Calculation of Transverse Hydrodynamic Coefficients Using Co

Technical note

Calculation of transverse hydrodynamic coefficients

using computational fluid dynamic approach

Amit Tyagi*, Debabrata Sen

Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur-721302,

West Bengal, India

Received 23 November 2004; accepted 30 June 2005

Available online 7 October 2005

Abstract

Computational Fluid Dynamic (CFD) based on Reynolds Averaged Navier–Stokes equation is

used for determining the transverse hydrodynamic damping force and moment coefficients that are

needed in the maneuverability study of marine vehicles. Computations are performed for two

geometrical shapes representing typical AUVs presently in use. Results are compared with available

data on similar geometries and from some of the available semi-empirical relations. It is found that

the CFD predictions compares reasonable well with these results. In particular, the CFD predictions

of forces and moments are found to be nonlinear with respect to the transverse velocity, and therefore

both linear and nonlinear coefficients can be derived. A discussion on the sources of the component

forces reveal that the total force and moment variations should in fact be nonlinear.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: CFD; Linear coefficient; Non-linear coefficient; Sway force; Yaw moment

1. Introduction

Application of computational fluid dynamic (CFD) to the maritime industry continues

to grow as this advanced technology takes advantage of the increasing speed of computers.

Numerical approaches have evolved to a level of accuracy, which allows them to be

applied for practical ship resistance and propulsion computations by industry. In the recent

Ocean Engineering 33 (2006) 798–809

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2005.06.004

* Corresponding author. Tel.: C91 94341 38579; fax: C91 3222 255303.

E-mail address: [email protected] (A. Tyagi).

http://www.elsevier.com/locate/oceaneng

A. Tyagi, D. Sen / Ocean Engineering 33 (2006) 798–809 799

past, CFD is also being applied to determine some of the hydrodynamic coefficients that

are needed in evaluating maneuvering characteristics of marine vehicles.

The hydrodynamic forces and moments on the body results from the distribution of

pressure (p) and shear stress (t) over the surface. The resulting force R can be resolved into

lift (L) and drag (D) forces perpendicular and parallel to the velocity, respectively. The

longitudinal force X along the longitudinal axis and the transverse force Y normal to the

longitudinal axis of the body are related to L and D as follows:

X 0 Z CD cos aKCL sin a;Y 0 Z CD sin a CCL cos a (1)

where X 0, Y 0, CD (drag coefficient) and CL (lift coefficient) are the non-dimensional form of

X, Y, D and L, respectively, non-dimensionalisation by ð1=2ÞrL2V2. a here is angle of

attack defined as the angle between velocity and longitudinal axis of the body. The

transverse force Y can then be determined once CD and CL are known.

Computation of CD and CL can be performed in several ways. For attached flow,

pressure can be determined using inviscid flow theory. Skin friction in a viscous fluid can

be determined from a boundary layer solution coupled with inviscid flow solution.

However, boundary layer solution is not suitable when flow separates, since the solution

tend to “blow up” in the region of flow separation. For the separated flow, therefore, the

complete solution is achieved by Navier–Stokes equation.

The main objective of the present work is to determine the transverse velocity

dependent force and moment coefficients using Reynolds Averaged Navier–Stokes

(RANS) solver. This is equivalent to determining variation of Y 0 and N 0 against a, where

N 0 is the non-dimensional moment about some reference point on the body. Majority of the

available empirical relation provide only the linear damping coefficients, which are

presumed to hold good for small value of a. However, the present computations show that

the nature of these forces and moments are essentially nonlinear and therefore for adequate

representation for these forces and moments a linear model is insufficient even over a

small-velocity range. To gain confidence in CFD computed results, comparison with

available data in literature based on computation and semi-empirical formulae for

geometries of comparable shapes are also reproduced and results are discussed in detail.

2. Computational details

Two geometries have been used in this study. The first model is axisymmetric with

diameter D, a hemispherical nose and a sinusoidal stern with tail radius 0.1 D. The stern

profile is generated using the relation:

r

RZ sin q Z sin

px

2LR

(2)

where, LR is the length of the rear section over which the stern is tapered, RZD/2, x is the

longitudinal coordinate measured from the aft perpendicular and r is radius of the stern

section. This shape and profile, first used by Kempf (White, 1977), is preferred over a

pointed tail since most AUV shapes have a finite diameter tail section

Fig. 1. Profile and midship section of Kempf Model.

