17_11_2008
turbulent kinetic energy (tke) equation
► velocity and pressure are decomposed as:
vi = Vi + vi′
i = 1,2,3
p = P + p′
turbulent kinetic energy (per unit volume):
1
2
2
2
ρ (v'1 ) + (v'2 ) + (v'3 )
2
. = . = turbulence (ensemble) average
k=
1=streamwise
2= vertical
3=spanwise
The equation for the turbulent kinetic energy
∂k
∂k
dk
≡
=
+ Vj
∂x j
dt
∂t
− ρ vi'v'j
∂Vi
∂
−
∂x j ∂x j
ρ
∂ 1
 vi′vi′v′j + v′j p′ − µ
 vi′vi′
2
∂
xj  2

 − µ ∂vi′ ∂vi′

∂x j ∂x j

is obtained from Navier-Stokes equation for incompressible flows.
Turbulent kinetic energy equation- physical interpretation
dk
∂Vi
∂
= − ρ vi'v'j
−
∂x j ∂x j
dt
ρ
∂k 
 vi′vi′v′j + v′j p′ − ν
 −ε
∂
2
x j 

In the turbulent kinetic energy equation different terms can be
recognized:
dk
dt
− ρ vi'v'j
∂
∂x j
variation (material derivative) of t.k.e with time
∂Vi
∂x j
production term
ρ
∂k 
 vi′vi′v′j + v′j p′ − ν

∂x j 
 2
redistribution term
ε DT pseudo- dissipation term (related to dissipation due to
turbulent fluctuations)
17_11_2008
k-ε turbulence model
k-ε turbulence model (Launder & Sharma, 1974) is one of the
most widely used 2-equation turbulence model
(
)
1
ρ v'1 v'1 + v'2 v'2 + v'3 v'3
turbulent kinetic energy (per unit volume)
2
∂v′ ∂v′
ε =µ i i
pseudo - dissipation
∂xn ∂xn
k=
two equations are introduced to compute space and time
development of k and ε.
then:
µ = ρClu 0 ⇒ l =
where Cµ is an
ρ
− 12
ε
k
3
2
1
;u0 = k 2 ρ
− 12
; C = Cµ
⇒ µT = C µ
k2
ε
empirically determined constant
Model equation for k
The equation for k is derived from that for turbulent kinetic energy:

ρ
dk
∂Vi
∂ 
∂ 1
∂vi′ ∂vi′
= − ρvi′v′j
−
vi′vi′  − µ
 v′j vi′vi′ + v′j p′ − µ
dt
∂x j ∂x j 
2
∂x j 2
∂x j ∂x j

14
4244
3
1444444
424444444
3 14243
T
j
ε
ˆ
P
The production term P is modeled on the basis of Boussinesq hypothesis:
 ∂V ∂V j
− ρ vi′v′j = µT  i +
 ∂x
 j ∂xi
⇓
− ρ vi′v′j
 2
 − δ ij k
 3

∂Vi ˆ   ∂Vi ∂V j
= P =  µT
+

∂x j
  ∂x j ∂xi

 2
 − δ ij k  ∂Vi
 3

 ∂x j
The term T is modeled on the basis of empirical arguments
r
ν
µ
T = − T ∇k where
σ k = 1 and ν T = T
σk
ρ
17_11_2008
Model equations
Model equation for k
 ∂V
dk   ∂Vi ∂V j  2
∂  µT ∂k 
=  µT
+
− δ ij k  i +

 −ε


dt   ∂x j ∂xi  3
 ∂x j ∂x j  σ k ∂x j 
1444442444443
ˆ
P
or equivalently
µ

dk ˆ
= P − ε + ∇. T ∇k 
dt
σ k

The model equation for ε is similar to that for k and is entirely
based on empirical arguments:
ˆε
µ

dε
P
ε2
= cε 1
− cε 2
+ ∇ . T ∇ ε 
dt
k
k
σ ε

with
c ε 1 = 1 . 44 , c ε 2 = 1 . 92 σ ε = 1.3 σ k = 1
k-e model: summary of equations to be used
Model equation for k
∂Vi
= 0 ; i = 1,2,3 (different directions) sum for m, j = 1,2,3
∂xi
 ∂Vi
ρ
 ∂t
+ Vj

