Modelling Viscous Free Surface Flow

Chapter 8

Free surface flows, where a boundary of a fluid body is free to move constrained only by forces across the surface, are possibly the most commonly observed flow phenomenon, with the motion of the free surface readily allowing the observation of flow of the fluid. Flows that are commonly encountered by the layperson include the motion of the surface of a river, the waves on the surface of the ocean, and the more personal flow of that in a cup of tea. From an engineering perspective, flows of interest include the open channel flow of rivers and canals, the erosive forces of waves on the shoreline, and the wakes generated by ships when under way.

When modelling these flows, the problems associated with the solution of the Navier–Stokes equations are compounded by the free motion of the surface of the fluid, with the boundaries of the flow domain being a function of the flow structure, and thus an unknown which must be calculated along with the flow field. The movement of the free surface therefore both aids the observation of the flow, and hinders it’s modelling.

In this chapter an attempt is made to model the steady flow around a ships hull. The modelling of such a flow is of great interest to Naval Architects, with the ability to predict the resistance of ships allowing the design of more efficient hull forms, whilst the accurate modelling of the ships wake allows the design of hulls that create a smaller disturbance in confined waterways. A brief overview of the modelling of ship flows is given, with emphasis being given to the use of CFD, being followed by the derivation of the numerical method used in this study. The implemented code is then used to model two test problems, open channel flow and flow in a free surface driven cavity, before being applied to the flow around a ships hull.

8.1 The Modelling of Flow About Ships

The ability to predict the resistance of ships hulls is of interest to the ship building industry. With the ability to predict ships speeds the designers can hopefully design a ship that meets the performance specifications of the clients, whilst an understanding of ship resistance enables the designers to pro- duce a more efficient hull form or to reduce the wave generation, an issue of interest when running ships in confined waterways.

Three methods are commonly used to predict the resistance of a ship. The earliest technique, as used by Froude, uses empirical data gained from testing a model in a towing tank. This process is not as straight-forward as modelling vehicles that are wholly immersed in a fluid, such as the case of aircraft models tested in a wind tunnel, since there are two length scales of interest in a ships flow, the Froude number ¢¡ , which is the ratio of the characteristic velocity and the wave speed for the flow, and the

195 CHAPTER 8. VISCOUS FREE SURFACE FLOW 196

Reynolds number £¥¤ , the ratio of the dynamic and viscous forces. These dimensionless groups are

deﬁned as1

¦¡¨§ ©

(8.3)

§ ©

£¥¤ (8.4)

When testing a model it is practically impossible to match both the Froude number and the Reynolds number of the full scale ﬂow. The resistance of the full scale ship is therefore estimated by assuming that the total resistance of the ship can be broken down into two independent components, the wave drag, due to disturbances to the free surface and characterised by the Froude number, and the frictional resistance, due to skin friction and characterised by the Reynolds number. The frictional resistance of the model is traditionally approximated by the drag of a ﬂat plate of the same cross section as the ship model, and so the wave drag of the model can be estimated as the total resistance of the model less this estimated frictional resistance. The resistance of the full scale ship can then be calculated as the sum of the scaled wave resistance of the model and the frictional resistance calculated for the Reynolds number of the full scale ship.

A second class of techniques models the flow about the ships hull using a distribution of source singularities to perturb the free stream flow. The initial method for such a flow was developed by Mitchell[112]2, a summary of the methods being given by Wehausen[177]. The neglect of viscous effects from the models however becomes problematical when modelling ships with blunt sterns, propeller wakes, and vessels at an angle of attack to the free stream flow, where viscosity can effect the flow topology by causing and controlling flow separation.

In more recent years CFD codes have been applied to modelling ship flows. These methods overcome some of the problems of the previous two techniques, but suffer from deficiencies of their own. Due to the large range of scales, varying from the length of the ships wave train to the smallest turbulent eddy, and the slow speed of the techniques, the resolution of the model used is a compromise between the physics of the problem and the computer resources and time available. Typically some form of turbulence model must be relied upon to model the finer structures of the flow. In addition, the discretisation schemes are dispersive and dissipative to varying degrees, affecting the accuracy of the solution. Worst of all, the methods are slow.

