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OCR for page 607
Numerical Solution of Viscous Flows about Submerged and
Partly Submerged Bodies
P. G. Esposito
Istituto Nazionale di Studi ed Esperienze di Architettura Navale
Roma, Italy
G. Grazian~ and P. Orlandi
University di "La Sapienza"
Rome, Italy
Abstract
In this paper a finite difference method based on the
NavierStokes equations in generalized curvilinear co
ordinates is applied to the simulation of unsteady vis
cous flows with free surface. An accurate analysis of
the method is performed by comparing numerical and
experimental results for several physical cases related
to ship hydrodynamics.
~ Introduction
The flow of a viscous fluid past a body shows a very
complex behaviour when a free surface is involved.
In fact the free surface strongly affects the flow field.
The detailed analysis of the interactions between body
and fluid is very important in the field of naval hydro
dynamics in order to quantify the viscous and wave
resistance. Experimental investigation can provide
global information at reasonable cost, while velocity
and pressure measurement are difficult or even impos
sible and very expensive, instead numerical simulation
gives the complete flow field quantities provided par
ticular care is devoted to implement the numerical
scheme. The validation of the numerical solution by
a comparison with the experimental findings is nec
essary to verify the accuracy of scheme. The best
results, however, are obtained when the two fields of
analysis compenetrate each other.
In this paper a numerical method for the solution
of the NavierStokes equations in general curvilin
ear coordinates based on an implicit finite difference
scheme is presented. The physical domain bounded by
the moving free surface is mapped onto a fixed com
putational domain. The time dependent coordinate
transformation introduces extra terms that account
for the grid velocity. For a complete test of the nu
merical method, the scheme has been applied to dif
ferent physical problems with increasing complexity.
As a first case the 2D flow field past a wallmounted
607
square obstacle in a channel has been considered [1~.
When the geometry is mapped onto a computational
domain, the coordinate lines turn out to be highly
distorted. The comparison with experimental results
suggests that very fine meshes are required for an ac
curate resolution of flow reversal.
To verify the treatment of the time dependent ge
ometry, the flow field inside a rectangular enclosure
with a moving indentation has been studied and the
temporal evolution has been compared with the ex
perimental findings.
The 2D free surface flow over a semicircular bump
on a straight wall has been considered because it is
closely related to ship design. This case is relevant for
the presence of large recirculating regions and of the
free surface. Experimental results are also available
[2~.
2 Governing Equations
The simulation of a two dimensional incompressible
viscous flow has been considered as a first step to
build a numerical scheme for 3D flows. The gov
erning equations are:
, . . ~
ovi
Ski
ovi o
+
tat {3x
~2
s . 't
Ire t3xj[3xj ~ ~
(1)
(2)
where valve are the Cartesian velocity components
and fi = (O.  1/Fr2) is the gravitational force. The
equations are nondimensionalized by the maximum
inlet velocity U and a length L characteristic of each
case examined. The Reynolds and the Froude n~,m
bers are Re = UL/zJ and Fr = U/~fi~, where ~ is the
kinematic viscosity and 9 the acceleration of gravity.
When the free surface is present and is adverted
by the unknown velocity field, the physical domain
varies in time. The numerical solution of the gov
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erning equations describing such complex phenomena
requires an accurate computational scheme. Among
the several possibilities to represent the behaviour of
the free surface, such as the Marker and Cell method
[3], in this paper it has been chosen to map the phys
ical domain (x`) into a Cartesian computational do
main (6j) at each time step. Interpolations to enforce
boundary conditions are thus avoided.
The choice of Cartesian velocity components as
dependent variables allows a simple form of the equa
tions in the (I coordinate system. The adoption of
other velocity components as unknowns require the in
troduction of higher order metric coefficients, as shown
in Ref. A.
Introducing the time dependent coordinate trans
formation xi = xi((j,t) into equations (1) and (2), it
follows
1 8~jVi) _ 1 ski _ o
J B6i J B(i
~ + ~ JO (qjoi) =
__~j p + ~ ~ ~ ak! ~ Vi + fi t4y
J
(3)
where qi = Levi are the fluxes and qi = qi_)j(~` ac
counts for the time dependence of the mapping. The
notation is given in Appendix A.
Eq. (3) is a very important constraint for incom
pressible flows, which is fulfilled in a simple way by
locating the fluxes qi at the cell sides.
