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Further enhancements of Boussinesq-type equations
COASTAL ENGINEERING ELSEVIER
Coastal Engineering
26 (1995)
l-14
Further enhancements of Boussinesq-type
equations
Hemming A. Schtiffer, Per A. Madsen International
Research
Centre
forComputational
Hydrodynamics,
5, DK-2970
Received 5 September
ICCH Danish Hydraulic
Institute Agem A&
H#rsholm, Denmark
1994; accepted
16 June 1995
Abstract On the basis of two recently published new forms of the Boussinesq equations, yet another set of is derived. These new equations represent another step forward in the improvement of the linear dispersion and shoaling characteristics. The linearized version of this new set of Boussinesq equations
equations corresponds to a higher order Pad6 expansion in kh of the Stokes linear dispersion relation for waves on arbitrary depth. The formulation also allows for an optimization of the linear shoaling properties giving excellent results. The linear dispersion and shoaling characteristics of the new equations are acceptable even for wavelengths as small as the water depth.
1. Introduction
A major breakthrough in the applicability of Boussinesq-type equations was made by Madsen et al. ( 1991) and Madsen and Sorensen ( 1992), who derived a set of equations corresponding to a [ 2/2] -Pad6 expansion in kh of the Stokes linear dispersion relation for waves on arbitrary depth. For comparison the most accurate of the original forms of the Boussinesq equations (the formulation in the depth averaged velocities) corresponds to a [O/2] -Pad& expansion. This gave a major improvement of the dispersion properties. A review of the phase relations for the traditional forms of the Boussinesq equations is given in Madsen et al. ( 199 1) The equations by Madsen et al. are only valid for constant depth and Madsen and Serensen ( 1992) generalized these equations to mildly sloping bottoms, thus retaining only the lowest order in spatial derivatives of the depth. A derivation similar to that of Madsen and Serensen but without this explicit mild-slope assumption is given in $2 as an introduction to the more complicated derivations that follow. Recently, Nwogu ( 1993) has derived another set of Boussinesq-type equations which have exactly the same linear dispersion characteristics as the equations derived by Madsen et al. While Madsen et al. obtained their results by consistent manipulations of the classical 037%3839/95/$09.50
0
SSDlO378-3839(95)00017-8
1995 Elsevier Science B.V. All rights reserved
equations, Nwogu derived his version in terms of the horizontal velocities taken at an arbitrary level. In $3 both of these ideas are combined leading to yet another class of Boussinesq-type equations, for which a suitably chosen subset gives highly accurate linear dispersion characteristics corresponding to a ]4/4]-PadC expansion in kh, as shown in $3.2. The linear shoaling characteristics are derived in $3.3 and again suitable choices are shown to give excellent results. A common feature of the equations by Madsen et al., by Nwogu, and the present ones is that it is the combined set of equations for mass and momentum that represents the improved dispersion properties. Each of the equations are not developed to a higher order than the classical Boussinesq equations. Thus the underlying expressions for the distribution of vertical and horizontal velocity cannot be trusted in the extended depth range in which the surface elevation is still very well described. Furthermore, since only lowest order nonlinearity is included, the present equations are effectively linear in deep water. Various types of higher order Boussinesq equations are now being investigated in order to mend these shortcomings.
2. Generalization 2.1. Derivation
of Madsen and Serensen (1992)
ofequations
In this section we derive a set of Boussinesq-type equations which is similar to that of Madsen and Sorensen ( 1992), but without the explicit restriction of small bottom slopes. For small bottom slopes a special case of these equations reduce to the Madsen and Sorensen equations. A commonly used scaling of variables is adopted. Let a prime denote dimensional variables then
and
Here (x,~,z) are Cartesian coordinates, h = h(x,y) is the depth, 77= q(x,y,t) is the surface elevation, (u,I’,w) is the particle velocity vector, p is the pressure, p is the density of the fluid and a, 1 and ho are typical values of wave amplitude, wavelength and water depth, respectively. As usual 0( E) = 0( p2) -=K1 is assumed, where E = a/h,, and p = h,,ll are the measures of nonlinearity and frequency dispersion. Let Q denote the vector of depth integrated velocity then the classical Boussinesq equations derived by Peregrine ( 1967) may be written in nondimensional form as
H.A. Schiiffer, P.A. Madsen /Coasial
Engineering 26 (1995) 1-14
and
Q,+(hf~v)Vq+~
(Q.V> &+&
(V.Q)
1 (2)
where V = (a/k, a/ay) is the horizontal gradient operator. This was the starting point of Madsen and Sorensen ( 1992). We now introduce two free parameters (B,, B2) 5 0( 1) . Dividing (2) by h + q (or just by h) and using the operator - Bzp2h3V( V. ) on the result, we obtain -B,~2h3(V(V~~)+V(V-V~))=O(&
E,u2, p4)
(3a)
A slightly different expression is obtained by using the operator - B,p2h2V( V. on both sides of (2)) which yields -B,~2h2(V(V.QJ
+V(V.(hVn)))
=O(z,
E/L’, p4)
(3b)
Adding (3a) and (3b) to (2) yields the following class of momentum (Q.V)&+&Q.V)Q
+p2[(:-B2b ( v 0.’
