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Representative terms from entire chapter:
viscous boundary
The Calculations of Fluid Actions
on Arbitrary Shaped Submerged Bodies
Using Viscous Boundary Elements
W. Price, M. Tan (Brunei University, United Kingdom)
ABSTRACT
A time dependent viscous boundary element method
is developed to calculate the unsteady fluid forces acting on
a rigid body moving in a stationary, unbounded,
incompressible viscous fluid. The theoretical approach
presented is analogous to a potential flow singularity
distribution panel method. Here the singularity is replaced
by a time dependent fundamental solution derived from a
modified Oseen equation developed through an integral
equation formulation involving convolution time
integrations. Analytical expressions are given for the
unsteady and steady state fundamental solutions
appropriate to two- and three-dimensional fluid-structure
interaction problems.
By distributing the viscous panels over the wetted
surface of the body and into the fluid domain, a numerical
scheme of study is devised to solve the non-linear time
dependent integral equations. To illustrate the method,
preliminary results are presented of the unsteady and
steady state fluid actions and flow fields associated with
arbitrary shaped, submerged bodies moving through
prescribed manoeuvres.
INTRODUCTION
When a rigid body departs from steady motion in a
straight line the surrounding fluid exerts a resultant force
and a resultant moment about the centre of gravity of the
body as a consequence of the disturbance. In
manoeuvring and control studies the variations of fluid
actions to disturbances (i.e. displacement, velocity,
acceleration), referred to as slow motion derivati~
In these steady flow studies, the analytically defined
fundamental solution adopted to describe the viscous av 2 .
element was derived from a modified Oseen's equation. at + (V.V jV = V V - Vp + f + u (1)
This was chosen because Oseen's equation and
Navier-Stokes' equation are qualitatively similar, so that
solutions of the former are expected to yield qualitative
information about solutions of Navier-Stokes equation for
all Reynolds number. In reality, however, steady flow
around a body exists only in the small Reynolds number
flow regimes and it becomes academic (though remaining
of great interest) to extend the investigation into the high
Reynolds number flows where unsteady influences
dominate.
The previous investigations showed that by
distributing viscous boundary panels over the wetted
body's surface only an Oseen flow-linearised model was
created, whereas distributing panels over the body's
surface and into the wake a non-linear convective model
resulted. The method could be readily applied to describe
the forces acting on bodies of arbitrary shape, the flow
field around such bodies and it was suitable to use in
multi-body structure - fluid interaction problems, e.g. the
flow around clusters of cylinders such as occur in a
cross-section of a leg of a lattice jack-up structure.
Further, after checks on convergence of solution, panel
distribution idealizations etc. of the numerical procedures,
in low Reynolds number flows Re < 100 theoretical
predictions of the fluid actions and experimental
datat14,15] showed good agreement and for higher
Reynolds number flows (Re ~ 103) the results confirm the
qualitative and quantitative trends observed by
othersE16,171. In comparison with the non-linear model,
the linear model overestimates the values of the fluid
actions. The global solutions (i.e. pressure distribution,
fluid actions, etc) derived by the non-linear model are
relatively insensitive to panel distribution, idealization etc.,
allowing simplifications in the numerical procedures;
however, the evaluation of a detailed flow velocity field is
more sensitive to idealization and the wake fluid domain
must be modelled to allow for the complete formation of
the vortex wake pattern.
Building on these previous studies, here we present a
viscous boundary element panel distribution method to
evaluate the unsteady fluid actions experienced by two
and three-dimensional arbitrary shaped bodies moving in a
stationary, unbounded, incompressible viscous fluid. The
final objective of this study is to describe the fluid actions
on a rigid body undergoing a prescribed, though arbitrary,
manoeuvre, e.g. a sequence of commands involving an
acceleration, constant forward speed, a deceleration etc.,
rather than an oscillatory motion or steady state condition.
For this reason the problem is described in a body fixed
frame of reference and an integral equation formulation
derived involving convolution time integrations. The
analytically defined time dependent fundamental solution
or oseenlet adopted to describe the viscous element is
derived from a modified time dependent Oseen's equation.
The numerical scheme of studies devised previously for
the steady state problem are adapted for this unsteady
problem and preliminary findings are presented of the fluid
actions and flow fields associated with a two-dimensional
circular cylinder moving through prescribed manoeuvres.
EQUATIONS OF MOTION
In a body fixed coordinate system translating with
velocity -unto, [u(0) = 0], the flow of an incompressible
fluid of constant viscosity described by Navier-Stokes'
equations, expressed in terms of non-dimensional Oseen
variables, are given in the form
802
V.V = 0 = div V
(2)
where V, p, f represent non-dimensional expressions for
the fluid flow velocity, pressure in the fluid and external or
body force respectively. The corresponding dimensional
quantities, denoted by a prime, are
V'=UV U'=UU p'=p'U p, P=(Re U /L'jf,
= (Lt/Re) A, t'= (L'/Re U)t, V'= (Re/L'jV,
a/0t' = (Re UlL')a/0t ,
where L' and U denote characteristic length and velocity
parameters respectively, p' represents the fluid density,
A', ~ denote spatial variables and the Reynolds number
Re = U L'/v' where lo' represents the kinematic viscous
coefficient.