Nevertheless, there is some hope for these techniques. With much research being performed into turbulence modelling and discretisation techniques it is hoped that the overall accuracy of the methods will improve. In addition, the rise in computer speeds and the use of parallelisation methods reduces the overall time taken for a computation and allows for a reﬁnement in the mesh used to model the ﬂow.

8.1.1 A Review of CFD Schemes Described in the Literature

In the earliest paper to apply CFD techniques to the modelling of ship flows, Miyata and Inui[113] described a three dimensional finite difference model of the flow past a ship’s bow, their TUMMAC code. A laminar Navier–Stokes code using primitive variables, the differencing was done on a regular

1There is some variation in the definition of the Froude number. Most fluids texts define it as

(8.1)

However, some Authors (such as Lighthill[102]) use

"! (8.2) as their deﬁnition. White[179] equivocates and gives both forms in different chapters of his book, without drawing attention to the inconsistency. 2An earlier model of the far ﬁeld wake of a ship in deep water was developed by Lord Kelvin[171]. CHAPTER 8. VISCOUS FREE SURFACE FLOW 197

Cartesian mesh, with the bow of the ship being modelled by blocking out mesh nodes that lay within the ships volume.

The surface of the ﬂow was modelled using a “Marker and Cell” approach as developed by Harlow and Welch[57, 178], using a particle tracking approach to track the free surface, using the assumption that particles on the free surface remain on the surface. To model the steady state wave patterns, the solution is advanced using pseudo time stepping, with the positions of the particles on the surface being extrapolated at each time step to get the new position of the free surface using a ﬁrst order explicit scheme. The vertical spacing of the mesh points at the free surface is irregular, with points being added and removed from the domain as the free surface rises and falls.

Later developments to the TUMMAC code are described by Miyata, Nishimura and Masuko[116, 115], with the code being used to model a full ships geometry. For some calculations the viscosity was set to zero, reducing the Navier–Stokes equations to the Euler equations. A later paper applied the scheme to the modelling of a ship advancing into head seas[114].

To overcome problems with the original model, Miyata, Sato and Baba[117] developed the WISDAM code, using a non-orthogonal curvilinear mesh that ﬁtted both the ship geometry and the free surface. At each time step a new wave pattern was calculated, and the mesh was regenerated, the equations being formulated to account for mesh movement from the remeshing. The Sub Grid Scale (SGS) turbulence model was incorporated into the momentum equations, which were applied in Reynolds averaged form.

Further development by Zhu, Miyata, Kajitani and Watanabe[186, 118], WISDAM-V , reworked the code using a staggered grid ﬁnite volume discretisation. Initially the method did not account for movement in the free surface[186], but this was later implemented into the code[118].

Hino[61] developed a finite difference method that used a body fitted mesh that was fitted to the hulls surface. The method did not align with the free surface however, and a marker and cell scheme was used with mesh points being added and subtracted as the free surface rises and falls, in a manner similar to the original method of Miyata[113]. Later the author[62] switched to using a finite volume artificial compressibility method coupled with a Baldwin–Lowmax[6] turbulence model and a free surface fitted mesh.

Kodama used an Implicit Approximation Factorisation scheme (IAF, a pseudo-compressibility method) to model the flow around a Wigley hull at zero Froude number[80], using a Baldwin–Lowmax turbulence model. A later paper discusses the details of the mesh generation used[81]. The method was then reworked to use a finite-volume scheme, and was applied to modelling double body flows around a flat plate and a series 60 ship hull[82]. Free surface flows were added in the 1993 paper of Liu and Kodama[106], with the free surface being tracked using a marker and cell scheme.