h)
-
‘+B (2
I)
h’V(V.Q,)
) directly
equations:
1 -B,h3(V(V.Vq))
-B,h2(V(V.(hV~)))
=o
1 (4)
retaining only terms up to 0( E, p2). Eqs. ( 1) and (4) give the desired result. The dispersion terms with a factor B, are essentially the ones introduced by Madsen and Sorensen ( 1992)) improving the dispersion properties of the equations. The additional terms with a factor B2 provide a means of adjusting the bottom slope terms in order to improve the linear shoaling characteristics. Returning to dimensional variables but omitting the primes ( 1) and (4) read in Cartesian form
(5)
~,+9r+Q.v=0 and
h2UL+Q,xJ
-Bzgh3hr,+xJ -B,gh2(thxlxx+
[hv,l,)
(6a)
h*(Q,v+W
-&gh3(~y,y.y+77x,v) -B,gh2Wrl?,ly,.+
thrlxl,,)
(6b)
H.A. Schtiffer, P.A. Madsen /Coastal
4
Engineering 26 (1995) l-14
where (P,Q) = Q. 2.2. Linear dispersion characteristics For constant depth B, and B2 only appear as B = B, + B2 and (5) and (6) become identical to the Madsen and Sorensen ( 1992) equations. Thus the same linear dispersion relation can be obtained: cZ
1 + Bk?h*
gh
1+(B+1/3)k2h2
(7)
2.3. Linear shoaling characteristics Madsen and Sorensen also derived the linear shoaling gradient corresponding to their equations and found excellent agreement with results obtained by combining Airy theory with conservation of energy flux. The first step in their approach was to eliminate the depthintegrated velocity from the linearized conservation equations to get a wave equation in the surface elevation alone. This is generally not possible for the linearized versions of (5) and (6). Only in the special case of B2 = 0 the flux can be eliminated and it can be shown that the wave equation obtained then reduces to that of Madsen and Sorensen for small bottom slopes. Thus Eqs. (5) and (6) with B, = 0 and B, = B represent an extension to other than small bottom slopes of the Madsen and Sorensen equations retaining their attractive results for linear dispersion and shoaling. The dispersion relation gives the constraint B, + B2 = B ( = 1 / 15 as shown by Madsen et al., 1991) on the parameters, but we still have one free parameter, say B, = B - B,, which can be chosen as to optimize some measure of the shoaling characteristics. Since, however, (B,, B2) = ( 1 / 15,O) gives satisfactory results, there is no need for further optimization and as shown below it gives only a marginal improvement of the shoaling gradient. Thus, the exercise of this section is rather to show a simple example of the procedure for the linear shoaling analysis when the velocity variable is not eliminated (although this could be done also for B, # 0 when the bottom slope is assumed to be small). This procedure is used again in 93.3 for the more complicated equations derived in 53.1. The linearized versions of (5) and (6) read in one dimension: qr+hu,+hxi=O and (dividing
(8)
by h)
r*,+g7),f(~-B2)h2~~~~~-
(
;+B,
h[h~,l,,-B,gh2rl,,-B,gh[h-rl,l.,=0 1
where U is the mean over depth of the horizontal becomes
(9)
velocity. For small bottom slopes (9)
h2ii,l, - Bgh’r/,*, -Zh,h((~+B,)u,,+B,jill=O
(10)
H.A. Schbffer, P.A. Madsen /Coastal
Engineering 26 (1995) 1-14
5
We now introduce solutions of the form v(~,t) =A(x)e’(“-lk(“)d’) u(x,t)
(lla)
= (D(X) +i~‘,(~))e”“~‘~‘“‘*
(llb)
where A(x), D(x), bJx>, k(x), and h(x) are slowly varying functions of x and where r?, = 0( h,) as marked by the differentiation subscript, the tilde indicating a distinction between fiX and D, (the derivative of D(x) ) . The justification of ( 1 lb) is that for constant depth U is in phase with 7, while for small bottom slopes a small phase shift should be permitted. Using ( 11) in (8) and ( 10) two equations are obtained by retaining only the terms free from derivatives of the slow variables. These are -khD+Aw=O
(12)
-gk&BgkAk?h’+wD+(B+l/3)wD~h*=O
(13)
and
for which non-trivial solutions for (A, D) require the dispersion relation (7) to be satisfied. Two other equations are obtained by collecting terms proportional to the first derivatives of the slow variables: Dh.,+h(D,+kb,)
=0
(14)
and gA,+3Bgkh*(kA,+Ak,)-coti,-w(B+1/3)h2(2kDx+k%-Dk,) -h&h(
(1+2B,)wD-2B,gkA)
=0
Solving (12) for D and differentiating
(15)
the result yields
(16) Solving ( 1.5) for OX, substituting the result in ( 14) and replacing (D, 0,) by their solutions obtained from ( 12) and ( 16), we get an expression which multiplied by ( 1 + Bph’) (1 + (B+ l/3)k2h2)kl(mA) can be written (17) where CY,=2( 1 +2Bk2h2+B(B+ CY~=- 1 +2Bk2h2+3B(B+
l/3)k4h4) 1/3)k4h4
a,=k2h2(2B-1/3+(2B*-B/3+2B1/3)ph2)
( 1W (18b) (18~)
where w has been eliminated using the dispersion relation (7). This expression is quite different from that of Madsen and Sorensen even in the shallow water limit, where the terms involving the parameters vanish. Thus for kh = 0 we get ( (Y,,(Y~,cx~)= (2, - 1,O) while
6
H.A. Schtifier, P.A. Madsen/Coastal
Engineering 26 (1995) 1-14
they have ( CX,,CQ,(Y~)= (2,1,1) (their Eqs. 3.7 and 3.8). However, both results are quite consistent. This is possible, since ( LY,,(Y~,(Y~) are not uniquely determined by ( 17) alone, and the seeming discrepancy vanishes when k,lk is eliminated from ( 17). Following Madsen and Sorensen in the last steps we obtain by differentiation of the dispersion relation (7): (19) where 1 @4=2-6( and elimination
k2h” 1 +2Bk2h2+B(B+
(20)
l/3)k4h4)
of k,lk from ( 17) and ( 19) finally yields the linear shoaling relation
A _I- - -a!- hx A h
(21)
where
is the shoaling gradient, which for Airy waves combined reads A,ry= G 1+ fG( 1 -cash ffs (1 +G)’
2kh)
,
GC2kh sinh 2kh
with conservation
of energy flux
(23)
For B, = B and B2= 0, as can be shown to be identical with Madsen and Sorensen’s expression. For kh = 0, where CX,= l/2, the well known shallow water result CX~ = l/4 appears. For B = 1/ 15 the Taylor expansion of CQfrom kh = 0 gives 4( 1+ 15B,)k4h4+ and a similar expansion
26 - 120B 15
(24)
of atiry reads
I-k2h2+$k4h4+&k”h”+...
(25)
Matching to 0( k4h4) yields B, = 1 / 10, by which B, = B - B, = - l/30. It is interesting to realize that for this optimization the relative weight of the Ur,, and [ hii,] *x terms in (9) is conserved relative to classical Boussinesq theory for which B, = B, = 0. In the derivation of the classical Boussinesq equations this weighting is a consequence of the sloping bottom boundary condition. Thus it is peculiar that it reappears from the above comparison with a combination of Airy theory (for which the depth is constant) and conservation of energy flux, since no sloping bottom boundary condition is included in this theory. For kh < 1.2, B, = 1/ 10 does give the best fit for the shoaling gradient. However, since terms of much higher order than the 0( k4h4) coefficient matched above are inevitable in
H.A. SchiifSer, P.A. Madsen /Coastal Engineering 26 (1995) 1-14
7
as, I?(
a good fit to larger &-values requires minimization of, for example, (y$iry_ (Ye)2d( kh) / (k,h,) with e.g. k,,h, = 3 corresponding to the traditional deep water limit. Such a minimization yields B, = 0.0653 which is only marginally different from the excellent result of Madsen and Sorensen, which corresponds to B, = l/15 =0.0667 and BI!=O. Three conclusions are to be drawn from this section: ( 1) A slightly different form (5) and (6) of the Madsen and Sorensen equations is valid without the explicit mild-slope condition. (2) In these equations B, = B = 1/ 15 and B2 = 0 should be used, since the marginal shoaling improvement possible for B2 f 0 is not worth the extra term in the equation. (3) The choice of the values of free parameters in connection with the improvement of shoaling characteristics should be based on an error minimization for example over the desired kh range rather than on matching coefficients of expansions. This last conclusion is used in the more complicated situation in $3.3.
