10-11-2008

1

Boundary layers in oscillatory flows (waves and tides)

Relevance: sediment transport and formation of bedforms

Flow field

Sediment transport

Bottom evolution

Coupling between sediment transport, bottom evolution and hydrodynamics

In many cases the problem is decoupled (see stability analyses discussed in lecture 3)

Modeling oscillating transitional and turbulent flows

flow induced by an oscillating pressure gradient is a prototype flow for example for:

bed shear stress, bottom friction

WAVE BOTTOM sediment pick-up and transportBOUNDARY LAYER

formation of small scale bedforms(roughness)

TIDAL FLOW (local study) large scale bedforms (sand banks and sand waves)

10-11-2008

2

the boundary layer at the bottom of sea waves

Following a well-established procedure we model the bottom boundary layer by considering a monochromatic gravity wave propagating over a constant depth.

Relevant parameters for sea waves : ρ, ν, a, ω, l(ω), h0,g

Typically

Relevant parameters in the bottom boundary layer: U0, ω,ν

length scale: U0/ω( )

number Reynolds UU

νω00=Re

Irrotational flow regime

ων>>0h

a) Irrotational flow

b) bottom boundary layer

h0

density

viscosity dynamic

viscosity kinematic

===

ρρνµ

ν

laminar/turbulent flows

In the turbulent regime, random velocity fluctuations are observed both in the velocity field and in the pressure field

laminar turbulent

In the turbulent regime it is meaningful to predict only mean quantities (to be defined more precisely later)

10-11-2008

3

modeling of the flow field

The model to be used depends on

1) the regime of the flow which can be

laminar (actual values)

turbulent (actual values are not significant, better to use averaged values)

2) the level of information/accuracy required (extreme examples:a) circulation model of a large coastal region, b) evaluation of the pick-up rate of sediment from the bottom) which influences the level of sophistication of the flowmodel

modeling of the flow field

In the first part of the lecture we will:

1) determine the parameters which influence the flow regime

2) describe, on the basis of experimental evidence, the difference between laminar and turbulent regime(smooth and rough wall)

and

look at the problem from the point of view of the engineer (aim:predict and design) and introduce appropriate quantities

10-11-2008

4

transition to turbulence

Question: in steady pipe flow what is the main parameter controlling transition to turbulence?

Answer: Reynolds number

This has been evidenced by the well known experiment by Reynolds (1883)

Question: What are the parameters controlling transitionin the oscillatory bottom boundary layer (BBL)?

Boundary layer at the bottom of sea waves: controlling parameters

Answer: let us use Pi theorem to discover

In the bottom boundary layer many dimensional parameters influence any dependent quantity Q0:

the roughness yr can be due to the presence of sand grains (flat bottom) or to the presence of small scale bedforms (ripples) and is related either to grain diameter or to the dimensions of ripples

Applying the Pi theorem with ρ, U0 and ω dimensionally independent variables:

where:parameter roughness

U

y

number Reynolds Re

0

r =

=

ω

ων

20U

roughnessfrequency angularvelocityviscositydensity ===== ryU ωµρ 0

( )

=

=⇒=

ωω

ρµω

ωρµωρ γβα

002

00

000

1000 U

y,

Ref~

Uy,

Uf~

UQ

y,,,U,fQ rrr

density

viscosity dynamic

viscosity kinematic

===

ρρνµ

ν

10-11-2008

5

Boundary layer at the bottom of sea waves: controlling parameters

NOTE that if you apply Pi theorem and choose ρ, ω and µ as dimensionally independent variables

It emerges a length scale which is related to the thickness of the laminar bottom boundary layer

roughnessfrequency angularvelocityviscositydensity ===== ryU ωµρ 0

( )

=⇒=

ωννων

µωρµωρ γβα

rr

yUf

QyUfQ ,

~~,,,, 00

00 000

ων

Boundary layer at the bottom of sea waves: controlling parameters

δ = order of magnitude of the thickness of the laminar boundary layer

Reynolds number:

Therefore the problem can be studied by using either Re or Rδsince:

with νδU

RR νδU

νδU

ωνννU

ωνU

Re δδ02

2

02

220

20

20

2

1

2

1

22

2 ==

====

2

220 δ

νωRU

Re ==

ων2=δ

( )

≈⇒∂∂=

∂∂

22

2

δνω 0

0

UO UO

y

Uν

t

U

10-11-2008

6

Boundary layer at the bottom of sea waves: controlling parameters

Roughness parameter:

Therefore the problem can be studied by using either yrω/U0 or yr/δ

IN SUMMARY:

•It is possible to take U0/ω as length scale and express the dimensionless dependent

quantities as function of

•Or take δ as length scale and express the dimensionless dependent quantities as function

of

δ

r

δ

r

RUUUy

RUy 222

00

2

00

====δω

ων

δωδδω

δω since

ω0Uyr and Re

δδryR and

From the discussion above we expect the critical value of Reynolds number for transition to depend on the (non-dimensional) roughness size

There is no objective criterion for transition

EMPIRICAL FORMULAS FOR TRANSITION:

diametergraind −=

Criteria for transition to turbulence

From Sleath “Sea Bed Mechanics” Wiley, 1984

turbulence developed fully for U

1000Re Kajiura

sediments fine U

104Re Manohar

0c

0c

d

d

ω

ω

=

=

Kajiura

Manohar

dUω

0

cRe

ω0U

yr

10-11-2008

7

νURe 20 ω=

Values of the parameters for field conditions

Typical values of the parameters in the field (Nielsen, 1992):

movement sand little with beds flat .

0.01

440ν

δUR 10

ν

URe 0

δ5

20

0800

0

<≈

>

>≈=>≈=

ω

ω

ω

Uy

Uy

r

r

ω0Uyr

where

a= U0/ω =amplitude of fluid displacement oscillations outside the boundary layerr = roughness length (due to sand grains or bedforms)

Actual velocity and pressure fields can be decomposed as:

in the following, when referring to turbulent flow, reference is made to mean quantities, if not stated otherwise

turbulent flows: background info- mean quantities

turbulent flows are often studied by considering averaged quantities (averaged velocity components and averaged pressure)

( ) ( ) ( )

entsts/experimmeasuremen of number N

txfN

txftxfN

jj

N

=

== ∑=∞→ ,

,,, lim1

1 rrr

pPpppvVvvv ′+=′+=′+=′+=rrrrr

10-11-2008

8

Near-wall turbulence (steady flows)

• no viscous sublayer !

( )

( ) tv

turbulentviscous

vudydu ττρµτ

ττ

+=′′−+=

≈

43421321

constant 0

Smooth wall Rough wall

Hydraulic roughness of natural sand beds

Bed roughness for a flat bed of sand (Nikuradse roughness):

Close to the bottom under a steady current the velocity profile is logarithmic:

Where

NOTE THAT y0≠yr

While the Nikuradse roughness is a geometrical quantity (for plane bed is related to d50), the roughness length is related to the velocity profile.

5052 dky sr .==

( )

=

0yyu

yu ln*κ

0.4constant Karman's von

length roughness bed

velocity friction 0*

≅==

==

κ

ρτ

0y

u

Log-law

10-11-2008

9

Hydraulic roughness of natural sand beds

relationships between yr and y0, mainly based on fit of experimental data, are available:

≤≤+

−=

>⇒>=

<⇒<=

70592730

707030

559

νν

ν

νν

ννν

rrr0

rr

r0

rr0

yu flow altransition for

u

yuexp-1

yy

yu

uy flow roughically hydrodynam for

y y

yu

uy flow othically smohydrodynam for

uy

*

*

*

*

*

*

**

Flow regimes – smooth wall

• The flow in the boundary layer over a plane smooth bottom depends only on Re

and it can be:

Typical field values:

Disturbed-laminar Intermittently-turbulent Turbulent

53

δ

101.8 Re 105

600 R 100

⋅<≈<≈⋅<≈<≈

5

δ

101.8 Re

600 R

⋅>≈>≈

Hino, Sawamoto & Takasu. J.F.M. 75, 1976) Rδ= 360 (Hino, Sawamoto & Takasu. J.F.M. 75, 1976) (Jensen, Sumer & Fredsoe. J.F.M. 206, 1989)