Letting V = U ~ v, where U denotes a constant
velocity, it follows from equations (1) and (2) that the
pressure p and velocity v satisfy the equations
v + (U+v).Vv = V v - Vp + f + u
V.v =0
(3)
where an overact denotes differentiation with respect to
time.
This equation of motion is non-linear, but by
invoking the usual assumptions, Oseen's equation is
obtained in the form
v+ U.Vv=V2v-Vp+f+u (4)
V.v =0
subject to the initial time condition v (t,0) = - U.
INTEGRAL EQUATION FORMULATION
(5)
Before transforming the Navier-Stokes' equations of
motion described in equation (3) into an integral equation
using Gaussian integral formulae, let us introduce the
linear, time dependent Oseen matrix operator
e, ~(a/at+u.v-v2~-vv. Vl (6)
V. 0]
and its linear adjoins time dependent operator
'9*_ ~ ~ (a/at - A. v -v2) - TV. _v ~ `7'
-V. O
Here I denotes the unit matrix, U = (U. V, W) and these
operators are applicable to either two- or three-dimensional
problems. Continuing this use of matrix and tensor
notation we may define
V [v] F ~f+u-V.(vv)
Vs =| ps |, Fs =l 0s ] (8)
and the components
Plj = -P6lj+ Vl,j + Vj,
PSlj PS51j + VS1J + Vsj,1 ~
In these expressions vl j = avl/axj, v*sl j = av*Sl/ax, etc.
and the subscripts s, l, j, etc. take values 1,2 in a
two-dimensional problem and values 1,2,3 in a
three-dimensional problem.
This notation allows the Navier-Stokes' equations to
be written as
49 . V = F
(10)
in which the non-linear convective term is included in F.
By introducing a convolution integration over the
time domain 0~ < ~ < t and using Gaussian integral
formulae we find that a Green's function identity can be
constructed. Thatis
; do j{V (x,t-~.~. Vex, ~) - Vs (x,t-~. ~.V(x,~ldQ
o- Q
_~{VT*~*.V - V T* O.V)dQ
Q
where
= | {Vsj * (Rj - U.n Vj) - Vj * Rsj } dE
£
- | {vsj~xto-) vj~x,t) + Uj vsj~x,t) } dQ (11)
Q
R. =p n.=~-p8 +v +v ~n. (12)
~ 13 1 1] 1,] ],1 1
denotes the force component in the jth direction and
* *
R . = p .. n.
S] SI] 1
(13)
where ni is redefined as the ith component of the unit
normal vector at the boundary surface ~ pointing into the
fluid domain Q. In equation (11), the surface £ encloses
the domain Q. the superscript T denotes a transposed
matrix, a summation convention holds i.e.
Ujnj = ~ Ujnj = U.n
j=1
etc. and the multiplication symbol * implies a convolution
operation (e.g. a * b = b * a).
Now the function Vs* is an undefined vector which
we are at liberty to choose. We shall make this the time
dependent fundamental solution of the linear adjoins Oseen
operator subject to a unit impulsive loading in the jth
direction acting at the position ~ = x, i.e. where the field
point ~ coincides with the source point, x. That is Vs*
sat~sfies the equat~ons
E) Vs = Fs in Q
Vs ~ 0 at infinity
~(14)
together with the imposed initial condition vs * (x,0) = 0.
Here the jth component of the external impulsive
force Fs* is given by fsj = dsj ~ (~-x) d~t), such that
Q
VT* ~ . Vs dQ = C(~) vs(t,t) (15)
where C(~) = 0, 0.5 or 1 depending on ~ ~ (Q U I),
or ~ ~ Q respectively.
Substituting equation (15) into equation (11) and
using equations (8) and (10) we find that the integral
equat~on becomes,
C(~) vs(t,t) = ,[ { vs; * (Rj - vj U.n) - vj * Rsj } d£
+ v * u - v v dQ - U v dQ 16
| sj ( j k j,k) | j si ~ )
Q Q
assuming a zero body force i.e. f = 0.
FUNDAMENTAL SOLUTIONS OR OSEENLETS
TIME DEPENDENT SOLUTIONS
Analogous to the steady state solutions (v = 0)
derived previouslyE 12,13], the time dependent
fundamental solutions or oseenlets satisfy the equations
* * 2 * *
Vsj - U.V Vsj = V Vsj + PS j + ~ j/` (t - x) ~t)
v .. = 0
s~,~
Vs*j(t,O)=O.