A. Tyagi, D. Sen / Ocean Engineering 33 (2006) 798–809800

The second model has the same profile but with an elliptic cross-section of major and

minor axis 2b and 2cZD, respectively. These two geometries are referred as Kempf model

and BlueEyes model in the sequel. Figs. 1–3 show a schematic plot of the two geometries

and stern shape.

Computations for CD, CL and CM are performed for both the geometries over a range of

angle of attack a in deeply submerged condition using the commercially available CFD

solver FLUENT. This code solves the Reynolds Averaged Navier–Stokes equation based

on SIMPLAC (Ferziger and Peric, 2002) algorithm using finite volume method. The

realizable kK3 model (Shih et al., 1995), with an enhanced wall treatment, is used for

modelling the Reynolds stresses. The length L of the vehicle is 1.5 m with LRZ2.5 D and

computations are performed at velocity VZ2 m/s (these numbers are typical to AUV’s).

Using Eq. (1) the non dimensional transverse force Y 0 can be calculated from the

computed values of drag and lift coefficients at various angles of attack. Note that the non-

dimensional moment N 0 is same as CM (all computation are performed with the body in

horizontal plane XY; the transverse force Y 0 and moment N 0 are then the sway force and

yaw moment, respectively).

To determine the hydrodynamic coefficients for the computed force and moment, the

following quadratic models are used:

Y 0 Z Y 0v,v0 CY 0

vv,v0jv0j (3)

N 0 Z N 0v,v0 CN 0

vv,v0jv0j (4)

where, v 0Zsin a. Unlike in many applications, no approximation related to the smallness

of the angle of attack is used, therefore v 0 can be reasonably large. By using appropriate

curve-fitting to the computed data, the coefficients ðY 0v;Y

0vv;N

0v;N

0vvÞ can easily be

determined.

Successful computations of turbulent flows require some consideration during the mesh

generation. Since turbulence (through the spatially varying effective viscosity) plays a

dominant role in the transport of mean momentum and other parameters, one must

ascertain that turbulence quantities in complex turbulent flows are properly resolved if

high accuracy is required. Due to the strong interaction of the mean flow and turbulence,

Fig. 2. Profile and midship section of BlueEyes Model.

Fig. 3. Sketch of stern section and imaginary tail Extension.


the numerical results for turbulent flows tend to be more susceptible to grid dependency

than those for laminar flows. To resolve the regions where the mean flow changes rapidly

and there are shear layers with a large mean rate of strain, one must generate sufficiently

fine mesh.

Three of the parameters characterizing a computational grid are total number of grid

points, location of outer computational boundaries, and minimum spacing (initial spacing

normal to body surface). The minimum spacing is generally based on yC, a dimensionless

parameter representing a local Reynold’s number in the near-wall region. This parameter

is defined as (Schlichting, 1966)

yC Zyu�

n(5)

where, yZdistance from the wall surface, u�Zffiffiffiffiffiffiffiffiffitw=r

p, twZshear stress at the wall, rZ

density and nZkinematic viscosity.

Using flat-plate boundary layer theory, this parameter can be derived as

yC Z 0:172y

L

� �Re0:9 (6)

where, ReZReynold’s number based on body length.

An estimate of the minimum grid spacing can be determined by setting yCZ1 and

solving for the value of y/L using Eq. (6). It should be noted here that the yC value from

Eq. (5) is based on a turbulent boundary layer on a flat plate. Therefore, it is used only as an

estimate in cases where the geometry is not actually a flat plate. The real yC is not constant

but varies over the wall surface according to the flow in the boundary layer.

While no systematic grid-sensitivity study is reported here, computations are

performed using sufficiently fine structured mesh. A schematic of this arrangement is

shown in Fig. 4.

Fig. 4. Schematic of grid structured around the body (not to scale).


3. Results and discussion

The variation of the force and moment against transverse velocity computed for the

Kempf geometries with L/DZ10 (K1) & L/DZ6 (K2) and the BlueEyes (BE) geometry

are displayed in Figs. 5 and 6. In order to have confidence on the present CFD results, it is

necessary to compare these values with available results. Unfortunately, available data on

hydrodynamic coefficients for submerged vehicles is extremely scarce in literature.