Farmer, Martinelli and Jameson[40, 41] used a finite volume Euler scheme to model the free surface flow about a ships hull using a surface tracking body fitted mesh. The code was compared against solutions calculated with linearised and non-linear potential flow techniques. The code was later modified to model the full Reynolds averaged Navier–Stokes equations using the Baldwin–Lowmax turbulence model[42].

Lilek and Peric[104´ , 103] developed a finite volume method for solving the full Navier–Stokes equations using a SIMPLE type scheme on a non-staggered (or collocated) mesh. A body fitted surface tracking mesh was used, with the true transient solutions being obtained for a variety of free surface problems, including a sloshing flow in a container, and the flow about a Wigley hull. For those with imperfect German, a brief summary of the method is given in English in the book by Ferziger and Peric[43´ ]

Tahara, Stern and Rosen[167] developed a matching scheme that coupled a Reynolds averaged Navier– Stokes method, used to model the boundary layer and near ﬁeld, with an inviscid ﬂow method being

used to model the far field of the flow. The k– # [87] turbulence model was used for the near field flow, CHAPTER 8. VISCOUS FREE SURFACE FLOW 198

being replaced with a two layer model in a later paper by Tahara[166]. Lungu and Mori[107] describe many shortcomings in existing Navier–Stokes free-surface schemes, concentrating on reﬂection at open boundaries and the ﬁrst order nature of the marker and cell free surface tracking scheme.

In two papers[12, 11] Bertram, Laudan and Jensen outline work done using the commercial “STAR” CFD code to model free surface ﬂows about a number of Tanker models. A SIMPLE ﬁnite volume

scheme is used in conjunction with a k- # turbulence model. The work was done for a shipbuilder, with few results being given in the paper.

In addition to the above papers modelling the full free surface problem, a number of researchers have applied various numerical codes to studying the “double body” problem, where no movement of the free surface is allowed and the free surface is treated as a symmetry plane. This is most commonly used to model the complex ﬂows around the sterns of tankers, which feature separated ﬂow and “bilge vortices”.

Papers using the staggered mesh SIMPLE algorithm coupled with the the k- # turbulence scheme to model the double body problem include those by Masuko et al.[109], Patel, Chen, and Ju[22, 125], Patel and Sotiropoulos[160], and Tzabiras[173]. Nonstaggered SIMPLE type schemes have been similarly used in the papers of Hochbaum[64], Graf[52], and Sames[146]. The fact that many of these papers use a ﬁrst order upwind differencing scheme casts some doubt on the usefulness of their results.

8.2 A Numerical Method to Model Free Surface Flow

To model free surface ﬂows there needs to be a method to track the movement of the free surface, and a means to implement the boundary conditions at that surface. With CFD codes there are two broad classes of methods used, and following the nomenclature of Ferziger and Peric´ we classify them as

$ interface capturing methods, where the equations are solved upon a ﬁxed mesh, with the free surface being found within the solution domain. Two commonly used methods are the Marker– And–Cell (MAC) method of Harlow and Welsh[57, 178], where the equations are solved upon a ﬁxed mesh and the free surface is tracked by following particles distributed on the surface, and the Volume-of-Fluid (VOF) method of Hirt and Nichols[63], where the equations are solved on

a ﬁxed mesh, with the surface being located via the use a void fraction, a fraction of % signifying &

a full cell, a fraction of & an empty cell, and a fraction of signifying a cell that is half full

(or half empty depending on your personality). The free surface is taken to be the & contour (' of the void fraction.

$ interface tracking methods, where the free surface is located at one boundary of the mesh, and the mesh deforms as the free surface moves.

For this study an interface tracking method was used. Such schemes tend to suffer less from the smearing of the interface that occurs with the VOF methods, whilst the surface tracking methods used in the MAC scheme also distort the free surface solution through the upwinding nature of the particle tracking. Unfortunately, the remeshing approach entails the regeneration of the grid properties each time the surface is moved, which can be an expensive operation.

