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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 540
On Submerged Stagnation Points and Bow Vortices Generation
L.Raheja (Indian Institute of Technology, Kharagpur, India)
ABSTRACT
The mechanism of the generation of bow vortices in two dimensions, in laboratory scale, is explained on the basis of
the existence of a submerged stagnation point below the free surface. The stream which originates from the submerged
stagnation point in the free surface direction reverses and neutralizes the momentum of incoming flow resulting in a free
surface separation point. The initial location of the submerged stagnation point and subsequently the free surface
separation point is calculated for the case of a semisubmerged horizontal circular cylinder, and the latter is compared with
the experimental results. The agreement is reasonably good. The generation of bow vortices is discussed as a balance
between inertial and gravitational effects. The loss of pressure at the bow and the consequent drag due to bow vortices
phenomena is calculated and is found to agree well with the value found by experiment. A methodology for the design of
efficient bow contour shape in two dimensions where the submerged stagnation point is used as a control handle, is
presented. The theory is also applied to non-regular shapes like vertical step and bulbous bow. The results are compared
with those obtained by flow computation and found to be in reasonably close agreement. Finally, a conjecture is advanced
to explain the generation of bow vortices in three dimensions, i.e. the necklace vortex around a ship's load waterline, on
the same basis.
INTRODUCTION
The vortical motion observed ahead of a partially submerged object towed in a hydrodynamic tank is known as bow
vortices. The bow vortices region is separated from the main potential flow by a sharp boundary termed as the free
surface separation point (FSSP), when the flow is two dimensional. The phenomena have also been observed in
hydrodynamic flumes. The understanding of this phenomenon is of direct consequence to bow-wave-breaking, which is
responsible for a substantial component of a ship's resistance (Baba 1969). The phenomenon depicts itself in the form of
white-water at the bow continuing all around the ship's load waterline, and is also called as necklace vortex.
Several authors have visualised bow vortices ahead of two-dimensional as well as three dimensional shapes, e.g.
Eckert and Sharma(1970), Suzuki (1975), Honji, (1976), Shahshahan (1981), Kayo and Takekuma (1981), Kayo,
Takekuma, Eggers, Sharma (1982) and Mori (1984). But, here we shall be primarily concerned with the experiments of
Kayo, Takekuma, Eggers and Sharma (1982) on a horizontal semi-submerged circular cylinder conducted at Institut fur
Schiffbau, Hamburg. It may be relevant to mention that the author has viewed the videotape of these experiments. In
these experiments, a circular cylinder was towed in a horizontal semisubmerged condition (two-dimensional flow) and the
bow vortices were visualised using a watercolour dye. The FSSP was measured for different values of draft Froude
number (Fd). The primary objective of this paper is to explain these results qualitatively and quantitatively. But before
that, we shall present a brief review of the attempts made so far in this direction.
Dagen and Tulin (1972) solved the gravity flow past a blunt body by using two perturbation expansions. A small
Froude number solution was obtained for the flow under the unbroken free surface upto second order, while a high
Froude number solution was obtained based upon the model of a jet detaching from the bow and not returning to the flow.
The breaking of the wave was assigned to Taylor instability due to the steepening of the streamlines. A
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 541
critical Fd was obtained to characterise the onset of wave breaking. The associated drag due to breaking of waves was also
calculated and was found to be twice the value estimated by Baba (1969) experimentally.
Mori (1984) assigned the white water generation phenomenon to shear flow instability and subsequent breaking of
the bow flow, and studied the same theoretically and experimentally in greater detail. The free surface curvature was
concluded to be one of the sources of shear flow beneath the free surface. Stability analysis, vorticity stretching theory,
and free surface boundary layer theory were involved to explain the experimental results, e.g. velocities, Reynolds
stresses and bow wave heights. Patel, Landweber and Tang (1984) attempted to explain the bow vortices generation on
the basis of the existence of a free surface boundary layer, which is a layer of concentrated vorticity occurring due to the
curvature of the free surface in the flow of a real fluid. The authors speculated that the free surface would move slower
than the layer beneath it and this velocity defect would lead to a FSSP ahead of the body and subsequently the bow
vortices. Further, by assuming that at the free surface the surface tension is balanced by normal viscous stress force, an
expression for FSSP location was obtained by direct integration of the boundary condition using the continuity equation.
