Viscous Drag Calculations For Ship Hull Geometry

Cheng-Wen Lin, Scott Percival, and Eugene H. Gotimer
Design
Evaluation Branch, Hydromechanics Directorate, David Taylor Model Basin,
Carderock Division Naval Surface Warfare Center, Bethesda, MD USA.

Abstract

Introduction

Axisymmetric
Body of Revolution

Geometry

Grid
Parameters

Surface Ship
Hullform

Multi-Zonal
Flow Solver

Results and
Analysis

Axisymmetric
Body of Revolution

Without
Angle of Attack

With Angle
of Attack

Effect of
Grid Parameters

Surface
Ship

Conclusions

References

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Table of Contents

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1.0 Abstract

This paper focuses on viscous drag computations based on Reynolds-averaged,
Navier-Stokes (RANS) equations. Computations were performed on axisymmetric
bodies of revolution at angles of attack ranging from 0 to 18 degrees.
Calculations were also performed on three surface ship hullforms. The results
were compared with experimental data and were found to be in good agreement.

A parametric study was performed on the sensitivity of the flow solutions to
changing grid parameters. Minimum grid spacing, distance from the body to the
outer grid boundary, and number of grid points were examined as parameters. The
results were equally as sensitive to the number of grid points as to the minimum
spacing. Extending the location of the outer boundary had a lesser effect on the
result. This investigation was based on a single flow solver, ISFLOW, and may
not be directly applicable to other RANS codes.

2.0 Introduction

Applications of computational fluid dynamics (CFD) to the maritime industry
continue 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 used during the design process to predict ship
resistance. Significant progress has been made in predicting flow
characteristics around a given ship hull. Ship designers can use this
information to improve a ship's design. However, not much effort has been
dedicated to determining viscous drag, an important element in the development
of a new design.

Prediction of drag on a ship hull is always a challenging task for a naval
architect. At the start of the design process, hull forms are developed given
certain requirements. One of the major design tasks is to estimate the powering
performance so that propulsion requirements can be determined. Early estimates
of resistance and power are often based on simple empirical formulas derived
from data for similar ships. As the design process proceeds, a more reliable
approach becomes necessary to predict resistance; scale-model testing has been
generally adopted for this purpose.

The accuracy of a prediction can be assessed by comparison with model-scale
experimental results. Full-scale ship resistance prediction is the final goal.
Although CFD methods can be used to calculate full-scale ship resistance, it is
quite difficult to assess the accuracy of these results without a reasonable
database of full-scale resistance. Such a database is not currently
available.

It is beyond the text of this paper to consider full-scale calculations. The
focus of the paper is to examine the effects of several uncertain/vague factors
in CFD technology concerning grid generation: minimum grid spacing, grid
distribution, and location of outer grid boundaries.

3.0 Axisymmetric Body Of Revolution

3.1 Geometry

The first set of bodies studied was a systematic series of
mathematically-defined bodies of revolution. Axisymmetric bodies are ideal
candidates for a parametric study with their easily defined geometry,
straightforward grid generation, and available experimental data.

Each body was defined by a sixth-degree polynomial [1]. Six axisymmetric
bodies were generated with length-to-diameter ratios (L/D) ranging from four to
ten. All six bodies were evaluated at zero angle of attack. The two bodies with
a L/D of four and ten were also evaluated at angles of attack up to 18 degrees.
(See Table
1.)

Table 1. Configurations examined for bodies of
revolution.

3.2 Grid Parameters

A computational grid for each body was generated by a transfinite
interpolation routine [2]. 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). Prior experience with RANS computations on similar
geometry usually guides the determination of these parameters. The intent of
this study is to qualify these parameters, as much as possible, and to begin to
develop some guidance for their selection.

The minimum spacing is generally based on , a
dimensionless parameter representing a local Reynold's number in the near-wall
region. This parameter is defined as

,
(1)
where y = distance from wall surface,
u* =
frictional velocity,
= shear
stress at the wall,
=
density, and
=
kinematic viscosity.

Using flat-plate boundary layer theory [3], this parameter
can be derived as

,
(2)
where L = body length, and
Re = Reynold's number based on body
length.

An estimate of the minimum grid spacing can be determined by setting =
1 and solving for the value of using equation (2). It should
be noted here that the value
from equation (2) 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
actual value of for the
hullform is obtained with the viscous flow solution. Furthermore, the real is
not a constant but varies over the wall surface according to the flow in the
boundary layer.

