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Particle dynamics class, SMS 618, (Emmanuel Boss 11/19/2003) Van Rijn’s TRANSPOR lab: computation of sediment transport in current and wave direction. Handout: Appendix A in van Rijn, 1993, Principles of sediment transport in rivers, estuaries, and coastal seas, Aqua Publications. Note: in this write-up log is log 10 , while ln is log e . Inputs (see Fig. A.1 in appendix for directions): h-water depth. [m] r v - depth-averaged velocity in main current direction. [m/s] r u - time-averaged and depth-averaged return velocity below wave trough compensating the mass transport between wave crest and trough. [m/s] u b - time-averaged near-bed velocity due to waves, wind or lateral density gradient. [m/s] Hs- significant wave height (significant wave height, Hs, is approximately equal to the average of the highest one-third of the waves. Hs is calculated using: Hs = 4.0 * sqrt(m 0 ) where m 0 is the variance of the wave displacement time series acquired during the wave acquisition period. Variance can also be calculated using the nondirectional wave spectrum according to the following relationship: m 0 = sum(S(f)*df) where the summation of spectral density, S(f), is over all frequency bands, f, of the nondirectional wave spectrum and df is the bandwidth of each band. Wave analysis systems typically sum over the range from 0.03 to 0.40 Hz with frequency bandwidths of 0.01 Hz. (see http://www.ndbc.noaa.gov/wavecalc.shtml ). [m] Tp- Dominant, or peak, wave period. (the period corresponding to the frequency band with the maximum value of spectral density in the nondirectional wave spectrum. It is the reciprocal of the peak frequency, fp: Tp = 1/fp. Dominant period is representative of the higher waves encountered during the wave sampling period). [s] φ-angle between the directions of waves propagation and main current direction. [°] d 50 - median diameter of bed material. [m] d 90 - 90 th percentile diameter of bed material. [m] d s -representative diameter of suspended material (will be in the range of (0.6 1) d 50 ). [m] k s,c - current-related bed roughness height (minimum 0.01m). (likely range 0.01 1m). [m] k s,w - wave-related bed roughness height (minimum 0.01m) (likely range 0.01 1m, for sheet flow, 0.01m). [m] Te- temperature of fluid. [°C] SA- salinity of fluid. [ppt] Assumptions: ρ s =2650kg/m 3 – sediment’s density. κ=0.4 – Von Karman’s constant. General parameters computed: Chlorinity: CL=(SA-0.03)/1.805 Fluid density: ρ=1000+1.455CL-0.0065(Te-4+0.4CL) 2 [Kg/m 3 ] Kinematic viscosity: ν=(4/(20+Te))10 -5 [m 2 /s]

Particle dynamics class, SMS 618, (Emmanuel Boss 11/19 ...misclab.umeoce.maine.edu/boss/classes/SMS_618_2003/... · Handout: Appendix A in van Rijn, 1993, Principles of sediment transport

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Particle dynamics class, SMS 618, (Emmanuel Boss 11/19/2003) Van Rijn’s TRANSPOR lab: computation of sediment transport in current and

wave direction. Handout: Appendix A in van Rijn, 1993, Principles of sediment transport in rivers, estuaries, and coastal seas, Aqua Publications. Note: in this write-up log is log10, while ln is loge. Inputs (see Fig. A.1 in appendix for directions): h-water depth. [m]

rv - depth-averaged velocity in main current direction. [m/s]

ru - time-averaged and depth-averaged return velocity below wave trough compensating the mass transport between wave crest and trough. [m/s] ub- time-averaged near-bed velocity due to waves, wind or lateral density gradient. [m/s] Hs- significant wave height (significant wave height, Hs, is approximately equal to the average of the highest one-third of the waves. Hs is calculated using: Hs = 4.0 * sqrt(m0) where m0 is the variance of the wave displacement time series acquired during the wave acquisition period. Variance can also be calculated using the nondirectional wave spectrum according to the following relationship: m0 = sum(S(f)*df) where the summation of spectral density, S(f), is over all frequency bands, f, of the nondirectional wave spectrum and df is the bandwidth of each band. Wave analysis systems typically sum over the range from 0.03 to 0.40 Hz with frequency bandwidths of 0.01 Hz. (see http://www.ndbc.noaa.gov/wavecalc.shtml). [m] Tp- Dominant, or peak, wave period. (the period corresponding to the frequency band with the maximum value of spectral density in the nondirectional wave spectrum. It is the reciprocal of the peak frequency, fp: Tp = 1/fp. Dominant period is representative of the higher waves encountered during the wave sampling period). [s] φ-angle between the directions of waves propagation and main current direction. [°] d50- median diameter of bed material. [m] d90- 90th percentile diameter of bed material. [m] ds-representative diameter of suspended material (will be in the range of (0.6 1) d50). [m] ks,c- current-related bed roughness height (minimum 0.01m). (likely range 0.01 1m). [m] ks,w- wave-related bed roughness height (minimum 0.01m) (likely range 0.01 1m, for sheet flow, 0.01m). [m] Te- temperature of fluid. [°C] SA- salinity of fluid. [ppt] Assumptions: ρs=2650kg/m3 – sediment’s density. κ=0.4 – Von Karman’s constant. General parameters computed: Chlorinity: CL=(SA-0.03)/1.805 Fluid density: ρ=1000+1.455CL-0.0065(Te-4+0.4CL)2 [Kg/m3] Kinematic viscosity: ν=(4/(20+Te))10-5 [m2/s]

