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A note on the three-dimensional shear stress distribution in a surf zone
Coastal Engineering, 20 ( 1993 ) 157-171 Elsevier Science Publishers B.V., A m s t e r d a m
157
A note on the t h r e e - d i m e n s i o n a l shear stress distribution in a surf zone Rolf Deigaard Institute of Hydrodynamics and Hydraulic Engineering (ISVA), Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 4 August 1992; accepted after revision 8 January 1993 )
ABSTRACT The three-dimensional t i m e - m e a n shear stress distribution in a surf zone is analyzed, thereby determining the vertical distribution of the driving forces due to wave-breaking. It is found that on a uniform coast the longshore current can be described as driven by a surface shear stress equal to the cross-shore gradient in the shear component of the radiation stress. The near surface shear stress is found to be in the direction of wave propagation, having a magnitude determined by the dissipation of wave energy. It can be determined as the shear force acting on the waves from the surface rollers of spilling breakers or broken waves.
LIST O F S Y M B O L S
A c D ~t Err Efw
g H H0 k L n p P qa
qu
cross-sectional area of surface roller wave celerity mean water depth energy dissipation rate per unit bed area energy flux associated with the surface rollers energy flux associated with the wave motion acceleration of gravity wave height wave height at x = 0 wave number wave length horizontal coordinate, parallel with the wave crests pressure pressure force acting on the wave from the roller the downward moving part of the exchange of fluid between the surface roller and the wave the upward moving part of the exchange of fluid between the surface roller and the wave
0 3 7 8 - 3 8 3 9 / 9 3 / $ 0 6 . 0 0 © 1993 Elsevier Science Publishers B.V. All rights reserved.
158
s S
S×y t T Ts u fi us v vn V w x y z o~ r/ ~/+ 0 p r~v r=x r:y rby rs r-~ 09
R. DEIGAARD
horizontal coordinate in the direction of wave propagation slope of mean water surface shear c o m p o n e n t of the radiation stress time wave period shear force acting on the wave from the surface roller orbital velocity in the x-direction flow velocity vector orbital velocity in the s-direction orbital velocity in the y-direction orbital velocity in the n-direction time and depth-averaged velocity in the y-direction vertical orbital velocity horizontal coordinate in the onshore direction horizontal coordinate in the longshore direction vertical coordinate with origin at the bed angle of wave approach m o m e n t u m exchange coefficient water surface elevation surface roller thickness angle between the pressure force P and the vertical the density of water shear stress shear stress shear stress mean bed shear stress in the y-direction the magnitude of the near surface shear stress the near surface shear stress vector the angular frequency
INTRODUCTION
When waves break in a surf zone the energy flux and radiation stress decrease towards the shoreline. The depth-integrated force balance in the surf zone of a long uniform coast has been understood well since the work of Longuet-Higgins and Stewart ( 1962 ), and Longuet-Higgins (1970a,b). The crossshore gradient in the shore-normal radiation stress is balanced by a shoreward slope S of the mean water surface, the wave set-up. The gradient in the shore parallel shear c o m p o n e n t of the radiation stress must be balanced by a time mean bed shear stress, which is associated with a wave-driven longshore current. The vertical distribution of the driving forces due to wave-breaking in the
THREE-DIMENSIONAL SHEAR STRESSDISTRIBUTION IN A SURF ZONE
159
surf zone is less well described. It must, however, be known in order to determine the vertical shear stress distribution and thus the velocity distribution of a cross-shore circulation current or a longshore current, Svendsen and Lorenz (1989). The shear stress distribution in a surf zone was analyzed by Deigaard and Fredsoe (1989) for waves approaching perpendicularly on a long straight uniform coast. The scope of the present work is to extend the analysis of Deigaard and Freds~e (1989) to three-dimensional conditions, i.e. to a situation with oblique wave approach. The flow velocities considered in this context are the organized wave-orbital velocities and the motion of the surface rollers. The velocity and pressure distribution are applied in the mom e n t u m conservation equation for a closed control surface. By time-averaging it is found that shear stresses must be introduced in order to conserve the m o m e n t u m . These shear stresses may be laminar shear stresses or turbulent Reynolds' stresses, and they may in the present formulation contain contributions from any m o m e n t u m transfer due to gradients in the wave-driven mean currents. In the former work by Deigaard and Fredsee it was found in the two-dimensional analysis that the vertical flux of m o m e n t u m is important for the shear stress distribution. For non-dissipative waves, described by potential flow theory, propagating over a horizontal bottom the vertical and horizontal orbital velocities are completely (zc/2) out of phase, and do not cause any vertical flux of horizontal m o m e n t u m . If the wave height varies in the direction of propagation due to dissipation of wave energy, a small phase shift is introduced between the surface elevation r/and the horizontal orbital velocity A:
Coast line
) Particle path Wove crests
/
/
% Fig. 1. The particle paths for dissipative waves a p p r o a c h i n g a coast. ( A ) N o r m a l approach. ( B ) Oblique approach.
160
R. DEIGAARD
u. Further, the vertical orbital velocity w and u are not totally out of phase, and a significant time-averaged vertical flux of m o m e n t u m puw is associated with the organized wave-orbital motion. In the present work it is demonstrated that in the case of oblique wave approach one more significant contribution to the vertical m o m e n t u m flux is introduced. For uniform longshore conditions the wave height is a function of the distance from the coast. When the waves approach the coast at an angle, the wave height thus varies along each wave crest. This gives a periodically varying pressure gradient normal to the direction of wave propagation. When the wave height decreases towards the shore, the pressure gradient is directed (positive) obliquely offshore at the crests and obliquely onshore at the troughs. The transverse pressure gradient gives a small transverse orbital velocity, and the particle paths due to the wave-orbital motion will appear as ellipses when viewed upon from above, Fig. 1. This weak transverse orbital motion is found to give a significant contribution to the vertical m o m e n t u m flux in the case of oblique wave approach. T H E SHEAR STRESS D I S T R I B U T I O N
The three-dimensional distribution of shear stresses in the surf zone is analyzed by considering the simplest possible situation: waves approaching a long uniform coast. At the location considered, the bed is horizontal, but the waves are breaking as spilling breakers or are broken when they pass through the section of interest. The waves approach the coast obliquely with an angle o~ between the wave crests and the coastline. The coordinate systems used for the analysis is shown in Fig. 2. The x - y system is fixed relative to the coast with the y-axis parallel to the coast and the x-axis normal to the coast, positive Coast line
~Y
jJ
X
Fig. 2. The coordinate systems applied.