!!! 440 Rδ >≈

stress shearbed maximum

yRF

yUF example for rr

=

=

==

τ

δωννων

ωµτ

µωρτ

δγβα

ˆ

,,ˆˆ

2

20

000

? Re>≈

10-11-2008

10

bottom boundary layer- smooth wall

From Jensen et al., 1989

the time of maximum bottom shear stress depends on Re (Jensen et al., 1989)

bottom shear stress= modulus of the force exerted by the flow on the bottom per unit area

laminar Intermittently turbulent

Mean (turbulent) velocity profile

( )ωtsinUU velocity outer

y

U

y

a bed, rough

bed smooth

R

0

r

0

r

=

===∇

=×=

≅

3700

106

35006

ω

δ

o

Re

Jensen et al. 1989

The bottom roughness causes a decrease in the near-bed velocity

The effect of the roughness starts to be evident for phases larger than 450

The roughness appears to increase the thickness of the boundary layer

The effect of the roughness becomes negligible for large distances from the bed

Bottom boundary layer: experimental results on smooth and rough walls

Question: what is the effect of roughness on the velocity profile?

10-11-2008

11

Jensen et al. 1989

The thickness of the logarithmic layer increases as U0/(ωyr) is increased ( i.e. for decreasing values of (ωyr) /U0 ).

● U0/(ωyr ) =124x U0/(ωyr ) =730◦ U0/(ωyr ) =3700

Bottom boundary layer: experimental results on rough walls

In the oscillating boundary layer the logarithmic profile is attained only at particular phases in the cycle

*uu

u

yk rs

=

=

+

Question: can velocity profile be approximated as logarithmic?(relevant for near-bed turbulence modeling, e.g. k-ε model)

A very simple model to express the complex hydrodynamic phenomena which take

place in the bottom boundary layer is obtained by introducing the friction factor

which, once estimated, gives the bottom shear stress (to be used to compute

circulation, sediment transport,…)

The wave friction factor fw (used by engineers) is defined in terms of the

maximum bed shear stress at the bottom as:

and is related to the drag coefficient (used by ocenographers):

On the basis of a Pi theorem

wfU 202

1 ρτ =ˆτ

=

ωUy

,ff rww

0

Re

Wave friction factor

202 Ucfc DwD ρτ == ˆ here since

10-11-2008

12

friction factor – smooth wall

From Fredsøe & Deigaard 1992Experimental results for friction factor Over a smooth bed by: ○ Jensen et al. (1979)

● Kamphuis (1975)X Sleath (1987)∆ Hino et al. (1983)

factor friction wave f

stress shear bed maximum

w

202

1Ufˆ wρτ =

The friction factor shows differences in the laminar and turbulent regimes

In the laminar regime:

νω2

22

afw ==

Re

See lecture 3 by Huib de Swart

Wave friction factor

from Fredsøe & Deigaard, 1992

The experimental measurements of the friction factor show four flow regimes:

• laminar regime (fw depends only on Re)

• smooth turbulent regime (fw depends only on Re)

• transitional (from smooth turbulent to rough

turbulent) turbulent regime

(fw depends on both Re and )

• rough turbulent regime (fw depends only on

)

Re

ωry

U0

0Uyrω

0Uyrω

10-11-2008

13

Wave friction factor

Empirical formulas for computing fw:

•Smooth turbulent flow (Kajura, 1968):

•Rough turbulent flow (Swart, 1976):

•Smooth, transitional and rough turbulent flows (Myrhaug, 1989):

Relog.log.

+−=+ 13501

18

1

ww ff

57130

571215002510

0

0

190

0

. if .

. if .exp.

.

≤=

>

=

−

rw

rrw

yU

f

yU

yU

f

ω

ωω

( )( )

641714

026201366320

2

210

0

21021 .

