An Investigation Into the Wave Making Resistance of a Submarine Travelling Below the Free Surface Ch2&3

10

2 THEORY

2.1 GENERAL THEORY

The drag of a submarine at a given speed is the fluid force acting on the hull in such a way as to

oppose its motion (Harvald, 1991). For a submarine deeply submerged the drag is due primarily to

viscous effects, which can be split into two categories.

• Skin Friction: The component of resistance obtained by integrating the tangential stresses

over the wetted surface of the ship in the direction of motion. The skin friction is related to

the surface area of the hull, therefore it is desirable to reduce the surface area for a given

volume (Burcher & Rydill, 1994); and

• Form Drag: The pressure resistance is the component of resistance obtained by integrating the

normal stresses over the surface of a body in the direction of motion. Due to viscosity, there

is a reduction of the fluid momentum resulting in a pressure difference between the bow and

the stern and thus a net drag in the direction of motion. The form drag can be minimized by

having very slow varying sections across the length of the submarine resulting in a high

surface to volume ratio (Burcher & Rydill, 1994).

It should be noted that induced drag due to lift not only on the appendages, but also on the hull itself

also exists when the submarine is travelling at an angle of attack to the flow. It can be seen that the

two forms have opposing requirements. The variation of the two kinds of drag and their summation

when plotted as a function of the length to diameter ratio produce a region at which the minimum drag

can be achieved as seen in Figure 2.1. It should also be noted that appendage drag is not considered.

Figure 2.1: Drag Components for Constant Volume (Joubert, 2004)

11

It can be seen that when determining the hydrodynamic characteristics of a submarine the length to

diameter ratio is critical. In 1948 the American designed Albacore set the bench mark when it opted

for optimal submerged performance at the expense of surface operation. The result was a length to

diameter ratio of 7.723; a figure most submarines have since strived for (Joubert, 2004). However,

submarines typically depart from this shape, mainly due to their diameter being set at a fixed number

of decks. That is, the diameter of a submarine is designed to accommodate a set amount of decks at a

set height, thus small changes are not feasible. Additionally larger, appendages relative to the size of

the boat are typically required for boats with lower L/D ratios; adding further to the appendage drag.

2.1.1 DRAG CLOSE TO A FREE SURFACE

Although the modern, conventional powered submarine (SSK) is designed with emphasis on

submerged performance, many occasions call for surface or near surface operation such as entering or

leaving harbour, lengthy transits to a dive area, reconnaissance and snorting (Burcher & Rydill, 1994).

When a submarine is operating close to, or on, the free surface, wave making resistance has an effect

on the total drag on the hull. Wave making resistance is the component of resistance associated with

the energy expended generating gravity waves (Lewis, 1988).

As water passes around a submarine it speeds up and slows down at various points and, according to

Bernoulli’s equation, the pressure will also change 9i.e. increased velocities result in a drop in

pressure). The forces acting on the submarine are of a considerable magnitude, however in an ideal

fluid they all act to cancel each other out (Rawson & Tupper, 2001).

As a submarine approaches a free surface, these pressure variations can manifest as elevations or

depressions of the water surface, that is, waves. This process results in an imbalance in the

distribution of pressure across the body and a drag force is created. The magnitude of the drag force

is related to the energy of the wave system (Rawson & Tupper, 2001).

There are two wave systems produced by a body moving close to or on the free surface divergent; and

transverse, as seen in Figure 2.2. As a submarine’s speed changes, so too does the wave pattern.

Therefore there will be a succession of speeds when the crests of the two wave systems reinforce one

another and vice versa, this process is known as interference (Lewis, 1988).

