3D structure of mean flows comparison with measurements

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3D structure of mean flows

comparison w i t h measurements

Report submitted to R I K Z

Ad Reniers

T U

Delft

Delft University of Technology

Section Fluid Mechanics

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CONTENTS 2

List of Figures

1 Plan view of field site at Duck, North Carolina (Picture with courtesy

of FRF) 4 2 Left panel: Listrument deploment on sled. Right panel: C R A B in

position to tow SLED into the water. (Pictures with courtesy of FRF) 5

3 Definition sketch (from Reniers, 1999) 7 4 Results for computed wave height transformation (upper left panel)

and set-up (upper middle panel) compared to measurements obtained from the pressure sensor array at a time corresponding to the

SLED-deploment at station 4 om day 1 9 5 Results for the computed cross-shore flow structure compared to

me-asurements obtained from the SLED on day 1 10 6 Results for the computed alongshore flow structure compared to

me-asurements obtained from the SLED on day 1 11 7 Results for computed wave height transformation (upper left panel)

and set-up (upper middle panel) compared to measurements obtained f r o m the pressure sensor arra}' at a time corresponding to the SLED-deploment at station 6 on day 2. The remaining panels show wave height, wave incidence angle, tidal elevation, wind speed and wind

direction as function of the SLED-deployment position 12 8 Results for the computed cross-shore flow structure compared to

me-asurements obtained f r o m the SLED on day 2 13 9 Results for the computed alongshore flow structure compared to

me-asurements obtained from the SLED on day 2 14 10 Wave directional spectrum on day 1 (left panel) and day 2 (right panel)

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1 INTRODUCTION 4

1 Introduction

This progress report refers to part the work done within the framework of the Dutch Center for Coastal Research ( N C K ) . The primary objective of our research is to develop knowledge and methods for the prediction of the hydrodynamic conditions for the Dutch coast taking into account the morphodynamic behaviour in the nearshore zone.

In general the progress reports descibe results, developments and new ideas ob-tained during our research. Most results have a preliminar}' status and should be not be used without prior consent of the authors. The various topics described in the progress reports are brought together to gain more insight in the proces to reach the primary objective. More detailed analysis of the various topics are or will be given in the form of journal papers. In addition attention is given to the (potential) links with work done by others.

2 Field experiment

2.1 Introduction

Recent field experiments under the name of Sandy Duck were performed at Duck (see Figure 2), North Carolina, to examine the local hydro-and morphodynamics. A large number of predominently american scientists participated in the project.

Figure 1: Plan view of field site at Duck, North Carolina (Picture with courtesy of FRF)

The topics varied f r o m large scale morphodjaiamics, monitored with the Argus-video system and aireal Argus-video, large scale hydrodynamics monitored by sonar and multiple measurement frames, to small scale measurements of wave boundary layer kinematics and sediment transport. Part of the experiment was to examine the ver-tical distribution of the surfzone currents. Due to a verver-tical imbalance between the various forces acting on the water column a curvature of the flow is to be expected.

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2 FIELD EXPERIMENT 5

This curvature was examined with a vertical stack of current velocity meters deployed on a sled (see Figure 2).

The curvature is a function of the forcing due to wind, waves, set-up gradients and the vertical distribution of the eddy viscosity. In most cases the flow conditions in the alongshore and cross-shore direction are examined separately, using different descriptions for the vertical distribution of forcing and eddy viscosity. Numerous models have been developed to describe the return flow (Svendsen, 1984, Stive and Wind, 1986, Stive and de Vriend, 1988 and others). The vertical distribution of the longshore current is often described with a logarithmic profile. In the following we have used a model (Roelvink and Reniers, 1994), describing the curvature of surfzone currents both in the alongshore and cross-shore direction as function of the various forcing mechanisms, in comparisons with some preliminary measurements available from the Sandy Duck field experiment (post-calibration of measurement data is in progress). The emphasis is on the description of the vertical distribution of the combined cross-shore and alongshore current velocity using a single depth-varying distribution for the eddy viscosity. The model is based on the concepts formulated by De Vriend and Stive (1989), defining a top layer above trough level, a middle layer and a bottom boundary layer.

