1 5 SEP, 1972
ARCH I
EF
DocumEtATATIE.. Es-eONO_
I Bibliotheek van de sbouwkunde or.... C)nderafideiin nische Hooeschoo 30 CUMEN r ATE DATUM: I 9 OP,13
Lab. v.
Scheepsbouwkunth
Technische HogeschoolDelft
d4,017-xi44,4.8-4-7Ge.124,/aded 4/72-6,z,
Transcritical Flow Past Slender Ships
by
G. K. Lea
National Science Foundation
Washington, D. C., U.S.A.
J. P. Feldman
Naval Ship Research and Development Center
Washington, D. C., U.S.A.
e
TRANSCRITICAL FLOW PAST SLENDER SHIPS*
G. K. Lea
National Science Foundation
Washington, D. C. U.S.A. J. P. Feldman
Naval Ships Research and Development Center
Washington, D. C. U.S.A.
The transcritical shallow water flow past slender ships is
analyzed using the method of matched asymptotic expansions.
A consistent first order approximation was derived which is
analogous to the non-linear transonic equation with the
Froude and Mach numbers playing similar roles. Solutions
are obtained for sinkage and trim in the transcritical region
and are compared with experimental results. An important
result is that both sinkage and trim are functions of Froude
number as well as beam to length ratio in the region where
Froude number based on undisturbed depth is close to unity.
Introduction
In a series of papers Tuck(1)(2) developed a systematic expansion procedure for the approximate solution to the
shallow water flow past slender ships. It was pointed out
that a close analogy exists between this problem and the
in-viscid slender body aerodynamics problem. In fact, Tuck's
solution contains the same type of singularity that is en-countered in aerodynamic theory and we present here an attempt to remove the singularity which occurs in the transcritical
region. Thus the shallow water problem that will be examined is concerned only with steady translational Motion of a slender ship and the associated surface waves so that viscous and
compressibility effects are neglected.
The Froude number
(Fh="iii-1,
where U., is the free streamspeed, g is the acceleration of gravity, h is the undisturbed depth) can be interpreted as the ratio of a characteristic speed to the propagation speed of small disturbances on the
water surface in shallow water theory. On the other hand in
aerodynamic theory, the Mach number
(14,=11.,/air,
where Uthe free stream speed and a., is the isentropic speed of sound) is the ratio of the characteristic speed to the propagation
speed of acoustic signals in the gas. Thus we see that Froude
and Mach numbers play similar roles. and this is reflected in
the mathematical fornulation of the two problems. For exanl,le,
Tuck(2) gave for the first approximation a hyperbolic equation
1
for supercritical flow Fh > 1 and an elliptic equation for
subcritical flow
Fh< 1.
We find the same situation inin-viscid compressible flow past slender bodies for supersonic 1 and subsonic L< 1 flows. His results for vertical
force, trim moment as as drag contains integrals which
relates the source sink distribution to the local hull area and multiplied by the following factor:
JDZ/
hliTITITI for subcritical flow and2 .-2
JoUco/
hva-Fh -1 for supercritical flow.This factor seem to indicate castatrophic failure at critical flow
Fh 1. However, it should be pointed out that aside
2
from the transcritical region, where IFh -
11
is small,Tuck's results appears to be more than adequate for most engineering purposes.
We shall seek a singular perturbation solution to the problem of shallow water flow past a slender ship with the requirement that the solution must be valid within the
trans-critical region. This approach is that followed by Tuck(1)(2)
(3)
and is well documented in books by Cole and Van Dyke(4).
The important difference between what follows and the works of Tuck is that two small parameters appear in the formulation,
slenderness ratio and Froude number parameter
IF
-
11 7inste3d of only the clendelmeSs ratio. Appearence of an
additional paraIniter drastically alters the matheinatical
representation of tne problem and the nonlinear effects ,<2)
Exact Statement of the Inviscid. Problem
Consider a ship immersed in a steady stream of inviscid,
incompressible stream with free stream velocity of
10x.