The location of the FSSP was obtained in terms of the slope of the free surface. The results so obtained were applied to a
circular cylinder and compared with the experimental values of Kayo et al (1982). The theoretical values showed an
increasing trend, i.e. the FSSP will move away from the body with increase in draft Froude number while experimental
results pointed to the reverse. However, the experiments of Grosenbaugh and Yeung (1985, 1989) reported good
agreement with their separation criterion. The idea of a free surface boundary layer leading to a FSSP was further
examined by Raheja (1995). The free surface boundary layer velocity and vorticity profiles were computed at various
stations upstream. It was observed that the free surface moved slower than the flow beneath it but this velocity defect was
not large enough to result in a FSSP ahead of the body. Yeung and Ananthakrishnan (1992), in their computational study
of the problem (to be discussed later), also concluded that the free surface vorticity is not intense enough to lead to bow
vortices. The boundary layer vorticity profiles computed by Raheja (1995) pointed towards instability of the boundary
layer flow owing to their nonmonotonic nature.
Vanden-Broeck & Tuck (1977), Vanden-Broeck, Schwartz and Tuck (1978) continued investigations into the
analytical solution of the bow flow problem by using a series expansion in Froude number, but concluded that it was not
possible to obtain a continuous bow wave profile because of non-uniqueness of the solution. However, the bow shape was
restricted to bows with a vertical or inclined flat-faced. They speculated that the possible form of solution for these shapes
is that of an overturning jet. Tuck and Vanden-Broeck (1984) showed that a continuous splashless bow flow was possible
for some different bow shapes such as bulbous bow. It was assumed all through that the stagnation point lies at the
intersection of the body and the free surface.
The difficulty in finding a closed for solution gave an impetus to computational studies. Miyata et al (1985) applied a
version of the MAC method to capture nonlinear wave breaking at the bow. The nonlinear waves breaking at the bow due
to steepness were termed as free surface shock wave (FSSW). The exact nonlinear free surface condition was used at the
free boundary and the no slip condition on the body. A computational study, which gives more insight into the bow flow,
is that of Grosenbaugh and Yeung (1989). These authors have used a boundary integral method to compute the two
dimensional free surface flow past a semi-infinite body in the time domain. The free surface computation is done
according to the method give by Longuet-Higgins and Cokelet (1976). The critical Froude number Fd for the onset of
wave breaking is found for bow shapes—vertical step, faired body and bulbous bow. It is found there that a bulb in the
bow shape delays the onset of wave breaking. In this study, the free surface flow is developed from the steady state
double body flow thus avoiding the impulsive start of the body which may result in uncontrolled bow wave elevation.
This is also supported by the finding of Dagen & Tulin (1972) where the lowest order asymptotic expansion of the free
surface flow past semi-infinite body is found to be a double body flow with the free surface replaced by a rigid plate.
Besides, it seems quite logical to consider a steady state double body flow as the initial condition; the suddenly removing
the upper half of the flow allows the flow to develop to a free surface flow. The authors also discuss the occurrence of
submerged stagnation point (SSP) on the body, which behaves differently for the breaking regime and non-breaking
regime of the flow. For the former case, the SSP remains below the free surface while the bow wave overturns, but for the
latter, the SSP is initially below the free surface and rises to the free surface as
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 542
the bow elevation reduces and settles at the stagnation height of We shall discuss these results latter in another
section of the work.
Yeung and Ananthakrishnan (1992) investigated the flow of a real fluid past a two-dimensional bow, one of the aim
being to examine the possibility of bow vortices observed at laboratory scale. The authors concluded that the occurrence
of bow vortices in the laboratory scale is due to the presence of surfactants which accumulate near the body and provide
rigid—boundary like behaviour to the free surface leading to boundary layer separation and subsequently vortices. The
N.S. equations coupled with the surfactant concentration equation were computed by a variational fractional step method
and using the no slip condition at the free surface, the occurrence of vortices was shown. The authors also investigated the
case of free slip and the exact nonlinear free surface boundary condition and arrived at the conclusion that vorticity
generated due to free suface curvature is not intense enough to lead to separation. As a recent development, Dong, Katz &
Huang (1997) have used PIV to visualise the bow flow and measured the flow velocities near the bow wave, upstream
and downstream. The laser sheet is visualised in different orientations ahead of the bow and at different stations
downstream of the bow.