A solution to these equations can be derived using
Laplace (L) and Fourier (F) transforms. That is, let ~y and
~ denote the Laplace and Fourier transform parameters
respectively and an application of the sequence of
operations LF~ ) to the previous equations gives
~ (17)
JVsj + i\.U Vsj = - \2Vsj - i~jPS + {8Sj ei~ X/~2~)nl2}
\. v.=0
~ s~
where i = ~1~-l), n = 2 for a two-dimensional flow problem
(s = 1, 2 = j) and n = 3, for the three-dimensional case (s =
1,2,3 = j).
From these equations it follows that
and
803
PS = - (il /~2) ei~ X/~2~)n/2
v* (6 (s\J) ei~ x/( 2~)nl2 [7 + il U + \2]
By reversing the transform operations, we obtain[18]
F(ps) = L { LF(pS) } = - i (\s/~2) e 6(t)/~2~)
F(vsi)=(6sj- s 3)ei\.x e-(\ +il.U)1~2~)nl2
or
and
Ps = n [,l >2 d) ].s ~ r = ~ - x (18)
00
~Sj f - 1~ t +iA (r+ut))
v .=- I e dX
(2~) 00
+ ~ [T e-{\t+ix (r+Ut)}
(211) 00
Three Dimensional Solution (n = 3)
,<2 - J,Sj (19)
For a three dimensional fluid-structure interaction
flow problem involving a field point ~ and a source point
x, such that the distance r = l; - xI and rs (s = 1,2,3)
denotes the sth component of r, the time dependent
fundamental solution is
Ps = 4 [1/r], s (20)
V . = Sj e~ (r + U t) /4 t
(4~t) I
1 [ 1-erfc ~ Ir+Utl/2
{l-e~( r)l2}{5s~ rsr~ (Us+rs/r) (Uj+rj/r) }
4~(U.r +r) r r~ U.r+r
The vorticity distribution is given by
C')sj= skj3 {2rk+r(Ukr+rk)}e~( r )/
8~r
where esk; is defined previously.
MODIFIED INTEGRAL EQUATIONS
Using the fundamental solutions derived in the
previous section, the integral equation described in
equation (16) can be written in alternative forms.
For the submerged body, the surface boundary );
(=~,oo ~ ~ b) consists of the body boundary ~,b and an
outer boundary ~0 at infinity. It can be shown that
(i) on £oo, Rj is constant, Vj = u; - Uj and
When r ~ O. the velocity soluiion reduces to vj *l Rsi dE = - (uS-Us)/n,
Vsj= ~ (6sj + 2 ~
corresponding to the equivalent Stokes flow solution and
when r~oOwe have
* ~ 0 (l/r2) outside the wake region,
sj ~ 0~1/r) inside the wake region.
Two Dimensional Solution (s = 1,2 = j = k)
The oseenlet solution takes the form
* 2
Ps = ~ rS / 2~r ,
* ~s~ - U.r/2 K ( /2)
- [U.r e '2K0(r/2) + U.r ln r - r e / Kl(r/2)
2~
2 rl2
(Uxr) I ~(~)e-(U.r)~4d~] sj
o
= Si e~Ur/2 Ko (r/2)
+ { Usrj + UjrS U ~Sj } { 1-0.5r e~ U r/2Kl(r/2) ~ (28)
2~r
where Ko, K~ are modified Bessel functions of the zero
and first ordertl83. When U = (1,0,0), this solution
agrees with the form presented by Besshotl9] using an
alternative approach.
When r ~ 0, the previous solution reduces to
v =- (- ~ . lnr+ s ~ ~or
s~ 4~ s~ r
and for r ~ oo, we have
* r 0(1/r ~ outside the wake region,
Vsj ' 0(l/r1/2) inside the wake region.
In this case, the vorticity distribution is described by or
the expression
Cl) 3 = sk3 {Uk Ko (r/2) +-K~(r/2) } e~U r/2.
r *
J Vsj* (Rj-vjU.n)d~=O
where n = 2,3 for two- and three-dimensional problems
respectively.
(ii) on 2,b, Vj= - U; and
J vj * Rsj d):
(27) = f 1 C(~ jus - J Uj*vsi dQ + J Uj*vsiU.n d~
and,
Qb [b
J u * v .dQ+lU. v .dQ
~s ~ s~
Qb
-
*
= U.rO vs; nj d~ = (n-l) Us/n
| vs; * Uj dQ = | vSj*u.r0 nj d~ - J vSj*u.r0 nj d~
Q ~o ~b
= (n-l) Us/n - | vSj * u.rO n; d~
where r0 = x-y and y is any point in the domain and Qb
denotes the internal volume of the body.
The substitution of these expressions into equation
(16) gives the integral equation relationships,
C(~) VS (t,t) = US ~ C(~) Us ~ | vSj * (Rj+u r0nj ~d~
- | Vsj * (Vkvj k) dQ
Q
C(~) VS (t,t) = US - C(~) Us Jvsj * (Rj+ u.rOn j - Uj U.n~d~
+| Vsj k * (VkVj ~ dQ (29,
Q
C(~) VS (t,t) = US - C(t,) US-Ivsj * (Rj+u.rOn j - 0. 5U2nj~d~
*
+ vsj*(vx03)jdQ
Q
805
where ~ represents the vorticity of the flow.