Re

.Re.expln.ln

. +

+

−−−

=

w

r

r

w

rw

w fyU

yUf

yU

ff

ωωω

turbulent flows: turbulence modeling

In many cases the simple models seen previously do not give accurate predictions (for example of sediment transport), therefore more refined modeling is required

Refined modeling is based on more advanced hydrodynamics and on deeper understanding of the physical processes controlling turbulent flows

In this lecture I will describe models which predict averaged quantities. Characteristics of the described models:

• Based on Reynolds equations (called Reynolds Averaged Navier-Stokes or RANS models)

• Based on empirical assumptions

• Implemented in many commercial codes

In the following lecture I will describe more advanced models

10-11-2008

14

turbulent flows: mean quantities

in order to understand how turbulence models work it is necessary first to introduce background information:

1) turbulent flows can be studied by considering averaged quantities (average velocity components and averaged pressure)

therefore actual velocity and pressure fields can be decomposed as:

( ) ( ) sexperiment of numberN txpN

txpN

jj

N

== ∑=∞→ ,

,, lim1

1 rr

pPpppvVvvv ′+=′+=′+=′+=rrrrr

Continuity equation – incompressible flow

( ) ( ) ( )

(spanwise) horizontalx

verticalx

e)(streamwis horizontalx that note

flow mean for equation continuity x

V

x

V

x

V

:obtain we

0vvv that account into taking and averaging after

x

vV

x

vV

x

vV

values actual x

v

x

v

x

v

x

vv

1

2

1

i

i

===

=∂∂+

∂∂+

∂∂

=′=′=′

=∂

′+∂+∂

′+∂+∂

′+∂

=∂∂+

∂∂+

∂∂=

∂∂=∇

0

0

0

3

3

2

2

1

1

321

3

33

2

22

1

11

3

3

2

2

1

1r.

10-11-2008

15

Reynolds equations

pressure dynamicgxPP where

x

vv

xx

Vµ

x

P

x

VV

t

Vρ

:average and pPp ; vVv substitute

1,2,3jk, for m su

1,2,3i xx

v

x

pf

x

vv

t

v

field) flow actual (for equation Stokes-Navier

d

k

ik

kk

i

i

d

j

ij

i

iii

kk

i

ii

j

ij

i

2

2

2

ρ

ρ

µρρ

+=

∂′′−∂

+∂∂

∂+∂∂−=

∂∂+

∂∂

′+=′+==

=∂∂

∂+∂∂−=

∂∂+

∂∂

Osborne Reynolds (1842-1912)

turbulent flows: closure problem

Reynolds equation can be written in terms of the Reynolds stresstensor:

where:

Reynolds stresses are the results of momentum fluxes associatedto the random fluctuations of small scale (turbulent eddies)

Indeed mathematically the Reynolds stresses are derived from theadvective terms of Navier-Stokes equation

1,2,3i =∂∂

+∂∂

∂+∂∂−=

∂∂+

∂∂

k

Rki

kk

i

ii

j

ij

i

xT

xxV

xP

fxV

VtV 2

µρρ

tensor stressReynolds of components vv -T jiR

li ′′= ρ

10-11-2008

16

turbulent flows: closure problem

The pequations to solve for tubulent flows are:

Reynolds equations present the well known closure problem

The number of equations available (4) is smaller than that of the unknowns (10) which are:

V1, V2,V3,P, -ρ<v’1v’1>, -ρ<v’1v’2>, -ρ<v’1v’3>, -ρ<v’2v’2>, -ρ<v’2v’3>, -ρ<v’3v’3>

To close the problem it is necessary to introduce an appropriateturbulence model !

1,2,3i

0

2

=∂∂+

∂∂∂+

∂∂−=

∂∂+

∂∂

=∂∂

k

Rki

kk

i

ii

j

ij

i

i

i

x

T

xx

V

x

Pf

x

VV

t

V

x

V

µρρ

not to be confused with Boussinesq approximation (buoyancy) or with Boussinsesq approximation

(water waves) !!