161
THREE-DIMENSIONALSHEAR STRESS DISTRIBUTION IN A SURF ZONE
in the onshore direction. The s-n system is aligned with the waves, the s-axis is in the direction of wave propagation and the n-axis is parallel to the wave crests. The water motion The wave m o t i o n is described by using assumptions which in many respects are similar to the model introduced by Svendsen (1984a,b) for waves in the surf zone. The pressure is assumed to be hydrostatic, and the horizontal orbital velocity u is constant over the water depth, the wave celerity c is given by the linear shallow water wave theory
c=v/
(1)
where g is the acceleration of gravity and D is the water depth. At each wave front there is a surface roller that follows the wave having a velocity of c. The volume of the surface roller per unit length of wave crest is equal to the crosssectional area A. The local thickness of the surface roller is ~+. The geometry of the broken wave is illustrated in Fig. 3. The main difference between the present formulation and the model of Svendsen (1984a) is that the variation in wave height is taken into account when calculating the wave-orbital motion. This is of significance for determining the correct shear stress distribution as demonstrated by Deigaard and Fredsoe (1989) and by De Vriend and Kitou (1990). The wave profile is described as a cosine. This is a simplification, but other wave profiles may be introduced by use of Fourier series. The water surface elevation 17can then be written H(x) r / = ~ - - - c o s ( k s - cot)
(2)
where k is the wave n u m b e r and co is the angular wave frequency. Here and in the following only the lowest non-zero term in (H/D) is maintained. It follows from the assumption of uniform conditions along the coast that the
Iz
Surface S
////I/I/I///////////////////////////////,.'////,¢/,¢/////////////
Fig. 3. The profile of a broken wave.
162
R. DEIGAARD
wave height is a function of x only. Due to the energy dissipation the wave height decreases towards the shore. Within a short distance a linear approximation can be applied
H=Ho+xH',-
(3)
where we take x = 0 at the location considered. The horizontal wave-orbital velocity in the s-direction Us is determined from the continuity equation
DOus _
O~l
0s
0t
(4)
Equations 2, 3, and 4 give the following expression for us
us=2D 1 + ( H ; / k H ) 2 c o s ( k s - c o t ) - ~ s l n ( k s - c o t )
(
2HD c o s ( k s - c o t )
~ s ,l n ( k s - c o t Ms
.
(5)
,)
where it has been assumed that the variation in the wave height is weak H;
cosaHx
kH-
kH
<<1
(6)
The vertical orbital velocity is given by
z Hcoz . w = ~ t D -- 2 ~sln(ks--cot)
(7)
where z is the vertical coordinate with origin at the bed. These two orbital velocity components are similar to the results from a twodimensional analysis with waves approaching the coast perpendicularly, see e.g. Deigaard and Fredsoe (1989). As described above the three-dimensionality gives, however, a small, but significant, velocity component in the n-direction. Due to the uniform conditions the wave height H is constant for a given value o f x . Because of the angle a between the wave crests and the yaxis this means that the wave height varies along the crest of the waves. The water surface slope along the wave crests gives a velocity component v,, which can be determined by the inviscid flow equation
Ov,, Or/ ---2g -Onn OHc°s( ks-cot ) =g'"sin2 H Ot -- --go-n= a cos(ks-cot)
(8)
giving vn =
sin a sin(ks-cot)
(9)
163
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
The particles paths are thus seen to be ellipses in the horizontal plane. The orbital velocities in the x - y coordinate system are determined by projection of us and v. u = u s c o s a - v . s i n o~=~-D cos(ks-tot)-~--HCOS o~ sin(ks-tot)
cos o~
+ g ~ sin2a sin ( ks - tot)
(lO)
and V=UsSin o l + v . c o s ol=~-~ c o s ( k s - t o t ) - ~ c o s
-
sin a cos a sin ( k s - t o t )
o~ s i n ( k s - t o t )
sin oz
( 11 )
In addition to these wave-induced orbital motions the water in the surface rollers moves with the wave celerity c in the direction of wave propagation.
The shear stress distribution The shear stresses are determined by applying the conservation equation for m o m e n t u m on the box-shaped control surface shown in Fig. 4. The m o m e n t u m equation is a vector equation which contains integrals over the volume F a n d the surface A of the control surface
where dA is the area vector of a surface element directed out of the surface, and d~ris the force from the shear stress acting on a surface element. Equation 12 expresses that the acceleration of the mass in the control volume is equal to the sum of the m o m e n t u m flux through the surface and the pressure force, gravity and shear stress force acting on the control volume. It should be noted that the control surface extends into the air, and that all terms in Eq. 12 become zero above the instantaneous water surface. We are interested in the time-averaged shear stresses, and Eq. 12 is time-averaged over a wave period. The left-hand side of Eq. 12 becomes zero after the time averaging. Because the waves are periodic no m o m e n t u m is added to the system during a wave period.