Summarising the above review, one may mention that the ideas of Taylor instability, free surface boundary layer and
surfactant concentration have been examined but the mechanism of bow vortices generation and the occurrence of the
FSSP are still not well understood and to the best of our knowledge the results of the experiments by Kayo et al (1982)
have not yet been explained. A two-dimensional study is considered almost a necessary step in the development of a
theory for the three—dimensional case, as it provides a valuable gain in insight at the expense of relatively simple
computation. Therefore, it is desirable to concentrate on finding a theory for explaining the experimental results of Kayo
et al (1982) before discussing the three-dimensional generation of white water, necklace vortex or bow-wave breaking in
an ocean going ship. In the present work, we propose a theory to explain the bow vortices generation in the laboratory
scale in two-dimensional flow on the basis of the occurrence of a submerged stagnation point (SSP). This is analytically
found for the flow past a semisubmerged horizontal circular cylinder. Subsequently, the approximate values of the FSSP
for different draft based Froude numbers are calculated and compared with the experimental values of Kayo et al (1982).
Further, the drag due to loss of pressure at the bow is calculated and compared with the value estimated by Baba (1969)
experimentally. Besides a circular cylinder, the theory is also applied to a vertical step and a bulbous bow shape. The
design of an efficient bow contour in two dimensions is discussed. Finally, a conjecture is advanced for explaining the
three-dimensional bow vortices generation. The subsequent sections of this paper develop the idea and the relevant
expressions in a step by step manner ending with conclusions and future scope of work.
SSP THEORY AND BOW VORTICES GENERATION
The mechanism of bow vortices generation and the occurrence of the free surface separation point can be explained
under the framework of the proposed SSP theory as follows. A stagnation point in the two-dimensional flow past a fully
submerged object is always the intersection of the dividing streamline and the body. The stream divides itself into two
parts at this point. The pressure on both the sides of the stagnation point decreases. The point is a maximum of the
pressure distribution. In the case of the flow past a partially submerged object, a stagnation point may exist below the free
surface and may be rightly called a submerged stagnation point (SSP) as shown in Fig. 1. It may be conjectured that in
case an SSP does exist it should similarly be the intersection of the dividing streamline and the body. Accordingly, the
stream should divide itself into two parts at this point and if the flow is two-dimensional, one part will flow below the
object and the other should move upward in the direction of the free surface. This latter part of the stream, owing to the
obvious limitation in moving upward due to gravity, reverses and neutralises the velocity of the incoming flow resulting
in an FSSP where the two velocities come in balance. The reversing flow traps a region of the fluid (Fig. 1) extending
horizontally from body to the FSSP and vertically from the SSP to the free surface F, duly bounded by the dividing
streamline and the free surface such that the fluid is moving along the boundary of this region, i.e. from FSSP to SSP
along the dividing streamline, from SSP to F along the body and finally towards FSSP along the free surface. The
circulation of the fluid along the boundary of this region sets the entire mass of the trapped fluid into cyclic motion giving
rise to the first vortex with positive i.e. anti-clockwise vorticity at the centre which may be due to only internal shear. The
vortex grows in size and for stability reasons, a second vortex starts from the body side with rotation in the opposite
direction. Thus the fluid keeps on
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 543
entering in this region from the body side due to the upward moving stream, and the vortices are continuously produced
from body side. The boundary FSSP thus keeps on shifting slowly towards upstream as the region is filled. Further, owing
to this fluid filling, the bow wave height continuously increases and upsets the inertial-gravitational balance which results
in expelling the first vortex out of the region to join the main flow with the FSSP moving backward and the bow wave
height coming down simultaneously, and the process of filling the region restarts. The oscillation in bow wave height and
the corresponding back and forth movement of the FSSP have been observed experimentally by Grosenbaugh and Yeung
(1985, 1989) and also by the present author in an open-air circulating water channel (unpublished). Thus, the fluid keeps
on entering this region from the upward moving stream and the vortices are continuously produced from body side and
expelled from the other side to join the main stream. This, in a nutshell explains the mechanism of bow vortices
generation and the occurrence of the FSSP. The situation described above corresponds to the case when the free surface is
not broken. In case the free surface breaks, the upward moving stream from the SSP forms a bow jet, which entrains air
and disintegrates to form bubbles, reversing into the main flow again resulting in the FSSP. Vanden-Broeck and Tuck
(1977) had speculated a similar scenario as one of the possibilities while attempting to find an analytical solution for
vertical or inclined flat-faced bow shapes.
