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The Experimental and Numerical Study of Flow Structure and
Water Noise Caused by Roughness of the Body
Lijin Gao and Lian-di Zhou
(China Ship Scientific Research Center, P.O. 116, Wuxi, Jiangsu 214082, China)
ABSTRACT
It is well known that eddies shed from rough surface is a main source of water noise. In this paper, the water noise
caused by the roughness of the surface of the foil (NACA0020) is numerically and experimentally studied. In the
numerical simulation, a mathematical roughness, which comes from the no-slip condition of vortex method, is introduced.
The numerical results indicate that the vortex's shedding could be avoided under some mathematical roughness. It
corresponds to forward multi-layer in dynamic phenomena. The experiment tried to find how to realize the mathematical
roughness in physics. The experiment results showed that certain riblets or slots on the surface could be related to
different kinds of mathematical roughness.
KEY WORDS: mathematical roughness, FMM, forward multi-layer, drop off, breakdown
INTRODUCTION
The quieter submarine is very important nowadays. So how to decrease the noise of the submarine becomes more
and more important. As engine and other noises have been reduced in these days, the internal noise levels of many
speeding submarines are significantly affected by the water noise. The main sources of water noise are as follows: 1)
pressure fluctuations in the turbulent boundary layer; 2) turbulence-excited wall vibrations; 3) eddies generated by surface
roughness; 4) eddies shed at tail fins.
Drag and water noise all consume energy, so how to reduce drag and lower water noise are the same thing in the
essence of consuming energy. Surface roughness has been used to reduce the drag for many years. But few people make
relation of surface roughness with water noise. Now there exists a new way to reduce drag and lower water noise, which
arranges slots and riblets on the surface of the body.
Study of viscous drag reduction using riblets has been a field of significant research during the last decade. Drag
reduction of as much as 4–8% has been measured in a variety of simple two-dimensional flows at low speed, see e.g.
Coustols & Savill (1992), Choi (1989), or Sundaram (1992). Riblets with symmetric V grooves (height equal to spacing)
with adhesive-backed film manufactured by the 3M Company have been widely used in most early work that has revealed
enormous consistency in the degree of drag reduction observed as well as many aspects of flow structure. Coustols (1989
& 1991) reported his results of drag reduction of 2.7% on LC100D airfoil and 6% on an ONERA D swept airfoil model at
α=0deg respectively. Caram & Ahmed (1992) studied the near and intermediate wake region of a NACA0012 airfoil
covered with 3M riblets at α=0deg. Total drag reduction determined from the wake survey indicated a
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maximum of 13.3%. Recent measurements by Viswanath & Mukund (1995) on a supercritical airfoil covered with 3M
riblets have showed skin friction drag reduction of 6–12% (in the α range of −0.5–1deg) at transonic Mach number.
But all the above studies are focused on testing and drag reduction, there is little efforts on theory and lowering water
noise. In this paper, how to lower the water noise level by surface roughness is investigated. The concept of mathematical
roughness is introduced, which deduced from the no-slip condition of vortex method. The results show that mathematical
roughness could keep in the vortex's shedding, and it can be considered as forward multi-layer in dynamic phenomena
(Gao Lijin & Zhou Liandi 2000). But how to realize the mathematical roughness in physics is a challenge. The
experiment results show that certain riblet or slot on the surface has some relation with the mathematical roughness.
In this paper, the influence of the roughness is investigated. An adaptive numerical scheme, based on vortex method,
is used to solve a time-dependent full Navier-Stokes equation. The analyses of flow field data and sound field data
indicate that there are some strong relations among the vortical motions and structures associated with the surface
roughness. The change of surface roughness might be a new way to lower water noise.
NUMERICAL METHOD
Fast vortex method
The flow was governed by unsteady two-dimensional incompressible Navier-Stokes equations, which can be
expressed in the vorticity transformation as:
(1)
where u(x, t) was the velocity, ω was the vorticity, v denoted the kinematics viscosity, Ub was the velocity of the
was the infinite velocity, D was the domain of the flow and ∂D denoted the boundary of the domain. Using the
body,
definition of the vorticity and the continuity ( ∇ · u=0), it could be shown that u is related to ω by the following Poisson
equation:
(2)
The velocity-vorticity formulation helped in eliminating the pressure from the unknowns of the equations. However,
for bounded domain it introduced additional constrains in the kinematics of the flow field and required the transformation
of the velocity boundary conditions to vorticity forms. In this paper, equation (1) was discreted in a Lagrangian frame
using particle (vortex) methods. The following formulation was used:
(3)
where, xa was vorticity-carrying fluid elements. In vortex method, the vorticity field was considered as a discrete
sum of the individual vorticity fields of the particles, having core radius ε, strength Γ(t) and an individual distribution of
vorticity η(x, t), so:
(4)
Here η(x, t) was defined as a Dirac-Delta function in a board sense on the nonstandard analytical space, the integral
of this function over the full space must be 1. In vortex method, at last the following equation would be solved:
(5)
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Here, U0 was the solution of the homogeneous equation with the no-through boundary condition; K(Z)=Z/|Z|2; G(x,
t) was the Green's function kernel in an unbounded domain; H(xa, t) was the influence function of the body surface; K1(x,
t) and G1(x, t) were the convolutions of K and G with η. In vortex method, the no-slip boundary condition accounted for
the generation of vorticity on the surface of the body. The surface of the body acted as a source of vorticity and the task
was to relate this vorticity flux on the surface of the body to the no-slip condition. In the present formulation, equation (5)
were not integrated simultaneously but instead of a fractional step algorithm was employed. We solved successively for
the convective and the viscous part of the equations. At each sub-step the relevant kinematic (no-through) and dynamic
(no-slip) boundary conditions were enforced. The more information about vortex method could be seen in Chorin's (1989)
and Koumoutsakos's (1994, 1995, 1996).