164
R.DEIGAARD
zy_
dx
u
U
y
X
Y
dx
z×l [ ~ -~--~,~zy
dy
V
i X
Fig. 4. The control volumefor the momentumequation. The horizontal shear stress in the x-direction Zzx is found by projecting Eq. 12 on the x-axis. Due to the uniform conditions along the coast there is no time-averaged contribution across the surfaces in the x-z plane, and the calculations are quite similar to the situation with normally incident waves. This calculation is described in detail by Deigaard and Fredsoe ( 1989 ), and the momentum equation is found to read 0=
d ( u 2 ) ( D - z ) - c o s o l ~ x - ~ +uw-g-~ Ox dx
---
p
(13)
where S is the slope of the mean water surface, the wave set-up. In Eq. 13 only the lowest order of H/D (second order) has been maintained. The mean water surface slope S is not known a priori, but it is assumed that S is of order
O((H/D)e). Insertion of the expressions for the wave kinematics is straightforward, giving the vertical distribution of the mean shear stress ( z ) g d ~ - - 2 ) c ° s 2 c ¢ - g d(H2)~ z g d ( H 2) Zx--s-~=- 1-~-D ~ 16 dx D 16 dx sin2°t P d /"mc'~
(14)
2
This expression is analogous to the result of two-dimensional analysis, except for the term with sinZa which is due to the contribution from the small transverse orbital velocity vn (cf. Eq. 9) to the uwterm in Eq. 13. For a equal zero
165
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
Eq. 14 is identical to the result obtained by Deigaard and Fredsoe (1989). The mean surface shear stress is different from zero, and it is therefore not possible to obtain a force balance in the x-direction without introducing shear stresses (%x # 0 ). This means that a circulation current, the undertow, will be generated for all angles of incidence. The vertical distribution of the driving forces (i.e. the momentum delivered by the waves due to breaking) in the longshore direction is determined by projecting the momentum equation on the y-axis. Due to the uniform conditions there is no contribution from the pressure or from the momentum flux across the surfaces in the x-z-plane, and the time-averaged momentum equation reads -d { cA'~ d(uv~) ( D - z ) - c ° s ° l s i n a d x ~ T )
0=
Vzy I D l d ( rxy ) dz p +Oz p dx
(15)
The last term in Eq. 15 is due to the shear stresses acting on the surfaces in the y - z plane. By inserting the expressions for the wave kinematics (Eqs. 2, 7, 10 and 11 ) the vertical distribution of the driving force from the wave motion is found to be zz_Zy_ ~f ° 1 dzxy d z = p pdx -
d [Hc . Hc (D-z) dx \ L,i c°s(ks- tot)cos afb c o s ( k s - t o t ) s i n s
-dx +
d ( u- -v ) ( D _ z ) _ c o s a s i n a ~ x k yd) +{cA~ vw dx "X
)
cos ot sin a 2Dk
= - (O-z)~D
cos a sin ot s i n ( k s - t o t ) ~ T ~ cos a sin a - ~
~
~ sin(ks-tot) cos a sin a
cH" He; z cos a sin a D k 2 2D = _ (D_z)d
{H2g'~ ~-)cos
d {cA'~ °t sin ° t - ~x ~-~)cos a sin a
z c 2 d {H2"~ D 4D ~ - ]cos ot sin a
=-
dx
~ - ~ -~
coso~sina
(16)
166
R. DEIGAARD
where it has been assumed that the dissipation was weak, cf. Eq. 6, and only first order terms in H ' ~ / ( k H ) has been maintained. This distribution of the driving force contains significant contributions to the vw-term from us as well as from vn. It can be noted from Eq. 16 that the driving force is constant over the depth, which means that the longshore current can be described as driven by a time-mean shear stress zsY=
dSxy dx
(17)
applied at the surface, where Sxy is the shear component of the radiation stress, including the contribution from the surface rollers. _
_
m
The relative importance Of Zzy and z~y It is not possible a priori to know the relative importance of zzy and Zxy over the depth, as it depends on the three-dimensional distribution of the velocities. Often Zxy is modelled by use of an eddy viscosity term, and it is recognized in the momentum exchange term in the depth-integrated flow equation for the longshore current, cf. Longuet-Higgins (1970a,b) dSxy d dV 1"~ 0 -- dx = 'rby- p - ~ D ~ ~ Zby -- Jo ~xZXYdz
( 18 )
where Sxy is the shear component of the radiation stress, rby is the mean bed shear stress in the y-direction, e is the eddy viscosity and Fis the depth-integrated longshore___velocity. The solution to Eq. 18 gives an indication of the significance of zxy. In the inner surf zone, far inshore of the breaker point, it is found that the bed shear stress is very close to the driving force, and the horizontal momentum exchange is insignificant. Close to the point of wavebreaking the ~ term may dominate over Zby in the case of regular waves, where the driving force often is taken to be changing abruptly from zero outside the breaker point to a finite value inshore of it. In the case of irregular (mY/s) ' 1.0-
--
With mixing
....
Without mixing
~---
0.5-
0.0 -600
-400
-200
0
x(m)
Fig. 5. The longshore current profile for irregular breaking waves on a plane beach, after Zyserman (1989).
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
167
waves the time-__averaged driving force is varying continuously, and the significance of the zxy term becomes less. Figure 5 shows an example of longshore current velocity profiles calculated for a plane beach with a slope 1 : 100 and irregular Rayleigh distributed waves with a deep water wave height H = 1.2 m and direction or0= 45 o, the wave period is 7.5 s (Zyserman, 1989 ). The longshore current is calculated for e = 0 and for an ~ calculated by the model of Battjes (1975). It is seen how the m o m e n t u m exchange gives a slightly smoother velocity profile. The difference between the velocities is, however, small except far from the coastline, where the driving forces are very small. At the location of m a x i m u m velocity the difference is only about 10%, which means that f~ (OZzy/Ox)dzis about 20% of Zby. THE NEAR SURFACE SHEAR STRESS
The near surface shear stress z~ (i.e. at wave trough level) can be found from Eqs. 14 and 16 by inserting z = D
(-c°s2°t-l+sin2°t ~gdx-~-~(7)c°s2°t
-
cos ot sin a ~ g dx
1 dE
d fac'
dx T
cos c~sin c~
" fcos, )
= -~g---~-coso~--~y)coso~)lsina ~
( 1 = (
=
a ( c55 cos.