Fig. 1 Submerged stagnation point (SSP) and the associated bow vortices generation in the flow past a horizontal
semisubmerged circular cylinder
RESULTS AND DISCUSSION
SSP-Existence and Location
To confirm the existence of an SSP and subsequently find its position should require, in general, solving the gravity
flow problem, which of course, is very difficult. But it is possible to know about the SSP and even find the initial location
of the SSP by analysing the initial condition of the problem. Consider the two dimensional flow past a horizontal semi-
submerged circular cylinder. It is assumed that the free surface flow is obtained by initially having potential double body
flow and removing the upper flow suddenly at an instant (t=0+) which may be taken as the onset of gravity flow
(Grosenbaugh and Yeung 1989). Hence the pressure and the velocity field prevailing initially, i.e. at t=0+, are thus known
and it is possible to see whether an SSP exists and calculate the location of the SSP which will then be the point where the
total pressure will have its maximum on the body. It may be mentioned that the free surface flow at t=0+ is double body
flow solution of the Laplace equation with the pressure replaced by total pressure, i.e. including gravity term. Referring to
the axis system as shown in Fig. 1, for the stream flowing in the positive x-direction, the initial pressure distribution, i.e.
at time t=0+, at any point (x, y) in the flow will be given by total pressure as
(1)
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 544
where U is the velocity of the flow, and U∞ is the velocity far upstream and patm is atmospheric pressure.
For the points on the cylinder, we know from the results of potential double body flow that
(2)
where θ′ is the polar coordinate of the point measured from the positive x-axis.
Further,
(3)
where R is the radius of the cylinder.
For the present problem, the domain of interest for
Therefore, we define a new variable θ as
(4)
so that the domain of interest of θ correspondingly becomes 0≤θ≤π/2 which is more convenient for analysis.
Introducing (2) & (3) into (1), we obtain
(5)
Now non-dimensionalising the distances with R and the pressures with and retaining the same notations for
non-dimensionalised variables, we get
(6)
where Fd is draft based Froude number defined as
(7)
Equation (6) gives the prevailing pressure distribution on the submerged part of the cylinder contour at the onset of
the gravity flow. The maxima and minima of this pressure distribution can be obtained by the conventional method i.e.
putting
which leads to
The points of optimum pressure will be given by
(8)
It is important to note that in the second case, for θ to be meaningful, one must have
(9)
However, there is no restriction on Fd being large even upto ∞ Proceeding further to check the maxima and minima,
we find the second derivative,
(10)
Evaluating (10) for the values of θ obtained in (8), we get
(11)
(12)
Keeping the restriction given by (9) in mind, we find that (11) gives a positive-definite and (12) gives a negative-
definite value for Fd>0.5. In other words, the pressure has a maximum at and a minimum at the bottom
most point θ=π/2 in the domain 0≤θ≤π/2. It may be pointed out that for Fd=0.5, (11) and (12) both give zero, confirming
(9). In fact for Fd=0.5, only one maximum occurs at θ=π/2 in the domain 0≤θ≤π, while for Fd more than 0.5, there is a
corresponding maximum at π−θ. Hence we conclude that for Fd>0.5, there is maximum in the pressure distribution at
and obviously this point is the
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 545
submerged stagnation point. The pressure decreases on both sides of it, so the velocity must be zero. Table 1 (placed in
the section on Bow Drag) gives the location of the SSP and the corresponding value of nondimensional pressure
coefficient defined in the conventional way as for different values of Fd. It may be observed that for
Fd=0.5, the SSP is at θ=π/2, the lowest point of the cylinder. As Fd increases, the SSP moves upward towards the free
surface and finally for Fd=∞, θ=0, i.e. SSP coincides with the stagnation point of double body flow, then patm, is duly
replaced by p∞ the definition of Cp. This result will be discussed further in a latter section on inertial-gravitational balance.
Pressure Profile and Flow with SSP
Fig. 2 Pressure distribution before (t=O−) and at the onset of gravity flow (t=O+) for the flow past a horizontal
semisubmerged circular cylinder. The radial distance measured from the body gives the value of Cp
A typical pressure distribution before and at the onset of gravity flow, for Fd=0.8 and 1.0 is shown in Fig. 2. The
pressure coefficient at the points on the contour is given by the radial distance from the point to the curve. It may be
observed that in the case of double body flow (t=0−), the pressure uniformly decreases from A to D, but at the onset of
gravity flow i.e. t=0+, the pressure increases from A to SSP and then decreases to D. The SSP occurs at θ =14.48° for
Fd=1.0. The pressure decreases on both sides of the SSP. Consequently, at the onset of gravity flow, the fluid particles,
above the SSP will move upwards and below the SSP will move downwards. The upward moving stream will reverse
owing to the limitation due to gravity, resulting in the FSSP and bow vortices provided the surface is not broken, as
explained in an earlier section.