Therefore, a direct comparison of the computed derivatives with available results is

difficult. As such, since even for same geometry, the coefficient values obtained from

different sources show large scatter (Roderick, 1993), it can only be said that the estimated

coefficient values presently are at best accurate within some bound. Since the main

objective of the present work is to show that the CFD results are within the acceptable

range and more importantly to show that CFD results essentially predict the forces/

moments as nonlinear variation of the velocity, comparison is made with the limited

available results on axisymmetric bodies as well as with data determined from available

semi-empirical regression based relations. Two of such relations used here are one given

by Clarke based on a combination of slender body theory and regression analysis of ship

data and the low aspect ratio wing theory of Jones, both of them are given in (Lewis,

1989). Both these relations were developed for ship hull shapes. However, for comparable

values of breadth-draft ratios, it is assumed here that these may also be applied within

reasonable accuracy for submerged hulls. This is not unreasonable, since for a slow

moving surface-piercing vehicle for which the free-surface effects are negligible, the flow

over a hull can be approximated within reasonable accuracy by a double hull in infinite

fluid. In the present work, the CFD results are for a slow moving AUV-type configuration

and therefore comparison of results from these relations are expected to be at least

indicative of the range and trend.

A description of the different sources and the associated symbols is given in Table 1

while Table 2 shows the numerical values of the results from these different sources.

Fig. 5. Non-dimensional transverse (sway) force for zero yaw rate.

Fig. 6. Non-dimensional transverse (yaw) moment for zero yaw rate.


The transverse forces and moments can in general be divided in two parts: an inviscid

(potential flow) part, and a part arising from viscous or real fluid effects. At small angle of

attack, there is no boundary layer separation, and the flow remains attached. In this

situation, the body essentially behaves like a lifting surface and the forces/moments can be

Table 1

Sources

Symbols Description

K1 Present computation using FLUENT for Kempf model with L/DZ10

K2 Present computation using FLUENT for Kempf model with L/DZ6

BE Present computation using FLUENT for BlueEyes model with 2b/cZ2.8 and L/2cZ6

A1 Existing CFD results for an axixymmetric body of revolution (6 degree polynomial (Lin et al.,

2004)) with L/DZ10

A2 Existing CFD results for an axixymmetric body of revolution (6 degree polynomial (Lin et al.,

2004)) with L/DZ4

Sub Experimental results for a submarine (Sen, 2000)

EO1 Experimental results for ESSO OSAKA model (Hydronautics Inc. (Post Test) (Roderick, 1993))

EO1A Experimental results for ESSO OSAKA model (Hydronautics Inc. (Pre-Test) (Roderick, 1993))

C(K1) Computation using empirical relation based on Clarke regression (Lewis, 1989) for Kempf model

with L/DZ10 (TZD/2)

C(K2) Computation using empirical relation based on Clarke regression (Lewis, 1989) for Kempf model

with L/DZ6 (TZD/2)

C(BE) Computation using empirical relation based on Clarke regression (Lewis, 1989) for BlueEyes

model with 2b/cZ2.8 L/2cZ6 (TZc)

WT(K1) Computation using empirical relation based on Jones low aspect ratio wing theory (Lewis, 1989)

for Kempf model with L/DZ10 (TZD/2)

WT(K2) Computation using empirical relation based on Jones low aspect ratio wing theory (Lewis, 1989)

for Kempf model with L/DZ6 (TZD/2)

WT(BE) Computation using empirical relation based on Jones low aspect ratio wing theory (Lewis, 1989)

for BlueEyes model with 2b/cZ2.8 and L/2cZ6 (TZc)

Table 2

Transverse hydrodynamic coefficients

Source Y 0v Y 0

vv N 0v N 0

vv

K1 0.0073 0.0363 0.0102 K0.0031

K2 0.0094 0.0767 0.0284 K0.0185

BE 0.0109 0.0454 0.0271 K0.0136

A1 0.0021 0.0519

A2 0.0148 0.1045

Sub 0.0456 0.0736 0.0138 K0.0199

EO1 0.0205 0.0338 0.0079 0.0000

EO1A 0.0157 0.0580 0.0092 K0.0034

C(K1) 0.0107 0.0049

C(K2) 0.0276 0.0153

C(BE) 0.0379 0.0153

WT(K1) 0.0078 0.0039

WT(K2) 0.0218 0.0109

WT(BE) 0.0218 0.0109


determined from inviscid flow theory. These potential flow forces have a linear relation

with angle of attack. Thus it may be stated that the linear coefficients Y 0v and N 0

v in (3) and

(4) represent the force/moments arising from the inviscid part of the flow. At large angles

of attack, however, there exists a cross-flow separation which has a nonlinear relation with

angle of attack. (This nonlinearity may be approximated by quadratic or cubic variation

with angle of attack.) Therefore the nonlinear part of the forces/moments in (3) and (4), i.e.