The basic unknowns in these formulations are the
fluid velocity v and fluid force R which can be determined
directly from the discretised forms of these equations. In
fact, although the details of the terms in these equations
differ from those occurring in the steady state problem, the
general method of solution adopted in the latter can be
suitably modified to solve the present integral equations.
Further, since u, rO, U and n are all prescribed
quantities we may treat Xj = Rj + u.rO nj - Uj U.n in
equation (29) as the unknown and this integral equation
reduces to
C(~) VS (t,t) = US ~ C(~) Us-| Xj * vSj did
job
+ | (VkVj) * VSj k dQ . (30)
Q
NUMERICAL SCHEME
By discretising the continuous integral equation
expressed in equation (29 or 30), the unknown X (or R)
and v can be determined. Apart from the additional
complication of the convolution time integral, the approach
adopted here is similar to the one developed previously to
evaluate solutions of the steady state flow problemEl2,131.
That is, in the spatial idealization, the body boundary ~ b
and fluid domain Q are discretised into mb elemental
surface panels and ma elemental surface panels or volumes
respectively and to assume that the unknowns satisfy
prescribed distributions. Integrations are performed over
each idealised element and the integral equation
transformed into a set of simultaneous algebraic equations
from which the unknowns on each panel and in each
volume (or panel) are determined. Thus equation (30) can
be written as
mb
C(E,) VS (`,t) = unit) - C(~) Us- 2, Xjm) * | Vsj did
m=1 i)
md
+ ~ (VkVj) * | VSj kdQ
m=1 Q(,~
mb
= unto - Ceil Us- ~ Xjm) * | Vsj dI
m=1 an)
md
+ ~ ~Vkvj, *f vsj nkdz
m=1 2(m)
(31)
where );Q(m) denotes the boundary of the mth fluid
domain panel. As can be seen, this expression involves an
integration of the time dependent fundamental solution
over defined surface boundaries and this can be
represented analytically, as shown in the appendix.
Following previously described proceduresEl2], the
continuous integral equation represented by the discretised
form of equation (31) can be expressed at each time step
by general matrix equations written in the form
- A X + B + C(V) = 0
- V - A'X + B'+ C'~V) = 0
} (32)
where X and V are the unknowns to be solved. Although
the individual descriptions of the matrices, A, A', B. B',
C, C' are omitted, the matrices A, A' consist of linear
elements, B contains linear and non-linear (i.e. products of
terms) elements whilst B', C, C' represent totally non-
linear expressions in the unknown V.
In the steady state case, when the Reynolds number
Re << 1, the non-linear contributions are negligible and
can be ignored. Thus the unknowns X and V are obtained
by direct solution from equation (32~. When Re ~ 1 and
the non-linear contributions retained, a simple iterative
scheme can be devised but when the Reynolds number is
large, a more refined iterative scheme is required. This is
based on the Levenberg-Marquardt algorithmEl2,20-22]
which is a hybrid algorithm combining Newton's iteration
method and the method of steepest descent.
COMPUTATIONS
STEADY STATE PREDICTIONS (V = 0)
Since details of the steady state calculations are
described elsewhereLl2,13], they are omitted here and the
example included serves to illustrate the method.
For a cylinder of diameter D' (=2A'), figure 1
illustrates the variation of the drag coefficient Cd
(=Force/0. 5 p'U2D') with Reynolds number Re.
Presented are predictions derived from an oseen
flow-linearised model and a non-linear convective model
as well as experimental resultsEl41. As can be seen, the
linear results overestimate the other two data sets, with the
non-linear results showing the better agreement with the
experimental data.
.
NONL I NERR
- - - - - - L I NERR
X EXP. ~ TR I TTON )
3 _
-
~-
o
LOG ( Re )
~-
Icy - _ _
Figure 1 A comparison between measuredly] and
calculated (i.e. linear and non-linear flow
models) steady state drag coefficient data Cd
for a circular cylinder in uniform flows.
In the linear model, the cylinder's surface was
idealised by a distribution of 100 equally spaced viscous
boundary elements though it was shown that a 10 element
distribution produced similar convergent solutions and
both sets of numerical results compared very well with
Oseen's theoretical prediction as given by LambE23, Re
13. Further, in the limit Re ~ on, Cd ~ 2.28 again
indicating an overestimation of the experimental data.
806
In the non-linear model, 100 equally spaced elements
were distributed over the cylinder's surface as well as a
similar number over an imposed surrounding outer
boundary six diameters (R=6D) away from the surface.
The enclosed fluid domain was idealised by 400 panels
with the distribution showing a greater density near the
cylinder and decreasing radially. Figure 2 shows the
computed flow field around the cylinder (Re = 200) with
the vortex wake pattern clearly defined.
..