Figure 2: Left panel: Instrument deploment on sled. Right panel: CRAB in position to tow SLED into the water. (Pictures with courtesy of FRF)

The bulk of the experiments was performed from the beginning of september 1997 until the end of October 1997. A d Reniers (AR in the following) attended the experiments in the period trough the month of October, collaborating with Prof. E.B. Thornton (EB in the following) of the Naval Postgraduate School (NFS).

2.2 Instrument deployment

A n overview of the instrument deployment is shown in Figure 2. The SLED was deployed at line north of the F R F pier. Early in the morning the CRAB, see the right panel of Figure 2, is used to tow the SLED to a particular position offshore at

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3 MODEL DESCRIPTION 6

which the measurements are to be performed for a duration of approximately 1 hour. After this period of time, the SLED is pulled inshore to a new measurement position, after which another 1 hour measurement is performed. Thus by pulling the sled a step at a time inshore a cross-subsection of the beach is monitored. The sampling frequency for all instruments is 48 Hz. A l l data is instantaneously transferred from the SLED to a data-station via a fiber optic cable, where i t can be monitored for any anomalies. Once the data is recorded calibration results in time series which can be used for analysis.

The idea was to compare measurements of the mean flow velocities in the alongs-hore and cross-salongs-hore direction with model results. To that end a model set-up was chosen that enabled an almost on-line comparison with the measurements. Once the measurement data was obtained, a model run could be started for intercompari-son, generating results at locations corresponding to the deployment positions of the SLED. For presentation purposes a limited set of comparisons is included.

In addition to the SLED, a cross-shore array of pressure transducers was deployed adjacent to the SLED deployment line. Using spectral transfer functions to account for the vertical variation of wave dynamics, a cross-shore distribution of the wave height was obtained. Taking the mean values of the pressure and taking into account the vertical reference level, a measure of the set-down and set-up could be obtained and compared to the model results.

3 Model description

3.1 Wave model

The wave transformation model is analogous to Reniers and Battjes (1997) which will be described briefly here. The wave energy balance is given by:

dEyjCg cos(ö)

5'

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where, Eyj, represents the wave energy, Cg, the group velocity and 9 the wave inci-dence angle with respect to the coast normal, x, being positive onshore. The wave dissipation, S, is modelled according to Thornton and Guza (1983), given by:

1

-

{l+{Hrinshhr)V-(2) where fp is the peak frequency, B a coefficient of 0 ( 1 ) , p water density, g, the gra-vitational acceleration, Hrms the root mean square wave height, h the total water depth and the weighting function, M , is given by (Whitford, 1988)

M = 14- tanh Hrms

jh - 1. (3)

Here j is a wave breaking parameter representing saturation. Given the alongshore uniformity the wave incidence angle can be obtained from Snell's law. The balance of roller energy, E,., is given by (Stive and de Vriend, 1994):

d{2ErCcos{0)) -S

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where c represents the phase speed of the waves and fg the shear stress at the roller interface. The wave and depth averaged momentum equation in the cross-shore di-rection used to compute the set-up of the mean water level, taking into account the zero mean flux in the cross-shore direction, is now given by:

^ ^ + ^ + K . = o (5)

ox ox

The first term represents the gradient in radiation stress associated with the wave,

Sxx,w, the second term the contribution due to roller motion, Sxx,f , and the third term a pressure gradient associated with the set-up.

3.2 Flow model

The flow model used in the comparison with measurements is described previously by Roelvink and Reniers (1994). For convenience a brief description will be given here as well. The conditions are assumed to be alongshore uniform. In the cross-shore direction there is an imbalance between the forcing due to wind and waves and the balancing pressure gradient associated with the slope of the mean water level which results in a circulation of the flow (Dyhr-Nielsen and Soren, 1970). In addition there is the mass-flux associated with the incident waves which has to be balanced from the point of continuity. The longshore flow is less constrained, where the alongshore forcing due to wind and waves is balanced by the bottom shear stress. A deflnition sketch is shown in Figure 3.

