Aerodynamics of an inverted wing in ground effect: the influence of cornering

6
Chapter 2
Turbulence modeling
All fows of engineering interest, such as the fow around an aircraft or a
vehicle, are turbulent. Therefore, it is necessary to make a theoretical digression
on turbulence and how it is modeled. In this chapter, a brief description of
turbulent fows will be given, with a focus on the Reynolds Averaged Navier-
Stokes (RANS) equations and the mixing length theory developed by Prandtl.
Additionally, two of the best-known turbulence models will be discussed: the
k-ω model, which will be used in the present work, and the k-ε model.
2.1 Turbulent fows
For "small enough" scales and "low enough" velocities, in the sense that
the Reynolds number is not too high, the equations of motion for a viscous
fuid have well-behaved, steady solutions. Such fows are controlled by viscous
difusion of vorticity and momentum and the resulting motion is called laminar.
Under this regime, if we consider the motion of a fuid in a straight duct, each
fuid particle moves with uniform velocity describing a straight trajectory.
As the Reynolds number increases, the fuid’s inertia overcomes the viscous
stressesandthelaminarmotionbecomesunstable. Theorderedbehaviortypical
of the laminar regime disappears: rapid velocity and pressure fuctuations
occur and the motion becomes inherently three dimensional and unsteady.
When this happens, we describe the motion as being turbulent.
A turbulent fow can be considered as the superposition of a mean and a
fuctuating motion, which produces a momentum transfer in the transversal
direction and gives rise to a strong mixing. The size of the vortices that con-
7 2.2 CFD methods
tinuously form and disintegrate determines the scale of turbulence, and for the
same fuid it can be diferen t depending on velocity, boundary conditions, and
fuid history. Therefore, turbulent transport properties are not a characteristic
of the fuid, but depend on the kind of motion.
Turbulence consists of a continuous spectrum of scales ranging from largest
to smallest (however much larger than any molecular length scale), that are
the Kolmogorov scales. In order to visualize a turbulent fow with a spectrum
of scales we often cast the discussion in terms of eddies. A turbulent eddy can
be seen as a local swirling motion whose characteristic dimension is the local
turbulence scale. We observe that eddies overlap in space, large ones carrying
smaller ones.
Turbulence features a cascade process whereby its kinetic energy transfers
from larger eddies to smaller eddies. Ultimately, the smallest eddies dissi-
pate this kinetic energy into heat through the action of molecular viscosity.
Therefore, we observe that, like any viscous fow, turbulent fows are always
dissipative.
The most important feature of turbulence from an engineering point of view
is its enhanced difusivity. Turbulent difusion greatly enhances the transfer of
mass, momentum and energy. Apparent stresses in turbulent fows are often
several orders of magnitude larger than in corresponding laminar fows [5].
2.2 CFD methods
Given the complexity of turbulent fows, their solution can only be obtained
numerically, by means of Computational Fluid Dynamics (CFD). Diferen t
methods have been developed in order to solve the Navier-Stokes equations.
In descending order of accuracy they are:
• Direct Numerical Simulation (DNS). It is the most general method
and consists of numerically solving the Navier-Stokes equations with an
extremely fne discretization so as to represent all scales of turbulence,
from largest to smallest (the Kolmogorov scales). This is the approach
that provides the most accurate results but has a very high computational
cost because it requires a huge number of calculation points. This method
allows only simple fows to be solved, but not complex fows of engineering
interest such as the fow around a wing.
8 2.3 Reynolds Averaged Navier-Stokes equations
• Large Eddy Simulation (LES). This method has lower computational
costandwiderrange ofapplication withrespecttoDNS.Itconsistsofsolv-
ing only the largest scales, which require a less demanding discretization,
while fner scales are simulated by means of empirical models.
• Reynlods Averaged Navier-Stokes equations (RANS). Finally,
the most widely used methods for problems of application interest are
based on the solution of the Reynlods Averaged Navier-Stokes equations.
In these methods, the turbulence is modeled in all its scales by means
of correlations of experimental data and we simply determine the time-
averaged behavior of the fuid dynamic quantities. However, they require
the use of additional equations to close the problem. This is the approach
on which we will focus in this chapter.
2.3 Reynolds Averaged Navier-Stokes equations
As already mentioned, a turbulent fow can be considered as the superpo-
sition of a mean and a fuctuating motion. Therefore, when we talk about a
steady turbulent fow we refer to the time invariability of the velocity averaged
over a sufciently long time.
Following the Reynolds’ approach, the velocity component u
i
(x,t) can be
′
split into a mean value u
i
(x) and a fuctuating value u
i
(x,t), so that:
′
u
i
(x,t) =u
i
(x) +u
i
(x,t) (2.1)
where the mean part is given by the time average:
t+T
1
u
i
(x) = lim u
i
(x,t) dt (2.2)
T→∞
T t
From this defnition, it follows that the mean value of the fuctuating part is
zero.
The dynamics of a viscous fow is governed by the Navier-Stokes equations:
∂ρ
+
∂
(ρu
i
) = 0 (2.3)
∂t ∂x
i
∂ ∂ ∂p ∂τ
ij
(ρu
i
) + (ρu
j
u
i
) =− + (2.4)
∂t ∂x
j
∂x
i
∂x
j
9 2.3 Reynolds Averaged Navier-Stokes equations
∂ 1 ∂ 1 ∂ ∂q
j
ρ e +
2
u
i
u
i
+ ρu
j
h +
2
u
i
u
i
= (u
i
τ
ij
)− (2.5)
∂t ∂x
j
∂x
j
∂x
j
where τ
ij
is the viscous stress tensor defned by:
∂u
i
∂u
j
τ
ij
=µ + (2.6)
∂x
j
∂x
i
For an incompressible fow, mass and momentum conservation equations can
be derived from (2.3) and (2.4) with the assumption thatρ is constant in time
and space:
∂u
i
= 0 (2.7)
∂x
i
∂u
i
∂ ∂p ∂τ
ij
ρ +ρ (u
j
u
i
) =− + (2.8)
∂t ∂x
j
∂x
i
∂x
j
The energy conservation equation is not used for incompressible fows.