Acartesian coordinate system is afixed to the ship with its
origin at the bow and at the undisturbed waterline. The
positive x-axis is directed toward the stern of the ship and
z-axis is directed vertically upward. The total velocity in
the
flow
field is given by:q
=x
+ grad pwhere p is the disturbance potential due to the presence of
the ship. The dimensional governing equations are the LaPlace equatibn, free surface kinematic and pressure equations,
bottom condition and hull tangency condition. These are as
follows: (4) + + = 0 (1-a) xx yy zz (Pz = (1 + (40x)-pc + (4)3,47,0, (1 .10) _.(2_ _2 _2 _2), (1-c) 'x 'vx (q)z)z=-H = ° (1-d) p = (1 + p )A
xx
+ p A (1-e)zz
where ;-(x,y) is the unknown free surface and A(x,z) is the z=3-(x,y)
given surface of the ship hull. If we are to proceed in a systematic fashion the relative orders of magnitude of the
various terms must be established. One way of accomplishing
this is by selecting proper scales for all the dependent as well as independent variables and thereby introduce
non-di-tensional variables of order unity. This does not mean thaT,
all quantities will have its maximum of one, but rather that if we choose the correct scale the maximum value could be
large as ten units but not ohe thousand units. We take U,,as
the velocity scale and the undisturbed depth, H, as the
ver-tical length scale. The selection of a horizontal length
-scale is a bit more involved as it must reflect the.
shallow
water approximation include the transcritical nonlinearities 'produced by a slender hull.
Now, shallow water theory assumes as a first approxima-tion that the vertical pressure variaapproxima-tion is purely
hydro-static
Or
that vertical accerlations are negligible compareto horizontal accelerations.
This
can be derived in asys-tematic manner assuming that the depth to characteristic
wave length
(H /
Lw 1) is small and utilize Lw as thelength scale in x and y directions and expand w as a power
series in H/Lw We note that shallow water theory is not
necessarily linearization and the latter results from restric-tions that we place on the type of "wave maker" present in the
generated by a ship at critical speed is a single wave of
translation perpendicular to the free stream. In the absence
of viscous dissipation this wave extend to infinity so that
the disturbance in the lateral, y, direction is greater than
in the axial, xl direction. Thus it seems logical that we
should have x = 0(1) and y 0(s-n) at large distances from
the ship where E is a small parameter related to the
"shal-lowness" of the water.
Shallowness implies that depth is small relative wave length (H/Lw<< 1) and slenderness implies that the wave
maker, the ship, must be longer than it is either wide or
deep. If we define B as the maximum beam, T as the maximum
draft, then slenderness means
B/L -<< 1, T/L 1
where L is the length of the ship. In order to proceed in an
orderly manner some estimates must be placed on the relative
orders of magnitude between Lw and L. We note that the
dis-persion relationship for steady progressive free waves in two dimensions is
tanh(2tH/Lw) / 271H/Lw = 1.1/gH (2)
which can be approximated by the following expression after
making use of the long wave assumption (H/L.= 1)
2 2 2
LA
The behavior Of this expression. in the transcritical, 'region
is, estimated as
-H/Lw, = (4)
Furthermore, if we take the depth to ship length ratio H/L.
as gauged by the slenderness of the hull, i.e. H/L.. 0(5a),
then
Lw/L
0(ca/T17*.
(5)
As the ship approaches the critical flow condition, the
char-acteristic wave length of the surface wave decreases th) that
in order to retain the transcritical effects and at the same time impose slenderness assumption we take
1w/
.o(
,
. 0(1).
We note that in Tuck 'S analysis(2) it was assumed that as the
ship approaches a line (E 7-4- 0) that, il
4)
remains fixedand of Order Unity which implies that Lw/L 0(E) -c-c 1 and
is equivalent to the condition. Ft = .16/17gT
0((7).
