Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
17_11_2008
velocity and pressure are decomposed as:
turbulent kinetic energy (per unit volume):
The equation for the turbulent kinetic energy
is obtained from Navier-Stokes equation for incompressible flows.
turbulent kinetic energy (tke) equation
j
i
j
iii
jjjii
jj
iji
jj
x
v
x
vvv
xpvvvv
xx
Vvv
dt
dk
x
kV
t
k
∂′∂
∂′∂−
′′∂∂−′′+′′′
∂∂−
∂∂−
=≡∂∂+
∂∂
µµρρ2
1
2''
pPp 1,2,3i vVv iii ′+==′+=
( ) ( ) ( )average (ensemble) turbulence
vvvk
==
++=
..
'''2
3
2
2
2
12
1 ρ 1=streamwise2= vertical3=spanwise
In the turbulent kinetic energy equation different terms can berecognized:
variation (material derivative) of t.k.e with time
production term
redistribution term
pseudo- dissipation term (related to dissipation due to turbulent fluctuations)
Turbulent kinetic energy equation- physical interpretation
ενρρ −
∂∂−′′+′′′
∂∂−
∂∂−=
jjjii
jj
iji x
kpvvvv
xx
Vvv
dt
dk
2''
dt
dk
j
iji x
Vvv
∂∂− ''ρ
∂∂−′′+′′′
∂∂
jjjii
j x
kpvvvv
xνρ
2
DTε
17_11_2008
k-ε turbulence model
( )
ndissipatio-pseudo x
v
x
v
volume) unit (perenergy kinetic turbulent vvvvvv2
1k
n
i
n
i
∂′∂
∂′∂=
++=
µε
ρ 332211 ''''''
k-ε turbulence model (Launder & Sharma, 1974) is one of the most widely used 2-equation turbulence model
two equations are introduced to compute space and time development of k and ε.
then:
εµρ
ερρµ µµ
22
12
1
0
23
21
0
kC CC ; ku
k uC T =⇒===⇒= −
−
;ll
where Cµ is an empirically determined constant
Model equation for k
x
v
x
vvv
xpvvvv
x
P
x
Vvv
dt
dk
j
i
j
iii
jjiij
jj
iji
434214444444 34444444 214434421ε
µµρρ∂
′∂∂
′∂−
′′
∂∂−′′+′′′
∂∂−
∂∂′′−=
jTˆ
2
1
2
The equation for k is derived from that for turbulent kinetic energy:
The production term P is modeled on the basis of Boussinesq hypothesis:
j
iij
i
j
j
iT
j
iji
iji
j
j
iTji
x
Vk
x
V
x
VP
x
Vvv
kx
V
x
Vvv
∂∂
−
∂∂
+∂∂==
∂∂′′−
⇓
−
∂∂
+∂∂=′′−
δµρ
δµρ
3
2
3
2
ˆ
ρµνσ
σν T
Tkk
T and wherekT ==∇−= 1r
The term T is modeled on the basis of empirical arguments
17_11_2008
Model equations
kPdt
dk
lyequivalent or
x
k
xx
Vk
x
V
x
V
dt
dk
k
T
jk
T
j
P
j
iij
i
j
j
iT
∇∇+−=
−
∂∂
∂∂+
∂∂
−
∂∂
+∂∂=
σµε
εσµδµ
.ˆ
ˆ44444 344444 21
3
2
Model equation for k
The model equation for ε is similar to that for k and is entirely based on empirical arguments:
1.3 92c ,c with
kc
k
Pc
dt
d
k21
T
11441
2
21
====
∇∇+−=
σσ
εσµεεε
εεε
εεε
..