The equations for fluxes qs are obtained by a lin
ear combination of (4~. The momentum equations
become
it, ~ ~` LIZ q ~` + ~]j ~5 (q q Ink) =
_o~ij UP + ~ of ~ ~kZ ~ qa~j + pi f j (by
where
Eq. (5) can be rewritten as
aq —R—A2 (qi) = _ pi _ R
Ri = Eli _ Fs + G— ~ (D~ +D2) (7)
the convective term is
the pressure gradient is
Hi = .T1i,'~ (q q Ilk)
pi _

the body force term is
 ails IMP
(8)
(9)
F = ~ f (10)
and the quantity
G' = ~qT ,'`—zip phi (11)
accounts for the time dependence of the grid. The
following definitions have been used:
^2`qi; = ~ ~ ~~ Nikko: tip' no sum over i (12)
and
Ds = ~ Hi ~ C!ki—quip S ~ i (13)
D2 = ~ ~ ~~ 't ~~ q p~ (14)
where /\2 contains only second derivatives of qi along
Ok, while the remaining part of the diffusive term is
accounted for in D' and D2
3 Numerical Mode!
Eq. (6) are discretized on a staggered grid where the
pressure p is defined at the node (i, j), A at (i+1/2, j)
and q2 at (i, j + 1/2~. The discretized equations need
the metric coefficients at (i,j), (i + 1/2,j), (i,j +
1/2) and (i + 1/2, j + 1/2~. These metric quantities
are evaluated by second order centered finite differ
ences. Interpolation of the metric coefficients intro
duces truncation errors A. To avoid such problem,
in the present work, the metric quantities are eval
uated on a grid twice finer than the computational
grid. The governing equations are solved by a frac
tional step method [6,7,8], where a first step yields
the nonsolenoidal field ql.
q —(q ~ _ ~ ^2(qi _ Lint =
_ (, Kiln— L t3(Ri~n _ (Ri~n
(15)
To have a second order time accurate method the Ri
have been discretized by the AdamsBashforth scheme
requiring a restriction on the time step (Courant num
ber < 1 ). The direct inversion of Eq. (15) requires a
large amount of memory and CPU time. The approx
imate factorization scheme t9,10] reduces the number
of operations. It consists on the approximation of the
LHS by the product of two threediagonal matrices.
Eq. (15) becomes:
(1—A1) (1—A2) (qi _ (quiz) =
—i\t(Pi~n _ ~ [3(Ri~n—(Ri~nl] +
+ 2(A1 + A2~(qi~n
Where Ak accounts for the discrete differential oper
60g
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ators of the laplacian along Eke
The velocity field (qs~n+t is obtained by introduc
ing a scalar 4} at the second step,
~qi~n+l = qi _ Id §~ (17)
By applying the discrete divergence operator at
the grid point (i,j), the equation for 4} follows
1 ~ aims 2 {q the kinematic condition
J phi Ads ~/~` Phi (18)
A significant amount of the CPU time of the en
tire computation is devoted to the solution of Eq. (18~.
Direct methods, like LU decomposition, yield a solu
tion within roundoff error, but their application to
very fine grids (N = N' x N2) is impossible both for
memory requirements and for number of operations
CONS. Generallyiterative methods do not efficiently
reduce low wave number errors. For such reason a lin
ear Correction Storage multigrid method [11,12] has
been applied with a reduction of CPU time.
The pressure field p appearing in Eq. (6) is evalu
ated from 4)
2 ( 2Re ~ k—~ t~ Act)
After the second step the overall accuray remains
of O(~\t2~.
4 Boundary Conditions
Noslip boundary conditions have been assumed on
solid walls. An assigned profile of the Cartesian com
ponent vt together with v2 = 0 has been imposed at
the inlet, and the fluxes are derived by Eq. (A.6~.
At the outflow a developed Dow condition
Levi Gina =0 (20)
has been assumed when free surface problems are
treated. The assumption of radiative conditions, that
is done for the other cases, gives rise to instabilities in
presence of free surface. Work is in progress to avoid
this numerical instability.
The NavierStokes equations require one kinematic
and two dynamic conditions at the free boundary. The
balance of the forces between the inner and the outer
fluid gives the dynamic conditions. With zero external
pressure and stresses, the normal and shear stresses
must vanish. These conditions cannot be easily imple
mented by an implicit scheme. Therefore the simpler
conditions of ref. t15] haste been used:
Su' = Thai 0 (21)
For slightly deformed grids the previous conditions are
approximately equivalent to require the vanishing of
normal and tangential stresses.