3. “Deeper water” Boussinesq-type 3.1. Derivation
equations
of equations
Nwogu In terms of the horizontal velocity vector u, at an arbitrary level z=z,(x,y) ( 1993) derived the following set of nondimensional equations: conservation of mass
1
=O(CY?, E/L, /._L~)
and conservation
(26)
of momentum
u,,+V~+E(IC;V)zf,+/.L~
$V(Vu,)
1
+z,V[V-(hu,)]
=U(z,
E/_L~, ci’) (27)
We now introduce four free parameters (/3,,/32,y,,?/2) I O( 1) and proceed as shown in $2.1, only now also the mass equation contains small dispersion terms allowing for further refinements. Using each of the operators - p,p2V. ( h2V ) and p2p2V. V( h2 ) on (26) we obtain -/3,~2(V.(h2V~,)
+V.(h2V[V.(hu,)]))
=O(Z,
E/.L~,/.L~)
(2Sa)
and &L~(V.(V~~~,)
+V.V[h*V.(hu,)])
=0(8,
Similarly, we apply each of the operators sides of (27) to obtain -y,p2h2(V(V.U,,) and
+V(V.Vq))
E/L*,/_L’>
- y,p2h2V(V.
=O(g,
E/L’, @‘up,
) and y,p’hV(V.h
(2gb) ) on both
(29a)
H.A. Schiiffer, P.A. Madsen /Coastal Engineering 26 (1995) 1-14
y+‘h(V(V.hu,)
+V(V.VhVv))
=0(8,
E/_L~, p4)
(29b)
Adding (28a) and (28b) to (26) and (29a) and (29b) to (27) yields +p’V.
r],+V.[(h+q)uJ
+&V[h2V.(hu,)]
hV(V.u,)
-/3,h2Vn,+&V(h2n,)
+ (c+,h)hV,V-(hu,),
=0
(30)
I and
y,h')V(V.u,)+(-,,fy,h)V,V.(/Lu,)]
u,+V,ic(u;V)u..+a’[(~-
-y,h’V(V.Vq)
+ y,hV[Vh.Vn]
I
=0
(31)
where we have omitted terms of 0( 2, ep2, p”) . Eqs. 30 and 3 1 represent a new class of Boussinesq-type equations. We shall refer to (30) and (3 1) as the “deeper water” Boussinesq equations, since for a suitable choice of the free parameters (PI, /$, y,, yz) their linear dispersion and shoaling characteristics are acceptable even when the water depth is almost as large as the wave length. In (30) and (3 1) only terms up to 0( l ,~‘) are retained just like in (26) and (27) or in the equations by Madsen et al. or in the classical forms of the Boussinesq equations. This means that (30) and (3 1) are formally just as restricted to shallow water as any of the other forms of the Boussinesq equations. Returning to dimensional variables but omitting the primes the Cartesian forms of (30) and (31) read
v,+ [(h+v)d,+ xh( [hue],+
+
[(h+rl)CnlY+
[hc,l,,)
+~z~h21h~,l.,+h2~h~1,1.1.,-P,h2rl,+Pz~h2~~1~
+“av) +(+B,h)h(,hu,,,,+
+P~~h2~h~~,l.+h2~h~,l.,l,.-P,h2~,+P~~h2rl,l.
and
+Ls,,,.I +(‘“f;-P,h)
I .i
[hc,],,)
1L
=O
(32)
H.A. Schkiffer, P.A. Madsen /Coastal Engineering 26 (1995) l-14
+
+
(z,+
(z,+
[~4xl,+
y,h)(
y&1(
[h4v,l,f
[~~,I,)
9
- T5gh2(rlxxx+77xyy+ 72gh( [h%lxx+ [h&J
[hLf,l,) - wh2h,+
71vw) +-Y&t [bJ~+
=o (33a)
[~~~I,) =o (33b)
3.2. Linear dispersion characteristics In constant water depth the linearized
forms of (32) and (33) read in one dimension:
rlt+hu,,+(a-p+l/3)h3U,,-ph277,~~=0
(34)
u,+g77.1+
(35)
and (a-
Y)h*1*,,-_g@r*vxxx=O
where 1 _a z 2h0
*&+h
2
(36)
as defined by Nwogu ( 1993), and where P=PI-P2>
(37)
Y--h-Y2
From these equations the linear dispersion relation can be found in the usual way. Let w and k denote the angular frequency and the wavenumber, respectively, then operator corik. ) used in (34) and (35) results in two linear, respondence (a./& d./dx) =(iw.,algebraic, homogeneous equations in n and u, which only have nontrivial solutions for a vanishing determinant. The condition of a vanishing determinant gives the dispersion relation c2 gh
(1+yk?h2)(1-(LY-/3+1/3)kZh*) (38)
(l+pph*)(l-(a-y)k2h2)
Some special cases of (38) appear for p = 0: 1 c2 gh -=I
1+(1/3)k2h2 1 + yk2h2 1+ (y+ 1/3)/?h2 I-(a+1/3)k2h2 1 - cu?h2
if
((Y, p, y) = ( - l/3,0,0)
if
((Y,P, y)=(-l/3,0,
if
(a,P,
y)=(wO,O)
(39a) y)
(39b) (39c)
10
H.A. Schtiffer,
P.A. Mu&w
/ Cotrrtul Engineering
26 (1995) l-14
1.1_
,
2 /
/ /’ 1.05
/
*
/’ ,’ /’ /
I
0.95
1
-
3 Fig. I c/c,,,, versuskh for ( I ) present equations ( [ 414 I -PadC expansion in kh), (2) Madsen et al. ( I99 I 1. Madsen and Sgxensen ( 1992) and Nwogu ( 1993) ( [ 2/2 ] -PadC) and (3) Classical Boussinesq equations ( [Oi 2]-Pade).
where (39a) corresponds to the classical Boussinesq equations expressed in terms of the depth averaged velocities, (39b) is the relation obtained by Madsen et al. ( 199 I ), and (39~) is the equivalent one of Nwogu ( 1993). It is our desire to choose the parameters ( LY,p, y) in order to match a [ 4/4] -PadC expansion in kh of the Airy dispersion relation for waves on arbitrary depth c&l (gh) = tanh( kh) / (kh) which reads 2 cAlcy -zz gh
I + $k2h” + &k4h3 1 + ‘Ph’ 9
+ ‘kdh” 63
Although this involves exist. These are
%p>r=
I I
+ o(k’Oh”‘)
four nonlinear
(40)
equations
-3-&-2fi28-26 18
with three unknowns
105-3\/8o.r
’
126
’
1890
=
several solutions
( - 0.395,0.039,0.011)
I -3+@-2@28-2fi 18
105+3\/805
’
126
’
’
126
’
’
126
’
-3-&+2@28+2a 18
lOS-3&%i?
-3+&+2fi28+2fi I8
( ~0.305,0.039,0.101)
( -0.029,0.405,0.01
lOS+3\/805
(0.061,0.405,0.101)
1890
=
I)
(4lc)
(4]d)
I
The solution (41 d) is not attractive for physical reasons since ( 36) requires - l/2 I (Y5 0 in order to keep the level z, inside the fluid, i.e. - h I z, I 0. However, any of the solutions may in principle be used, but as shown in the next subsection, (41~) is preferable due to shoaling considerations. Thus, using one of the parameter sets given in (41), Eqs. 32 and 33 do yield the very accurate dispersion relation (40). Fig. I compares the celerity c/c,,,,
H.A. SchiifJer, P.A. Madsen /Coastal
Engineering 26 (1995) 1-14
11
2 / :
-.