∂Vi 
∂P
∂ 2Vi
∂ 
k 2  ∂Vi ∂Vm  2

 − δ mi k 
+µ
+
Cµ
−
 = ρf i −


ε  ∂xm ∂xi  3
∂x j 
∂xi
∂xm ∂xm ∂xm 

dk  k 2  ∂Vi ∂V j
= C µ
+
ε  ∂x j ∂xi
dt 

 2
 − δ ij k  ∂Vi + ∂
 3
 ∂x j ∂x j


k 2 ∂k 
C µ
 −ε
 εσ k ∂x j 
Continuity eq.
Reynolds eq.
(3 scalar eq.)
equation for k
2
2


∂V j  2
dε
k 2  ∂ V i
∂ε 
ε ∂Vi 
 − δ ij k  − c ε 2 ε + ∂  C µ k
= cε 1
+
C µ

dt
k ∂ x j 
∂ x i  3
k
∂
x
∂
x j 
εσ
ε  ∂ x j

ε
j 

with
C µ , c ε 1 , c ε 2 , σ ε , σ k empirical constants
equation for ε
Please note: 6 unknowns (V1,V2,V3, P, k, ε) and 6 equations !!!
17_11_2008
Boundary conditions
While the boundary condition to be used for the mean velocity is derived from
the no-slip condition, it is not obvious which condition should be imposed upon ε
(and k)
But
Assuming that the mean flow close to the bottom is nearly parallel to the wall
the law of the wall (log velocity profile, described in the previous lecture) is
used to derive the boundary conditions (empirical assumption)
It is further assumed that near the wall the budget of turbulent kinetic
energy is such that Production=dissipation
(empirical assumption based on channel flow results)
Therefore the turbulence model will be used only to compute the flow field in a
region located far from the wall (where we assume the log law to be valid)
while in the region closer to the wall the log-law is assumed to be valid. (is
this always true for BBL ?)
Boundary conditions for k and ε
Law of the wall (see lecture 1):
Generally assumed to be valid for
V1 ( y ) =
30
 y
ln  κ = 0.41
κ  y0 
y0 = roughness length
uτ = friction velocity
uτ
ν
uτ
< y < 0.3D
D = half channel width
If the first grid point is located at a distance from the wall such that
it is possible to assume that the log-law holds, boundary conditions for
k and ε can be derived
Further advantage: the region closer to the wall, where large gradients
are encountered and viscous effects are relevant is not computed
Other hypothesis introduced: production=dissipation in the turbulent
kinetic budget close to the wall
17_11_2008
Channel flow: near-wall region
y+ =
y
ν uτ
ν
= length scale
uτ
uτ =
∂V1
∂
− ρ v1′v2′
−
∂x2 ∂x2
14
4244
3
Production
τ
ρ


ρ r r
∂k 
∂vi′ ∂vi′
=0
 v′2 v′ ⋅ v′ + v′2 p′ − ν
−µ
1
2
3
2
∂
x
∂
x j ∂x j
 14243 press. trans. 1232  142
43
 turb. conv.
viscous diff.
 Pseudo dissipation
Turbulent kinetic energy in the BBL (unsteady flow)
Turbulent kinetic energy depends on time
Production term
Pseudo-dissipation term
x2
x2
δ
δ
− ρ vi′v′j
∂Vi
∂x j
(ρU ω )
2
0
−µ
∂vi′ ∂vi′
∂x j ∂x j
(ρU ω )
2
0
Rδ=800
• Production is maximum slightly after the maximum of velocity
• Dissipation is maximum slightly after the maximum of velocity
but later than production
17_11_2008
boundary conditions ε
The boundary conditions to be applied close to the wall (y=y0) for k and ε are
derived assuming that:
•the log law holds close to the bottom
•In the near-wall region the turbulent kinetic energy production = dissipation
ˆ = − ρ v1′v′2 ∂V1 ≈ τ 0 ∂V1 = ρuτ2 ∂V1
P
∂x 2
∂x 2
∂x 2
but V1 ( x2 ) =
uτ  x2 
ln  
k  ˆ
x2 
uτ =
τo
friction velocity
ρ
k = 0.41 ; τ o = bottom shear stress
u3
∂V1 uτ 1
=
⇒ ε =P=ρ τ
∂x 2
κ x2
κx2
Boundary condition for ε
boundary condition for k
since in the region closer to the wall :
τ ≈ τ 0 = µT
∂ V1
∂x 2
P = − ρ v1′v ′2
but µ T = C µ
y=0
∂ V1
∂V
≈ τ0 1
∂y
∂y
k
ε
2
⇒ P =
≈
y=0
τ 02 ε
Cµ k 2
=
τ 02
µT
ε (assumption )
therefore
k2 =
τ 02
Cµ
⇒ k =
ρ uτ2
Cµ
Boundary condition for k
N.B. initial conditions and conditions far from the wall depend on the
problem
17_11_2008
k-ε predictions of oscillatory BBL
The comparison of k-ε predictions with
experimental data (test13 of Jensen
et. al (1989), rough wall) and with the
prediction of other models (a.o. e-ω
model, to be described) shows that k-ε
model (medium thick line) has
difficulty to predict correctly
turbulent kinetic energy in the region
closer to the bed
From Puleo et. al. 2004
e-ω model: formulation for 1D flow (BBL)
Momentum equation
∂V1 ∂U ∞
∂
=
+
∂t
∂t
∂x2