The kinematic condition enforces the fluid parti
cles to remain on the free surface. If the free surface
configuration is described by the function
z2=~(=l,t) (22)
D(x2—rI) = 0 (23)
yields
`~77 = v2 _ v: 077 = v2—vi C1 (24)
At every time step the free surface is moved us
ing the previous equation. To eliminate the possible
occurrence of instabilities, the new configuration is
smoothed with the same filter used by [13,14,15~.
5 Results
To test the complexities associated with the described
numerical scheme, several cases have been chosen, each
one enlighting a particular complexity. The flow in
a steady domain and the flow in a domain where a
boundary moves with a prescribed law have been con
sidered as test cases before solving the free surface flow
past a semicylinder.
5.1 Flows in steady complex domains
The use of a general curvilinear coordinate system is
the main characteristic of the present method. There
fore to analyze the influence of the grid on the numer
ical solution the flow past an obstacle inside a channel
has been considered. The sketch of the domain and
the mesh obtained by an analytical function based on
a conformal transformation is given in Fig.1. Fig. 2
shows, by a stream function plot, that the numerical
method reproduces the main features revealed in the
experiment of Ref. A. To capture the relevant de
tails at least a 64 x 32 grid with refinements in the
regions of flow reversal is necessary. The reattach
ment length Xr of the main separation bubble, is the
significative parameter of this flow. The numerical
values of Xr, together with the experimental data, are
given in table 5.1 for two Reynolds numbers {,Re = 144
and Re = 288`J. The numerical simulations have been
Re Xr Num. Xr Exp.
144 5.5 6.7
288 9.5 12.1
Table 1: Values of Xr by present work and by Ref. [1]
609
OCR for page 607
done respectively by a 64 x 32 and a 96 x 32 grid. The
coarseness of the grid is the main reason to explain the
disagreement between the numerical and the experi
mental results. A second reason, to a lesser extent,
is ascribed to the fact that the calculation was inter
rupted before a real stationary solution was reached.
As it occurs for the backward facing step, numerical
solutions usually present a very steep growth of X,
within a short time. Later on Xr grows in time very
slowly and the steady state is reached using a large
amount of CPU time. However the obtained results
can be acceptable as a presentation of the method.
A more accurate computation to obtain a physically
interesting solution will be presented in the future.
5.2 Flows in complex domains mov
ing with a prescribed law
In order to verify the accuracy of the numerical scheme
to describe flows driven by an unsteady boundary, the
flow inside a channel with a moving indentation has
been considered. For such a case experimental [16]
and numerical [17] results allow to test the treatment
of the grid time dependence. The numerical results
of ref. [17] have been obtained by a vorticitystream
function method on a very fine grid (1440 x 48~. The
same time dependent law of the indentation consid
ered in Ref. [16,17] has been used. Although this flow
is not relevant to ship hydrodynamics, it has been con
sidered as a test case. A sequence of counter rotating
eddies close to the upper and lower walls and sepa
rated by a wavy core flow is generate downstream the
indentation. The eddies move downstream with a de
fined group velocity. This physical problem has been
solved for Re = 760 arid St = 0.025 and the same
flow time history reported by [17] has been obtained.
The time sequence of istant~neous streamlines is re
ported in Fig. 3 for a sequence of time steps within
one period and it shows the same position, shape and
temporal development of the eddies of Ref. t17~. The
values of the positions of the crests and troughs cor
responding to eddies B. C and D are shown in Fig. 4
in comparison with numerical t17] and experimental
[16] values. In this case the better agreement derives
from the much finer grid (384 x 32) used in this case
with respect to that used in the previous case.
These preliminary numerical results assess that
the numerical model provides good flow simulation in
presence of steady and unsteady irregular domains.
5.3 Free surface flows
free surface flows involve the difficulty of irregular
and time dependent domain, where boundary moves
according to the velocity field. The correct enforce
ment of freeslip conditions gives the deformation of
the free surface which strongly affects the inner veloc
ity field. To our knowledge few experimental results
concerning free surface flows are available. One ex
periment [2] deals with the flow over a semicylindrical
obstacle. We have chosen to verify the validity of the
approximated form of the free surface boundary con
dition specified by Eq. (21~. A proper treatment of
the free surface conditions is very difficult particularly
when implicit schemes are used. In this case the ex
act freeslip conditions require an iterative procedure
which increases the CPU time.