==~ ~ ~
~-
Figure 2 Computed Oseen flow field around a circular
cylinder, Re=200.
This investigation showed that calculations of the
fluid actions, pressure distributions over the cylinder's
surface etc. are satisfactorily determined by the non-linear
convective model and the global solutions are relatively
insensitive to the mathematical model and idealization.
However, calculations of the flow velocity field are far
more sensitive to idealization and the truncation distance R
of the surrounding outer fluid domain boundary. The
latter must be placed at a sufficient distance from the
cylinder to allow for the complete generation of the vortex
wake pattern. Therefore, if information on fluid actions is
required only, a much reduced computational model can be
adopted, significantly decreasing the computational effort
with a relatively small numerical error introduced if the
truncation distance R is taken to be some value R > 2D
(say).
NONSTEADY PREDICTIONS (V = 0)
Oseen Flow-Linearised Model
A linearised mathematical model can be developed if
the time dependent fundamental solutions are distributed
only over the cylinder's surface. The convolution time
integration is retained in the modified integral equation of
equation (29) but the troublesome non-linear convective
contribution is discarded.
For the time history predictions presented in figure 3,
the cylinder's surface was idealised by a distribution of 40
equally spaced viscous boundary elements each containing
at its centre a time dependent oseenlet. The co .mponent
velocities us, u2 defined in the body frame of reference are
shown in figure 3(a). That is, the motion in the
longitudinal direction of the body (i.e. surge, ups shows
the body accelerating to a fixed speed, remaining steady
until it experiences a slight blip before returning to the
previous steady condition. At the same time, in the
transverse direction the sway component u2 is sinusoidal.
This simple example serves to demonstrate the arbitrarily
selected motions (e.g. a prescribed manoeuvre) which can
be introduced into the mathematical model without
difficulty, though it bears a similarity to a PMM test
procedure.
The time histories of the drag coefficient CdX (=R~)
shown in figures 3(b-d) follow the variations of the motion
units in each of the Reynolds number flows Re = 2, 40,
200. At each transition the coefficient exhibits an over- or
undershoot tendency whereas the transverse force
coefficient Cdy (=R2) retains the oscillatory behaviour of
the prescribed motion input but with a phase shift
depending on Reynolds number.
in
~6
2
8
L) ~
\ ~
4
I ,'
, \
I,, \
\ , \
, ' ,
so " coo
(C)
In ~an
\ /
Cdu1 (d)
V
/ \
/
\ to
\ /
\ ,,
~\
\ / \
\ ~
\ 720
1!
\ to
\ /
\ /
\ /
\ ,
Figure 3 The time histories of the prescribed manoeuvre
of the circular cylinder and the associated
calculated Oseen drag coefficient CdX and sway
transverse force coefficient Cdy for different
Reynolds number flows. The abscissa denotes
the number of time intervals passed into the
calculation.
(a) the surge units and sway u2(t) motions,
(b) Re=2, (c) Re=40, (d) Re=200.
807
Figure 4 illustrates a series of snapshots of the flow
field behind the cylinder observed from a fixed reference
position. This sequence of frames at 10, 20, 3(),..., 120
times the time interval increment clearly shows the vortex
wake forming and decaying behind the cylinder as it
undergoes the prescribed manoeuvre shown in figure 3a.
The continuous line behind the cylinder indicates the path
of the cylinder, in a flow field associated with a Reynolds
number Re = 40.
10
20
to
30
40
50
60
70
80
Q-
o
i
o o
a
To
-
90
100
1 10
120
of
C:
-
Figure 4 A sequence of snapshot frames at 10, 20, ....
120 times the time interval increment
illustrating the formation of the vortex wake
behind the cylinder associated with the
calculation Re=40 in figure 3.
In both the steady and unsteady oseen flow
calculations, no difficulties were encountered in the
numerical scheme of study for any chosen value of
Reynolds number (e.g. tom, etc). However, it must be
emphasised that from the evidence available the results
derived from this linearised model are expected to
overestimate the experimental data and, as we shall further
show, the results obtained from the non-linear convective
model.
Non-linear Convective Model
In the results presented in figure 5, the non-linear
convective term in the modified integral equation, i.e.
equations (29,30), is included in the numerical scheme of
study. To do so requires distributing viscous panel
elements into the fluid domain and this greatly increases
the computational effort needed to provide solutions. In
fact, this necessitates the full solution of the non-linear
coupled matrix equations in equation (32) whereas the
oseen flow solutions are obtained from a much simpler
linear matrix modelEl2,131.