Thusoutside the transcritical region the proper scale length in
any horizontal plane
is
the length of thehull
L. However,the situation within the transcritical region is I.. 0(Lw) so
=
8
that we can choose t..ither L or Lw as a horizontal character-istic length with the retriction La / 0(1) or
F- 1 .1- EK (6)
where K is some similarity constant. which is of order unity. The_ particular form chosen here is guided by the transonic
aerodynamics analysis of the slender airfoil theory since we
anticipate a close analogy between it and the present shallow
water problem. It should be noted that in aerodynamic theory
K is not uniquely determind by
any analytical
approach butFar Field Approximation
Singular pertur tori s,;:utica 1,s a systematic procedure
by which succesive estiplates to the solutions can be made in
the various region of the domain of solution. If properly
applied, the dominate features of each of theae regions will be magnified and secondary features aurpressed by scaling of
variables. We expect that in the far field details of the
ship hull will be lost and that the dominate -feature of the
problem is that of the surface Wave system. As noted in the
previous section
Lave / Lship-W - 0(1) in transcritical region
so that for scaling purposes either one would be appropriate
and we shall refer to it as simply the characteristic length
I. The shallowness, parameter 8 and the slenderness parameter
are given by
e H/L
. B/L, T/L.
We shall restrict our attention to that class of problems in which the hull must te more slender than the water is shallow,
i.e, the maximum draft be less than the depth, thus
14m (6/e) . Q.
.A simple relation Which satisfies this condition is
e = 6m, 0 <, m 1.
where m is a constant which will be determined by the matching
of far field Solution to near field solution. It Must be
re-membered that this restriction placed
on
z and 6 does, not imply=
'See Van Dyke(4), pages 23-28
10
the existance of a functional relationship between
slender-ness of hull and depth of free stream. We have chosen the
shallowness parameter, E, as a convenient gauge functions
and use it as a standard for order of magnitude comparison. The following non-diensional and scaled variables are
introduced:
-p
-x = -xL, y = yLE , z
zLE,
= '6LEand the non-dimensional variables
q = qUa p = 11,01_4, F 1 + EK, K = 0(1).
The full inviscid equations become:
Potential Equation 2+2p (I) 4-E2(1)XX E (1) = 0 (7-a) zz YY Bottom Tangency (4) (x,v -1) = 0 (7-h) z '
Free Surface Kinematic
(4)z
= E2(1 + (4) ) +2+2p(i)
r
on z = Y(x,y) (7-c)x x Y Y
Free Surface Pressure
2 2p, .2
2
Er/
1 + EK = -(i)2 - E2[2(p + ((px) + ]xon z = (7-d)
We assume the following far field expansions for the dis-turbance potential and the free surface elevation
(4) En(i) (x,y,z),
n=1 Enc(x,y).n (8)
n=1 n
=
Free Surface Pressure:
(C1 flx)E3 (2-2 2f2x fTx + 2Kf
lx
*See Cole(3), page 46
Substituting these expansions into the full invisid equations and equating like powers of c results gives the following:
Disturbance Potential:
(;)
f1(x,y)6
+ f2(x,y)62 + [f3(x,y) - (z +l / 2)2f1]3+
624if4(x'y) - (z+1 / 2) 2(f2xx + f1yy)]
'3(65)
(9-a)
Free Surface Kinematic:
.+ f )63 + + f, + f + + f )64
lx lxx 2x 3c)( lx lx 1 lxx lyy
+ 0(65) = 0 (9-b)
(9-6
The bottom tangency condition is satisfied to 0(64 and the
"stretch" in the y-coordinate is.taken as
7
.
yL/6P = yL/117or p = 1/2. This is determind by the observation that if
p < 1/2, then the expansions should proceed as fractional powers of c which cannot be matched to the near field
solu-tion. On the other hand, if p > 1/2, then the term flyy would not appear to 0(64) and a degenerate case results. Thus the choice of p = 1/2 results in a "distinguished limit
process" as c -p- 0*. The governing equation for the first approximation to the disturbance potential ((p1 = fl(x,y)) is obtained by elimination of second order variables (f2 and between the free surface kinematic and pressure conditions to
0(s4) and is
(K + 3f
lxl
)f-xx - f = 0.lyy
The mathematical structure of this -equation could change
locally in the domain solution depending on the algebraic
sign of the term (K + 31.1x)* This equation can describe locally subcritical flow .(elliptic equation) when
(K 3f1x) supercritical flow (hyperbolic equation) when
(K 3f1x) > 0 and the local characteristics have the slope
(dy/dx) =
K 3f1x3-1/2
The expansion for the disturbance potential ((p) given by equation (9-a) is similar to Tuck's outer expansion; how-ever, it must be noted that our small parameter is based on
depth (E H/L) while Tuck's parameter is the slenderness
ratio (E) = B/L). We note that the LaPlacian operator in the
horizontal plane does not occur to 0(E4) thus in this respect the present expansion for the disturbance potential is
sim-pler than the linear theory. On the other hand, the free'
surface kinematic and pressure conditions for (pi are derived from higher order approximation which lead directly to the
nonlinearity in the problem. It would appear that in the
transcritical region the nonlinear free surface conditions
12 (10) of < 0, + =
3 3
are dominate and the potential natu/e of the flow i only
secondary. In passins we note that equation (10) is
mathe-maticalfy identical to the equation c.overnin2; itransonic flow
past two dimensional airfoils.