.ˆ
k-e model: summary of equations to be used
x
kkC
xx
Vk
x
V
x
VkC
dt
dk
jkjj
iij
i
j
j
i εεσ
δε µµ −
∂∂
∂∂+
∂∂
−
∂∂
+∂∂=
22
3
2
Model equation for k
constants empirical , ,c ,c,C with
x
kC
xkck
x
V
x
VkC
x
V
kc
dt
d
k21
jjij
i
j
j
i
j
i
σσ
εεσ
εδε
εε
εεεµ
εµεµε
∂∂
∂∂+−
−
∂∂
+∂∂
∂∂=
22
2
2
1 3
2
kx
V
x
VkC
xxx
V
x
Pf
x
VV
t
V
1,2,3jm, for sum)directions (different 1,2,3i ; x
V
mii
m
m
i
mmm
i
ii
j
ij
i
i
i
−
∂∂−
∂∂
∂∂+
∂∂∂+
∂∂−=
∂∂+
∂∂
===∂∂
δε
µρρ µ 3
2
0
22
Continuity eq.
Reynolds eq.(3 scalar eq.)
equation for k
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
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
( )
velocity frictionu
length roughnessy
0.41 y
yuyV
0
==
=
=
τ
τ κκ 0
1 ln
width channel half
.u
30
=
<<
D
Dy 30τ
ν
17_11_2008
Channel flow: near-wall region
ρτ
νν
τ
τ
τ
=
=
=+
u
u
uy
y
scale length
02
2222
121 =
∂′∂
∂′∂−
∂∂−′′+′⋅′′
∂∂−
∂∂′′−
43421321321
43421
rr
4434421ndissipatio Pseudo
j
i
j
i
diff. viscoustrans. press.
conv. turb.Production
x
v
x
v
x
kpvvvv
2xx
Vvv µνρρ
Turbulent kinetic energy in the BBL (unsteady flow)
Turbulent kinetic energy depends on time
Production term Pseudo-dissipation term
• Production is maximum slightly after the maximum of velocity• Dissipation is maximum slightly after the maximum of velocity
but later than production
δ2x
δ2x
( )ωρρ 20U
x
Vvv
j
iji ∂
∂′′− ( )ωρµ 20U
x
v
x
v
j
i
j
i
∂′∂
∂′∂− Rδ=800
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
( )
2
3
22
1
2
22
2
12
2
10
2
121
1
x
uP
x
u
x
V
stress shearbottom ; 0.41k x
x
k
uxV but
velocity friction u x
Vu
x
V
x
VvvP
o1
o
κρε
κ
τ
ρτρτρ
ττ
τ
ττ
==⇒=∂∂
==
=
=∂∂=
∂∂≈
∂∂′′−=
ˆln
ˆ
Boundary condition for ε
boundary condition for k
( )
µ
τ
µ
µµ
ρτ
εετε
µ
µττρ
µττ
C
uk
Ck
therefore
assumption kC
Pk
C but
y
V
y
VvvP
x
V
: wallthe to closer region the in since
2
T
Ty
y
T
220
2
20
2
20
0
10
121
02
10
=⇒=
==⇒=
≈∂∂≈
∂∂′′−=
∂∂=≈
=
=
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
From Puleo et. al. 2004
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-ωmodel, to be described) shows that k-εmodel (medium thick line) has difficulty to predict correctly turbulent kinetic energy in the region closer to the bed
( )
∂∂+
∂∂+
∂∂=
∂∂ ∞
2
1
2
1
x
V
xt
U
t
VTνν
( )
∂∂+
∂∂+−
∂∂=
∂∂
222
1
x
e
xe
x
Ve
t
eTeee νσνωβα
( )
∂∂+
∂∂+−
∂∂=
∂∂
2
2
2
3
2
122
xxx
V
t T
ωνσνωβωαωωωω
ωγ=ν e
T
Momentum equation
The eddy viscosity is given in terms of pseudo-energy (e), related to turbulent kinetic energy, and pseudo-vorticity (ω):
Equation for pseudo energy:
Equation for pseudo vorticity:
γ, αe,, βe, σe, αω, βω and σω are universal constants
e-ω model: formulation for 1D flow (BBL)
17_11_2008
At the wall, velocity and pseudo-energy vanish
While pseudo-vorticity is related to the roughness height yr by means of a universal function Ω:
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)
0 yfor === 00 eu
number Reynolds roughness uy
uy
r
r
=
Ω=
ν
νω
τ
τ
Eddy viscosity models: e-ω model – boundary conditions
smooth regime