_\
1
2
/
3
4
5
6
kh
0.95.
(, 3 0.9
-
Fig. 2. As Fig. I, but for cg/cg A5v.
takingcfrom (38) andusing (a) oneoftheparameterssetsin (41), (b) 3,0, l/15) corresponding results from Madsen et al. (1991) ( CX,p, y) = ( - 2/&O, 0)) Nwogu, 1993), and (c) the best result of the the Boussinesq equations (cy, p, y) = ( - l/3,0,0). As noted by Witting (1984) the dispersion relation (14) is correct
(a, p, r) = (- l/ (or equivalently classical forms of to fourth order in
(kh)‘.
of kh defined by
andf (‘) be functions
Letf,f’“,
(42) then the group velocity cg = bdak
s= +kh&= 1
C
may be expressed as
1 +khover2
(43)
2”f
where subscript kh denotes differentiation
with respect to kh. Using (38) this yields
-(a++f)-b’((~-@+f)k2h*_
/3-((~-~)-2P(a-~)k2h*
(l+@h2)(1-(a-/3+f)k2h2)
(l+@@h2)(1-(a-y)k2h2)
(44) Fig. 2 shows the same comparison +,
as Fig. 1, but for the group velocity instead of the celerity;
Airy.
3.3. Linear shoaling characteristics The linear shoaling analysis for the “deeper water” Boussinesq equations is made as shown in $2.3, only now for the one-dimensional, linearized forms of (32) and (33)) which for small bottom slopes (i.e. neglecting 0( h,$ hx,) read
12
H.A. Schbffer, P.A. Mu&n
rlr+h%Yr+ (a-P+
l/3M34m,
/Cnustal
Engineering 26 (1995) I-14
-~h2%x.x
+h,(u,+(3~2,/2+5z~+(-55P,+7P~+2)h2)~,-2(P,-2P2)hrltxJ=O (45) and
Naturally, the dispersion relation (7) is now replaced by (38) and u, takes the place of U in ( 1 lb). The result is expressed as before: (47) where
where (or, LYE,I+, CQ) are functions of ph2, which were evaluated using the symbolic calculation software Mathematics. Since these expressions are very lengthy they will be omitted here. From (18), (20) and (22) it is seen that in the simple case of $2.3, a5 appears as the ratio of two polynomia of order k8h8. The present result is similar, only now the order is k’6h’6 in both numerator and denominator, and the coefficients are lengthy expressions in the parameters. With the constraint of (4 1) we still have two free parameters to chose and we shall base this choice on a least mean square approach, minimizing koho r (PI,
yt;
W+,l=
(49)
(~~‘“-~s>2d(kh)I(koho) I
0
where kh = [ 0; k,h,] is the practical range of applicability. the traditional deep water limit, we get the minima
For k,h,, = 3 corresponding
to
r (-0.12919,
-0.07327;
3) =2.4X
10m6 for (41a)
(50a)
r (-0.13054,
-0.15013;
3) =2.4x
IOP6 for (41b)
(50b)
I- (0.20431, 0.01196; r (0.19074,
3) =4.4X
-0.11674;
IO-’
for (41~)
3) = 1.2X 10-h
for (41d)
of which the best choice is (50~). Thus, collecting and (33) or (45) and (46) our preference is ((Y, z,/h)
= ( - 0.02865,
(p, p,, p2) = (0.40528,
(5Oc) (50d)
all the parameters that appear in (32)
- 0.02907) -0.13054,
-0.53582)
(y, y,, rZ) = (0.01052,0.01196,0.00144)
(51)
H.A. Schiiffer, P.A. Madsen /Coastal
Engineering 26 (1995) I-14
13
0.2
0.1
\ \
\\\
0.5
i
?h
kh
1
Fig. 3. Shoaling gradient (I~ for ( I ) present equations using the parameters in (5 1) or (~OC), (2) equations by Madsen and Sorensen ( 1992), (3) equations by Nwogu ( 1993), and (4) Airy theory using energy flux conservation. ( 1) matches (4) almost exactly.
where z,/h is taken as the wet root to (36). The equations by Nwogu ( 1993) (taking (Y= - 2/5 to get the PadC fit to the dispersion relation) which appear as the special case (/3, /?,, &, y, y,. y2) = (O,O,O,O,O,O) givetheminimumr(O,O; 3) =0.0013.Asimilar test for the Madsen and Sorensen (1992) equations gives the excellent result r( k,ho = 3) = 0.00009, while the result for classical Boussinesq (Madsen and Sorensen, B=O) is horrendous: r(k&=3) = 1.8! Fig. 3 compares the shoaling gradient os using (5 1) with the reference, &IV. Also the curves corresponding to Madsen and Sorensen (1992) and to Nwogu (1993) are shown. The new (Yelies almost exactly on top of (Y?~.. Since the frequency dispersion is acceptable even for k/z = 6, i.e. twice the traditional deep water limit, the shoaling gradient is now optimized using k& = 6 in (49). This yields the minima r
(0.14453,0.02153;
r (-0.08891,
6) =5.7x
-0.17519;
lop3
6) =3.8x
for (41a) 1O-4 for (41b)
(52a) (52b)
r (0.20597,0.00953;
6) = 3.9 X lop6
for (41~)
(52~)
r (0.25733,0.07535;
6) = 2.8 X lop3
for (41d)
(52d)
Again the third choice is by far the best, and it is seen that (p,, 7,) are very close to the values in (50~). For 0 I kh 5 6 Fig. 4 compares the shoaling gradient using (52~) with c$“. Also the results of the equations by Madsen and Sorensen ( 1992) and Nwogu ( 1993) are shown. It appears that the new set of equations have acceptable linear shoaling characteristics all the way to kh = 6. 4. Summary
and conclusions
A slightly different version of the Boussinesq-type equations derived by Madsen and Sorensen ( 1992) has been developed. These have the same improved linear dispersion
14
H.A. Schifer,
P.A. Madsen /Coastal
Engineering 26 (1995) l-14
kh
. . -_ 3
Fig. 4. Shoaling gradient os for ( 1) present equations using the parameters in (52~)) (2) equations by Madsen and Sorensen (1992). (3) equations by Nwogu ( 1993), and (4) Airy theory using energy flux conservation.
characteristics and the same excellent linear characteristics, but they are not explicitly restricted to mildly sloping bottoms. Based on Nwogu’s ( 1993) Boussinesq-type equations another set of equations has been derived, which is highly accurate up to the traditional deep water limit with regard to linear dispersion as well as shoaling characteristics. Even at twice this depth, the properties of linear dispersion and linear shoaling are acceptable.
Acknowledgements This work was financed by the Danish National Research Foundation.
References Madsen, P.A., Murray, R. and Sorensen, O.R., 199 I. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 1. Coastal Eng., 15: 371-388. Madsen, P.A. and Sorensen, O.R., 1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coastal Eng., 18: 183-204. Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Water-w. Port Coastal Ocean Eng. ASCE, 119(6): 618-638. Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech., 27(4): 815-827. Witting, J.M., 1984. A unified model for the evolution of nonlinear water waves. J. Comput. Phys., 56: 203-236.