12

Figure 2.2: Kelvin Wave Pattern Patterns of a Point Disturbance

In the case of a ship, the Froude Number (a dimensionless parameter used to relate inertia and

gravitational forces) is typically defined as:

!" � #�� (2.1)

As submarines are relatively small vessels, obtaining any reasonable speed requires them to operate at

high Froude Numbers (Fr). Wave making resistance becomes the dominate force at these Froude

Numbers, and the form idealised for underwater performance is not suited to these operational

conditions on the surface and results in the submarine operating at a significantly reduce speed

(Burcher & Rydill, 1994). For a given Froude Number, the effect of wavemaking resistance will

reduce as the submarine moves further below the free surface until it is considered negligible, which

typically occurs at half the length of the body (Rawson & Tupper, 2001). It is clear that a surfaced

submarine will exhibit the highest wavemaking resistance for a given Froude Number; Figure 2.3

shows a typical resistance curve of a surfaced submarine, where the large proportion of wavemaking

resistance at higher Froude Numbers can clearly be seen.

Figure 2.3: Typical Surfaced Submarine Resistance Coefficient Curve (Burcher & Rydill, 1994)

13

2.2 COMPUTATIONAL FLUID DYNAMICS

2.2.1 FUNDAMENTALS

CFD is essentially the integration of fluid mechanics, mathematics and computer science. The key

advantage of using CFD over more conventional model testing is being able to ascertain the impact of

design modifications before a large investment in time or money has been made in model testing.

CFD is fundamentally based on the governing equations of fluid dynamics. They represent

mathematical statements of the conservation laws of physics. According to Tu (2008), the three

equations that govern all computations in CFD are:

• Continuity Equation: mass is conserved for the fluid:

D�DE ? %. �FG� � 0 (2.2)

• Newton’s Second Law: the rate of change of momentum equals the sum of forces acting on

the fluid:

X component: D��I�

DE ? %. �F/G� � J D�DK ? D�LL

DK ? D�MLD6 ? D�NL

DO ? FPK (2.3)

Y component: D��.�

DE ? %. �F1G� � J D�D6 ? D�LM

DK ? D�MMD6 ? D�NM

DO ? FP6 (2.4)

Z component: D��6�

DE ? %. �F1G� � J D�DO ? D�NL

DK ? D�MND6 ? D�NN

DO ? FPO (2.5)

• First Law of Thermodynamics: the rate of change of energy equals the sum of rate of heat

addition to, and the rate of work done on, the fluid:

DDE QF 9R ? �

� <S ? %. QFG 9R ? �� <S � F)* ? D

DK 9@ DTDK< ? D

D6 9@ DTD6< ? D

DO 9@ DTDO< J D�I��

DK J D�.��D6 J

D�U��DO ? D�I�LL�

DK ? D�MLD6 ? D�NL

DO ? D�LMDK ? D�MM

D6 ? D�NMDO ? D�NL

DK ? D�MND6 ? D�NN

DO ? FPG (2.6)

14

The above five equations known as the Navier Stokes Equations, represent seven unknowns. They

are completed by adding two algebraic equations; one relating density to temperature and pressure

(equation 2.7) and the other relating static enthalpy to temperature and pressure (equation 2.8):

F � F�V, X� (2.7)

Y � Y�V, X� (2.8)

2.2.2 NUMERICAL TECHNIQUES

2.2.2.1 Discretization

Discretization is essentially converting the partial differential equations and auxiliary conditions into a

system of discrete algebraic equations (Tu, 2008). The discretization process does however mean that

in all cases the solutions are approximate (MARNET-CFD, 2002).

In principle the finite-difference method can be applied to any type of grid system, however in

practice it is only applied to a structured mesh. The grid spacing does not need to be identical but

there are limits to the amount of distortion that the grid can undergo. It is important to note that the

numerical calculations do not need to be performed in the physical space but, rather they are

preformed in a transformed computational space. There are essentially two forms of these; known as

the forward and backward difference method, in reference to their particular bias in a given direction

(Tu, 2008).

In contrast, the finite-volume method discretizes the integral of the conservation equation directly in

the physical space. The computational domain is divided into finite number contiguous volumes. As

the finite-volume method works with the control volumes and not at grid intersection points, it has the

capacity to accommodate any type of grid. Thus, the finite-volume method is capable of computing

both structured and unstructured meshes. The finite-volume method is implemented in the majority of

calculations today (Tu, 2008).