Preliminary results were obtained at lower Re and
higher Fr then those used in Ref. [2] because a coarse
grid was employed. However also in this case the free
surface is largely deformed. The velocity vector plot,
shown in Fig. 5 for Re = 400 and Fr = 0.5, qual
itatively resembles the expected flow field. Also the
dynamic pressure reported in Fig. 6 and the develop
ment of the recirculating regions qualitatively agree.
However, it is to be noted that the flow field simu
lation is not possible up to the steady state because
the free surface conditions are not correctly enforced
when the deformation of the interface becomes rele
vant. A better treatment of such boundary conditions
is required to achieve a complete and accurate flow
simulation.
6 Conclusions
The purpose of this paper is restricted only to the
description of the numerical method, which has been
tested by using coarse grids, certainly larger than those
required by the physics of the flow. This is the main
reason for the discrepancies between the numerical
and experimental results in the two cases firstly con
sidered. More work should be done to define the
best grid transformation for the wall mounted obstacle
case. On the contrary, the case of the flow generated
by a moving indentation shows that the grid time de
pendence is very accurately handled by our scheme.
In fact our results computed on a relatively coarse
mesh are in very good agreement with the numerical
results obtained with a larger number of arid Nina.:
in Ref. [17~.
At the moment the lack of the method to accu
rately describe free surface flows is mainly related to
two intimately connected aspects: the choice of the
grid transformation and the approximations on the
boundary conditions at the free surface. The present
results show that the generation of a grid with co
ordinate lines orthogonal to the boundaries reduces
the truncation errors. If such effect is large for solid
boundaries, the effect is amplified at the free surface.
The generation of coordinate lines nearly orthogonal
610
O ~ _ _ a_ _ _ ~ _ Am
OCR for page 607
to the free surface is more difficult and requires a high
CPU time but avoids the approximations in at the free
surface boundary conditions. In the future our work
will be focused on these aspects and only afterwards
real ship related Bows will be successfully simulated.
Appendix
A Metric Notation
In this section a brief review of the tonsorial notations
used throughout our paper is given.
Let us consider the coordinate transformation be
tween a Cartesian reference system xi and a curvilin
ear system A, being the base vectors respectively e'
and gj. If we introduce the coefficients c, defined as
Ad (Aft)
the Cartesian components vi of a vector v are related
to the contravariant components uj [37
v = vied = ujgj (A.2)
by the transformation
. . .
V$ = CS u] (A.3)
If we consider the inverse transformation c~i then
pi = ~c~~. vi (A.4)
If the vector v introduced above represents a ve
locity field the fluxes qj are defined as
qj = Juj = J(c~'vi =~y~vi (A.5)
and
v = J 1c~qj = I3~.q' (A.6)
where J = detect) is the Jacobian of the transforma
tion, eye = J(c~' and,l'7 = J~c'.
Using the matrix c we can write the metric tensors
as
9ij = gi gj = cici. (A.7)
gij = g' . gj = (c~1)~(c~l)' (A.8)
9 = det(gij) = J2 (A.9)
The gradient and the Laplacian operators in curvilin
ear coordinate are
V¢)i = gi] 0¢
aci ( aci)
(A.10)
(A.ll)
the metric quantities aid which appear in the equa
tions for fluxes are simply given by
Hi} = Jgi] (A.12)
References
[1] Tropez C.D., Gackstatter R. UThe Flow Over
TwoDimensional SurfaceMounted Obstacles at
Low Reynolds Numbers J. Fluids Egg. 107
(1985), 109
Miyata H., Matsukawa C., Kajitani H. "Shallow
Water Flow with Separation and Breaking Wave"
read at The Autumn Meeting of The Society of
Naval Architects of Japan Nov. 1985
Welch J.E., Harlow F.H., Shannon J.P., Daly
B.J. "The MAC method: A computing tech
nique for solving viscous, incompressible, tran
sient Duidflow problems involving free surfaces",
LA 3425, Los Alamos Scientific Labs. (1966)
Levi Civita T., The absolute differential calcu
lus" Dover, 1977
Thompson J.F., Warsi Z.U.A., Mastin C.W.
"Numerical Grid Generation" NorthHolland,
1985
~ Chorin A.J., "A Numerical Method for Solving
Incompressible Viscous Flow Prblems" J. Comp.