Coastal Engineering
26 (1995)
l-14
Further enhancements of Boussinesq-type
equations
Hemming A. Schtiffer, Per A. Madsen International
Research
Centre
forComputational
Hydrodynamics,
5, DK-2970
Received 5 September
ICCH Danish Hydraulic
Institute Agem A&
H#rsholm, Denmark
1994; accepted
16 June 1995
Abstract On the basis of two recently published new forms of the Boussinesq equations, yet another set of is derived. These new equations represent another step forward in the improvement of the linear dispersion and shoaling characteristics. The linearized version of this new set of Boussinesq equations
equations corresponds to a higher order Pad6 expansion in kh of the Stokes linear dispersion relation for waves on arbitrary depth. The formulation also allows for an optimization of the linear shoaling properties giving excellent results. The linear dispersion and shoaling characteristics of the new equations are acceptable even for wavelengths as small as the water depth.
1. Introduction
A major breakthrough in the applicability of Boussinesq-type equations was made by Madsen et al. ( 1991) and Madsen and Sorensen ( 1992), who derived a set of equations corresponding to a [ 2/2] -Pad6 expansion in kh of the Stokes linear dispersion relation for waves on arbitrary depth. For comparison the most accurate of the original forms of the Boussinesq equations (the formulation in the depth averaged velocities) corresponds to a [O/2] -Pad& expansion. This gave a major improvement of the dispersion properties. A review of the phase relations for the traditional forms of the Boussinesq equations is given in Madsen et al. ( 199 1) The equations by Madsen et al. are only valid for constant depth and Madsen and Serensen ( 1992) generalized these equations to mildly sloping bottoms, thus retaining only the lowest order in spatial derivatives of the depth. A derivation similar to that of Madsen and Serensen but without this explicit mild-slope assumption is given in $2 as an introduction to the more complicated derivations that follow. Recently, Nwogu ( 1993) has derived another set of Boussinesq-type equations which have exactly the same linear dispersion characteristics as the equations derived by Madsen et al. While Madsen et al. obtained their results by consistent manipulations of the classical 037%3839/95/$09.50
0
SSDlO378-3839(95)00017-8
1995 Elsevier Science B.V. All rights reserved
equations, Nwogu derived his version in terms of the horizontal velocities taken at an arbitrary level. In $3 both of these ideas are combined leading to yet another class of Boussinesq-type equations, for which a suitably chosen subset gives highly accurate linear dispersion characteristics corresponding to a ]4/4]-PadC expansion in kh, as shown in $3.2. The linear shoaling characteristics are derived in $3.3 and again suitable choices are shown to give excellent results. A common feature of the equations by Madsen et al., by Nwogu, and the present ones is that it is the combined set of equations for mass and momentum that represents the improved dispersion properties. Each of the equations are not developed to a higher order than the classical Boussinesq equations. Thus the underlying expressions for the distribution of vertical and horizontal velocity cannot be trusted in the extended depth range in which the surface elevation is still very well described. Furthermore, since only lowest order nonlinearity is included, the present equations are effectively linear in deep water. Various types of higher order Boussinesq equations are now being investigated in order to mend these shortcomings.
2. Generalization 2.1. Derivation
of Madsen and Serensen (1992)
ofequations
In this section we derive a set of Boussinesq-type equations which is similar to that of Madsen and Sorensen ( 1992), but without the explicit restriction of small bottom slopes. For small bottom slopes a special case of these equations reduce to the Madsen and Sorensen equations. A commonly used scaling of variables is adopted. Let a prime denote dimensional variables then
and
Here (x,~,z) are Cartesian coordinates, h = h(x,y) is the depth, 77= q(x,y,t) is the surface elevation, (u,I’,w) is the particle velocity vector, p is the pressure, p is the density of the fluid and a, 1 and ho are typical values of wave amplitude, wavelength and water depth, respectively. As usual 0( E) = 0( p2) -=K1 is assumed, where E = a/h,, and p = h,,ll are the measures of nonlinearity and frequency dispersion. Let Q denote the vector of depth integrated velocity then the classical Boussinesq equations derived by Peregrine ( 1967) may be written in nondimensional form as
H.A. Schiiffer, P.A. Madsen /Coasial
Engineering 26 (1995) 1-14
and
Q,+(hf~v)Vq+~
(Q.V> &+&
(V.Q)
1 (2)
where V = (a/k, a/ay) is the horizontal gradient operator. This was the starting point of Madsen and Sorensen ( 1992). We now introduce two free parameters (B,, B2) 5 0( 1) . Dividing (2) by h + q (or just by h) and using the operator - Bzp2h3V( V. ) on the result, we obtain -B,~2h3(V(V~~)+V(V-V~))=O(&
E,u2, p4)
(3a)
A slightly different expression is obtained by using the operator - B,p2h2V( V. on both sides of (2)) which yields -B,~2h2(V(V.QJ
+V(V.(hVn)))
=O(z,
E/L’, p4)
(3b)
Adding (3a) and (3b) to (2) yields the following class of momentum (Q.V)&+&Q.V)Q
+p2[(:-B2b ( v 0.’