The closure problem can be solved by assuming that turbulent stresses can be modeled similarly to viscous stresses:

for example for a 1D flow this will give:

Boussinesq hypothesis (1877)

∂∂

+∂∂=−=

i

j

j

iTji

Rij x

V

x

Vvv µρ ''T

2

12112 x

Vvv T

R

∂∂=−= µρ ''T

Joseph Velentin Boussinesq (1842-1929)

10-11-2008

17

However it is necessary to correct the expression above to make the trace of the two matrices equal !

Boussinesq hypothesis (1877)

( ) ( ) ( )volume unit per

energy kinetic turbulent vvvk with

kx

V

x

Vvv ij

i

j

j

iTji

23

22

212

1

3

2

′+′+′=

−

∂∂

+∂∂=−

ρ

δµρ ''

Eddy viscosity is a function of time, position and of the considered problem !!!

The Boussinesq hypothesis is the starting point of many turbulence models commonly used in the engineering practice

The problem now is to prescribe (model) the dynamic eddy viscosity µT (or equivalently the kinematic eddy viscosity νT= µT/ρ)

Prandtl mixing length model

µT can be computed by means of different models, for example with the simple model proposed by Prandtl(mixing length model) used for wall bounded turbulent flows

this very simple model is able to provide the logarithmic velocity profile !

constant sKarman' von 0.4 withdx

dVxT ≅= κρκµ

2

122

Ludwig Prandtl(4 February 1875 – 15 August 1953)

kx

V

x

VvvT ij

i

j

j

iTji

Rij δµρ

3

2−

∂∂

+∂∂=−= ''

10-11-2008

18

limitations of Boussinesq hypothesis

It is important to keep in mind that Boussinesq hypothesis is just an hypothesis (or assumption) therefore in many situations may give inaccurate results.For example, consider a parallel shear flow and take B. hyp. :

It appears that the shear stress should be zero where the mean flow velocity has an extremum.However this is not true for a wall jet ( an approximation of a parallel flow)

THE INACCURACY OF BOUSSINSESQ HYPOTHESIS MAY HAVE IMPLICATIONSOF TURBULENCE MODELS BASED ON IT !!

2

121 x

Vuu T ∂

∂=− ν

turbulent eddies

Fluctuating turbulent quantities can be considered as the superposition of a large number of eddies of different sizes.

If L=spatial scale of the average flow fieldU= velocity scale of the average flow field ℓ= characteristic eddy size u(ℓ)= characteristic velocity of the eddies

Large scale eddies are such that ℓ≈L and u(ℓ) ≈U:

� their dynamics is not influenced by viscosity (large u(ℓ) ℓ/ν=Re i.e. negligible viscous effects)

Small scale eddies (such that ℓ<<L and u(ℓ) <<U):

�their dynamics is influenced by viscosity (small u(ℓ) ℓ/ν=Re i.e. relevant viscous effects)

10-11-2008

19

spectrum of velocity fluctuations

( )

( )

( )

=

=

=

∫

∫

∫

∞

∞

∞

0

333

0

222

0

111

dkkFvv

dkkFvv

dkkFvv

ρρ

ρρ

ρρ

''

''

''

Or equivalently:

( )

( )

( )

=′′

=′′

=′′

kFdk

vvd

kFdk

vvd

kFdk

vvd

333

222

111

Lecture 1: small and large eddies

Large eddies:

•Are anisotropic•Are influenced by the geometry•Contain most of the turbulent energy

Small eddies:

•Are isotropic•Are independent of geometry

Large eddies Small eddies

10-11-2008

20

the energy cascade- 3D turbulence

Large eddies are not influenced by viscosity and extract energy from the mean flow

Large eddies are unstable andbreak-up into smaller eddies

The turbulent spectra in the intermediate range is proportional to

the role of smallest eddies is that of dissipating energy

35−k

General remarks on µT

consider a turbulent plane unidirectional flow

Reynolds equations suggest that this flow field is influenced mainly by -<ρv’1 v’2> ���� large eddies contribute most to <ρv’1 v’2>

if l=length scale of large eddies; u0= velocity scale of large eddies, it is reasonable to expect that :