XX set-up ..^

^ y^l kxioshoro

/ current

u [

return TtowX"

Figure 3: Definition sketch (from Reniers, 1999)

The wave averaged momentum balance for the iBiddle layer is given by:

0(7

where the subscript denotes direction, a represents the vertical position scaled with the water depth, r the shear stress and F the forcing. A t the mean water level the

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forcing, Ts^i, is given by the shear stress due to wind and wave breaking. In the bottom boundary layer there is an additional forcing due to dissipation of wave energy:

dfi d < UjU) >

0 ^ = ^'- + ^ - ^ ^

with Ü and lu being the horizontal and vertical orbital velocities and p the water density. The shear stress is related to the curvature of the mean velocity by:

dui

n = vt-^ (8)

where Vf, represents the turbulent eddy viscosity. There is a no-slip condition at the bed and the velocity at the top of the wave boundary layer, us, acts as a boundary condition for the middle layer. Defining the vertical distribution of the turbulent eddy viscosity:

vt = (t>sVj.(r{as - a) (9)

for the middle layer and

Vt = 4>sV'i,a{ag - cr) cj)bvEa{5 - (T) = {(t>snf. + 4>bVjb){(ib - cr)o- (10)

for the bottom boundary layer where vï and vïb represent the depth averaged values of the eddy viscosity in the middle and boundary layer respective^, (f)s and (f)^ are shape functions describing the vertical distribution of the eddy viscosity and ai,:

(j^s^ïas + (t^bVtbS

0-6 = , _ , , (11)

Vertical integration of the momentum balance gives:

fi = fs,i - Fi{l - a) (12)

for the middle layer and

d<üiW>{S-a) . .

n = r s , - p ^ ^ r - ^''^

in the bottom boundary layer. The forcing due to wave energy dissipation in the boundary layer is estimated by:

d<üiw> IDf

where D/ represents the dissipation due to bottom friction and c the phase velocity of the waves. Using ecj. 8 the curvature of the flow velocity is given by:

duj ^ h {fs,i- Fi) + FiO da " p(f)sV't cr{as - a)

for the middle layer. In the wave boundary layer the following is obtained:

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4 COMPARISON WITH MEASUREMENTS 9

once the shape functions are defined and the depth-invariant forcing is known the vertical distribution of the flow velocity can be obtained. In the case where the depth-invariant forcing associated with the set-up, F^^, is not know, an iterative procedure is followed where the double vertical integration has to be balanced by the wave induced mass flux. Using parabolic expressions for the shape functions results in logarithmic velocity profiles.

4 Comparison with measurements

Figure 4: Results for computed wave height transformation (upper left panel) and set-up (upper middle panel) compared to measurements obtained f r o m the pressure seiisur array at a time corresponding to the SLED-deploment at station 4 om day 1.

In the following the measurement data obtained on two different days have been used in a comparison with the previously described combined wave transformation and IDV-flow model. The wave and wind conditions for those two days are significantly different. The first day can be characterised as a quiet day, with wind f r o m the south, parallel to the shore with a speed of 10 m/s. The second day corresponds to the onset of a storm where the wind is f r o m the east, i.e. inthe cross-shore direction, with a wind speed of 8 m/s.

Ofi'shore the forcing is predominantly by the wind given the fact that wave bre-aking does not occur. In the surf zone intense wave brebre-aking dominates. The forcing due to wave breaking is obtained f r o m the computed wave transformation. Using the breaker parameters as fit-coefficents, a good correspondence between the measured and computed wave height across the profile is obtained. Using the set-up data to calibrate the roller model, an accurate description of the wave forcing defined by the shear stress occurring between the roller and underlying water column is obtained. The wave transformation is shown in Figure 4, which shows a good match with the measurements, both for the wave height and the set-up of the mean water level. The optimal wave breaking parameters are given by a = 1.0 and j = 0.28 whereas the

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station 4 station 5 station 6

-0.4 - 0 , 2 0 U (m/s) - 0 . 4 - 0 . 2 0 0.2 station 8 3r -0.4 - 0 . 2 0 station 2 N 0 -0.4 O O c of -0.2 0 U (m/s) 0.2 - 0 . 4 - 0 . 2 0 0.2 station 1 3r - 0 . 4 - 0 . 2 0 0.2 station 3 3 t -0.4 - 0 . 2 0 0.2 U (m/s)

Figure 5: Results for the computed cross-shore flow structure compared to measure-ments obtained from the SLED on day 1.

roller model coeiïicient, /?, equals 0.1. The wave forcing is obtained f r o m the roller shear stress and applied at the trough level.