∂V1 
(ν + ν T )

∂x2 

The eddy viscosity is given in terms of pseudo-energy (e), related to
turbulent kinetic energy, and pseudo-vorticity (ω):
νT = γ
e
ω
Equation for pseudo energy:
∂e
∂V
∂ 
∂e 
= α e e 1 − β e eω +
(ν + σ eν T )

∂t
∂x2
∂x2 
∂x2 
Equation for pseudo vorticity:
∂ω 2
∂V
∂
= α ωω 2 1 − βω ω 3 +
∂t
∂x2
∂x2
γ, αe,, βe, σe, αω, βω and σω are universal constants

∂ω 2 
(ν + σ ων T )

∂x2 

17_11_2008
Eddy viscosity models: e-ω model – boundary conditions
At the wall, velocity and pseudo-energy vanish
u =0
e = 0 for y = 0
While pseudo-vorticity is related to the roughness height yr by means of a
universal function Ω:
ω = Ω 
y r uτ 

 ν 
y r uτ
ν
= roughness
Reynolds number
Note that it is not necessary (as done for the k-ε) model to assume the
validity of the log-law close to the wall (useful in unsteady flows or in flows
characterized by adverse pressure gradients)
Eddy viscosity models: two-equation models (Blondeaux, J. Hydr. Res, 1988)
smooth regime

ν
 y r << 5
uτ


uτ = averaged friction velocity 

Turbulence is generated
explosively near the wall at the
beginning of the decelerating
phase and it propagates far from
the wall
pseudo-energy (smooth regime)
The model correctly provides the friction factor
in the laminar regime
The model works well at high values of the
Reynolds numbers
The transition to turbulence is not predicted
accurately
17_11_2008
Eddy viscosity models: two-equation models
Rough regime