The cylinder's surface was again idealised by a
distribution of 40 equally spaced viscous boundary
elements, each containing a time dependent oseenlet and in
a similar procedure to the equivalent steady state
calculations, 400 panels were distributed into the fluid
domain contained within an imposed surrounding outer
boundary at R = 6D (say). The prescribed manoeuvre
displayed in figure 3(a) was again chosen and figures
S(a-c) show the calculated force coefficient components as
a function of the number of time step intervals and
Reynolds number Re = 2, 40, 100. It is seen that these
time histories show similar trends to the equivalent linear
predictions in figures 3(b-d) following the variations of the
808
input parasitic motions but, in comparison with the linear
findings, their values are reduced. Note, however, the ~ "
oscillatory variation now evident in the CdX time record , , ~ ~ ~ ~ '
created by the oscillatory sway motion. This again ~, ' , _ - _ ~ ~
illustrates the influence of the non-linear convective term in ~t , ~ _ _ _ ' ~ ~l l
the mathematicalmodel. ~ ~ ' __ ' ~ I, '
Figure 6 shows a limited sequence of pictures " "t If',,_ 'it i, ,
illustrating the creation and decay of the vortex wake flow ~ ~" ~, ~ ~ '~ ~ ,',, ~ I, i, ', ~
field around the cylinder as seen by a fixed observer.' ~ ~ ~ ~'~t ,' - ~ ~ _
16 -- - ' ~ ~ -
,, ~, , ~ \ , ~ _ - / , '
10
\\ // 40 \ ,/ 80 \ ,/1 20
'A ~ ,' " ,
~-` ~ ~ ~ ~ ~t ~
, 40 \ / 80 \ ~120 ~ ~ ~ ~ _ ' / I
20
3 _ _ Cdx I (C)
3, _ ' ~ i'" '. '
~, ,,~tt'x'~-~ hi,, "
Figure 5 For the manoeuvre illustrated in figure 3(a), ~ ~ ,, " `-~ ~ , ' ,
this figure shows the associated drag ' ~ ~ "a-- ,,
coefficient CdX and sway transverse force ' ~ ~" ~ _ - ,
coefficient Cdy for different Reynolds number ~ ~ _ ~
flows calculated from the non-linear convective ' \ -
model. The abscissa denotes the number of '
time interval increments passed into the
simulation. 30
(a) Re=2, (b) Re=40, (c) Re=100.
809
~ ~ - ~ -
t~ Air
it's Ott
/ l l ~- ~"
~- - ~- -
' ~- ~-
40 70
t _ 1 _
t ~_ ' t ~/ ~_ _
~ i~ ' I it ~ ,'- ~ ~
_ - - ; ~1. = _ _
~_ _ ~_
~_ ~_
50 80
~t _ ~_
t t _ t ~ ,
t t _ _ ~ t ~ ,, _
t ~ ~ ', - ~t , t ~ - -
__- As, A= --' -','~gT~J
, - ·,'. Ads ,' , , ~ , ~ ~ ~ ,~ __ _ _
, , ~_ - ~, , , , , ~_
I ~_ , , , ~_ _
_ ~, ~_
' ~_ ,
60 90
810
, ' ' ' - _ Figure 6
\ t ' ' ,
' t , , , ,, _ _
'` ' "`'!,',',,''," 2 "i
J
~ J
J I
t
1
l ~ ~ ~ ~ ~- , _
1 1 ~ ~ ~ ~
~_
_ _
_
-
-
100
t
t t
t t t ~ J
- ~ l ' ~ \~-\ ~- ~t
J l
J l
1
J J ~ ~ ~ ~- 1
1 ~ ~ _ _ ,
_ _ ,
1 ~_
_
_
110
t
t
l
~J ~ l
, ~ ~ ~ \~N _
~ ~ ~ ~ _
l ~ ~ ~ ~ '
~_
120
, ' ~
-
811
A sequence of snapshot frames at 10, 20, ....
120 times the time interval increment showing
the generation and decay of the vortex wake
behind the circular cylinder associated with the
calculation Re=100 in figure 5(c) and also
figure 3(a).
CONCLUSIONS
The hybrid analytical and numerical viscous
boundary element approach using time dependent oseenlets
described herein, allows the predictions of the fluid actions
and flow fields associated with arbitrary shaped bodies
moving in a prescribed manoeuvre in an incompressible,
viscous fluid. Although the method is demonstrated using
a two-dimensional simple shaped body, the concepts
introduced remain valid when tackling three-dimensional
fluid-structure problems. However, in the latter, the
presentation of information - especially a description of the
flow field - becomes more difficult and the computational
effort greatly increases.
The Oseen flow-linearised model produces over-
estimates of the fluid actions and flow fields but it is easy
to apply and, since it provides a 'broad brush' picture of
the fluid-structure interactions, in engineering terms, it
produces a reasonable first insight and solution to the
problem.
The non-linear convective model is computationally
more time consuming though the evaluation of the fluid
actions is obtained from a relatively robust numerical
scheme of study. However, because of the sensitivity of
the flow field calculation to panel idealization, truncation
distance etc., the preliminary calculations presented serve
to illustrate the applicability of the viscous boundary
element approach to evaluate the time dependent fluid
actions and flow fields associated with bodies
manoeuvring in an incompressible, viscous fluid.