With the interface tracking method, one boundary of the domain is located at the free surface. At this boundary there are two boundary conditions that may be imposed;

$ The kinematic condition, which states that the free surface is a sharp boundary, and no ﬂuid crosses the surface. CHAPTER 8. VISCOUS FREE SURFACE FLOW 199

$ The dynamic condition, which states that the forces on both sides of the boundary are in equi- librium.

For a cell at the free boundary, the kinematic condition can be imposed by forcing the mass ﬂux across

the boundary to be zero, )+*-,

& (8.5) *-, where the subscript signiﬁes a value at the free surface.

The dynamic condition is more complex, and requires the calculation of the forces in the ﬂuids on both sides of the boundary, in addition to the surface tension. For the modelling of water waves in still air the shear forces from the air and the surface tension can be neglected, and so the condition reduces to the free surface pressure being at the atmospheric pressure, whilst there is no shear in the surface

velocities. The dynamic condition therefore reduces to

*-,

. .¢/1032

(8.6)

*7,

& (8.7)

*-,

6 /8032 where . is the atmospheric pressure, and is the normal to the free surface.

Within the CFD code used here a non-hydrostatic pressure was used, which is the pressure less the :

hydrostatic component, and so assuming 9 is aligned with the negative axis, the pressure is

.<;>= .?A@

: (8.8)

At the free surface the pressure boundary condition is modiﬁed to

*-,

. ?¥@

(8.9) B where is the local elevation of the surface, relative to some arbitrary datum.

To solve for a free surface ﬂow the dynamic boundary condition is imposed at the free boundary, with the kinematic condition being used to move the free surface, the satisfaction of the kinematic condition signifying a converged solution3. The algorithm used is shown in Figure 8.1. The free surface

boundary conditions are set using the dynamic conditions, setting the gradients in the velocities nor- )

mal to the surface to zero and the pressure to be proportional to the surface elevation. The flow*7, is then EHG& solved, and the mass flux through the free surface is taken as a test of convergence, with DFE as the flow converges.

I set surface pressure I solve ﬂow ﬁeld repeat I move free surface and remesh

I set surface pressure I

solve ﬂow)+*-,ﬁeld

I E check DJE for convergence

Figure 8.1: The free-surface tracking algorithm.

The surface is tracked by calculating the mass ﬂux across the free surface for each cell. The centre of )+*-,

each cell is then displaced by K *-,

§ML

*-,

: (8.10) @ON+P1P

3 UT

An earlier free surface model imposed the kinematic condition at the boundary ( QR(S ) and used the dynamic condition

XWZY V to move the free surface (moving the surface so that []\ ). This method suffered from a Helmholtz instability, such as occurs in the ﬂapping of a ﬂag, and had to be abandoned.

CHAPTER 8. VISCOUS FREE SURFACE FLOW 200

*-, )^*-, L

with being*-, a relaxation parameter (or a pseudo-time step), the mass flux through the cell P_P face, and N the area of the face. The displacement of the cell vertices is found by interpolating the cell face centre values. At boundaries the cell vertex elevation is either fixed (such as at inlets), or is extrapolated from the first internal vertex. For confined flows, where there is no inlet or outlet,

the total volume of the ﬂuid should be conserved, and so the averageK surface displacement should be K zero. This can be enforced by calculating the average displacement : and then subtracting it from

the individual displacements : ,

P_P

P1P

K K K

D (8.11)

: :C` :

)+*-,

However, the sum of mass ﬂuxes across the surface D should equal zero, a non-zero value signifying a non-converged solution to the pressure correction equation. Nevertheless, it is prudent to enforce conservation of mass through Equation (8.11) for cases where the pressure equation isn’t fully converged.