For the L/D and angle of attack studies, a default set of grid parameters
typical for an axisymmetric body was used. In the j-direction
(normal to the body), 61 points were used. The outer boundary was located one
body-length from the body and two lengths aft. A schematic of this arrangement
is shown in Figure
1. The minimum spacing was based on = 1.

Note: In CFD analyses of marine vehicles, it is customary
to use i, j, and k to
describe the grid dimensions, where the i-direction is in the
axial direction, j is normal to the body, and
k is around the body's girth.

Figure 1.
Schematic of grid structure around an axisymmetric body.

4.0 Surface Ship Hullform

The second study examined surface ship hull-forms. The drag coefficient was
computed for three ship hullforms. Generation of the geometry and grid were far
more time consuming than for the bodies of revolution, due to the complex
curvature of the shapes.

The first hull was the HSVA tanker, shown in Figure 2, which has
been used extensively for CFD validation of the viscous flow field [4]. Detailed
measurements of the viscous flow velocity field were made in a wind tunnel. By
eliminating the free surface effect, the viscous drag on the double model of the
HSVA tanker was measured and published in Reference [5].

Figure 2.
Geometry of the HSVA tanker.

A multi-block, also known as multi-zonal, structured grid was generated
around the HSVA tanker with H-type grid topology. The minimum spacing was based
on a of 1. Viscous flow calculations were performed to find the double
body solution.

The two other ship hulls are similar to each other in shape, and differ only
that the second hull has an added a centerline skeg in the stern. Figure 3 shows the two
hulls. The complex curvature of these two hulls leads to a more complex grid
topology than used for the HSVA tanker. The grid generation package
GRIDGEN was used to generate the multi-block grids around the two hulls.
A view of the grid structure around one of the hulls is shown in Figure 4. The minimum
spacing was based on a of 1.
Again, the double body solution was calculated.

a. Surface ship
without a centerline skeg.

b. Surface ship
with a centerline skeg.
Figure 3. Surface ship geometries, with and without a
centerline skeg.

Figure 4. Grid
structure around a surface ship.

5.0 Multi-Zonal Flow Solver

A multi-zonal flow solver (ISFLOW) has been developed for computation
of multi-block structured grids. The numerical algorithm adopted was based on a
cell-centered, second-order central difference, finite volume formula for
spatial discretization with an explicit one-step Runge-Kutta time-stepping
scheme. The three-dimensional Reynolds-averaged Navier-Stokes equations with
pseudo-compressibility approach for incompressible viscous flow were written in
tensor notation as follows:

,
(3)

,
(4)

where t represents time, X represent the Cartesian coordinates, U the
corresponding velocity components, P the pressure, and the
viscous stress tensor. The parameter 1/ represents
the pseudo-compressibility. The fact that the transient-state solution obtained
by equations (3) and
(4) may not be
physically valid was not relevant, since only the steady-state solution was of
interest here. The goal is to choose a reasonable value in
order to reduce the disparity in propagating speeds during the transient state.
Numerical experiments on choosing indicate
that = 2 gives a good convergence rate.

A combination of finite volume discretization and the Runge-Kutta
time-stepping scheme was used to yield an effective method for solving equations
(3) and (4) in arbitrary
geometric domains. The domains are divided into quadrilateral cells. The
discretization scheme decouples the approximation of the spatial and temporal
terms in equations (3) and (4). In addition, the
spatial terms are discretized in a finite-volume formulation. This is
accomplished by defining the dependant variable at the center of a computational
cell and evaluating the flux at the midpoint of each cell surface using the
central differencing technique. A fourth-difference artificial dissipation was
included to damp high-frequency oscillation and stabilize the computation. Since
artificial dissipation is a purely numerical parameter, which does not appear in
the analytic form of the equation, it must vanish as the discrete equations
approach the analytic form of equations (a limitation of the mesh refinements).
In addition, the terms must vanish if a natural dissipation mechanism, such as
viscosity, is strong enough to stabilize the computation. To avoid the
interference of artificial damping terms with the physical transport phenomena,
a hyper-tangent distribution of a weighting function for reducing the magnitude
of the artificial dissipation is used for computations inside the viscous layer.
The use of such a dissipation model has proved to be successful in the
calculation of turbulent vortex flow around the juncture of an airfoil attached
to a flat plate [6].