h)
-
‘+B (2
I)
h’V(V.Q,)
) directly
equations:
1 -B,h3(V(V.Vq))
-B,h2(V(V.(hV~)))
=o
1 (4)
retaining only terms up to 0( E, p2). Eqs. ( 1) and (4) give the desired result. The dispersion terms with a factor B, are essentially the ones introduced by Madsen and Sorensen ( 1992)) improving the dispersion properties of the equations. The additional terms with a factor B2 provide a means of adjusting the bottom slope terms in order to improve the linear shoaling characteristics. Returning to dimensional variables but omitting the primes ( 1) and (4) read in Cartesian form
(5)
~,+9r+Q.v=0 and
h2UL+Q,xJ
-Bzgh3hr,+xJ -B,gh2(thxlxx+
[hv,l,)
(6a)
h*(Q,v+W
-&gh3(~y,y.y+77x,v) -B,gh2Wrl?,ly,.+
thrlxl,,)
(6b)
H.A. Schtiffer, P.A. Madsen /Coastal
4
Engineering 26 (1995) l-14
where (P,Q) = Q. 2.2. Linear dispersion characteristics For constant depth B, and B2 only appear as B = B, + B2 and (5) and (6) become identical to the Madsen and Sorensen ( 1992) equations. Thus the same linear dispersion relation can be obtained: cZ
1 + Bk?h*
gh
1+(B+1/3)k2h2
(7)
2.3. Linear shoaling characteristics Madsen and Sorensen also derived the linear shoaling gradient corresponding to their equations and found excellent agreement with results obtained by combining Airy theory with conservation of energy flux. The first step in their approach was to eliminate the depthintegrated velocity from the linearized conservation equations to get a wave equation in the surface elevation alone. This is generally not possible for the linearized versions of (5) and (6). Only in the special case of B2 = 0 the flux can be eliminated and it can be shown that the wave equation obtained then reduces to that of Madsen and Sorensen for small bottom slopes. Thus Eqs. (5) and (6) with B, = 0 and B, = B represent an extension to other than small bottom slopes of the Madsen and Sorensen equations retaining their attractive results for linear dispersion and shoaling. The dispersion relation gives the constraint B, + B2 = B ( = 1 / 15 as shown by Madsen et al., 1991) on the parameters, but we still have one free parameter, say B, = B - B,, which can be chosen as to optimize some measure of the shoaling characteristics. Since, however, (B,, B2) = ( 1 / 15,O) gives satisfactory results, there is no need for further optimization and as shown below it gives only a marginal improvement of the shoaling gradient. Thus, the exercise of this section is rather to show a simple example of the procedure for the linear shoaling analysis when the velocity variable is not eliminated (although this could be done also for B, # 0 when the bottom slope is assumed to be small). This procedure is used again in 93.3 for the more complicated equations derived in 53.1. The linearized versions of (5) and (6) read in one dimension: qr+hu,+hxi=O and (dividing
(8)
by h)
r*,+g7),f(~-B2)h2~~~~~-
(
;+B,
h[h~,l,,-B,gh2rl,,-B,gh[h-rl,l.,=0 1
where U is the mean over depth of the horizontal becomes
(9)
velocity. For small bottom slopes (9)
h2ii,l, - Bgh’r/,*, -Zh,h((~+B,)u,,+B,jill=O
(10)
H.A. Schbffer, P.A. Madsen /Coastal
Engineering 26 (1995) 1-14
5
We now introduce solutions of the form v(~,t) =A(x)e’(“-lk(“)d’) u(x,t)
(lla)
= (D(X) +i~‘,(~))e”“~‘~‘“‘*
(llb)
where A(x), D(x), bJx>, k(x), and h(x) are slowly varying functions of x and where r?, = 0( h,) as marked by the differentiation subscript, the tilde indicating a distinction between fiX and D, (the derivative of D(x) ) . The justification of ( 1 lb) is that for constant depth U is in phase with 7, while for small bottom slopes a small phase shift should be permitted. Using ( 11) in (8) and ( 10) two equations are obtained by retaining only the terms free from derivatives of the slow variables. These are -khD+Aw=O
(12)
-gk&BgkAk?h’+wD+(B+l/3)wD~h*=O
(13)
and
for which non-trivial solutions for (A, D) require the dispersion relation (7) to be satisfied. Two other equations are obtained by collecting terms proportional to the first derivatives of the slow variables: Dh.,+h(D,+kb,)
=0
(14)
and gA,+3Bgkh*(kA,+Ak,)-coti,-w(B+1/3)h2(2kDx+k%-Dk,) -h&h(
(1+2B,)wD-2B,gkA)
=0
Solving (12) for D and differentiating
(15)
the result yields
(16) Solving ( 1.5) for OX, substituting the result in ( 14) and replacing (D, 0,) by their solutions obtained from ( 12) and ( 16), we get an expression which multiplied by ( 1 + Bph’) (1 + (B+ l/3)k2h2)kl(mA) can be written (17) where CY,=2( 1 +2Bk2h2+B(B+ CY~=- 1 +2Bk2h2+3B(B+
l/3)k4h4) 1/3)k4h4
a,=k2h2(2B-1/3+(2B*-B/3+2B1/3)ph2)
( 1W (18b) (18~)
where w has been eliminated using the dispersion relation (7). This expression is quite different from that of Madsen and Sorensen even in the shallow water limit, where the terms involving the parameters vanish. Thus for kh = 0 we get ( (Y,,(Y~,cx~)= (2, - 1,O) while
6
H.A. Schtifier, P.A. Madsen/Coastal
Engineering 26 (1995) 1-14
they have ( CX,,CQ,(Y~)= (2,1,1) (their Eqs. 3.7 and 3.8). However, both results are quite consistent. This is possible, since ( LY,,(Y~,(Y~) are not uniquely determined by ( 17) alone, and the seeming discrepancy vanishes when k,lk is eliminated from ( 17). Following Madsen and Sorensen in the last steps we obtain by differentiation of the dispersion relation (7): (19) where 1 @4=2-6( and elimination
k2h” 1 +2Bk2h2+B(B+
(20)
l/3)k4h4)
of k,lk from ( 17) and ( 19) finally yields the linear shoaling relation
A _I- - -a!- hx A h
(21)
where
is the shoaling gradient, which for Airy waves combined reads A,ry= G 1+ fG( 1 -cash ffs (1 +G)’
2kh)
,
GC2kh sinh 2kh
with conservation
of energy flux
(23)
For B, = B and B2= 0, as can be shown to be identical with Madsen and Sorensen’s expression. For kh = 0, where CX,= l/2, the well known shallow water result CX~ = l/4 appears. For B = 1/ 15 the Taylor expansion of CQfrom kh = 0 gives 4( 1+ 15B,)k4h4+ and a similar expansion
26 - 120B 15
(24)
of atiry reads
I-k2h2+$k4h4+&k”h”+...
(25)
Matching to 0( k4h4) yields B, = 1 / 10, by which B, = B - B, = - l/30. It is interesting to realize that for this optimization the relative weight of the Ur,, and [ hii,] *x terms in (9) is conserved relative to classical Boussinesq theory for which B, = B, = 0. In the derivation of the classical Boussinesq equations this weighting is a consequence of the sloping bottom boundary condition. Thus it is peculiar that it reappears from the above comparison with a combination of Airy theory (for which the depth is constant) and conservation of energy flux, since no sloping bottom boundary condition is included in this theory. For kh < 1.2, B, = 1/ 10 does give the best fit for the shoaling gradient. However, since terms of much higher order than the 0( k4h4) coefficient matched above are inevitable in
H.A. SchiifSer, P.A. Madsen /Coastal Engineering 26 (1995) 1-14
7
as, I?(
a good fit to larger &-values requires minimization of, for example, (y$iry_ (Ye)2d( kh) / (k,h,) with e.g. k,,h, = 3 corresponding to the traditional deep water limit. Such a minimization yields B, = 0.0653 which is only marginally different from the excellent result of Madsen and Sorensen, which corresponds to B, = l/15 =0.0667 and BI!=O. Three conclusions are to be drawn from this section: ( 1) A slightly different form (5) and (6) of the Madsen and Sorensen equations is valid without the explicit mild-slope condition. (2) In these equations B, = B = 1/ 15 and B2 = 0 should be used, since the marginal shoaling improvement possible for B2 f 0 is not worth the extra term in the equation. (3) The choice of the values of free parameters in connection with the improvement of shoaling characteristics should be based on an error minimization for example over the desired kh range rather than on matching coefficients of expansions. This last conclusion is used in the more complicated situation in $3.3.