Turbulence is generated explosively near the wall at the beginning of the decelerating phase and it propagates far from the wall
Eddy viscosity models: two-equation models (Blondeaux, J. Hydr. Res, 1988)
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
=<< velocity friction averaged τ
τ
νu
uyr 5
17_11_2008
Rough regime
The time development of the wall shear stress is not sinusoidal
experimental data by Jonsson & Carlsen (1976)
Good predictions of the friction factor
>
τ
νu
y r 5
Eddy viscosity models: two-equation models
20
0xy Uρ
τ=τ
ryUω
0
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 (Navier-Stokes 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
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
DNS solves for the actual values of velocity and pressure therefore computes exactly the turbulent eddies at all scales
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:
where K = eddy wavenumberL=spatial scale of the average flow field≈ spatial scale of macrovorticesU= velocity scale of the average flow field ≈ velocity scale of macrovortices
Therefore any quantity F (influenced by the small eddies) can be expressed as:
which in dimensionless form becomes: with
where η = length scale of the small eddies (Kolmogorov scale)
LU 3
( )νε,,KFF =1
( )ηνε
KfF =
45411 4
13
=ε
νη
(Andrej Nikolaevič Kolmogorov1903-1987)
17_11_2008
scales of small turbulent eddies
( ) L
U since
ULLL 343
43
41
3≈=
== ενενη
Re
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)
•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:
and the number of grid-points Nx1, Nx2, Nx3 (example Nx1=64, Nx2=64, Nx3=32)
The equations (Navier-Stokes and continuity) are solved, to compute actual velocity and pressure, numericallyon a regular grid introduced in the computational domain
DNS of the bottom boundary layer (Vittori & Verzicco, J.Fluid Mech. 371, 1998)
δ13251 .Lx = δ13252 .Lx = δ57123 .Lx =
17_11_2008
DNS- Fractional-step scheme (Kim & Moin (1985), Orlandi(1989) and Rai & Moin(1991)).
terms viscousterms pressuretermslinear non
LGHt
v
0vrrrr
r
+=+∂∂
=⋅∇Governing equations:
Assume the pressure and velocity fields known at time n ( ) and compute the fields at time n+1
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.
nn vpr
,
2nd step: define the scalar function Φ :
which should satisfy the equation :
Fractional-step scheme
terms viscousterms pressuretermslinear non
LGHt
v
0vrrrr
r
+=+∂∂
=⋅∇
( )nii
ni
1ni
ni
nii LL
2
1GH50H51
t
vv ++−−=∆− − ˆ..
ˆ
1ni
i1n
i Gt
vv ++
Φ−=∆
− ˆ
t
v1n2
∆⋅∇=Φ∇ + ˆ
∆Φ−=
Φ∇∆+Φ+=
++
+++
tGvv
2
tpp
1nii
1ni
1n21nn1n
ˆ
Governing equations
1st step: compute an intermediate velocity field (non solenoidal)
3rd step: compute velocity and pressure at the new time level:
17_11_2008
DNS: numerical algorithm
( ) ( )∑ ∫ ∫=
+=P x xN
n
L L
xxP
dxdxntxxxfLLN
txf1 0 0
3132131
2
1 3
211 π,,,,
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:
DNS results
Disturbed laminar regimeIntermittently turbulent regime
Re=1.25 105 Rδ=500 Re=5 105 Rδ=1000Vittori & Verzicco, JFM, 371, 1998
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