The results for the flow computations at the various stations are shown in Figures 5 and 6. For the offshore stations (see upper panels Figure 5) the roller shear stress is not present because wave breaking does not yet occur and the cross-shore flow is gouverned by compensation of the wave mass-flux in the cross-shore direction. The comparison with the measurements for these stations is favourable. Deviations occur close to the bed and near the surface. The latter is a result of the intermittent surfacing of the uppermost flow meters when a wave trough passes. In the alongshore direction the wind forcing generates a flow which compares well to measurements. Excluding the the wind stress clearly results in a underprediction of the longshore current (see upper panels Figures 6).

Proceeding towards the outer surfzone, indicated by the results on the middle panels, wave breaking becomes important. The return flow magnitude has increased with respect to the offshore stations. Also the curvature of the cross-shore flow is now

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d 1

Q

a

0 station 7 O O c 1 i N 0 - 1

d

6 A

9 Q

0) 0 1 station 8 0 1 station 3 O O O D 1 O 1 O ' O 1 n -1 O 1 V (m/s)

Figure 6: Results for the computed alongshore flow structure compared to measure-ments obtained from the SLED on day 1.

becoming apparant in both measurements and computations, where the two upper-most measuring points are frequently out of the water as wave troughs pass by the instruments. The comparison for the cross-shore flow is reasonable (see middle panels Figures 5). In the alongshore direction the forcing is now generated by a combination of wind and wave breaking. Including the wind forcing in the computations results in an overprediction of the magnitude of the measured longshore current velocity. Ho-wever, neglecting the wind forcing does not lead to signiflcantly better results, except for station 8.

In the inner surfzone, wave breaking is the predominant forcing of both cross-shore and alongshore currents. The computed return flow compares well to the measure-ments (see lower panels Figures 5), which now show a clear curvature with maximum flow velocities in the order of 20 cm/s. In the alongshore direction a good match is ob-tained for the longshore current, except for station 3. The latter can be explained by the fact that in the flow computations only the local momentum balance is used. As can be seen f r o m the wave transformation (upper left panel of Figure 4), there is little

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wa.ve dissipation at station 3, so based on tlie local balance the expected alongshore flow velocities are small. In reality the longshore current driven by the dissipation of the remaining wave energy near the water leads to signiflcant flow velocities at station 3 due to lateral mixing.

The same procedure is used in comparison with the measurement data obtained two days later. Compared to the previous day, the wave height nearly doubled whereas the angle of incidence is close to normal. The wind-direction is now perpendicular towards the coast. The results for the wave transformation and set-up are shown in Figure 7.

Figure 7: Results for computed wave height transformation (upper left panel) and set-up (upper middle panel) compared to measurements obtained f r o m the pressure sensor array at a time corresponding to the SLED-deploment at station 6 on day 2. The remaining panels show wave height, wave incidence angle, tidal elevation, wind speed and wind direction as function of the SLED-deployment position.

The computed wave transformation and set-up again show reasonable agreement with the measurements. A l l the measuring stations are now within the surf zone (see upper left panel of Figure 7).

The comparison of the computational flow-results with measurements is shown in Figures 8 and 9. Starting with the cross-shore flow in the outer surfzone (given by the upper panels of Figure 8), the agreement is reasonable with moderate curvature in the vertical and flow velocities in the order of 10 cm/s. Excluding the windforcing results in negligeble differences in the computed flow conditions. In the alongshore direction both computed and measured flow velocities are relatively small.

Proceeding towards the coast, wave forcing becomes more important, with an expected increase in curvature for the computed return flow proflles (middle panels of Figure 8) which is not present in the measurements. The comparison with the measurment data is not good, showing signiflcant discrepancies between computed and measured return flow profiles. In the alongshore direction the mismatch

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bet-5 DISCUSSION 13

-0.2 0 station 9 0 - 0 . 4 O o / O / O /

ol

-0.2 0 Station 3 -0.2 0 U (m/s) 0.2 1. 4 - 0 . 2 0 U (m/s) 0.2 - 0 . 4 - 0 . 2 0 0.2 station 2 3r - 0 . 4 - 0 . 2 0 0.2 station 5 3r - 0 . 4 - 0 . 2 0 0.2 U (m/s)

Figure 8: Results for the computed cross-shore flow structure compared to measure-ments obtained from the SLED on day 2.

ween computations and measurements is also significant. The computations predict a longshore current whereas the measurements show a near zero flow, except for station 9, which displays a rather peculiar vertical flow distribution.