15

2.2.3 TURBULENCE MODELS

Whilst theoretically the above equations can sufficiently describe all incompressible flows they are

inherently non-linear and as a result are subject to instabilities, which physically grow to form a

mechanism to describe turbulence (MARNET-CFD, 2002). All flows can be described as either

turbulent or laminar, see Figure 2.4. Laminar flow has a smooth velocity gradient. It moves along in

streamlines, or sheets. The velocity at the wall is zero and there is a slow increase to the free stream

velocity. The velocity of each sheet is affected by the shear stress or friction between the molecular

momentums (Gerhart, Gross, & Hochstein, 1992). Most flows in engineering are turbulent;

turbulence can originate from the free stream or be induced by surface roughness. Turbulence is

associated with random fluctuations in the fluid, making computations based on equations that

describe fluid motion exceptionally difficult. As a result it is assumed that turbulence essentially

causes a variance in velocity with respect to time; this variance is expressed as a mean velocity /Z with

a fluctuating component of /0�[�. Accordingly, time averaged calculations, commonly known as the

Reynolds-Averaged Navier-Stokes (RANS) equation, are preformed. (Tu, 2008).

Figure 2.4: (a) Laminar Flow; (b) Transition Flow; (c) Turbulent Flow (White, 2007)

16

Figure 2.5: Laminar and Turbulent Flows Within a Pipe (White, 2007)

After a century of intensive theoretical and experimental research, it is accepted that no single

turbulence model can be considered adequate for all flow situations (Tu, 2008). A large number of

models exist today, and the selection of the best one to use for a particular application must not only

take into account questions of accuracy but also computational requirements. It is acknowledged that

more advanced techniques such as Large Eddy Simulation (LES) and Detached Eddy Simulation

(DES) exist however they will not be considered for the purposes of this study due to their excessively

large computational requirements. The turbulence models mentioned, k-ω, k-ε and the Shear Stress

Transport (SST), all rely on developing a relationship between the Reynolds Stresses and the mean

velocity.

2.2.3.1 Equation for Kinetic Energy of the Turbulent Fluctuation

As the complete Navier-Stokes system of equations is not closed, one such equation to solve the

Navier-Stokes equations is the kinetic energy equation for turbulent flow. This equation is obtained

through the balance of kinetic energy of the turbulent fluctuations. The kinetic equation of the

turbulent fluctuations is given by Versteeg & Malalasekera (2007) as:

@ � \� �/0� ? 10� ? 20��ZZZZZZZZZZZZZZZZZZZZZZ (2.9)

The k equation describes a balance between the four contributions to the energy of turbulent

fluctuations. These energies are convection, diffusion, production and dissipation. Therefore, the

change in turbulent energy due to convection is compensated by an energy source (production), an

energy sink (dissipation) and energy transportation (diffusion). However, the Navier-Stokes equations

17

cannot be closed without further assumptions being made between the Reynolds stresses and

quantities of mean motion (Tu, 2008). In using this kinetic equation as a base, and assuming the

Reynolds stress or the energy fluctuation, the Navier-Stokes equations can be solved.

2.2.3.2 k-ε Turbulence Model

The two-equation k-ε model is the most widely used and validated turbulence model (Tu, 2008). k is

as described in section 2.2.3.1 and ε is the rate at which turbulent energy is dissipated by the action of

viscosity on the smallest eddies (Launder & Spaulding, 1974). The one dimensional k-ε model is

expressed as follows (Anderson, 1995):

@ � \� /]0/]0 (2.10)

^ � 1V _`Iab`Kc

d _`Iab`Kc

d (2.11)

where; i,j = 1,2,3

The k-ε model offers a good compromise between versatility and accuracy for most general purpose

calculations (ANSYS CFX, 2008). The k-ε model is typically inadequate in adverse pressure

gradients such as those found in the boundary layer of a surface vessel (MARNET-CFD, 2002).

Other applications where the k-ε model is not suitable according to ANSYS CFX (2008) are flows:

• with boundary layer separation;

• with sudden changes in the mean strain rate;

• in rotating fluids; and

• over curved surfaces.