Let us now substitute (2.1) into the Navier-Stokes equations written in the
steady incompressible case. For the mass conservation equation we get:
∂u ∂v ∂w ∂u
′
∂v
′
∂w
′
+ + + + + = 0 (2.9)
∂x ∂y ∂z ∂x ∂y ∂z
Taking the time average of the entire equation, and considering that the mean
value of the last three terms is zero, we obtain:
∂u ∂v ∂w
+ + = 0 (2.10)
∂x ∂y ∂z
It can be noticed that (2.10) is (2.7) written for the mean velocity.
Let us then consider the x component of the momentum conservation
equation written in the steady incompressible case:
∂u
2
∂uv ∂uw ∂p
ρ + + =− +µ∇
2
u (2.11)
∂x ∂y ∂z ∂x
Taking the time average of the equation we obtain:
∂u
2
∂uv ∂uw ∂p ∂u
′2
∂u
′
v
′
∂u
′
w
′
ρ + + =− +µ∇
2
u−ρ + + (2.12)
∂x ∂y ∂z ∂x ∂x ∂y ∂z
Except for the last term containing the fuctuating velocities, (2.12) is formally
identical to (2.11), where the various quantities are replaced with their mean
values.
10 2.4 The closure problem
2.4 The closure problem
We can write the equation (2.12) in terms of stresses (including also the
unsteady term) so as to get:
Du ∂p ∂ ∂ ∂
ρ =− + τ
xx
−ρu
′2
+ τ
xy
−ρu
′
v
′
+ τ
xz
−ρu
′
w
′
(2.13)
Dt ∂x ∂x ∂y ∂z
This can be done in a similar way with the other two components of the
momentum conservation equation.
We can observe that the terms containing the fuctuating velocities may be
interpreted as additional stresses that are called turbulent stresses or Reynolds
stresses. Thus in turbulent motion, in addition to the stress tensor present in
laminar motion, we must consider the Reynolds stress tensor:
′ ′
u
′2
u
′
v u
′
w
′
T =−ρ u
′
v
′
v
′2
v
′
w
′
(2.14)
′ ′ ′ ′ ′2
uw vw w
Therefore, a turbulent motion can be described by the same equations used for
laminar motion, provided we replace the instantaneous quantities with their
time-averaged values and include the additional turbulent stresses.
The Reynolds stress tensor is a symmetric tensor and thus has six indepen-
dent components. Hence, as a result of Reynolds averaging, we have produced
six new unknown quantities but no additional equations. In fact, for general
3D incompressible fows, we have four unknown mean-fow properties, i.e. pres-
sure and the three velocity components. Along with the six Reynolds-stress
components, we thus have ten unknowns. Our equations are mass conservation
and the three components of momentum conservation, for a total of four. This
means that our system is not closed and the problem is not mathematically
solvable except through the introduction of additional closing equations.
2.4.1 Boussinesq hypothesis
Most of turbulence models are based on the hypothesis made by Boussinesq,
who exploited the analogy between turbulent and viscous stresses and proposed
to write the turbulent stresses as a function of the mean velocity gradients:
∂u
i
∂u
j
2
′ ′
−ρu
i
u =µ T
+ −
3
ρkδ
ij
(2.15)
j
∂x
j
∂x
i
11 2.5 Mixing length theory
where µ T
is the turbulent viscosity, also known as eddy viscosity, and k is the
turbulence kinetic energy, defned as:
′ ′
k =
2
1
u
i
u
i
=
1
u
′2
+v
′2
+w
′2
(2.16)
2
Unlike the molecular viscosity µ , the eddy viscosity is not a property of the
fuid but depends itself on the mean velocity and the nature of the fow.
The Boussinesq hypothesis allows us to reduce the number of additional
unknowns: the six components of the Reynolds stress tensor are replaced by
only one variable, the eddy viscosity µ T
, which, however, must be properly
modeled.
2.5 Mixing length theory
An early modeling of the turbulent viscosity was proposed by Prandtl, who
introduced the additional concept of mixing length [9].
In order to better understand Prandtl’s theory, let us consider the two-
dimensional problem of a fat plate invested by a uniform fow, for which the
conservation equations in the steady incompressible case are written as:
∂u ∂v
+ = 0 (2.17)
∂x ∂y
∂u ∂u 1∂p ∂ ∂u
′ ′
u +v =− + ν −uv (2.18)
∂x ∂y ρ∂x ∂y ∂y
According to Boussinesq hypothesis, the last term of (2.18) can be written as:
′ ′
∂u
−uv =ν
t
(2.19)
∂y
=
µ T
where ν
t
ρ
is the kinematic turbulent viscosity.
Prandtl visualized a simplifed mechanism for turbulent motion, which can
be schematized as follows. Fluid particles coalesce into lumps that move as a
unit. With reference to Figure 2.1, a fuid lump located at height y
0
+l with
velocity u(y
0
+l) moves crosswise to the main motion and goes from y
0
+l to
y
0
displacing the fuid that was there and preserving his velocity.
Length l was called mixing length by Prandtl and it represents the length
scale of turbulent eddies, that are assumed to be isotropic.