3. “Deeper water” Boussinesq-type 3.1. Derivation
equations
of equations
Nwogu In terms of the horizontal velocity vector u, at an arbitrary level z=z,(x,y) ( 1993) derived the following set of nondimensional equations: conservation of mass
1
=O(CY?, E/L, /._L~)
and conservation
(26)
of momentum
u,,+V~+E(IC;V)zf,+/.L~
$V(Vu,)
1
+z,V[V-(hu,)]
=U(z,
E/_L~, ci’) (27)
We now introduce four free parameters (/3,,/32,y,,?/2) I O( 1) and proceed as shown in $2.1, only now also the mass equation contains small dispersion terms allowing for further refinements. Using each of the operators - p,p2V. ( h2V ) and p2p2V. V( h2 ) on (26) we obtain -/3,~2(V.(h2V~,)
+V.(h2V[V.(hu,)]))
=O(Z,
E/.L~,/.L~)
(2Sa)
and &L~(V.(V~~~,)
+V.V[h*V.(hu,)])
=0(8,
Similarly, we apply each of the operators sides of (27) to obtain -y,p2h2(V(V.U,,) and
+V(V.Vq))
E/L*,/_L’>
- y,p2h2V(V.
=O(g,
E/L’, @‘up,
) and y,p’hV(V.h
(2gb) ) on both
(29a)
H.A. Schiiffer, P.A. Madsen /Coastal Engineering 26 (1995) 1-14
y+‘h(V(V.hu,)
+V(V.VhVv))
=0(8,
E/_L~, p4)
(29b)
Adding (28a) and (28b) to (26) and (29a) and (29b) to (27) yields +p’V.
r],+V.[(h+q)uJ
+&V[h2V.(hu,)]
hV(V.u,)
-/3,h2Vn,+&V(h2n,)
+ (c+,h)hV,V-(hu,),
=0
(30)
I and
y,h')V(V.u,)+(-,,fy,h)V,V.(/Lu,)]
u,+V,ic(u;V)u..+a’[(~-
-y,h’V(V.Vq)
+ y,hV[Vh.Vn]
I
=0
(31)
where we have omitted terms of 0( 2, ep2, p”) . Eqs. 30 and 3 1 represent a new class of Boussinesq-type equations. We shall refer to (30) and (3 1) as the “deeper water” Boussinesq equations, since for a suitable choice of the free parameters (PI, /$, y,, yz) their linear dispersion and shoaling characteristics are acceptable even when the water depth is almost as large as the wave length. In (30) and (3 1) only terms up to 0( l ,~‘) are retained just like in (26) and (27) or in the equations by Madsen et al. or in the classical forms of the Boussinesq equations. This means that (30) and (3 1) are formally just as restricted to shallow water as any of the other forms of the Boussinesq equations. Returning to dimensional variables but omitting the primes the Cartesian forms of (30) and (31) read
v,+ [(h+v)d,+ xh( [hue],+
+
[(h+rl)CnlY+
[hc,l,,)
+~z~h21h~,l.,+h2~h~1,1.1.,-P,h2rl,+Pz~h2~~1~
+“av) +(+B,h)h(,hu,,,,+
+P~~h2~h~~,l.+h2~h~,l.,l,.-P,h2~,+P~~h2rl,l.
and
+Ls,,,.I +(‘“f;-P,h)
I .i
[hc,],,)
1L
=O
(32)
H.A. Schkiffer, P.A. Madsen /Coastal Engineering 26 (1995) l-14
+
+
(z,+
(z,+
[~4xl,+
y,h)(
y&1(
[h4v,l,f
[~~,I,)
9
- T5gh2(rlxxx+77xyy+ 72gh( [h%lxx+ [h&J
[hLf,l,) - wh2h,+
71vw) +-Y&t [bJ~+
=o (33a)
[~~~I,) =o (33b)
3.2. Linear dispersion characteristics In constant water depth the linearized
forms of (32) and (33) read in one dimension:
rlt+hu,,+(a-p+l/3)h3U,,-ph277,~~=0
(34)
u,+g77.1+
(35)
and (a-
Y)h*1*,,-_g@r*vxxx=O
where 1 _a z 2h0
*&+h
2
(36)
as defined by Nwogu ( 1993), and where P=PI-P2>
(37)
Y--h-Y2
From these equations the linear dispersion relation can be found in the usual way. Let w and k denote the angular frequency and the wavenumber, respectively, then operator corik. ) used in (34) and (35) results in two linear, respondence (a./& d./dx) =(iw.,algebraic, homogeneous equations in n and u, which only have nontrivial solutions for a vanishing determinant. The condition of a vanishing determinant gives the dispersion relation c2 gh
(1+yk?h2)(1-(LY-/3+1/3)kZh*) (38)
(l+pph*)(l-(a-y)k2h2)
Some special cases of (38) appear for p = 0: 1 c2 gh -=I
1+(1/3)k2h2 1 + yk2h2 1+ (y+ 1/3)/?h2 I-(a+1/3)k2h2 1 - cu?h2
if
((Y, p, y) = ( - l/3,0,0)
if
((Y,P, y)=(-l/3,0,
if
(a,P,
y)=(wO,O)
(39a) y)
(39b) (39c)
10
H.A. Schtiffer,
P.A. Mu&w
/ Cotrrtul Engineering
26 (1995) l-14
1.1_
,
2 /
/ /’ 1.05
/
*
/’ ,’ /’ /
I
0.95
1
-
3 Fig. I c/c,,,, versuskh for ( I ) present equations ( [ 414 I -PadC expansion in kh), (2) Madsen et al. ( I99 I 1. Madsen and Sgxensen ( 1992) and Nwogu ( 1993) ( [ 2/2 ] -PadC) and (3) Classical Boussinesq equations ( [Oi 2]-Pade).
where (39a) corresponds to the classical Boussinesq equations expressed in terms of the depth averaged velocities, (39b) is the relation obtained by Madsen et al. ( 199 I ), and (39~) is the equivalent one of Nwogu ( 1993). It is our desire to choose the parameters ( LY,p, y) in order to match a [ 4/4] -PadC expansion in kh of the Airy dispersion relation for waves on arbitrary depth c&l (gh) = tanh( kh) / (kh) which reads 2 cAlcy -zz gh
I + $k2h” + &k4h3 1 + ‘Ph’ 9
+ ‘kdh” 63
Although this involves exist. These are
%p>r=
I I
+ o(k’Oh”‘)
four nonlinear
(40)
equations
-3-&-2fi28-26 18
with three unknowns
105-3\/8o.r
’
126
’
1890
=
several solutions
( - 0.395,0.039,0.011)
I -3+@-2@28-2fi 18
105+3\/805
’
126
’
’
126
’
’
126
’
-3-&+2@28+2a 18
lOS-3&%i?
-3+&+2fi28+2fi I8
( ~0.305,0.039,0.101)
( -0.029,0.405,0.01
lOS+3\/805
(0.061,0.405,0.101)
1890
=
I)
(4lc)
(4]d)
I
The solution (41 d) is not attractive for physical reasons since ( 36) requires - l/2 I (Y5 0 in order to keep the level z, inside the fluid, i.e. - h I z, I 0. However, any of the solutions may in principle be used, but as shown in the next subsection, (41~) is preferable due to shoaling considerations. Thus, using one of the parameter sets given in (41), Eqs. 32 and 33 do yield the very accurate dispersion relation (40). Fig. I compares the celerity c/c,,,,
H.A. SchiifJer, P.A. Madsen /Coastal
Engineering 26 (1995) 1-14
11
2 / :
-.