The discretization procedure on spatial terms in equations (3) and (4) leads to a set of
coupled ordinary differential equations, which has the form

,
(5)

where W is the vector of the flow variables and R(W) is the vector of the
residuals, consisting of the flux balance from the convective and diffusive
terms, as well as the artificial dissipation terms. The time integration is
performed by using an explicit one-step multi-stage scheme derived from a
generalization of the Runge-Kutta formulation [7]. The multi-stage
scheme of the Runge-Kutta method can be tailored to give the desired stability
properties. The Courant-Friedrichs-Lewy (CFL) number applies a conditional
stability requirement on the time step for the numerical solution. It has been
determined that the maximum stability bound on the CFL number for an m-stage
scheme is (m-1) along the imaginary axis. The optimal choice of the scheme
depends on a trade-off between the extent of the stability region needed and the
cost of the integration of an extra stage. An implicit residual smoothing
technique was implemented to relax the restriction on the time step imposed by
the CFL condition. To accelerate the numerical solution to a steady-state, local
time-stepping has also been adopted. The CFL number is increased from 3.0 to 4.5
when the 4-stage Runge-Kutta scheme is used with 2 evaluations of implicit
residual smoothing. In addition, the algebraic turbulence model proposed by
Baldwin and Lomax [8] for separated
turbulent flows is used.

The multi-zonal capability in ISFLOW was designed specifically for the
non-overlapping grid structure [9]. This multi-zonal
approach allows the user to divide the whole flow field into a number of smaller
regions. By defining these regions in a logical manner, the coding of the flow
solver can be greatly simplified. In addition, a different set of governing
equations and boundary conditions can be used independently within each zone.
The grid distribution can also vary between zones, depending on the desired
resolution.

Compared to the single zone approach, the multi-zonal approach has played a
significant role in computing the flow around complicated geometry. Besides the
substantial advantage for grid generation, a significant saving in computer
memory and cpu time can be achieved. The treatment of the interface boundary
between neighboring zones is also important. Generally, the interior points of
each zone are updated using the regular integration method similar to the single
zone scheme. The interface boundary points are updated through the interpolation
between interior points of neighboring zones. It is important to make the zonal
boundary scheme conservative enough so that a distortion-free flow movement is
achieved across the zonal interface.

6.0 Results and Analysis

6.1 Axisymmetric Body of Revolution

6.1.1 Without Angle of Attack

The computed results for the six axisymmetric bodies are shown in Table 2. The total
drag coefficient (Cd) is composed of viscous pressure (Cvp), and frictional drag
(Cf), which are obtained by integrating the pressure distribution and viscous
shear stress, respectively, over the body surface. The computed results are in
agreement with the experimental measurements. The calculations and experiments
were performed for a Reynold's number of 2.0 x 10e7.

Table 2. Computational versus experimental
results for bodies of revolution at zero angle of attack.

Several prediction methods have been investigated to calculate the drag for
this series of bodies [10]. Table 3 provides a
comparison of the current results with two methods in Reference [11], one
based on a differential boundary layer formulation (theory of Cebeci and Smith)
and the other based on a simple drag formula by White [10]. Although not as
close as the RANS prediction, the two methods show good correlation with
experimental data.

Table 3. Drag coefficients (Cd x 10e-3) from
different prediction methods.

6.1.2 With Angle of Attack

The computed results for the two axisymmetric bodies at angles of attack are
shown in Tables 4
and 5. The only
experimental data available was the coefficient of force normal to the body axis
(Cn) in the y = 0 plane. Drag (Cd), lift (Cl), and normal force (Cn)
coefficients from ISFLOW were tabulated with experimental data. For the
slender body (L/D = 10), the computed normal forces show reasonable correlation,
within 5% to 8% of the experimental values.The fat body (L/D = 4) shows slightly
better correlation at angles of attack up to 12 degrees, within 1% to 6% of the
experiment. However, at 18 degrees the discrepancy is 13.5%.

Table 4. Results for slender body (L/D = 10) at
angles of attack.

Table 5. Results for fat body (L/D = 4) at angles
of attack.