The k-ε model is particularly suited for the following situations:

• modelling the free stream around a hull form; and

• modelling the turbulence aft of the hull form.

18

2.2.3.3 k-ω

The second most widely used type of two equation model is the k-ω model, where ω is the frequency

of the large eddies (Wilcox, 1986). The k-ω model performs very well close to walls in boundary

layer flows, particularly under strong adverse pressure gradients. The main problem with the Wilcox

model is its strong sensitivity to free stream conditions (Menter, 1994). Depending on the value

specified for ω at the inlet, a significant variation in the results of the model can be obtained. (ANSYS

CFX, 2008).

2.2.3.4 Shear Stress Transport

The Shear Stress Transport (SST) model, developed by Menter (1994), is one of the most effective

turbulence models commercially available. To overcome the shortcomings of both of the previous

methods Menter proposed a model that combined the accuracy of the k-ω model near the wall and the

accuracy of the k-ε in the free stream by the use of blending functions and limiters. Essentially a

hybrid of both models is used; k-ω at the wall which is then blended into the k-ε as the flow

approaches the free stream. Blending functions are introduced to achieve a smooth transition between

the two models (Versteeg & Malalasekera, 2007). To improve the flow in the region of adverse

pressure gradients, viscosity limiters are used. Turbulent kinetic energy is limited to prevent the build

up of turbulence in stagnation regions (Versteeg & Malalasekera, 2007). In a NASA Technical

Memorandum, the SST was rated the most accurate model for aerodynamic applications (Sulficker &

Murali, 2006). ANSYS CFX (2008) recommends the SST model for high accuracy boundary layer

simulations. To benefit from this model, a resolution of the boundary layer of more than 10 points is

required. The SST model is significantly more computationally demanding than the k-ω and k-ε

turbulence models, as twice as many equations are being solved in every iteration.

2.2.4 NEAR WALL MODELLING

The modelling of the flow near the wall is important as most engineering flows contain solid

boundaries. Due to the presence of the solid boundary, the flow behaviour and turbulence structure

are considerably different from free turbulent flows and these difference must be accounted for when

using CFD (Malalasekera & Versteeg 2007). To understand how to model this correctly an

understanding of the flow associated within this region is paramount.

19

In the thin wall layer, the flow is dominated by viscous stresses and is not yet influenced by the free

stream. Therefore the mean flow velocity depends on the distance y from the wall, the fluid density ρ,

viscosity µ, and the wall shear stress τw (Malalasekera & Versteeg 2007). Through dimensional

analysis it is shown that,

/4 � Ie

� P 9�Ie6f < � P�34� (2.12)

This equation is called the law of the wall and introduces two important non-dimensional terms, u+

and y+, where ut is called the frictional velocity and defined as:

/E � 9T�� <

&� (2.13)

Karman in 1933 found that u in the outer layer is independent of viscosity, but must depend on other

properties. He deduced that,

�G J /��IE," � g�h, iUF3� (2.14)

by dimensional anlysis

�IIe

� g 96j< (2.15)

This equation is called the velocity defect law.

As seen from Figure 2.6, there are 3 regions in the flow near the wall;

1. wall layer – thin layer dominated by viscous stresses;

2. overlap layer – viscous and turbulent stresses; and

3. outer Layer – dominated by turbulent stresses.

20

Figure 2.6: Velocity Distribution Near a Solid Boundary (White, 2007)

The law of the wall and the velocity defect law are used to model the flow in each layer

2.2.4.1 Thin Viscous Sub-layer

The viscous sub-layer is in practice extremley thin (y+<5) and it may be assumed that the shear stress

remains approximatelyy constant and equal to the wall shear stress, τw, throughout the layer (Versteeg

& Malalasekera, 2007); Therfore:

i�3� � k DD6 l iU (2.16)

Upon integration with boundary conditions and then making the equations non-dimensional it is seen

that

/4 � 34 (2.17)

This shows that there is a linear relation between the velocity u and the distance from the wall y, and

hence is also known as the linear sub layer.