Y 0vv,vjvj and N 0

vv,vjvj can be considered as forces/moments due to viscous effect of the

flow, which becomes increasingly important as the angle of attack increases. This

therefore suggests that the total force, having a linear and nonlinear component as given by

Fig. 7. Linear component of non-dimensional transverse (sway) force for zero yaw rate: comparison with existing

data.


(3) and (4) have their origin from two different sources: the linear term representing the

inviscid part of the force and the nonlinear term representing the viscous part. The former

dominates at smaller angle of attack while reverse is the case at larger angles of attack.

These two forces, i.e. the linear part ðY 0v,v;N 0

v,vÞ and the nonlinear part ðY 0vv,vjvj;N 0

vv

,vjvjÞ are shown separately in Figs. 7–10.

In order to discuss the magnitude and trend of the coefficients as determined from CFD,

consider the semi-empirical expression of Clarke for the sway-velocity derivatives:

Y 0v Z p

T

L

� �2

1 C0:40 CB

B

T

� �; N 0

v Z pT

L

� �2 1

2C2:4

T

L

� �(7)

The above indicates that for a given T/L, Y 0v increases with B/T. In the present

computations, both BE and K2 models have same T/L but the former has a larger B/T,

according to (7), Y 0v for BE should be larger than for K2. This trend is seen in Fig. 7.

Similarly, comparing the models K1 and K2, since K2 has relatively larger T/L, Y 0v for this

should be larger. This characteristic is also reproduced in the computations.

With regard to the nonlinear coefficient Y 0vv, if this is considered to be a result of viscous

effect related to cross-flow separation, than the magnitude of this should be larger for the

model where the separation occurs relatively upstream (in the cross-sectional plane). The

BE model with elliptic section clearly will have a separation point moved relatively further

downstream compared to a circular cross-section, and therefore the cross-flow drag for the

circular cross-section is expected to be larger. This is what is found in Fig. 9.

Consider now the moment coefficients. It is well known that any shape other than a

sphere generates a moment when inclined in an inviscid flow while the force generated is

zero (according to D-Alembert’s paradox). The Munk moment (Newman, 1977) is always

destabilizing as it tends to turn the vehicle perpendicular to the flow. On the other hand, for

Fig. 8. Linear component of non-dimensional transverse (yaw) moment for zero yaw rate: comparison with

existing data.

Fig. 9. Non-linear component of non-dimensional transverse (sway) force for zero yaw rate: comparison with

existing data.


viscous flow over the body, a boundary layer is formed which eventually separates over a

region near the trailing edge. In this region, a helical vortex forms and convects

downstream. While this causes an additional drag, the nature of the resulting moment is

generally stabilizing and thus opposite to the nature of the Munk moment. These two

competing moments eventually decide the overall direction of the moment. As can be seen

here, the linear coefficient N 0v and the nonlinear coefficient N 0

vv show opposite sign, and

therefore support the conjecture that the former (the linear part) is the Munk moment

Fig. 10. Non-linear component of non-dimensional transverse (yaw) moment for zero yaw rate: comparison with

existing data.


arising from inviscid flow effect, and the nonlinear part represents the effect due to

viscosity.

The viscous effects are expected to be relatively less for the BE model for the same

reason mentioned in connection to force: due to the fact that the flow separation occurs

relatively further downstream in the cross plane. Thus, similar to the force case, absolute

magnitude of N 0vv for BE should be less than that for K2 (both having same T/L ratio). This

is evident in Fig. 10. With regard to the linear coefficient, the empirical relation (7)

indicates that this coefficient depends only on T/L but not on B/T. This nature is clearly

reproduced by the present computations: both BE and K2 models having same T/L but

different B/T show almost same value as shown in Fig. 8. Comparative results for models

K1 and K2 also show the trend according to (7): model K1 with smaller T/L has a lower

value.

With respect to the magnitudes of the coefficients, it is found that the coefficients are

generally of the same order of magnitudes as from other source (see Table 2) except for the

linear transverse coefficient Y 0v, which is generally smaller in the present computations.