Calculation of FSSP
The FSSP is created when the flow from SSP to F (Fig. 1) reverses and its velocity comes in balance with the
incoming flow velocity along the free surface. The velocity of the reverse flow at the point A (the stagnation point of the
double body flow) at the onset of gravity flow can be calculated by applying Bernoulli's equation at the SSP and at the
point A. The former being a stagnation point of the gravity flow, the velocity is zero there and therefore we get
(13)
where V is the velocity of the upward flow at A. The pressure (p)ssp and (p)A can be obtained by substituting θ=θssp
and θ=0 respectively in equation (6). Accordingly, we obtain
(14)
So far we have been obtaining the results just by analysing the initial condition, as it was pertaining to the onset of
gravity flow or immediately thereafter; but the next step i.e. balancing the reverse flow against the incoming flow along
the free surface, is a result belonging to the final steady state, which can be achieved only after several smaller time steps
from the time t=0+ onward. In these steps, the free surface condition is to be applied on the free surface, the incoming and
reverse flow is to be calculated along the free surface which itself is an unknown of the problem. The non-linearity of the
free surface condition adds a further complication. However, since the aim of the paper is to present the theory and
explain the mechanism of bow vortices generation, and not to go into detailed computation, we shall try to obtain some
approximate results so as to get an insight into the phenomenon. Accordingly if ζx is the slope of the free surface we can
write the balancing process as
(15)
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 546
where Ux is the x-velocity component of the double body flow along the x-axis.
Substituting Ux=1−(1/x2) for the case of the circular cylinder, (15) leads to
(16)
As pointed out, it is not possible to know the correct value of ζx, so we shall assume here that ζx=0, which may be
valid for relatively small values of Froude number. Consequently we obtain
(17)
For obtaining real values of x from (17) we must have
(18)
Hence the validity of (17) starts from 0.71 instead of 0.5 which is the natural restriction for the case of a circular
cylinder (Eqn. 9). It may be noted that the validity rightly improves if (16) is used in place of (17). In this case, we obtain
in placed of (18)
(19)
Since cosζx<1 anywhere on the free surface, the r.h.s. will always be less than and for ζx=60°, the result
Fd>1/2 is obtained. But, as mentioned earlier, the free surface slope and the velocities of the incoming and reverse flows
are to be calculated together, so anything less than the full flow computation will not do and any approximate value of
free surface slope will lead to an erroneous result. Therefore, we accept the reduced validity of the formula (17) for the
FSSP for the present.
Comparison with Experimental Results
The numerical values of xFSSP as obtained from (17) are duly converted to β (=x−1) Fig. 1, and plotted along with the
experimental values in Fig. 3. A curve has been passed through the experimental points of Kayo et al and is shown against
the curve obtained from present theory. The observations can be summed up as follows:
Fig. 3 Free surface separation point (FSSP) in the flow past a semisubmerged horizontal circular cylinder-Experiment and
Theory
(i) The FSSP moves closer to the body in confirmation with the experimental results qualitatively.
(ii) The experimental values for Fd<0.5 seem to be fluctuating and do not behave in a regular fashion as the
values for Fd>0.5. This may be related to the fact that a well defined SSP does not exist for Fd≤0.5 as it does
for Fd>0.5 (c.f. Eqn. 9).
(iii) The theoretical values for Fd=0.72 to 1.2 can be said to be reasonably close to the experimental curve keeping
the crude basis of their derivation in mind.
Inertial-Gravitational Effect
It is interesting to observe the change in the location of the SSP and the FSSP with the increase in Fd. As Fd is
increased, the SSP moves closer to the free surface and the FSSP moves closer to the body, Fig. 4. The region, which is
obtained by joining the SSP, the FSSP and F (obtained by taking bow wave
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 547
height as ) which is a region of trapped fluid containing bow vortices, shrinks with the increase of Fd.