We can now define the relative orders of machitude
be-tween the shallow water parameter
(E)
and the slender bodyparameter (b) by examination of the behavior of the far field
solution on the body surface. While this
can actunlly
bedone by the formal matchin6 process, we choose to do it here
to simplify the algebra. Substituting the far field
varia-bles and expansion into the hull tangency condition, we ob-tain for the leading terms
E3/2fly(x,bA) +5/2f2y(x,6A) bAx + EtAflx(x't))k)Ax
-(z+1)f, (x,6A)Az], (12)
where the "slenderness" of the ship hull is exhibited
expli-citly through 6A with A = 0(1). Guided by the two-dimensional
aerodynamic slender air foil theory,. we take E2 which
satisfies our earlier requirement that urn0.
E 0
Thus it seems to imply that the shallow water problem is
ana-logus to high .aspect ratio airfoil problem while the deepwater problem is analogus to the low aspect ratio problem.
Near Field k)nroximl:ition
The nonlinear effects are not expected tobe important in the near field region where the basic flow pattern is
strongly influenced by the hull form. As a result, one would
expect that the near field expansion would yield a sciries of
Neumann problems in the (y-z) cross flow plane similar to
those derived by Tuck(2). The following non-dimensional and
scaled variables are introduced:
.
-x = XL, y = ELY, z ELZ, = ELr, N = ELN.
All the remainins variables are as in the case of far field
approximation. An additional variable 17; is introduced such
that the unit vectOr in the N direction is normal to
and the hull contour (6A(Y,Z)) at any given cross section. We assume the following expansions for the disturbance poten-tial and the free surface elevation:
(1) --
anwon,
Pn(c);;.(13)
FollowinE; the same procedure as for the far field so-lution, we obtain a consistant approximation in the near
field to 0(c5/2) given by
al(c) = c a2(F,) = c5/2, Pi(c) = c
c-72 n
n (n = 1,2) (14-a)
nZ(X,Y,-1) = 0 (all n's) (14-b)
15
Onz(X,Y,O) = 0 (n 1,2) (14-c)
onN(X,A) = on, ol = 0, = Ax/i171 . (14-d)
To the second approximation (n=2), the boundary vaThe
prob-lems derived are identical to those of Tuck(2) as well as
the order of magnitude estimate pliced on the disturbance
potential*. However, we note there is a difference in the
estimate placed on the elevation of the free surface and is
( 2 )
n(
,(
6/2n
-7Tuck sj'ETucki = 'present) = '''Epresent)
n(
present' 2
Dpresent 0(present)
"prese
°(64/3 )TuckThus we see that the surface disturbance is stronger here than in the linear case.
Since the Neumann problems defined by equations (14) have already discussed in detailed by Tuck(1)(2), we shall make use of his results and using the restricted matching
technique of Van Dyke(4) to match one term far field to the
two terms near field ai,Troximation. The important result is the hull tancency condition for the far field equation on a equivalent body and
fly(x,e)
= S'(x)/2, (15)'where S(x) is the cross sectional area of the ship hull im-mersed in the water.
de note here the difference in notation cr =c
2uck=bpres. pres.