(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
Rδ=740 Rδ=1120
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)
the study of coherent structures is important:
a. to aid predictive modeling of turbulence statisticsb. to understand the mechanism of sediment pick-up and improve sediment
transport predictors
elocityfriction v ==+τ
τ
νu
yuy
Experimental visualizations of coherent structures in oscillatory boundary layers(Sarpkaya JFM 253, 1993)
Rδ ≈ 420-460Re ≈ 8.8 104- 1.1 105
BBL: coherent structures
17_11_2008
Moderate Rδ (Re) 600 <≈ Rδ <≈ 1000 - 1.8 105 <≈ Re <≈ 5 105
t = 42.17 π t = 42.47 π t = 42.52 π
Similar dynamics has been observed in steady boundary layers and by Sarpkaya (1993) for oscillatory BL
( Rδ=800 Re=3.2 105 x1=25.13 δ x2=0.1 δ )
Coherent structures in BBL (Costamagna, Vittori & Blondeaux JFM 474, 2003)
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
RANS
Large eddy simulation
LES is a turbulence model which stands between eddy viscosity models and DNS
Inertial subrange
Viscous cut_off
Large eddy simulation
( ) ( ) ( ).... pressure component, velocityU
rdtrxUrGtxU
=
−= ∫rrrrr
,,
width filter=∆
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
( )xUr
( ) ( ) ( )
( ) 0≠′
′+=
txu
that note
txutxUtxU
,
,,,
r
rrr
17_11_2008
Filtered equations:
Note that therefore we introduce the residual stress tensor:
DEFINE:
The filtered momentum equation becomes:
Large eddy simulation
momentum x
p
xx
v
x
vv
t
v
continuity x
v
jii
j
i
jij
j
i
∂∂−
∂∂∂
=∂
∂+
∂∂
=∂∂
ρν 1
0
2
( )i
Rij
jiii
jijiji
Rij x
vvxx
vv vvvv
∂∂
+∂∂=
∂∂
⇒−=τ
τ
−=
=
tensor stress-residual canisotropi
energy kinetic residual
ijrRij
rij
Riir
k
k
δττ
τ
3
22
1
( )
xx
v
x
k
x
p
xx
vv
t
v
ii
j
kpx
j
r
j
modeled be to
i
rij
Dt
vD
i
jij
rj
j
∂∂∂
+∂∂−
∂∂−=
∂∂
+∂
∂+
∂∂
+∂∂−
2
3
21
3
21 νρ
τ
ρρ
44 344 214434421
vvvv jiji ≠
2) Introduce an appropriate model for the anisotropic residual stress tensor
The momentum equation becomes
(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
Large eddy simulation: Smagorinsky model (1963)
( )( )
widthfilter
strainof rate filtered sticcharacteri SS
widthfilter the to alproportion tcoefficieny SmagorinskC th wiC
strainof rate filtered x
v
x
vS
S
1
ijij
ssr
i
j
j
iij
ijrrij
=∆≡
∆=∆=
∂∂
+∂∂≡
−=
2
2
2
2
1
2
S
Sν
ντ
i
ji
jj
i
rij
jii
jj
x
vv
t
v
tD
vD with
xx
p
xx
v
tD
vD
∂∂
+∂
∂=
∂∂
−∂∂−
∂∂∂
=τ
ρν 1
2
17_11_2008
Velocity (averaged) profiles
LES profiles during decelerating phases perform better than k-ε profiles
During the accelerating parts k-ε model performs better than LES
Large eddy simulation: plane bed velocity profiles
(From Lohmann et al., J. Geophys. Res., 111 2006)
Re=6 x 106 Rδ≈3500
Dots: experimental results by Jensen et al., 1989
Solid line : LES
Dashed line: k-ε model
Small scale bedforms formed under oscillatory flow
2D BEDFORMS
h/l<0.1h/l<0.1Rolling-grain ripples
Vortex ripplesh/l>0.1h/l>0.1
Vortex ripples start to appear
Sleath (1984)
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
•
Note that pressure term is not present !!!