- 8 g ds - d s \ T / / ( s i n o ~ ) 1 dEfw
c ds
d (Ac~cosa~ ds \T/J(sinot J
(19)
where Efw is the energy flux in the direction of wave propagation, associated with the wave motion only (i.e. not including any roller contribution). It is seen that the near surface shear stress is in the direction of wave propagation and has a magnitude which can be expressed as the loss of wave energy and the change in m o m e n t u m of the surface rollers, similarly to the expression found by Deigaard and Fredsoe (1989) for normally incident waves. It should be noted that this analysis is only valid for surf zone conditions, where the energy dissipation takes place near the surface. Outside the surf zone quite different results are obtained. If energy dissipation is neglected the wave motion can be described by potential flow theory, in which case no shear stresses would be found. If the energy dissipation takes place in the near-bed
168
R. DEIGAARD
oscillatory boundary layer, the vertical motion due to the spatial and temporal variation of the displacement in the oscillatory boundary layer must be included in the analysis, cf. the streaming analyzed by Longuet-Higgins (1953). The vertical transfer of horizontal m o m e n t u m due to this motion will cancel out other contributions to the longshore driving forces outside the boundary layer. T H E S H E A R STRESS F R O M T H E S U R F A C E R O L L E R S
The analysis of the distribution of the driving force from spilling breakers or broken waves, outlined in the previous section, covers the simplest possible three-dimensional situation. It will be far more complicated to extend this analysis to include effects of a sloping bed, a three-dimensional bed topography or waves which are not described by linear shallow water theory. In order to avoid these complications, the near surface shear stress is determined from a mass and m o m e n t u m balance for the surface rollers, using a consistent, but simplified, model for the roller dynamics. The mean shear stress is determined along the wave surface at z equal to D + r/, and the result may therefore deviate from z~ which is at the trough level, by a higher order term, but they shall be identical at the order considered presently. The force balance of a surface roller is considered, Fig. 6. The surface roller acts on the surface of the wave with a force that can be composed of a pressure force P and a shear force Ts, both are integrated over the length of the roller, cf. Duncan ( 1981 ). The angles of P (and Ts) with the vertical (horizontal) are taken to be small sin0 ~ tan0 ~ 0 COS0~ 1
(20)
The roller moves with a velocity equal to the wave celerity c, which is assumed to be large compared to the orbital velocities in the wave motion. The shear force - - or exchange of horizontal m o m e n t u m over the boundary - - is represented by an upward flux of water qu into the roller and a downward flux qa out of the roller, the water of the downward flux has a horizontal velocity of c, and the water in the upward flux have a horizontal velocity equal to the wave-orbital velocity, which is small compared to c. c 0
Fig. 6. The forcesactingon the wave from the surfaceroller.
THREE-DIMENSIONALSHEARSTRESSDISTRIBUTION1NASURFZONE
169
The continuity equation for the roller reads dA =qu--qo dt
(21)
The momentum equation for the roller reads dAc P ~ - t = P sin0-pcqd
(22)
The shear force from the roller, acting on the water below it, is equal to the downward flux of horizontal momentum Ts =pcqo
(23)
giving a mean surface shear stress of _pcqd _Pqd L T
(24)
where L is the wave length. The conservation of energy is expressed by dgfw ds
dgfr ds = ~ t
(25)
where Efw is the wave energy flux and Err is the energy flux associated with the surface rollers, cf. Svendsen (1984a) dEft _ ds
d ..fpAc2'~= d s \ 2T J
p c 2 dA 2T ds
pAc dc T ds
( 26)
~t is the total rate of energy dissipation per unit bed area, which can be expressed as the loss of kinetic energy due to the exchange of water between the rollers and the wave 1 ~t=~(2
pquc 2+~l
flqd¢2) = ~pc( q u
+q.)
(27)
Inserting Eqs. 27 and 26 into 25 gives
pc/ldAc 1Adc) -~ ~ - ~ - + ~ ~ - )
=
~
pc
~-~(qu + qd)
(28)
By use of the continuity equation, Eqs. 27 and 28 can be written -- Pqd 1 p dA 1 ~t zs = r = c @ t - 2 - T d---t-=c or
pc da 2T ds
(29)
170
R. DEIGAARD
-- Pqd z~- T-
1 dEfw c ds
1 dEfw -
-
c
ds
p ( I d(Ac) 1 dc'~ pc dA 7",2 ds ~-2Adss) 2 T d s
(30)
p d(Ac) T ds
where it has been used that the surface roller travels with the velocity c dA dA dt=C-d~s
(31)
Equation 30 is identical to the result of the Eulerian analysis based on the momentum equation, Eq. 19. This derivation has been made without any assumptions for the wave motion, except that the wave-orbital velocities are neglegible compared to c, and that 0 is small. CONCLUSION
The driving force from obliquely incident waves has been analyzed, and it is found that a longshore current can be described as driven by a mean shear stress applied at the surface. In the cross-shore direction, shear stresses and thus a circulation (the undertow) will be present for all angles of wave incidence. The near surface shear stress caused by the spilling breakers or broken waves can be determined by considering the momentum balance for the surface rollers. ACKNOWLEDGEMENT
This work was undertaken as part of the MAST G6 and G8 Morphodynamics research programme. It was founded jointly by the Danish Technical Research Council under the programme "Marin Teknik" and by the Commission of the European Communities, Directorate General for Science, Research and Development under MAST contract No. 0035-C and MAS2-CT920027.
REFERENCES Battjes, J.A., 1975. Modelling of turbulence in the surf zone. In: Proc. Symp. Modelling Techniques, San Francisco. ASCE, pp. 1050-1061. De Vriend, H.J. and Kitou, N., 1990. Incorporation of wave effects in a 3D hydrostatic mean current model. In: Proc. 22nd Coastal Eng. Conf., Delft. ASCE, pp. 1005-1018. Deigaard, R. and Fredsoe, J., 1989. Shear stress distribution in dissipative water waves. Coastal Eng., 13: 357-378. Duncan, J.H., 1981. An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. London, Ser. A, 377: 331-348.
THREE-DIMENSIONALSHEAR STRESS DISTRIBUTION IN A SURF ZONE
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Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser. A, 245: 535-581. Longuet-Higgins, M.S., 1970a. Longshore currents generated by obliquely incident waves 1. J. Geophys. Res., 75: 6778-6789. Longuet-Higgins, M.S., 1970b. Longshore currents generated by obliquely incident waves 2. J. Geophys. Res., 75: 6790-6801. Longuet-Higgins, M.S. and Stewart, R.W., 1962. Radiation stress and mass transport in gravity waves, with application to surf beats. J. Fluid Mech., 13:481-504. Svendsen, I.A., 1984a. Wave heights and set-up in a surf zone. Coastal Eng., 8: 303-329. Svendsen, I.A., 1984b. Mass flux and undertow in a surf zone. Coastal Eng., 8: 347-365. Svendsen, I.A. and Lorenz, R.S., 1989. Velocities in combined undertow and longshore currents. Coastal Eng., 13: 55-79. Zyserman, J., 1989. Characteristics of stable rip-current systems on a coast with a longshore bar. Series Paper No. 46, Inst. of Hydrodynamics and Hydraulic Engineering, ISVA, Techn. Univ. Denmark, 178 pp.