ν
 y r > 5
u
τ




The time development of the wall shear stress is not sinusoidal
τ xy =
τ0
ρ U 20
○ experimental data by Jonsson & Carlsen (1976)
Good predictions of the friction factor
U0
ωy r
Why is e-ω model popular for modeling BBL ?
Main reasons:
It works also in the phases when the flow
becomes laminar
It does not assume the log law to be valid close
to the wall
17_11_2008
Concluding remarks on RANS models
RANS MODELS PROVIDE:
Averaged velocity/pressure
Turbulent kinetic energy
Reynolds stresses …….
BUT THEY DO NOT PROVIDE
information on quantities which depend on the particular realization
considered (e.g. turbulent eddies/vortex structures important for
suspension events)
MOREOVER
They are based on empirical assumptions and may be inaccurate
(particularly close to the walls)
Direct Numerical Simulation (DNS)
The most accurate (and computationally expensive) “model”
is Direct Numerical Simulation (DNS)
DNS uses the most general governing equations (NavierStokes equations + continuity equation + boundary
conditions) and solves them numerically without empirical
assumptions. It is in many ways similar to an experiment !!
ADVANTAGES:
• no empirical assumption is introduced
• detailed knowledge of all the flow quantities
• access to all derived quantities (e.g. vorticity, turbulent kinetic
energy…..) and to coherent turbulent vortex structures
17_11_2008
length scales of turbulent eddies
DNS solves for the actual values of velocity and pressure
therefore computes exactly the turbulent eddies at all scales
Large eddies are characterized by
scales comparable to the mean flow
small eddies (dissipative) have scales
unrelated to those of the mean flow
PROBLEM: determine the scale of the
smaller eddies
RELEVANCE FOR DNS: determination
of the computational grid
small eddies: Kolmogorov scale
Hypothesis: the characteristics of small eddies are not influenced by the geometry of
the particular flow considered but they are determined by:
the kinematic viscosity of the fluid (ν)
the energy dissipation rate (per unit mass) (ε)
the order of ε is:
U3 L
(Andrej Nikolaevič Kolmogorov 1903-1987)
where K = eddy wavenumber
L=spatial scale of the average flow field≈ spatial scale of macrovortices
U= velocity scale of the average flow field ≈ velocity scale of macrovortices
Therefore any quantity F (influenced by the small eddies)
can be expressed as: F = F ( K , ε ,ν )
1
which in dimensionless form becomes:
F1
ε 1 4ν 5 4
= f ( Kη )
1
with
where η = length scale of the small eddies (Kolmogorov scale)
ν 3  4
η =  
ε 
17_11_2008
scales of small turbulent eddies
L
η
=
L
(ν ε )
3
1
4
UL 
= 

ν 
34
= Re 3 4
since ε ≈
U3
L
therefore it is easy to guess which of the following flows has the
highest Reynolds number:
Re=2300
Re=11000
Flows with higher Re require a finer computational grid (i.e. more
gridpoints and larger computational times)
DNS of the bottom boundary layer (Vittori & Verzicco, J.Fluid Mech. 371,
1998)
The equations (Navier-Stokes and continuity) are solved, to compute actual velocity and
pressure, numerically on a regular grid introduced in the computational domain
•Different numerical algorithms can be used to integrate the equations
•To solve the problem numerically it is necessary to fix the size of the computational box.
Example:
L x1 = 25 . 13 δ
L x 2 = 25 . 13 δ
L x 3 = 12 . 57 δ
and the number of grid-points Nx1, Nx2, Nx3 (example Nx1=64, Nx2=64, Nx3=32)
17_11_2008
DNS- Fractional-step scheme
Governing equations:
(Kim & Moin (1985), Orlandi(1989) and Rai & Moin(1991)).
r
∇⋅v = 0
r
r
∂v
+
H
{
∂ t non linear
r
G
{
=
terms
+
pressure terms
r
L
{
viscous terms
r
n
n
Assume the pressure and velocity fields known at time n ( p , v )
compute the fields at time n+1
and
The method consists of three steps
1st step: compute an intermediate velocity field (non divergence free)
2nd step: force continuity equation (Poisson equation)
3rd step: compute velocity and pressure at the new time level.
Fractional-step scheme
Governing equations
r
∇ ⋅v = 0
r
r
∂v
+
H
{
∂ t non linear
=
terms
r
G
{
pressure terms
+
r
L
{
viscous terms
1st step: compute an intermediate velocity field (non solenoidal)
(
ˆv i − v in
1 ˆ
= 1 . 5 H in − 0 . 5 H in −1 − G in +
L i + Lni
∆t
2
2nd step: define the scalar function Φ :
which should satisfy the equation :
3rd step: compute velocity and
pressure at the new time level:
v in +1 − ˆv i
= − G i Φ n +1
∆t
∇ 2 Φ n +1 =
∇ ⋅ ˆv
∆t
∆t 2 n +1
 n +1
n
n +1
p = p + Φ + ∇ Φ
2

v n +1 = ˆv − G Φ n +1∆t
i
i
 i
)
17_11_2008
DNS: numerical algorithm
MOREOVER
• spatial derivatives are approximated by 2nd order finite differences
• symmetry condition is forced at the upper wall (free-slip boundary)
• no-slip condition is forced on the lower wall
• periodicity conditions are forced in the two horizontal directions
In order to filter out random turbulent fluctuations from computed
quantities a phase-average procedure is used:
f ( x2 , t ) =
1
NP
NP