ACKNOWLEDGEMENT
We gratefully acknowledge the support of the
Science Engineering Research Council, the Ministry of
Defence Procurement Executive and the encouragement of
the staff at Admiralty Research Establishment (Haslar).
We are indebted to Mrs Christa Steele for her typing (and
retyping) of this manuscript.
REFERENCES
1. Duncan, W.J., The Principles of the Control and
Stability of Aircraft, Cambridge University Press,
Cambridge, 1952.
2. Etkin B. The Dynamics of Flight Wiley New
, . . .
York, 1959.
3. Mandel, P., "Ship Maneuvering and Control,"
Principles of Naval Architecture, (ed.J.P.Comstock),
Society of Naval Architects and Marine Engineers, New
York, 1967, pp. 463-606.
4. Bishop, R.E.D., Burcher, R.K. and Price, W.G.,
''The Uses of Functional Analysis in Ship Dynamics,'
Proceedings of The Royal Society London, Vol. A332,
1973, pp. 23-35.
5. Bishop, R.E.D., Burcher, R.K. and Price; W.G.,
reapplication of Functional Analysis to Oscillatory Ship
Model Testing,ll Proceedings of The Royal Society
London, Vol. A332, 1973, pp. 37-49.
6. Booth, T.B. and Bishop, R.E.D., "The Planar APP
Motion Mechanism," Admiralty Experiment Works, ENDS
Haslar, 1973.
7. Burcher, R.K., "Developments in Ship
Manoeuvrability," Transactions Royal Institution of Naval
Architects, Vol. 114, 1972, pp. 1-32.
8. Clarke, D., "A Two-Dimensional Strip Method for
Surface Ship Hull Derivatives: Comparison of Theory
with Experiments on a Segmented Tanker Model," Journal
Mechanical Engineering Science, Vol. 14, 1972, pp.
53-61.
9. Mikelis, N.E. and Price, W.G., "Calculation of
Hydrodynamic Coefficients for a Body Manoeuvring in
Restricted Waters Using a Three-Dimensional Method,"
Transactions Royal Institution of Naval Architects, Vol.
123, 1981, pp. 209-216.
10. Mikelis, N.E. and Price, W.G., "Calculations of
Acceleration Coefficients and Correction Factors
Associated with Ship Manoeuvring in Restricted Water:
Comparison between Theory and Experiments,"
Transactions Royal Institution of Naval Architects, Vol.
123, 1981, pp. 217-232.
11. Price, W.G. and Tan, M., "A Preliminary
Investigation into the Forces Acting on Submerged Body
Appendages," Proceedings of the Conference on Ship
Manoeuvrability. Prediction and Achievement, The Royal
Institution of Naval Architects, 1987, paper 13.
12. Price, W.G. and Tan, M., "The Evaluation of Steady
State Flow Parameters Around Arbitrarily Shaped Bodies
Using Viscous Boundary Elements," Report 1/89, 1989,
Department of Mechanical Engineering, Brunel University.
13. Price, W.G. and Tan, M., "The Evaluation of Steady
Fluid Forces on Single and Multiple Bodies in Low Speed
Flows Using Viscous Boundary Elements," International
Union of Theoretical and Applied Mechanics Symposium
on The Dynamics of Marine Vehicles and Structures in
Waves, June 1990, Brunei University, (Also Elsevier
Press, 1991~.
14. Tritton, D.J., "Experiments on the Flow Past a
Circular Cylinder at Low Reynolds Number," Journal
Fluid Mechanics, Vol. 6, 1960, pp. 547-567.
15. Thom, A., "The Flow Past Circular Cylinders at
Low Speeds," Proceedings of The Roval Society London,
Vol. A141, 1933, pp.651-669.
16. Fornberg, B., "A Numerical Study of Steady
Viscous Flow Past a Circular Cylinder," Journal Fluid
Mechanics, Vol. 98, 1980, pp. 819-855.
17. Fornberg, B., "Steady Viscous Plow Past a Circular
Cylinder up to Reynolds Number 600," Journal
Computational Physics, Vol. 61, 1985, pp.297-320.
18. Abramowitz, M. and Stegun, I.A., ea., Handbook
of Mathematical Functions, Dover, New York, 1972.
19. Bessho, M., "Study of Viscous Flow by Oseen's
Scheme, (Two Dimensional Steady Flow)," Journal
Society of Naval Architects Japan, Vol. 156, 1984,
20. Levenberg, M., "A Method for the Solution of
Certain Non-linear Problems in Least Squares," Ouarterlv
Journal of Applied Mathematics, Vol. 2, 1944,
pp. 164- 168.
21. Marquardt, D.W., "An Algorithm for Least Squares
Estimation of Non-linear Parameters," Journal of
Industrial Applied Mathematics, Vol. 11, 1963, pp.
22. Twizell, E.H., "Numerical Methods. with
Applications in the Biomedical Sciences," Ellis Horwood
and John Wiley, Chichester, 1988.
23. Lamb, H., "Hydrodynamics," (6th ed.), Cambridge
University Press, Cambridge, 1932.