( )210 dxdVufvv ji ,,,'' lρρ =−

10-11-2008

21

General remarks µT

constant empiricalC ith w

dx

dV

uC

dx

dV

uf assuming

dx

dV

uf

u

vv ji

=

≅

=

−

2

1

02

1

02

1

020

lll

ρρ ''

Pi theorem gives:

but according to Boussinesq hypothesis:

which gives:

the problem now is to prescribe u0 and ℓ

2

1

0

20

2

1

0

20

2

1

dxdV

uCu

dxdV

ufu

dxdV

'v'v Tji

ll ≅

==− ν

uCT 0l=ν

General form for the eddy viscosity:

the length scale ℓ and velocity scale u0 are computed by means of:

� algebraic relationships (0-equation models)

Example: for a near-wall flow taking:

ℓ=x2 ; u0=x2|dV1/dx2| and C=κ,

we obtain the mixing-length expression by Prandtl

� differential equations (1-equation model, 2-equation model, ….)

form of the eddy viscosity

0 uρC µT l=

2

122

2

1220 dx

dVx

dx

dVxxuCT ρκρκρµ === l

10-11-2008

22

Eddy viscosity according to Kajiura (1968) :

Smooth wall for � the overlap layer exists

Smooth wall for � the overlap layer does not exist

420 1074 ⋅>= .Re

ωνU

420 1074 ⋅<= .Re

ωνU

∆>∆=

∆<<=

<<=

2 for

x12 for

120 for

xuK

uxuK

ux

T

T

T

τ

ττ

τ

ν

νν

ννν

22

2

>∆=

<<=

ττ

τ

νν

ννν

uxuK

u

T

T

12 for

12x0 for 2

2

stressshear bottom theof amplitude ˆ

u05.0

ˆu

constantKarman von 41.0

=

=∆

=

=

τω

ρτ

τ

τ

K

0-equation models: Kajiura (1968)- smooth wall

Rough wall for

Rough wall for

exist layer overlap the ⇒> 1150

ryU

ω

∆>∆=

∆<<=

<<=

for

for

x0 for

2

22

2

2

21850

xuK

xy

xuK

yyuK.

T

rT

rrT

τ

τ

τ

ν

ν

ν

exist not does layer overlap the ⇒< 1150

ryU

ω

>∆=

<<=

2

21850

2

2

r*T

rr*T

yuK

yyuK.

x for

x0 for

ν

ν

0-equation models: Kajiura (1968)- rough wall

ryˆ

.

ˆˆ

.K

==∆

=

=

sk u

u

constant Karman von

ω

ρτ

τ

τ

050

410

ks=yr

10-11-2008

23

The model by Kajiura gives reasonable results for the friction factor fw:

ων

20UReR ==

Eddy viscosity models: Kajiura (1968)

from Sleath, 1984

Smooth laminar

21

2

Re=wf

Experimental measurements by Sleath (1987) have shown that

• The variation of the eddy viscosity with time is significant while Kajiura’s model assumes a time-independent eddy viscosity

• The profile of νT is different from that assumed in the Kajiura’s model and of that assumed in steady flows

x-x-x y=3.5 mmo-o-o y=30 mmRe=2.3 105

U0/(ωyr)=138

d=1.63 mm → yr=2d=3.26 mm○ Re=2.8 105 U0/(ωyr)=151● Re=2.3 105 U0/(ωyr)=138x Re=1.5 105 U0/(ωyr)=115

T

cycle the during viscosity) (eddy of value mean

velocity friction of amplitude

νεεε

===*u

Eddy viscosity models: experimental measurements of νt

sm 2510×ε

εε

10-11-2008

24

Better predictions are given by (N-equation) turbulence models which are based on a number N of differential equations to compute the length-scale l and velocity scale u0:

example: k-ε model

model PDE equations for k and ε are based on the equation for the turbulent kinetic energy

N-equation eddy viscosity models

0T u lρµ =

PDE) model solving (computed

energy kinetic turbulent of

ndissipatio to related quantity

PDE) model solving (computed

volume unit per energy kinetic urbulent

u 1-0

1-

=

=

===

ε

ρε

ρε

µ µ

tk

kkk

CT212

2322

l