The total forces compare fairly well, as can be seen from Figs. 11 and 12 where the total

force and moments from all sources are plotted, although the transverse force in general

appear to be on the lower side and the moment appear to be on the higher side of the

available results. To be more specific, the computed N 0v=Y

0v appears to be larger than the

N 0v=Y

0v from the other sources. For example, the semi-empirical relations (Clarke and Jones)

give this ratio as about 0.5 (i.e. a lever arm of 0.5L, NZ0.5Y!L), while present

computations show this to be of the order of 2–3L indicating that there can be no

physically possible lever arm. A possible explanation of this is as follows: at small angles

of attack where the inviscid nature of the flow dominates, the transverse force is generally

small and the moment is predominantly Munk moment. This Munk moment is a pure

couple, having no associated lever arm. In other words, there is a finite moment, but the

associated Y force is small (theoretically zero), and therefore N 0v=Y

0v can be large. At large

Fig. 11. Non-dimensional transverse (sway) force for zero yaw rate: comparison with existing data.

Fig. 12. Non-dimensional transverse (yaw) moment for zero yaw rate: comparison with existing data.


angle of attack, however, the viscous effects in both for forces and moment begin to

dominate, and therefore the ratio of the two (the total N moment and the total Y force)

should now reduce. This means, the linear coefficient Y 0v, which represents inviscid part of

the transverse force and is the main contribution to total force at small angles of attack, is

as such small and therefore the presently computed Y 0v are somewhat smaller than those

from existing literature. However, present computations show a more dominating

nonlinear part (and a somewhat greater nonlinear nature of the force) at higher angles of

attack.

4. Concluding remarks

In this note, a Reynolds Averaged Navier–Stokes equation based CFD solver is used to

determine the linear and nonlinear sway-velocity derivatives for some typical submerged-

vehicle geometries. Results are compared with available data for similar geometries. It is

seen that the CFD computations are capable of producing the force and moment variations

within reasonable accuracy. The magnitude and trend of the results appear to conform to

the existing data. In particular, the results show that the force and moment variations are

essentially nonlinear and therefore both linear and nonlinear coefficients can be derived

from them. This is particularly useful, as majority of the regression-based semi-empirical

formula or formula derived from slender body or thin-wing theory only provide estimate

for the linear coefficients. It may also be noted that the experiments such as PMM from

which both linear and nonlinear derivatives can be determined, are generally conducted

covering a range of small angle of attack. From this point, the CFD computations can be

performed even for a relatively wider range of angle of attack as no assumption with

regards to the smallness of angle of attack need to be used here in getting the coefficients.

Some other advantage of CFD computation is that it can be applied for determining


the coefficients under different conditions such as in shallow water and in the vicinity of

other boundaries like sidewalls.

References

Ferziger, J.H., Peric, M., 2002. Computational Method for Fluid Dynamic, 3rd ed. Springer, Berlin.

Lewis, E.V., 1989. Principles of Naval Architecture, Volume III: Motions in Waves and Controllability. The

Society of Naval Architects and Marine Engineers, New York, USA.

Lin, C.W., Perival, S., Gotimer, E.H., 2004. Viscous Drag Calculation for Ship Hull Geometry (Technical report),

Design Evaluation Branch, David Taylor Model Basin, MD USA.

Newman, J.N., 1977. Marine Hydrodynamic. MIT Press, Cambridge, Massachusetts, USA.

Roderick Barr, A., 1993. Review and comparison of ship maneuvering simulation methods. SNAME Transaction

101, 609–635.

Schlichting, H., 1966. Boundary Layer Theory. McGraw-Hill, New York, NY.

Sen, D., 2000. A study on sensitivity of maneuverability performance on the hydrodynamic coefficients for

submerged bodies. Journal of Ship Research 44 (3), 186–196.

Shih, T.H., Liou, W.W., Shabbir, A., Yang, Z., Zhu, J., 1995. A new—eddy-viscosity model for high reynolds

number turbulent flows—model development and validation. Computers Fluids 24 (3), 227–238.

White, N.M., 1977. A Comparison Between a Single Drag Formula and Experimental Drag Data for Bodies of

Revolution, DTNSRDC Report 77-0028.

Calculation of Transverse Hydrodynamic Coefficients Using Co

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