The straight forward method of computing the right-hand side of (5) required O(N2) operations for N vortex
elements. If N was very large, it must be a time-consuming procedure. This was the N-body problem, which was how to
calculate the interactions among the N discrete points in a system. It was very important in the fields of molectroics,
celestial mechanics and so on. Traditionally, it required O(N2) calculation operations, Barnts-Hut introduced the concept
of tree and reduced the computation to the order of O(NlogN). Now the FMM (fast multi-pole method) method was
introduced in this paper, and it reduced the work to O(N).
The vortex method based on the FMM was the main frame of our scheme, called fast vortex method. Note that the
number of particles N was a function of time. As the particles adapt to resolve the vorticity field that was guaranteed
around the surface of the body was subsequently convected and diffused. New computation elements were added as
needed.
Mathematical roughness
The charge of the roughness was enforced in the second sub-step of the fractional step. In this paper, the effect of the
flow structure, vortex structure near the surface of the body and the water noise are observed when the roughness on the
surface of the body was changed.
After the enforcement of the no-through boundary condition, a tangential vortex sheet on the surface of the body
could be observed, and the strength was γ(s). The mathematical roughness was introduced by the following formula:
(6)
here, ∂ ω/∂n was the normal vorticity flux on the surface the body; δt was the time step of the fractional method; α(s)
was the function which embodied the influence of the roughness. The function α(s) could be continuous function,
constant, discrete points and so on. In general, different values of α(s) expressed different roughness of the surface of the
body. It was briefly described below:
0: Slip boundary condition;
0~1: Partial-slip boundary condition;
1: Standard no-slip boundary condition;
>1: Backward multi-layer slip boundary condition;
<0: Forward multi-layer slip boundary condition;
Hydroacoustic theory
When the Mach Number M=u/C0≪1, the Powell equation could be used to predict the water noise:
(7)
It was a typical hyperbolic equation. Water noise in far field could be gotten.
EXPERIMENT
The experiment was carried on in water tunnel (Fig. 1), the model is a NACA0020 airfoil with a chord 131mm. Two-
dimensional LDV was used to measure
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the velocity, BK8103 and BK8105 were used to measure the pressure fluctuation to get the water noise.
Fig. 1: Equipment of the Experiment
In order to investigate the influence of the roughness of the surface and how to realize the mathematical roughness in
physics, two models were considered. One model had normal surface, the other had rough surface on which some little
cylinders of 2–3mm diameter and 1–2mm height were distributed orderly. The rough surface model could be shown in the
Fig. 2:
Fig. 2: Model of Roughness Surface
RESULTS
To investigate the influence of roughness, four cases were calculated under the same Reynolds number about 105. In
the experiment, horizontal velocities were measured by LDV at several velocity sections. The horizontal velocities were
given in this paper at these same sections. The numerical and experimental horizontal velocities were compared in Fig. 3-
Fig. 6. In these figures, (a) described the numerical velocity field, (b) described the experimental velocity field.
Fig. 3: Velocity field compared at 00 attack angle (full wing with normal roughness)
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Fig. 4: Velocity field compared at 00 attack angle (full wing with relative roughness)

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Fig. 5: Velocity field compared at 150 attack angle (full wing with normal roughness)
Fig. 6: Velocity field compared at 150 attack angle (full wing with relative roughness)
In Fig. 3, a 00 attack angle foil with normal roughness (no-slip boundary condition) was investigated. The results
show that the computed velocity field agrees with the experimental ones. In Fig. 4, a 00 attack angle foil with relative
roughness, which was used in experiment, was investigated. The mathematical roughness of α(s)=1.5 corresponded to the
relative roughness in the experiment. The boundary layer thickened because of the relative roughness under 00 attack
angle. In Fig. 5, a 150 attack angle foil with normal roughness (no-slip boundary condition) was investigated. The results
show that the computed velocity field agrees with the experimental results. In Fig. 6, a 150 attack angle foil with relative
roughness, which was used in the experiment, was investigated. The mathematics roughness of α(s)=1.5 corresponds to
the relative roughness in the experiment. The results indicated that the seperation of the flow was strengthened and the
seperated position moved ahead. It means that the positive mathematical roughness (backward multi-layer slip boundary
condition) made the breakdown of vortex appear early. But the numerical flow field did not agree with the experiment
well. There were some problems in the experiment.
