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