21

2.2.4.2 Overlap Layer

In this region the viscous and turbulent forces are both considered. Versteeg and Malalasekera (200)

explain that the shear stress τ varies slowly with the distance from the wall. And using the mixing

length equations, it is possible to derive an equation relating y+ and u+.

/4 � \m ln�34� ? p (2.18)

where: k=0.4

B=5.5

This equation is called the Log Law and for this reason the overlap layer is often called the log law

layer (Versteeg & Malalasekera, 2007).

2.2.4.3 Outer Layer

For larger values of y (y/δ>0.2) the velocity defect law is applicable. However this equation needs

modification so that at the overlap region the velocity defect law and the log law are equal, Tennekes

and Lumley (1972) show that a matched overlap is obtained by assuming the following logarithmic

form:

�I

Ie� J \

m qr 96j< ? s (2.19)

22

Figure 2.7 shows a boundary layer profile with the visous or linear sub-layer transitioning into the

overlap region and the outer turbulent region.

Figure 2.7: Boundary Layer Profile (White, 2007)

A major difference that separates the turbulence models discussed in the previous sections is how they

resolve the boundary layer. k-ε uses a wall function that completely resolves the viscous sub layer of

the boundary layer while the k-ω and SST turbulence models do not use wall functions and solve the

boundary layer completely. This means that the k-ε model needs the first node to be on the edge of

the boundary layer while the k-ω and SST models need to have the first node very close to the wall to

fully resolve the boundary layer. Turbulence models require the use of a non dimensional parameter

y+ to determine the height of the first node. y+ is essentially a function of Reynolds number and

allows the CFD user the ability to determine the required height of the first node based on what

turbulence model is being used, see Figure 2.8. The recommended y+ values for three of the most

common turbulence models employed in commercial CFD codes can be seen in Table 2.1.

23

Figure 2.8: First Layer Height (ANSYS CFX, 2008)

Table 2.1: Required y+ for Various Turbulence Models (Ranmuthugala, 2008)

Turbulence Model Required y+

k-ε 30-100

k-ω <1-2

SST <1-2

The y+ value can be found from;

34 � 6Ie. (2.20)

For simplicity, ANSYS CFX (2008) recommends using the following to calculate initial y+ values:

34 � ∆6�√8��,9:&'

&; < (2.21)

where,

∆y = First prism height

L = Reference length

Re = Reynolds number

24

2.2.5 GRIDDING TECHNIQUES

Grid generation presents an important consideration in computing numerical solutions to the

governing partial differential equations of the CFD problem. The quality of the grid can have a

significant influence on the overall accuracy of the solution. As a general rule the accuracy of the

simulation increases with increasing number of cells, i.e. with decreasing cell size. However, with

increased mesh size the computational demands also increase resulting in longer run times

(MARNET-CFD, 2002). Currently, body fitted grids are used almost universally. The above

publication describes the various forms of mesh topology commonly used as follows:

Structured grid

The points of a block are addressed by a triple of indices. The connectivity is straight-forward because

cells adjacent to a given face are identified by the indices. Cell edges form continuous mesh lines

which start and end on opposite block faces. Cells have the shape of hexahedral, but a small number

of prisms, pyramids and tetrahedral with degenerated faces and edges are sometimes accepted.

Block structured grid

For the sake of flexibility the mesh is assembled from a number of structured blocks attached to each

other. Attachments may be regular, i.e. cell faces of adjacent blocks match, or arbitrary (general

attachment without matching cell faces). Figure 2.9 demonstrates a simple multi-block structured grid

Figure 2.9: Block Structured Grid (Wyman, 2001)

25

Chimera grid

Structured mesh blocks are placed freely in the domain to fit the geometrical boundaries and to satisfy

resolution requirements. Blocks may overlap, and instead of attachments at block boundaries

information between different blocks is transferred in the overlapping region. Figure 2.10

demonstrates the typical way a chimera grid is developed.

Figure 2.10: Chimera Grid (Wyman, 2001)

Unstructured grid

Meshes are allowed to be assembled cell by cell freely without considering continuity of mesh lines.