Fig. 7–Fig. 10 describe the vortex structure in four different cases. In these figures, (a) and (b) describe the unsteady
vortex structure of same case.
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Fig. 7: Vorticity contours at 00 attack angle with α(s)=1.0

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Fig. 8: Vorticity contours at 00 attack angle with α(s)=1.5
Fig. 9: Vorticity contours at 150 attack angle with α(s)=1.0
Fig. 10: Vorticity contours at 150 attack angle with α(s)=1.5
In Fig. 7 and Fig. 9, a Karman vortex structure was observed under normal roughness under 00 and 150 attack angle
repectively. In Fig. 8, it shows that the vortex structure was decreased under mathematical roughness of α(s)=1.5 at 00
attack angle. In Fig 10, the mathematical roughness of α(s)=1.5 made the vortex strengthen and the breakdown of vortex
appear early.
Fig. 11 describes the numerical water noise under 150 attack angle in different mathematical roughness. The positive
mathematical roughness increases the water noise.
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Fig. 11: Water noise under 150 attack angle
CONCLUSION
The numerical and experimental results show that the introduced mathematical roughness can be corresponded by
the real physical roughness and could be used as useful tool in investigation of reducing noise. The positive mathematical
roughness makes the water noise enhanced, and a negative mathematical roughness might reduce the water noise (Gao
Lijin & Zhou Liandi 2000). How to realize the negative mathematical roughness in physics will be further studied.
REFERENCE
E.Coustols & A.M.Savill, ‘Turbulent Skin Friction Drag Reduction by Active and Passive Means', Special Course on Skin Friction Drag Reduction,
AGARD Rept. 786, London, pp. 8.1–8.80, March 1992
K.S.Choi, ‘Near Wall Structures of a Turbulent Boundary Layer with Riblets', Journal of Fluid Mechanics, Vol. 208, pp. 417–458, 1989
S.Sundaram & P.R.Viswanath, ‘Study on Turbulent Drag Reduction Using Riblets on a Flat Plate', National Aeronautical Lab., NAL Rept. PDEA
9209, Bangalore, India, Nov. 1992
E.Coustols, ‘Control of Turbulence by Internal and External Manipulators', Proceedings of 4th International Conference on Drag Reduction, A.M. Savil
ed., Applied Scientific Research, Vol. 46, No. 3, Kluwer Academic, Norwell, MA, 1989
E.Coustols, ‘Performance of Internal Manipulators in Subsonic Three Dimensional Flows', Recent Developments in Turbulence Managements,
K.S.Choi ed., Vol. 6, Kluwer Academic, Norwell, MA, pp. 43–64, 1991
J.M.Caram & A.Ahmed, ‘Development of Wake of an Airfoil with Riblets', AIAA Journal, Vol. 30, No. 12, pp. 2817–2818, 1992
E.Coustols & V.Schmitt, ‘Synthesis of Experimental Riblet Studies in Transonic Conditions', Turbulence Control by Passive Means, edited by
E.Coustols, Kluwer Academic, Dordrecht, The Netherlands, pp. 123–140, 1990
P.R.Viswanath & R.Mukund, ‘Turbulent Drag Reduction Using Riblets on Supercritical Airfoil at Transonic Speeds', AIAA Journal, Vol. 33, No. 5,
pp. 945–947, 1995
Gao Lijin & Zhou Liandi, ‘the Numerical Study of the Influence on the Flow Structure caused by Roughness of the Cylinder', Journal of
Hydrodynamics, Ser. B, to be published
Gao Lijin & Zhou Liandi, ‘The Numerical Study of Water Noise Caused by Roughness', 4th International Conference on Hydrodynamics, Yakahama,
Japan, Sep. 2000
A.J.Chorin, ‘Computational Fluid Mechanics Selected Papers', Academic Press Inc., 1989
Ding Li & Allen T.Chwang, ‘Time domain analysis of ship-generated waves in harbor using a fast hierarchical method', ISOPE'97, Hawaii, 1997
Gao Lijin & Zhou Lian-di, ‘Numerical Study of the Mechanism of Water Noise Generation using the Fast Vortex Method', ICHD'98, 1998
L.Greengard & V.Rokhlin, ‘A fast algorithm for particle simulations', Journal of Computational Physics 73, pp. 325–348, 1987
P.Koumoutsakos, A.Leonard, F.Pepin, ‘Boundary Condition for Viscous Vortex Methods', Journal of
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Computational Physics (1994a), Vol. 113, pp. 52–61, 1994
P.Koumoutsakos & A.Leonard, ‘High-resolution simulations of the flow around an impulsively started cylinder using vortex method', J. Fluid Mech.,
Vol. 296, pp. 1–38, 1995
P.Koumoutsakos & D.Shiels, ‘Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate', J. Fluid Mech.,