6.1.3 Effect of Grid Parameters

The results of varying the grid parameters for the slender body at zero angle
of attack are shown in Table 6. Based on equation (1), the
values for the first, tenth, and twentieth cell off the body were also shown.
Looking at the three (1) = 1
cases, 61 and 71 j-points were satisfactory, but there was a
loss in accuracy going from 61 to 51 j-points. Similarly, for
the three 61 j-cases, values for (1) of 1.0
and 0.5 were satisfactory, but 2.0 has a much larger error.

Table 6. Effects of varying grid parameters on
the solution for the slender body (L/D = 10) at zero angle of attack.

The results of varying the grid parameters for the fat body at 18 degrees
angle of attack were shown in Table 7. Both
increasing the number of j-points and extending the outer
boundary make small improvements in the accuracy. The reduction of has a
much more significant effect in improving the accuracy of the results for these
configurations.

Table 7. Effects of varying grid parameters on
the solution for the fat body (L/D = 4) at 18 degrees angle of attack.

6.2 Surface Ship

The drag calculations for the three surface ship hullforms are compared with
experimental data in Table 8. Both the
experiment and computations for the HSVA tanker hullform were performed for a
Reynold's number of 5.0 x 10e6. The computations and experiment for the other
surface ships (with and without the centerline skeg) were performed for a
Reynold's number of 4.5 x 10e6. A good agreement was found for the HSVA tanker
and ship without the centerline skeg. However, there was a large (6.9%)
difference for the ship with the centerline skeg. Consequently, the flow
solution around the ship with the skeg was more thoroughly examined. A vortex
along the bilge (the transition between the bottom and side of the ship) showed
rapid and unusual diffusion near the ship stern. The grid spacing around the
girth of the ship, especially at the bilge, is suspected to have been too
large.

Table 8. Computational versus experimental
results for surface ships.

7.0 Conclusions

The lift and drag forces on both the axisymmetric bodies and the surface
ships predicted by ISFLOW show good agreement with experimental results.

For the axisymmetric bodies at zero angle of attack a minimum spacing based
on = 1, an outer boundary 1 body length away, and 61
j-points were sufficient to ensure good agreement with
experiment. When increasing the number of grid points for a given value of ,
there appears to be a threshold up to which the accuracy increases rapidly, and
beyond which there was little improvement.

For examining both the slender and fat bodies at low angles of attack, these
grid parameters seem to be on the threshold of accuracy. The fat body clearly
requires a different set of grid parameters at high angles of attack.

Overall, the results were equally as sensitive to the number of
j-points as to the minimum spacing. Extending the location of
the outer boundary beyond one body length has a lesser effect on the result.

8.0 References

GERTLER, M - Resistance Experiments on a Systematic Series of
Streamlined Bodies of Revolution For Application to the Design of High Speed
Submarines, DTMB Report C-297 (1950)

SMITH, R E and ERIKSSON, L E - `Algebraic Grid Generation',
Computer Methods in Applied Mechanics and Engineering (1987)

SCHLICHTING, H - Boundary Layer Theory, McGraw-Hill,
New York, (1966)

WIEGHARDT, K - Kinematics of Ship Wake Flow, DTNSRDC 81-093
(1982)

Proceedings of the CFD Workshop Tokyo (1994)

SUNG, C H and LIN, C W - `Numerical Investigation on the
Effect of Fairing on the Vortex Flow around Airfoil/Flat-Plate Junctures',
AIAA 26th Aerospace Sciences Meeting (1988)

JAMESON, A, SCHMIDT, W, and TURKEL, E - Numerical Solution of
the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping
Schemes, AIAA Paper 81-1259 (1981)

BALDWIN, B S and LOMAX, H - `Thin Layer Approximation and
Algebraic Model for Separated Turbulent Flows', AIAA 16th Aerospace Sciences
Meeting (1978)

LIN, C W, SMITH, G D, and FISHER, S C - `Numerical Viscous
Flow Simulation of Wind Tunnel Model-Scaled Experiments', The 2nd Osaka
International Colloquium on Viscous Fluid Dynamics in Ship and Ocean
Technology (1991)

WHITE, N M - A Comparison Between a Single Drag Formula and
Experimental Drag Data for Bodies of Revolution, DTNSRDC Report 77-0028
(1977)

WHITE, N M - A Comparison Between the Drags Predicted by
Boundary-Layer Theory and Experimental Drag Data for Bodies of Revolution,
DTNSRDC SPD-784-01 (1978)