The comparison for the computed and measured return flows in the inner surfzone is shown i n the lower panels of Figure 8. Significant differences in computed and measured values are present. This is even more so for the computed and measured longshore current velocity profiles shown in Figure 9.

5 Discussion

The comparison of computational results with measurements shows shows strikingly different results depending on which data is us uesd. Overall the correspondance with data obtained on day 1 is good for both cross-shore and alongshore flow profiles. However, the comparison with the data obtained for day 2 is not good at all, where

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5 DISCUSSION 14

station 9 O O O O O O

9. )

O station 3 O - 1 O O O O O O

B

O - 1 O 1 station 1 O O O O O O \8 O 1 station 4 E, N O l

o |

O O 1 O 1 O 1 E, N -1 O 1 station 2 O O O

V 8

Figure 9: Results for the computed alongshore flow structure compared to measure-ments obtained from the SLED on day 2.

large differences between computed and measured flow velocities occur, especially in the alongshore direction.

There are numerous possible causes which may explain the observed discrepancies. Sofar we have assumed that the conditions are uniform in the alongshore direction. In addition, the wave field has been parameterised to a single root mean square wave height and mean wave direction. That this is not nescessarily the case becomes clear if we look at the directional wave spectra for the two days under consideration. On both days there is a bi-modal spectrum present, which can not be properly parametrised as a single wave heigth with a mean direction. Looking in more detail at the directional spectra, we can see that on day 1 the bi-modality in direction is dominated by the energy present in the sea waves (compare energy density as function of direction). On day 2 this is less the case, where energy is more or less equally distributed over swell and sea waves which are of opposite direction. Depending on the breaking charaterisctics of combined sea and swell the longshore current forcing can become significantly different from the results obtained for a parameterised wave height with

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5 DISCUSSION 15

Figure 10: Wave directional spectrum on da}' 1 (left panel) and day 2 (right panel) at 16 h obtained from the 8 m FRF pressure sensor array.

a mean direction.

Acknowledgements

The present progress report results from a collaboration of A R with EB during the Sandy Duck field measurement campaign in 1997. AR's visit was co-funded by the NICOP programme. The wave directional data were provided by the Field Research Facility of the U.S. Army Engineer Waterways Experiment Station's Coastal Engi-neering Research Center. Permission to use these data is appreciated. Additional sponsoring by the National Institute for Coastal and Marine Management (RIKZ) through the Netherlands Centre for Coastal Research (NCK) is also greatly appreci-ated.

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References

Dyhr-Nielsen, M . and T. S0rensen, 1970: Some sand transport phenomena on coasts with bars. Proc. 12th Int. Conf. Coastal Eng., Washington, pp 855-865.

Reniers, A.J.H.M., 1999: Longshore curent dynamics. Ph.D. thesis in preparation. Reniers, A . J . H . M , J.A. Battjes, 1997: A laboratory study of longshore currents over barred and non-barred beaches. J. of Coastal Eng., vol 30, pp. 1-22.

Roelvink, J.A. and A . J . H . M . Reniers, 1994: Upgrading of a quasi-3D hydrodynamic model. Abstracts-in-depth, M A S T G8-M overall workshop, Gregynog.

Stive M.J.F. and H.G. Wind, 1986: Cross-shore mean flow in the surf zone. Coastal. Eng., 10, pp. 325-340.

Stive M.J.F. and H.J. de Vriend, 1994: Shear stresses and mean flow in shoaling and breaking waves. Proc. 24th Int. Conf. Coastal Eng., Kobe, pp. 594-608.

Svendsen, L A . , 1984: Wave heights and set-up in a surfzone. Coastal Eng., 8, pp. 303-329.

Svendsen, L A . , 1985: On the formulation of the cross-shore wave-current problem. Proc. Workshop 'Euripean Coastal Zones', Athens, pp. 1.1-1.9.

Thornton, E.B. and R.T. Guza, 1983: Transformation of wave height distribution. JGR, 84, C8, pp. 4931-4938.

Vriend, de H.J and M.J.F. Stive, 1987: Quasi-3D modelling of nearshore currents. In: P.P.G. Dyke (ed), JONSMOD'86, Coastal Eng. 11, pp. 565-601.

W h i t f o r d , D.J., 1988: Wind and wave forcing of longshore currents across a barred beach, Ph.D. Thesis, Navala Postgraduate School, Monterey, CA, pp. 205.

3D structure of mean flows comparison with measurements