Fig. 4 Variation in the bow vortices region with Fd. The region is formed by joining SSP locations to the corresponding FSSP
locations and taking the rise of free surface= at the cylinder
This can be explained as follows: The origin of this trapped vortex region is the effect of gravity. The Froude number
represents the ratio of inertia force to gravity force. When Fd is increased, the gravity effect relatively reduces and thereby
the region shrinks. In the limiting case of Fd=∞ the region finally vanishes with the SSP coming at θ=0 and the FSSP also
coming at β=0, i.e. on the body. The flow then behaves like double body flow. This is quite logical, as in the absence of
gravity, the flow should behave like double body flow. The bow wave height becomes infinite but this is due to the
fact that Fd is the draft based Froude number and for Fd → ∞, we must have draft becoming zero, which indicates that
there is no gravity acting any more and so no bow wave.
Bow Drag
Since there is a reverse flow between the point A and the SSP (Fig. 1), there will be a loss of pressure and associated
bow drag, which can be obtained by integrating the x-component of the force due to pressure from A to the SSP.
Accordingly, the bow drag D is given by
where nx, is the x-direction cosine and ds is the arc element.
Here nx=cosθ and the value of p−patm can be substituted from (5). By doing so we obtain
(20)
which, on performing the required integration and using (8) for θSSP, gives
(21)
The above formula can also be written in terms of the SSP and the Froude number, i.e.
(22)
The above form is, in a way, comparable to the one obtained by Dagen and Tulin (1972, Eqn. 72). One can also
express CD as a function of the SSP only and obtain
(23)
Table 1 gives the values of CD for different Fd values and also the corresponding location of the SSP. The
observations are as follows:
Table 1. The location of submerged stagnation point (θSSP) with the corresponding value of Cp at the onset of gravity flow and the
non-dimensional value of bow drag (CD) for different values of Fd for the case of semi-submerged horizontal circular cylinder.
Fd θSSP(deg) Cp CD
0.5 90.00 5.00 3.667
0.6 44.00 2.93 1.588
0.7 30.67 2.04 0.864
0.8 23.00 1.61 0.552
0.9 18.00 1.38 0.387
1.0 14.47 1.25 0.292
1.1 11.93 1.17 0.234
1.2 10.00 1.12 0.188
1.5 6.37 1.05 0.115
1.7 4.97 1.03 0.088
0.063
2.0 3.58 1.02
∞ 0.00 1.00 0.000
1 The numerical values in Table 1 show that CD decreases when the draft Froude number increases, which
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would mean that higher the draft Froude number, the lower the bow drag. At the first instant this does not
appeal to the common understanding of drag vs speed relationship. But, it can be interpreted simply in

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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 548
terms of inertial-gravitational effect as discussed earlier. As Fd increases, the inertia effects relatively rise
over gravity effects and bow drag, being a consequence of gravity effect, reduces. This is further clear from
the location of the SSP which moves closer to the free surface with increase of Fd and thus reduces the range
SSP to A (Fig. 4) which is responsible for increasing the kinetic energy of the surrounding water by
generating reverse flow and subsequently bow vortices/bow jet/white water.
2. There is no elaborate experimental data available to compare the values in Table 1. However, Baba (1969)
has suggested a two-dimensional representation of breaking waves for his experiments on a tanker, as if it
was uniform and normal to the bow, and has estimated its equivalent length as roughly half the beam (Dagen
& Tulin 1972). The drag coefficient, per unit length, corresponding to a two dimensional flow across the
breaking wave for Fd=1.7 is given as where D is the drag and T the draft (Baba 1969 §
7.3). Interestingly, we find that our value of CD for Fd =1.7 comes out as 0.088 (Table 1), which is very close
to the estimate of Baba. This closeness of CD value obtained by the present theory with the one estimated
from experimental results further provides support to the SSP theory explaining the mechanism of bow
vortices generation. It may be relevant to mention that Dagen & Tulin (1972) obtained this value of CD=0.17
(about two times) for a vertical step by solving the gravity flow using the method of two perturbation
expansions.
3. Bow drag depends upon the location of the SSP, the lower the SSP the more is the bow drag. The location of
the SSP on the other hand is, directly related to the pressure distribution of double body flow on the bow,
which in turn depends upon its shape. Therefore, the double body pressure distribution on the bow and the
location of the SSP provide the key to the design of a bow contour for minimum bow drag.
Bow Contour Design-two dimensional case
A bow contour for the two dimensional case can be designed now for minimum bow drag as follows. The total
pressure at any point of the bow is the sum of double body pressure and the gravity pressure, i.e.
