t) is the pressure. Applying the divergence to Eq.
(1.5) and considering that u is a solenoidal , as expressed in 1.2, it is possible
to write an equation for p:
∇2p = −ρ0∇⊗∇ : u⊗ u+ ρ0∇·f (1.6)
In the momentum equation 1.5 each term has a proper physical meaning:
CHAPTER 1. INTRODUCTION TO TURBULENCE 11
• the first term on LHS represents the temporal evolution of an instan-
taneous field of velocity;
• the second term on LHS is the convection of the velocity field, which
is nonlinear and responsible for fluctuations;
• the first term on RHS describes the effect of the gradient of the pressure
on fluid;
• the second term on RHS represents the effect of viscous forces, that
is the momentum diffusion and the dissipation of kinetic energy into
thermal energy
• the last on RHS is the mass force term.
Obviously, this system of equations 1.2-1.5 needs specific initial and
boundary conditions to be resolved. The initial conditions, which should
be necessarily a solenoidal field, can be generally expressed, in the domain
D, as:
u (x, t) = u0 (x) per x ∈ D (1.7)
Instead, concerning boundary conditions, both velocity or strain can be
assigned. It is in fact possible to divide the boarder of the domain in a
part where the quantity assigned is the velocity and another where it is the
strain:
u (x, t) = u∂ (x, t) per x ∈ ∂DV , (1.8)
t (x, t) = t∂ (x, t) per x ∈ ∂DT (1.9)
The system of equations (1.2)- (1.5), together with the boundary con-
ditions (1.8), (1.9), and initial conditions (1.7), is said to be a well posed
problem, according to Hadamard, if:
• the solution exists,
• the solution is unique,
• the solution depends continuously on the data
CHAPTER 1. INTRODUCTION TO TURBULENCE 12
For Navier-Stokes equations, none of these three points has been mathe-
matically demonstrated so far, although experimental data suggest that first
two points are generally verified. The third condition,in particular, is strictly
related to the sensitivity to the initial data. Let us consider two generic ini-
tial states uo (x) and u′o (x), which differ for a small value less than an
assigned arbitrarily small δε, and their respective solutions u = u(x, t) and
u′ = u′(x, t); if the maximum of the difference between these last solution
is smaller than ε, then the third point is verified.
if ‖ uo (x)− u′o (x) ‖ ≤ δε (1.10)
⇒ max
t
‖ u (x, t)− u′ (x, t) ‖ ≤ ε ∀t ≥ 0. (1.11)
This is the typical behavior of laminar flows: two “near” initial condi-
tions evolve into two “near” solutions. In a turbulent flow, on the contrary,
this condition takes place only for t ≤ Tε , where Tε is a typical time, called
separation Time, dependent on δε : smaller δε are associated to longer time
length, Tε , beyond which it is not possible to define ε anymore. There-
fore laminar flows are said to be stable; on the contrary for high Reynolds
numbers, after time Tε , which decreases with Re, the solutions are instantly
totally different, so it is said they diverge.
The phenomenon that is observed at a generic instant t > Tε is called devel-
oped turbulence phenomenology ; this fact introduces the existence of stochas-
tic solutions, although it is better to repeat that Navier-Stokes equations are
purely deterministic.
1.2 Statistical approach to turbulent flows
In order to analyze a turbulent flow using a statistical approach as previously
described, it is useful to split a generic fluid dynamic field in two parts: an
average value and a fluctuation term. Let us apply this method, called
Reynolds decomposition, to the velocity field.
ui (x, t) = Ui (x, t) + u′i (x, t) (1.12)
The mean term is defined by the ensemble average1 :
1
sometimes ensemble average is indicated as U i
CHAPTER 1. INTRODUCTION TO TURBULENCE 13
Ui = 〈ui (x, t)〉 = limN→∞
1
N
N∑
n=1
uni (x, t) (1.13)
where N is the number of used realizations. A direct consequence of this
definition is the zero value of the ensemble average of the fluctuating term:
⟨
u′i (x, t)
⟩
= 0 (1.14)
Anyway it is important to underline that the ensemble average of the
square of the fluctuation is generally not null.
⟨
u′i
2
(x, t)
⟩
6= 0 (1.15)
In fact, the root square of this quantity can be used as a significant
reference value of the fluctuations
urms =
√⟨
u′
i
2
⟩
(1.16)
Thus Reynolds decomposition can be applied, in the same way, to every
physical quantity, for example pressure p , the stress tensor T and forces f .
Moreover it is easy to demonstrate that ensemble average commutes with
derivatives and integrals, which leads to the following relations:
〈Ui〉 = Ui (1.17)
⟨
u
′
iUi
⟩
=
⟨
u′i
⟩
Ui (1.18)
1.2.1 Reynolds Averaged Navier-Stokes equations
Substituting Reynolds decomposition for every fluid dynamics fields in the
Navier-Stokes equation and then effectuating the ensemble average, it is
possible to obtain the so-called Reynolds Averaged Navier-Stokes Equations
(R.A.N.S.) for an incompressible flow2
2
from this moment on, the Einstein notation will be used. According to this convention,
when an index variable appears twice in a single term, it implies that we are summing
over all of its possible values