1
n =1  x1 Lx 3
∑L

L x 1 Lx 3
∫ ∫ f (x , x , x , t + 2πn )dx dx 
1
2
0 0
3
1
3

DNS results
Disturbed laminar regime
Intermittently turbulent regime
Vittori & Verzicco, JFM, 371, 1998
Re=1.25 105 Rδ=500
Re=5 105 Rδ=1000
17_11_2008
DNS: vertical profiles of the shear stress
_____ viscous stress
-- - - turbulent component
● experimental results by
Akhavan et al. (1991)
• Viscous component is relevant
only in the region closer to the
wall (O~δ)
• Turbulent component is relevant
in region of O~15 δ
• Turbulent stresses start to grow
at the end of the accelerating
phase and reach the maximum
during the decelerating part
Modeling of the bottom boundary layer: DNS
Shear stress at the wall
Rδ=740
Rδ=1120
(measurements by Jensen et al., 1989)
turbulent stresses are larger during the decelerating part of the cycle and their intensity
changes from one cycle to the next particularly for smaller Re
17_11_2008
Coherent structures in the near-wall region
Turbulence can be seen as tangles of vortex filaments
Turbulent shear flows have been found to be dominated by spatially coherent,
temporally evolving, vortical motions called coherent structures
A coherent motion (structure) can be defined as: a three-dimensional region
of the flow over which at least one fundamental flow variable exhibits
significant correlation with itself or with another variable over a range of
space and/or time that is significantly larger than the smallest local scales
of the flow.
For steady flows coherent structures are observed in the “wall region” (y+ <
100) which includes the viscous sublayer (y+< 5 ), the buffer region
(5<y+<30) and part of the logarithmic region (y+> 30)
y+ =
the study of coherent structures is important:
yuτ
ν
uτ = friction velocity
a. to aid predictive modeling of turbulence statistics
b. to understand the mechanism of sediment pick-up and improve sediment
transport predictors
BBL: coherent structures
Experimental visualizations of coherent structures in oscillatory
boundary layers
(Sarpkaya JFM 253, 1993)
Rδ ≈ 420-460
Re ≈ 8.8 104- 1.1 105
17_11_2008
Coherent structures in BBL
Moderate Rδ (Re)
(Costamagna, Vittori & Blondeaux JFM 474, 2003)
600 <≈ Rδ <≈ 1000
t = 42.47 π
t = 42.17 π
-
1.8 105 <≈ Re <≈ 5 105
t = 42.52 π
( Rδ=800 Re=3.2 105 x1=25.13 δ x2=0.1 δ )
Similar dynamics has been observed in steady boundary layers and by
Sarpkaya (1993) for oscillatory BL
DNS: advantages/disadvantages
ADVANTAGES (already mentioned):
• no empirical assumption is introduced
• detailed knowledge of all the flow quantities
• access to all derived quantities (e.g. vorticity, turbulent kinetic energy…..) and to
coherent turbulent vortex structures
DISADVANTAGE:
Computationally expensive:
• Large number of grid points necessary (also the small vortices should be
computed !!)
•The number of gridpoints necessary increases with the Reynolds number
(only moderate Reynolds numbers can be simulated)
• Limited to smooth wall
• It is prohibitively expensive for large domains (already very expensive for
computing flow over vortex ripples)
17_11_2008
Large eddy simulation
LES is a turbulence model which stands between eddy viscosity
models and DNS
Inertial subrange
RANS
Viscous cut_off
Large eddy simulation
The main steps of a LES model are:
Introduce an appropriate filter function G(r) so that it is possible
to filter velocity, pressure …. and equations
r
r
r r
r
U ( x, t ) = ∫ G (r )U ( x − r , t )d r
U = velocity component, pressure ....
U (x )
r
r
r
r
U ( x, t ) = U ( x, t ) + u ′( x, t )
note that
r
u ′( x, t ) ≠ 0
∆ = filter width
17_11_2008
Large eddy simulation
∂vi
=0
∂x j
Filtered equations:
∂v j
∂t
Note that
vi v j ≠ vi v j
+
continuity
∂ vi v j
∂xi
=ν
−
∂xi ∂xi
1 ∂p
ρ ∂x j
momentum
therefore we introduce the residual stress tensor:
τ ijR = vi v j − vi v j ⇒
DEFINE:
∂ 2v j
∂ vi v j
=
∂xi
R
∂
(vi v j ) + ∂τ ij
∂xi
∂xi
k = 1 τ R residual kinetic energy
 r 2 ii