Vol. 328, pp. 177–227, 1996
A.Powell, ‘Theory of Vortex Sound', J. Acoustic. Soc. Am. 36, pp. 177–195, 1969
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Dr. Ulderico “Paolo” Bulgarelli
Institute Nazionale per Studied Esperienze di Architettura Navale
Via di Vallerano 139
00128 Rome
Italy
Tel: +39–6–50299–270/222
Fax: +39–6–5070619
E-mail: Surfer@rios3.insean.it
upbulgu@rios6.insean.it
Prof. Luis Perez-Rojas
Escuela Tecuica Superior de Ingenieros Navales
Avda Arco de la Victoria S/N
28040 Madrid
SPAIN
Tel: +3491–3367154
Fax: +3491–5442149
E-mail: lperezr@etsin.upm.es
Prof. Toshio Suzuki
Osaka University
Department of Naval Architecture & Ocean Eng.
2–1 Yamadaoka, Suita
Osaka 565–0871, Japan
Tel: +8168797579
Fax: +8168797594
E-mail: Suzuki@naoe.eng.osaka-u.ac.jp
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DISCUSSION
Luis Perez Rojas
Escuela Tecnica Superior de Ingenieros Navales
I really congratulate to the authors for this interesting paper that is dealing with a topic, the roughness, that need a
very deep study as it was stated in 22nd. ITTC (1). Nevertheless, I would like to indicate some comment and questions.
The roughness is represented in this paper by the function α(s) and its action is expressed in the boundary condition,
equation (6) of the paper. Do the authors think that this expression is enough for representing the effect of roughness
changing the velocity and turbulence distribution near the body surface as it was stated by Patel (2)?. Do you think that
this model is capable of describing the three roughness regimes, namely, hydrodynamically smooth, transitional and full-
rough surfaces?.
The figure 3 of the paper represents the standard no-slip boundary condition, both figures (a) and (b) seems not
verify this condition. Are the measurements done in the experimental work only outside the boundary layer?.
In order to evaluate the validation of the numerical calculation with the experimental work, the uncertainty
assessment must be done as it is indicated by the Resistance Committee of the ITTC (1), based in the AIAA Standard (3).
In this aspect, has been done any uncertainty analysis not only in the experimental data but also in the numerical results?.
In the conclusion of the paper, the positive mathematical roughness is related with an increasing water noise what
can conduct to a decreasing water noise through a negative mathematical roughness. Can it be considered that this
negative mathematical roughness is related to smoother surfaces than what is defined by the authors as normal roughness?.
REFERENCES.
1. ITTC, “Report of the Resistance Committee”, Proceedings of the 22nd. International Towing Tank Conference, Seoul, Korea & Shanghai, China,
1999, pp. 174–181.
2. Patel, V.C. “Flow at High Reynolds Number and over Rough Surfaces—Achilles Heel of CFD”. ASME J. Fluid Engineering, Vol. 102, 1998.
3. AIAA, “Assessment of Wind, Tunnel Data Uncertainty”. AIAA S-071–1995.
AUTHOR'S REPLY
Thanks for Professor Luis Perez Rojas's comments.
1. Yes, the equation (6) can be used to represent the effect of roughness as Patel stated. By using proper vortex
layers near wall, the model can describe difference surface roughness regimes. As Patel's definition, different y
+ represent different surface roughness, which determine the strength of the vortex layer near wall. So we can
get the following conclusion that, hydrodynamically smooth regime:
transitional and full-rough surfaces:
2. One-dimensional LDV was used to measure the horizontal velocity, which could not get the velocity in the
boundary layer. We only compared the velocity at the points where the experimental velocities were achieved.
3. Because of the time, we haven't made uncertainty analysis about our numerical method. However we will do
this work latter.
4. The negative mathematical roughness is not related to smoother surfaces than normal rough surface. The
smoother rough surfaces correspond to 0<α(s)<1. The negative mathematical roughness will add extra energy
to the flow field. It may correspond to blowing or suction surfaces or any other energy-adding surfaces. But
the corresponding relationship and the realization of these surfaces should be studied further.
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