Sinking velocity:

( )

( )

2

3

s2

1 1 10018

100 10000.01 110 1 1 ; s=

10001.1 ( 1)

s

s gd d m

d ms gdw

dm ds gd

µν

µν ρ ρν

µ

− < < < <− = + −

<

Sediment characteristics computed: s= ρs/ρ – relative density. D*=d50[(s-1)g/ν2]1/3 - Particle parameter.

Shields parameter:

1**

0.64**

0.1**

0.29**

*

1 40.244 100.14

10 200.0420 1500.013

1500.055

cr

DDDDDDDD

D

θ

< < < <= < < < <

<

τcr=( ρs -ρ)gd50θcr – critical shear stress. ( )50 905.75 ( 1) log 4cr cru s gd h dθ= − – Critical depth-averaged velocity.

( )( )

2/3

50 500.5710.75 50

50

0.12( 1) 0.0005ˆ 0.00051.09( 1)

p

cr

p

s g d T d mU

m ds gd T

− <= < −

-- Critical peak orbital velocity.

Wave parameter computed L’–Doppler shifted wavelength (change in wavelength due to presence of mean flow):

find L’, the solution of: 2

' ' 2cos tanh2 'R

p

L gL hvT L

πφπ

− = .

Tp’- Doppler shifted wave period: ( )

'1 cos / '

pp

R p

TT

v T Lφ=

−.

Near-bed peak orbital excursion: ( )

ˆ2sinh 2 '

sHAh Lδ π

= .

Near-bed peak orbital velocity: ( )

ˆ' 2sinh 2 '

s

p

HUT h Lδ

ππ

= .

Wave-boundary layer thickness: ( )0.25

,ˆ ˆ0.072w s wA A kδ δδ =

Near-bed peak orbital velocity in forward direction:

( )( )2 2

24

,2

3ˆ 0.014 ' ' sinh 2 'ˆ

ˆ 0.011 0.3

sp

pf

sp

HU gT hT L h L

UH gT hUh

δ

δ

δ

π

π

+ <

= >+

.

Near-bed peak orbital velocity in backward direction:

( )( )2 2

24

,2

3ˆ 0.014 ' ' sinh 2 'ˆ

ˆ 0.011 0.3

sp

pb

sp

HU gT hT L h L

UH gT hUh

δ

δ

δ

π

π

− <

= >−

.

Return velocity mass transport: ( )

20.1250.95 0.35

sr

s

g Hu

h h H= −

−.

Near bed wave induced mean velocity: , ,

, ,

ˆ ˆˆ0.05 0.5 ˆ ˆ

f bb

f b

U Uu U

U Uδ δ

δδ δ

−= − +

The last two are input parameter. To have them computed by the program insert 9 at the input line. Bed parameters computed: Apparent bed roughness:

( ),

, 22 2

,

ˆ 10exp 2 , =0.8+ -0.3 ,

360otherwise

10

a s cs c

a R R

s c

k kUkk v u

k

δγ πγ β β β φ

< = = +

o.

Effective time-averaged bed-shear stresses computed:

Efficiency factor for current: ( )( )

2

90

,

log 12 3log 12c

s c

h dh k

µ =

.

Efficiency factor for waves: ( )( )

0.19

90

0.19

,

ˆexp 6 5.2 3

ˆexp 6 5.2w

s w

A d

A k

δ

δ

µ

− + = − +

.

Wave-current interaction coefficient: ( )( )

( )( )

2 2

,

,

1 ln 30ln 901

1 ln 30ln 90s cw a

cwaw s c

h kkh kk

δα

δ

− += ≤

− + .

Bed shear stress due to current: ( )( )

( )2 22

,

1 0.248 log 12

c R R

s c

v uh k

τ ρ = +

Bed shear stress due to waves: ( ) 0.19 2,

1 ˆ ˆexp 6 5.28w s wA k Uδ δτ ρ

− = − +

The total shear is the sum of the shears: τwc= τc + τw. Effective bed shear velocity current: ( )*,' c cw c cu α µ τ ρ= Bed shear stress parameters computations:

Dimensionless bed-shear stress for bed load transport: ( )cw c c w w cr

cr

Tα µ τ µ τ τ

τ+ −

= .