τ ijr = τ ijR − 2 kr δ ij anisotropic residual - stress tensor

3
The filtered momentum equation becomes:
∂ (vi v j )
+
+
∂t
∂xi
14
4244
3
∂v j
∂τ ijr
∂xi
{
to be modeled
Dv j
∂ 2v j
1 ∂p 2 ∂k r
=−
−
+ν
ρ ∂x j 3 ∂x j
∂xi ∂xi
1442
443
−
Dt
1 ∂  2
p + ρ k r 
3

ρ ∂x j 
Large eddy simulation: Smagorinsky model (1963)
2) Introduce an appropriate model for the anisotropic residual stress tensor
τ ijr = −2ν r Sij
1  ∂vi ∂v j 
+
filtered rate of strain
2  ∂x j ∂xi 
with Cs = Smagorinsky coefficient proportional to the filter width ∆
ν r = (Cs ∆ )2 S
S ij ≡
S ≡ (2 Sij S ij )
12
characteristic filtered rate of strain
∆ = filter width
The momentum equation becomes
Dv j
Dt
=ν
∂ 2v j
∂xi ∂xi
−
r
1 ∂p ∂τ ij
−
ρ ∂x j ∂xi
with
Dv j
Dt
=
∂v j
∂t
+ vi
∂v j
∂xi
(the term related to the residual kinetic energy has been included in the dynamic
pressure term)
3) Substitute the modeled anisotropic residual stress tensor into the equations and
obtain a set equation for the filtered velocity and pressure fields which can be
solved numerically
17_11_2008
Large eddy simulation: plane bed velocity profiles
Velocity (averaged) profiles
Re=6 x 106
Rδ≈3500
Dots: experimental results by Jensen et al.,
1989
Solid line : LES
Dashed line: k-ε model
(From Lohmann et al., J. Geophys. Res., 111 2006)
LES profiles during decelerating phases perform better than k-ε profiles
During the accelerating parts k-ε model performs better than LES
Small scale bedforms formed under oscillatory flow
2D BEDFORMS
h/l<0.1
h/l>0.1
Rolling-grain ripples
Vortex ripples
Sleath (1984)
Vortex ripples start to appear
17_11_2008
2D vortex ripples
The flow field can be modeled using:
• ψ-ω model (only for laminar flow)
•Discrete vortex model (irrotational flow + empirical model for the
shear layers) (see M.S. Longuet-Higgins , J Fluid Mech 107 (1981)).
• DNS (for laminar/transitional flow)
• LES/RANS models for fully turbulent flow
THINGS TO REMEMBER ON VORTICITY AND 2D FLOWS
•
r
r
ω = ∇ × v = curl (v ) vorticity
r
r
r
r
∂ω r
r
+ v ⋅ ∇(ω ) = ω ⋅ ∇(v ) + ν∇ 2 (ω )
∂t
Note that pressure term is not present !!!
• if the flow is 2D (x-y plane) r
ω = (0,0, ω z )
• if the flow is 2D and incompressible:
r
v = ∇ × (0,0,ψ )
ψ streamfunction
r
ω = ∇ × ∇ × (0,0,ψ ) = −∇ 2ψ
17_11_2008
2D vortex ripples: ψ-ω model (Blondeaux & Vittori, J. Fluid Mech, 1991)
As the flow is 2D it is convenient to solve the problem in terms of the
span-wise component of vorticity ω and of the streamfunction ψ
ψ = stream function
u =
∂ψ
,
∂Y
2 ∂ω ∂ψ ∂ω ∂ψ ∂ω
1  ∂ 2ω ∂ 2ω 
+
−
=
+
Rδ ∂t
∂y ∂x
∂x ∂y Rδ  ∂x 2 ∂y 2 
∂ 2ψ ∂ 2ψ
+ 2 = −ω
∂x 2
∂y
boundary conditions :
∂ψ ∂ψ
=
= 0 at the ripple profile
∂x
∂y
∂ψ
∂ψ
→ 0,
→ sin(t ) for y → ∞
∂x
∂y
v =−
∂ψ
∂x
t = t*
2π
,
T
*
*
(x, y ) = (x , *y ),
δ
......
* = dimensional quantity
Note: laminar flow !!!
2D vortex ripples: laminar flow-field
The coordinate transformation: ξ* = x* +
maps the ripple profile
(
)
(
)
(
)
h* − k* η*
e
sin k* ξ* ;
2
h*
h*
cos k *ξ * ;
x* = ξ * − sin k *ξ *
2
2
2π
with k * = * wavenumber
l
y* =
η* = y* −
into the line η=0, and the fluid domain into a rectangular domain
physical plane
(
h* − k* η*
e
cos k* ξ*
2
transformed plane
Vorticity equation
∂ω Rδ  ∂ψ ∂ω ∂ψ ∂ω 
1  ∂ 2ω ∂ 2ω 
+
−
=
+