Hence, the connectivity information for each cell face needs to be stored in a table. The most typical

cell shape is the tetrahedron, but any other form including hexahedral cells is possible. Figure

2.11demonstrates a typical unstructured grid.

Figure 2.11: Unstructured Grid (Wyman, 2001)

26

Hybrid grid

This grid combines structured with unstructured meshes. In Figure 2.12 a hybrid mesh can be seen.

Figure 2.12: Hybrid Grid -Structured Left, Unstructured Right (Wyman, 2001)

2.2.5.1 Grid Topology

Typically in structured multi block meshes there are several topologies adopted. A topology is

essentially a way of arranging the blocks to ensure optimum grid characteristics. Each topology has its

own advantages and the choice of which to use is very much dependent on the situation. It is indeed

not uncommon to employ multiple topologies in the one grid.

A H type mesh, as seen in Figure 2.13, is the standard meshing method used in ANSY-ICEM CFD. A

H mesh can achieve good results for a simple geometry, however to maintain accuracy for complex

shapes the blocking becomes complex.

Figure 2.13: 2D H Grid around a Cylinder (Widjaja 2009)

Unstructured

Grid

Structured

Grid

27

An O type mesh, as seen in Figure 2.14 is ideally suited for circular or curved surfaces; Figure 2.15

shows that when an H mesh is used on a circular geometry highly skewed elements exist at angles of

45˚ around the geometry; an O type mesh removes this skewness. O type meshing is not well suited

to wake flows. Figure 2.14 shows that as the O expands to outer edges of the geometry the elements

become quite large, and would not accurately capture the wake region of the flow.

Figure 2.14: 2D O Grid around a Cylinder (Widjaja, 2009)

Figure 2.15: O-Grid inside a Cylinder (Widjaja, 2009)

A C mesh, as seen in Figure 2.16, is a combination of an H and C grid, it has the benefit of the O grid

where it accurately models a curved surface, but also allows for refinement of the mesh in the leeward

edge of the geometry. C type meshing is ideally suited for flows where a wake needs to be captured

and anything that has a bluff leading edge and small finite to infinite trailing edge such as foils and

wings as the mesh reduces to H mesh at these sections allowing for mesh edges to fully capture the

geometry of these critical regions.

28

Figure 2.16: 2D C Grid around a Cylinder (Widjaja, 2009)

2.2.6 NUMERICAL SOLUTION OF A FREE SURFACE

A free surface is an interface between a gas and a liquid where the difference in densities is

significant. The gas’s inertia is ignored; its only effect is to apply a static pressure on the liquid

surface. Two commonly adopted approaches for computing a free surface in CFD are Eularian-

Lagrangian and Eulerian-Eulerian methods, however Eulerian-Elurian methods allow for more

complex free surfaces to be tracked. (Senocak & Iaccarino, 2001)

When simulating a flow using the RANS equations there are essentially two methods that can be

adopted; interface tracking and interface capturing. Interface tracking involves modelling only the

fluid domain. One of the domain boundaries is then assigned to be the free surface and the grid is

adapted to the position of the free surface. This method is not suitable for conditions of steep or

breaking waves as the entrapment of air cannot be accounted for. Interface capturing involves solving

the governing equations for both air and water, which makes this approach typically more versatile

(Senocak & Iaccarino, 2001).

2.2.6.1 Volume-of-Fluid

If the amount of fluid in each cell is known then the surface can be tracked. The Volume-of-Fluid

method (VOF), first introduced by Hirt and Nichols (1981), involves tracking the volume fraction of

each cell. Initially each cell is prescribed a volume faction of either one or zero. The free surface is

then defined as an area of rapid change in the volume fraction. The volume fraction is solved for one

of the phases by means of an extra transport equation, having generally the same mass. One common

algorithm used for this solution is the semi-implicit method for pressure linked equations (SIMPLE).

Interface sharpening algorithms can be used to refine the cells with a value of between one and zero.

29

The main disadvantage of VOF, and indeed all surface capturing techniques, is that the mesh is

required to be more refined at the free surface interface than surface tracking methods, resulting in a

larger grid (Senocak & Iaccarino, 2001). However, VOF remains the most viable option for accurate

three dimensional free surface calculations.