(25)
The condition for finding SSP is given by
which gives
(26)
In general Cp(d.b) is maximum at the double body stagnation point (DBSP) and decreases towards the bottom as the
fluid accelerates. The gravity pressure, on the other hand, is zero at the undisturbed free surface (i.e. at DBSP) and
increases towards the bottom. Owing to the opposite signs of the rate of change of these two pressures, a maximum in Cp
(total) (i.e. the SSP) occurs at the point where the two rates of change are equal in magnitude, making the rate of change of
Cp (total) to be zero.
The rate of change of double body pressure is independent of Fd and depends purely upon the geometry or the slopes
of the bow contour at different points. On the other hand, the rate of change of gravity pressure is which is constant
depending upon draft based Froude number. The SSP location is given by the intersection of the two sides of (26) when
plotted with respect to y. Now by using a double body flow calculation programme by a suitable panel method and
coupling it with a curve design programme using Bezier curve/B-spline with a visual interface, it should be possible to
design a bow contour which gives the SSP location as near the free surface as possible for a given value of Fd. It may be
desirable to tag the process with the selection of design draft of the ship. This will give the scope of manipulation of the
rate of change of gravity pressure additionally. Thus the location of SSP acts as a control handle for the design of an
efficient bow contour and also serves as a measure of performance with regard to bow drag.
For a circular cylinder (Fig. 1), we can write (25) as
(27)
and applying the procedure as described above based on (26) the location of SSP is straightaway obtained
analytically as
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 549
(28)
The application of the above to other shapes is shown below.
SSP for Vertical Step and Bulbons bow
So far we have been using a regular shape, i.e. a circular cylinder, for the application of the proposed theory. It is so
because we have the experimental results for the circular cylinder. But the theory can be applied to other bow shapes
equally well and whatever results are obtained analytically in the case of the circular cylinder, can be obtained
numerically in the case of nonregular two dimensional bow shapes. As an example, we present here the application of the
theory to a vertical step and a bulbous bow shape. The geometry of the vertical step and the bulbous bow has been taken
from Grosenbaugh and Yeung (1989). The double body flow is calculated using a low order panel method. Fig. 5 shows
the curves of Cp (d.b.) and the Cp(total) for Fd=1.
Fig. 5. The variation of Cp(d.b.) and Cp(total) vs Draft for vertical a step and Bulbous bow.
The sharp changes in the slope and its direction in the geometry are duly reflected in the curve of Cp(d.b.) at the
corresponding points. The maxima occurring in the curve of total pressure can be easily marked.
Fig. 6 shows the location of SSP in the case of the vertical step for different draft Froude numbers determined by the
procedure based on (26) and described in the earlier section i.e. by drawing the slope curve of Cp(d.b.) and the constant
slope lines of Cpg for different values of Fd. The intersection of the two slope curves then gives the corresponding SSP
location. It can be seen that as Fd increases, the SSP in the vertical step moves closer to the free surface, similar to the
case of a circular cylinder. For Fd=1, the SSP lies a little above half the draft, which is comparable to the value as shown
by Grosenbaugh and Yeung (1989) with the flow computation. Fig 7 shows the same for the case of bulbous bow shape.
Owing to the frequent changes in the direction of slope in the shape, the maxima are not as well
Fig. 6 Determination of submerged stagnation points for Fig. 7 Determination of submerged stagnation points for
a vertical step. The vertical lines show the slope of a bulbous bow. The Vertical lines show the slope of
gravity component of pressure. gravity component of pressure.
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 550
defined as in the case of the vertical step. For example, for Fd=1, there are two maxima and two minima. Accordingly the
flow picture will be more complicated than as has been described for the case of the circular cylinder or vertical step. But,
since the second maxima has the higher value of Cp(total) that should be taken as the true maximum and be the SSP in the
current context. It is then observed that the SSP for the bulbous bow shape is lower than in the case of the vertical step for
the same value of Fd=1. This provides an explanation for the observation reported by Grosenbaugh and Yeung (1989) that
the wave breaking is delayed for a bulbous bow relative to a vertical step, since lower is the SSP the less dominant is the
inertia effect and wave breaking which is directly linked with speed is delayed accordingly.
Bow Vortices—three dimensional case
The foaming motion or white water observed at the bow of a ship and all along the load waterline known as necklace
vortex, is the three—dimensional picture of bow vortices. If the bow vortices are generated in the two dimensional case
by the presence of an SSP below the free surface, it will be just right to conjecture that an SSP may exist at the ship's bow
and at the sections below the free surface. In other words, the curve of Cp(total) vs draft may have a maximum at the bow
and at the sections.