Phys. 2 (1967),12
t7] Kim J., Main P. Application of a Fractional
Step Method to uncompressible NavierStokes
Equations" J. Comput. Phys. 59 (1985), 309
t8] Orlandi P., Kim J., Main P. Numerical Solution
of 3D Flows Periodic in One Direction and with
Complex Geometries in 2D" unpublished
t9; Beam R.M., Warming R.F., "An Implicit Finite
Difference Algorithm for Hyperbolic Systems in
ConservationLaw Form" J. Comp. Phys. 22
(1976), 87
[10] Briley W.R., McDonald H., "Solution of the Mul
tidimensional Compressible NavierEquations by
a Generalized Implicit Method" J. Comp. Phys.
24 (1977), 372
611
OCR for page 607
[11] Brandt A. "Multilevel Adaptive Grid Solutions
to Boundary Value Problems" Math. Comp 31
(1977), 330
[12] P.G. Esposito, P. Orlandi "A Multigrid Solver
for the Pressure Equation for Idcompressible
NavierStokes Computations in Curvilinear Co
ordinates" unpublished
[13) LonguetHiggins M.S., Cokelet E.D. ~The Defor
mation of Steep Surface Waves. I. A Numerical
Method of Computation" Proc. Roy. Soc. A 350
(1976), 1
[14] Haussling H.J., Coleman R.M. "Finite Differ
ence Computations Using Boundaryfitted Coor
dinates for Freesurface Potential Flows Gener
ated by Submerged Bodies" in Proceedings of the
2r~d Ir~tern. Conf. or~ Numerical Ship Hydrody
namics, Berkeley, 1977, 221
.
H
[15] Miyata H., Sato T., Baba N. "Difference Solution
of a Viscous Flow with Free Surface Wave about
an Advancing Ship" J. Comput. Phys. 72 (1987),
393
[16] Pedley T.J., Stephanoff K.D. ~Flow along a
Ch~nnel with a TimeDependent Indentation in
One Wall: the Generation of Vorticity Waves" J.
Fluid Mech. 160 (1985), 337
[17~ Ralph M.E., Pedley T.J. "Flow in a Channel
with a Moving Indentation~ J. Fluid Mech. 190
(1988), 337
I I I I I I I I t_1_~=~ J l ~L I I I I I I I I I I I I I I I I I I I l l l l
1 LL I I~ ~ 1 1 1 1 1 T 1 1 1 1 1 1 1 1 T I T 1 1 1 1
I I I I I ~! ~ ~ _ 1 1 1 1 1 1 1 T I I I 1 1 1 1 1 1 1 1 1 1
L I I I ~ Ih ~ I I I rl I T ~ T T I I T T rT 1 ~ I I I
,,,,,, ,~ ~ l · ,~ ,,,, I , I , , , , , I , , I , , , , ~
Fig. 1 A sample 32 x 16 grid for the solution of the Dow past a surfacemounted
obstacle in a channel (h/H = 0.5, l/h = 4)
xr
Fig. 2 Stream function plot for Re = 288; dashed lines correspond to negative
values. The insterval between contour lines is 0.02 for negative values and 0.1 for
positive ones.
612
OCR for page 607
t =0.3
t = 0.4
t=O.S
t=0.6
A
t =0.7
E
t =0.8
t =0.9
t = 1.0
Fig. 3 Instantaneous streamline plots within one period for the Dow in channel
with a moving indentation
613
OCR for page 607
1.00
0.80
0.60 
0.40
0.20 
0.00
1.00
O.BO
0.60
0.40
0.20
0.00
1.00
0.80 
0.60
0.40
0.20 
1 1 1 1 1 1 1 1 1 1 1 0.00
0 1 2 3 4 5 6 7 8 9 10 11 0 1
· Exper.
~ Numer.
Am Pres. Work
x
(a) Eddy B
A//
I'
1 1
6 7 8 9 10 11
· Exper.
Numer.
Pres. Work
0 1
x
(b) Eddy C
1 1
2 3 4 5
x
(c) Eddy D
Fig. 4 Time histories of the eddies B. C and D for the flow in a channel with a
moving indentation
614
~7
l
1 i 1 1
2 3 4 5 6
· Exper.
~ Numer.
+} Pres. Work
1 1
7 8 9
l
10 11
OCR for page 607
 =  = /
 ~ l
~ JO
:?ig. 5 Velocity vector plot at t=0.8, Re = 400, Fr = 0.5 for the flow over a
semicylindrical bump
Fig. 6 Pressure contour map at t=0.8, Re = 400, Fr = 0.5 for the flow over a
semicylindrical bump. The increment between two isolines is O.OS
615
OCR for page 607