2.2.6.2 Level Set

The level set technique involves enforcing a step-wise variation of the fluid properties. Initially it is

set equal distance from the free surface, positive in one direction and negative in the other. “At every

later instance, the function is computed from the condition that its total (material) derivative with

respect to time is zero” (MARNET-CFD, 2002); thereby ensuring that the function is constant with

time on all particles. The main issue with the level set technique is the need to arbitrarily induce some

finite thickness across the interface region to promote smooth but rapid change in properties (Sethian,

1996).

2.2.7 SOLUTION PROCEDURE

The most common solution procedure adopted in ANSYS CFX when solving free surface flows is the

SIMPLE algorithm. Essentially this is a pressure correction equation. The equation is obtained from

the discretized mass and momentum conservation equations. The momentum equations are solved

first for an estimate of velocity components. These are then corrected upon solving the pressure-

correction equation. The equation for the volume fraction is then solved, followed by the equations

for turbulent kinetic energy and its dissipation rate. The eddy viscosity is then computed and the

whole process is repeated until all non-linear equations are satisfied. The time is then advanced to the

next time step (Azcueta et al, 1999).

30

2.2.8 TIME STEPS

According to ANSYS 2008 simulations of a physical process can either be transient or steady-state. In

a transient simulation, the behaviour of a physical system as a function of time is investigated. In a

steady-state simulation, it is assumed that the physical system is moving towards an equilibrium or

steady-state solution

Time stepping refers to solving the governing equations for a different value of time (t). A time step

is the change in time per iteration. For steady-state problems, the ANSYS CFX-Solver applies a false

time step as a means of under relaxing the equations as they iterate towards the final solution (ANSYS

CFX, 2008). There are two ways that a time dependent flow can be solved, either implicitly or

explicitly. Essentially explicit methods compute the next time step solution directly based on the

known current time information, whilst implicit methods evaluate the solution at two or more time

spaces, one of which is the current time step. For stability in an explicit equation the time step ∆[ is

governed by:

∆[ t Fu �∆K��m (2.22)

This is a stringent rule and if not followed the solution will become unstable and result in divergence.

This represents a serious limitation for the explicit scheme as it becomes computationally expensive to

increase spatial accuracy (Versteeg & Malalasekera, 2007).

In theory the fully implicit scheme will always be stable, i.e. it will not accumulate errors or become

un-bounded, or diverge, (which is why most commercial codes employ this method). However, the

majority of CFD calculations are preformed in single precision or first order to avoid excessive

computational demands (Tu, 2008). As a result, if too large a time step is taken the round-off errors

will increase leading to an inaccurate answer. It is also possible to use a time step that is too small

leading to an increased discretization error. Versteeg and Malalasekera (2007) recommend the

implicit method due to its robustness and unconditional stability. ANSYS CFX (2008) notes that free

surface and multiphase flows typically require very small time scales to ensure an accurate solution.

When simulating a transient or time dependent run, the Courant-Friedrichs-Lewy (CFL) condition is

typically used as a form of monitoring the stability of the equations. For convection-type simulations

the Courant number is written as (Tu, 2008);

31

∆[ t ∆KI (2.23)

It is seen that the parameter relates the spatial and time steps to the velocity. For an accurate solution

between each time step the Courant number is required to be below one, however as in the explicit

method this can result in very small time scale for refined grids. As a result it is common to use a

larger Courant number and take time averaged results to account for the fluctuations.

3 APPROACH

3.1 MODEL DESIGN

In order to decide on a suitable geometry for the project, an extensive study of many of the world’s

current operational submarines was conducted. It should be noted that no Ballistic Submarines

(SSBNs) were considered. Of particular interest was the length to diameter ratio of vessels, as it has

been highlighted by Vine (1991) that notable discrepancies between nuclear powered submarines

(SSNs) and SSKs exist. As can be seen in Figure 3.1 and Figure 3.2, no discernable difference

between SSKs and SSNs can be noted in regard to the L/D ratio and, as a result, the DARPA

SUBOFF geometry was adopted as there is a considerable number of studies, both experimental and

numerical, that have been undertaken on this particular hull form. These include the experimental

studies of Roddy (1990) and Huang et al (1992) and the computer modelling studies of Bull (1996),

Toxopeus (2008) and Widjaja et al (2007).