157
A note on the t h r e e - d i m e n s i o n a l shear stress distribution in a surf zone Rolf Deigaard Institute of Hydrodynamics and Hydraulic Engineering (ISVA), Technical University of Denmark, DK-2800 Lyngby, Denmark (Received 4 August 1992; accepted after revision 8 January 1993 )
ABSTRACT The three-dimensional t i m e - m e a n shear stress distribution in a surf zone is analyzed, thereby determining the vertical distribution of the driving forces due to wave-breaking. It is found that on a uniform coast the longshore current can be described as driven by a surface shear stress equal to the cross-shore gradient in the shear component of the radiation stress. The near surface shear stress is found to be in the direction of wave propagation, having a magnitude determined by the dissipation of wave energy. It can be determined as the shear force acting on the waves from the surface rollers of spilling breakers or broken waves.
LIST O F S Y M B O L S
A c D ~t Err Efw
g H H0 k L n p P qa
qu
cross-sectional area of surface roller wave celerity mean water depth energy dissipation rate per unit bed area energy flux associated with the surface rollers energy flux associated with the wave motion acceleration of gravity wave height wave height at x = 0 wave number wave length horizontal coordinate, parallel with the wave crests pressure pressure force acting on the wave from the roller the downward moving part of the exchange of fluid between the surface roller and the wave the upward moving part of the exchange of fluid between the surface roller and the wave
0 3 7 8 - 3 8 3 9 / 9 3 / $ 0 6 . 0 0 © 1993 Elsevier Science Publishers B.V. All rights reserved.
158
s S
S×y t T Ts u fi us v vn V w x y z o~ r/ ~/+ 0 p r~v r=x r:y rby rs r-~ 09
R. DEIGAARD
horizontal coordinate in the direction of wave propagation slope of mean water surface shear c o m p o n e n t of the radiation stress time wave period shear force acting on the wave from the surface roller orbital velocity in the x-direction flow velocity vector orbital velocity in the s-direction orbital velocity in the y-direction orbital velocity in the n-direction time and depth-averaged velocity in the y-direction vertical orbital velocity horizontal coordinate in the onshore direction horizontal coordinate in the longshore direction vertical coordinate with origin at the bed angle of wave approach m o m e n t u m exchange coefficient water surface elevation surface roller thickness angle between the pressure force P and the vertical the density of water shear stress shear stress shear stress mean bed shear stress in the y-direction the magnitude of the near surface shear stress the near surface shear stress vector the angular frequency
INTRODUCTION
When waves break in a surf zone the energy flux and radiation stress decrease towards the shoreline. The depth-integrated force balance in the surf zone of a long uniform coast has been understood well since the work of Longuet-Higgins and Stewart ( 1962 ), and Longuet-Higgins (1970a,b). The crossshore gradient in the shore-normal radiation stress is balanced by a shoreward slope S of the mean water surface, the wave set-up. The gradient in the shore parallel shear c o m p o n e n t of the radiation stress must be balanced by a time mean bed shear stress, which is associated with a wave-driven longshore current. The vertical distribution of the driving forces due to wave-breaking in the
THREE-DIMENSIONAL SHEAR STRESSDISTRIBUTION IN A SURF ZONE
159
surf zone is less well described. It must, however, be known in order to determine the vertical shear stress distribution and thus the velocity distribution of a cross-shore circulation current or a longshore current, Svendsen and Lorenz (1989). The shear stress distribution in a surf zone was analyzed by Deigaard and Fredsoe (1989) for waves approaching perpendicularly on a long straight uniform coast. The scope of the present work is to extend the analysis of Deigaard and Freds~e (1989) to three-dimensional conditions, i.e. to a situation with oblique wave approach. The flow velocities considered in this context are the organized wave-orbital velocities and the motion of the surface rollers. The velocity and pressure distribution are applied in the mom e n t u m conservation equation for a closed control surface. By time-averaging it is found that shear stresses must be introduced in order to conserve the m o m e n t u m . These shear stresses may be laminar shear stresses or turbulent Reynolds' stresses, and they may in the present formulation contain contributions from any m o m e n t u m transfer due to gradients in the wave-driven mean currents. In the former work by Deigaard and Fredsee it was found in the two-dimensional analysis that the vertical flux of m o m e n t u m is important for the shear stress distribution. For non-dissipative waves, described by potential flow theory, propagating over a horizontal bottom the vertical and horizontal orbital velocities are completely (zc/2) out of phase, and do not cause any vertical flux of horizontal m o m e n t u m . If the wave height varies in the direction of propagation due to dissipation of wave energy, a small phase shift is introduced between the surface elevation r/and the horizontal orbital velocity A:
Coast line
) Particle path Wove crests
/
/
% Fig. 1. The particle paths for dissipative waves a p p r o a c h i n g a coast. ( A ) N o r m a l approach. ( B ) Oblique approach.
160
R. DEIGAARD
u. Further, the vertical orbital velocity w and u are not totally out of phase, and a significant time-averaged vertical flux of m o m e n t u m puw is associated with the organized wave-orbital motion. In the present work it is demonstrated that in the case of oblique wave approach one more significant contribution to the vertical m o m e n t u m flux is introduced. For uniform longshore conditions the wave height is a function of the distance from the coast. When the waves approach the coast at an angle, the wave height thus varies along each wave crest. This gives a periodically varying pressure gradient normal to the direction of wave propagation. When the wave height decreases towards the shore, the pressure gradient is directed (positive) obliquely offshore at the crests and obliquely onshore at the troughs. The transverse pressure gradient gives a small transverse orbital velocity, and the particle paths due to the wave-orbital motion will appear as ellipses when viewed upon from above, Fig. 1. This weak transverse orbital motion is found to give a significant contribution to the vertical m o m e n t u m flux in the case of oblique wave approach. T H E SHEAR STRESS D I S T R I B U T I O N
The three-dimensional distribution of shear stresses in the surf zone is analyzed by considering the simplest possible situation: waves approaching a long uniform coast. At the location considered, the bed is horizontal, but the waves are breaking as spilling breakers or are broken when they pass through the section of interest. The waves approach the coast obliquely with an angle o~ between the wave crests and the coastline. The coordinate systems used for the analysis is shown in Fig. 2. The x - y system is fixed relative to the coast with the y-axis parallel to the coast and the x-axis normal to the coast, positive Coast line
~Y
jJ
X
Fig. 2. The coordinate systems applied.