_\
1
2
/
3
4
5
6
kh
0.95.
(, 3 0.9
-
Fig. 2. As Fig. I, but for cg/cg A5v.
takingcfrom (38) andusing (a) oneoftheparameterssetsin (41), (b) 3,0, l/15) corresponding results from Madsen et al. (1991) ( CX,p, y) = ( - 2/&O, 0)) Nwogu, 1993), and (c) the best result of the the Boussinesq equations (cy, p, y) = ( - l/3,0,0). As noted by Witting (1984) the dispersion relation (14) is correct
(a, p, r) = (- l/ (or equivalently classical forms of to fourth order in
(kh)‘.
of kh defined by
andf (‘) be functions
Letf,f’“,
(42) then the group velocity cg = bdak
s= +kh&= 1
C
may be expressed as
1 +khover2
(43)
2”f
where subscript kh denotes differentiation
with respect to kh. Using (38) this yields
-(a++f)-b’((~-@+f)k2h*_
/3-((~-~)-2P(a-~)k2h*
(l+@h2)(1-(a-/3+f)k2h2)
(l+@@h2)(1-(a-y)k2h2)
(44) Fig. 2 shows the same comparison +,
as Fig. 1, but for the group velocity instead of the celerity;
Airy.
3.3. Linear shoaling characteristics The linear shoaling analysis for the “deeper water” Boussinesq equations is made as shown in $2.3, only now for the one-dimensional, linearized forms of (32) and (33)) which for small bottom slopes (i.e. neglecting 0( h,$ hx,) read
12
H.A. Schbffer, P.A. Mu&n
rlr+h%Yr+ (a-P+
l/3M34m,
/Cnustal
Engineering 26 (1995) I-14
-~h2%x.x
+h,(u,+(3~2,/2+5z~+(-55P,+7P~+2)h2)~,-2(P,-2P2)hrltxJ=O (45) and
Naturally, the dispersion relation (7) is now replaced by (38) and u, takes the place of U in ( 1 lb). The result is expressed as before: (47) where
where (or, LYE,I+, CQ) are functions of ph2, which were evaluated using the symbolic calculation software Mathematics. Since these expressions are very lengthy they will be omitted here. From (18), (20) and (22) it is seen that in the simple case of $2.3, a5 appears as the ratio of two polynomia of order k8h8. The present result is similar, only now the order is k’6h’6 in both numerator and denominator, and the coefficients are lengthy expressions in the parameters. With the constraint of (4 1) we still have two free parameters to chose and we shall base this choice on a least mean square approach, minimizing koho r (PI,
yt;
W+,l=
(49)
(~~‘“-~s>2d(kh)I(koho) I
0
where kh = [ 0; k,h,] is the practical range of applicability. the traditional deep water limit, we get the minima
For k,h,, = 3 corresponding
to
r (-0.12919,
-0.07327;
3) =2.4X
10m6 for (41a)
(50a)
r (-0.13054,
-0.15013;
3) =2.4x
IOP6 for (41b)
(50b)
I- (0.20431, 0.01196; r (0.19074,
3) =4.4X
-0.11674;
IO-’
for (41~)
3) = 1.2X 10-h
for (41d)
of which the best choice is (50~). Thus, collecting and (33) or (45) and (46) our preference is ((Y, z,/h)
= ( - 0.02865,
(p, p,, p2) = (0.40528,
(5Oc) (50d)
all the parameters that appear in (32)
- 0.02907) -0.13054,
-0.53582)
(y, y,, rZ) = (0.01052,0.01196,0.00144)
(51)
H.A. Schiiffer, P.A. Madsen /Coastal
Engineering 26 (1995) I-14
13
0.2
0.1
\ \
\\\
0.5
i
?h
kh
1
Fig. 3. Shoaling gradient (I~ for ( I ) present equations using the parameters in (5 1) or (~OC), (2) equations by Madsen and Sorensen ( 1992), (3) equations by Nwogu ( 1993), and (4) Airy theory using energy flux conservation. ( 1) matches (4) almost exactly.
where z,/h is taken as the wet root to (36). The equations by Nwogu ( 1993) (taking (Y= - 2/5 to get the PadC fit to the dispersion relation) which appear as the special case (/3, /?,, &, y, y,. y2) = (O,O,O,O,O,O) givetheminimumr(O,O; 3) =0.0013.Asimilar test for the Madsen and Sorensen (1992) equations gives the excellent result r( k,ho = 3) = 0.00009, while the result for classical Boussinesq (Madsen and Sorensen, B=O) is horrendous: r(k&=3) = 1.8! Fig. 3 compares the shoaling gradient os using (5 1) with the reference, &IV. Also the curves corresponding to Madsen and Sorensen (1992) and to Nwogu (1993) are shown. The new (Yelies almost exactly on top of (Y?~.. Since the frequency dispersion is acceptable even for k/z = 6, i.e. twice the traditional deep water limit, the shoaling gradient is now optimized using k& = 6 in (49). This yields the minima r
(0.14453,0.02153;
r (-0.08891,
6) =5.7x
-0.17519;
lop3
6) =3.8x
for (41a) 1O-4 for (41b)
(52a) (52b)
r (0.20597,0.00953;
6) = 3.9 X lop6
for (41~)
(52~)
r (0.25733,0.07535;
6) = 2.8 X lop3
for (41d)
(52d)
Again the third choice is by far the best, and it is seen that (p,, 7,) are very close to the values in (50~). For 0 I kh 5 6 Fig. 4 compares the shoaling gradient using (52~) with c$“. Also the results of the equations by Madsen and Sorensen ( 1992) and Nwogu ( 1993) are shown. It appears that the new set of equations have acceptable linear shoaling characteristics all the way to kh = 6. 4. Summary
and conclusions
A slightly different version of the Boussinesq-type equations derived by Madsen and Sorensen ( 1992) has been developed. These have the same improved linear dispersion
14
H.A. Schifer,
P.A. Madsen /Coastal
Engineering 26 (1995) l-14
kh
. . -_ 3
Fig. 4. Shoaling gradient os for ( 1) present equations using the parameters in (52~)) (2) equations by Madsen and Sorensen (1992). (3) equations by Nwogu ( 1993), and (4) Airy theory using energy flux conservation.
characteristics and the same excellent linear characteristics, but they are not explicitly restricted to mildly sloping bottoms. Based on Nwogu’s ( 1993) Boussinesq-type equations another set of equations has been derived, which is highly accurate up to the traditional deep water limit with regard to linear dispersion as well as shoaling characteristics. Even at twice this depth, the properties of linear dispersion and linear shoaling are acceptable.
Acknowledgements This work was financed by the Danish National Research Foundation.
References Madsen, P.A., Murray, R. and Sorensen, O.R., 199 I. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 1. Coastal Eng., 15: 371-388. Madsen, P.A. and Sorensen, O.R., 1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coastal Eng., 18: 183-204. Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation. J. Water-w. Port Coastal Ocean Eng. ASCE, 119(6): 618-638. Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech., 27(4): 815-827. Witting, J.M., 1984. A unified model for the evolution of nonlinear water waves. J. Comput. Phys., 56: 203-236.