Figure 3.1: Comparison of the Length to Diameter Ratio

of Existing Submarines (Hazegray, 2009)

4.00

6.00

8.00

10.00

12.00

14.00

45 50 55 60 65 70 75

L/D

L/V1/3

L/D as a function of L/V1/3

SSN

SSK

Collins

Class

Due to the available depth of the towing tank (1.5m), the standard size model of 4.356m

Huang

of the free surface.

when using a model of 1.25m in length and 0.25m diameter, a deep water run

surface effects

original model was adopted

form is given in


Huang



surface effects


form is given in

L/D


Huang et al



surface effects


form is given in

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00


et al



surface effects


form is given in

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00


(1992)



surface effects,


form is given in

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

20


(1992)



was not attainable


form is given in Figure

20

Figure


(1992) - was



was not attainable


Figure

Figure


was

of the free surface. Vine


was not attainable


Figure 3

Figure


was considered too l

Vine


was not attainable


3.3 and

Figure 3.

of Existing Submarines


considered too l

(1991)


was not attainable

original model was adopted for both the EFD and CFD

and

40

L/D as a Function of Length

.2: Comparison of the Length to Diameter Ratio



considered too l

1991)


was not attainable

for both the EFD and CFD

and the principal particulars are given in

40


: Comparison of the Length to Diameter Ratio



considered too l

1991) demonstrated in experiments conducted in the same facility that,


was not attainable at Froude Numbers upwards of


the principal particulars are given in





considered too l

demonstrated in experiments conducted in the same facility that,


at Froude Numbers upwards of



Figure





considered too large to facilitate the measurement of drag without the effects






Figure





arge to facilitate the measurement of drag without the effects






Figure 3

60











3.3: SUBOFF

Length (m)


32










: SUBOFF

Length (m)


32










: SUBOFF

Length (m)



of Existing Submarines (Hazegray, 2009)






for both the EFD and CFD work. A diagram of the


: SUBOFF

80

Length (m)



(Hazegray, 2009)






work. A diagram of the


: SUBOFF (mm)

80



(Hazegray, 2009)








(mm)



(Hazegray, 2009)





at Froude Numbers upwards of 0.6



(mm)



(Hazegray, 2009)





0.6.


the principal particulars are given in Table



(Hazegray, 2009)





. As a result


Table

100







As a result


Table 3.

100







As a result


.1.





when using a model of 1.25m in length and 0.25m diameter, a deep water run,

As a result,





or run with no free

, a 2.8:1 scale of the

work. A diagram of the axi




or run with no free

a 2.8:1 scale of the

axi-symmetric

120

Due to the available depth of the towing tank (1.5m), the standard size model of 4.356m-



or run with no free


symmetric

120

as used by



or run with no free


symmetric

SSK

SSN

Collins

as used by



or run with no free


symmetric

SSK

SSN

Collins

as used by



or run with no free


symmetric hull

Collins

as used by



or run with no free


hull

33

Table 3.1: Parameters of Scaled SUBOFF Model

Parameter Value

Length 1.55575 m

Diameter 0.181 m

Surface Area 0.3819 m2

The depths at which the model was tested both experimentally and computationally were expressed

non-dimensionally as D*, where;

�� (3.1)

3.2 TESTING RANGE

The depths tested both experimentally and computationally were; 1.1D*, 1.3 D*, 2.2 D*, 3.3 D*, 4.4

D*, 5.5 D* (see Figure 3.4). In order to ensure the current project would be relevant to future

submarine design, a full scale length corresponding to Australia’s existing Collins Class Submarines

(77.5m) was assumed. The speeds tested ranged from 0.5 to 2.5m/s corresponding to a Froude

Number range of 0.128 to 0.640 or a full scale speed range of 6.8 to 34knots.

Figure 3.4: Test Depths