16
Results and Discussion
The resultinc:, nonlinear problem, for the first
approxi-mation, can be solved numcrically or solved approximaLelw using methods of local linearization developed. in transonic
aerodynamic literature(5)(6). We have taken the latter
ap-proach due to limitations
on computer
time and the detailsof which are given by Fe1dman(7). Here, we shall present
some results for the sirka:L.e and trim of a seni-submerged
spheroidal hull. The cross sectional area of the hull is
R Bmax(x/L - x2/L2)1/2,
where Bmax is the maximum beam. The trim and sinkage are
computed at the bo with units of trim measured in terms of
ship length and slenderness ratio of 1:10. The results are
presented in Figure I where the Froude number is based on
the undisturbed depth.
In
Figure II, we have presented thesame curves but using a different scale so that the linear results computed from Tuck's(2) solution can be viewed simul-taneously for comparison.
The apparent discontinuity in slope at Fh 1.0 and
Fh = 1.09 is due to the method of solution and not the model
equation. We note that these solutions do indicate the
over-shoot as well as underover-shoot of sinka:Ee and trim respectively through the transcritical region which have been measured in experiments such as the works of Graff, Kracht and Weinblum(8)
17
as well as trim data have been computed for more realistic
hulls and these will be reported else-whergf.. However, one particular case with experimentai results of Graff
is given here for comparison. The hull chosen is Wodel A3
of D. W. Taylor's Standard Series and the flow condition is
exactly critical CFh 1). For computational purposes, the
cylindrical hull is approximated by a fourth degree polyno-mial-arc, we have
Experiment (Graff et al):
Trimbow = 2.0, Sinkage/Lengthhow -.015
Theory
Trimhow = 2.09, Sinkage/Lengthhow = -.0123
at Fh = 1.0.
The theory appears to be in fair agreement with experiment and indicates that this direction of research should be
fruitful.
Acknowledgment
We are greatly indebted to Professors Th. Y. Wu and J. N. Newman who encouraged us and nudged us alons the path to
a possible solution. One of us (G. K. L.) wishes to express
his sincere thanks to Mrs. Lea who typed and retyped this manuscript expertly and willingly.
References
18,
Tuck.,
.1-.0., 'A Systematic Asymptotic Expansion
proce-dure for Slender Ships", Journal ship Lesearch, 1964.
Tuck, E. 0., "ShallowjWater Flows
Past
Slender Bodies',Journal of Fluid
;Jechanics, Vol. 26,, Part 1, 1966,
Cole, J. D., "Perturbation ii:ethods in Applied
Mathemat-ics", Blaisdell Publishing Company, Waltham,
assachu-setts, 1968,
Van Dyke, M. D., 'Perturbation Methods in Rluid
Me--chanics", Academic Press,
New York, New York, 1964.
Spreiter, J. E. and Alksne, A. Y.,."ThSn Airfoil Theory Based on Approximate Solution of the Transonic Flow
Equation", NACA Technical Report
1359, 1958.
Hosokawa, I., "A Simplified Analysis for Transonic
Flows Around
Thin Bodies", Symposiun Transsonicum,
Aachen, September
3-7, 1962,
Springer-Verlag, Berlin,1964.
Feldman, J. P., nTranscritical Shallow 'Water Flow Past
Slender Ships", Ph.D. Dissertation, The George
Washing-ton
University,
Washington, D. C.,1971.
Graff, 15,, Kracht, A-, and ':;einblum, G.,1°Some
Exten-sions of D. W. Taylor's Standard Series", Transactions
of the
Society
of Naval Architects and 1.a.rine Engineer,Vol. 72,
1964.
Graff, W., and Binek, II., Untersuchung des
Modelltank--einflusses an einem Flachwasserschiff",
Forschungs-berichte des Landes Nordrhin-Westfalen, Nr. 1986,
West-deutscherverlagi
1971.
0
0.01
0 0 4-+ 0 '0)-0.02
0Zeam-Lenqth Patio 0.1
Beam-Depth Ratio 1.0
Trim
Sinkage
0 .I5
_.A,___ ___.i, -1)--0.84
0.88
0.92
0.96
1.00
1.04
1.08
1,12
1.16
Froude Number
Figure I - Semi-Submerged Speroidal Hull
.2
1