• if the flow is 2D (x-y plane)
• if the flow is 2D and incompressible:
( )
( ) ( ) ( )ωνωωωω
rrrrrr
rr
2∇+∇⋅=∇⋅+∂∂
=×∇=
vvt
vorticity vcurlv
( )zωω ,,00=r
( )
( ) ψψωψ
ψ
200
00
−∇=×∇×∇=
×∇=
,,
,,
r
r
nction streamfu
v
17_11_2008
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
2D vortex ripples: ψ-ω model (Blondeaux & Vittori, J. Fluid Mech, 1991)
( ) ∞→→∂∂→
∂∂
=∂∂=
∂∂
−=∂∂+
∂∂
∂∂+
∂∂=
∂∂
∂∂−
∂∂
∂∂+
∂∂
y for ty
x
profile ripple the at yx
:conditionsboundary
yx
yxRyxxytR
sin,ψψ
ψψ
ωψψ
ωωωψωψωδδ
0
0
12
2
2
2
2
2
2
2
2
xv
Yu
∂∂−=
∂∂== ψψψ , function stream
( ) ( )
quantity ldimensiona *
...... ,y,x
yx, ,tt*
***
=
==δ
πT2
Note: laminar flow !!!
The coordinate transformation:
maps the ripple profile
into the line η=0, and the fluid domain into a rectangular domain
2D vortex ripples: laminar flow-field
( ) ( )***
*****
** cos;sin****
ξ−=ηξ+=ξ η−η− ke2
hyke
2
hx kk
( ) ( )
ber wavenum2
22
**
***
*****
*
lkwith
ksinh
x;kcosh
y
π
ξξξ
=
−==
physical plane transformed plane
Vorticity equation
A finite difference approximation of vorticity equation is used to compute ω at time t+∆t once ω at time t is known
∂∂+
∂∂=
∂∂
∂∂−
∂∂
∂∂+
∂∂
2
2
2
2
2
1
2 ηω
ξω
ηω
ξψ
ξω
ηψω δ
JJR
t
17_11_2008
Given the periodicity in the x (and ξ) direction, the solution can be expressed in the form
The relationship between ψ and ω gives a second equation (Poisson equation) which is used to compute ψ.
2D vortex ripples: laminar flow-field
∑=
−−=∂∂+−
N
ssns
nnn Jk
12
22 ω
ηψψ
( ) ( ) ( )
( ) ( ) ( ) ξπ
ξπ
ηωηωηξω
ηψηψηξψ
n
n
ikN
nn
Nj
niN
nn
N
n
ikn
N
n
Nj
ni
n
etett
etett
∑∑
∑∑
=
=
==
==
==
1
2
1
11
2
,,,,
,,,,
The boundary condition for vorticity is first-order accurate (Thom(1933)).
Considering the boundary conditions at the wall:
and the relationship:
at the wall (grid-point k):
For a plain wall, it turns out that
therefore:
assuming ψk=0 it is obtained: boundary condition for ω
If the wall is not plane, the boundary conditions can be derived similarly as above
2D vortex ripples: laminar flow-field
....+
∂ψ∂∆+
∂ψ∂∆+ψ=ψ +
k2
22
k
k1k y2
y
yy
00 =∂∂==
∂∂=
kkxy
uψψ
-v
2
2
2
2
yx ∂∂+
∂∂=− ψψω
k
k y2
2
∂∂=− ψω
kkk
y ωψψ2
2
1
∆−=+
12
2+∆
−= kk yψω
17_11_2008
2D vortex ripples: laminar flow-field
ωt=π/4 ωt=π/2 ωt=3 π/4ωt=π
ωt=5 π/4 ωt=2 πωt=7 π/4ωt= 3 π/2
Vorticity contoursΔω=0.15
___ clockwise vorticity;______ counterclockwise vorticityRδ=50, h/l=0.15, a/l=0.75
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
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
•Considering a pulsating flow Chang & Scotti observed that:
Flow over 2D vortex ripples: LES results
____ LES
- - - - RANS
•VERTICAL VELOCITY COMPONENT
17_11_2008
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 effectsi.e. 3D flow patterns which may be related to 3D ripples (Scandura, Blondeaux & Vittori, J.Fluid. Mech. 2000)
2D vortex ripples: 3D effects