161
THREE-DIMENSIONALSHEAR STRESS DISTRIBUTION IN A SURF ZONE
in the onshore direction. The s-n system is aligned with the waves, the s-axis is in the direction of wave propagation and the n-axis is parallel to the wave crests. The water motion The wave m o t i o n is described by using assumptions which in many respects are similar to the model introduced by Svendsen (1984a,b) for waves in the surf zone. The pressure is assumed to be hydrostatic, and the horizontal orbital velocity u is constant over the water depth, the wave celerity c is given by the linear shallow water wave theory
c=v/
(1)
where g is the acceleration of gravity and D is the water depth. At each wave front there is a surface roller that follows the wave having a velocity of c. The volume of the surface roller per unit length of wave crest is equal to the crosssectional area A. The local thickness of the surface roller is ~+. The geometry of the broken wave is illustrated in Fig. 3. The main difference between the present formulation and the model of Svendsen (1984a) is that the variation in wave height is taken into account when calculating the wave-orbital motion. This is of significance for determining the correct shear stress distribution as demonstrated by Deigaard and Fredsoe (1989) and by De Vriend and Kitou (1990). The wave profile is described as a cosine. This is a simplification, but other wave profiles may be introduced by use of Fourier series. The water surface elevation 17can then be written H(x) r / = ~ - - - c o s ( k s - cot)
(2)
where k is the wave n u m b e r and co is the angular wave frequency. Here and in the following only the lowest non-zero term in (H/D) is maintained. It follows from the assumption of uniform conditions along the coast that the
Iz
Surface S
////I/I/I///////////////////////////////,.'////,¢/,¢/////////////
Fig. 3. The profile of a broken wave.
162
R. DEIGAARD
wave height is a function of x only. Due to the energy dissipation the wave height decreases towards the shore. Within a short distance a linear approximation can be applied
H=Ho+xH',-
(3)
where we take x = 0 at the location considered. The horizontal wave-orbital velocity in the s-direction Us is determined from the continuity equation
DOus _
O~l
0s
0t
(4)
Equations 2, 3, and 4 give the following expression for us
us=2D 1 + ( H ; / k H ) 2 c o s ( k s - c o t ) - ~ s l n ( k s - c o t )
(
2HD c o s ( k s - c o t )
~ s ,l n ( k s - c o t Ms
.
(5)
,)
where it has been assumed that the variation in the wave height is weak H;
cosaHx
kH-
kH
<<1
(6)
The vertical orbital velocity is given by
z Hcoz . w = ~ t D -- 2 ~sln(ks--cot)
(7)
where z is the vertical coordinate with origin at the bed. These two orbital velocity components are similar to the results from a twodimensional analysis with waves approaching the coast perpendicularly, see e.g. Deigaard and Fredsoe (1989). As described above the three-dimensionality gives, however, a small, but significant, velocity component in the n-direction. Due to the uniform conditions the wave height H is constant for a given value o f x . Because of the angle a between the wave crests and the yaxis this means that the wave height varies along the crest of the waves. The water surface slope along the wave crests gives a velocity component v,, which can be determined by the inviscid flow equation
Ov,, Or/ ---2g -Onn OHc°s( ks-cot ) =g'"sin2 H Ot -- --go-n= a cos(ks-cot)
(8)
giving vn =
sin a sin(ks-cot)
(9)
163
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
The particles paths are thus seen to be ellipses in the horizontal plane. The orbital velocities in the x - y coordinate system are determined by projection of us and v. u = u s c o s a - v . s i n o~=~-D cos(ks-tot)-~--HCOS o~ sin(ks-tot)
cos o~
+ g ~ sin2a sin ( ks - tot)
(lO)
and V=UsSin o l + v . c o s ol=~-~ c o s ( k s - t o t ) - ~ c o s
-
sin a cos a sin ( k s - t o t )
o~ s i n ( k s - t o t )
sin oz
( 11 )
In addition to these wave-induced orbital motions the water in the surface rollers moves with the wave celerity c in the direction of wave propagation.
The shear stress distribution The shear stresses are determined by applying the conservation equation for m o m e n t u m on the box-shaped control surface shown in Fig. 4. The m o m e n t u m equation is a vector equation which contains integrals over the volume F a n d the surface A of the control surface
where dA is the area vector of a surface element directed out of the surface, and d~ris the force from the shear stress acting on a surface element. Equation 12 expresses that the acceleration of the mass in the control volume is equal to the sum of the m o m e n t u m flux through the surface and the pressure force, gravity and shear stress force acting on the control volume. It should be noted that the control surface extends into the air, and that all terms in Eq. 12 become zero above the instantaneous water surface. We are interested in the time-averaged shear stresses, and Eq. 12 is time-averaged over a wave period. The left-hand side of Eq. 12 becomes zero after the time averaging. Because the waves are periodic no m o m e n t u m is added to the system during a wave period.