812
Before writing equation (31) in matrix form, we need
to Integrate the time dependent fundamental solution over
an elemental surface panel or volume. For the two
dimensional problem under discussion this may be
achieved as follows.
Uxt
~,~s
Figure 7 Schematic illustration of the panel, field point,
etc. and symbol definitions.
Figure 7 illustrates a panel with end points (1,2)
lying in the direction of a and ~ is a field point connected
to the source point x on the panel by the vector r = x - (.
The vectors dl, d2 are as shown and the unit normal n
points out of the panel. If we let
r=r+Ut, dl=dl - Ut, d2=d2-Ut
where
and
Because
r= 2 (BaO-£n), 86 [-
r 1 ~^ ~ nnN ~ a n 1 na ~ _ = fi_nn =
r2
O O _ O O _ O O
r
J
~1
-~52 - r 2/4t
e d~
t
= 0 S(~/t)ll2 e- (n dl) /4t [erf (aO d 1/2tl/2) - erf(a~' . d212t 12)]
then the surface integral involving the two-dimensional,
time dependent oseenlet is given by the analytical
expression
2
2~ | vSj d~ = | { t (6 --) e~ r /4t
~1
+2( 6+2rr)(1-e~r /4t)}dD
r2 r4
= ~-(nn-a~aA) + 26 (a n+na^)1 (1-e r2/4t) |
<52
r2 --U U' r2~ V u~-~ -~71
,[ t {~~ - 2 +(aOaO-nn) 2 ~ (aOn+naO)4E~ }dJ3
~1
= 0.5(~/t)l/2 e~ (n dl) /4t{erf (aO. dl/2tl/2
- erf (aO.d2/2t /2) } aOaO
aO. d2 dO2/4t
+ (aOaO-nn){ 2 (1-e
d2
~.
+(aOn+na`}) {-(l_e-d2/4ti
d2
-2
(l-e-d1 /4t)}
)- 2 1 (l-e~d1 / ))
dl
Although this is an unwieldly analytical expression,
it avoids the numerical integration of the oseenlet over the
surface panel but, unfortunately, its form does not readily
permit the convolution integration to be reduced to
analytical expressions.
813
l
DISCUSSION
Gerard Fridsma
General Dynamics, USA
The statement was made about the importance of Reynolds number on
model testing of submerged bodies. While indeed answers are being
obtained on the testing tunnel and towing tank, these are of a
qualitative nature to understand the nature of the flow and production
of loads. For small cross-flow angles (yaw), the loads are linear and
not Reynolds number dependent in this range. As one goes to higher
cross-flows (yaw angles from 10° to 20°), vortices are generated from
the hull which create serious nonlinearities in the trends of the loads.
These must be dealt with for a full maneuvering submerged body
which means Reynolds number must be dealt with, since separation
and vortex generation is very much dependent on Re.
W\\~\~: ~ ~ '
4.5
4ao
AUTHORS' REPLY
We thank Mr. Fridsma for his contribution to this paper and we agree
with his overall observations.
Our comments relating to a submarine hull were based on steady state
calculations performed on a two-dimensional cross-section with sail
plane or fin. This was simply idealized by a circular cylinder and
appendage as illustrated in the following figures. These also display
the steady flow field around the section and show the variation of the
steady drag and lift coefficients with Reynolds number. In the region
lee < 103 (say) the curves tend to flatten indicating that large changes
in the Reynolds number produce relatively small changes in these
steady forces. These preliminary findings may provide a simple
explanation why model scale and full scale submarine experiments
produce a measure of correlation even though true scaling is
impossible to apply in practice.
Although a free running or towed model is geometrically scaled
correctly, it must be of a size to contain the instrumentation
measurement packages and not too large for the towing tank or
maneuvering basin. Thus constant Reynolds number and viscosity
coefficient imply that the model's forward speed must be set at Um =
(LFS/L,,,) UFS, where subscript FS denotes full scale value. Practica
fly, in a towing tank, this relationship is impossible to fulfill and so
a model forward speed is chosen as high as safe powering allows
within the confines of the test facility. Thus the Reynolds number
for the model and full scale differ, but if both experiments are
performed at Reynolds numbers lying within the flat portion of the
drag and lift curves, then it can be expected that steady state
predictions for model and full scale would display reasonable
correlation. In fact, from the values of the steady state forces at the
appropriate Reynolds numbers for a model and full scale submarine
a simple correction factor could be deduced and incorporated into
maneuvering prediction mathematical models.
The calculated fluid flow
field around a body
appendage configuration Elm).
- - - DRRG
~ ~ F T
3.5
v)
an
~ 3.0
-
IL
~ 2.S
to
LJ
~ 2.0
LL
.5
1.0
0.5 /
\
/
\
-
. . .
2 3 4
REYNOLDS NU1118EQS ( LOG(Re) )
s 6
814
The calculated variation in
the lift and drag coefficients
with Reynolds number.