Dimensionless bed-shear stress for reference concentration at z=a: ( )( )*0.6 /cw c c w cr

acr

DT

α µ τ τ ττ

+ −= .

Both are zero if negative. Current velocity profile computations:

( )

( ) [ ][ ]

( ) [ ][ ]

[ ][ ]

, ,.

.

, ln 3031 ln 30

, , ln 90 ln 30 31 ln 30 ln 90

R R aw

aR z R z

R R w a s cw

a w s c

u v z kzh k

u vu v k z k z

h k k

δ

δ δδ

≥ − +=

< − +

Sediment eddy diffusion profile computations: Eddy diffusion due to mean current:

( )

( ) ( )( )

*,,

*,

2 2*,

*,90

1 0.5 ;

0.25 0.5

1 2 1.5where: ;

18log 41.5

cs c

c

r rs cc

u z z h z hu h z h

g u vw uu

otherwise h d

κβε

κβ

ββ

− <= ≥

+ + <= =

Eddy diffusion due to waves:

, *

, ,max

, ,max ,

s ,max ,min

ˆ0.0040.035 0.5 ;

0.50.5

=0.3 H but =0.2m & =0.05m

s bed s s

s w s s p

ss bed s s bed

ss

s s s

D U zh H T z h

zz hh

h

δε δ δε ε

δε ε ε δδ

δ δ δ

= <= = ≥ − + − < < −

Combined current and wave eddy diffusion: εs,wc= [ ε2s,c + ε2

s,w]0.5.

Sediment volume concentration computations: Reference level: a=maximum(ks,c, ks,w).

Bed concentration (for z≤a): 1.5

500.3*

0.015 aa

d TCa D

= .

For z≥a the concentration gradient is give by: ( )

( ) ( )( )5

0.8 0.4,

1

1 / 0.65 2 / 0.65s

s cw

c cwdCdz c cε

−= −

+ −.

Obtain C by integration to the surface from a (where concentration is known) to h. Time averaged suspended load transport computations:

,

,

current direction

wave direction

h

s R za

s h

s R za

v Cdzq

u Cdz

ρ

ρ

=

Time averaged and instantaneous bed load transport rate computations:

Velocities are computed at δ=max(3δw,ks,c):

[ ][ ]

[ ][ ]

,

,

ln 301 ln 30

ln 301 ln 30

R aR

a

R arR

r a

v kv

h k

v kuuv h k

δ

δ

δ

δ

− +=

− +

.

Instantaneous wave velocity: ( )ˆ cosU Uδ δ ϕ= , ϕ is the phase of the wave.

Instantaneous velocity along current: ( ), , ,cos cosx R b RU U v u uδ δ δ δφ φ= + + + .

Instantaneous velocity across current: ( ), ,sin siny b RU U u uδ δ δφ φ= + + .

Instantaneous velocity: 2 2, , ,R x yU U Uδ δ δ= +

Instantaneous friction coefficient: ,

R

R

v

v Uδ

δ δ

α =+

.

Instantaneous bed shear stress:

( )( ) ( )( )

( ) ( )2

0.19, 2, 90 ,2

, 90

1 ln 30 0.24 ˆ' 0.5 0.25 1 exp 6 5.2 3ln 30 log 4

s cb cw R

s c

h kA d U

h k h dδ δτ ρ α α

− − + = + − − + Instantaneous bed-load transport:

, , , , ,0.350 *

,

' ' '0.25

, max 0.3,1 otherwise0

b cw b cw b cr b cw b crs S

b b cr

d D Hqh

τ τ τ τ τγρ

γρ τ−

− > = = −

Time averaged values are obtained by averaging over a wave period (e.g. over its phase, ϕ). Bed form calculations: This is beyond the scope of our class. Class assignment: How waves affect the total transport. Use the TRANSPOR model with the following inputs (maybe appropriate to the Dock in front of the Darling Marine center): h=12m d50=150µm d90=400µm ds=0.8 d50. Te=12°C SA=30psu

0ru = 0.5 / srv m=

, , 0.02s c s wk k= = How do you expect the transport to change with wave parameters (Hs, Tp and φ) compared to having no waves? Write down your predictions. Investigate how the total transport varies with changes in the significant wave height (0 2m), wave period (1 8s), and angle (0 90) between wave and current. How do the results compare with prediction? How do they (qualitatively) compare with van-Rijn calculations attached to the handout? For one case, vary ds from 0.6d50 to d50. How would you expect the total transport to change? How much change do you observe in the total transport?