164
R.DEIGAARD
zy_
dx
u
U
y
X
Y
dx
z×l [ ~ -~--~,~zy
dy
V
i X
Fig. 4. The control volumefor the momentumequation. The horizontal shear stress in the x-direction Zzx is found by projecting Eq. 12 on the x-axis. Due to the uniform conditions along the coast there is no time-averaged contribution across the surfaces in the x-z plane, and the calculations are quite similar to the situation with normally incident waves. This calculation is described in detail by Deigaard and Fredsoe ( 1989 ), and the momentum equation is found to read 0=
d ( u 2 ) ( D - z ) - c o s o l ~ x - ~ +uw-g-~ Ox dx
---
p
(13)
where S is the slope of the mean water surface, the wave set-up. In Eq. 13 only the lowest order of H/D (second order) has been maintained. The mean water surface slope S is not known a priori, but it is assumed that S is of order
O((H/D)e). Insertion of the expressions for the wave kinematics is straightforward, giving the vertical distribution of the mean shear stress ( z ) g d ~ - - 2 ) c ° s 2 c ¢ - g d(H2)~ z g d ( H 2) Zx--s-~=- 1-~-D ~ 16 dx D 16 dx sin2°t P d /"mc'~
(14)
2
This expression is analogous to the result of two-dimensional analysis, except for the term with sinZa which is due to the contribution from the small transverse orbital velocity vn (cf. Eq. 9) to the uwterm in Eq. 13. For a equal zero
165
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
Eq. 14 is identical to the result obtained by Deigaard and Fredsoe (1989). The mean surface shear stress is different from zero, and it is therefore not possible to obtain a force balance in the x-direction without introducing shear stresses (%x # 0 ). This means that a circulation current, the undertow, will be generated for all angles of incidence. The vertical distribution of the driving forces (i.e. the momentum delivered by the waves due to breaking) in the longshore direction is determined by projecting the momentum equation on the y-axis. Due to the uniform conditions there is no contribution from the pressure or from the momentum flux across the surfaces in the x-z-plane, and the time-averaged momentum equation reads -d { cA'~ d(uv~) ( D - z ) - c ° s ° l s i n a d x ~ T )
0=
Vzy I D l d ( rxy ) dz p +Oz p dx
(15)
The last term in Eq. 15 is due to the shear stresses acting on the surfaces in the y - z plane. By inserting the expressions for the wave kinematics (Eqs. 2, 7, 10 and 11 ) the vertical distribution of the driving force from the wave motion is found to be zz_Zy_ ~f ° 1 dzxy d z = p pdx -
d [Hc . Hc (D-z) dx \ L,i c°s(ks- tot)cos afb c o s ( k s - t o t ) s i n s
-dx +
d ( u- -v ) ( D _ z ) _ c o s a s i n a ~ x k yd) +{cA~ vw dx "X
)
cos ot sin a 2Dk
= - (O-z)~D
cos a sin ot s i n ( k s - t o t ) ~ T ~ cos a sin a - ~
~
~ sin(ks-tot) cos a sin a
cH" He; z cos a sin a D k 2 2D = _ (D_z)d
{H2g'~ ~-)cos
d {cA'~ °t sin ° t - ~x ~-~)cos a sin a
z c 2 d {H2"~ D 4D ~ - ]cos ot sin a
=-
dx
~ - ~ -~
coso~sina
(16)
166
R. DEIGAARD
where it has been assumed that the dissipation was weak, cf. Eq. 6, and only first order terms in H ' ~ / ( k H ) has been maintained. This distribution of the driving force contains significant contributions to the vw-term from us as well as from vn. It can be noted from Eq. 16 that the driving force is constant over the depth, which means that the longshore current can be described as driven by a time-mean shear stress zsY=
dSxy dx
(17)
applied at the surface, where Sxy is the shear component of the radiation stress, including the contribution from the surface rollers. _
_
m
The relative importance Of Zzy and z~y It is not possible a priori to know the relative importance of zzy and Zxy over the depth, as it depends on the three-dimensional distribution of the velocities. Often Zxy is modelled by use of an eddy viscosity term, and it is recognized in the momentum exchange term in the depth-integrated flow equation for the longshore current, cf. Longuet-Higgins (1970a,b) dSxy d dV 1"~ 0 -- dx = 'rby- p - ~ D ~ ~ Zby -- Jo ~xZXYdz
( 18 )
where Sxy is the shear component of the radiation stress, rby is the mean bed shear stress in the y-direction, e is the eddy viscosity and Fis the depth-integrated longshore___velocity. The solution to Eq. 18 gives an indication of the significance of zxy. In the inner surf zone, far inshore of the breaker point, it is found that the bed shear stress is very close to the driving force, and the horizontal momentum exchange is insignificant. Close to the point of wavebreaking the ~ term may dominate over Zby in the case of regular waves, where the driving force often is taken to be changing abruptly from zero outside the breaker point to a finite value inshore of it. In the case of irregular (mY/s) ' 1.0-
--
With mixing
....
Without mixing
~---
0.5-
0.0 -600
-400
-200
0
x(m)
Fig. 5. The longshore current profile for irregular breaking waves on a plane beach, after Zyserman (1989).
THREE-DIMENSIONAL SHEAR STRESS DISTRIBUTION IN A SURF ZONE
167
waves the time-__averaged driving force is varying continuously, and the significance of the zxy term becomes less. Figure 5 shows an example of longshore current velocity profiles calculated for a plane beach with a slope 1 : 100 and irregular Rayleigh distributed waves with a deep water wave height H = 1.2 m and direction or0= 45 o, the wave period is 7.5 s (Zyserman, 1989 ). The longshore current is calculated for e = 0 and for an ~ calculated by the model of Battjes (1975). It is seen how the m o m e n t u m exchange gives a slightly smoother velocity profile. The difference between the velocities is, however, small except far from the coastline, where the driving forces are very small. At the location of m a x i m u m velocity the difference is only about 10%, which means that f~ (OZzy/Ox)dzis about 20% of Zby. THE NEAR SURFACE SHEAR STRESS
The near surface shear stress z~ (i.e. at wave trough level) can be found from Eqs. 14 and 16 by inserting z = D
(-c°s2°t-l+sin2°t ~gdx-~-~(7)c°s2°t
-
cos ot sin a ~ g dx
1 dE
d fac'
dx T
cos c~sin c~
" fcos, )
= -~g---~-coso~--~y)coso~)lsina ~
( 1 = (
=
a ( c55 cos.