∂t 2J  ∂η ∂ξ ∂ξ ∂η  2J  ∂ξ 2 ∂η 2 
A finite difference approximation of vorticity equation is used to
compute ω at time t+∆t once ω at time t is known
)
17_11_2008
2D vortex ripples: laminar flow-field
Given the periodicity in the x (and ξ) direction, the solution can be
expressed in the form
N
ψ (ξ ,η,t ) = ∑ψ n (η,t )e
 j 

N 
i 2πn 
n =1
N
ω (ξ ,η,t ) = ∑ ωn (η,t )e
N
= ∑ψ n (η,t )e ikn ξ
n =1
 j 

N 
i 2πn 
n =1
N
= ∑ ωn (η,t )e ikn ξ
n =1
The relationship between ψ and ω gives a second equation (Poisson
equation) which is used to compute ψ.
− kn2ψ n +
N
∂ 2ψ n
= −∑ J s ωn −s
2
∂η
s =1
2D vortex ripples: laminar flow-field
The boundary condition for vorticity is first-order accurate (Thom(1933)).
Considering the boundary conditions at the wall:
and the relationship:
−ω =
∂ψ ∂ψ
+
∂x 2 ∂y 2
2
2
− ωk =
at the wall (grid-point k):
For a plain wall, it turns out that
therefore:
ψ k +1 = ψ k −
∂ 2ψ
∂y 2
u =
∂ψ
∂y
=0
k
v=-
∂ψ
∂x
=0
k
k
 ∂ψ 
∆y 2  ∂ 2ψ 
ψ k +1 = ψ k + ∆ y 
 + 2  ∂ y 2  + ....
 ∂y  k

k
∆y 2
ωk
2
assuming ψk=0 it is obtained:
ωk = −
2
ψ k +1
∆y 2
boundary condition for ω
If the wall is not plane, the boundary conditions can be derived similarly as
above
17_11_2008
2D vortex ripples: laminar flow-field
Vorticity contours
Δω=0.15
___ clockwise vorticity;
______
counterclockwise vorticity
Rδ=50, h/l=0.15, a/l=0.75
ωt=π/4
ωt=5 π/4
ωt=π/2
ωt=3 π/4
ωt= 3 π/2
ωt=7 π/4
ωt=π
ωt=2 π
Flow over 2D vortex ripples: e-ω model (Fresdsoe et al., 1999)
right K-ω model by Andersen (1999)
(span-wise vorticity)
left experimental visualizations
17_11_2008
Flow over 2D vortex ripples: LES results
•Considering a pulsating flow Chang & Scotti observed that:
LONGITUDINAL VELOCITY
COMPONENT:
• Good agreement at maximum
forward or reverse flow
• Largest discrepancy at the
trough
• Worse agreement during the
decelerating phase
• Flow reversal occurs earlier
in the RANS than in LES
____ LES
- - - - RANS
Flow over 2D vortex ripples: LES results
•VERTICAL VELOCITY COMPONENT
____ LES
- - - - RANS
17_11_2008
2D vortex ripples: 3D effects
A flow filed oscillating in one direction can form 2D or 3D ripple patterns !
Generally 3D patterns appear after 2D vortex ripples are formed
Solving the fully non-linear 3D equations allows to observe further effects
i.e. 3D flow patterns which may be related to 3D ripples (Scandura, Blondeaux & Vittori,
J.Fluid. Mech. 2000)