Since the ship has a finite draft, its double body flow must acquire a component of velocity in the depth direction
right from the forward perpendicular down the hull. Accordingly, at the bow there is a decrease of double body pressure
from forward perpendicular to the keel and at each section from load waterline to the bilge, which in presence of gravity,
results in a SSP below the free surface giving rise to a flow from the SSP to the free surface. This upward flow may form
a jet at the free surface, entrain a lot of air and disintegrate into innumerable bubbles at the free surface reflecting more
light owing to their large surface area and form a white water or a foam like appearance.
CONCLUSIONS
The problem of bow vortices generation at laboratory scale is addressed here with the main aim to explain the results
of the experiments on a horizontal semisubmerged circular cylinder, and the mechanism of bow vortices generation
observed ahead of the cylinder.
1. It is shown that there exists a stagnation point below the free surface, i.e. a submerged stagnation point (SSP)
which is responsible for making a branch of the mainstream flow upward towards the free surface and thus
produce a reverse flow which results in bow vortices and a free surface separation point (FSSP).
2. The SSP is a function of draft based Froude number Fd and double body pressure distribution. The SSP
occurs when the double body pressure decreases along the draft and consequently, a maximum appears in the
total pressure below the free surface. It is further discussed that the location of the SSP represents a balance
between inertial effects and gravity effects. The SSP moves towards the free surface as Fd is increased and
only for Fd=∞ (i.e. no gravity, only inertia) SSP lies on the free surface coinciding with the stagnation point
of double body flow.
3. The values of FSSP calculated on the basis of this theory for flow past a horizontal semi-submerged circular
cylinder explain the experimental results of Kayo et al (1982) qualitatively and quantitatively to a reasonable
extent.
4. The bow drag is obtained as a function of draft based Froude number. The value of the bow drag for Fd=1.7
for the case of a circular cylinder, agrees very well with the value estimated by Baba (1969) by experiments
on a tanker and finding a two dimensional equivalent of the same.
5. It has been conjectured that the bow vortices in the three dimensional case, i.e. the necklace vortex in a ship,
are also produced by the same consideration. Owing to a finite draft, there is a component of velocity in the
depth direction in double body flow of the ship which leads to a decrease of double body flow pressure from
forward perpendicular to keel at the bow and from waterline to bilge at sections. This results in a SSP at the
bow and at the sections below the free surface. The flow from this SSP towards the free surface results in a
jet at the free surface, which entrains air and forms bubbles making white water or a foam like appearance.
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ON SUBMERGED STAGNATION POINTS AND BOW VORTICES GENERATION 551
SCOPE OF THE FUTURE WORK
1. Computational studies are required to be undertaken to verify the results of this theory. First, the flow past a
horizontal seniisubmerged circular cylinder should be computed and the SSP location, reversal of flow, the
FSSP location and the bow drag be calculated and checked with the results obtained by the theory.
2. The experiments on a horizontal semisubmerged circular cylinder should be repeated and the bow vortices
should be visualised and studied in the light of the present theory. The SSP location should be found and the
nature of reverse flow should be studied. The FSSP location should be found and compared with the theory.
The oscillations in the FSSP location and corresponding bow wave height should be studied. Also, the bow
drag should be found experimentally and checked with the theory.
3. The steps (1) and (2) should be conducted for other bow geometries with different slope distribution, e.g.
vertical step, conventional ships bow and bulbous bow etc.
4. The conjecture made for bow vortices generation in a three dimensional case is to be verified by using a three-
dimensional analytical body generated by a known combination of singularities. The flow past a sphere or
Rankine oval do not serve the purpose as these are axisymmetric. The combination should be such as to result
in a true three dimensional flow. This will require the distribution to be asymmetric. Alternatively, the double
body pressures may be computed on a Wigley hull using a higher order panel method so that the points at the
bow can be taken as nodes as well as collocation points and then the pressure distribution at the bow and at
the section should be obtained. Subsequently, the SSP at the bow and at the sections should be calculated to
confirm the generation of bow vortices in the three dimensional case. We have worked with the values of Cp
for double body flow for a series 60 hull obtained from “SHIP FLOW” but these were at the panel centroids
which were away from the points at the bow contour. Owing to the high tangential velocity in the
neighbourhood of bow, the Cp was far below the expected value of unity at DBSP. We used a four/five
degree surface fitting but the extrapolation was not satisfactory.
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