- 8 g ds - d s \ T / / ( s i n o ~ ) 1 dEfw
c ds
d (Ac~cosa~ ds \T/J(sinot J
(19)
where Efw is the energy flux in the direction of wave propagation, associated with the wave motion only (i.e. not including any roller contribution). It is seen that the near surface shear stress is in the direction of wave propagation and has a magnitude which can be expressed as the loss of wave energy and the change in m o m e n t u m of the surface rollers, similarly to the expression found by Deigaard and Fredsoe (1989) for normally incident waves. It should be noted that this analysis is only valid for surf zone conditions, where the energy dissipation takes place near the surface. Outside the surf zone quite different results are obtained. If energy dissipation is neglected the wave motion can be described by potential flow theory, in which case no shear stresses would be found. If the energy dissipation takes place in the near-bed
168
R. DEIGAARD
oscillatory boundary layer, the vertical motion due to the spatial and temporal variation of the displacement in the oscillatory boundary layer must be included in the analysis, cf. the streaming analyzed by Longuet-Higgins (1953). The vertical transfer of horizontal m o m e n t u m due to this motion will cancel out other contributions to the longshore driving forces outside the boundary layer. T H E S H E A R STRESS F R O M T H E S U R F A C E R O L L E R S
The analysis of the distribution of the driving force from spilling breakers or broken waves, outlined in the previous section, covers the simplest possible three-dimensional situation. It will be far more complicated to extend this analysis to include effects of a sloping bed, a three-dimensional bed topography or waves which are not described by linear shallow water theory. In order to avoid these complications, the near surface shear stress is determined from a mass and m o m e n t u m balance for the surface rollers, using a consistent, but simplified, model for the roller dynamics. The mean shear stress is determined along the wave surface at z equal to D + r/, and the result may therefore deviate from z~ which is at the trough level, by a higher order term, but they shall be identical at the order considered presently. The force balance of a surface roller is considered, Fig. 6. The surface roller acts on the surface of the wave with a force that can be composed of a pressure force P and a shear force Ts, both are integrated over the length of the roller, cf. Duncan ( 1981 ). The angles of P (and Ts) with the vertical (horizontal) are taken to be small sin0 ~ tan0 ~ 0 COS0~ 1
(20)
The roller moves with a velocity equal to the wave celerity c, which is assumed to be large compared to the orbital velocities in the wave motion. The shear force - - or exchange of horizontal m o m e n t u m over the boundary - - is represented by an upward flux of water qu into the roller and a downward flux qa out of the roller, the water of the downward flux has a horizontal velocity of c, and the water in the upward flux have a horizontal velocity equal to the wave-orbital velocity, which is small compared to c. c 0
Fig. 6. The forcesactingon the wave from the surfaceroller.
THREE-DIMENSIONALSHEARSTRESSDISTRIBUTION1NASURFZONE
169
The continuity equation for the roller reads dA =qu--qo dt
(21)
The momentum equation for the roller reads dAc P ~ - t = P sin0-pcqd
(22)
The shear force from the roller, acting on the water below it, is equal to the downward flux of horizontal momentum Ts =pcqo
(23)
giving a mean surface shear stress of _pcqd _Pqd L T
(24)
where L is the wave length. The conservation of energy is expressed by dgfw ds
dgfr ds = ~ t
(25)
where Efw is the wave energy flux and Err is the energy flux associated with the surface rollers, cf. Svendsen (1984a) dEft _ ds
d ..fpAc2'~= d s \ 2T J
p c 2 dA 2T ds
pAc dc T ds
( 26)
~t is the total rate of energy dissipation per unit bed area, which can be expressed as the loss of kinetic energy due to the exchange of water between the rollers and the wave 1 ~t=~(2
pquc 2+~l
flqd¢2) = ~pc( q u
+q.)
(27)
Inserting Eqs. 27 and 26 into 25 gives
pc/ldAc 1Adc) -~ ~ - ~ - + ~ ~ - )
=
~
pc
~-~(qu + qd)
(28)
By use of the continuity equation, Eqs. 27 and 28 can be written -- Pqd 1 p dA 1 ~t zs = r = c @ t - 2 - T d---t-=c or
pc da 2T ds
(29)
170
R. DEIGAARD
-- Pqd z~- T-
1 dEfw c ds
1 dEfw -
-
c
ds
p ( I d(Ac) 1 dc'~ pc dA 7",2 ds ~-2Adss) 2 T d s
(30)
p d(Ac) T ds
where it has been used that the surface roller travels with the velocity c dA dA dt=C-d~s
(31)
Equation 30 is identical to the result of the Eulerian analysis based on the momentum equation, Eq. 19. This derivation has been made without any assumptions for the wave motion, except that the wave-orbital velocities are neglegible compared to c, and that 0 is small. CONCLUSION
The driving force from obliquely incident waves has been analyzed, and it is found that a longshore current can be described as driven by a mean shear stress applied at the surface. In the cross-shore direction, shear stresses and thus a circulation (the undertow) will be present for all angles of wave incidence. The near surface shear stress caused by the spilling breakers or broken waves can be determined by considering the momentum balance for the surface rollers. ACKNOWLEDGEMENT
This work was undertaken as part of the MAST G6 and G8 Morphodynamics research programme. It was founded jointly by the Danish Technical Research Council under the programme "Marin Teknik" and by the Commission of the European Communities, Directorate General for Science, Research and Development under MAST contract No. 0035-C and MAS2-CT920027.
REFERENCES Battjes, J.A., 1975. Modelling of turbulence in the surf zone. In: Proc. Symp. Modelling Techniques, San Francisco. ASCE, pp. 1050-1061. De Vriend, H.J. and Kitou, N., 1990. Incorporation of wave effects in a 3D hydrostatic mean current model. In: Proc. 22nd Coastal Eng. Conf., Delft. ASCE, pp. 1005-1018. Deigaard, R. and Fredsoe, J., 1989. Shear stress distribution in dissipative water waves. Coastal Eng., 13: 357-378. Duncan, J.H., 1981. An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. London, Ser. A, 377: 331-348.
THREE-DIMENSIONALSHEAR STRESS DISTRIBUTION IN A SURF ZONE
171
Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser. A, 245: 535-581. Longuet-Higgins, M.S., 1970a. Longshore currents generated by obliquely incident waves 1. J. Geophys. Res., 75: 6778-6789. Longuet-Higgins, M.S., 1970b. Longshore currents generated by obliquely incident waves 2. J. Geophys. Res., 75: 6790-6801. Longuet-Higgins, M.S. and Stewart, R.W., 1962. Radiation stress and mass transport in gravity waves, with application to surf beats. J. Fluid Mech., 13:481-504. Svendsen, I.A., 1984a. Wave heights and set-up in a surf zone. Coastal Eng., 8: 303-329. Svendsen, I.A., 1984b. Mass flux and undertow in a surf zone. Coastal Eng., 8: 347-365. Svendsen, I.A. and Lorenz, R.S., 1989. Velocities in combined undertow and longshore currents. Coastal Eng., 13: 55-79. Zyserman, J., 1989. Characteristics of stable rip-current systems on a coast with a longshore bar. Series Paper No. 46, Inst. of Hydrodynamics and Hydraulic Engineering, ISVA, Techn. Univ. Denmark, 178 pp.