After moving the surface the ﬂow domain is remeshed (a non-orthogonal algebraic mesh generation

scheme being used), and the free surface pressure is set to

*-,

. ?¥@

(8.12) B where is the elevation of the centre of the cell faces lying on the free surface. The flow is solved using the standard SIMPLE pressure-velocity coupling scheme described in Chapter 4. For a fully converged steady flow the surface should be stationary, and the mass fluxes through the free surface should equal zero, satisfying the kinematic condition at the surface. For the numerical scheme the convergence of the free surface algorithm is flagged by the norm of the mass fluxes through the free surface being reduced below some tolerance.

With regards to flow boundaries apart from the free surface, wall boundaries pose no problems, with the elevation of the surface at the wall being extrapolated from the elevations of the interior points. At symmetry planes the gradient in the surface normal to the symmetry plane is zero. Inlet and outlet boundaries present problems however. For inlets the velocity and surface elevation are specified, but the specification of the elevation implies a specification of the inlet pressure, since the pressure at the free surface is a function of elevation only. Thus the conditions at the inlet are overprescribed and prevent the propagation of surface waves out of an upstream boundary.

As for outlet boundaries, the use of a prescribed pressure boundary forces the outlet elevation to be fixed, since the elevation is a function of pressure. Thus outlets in free surface flow calculations were defined using a zero velocity gradient normal to the outlet, the mass flux across the outlet being found using an interpolation of the velocities, with a correction being added to ensure global conservation of mass between the inlet and outlet flows.

8.3 Examples of Free Surface Flow

To test the free surface scheme the code was applied to model two simple free surface flow problems. The first, a driven cavity flow with a free surface, has the advantage of having simple boundary conditions with all boundaries being solid walls with the exception of the free surface, leaving the free surface as the single non-trivial component of the flow. The second problem, two dimensional open channel flow, has rather more troublesome boundary conditions with inlet and outlet boundaries having to be modelled. However it allows the comparison of the calculated solution with an analytic model.

After these two test the code was used to model the ﬂow past a ship hull, a standard Wigley hull CHAPTER 8. VISCOUS FREE SURFACE FLOW 201

section being modelled. The ﬂow was calculated for a range of Reynolds numbers, Froude numbers, and with several differencing schemes, and the results are compared with experimental data.

8.3.1 The Free Surface Driven Cavity

The simplest of the free surface problems modelled was that of a driven cavity with a free surface. A two dimensional cavity was modelled, with ﬁxed vertical side walls, a driven ﬂoor, and a free surface

at the upper boundary, the geometry being shown in Figure 8.2. Results were obtained for ﬂows

& % c &

with a Reynolds number of %a&b& , for Froude numbers varying between and . All results were

calculated using the ULTRA differencing scheme.

g h f

l l

e d l l

l l

k ©

Figure 8.2: The geometry for the free surface driven cavity problem.

The streamfunction for the free surface flow at a Froude number of 2.0 is shown in Figure 8.3, along- side the streamfunction for a conventional driven cavity, where the top surface is replaced by a no-slip stationary wall, at the same Reynolds number. It is readily seen that for the free surface problem the surface has been displaced by the flow and no longer lies on the horizontal. Also apparent is that the volume flow is greater for the free surface cavity, it extends further up the centre of the cavity, and the two recirculation bubbles in the top corners for the conventional cavity are missing. These last three features are all due to the lack of a restraining shear force upon the upper boundary.

The effect of Froude number on the free surface elevation is shown in Figure 8.4. As the Froude number increases the disturbance in the free surface position grows. This is more clearly shown in the

plot of surface elevation versus Froude number, where the elevation of the free surface is seen to be

¢¡ P proportional to the second power of the Froude number, npo .

This relationship can be explained by an energy conservation argument. The potential energy due

@Or @ q

to elevation is given as n , whilst the kinetic energy is . By equating the two energies, and

rts

recalling that the Froude number is deﬁned as h , then

P P

§vuxw

n (8.13)

To demonstrate the mesh convergence of the free surface ﬂow, the free surface proﬁle for a cavity

with Froude number c is shown in Figure 8.5, together with a plot of depth of the trough on the free surface plotted as a function of mesh size. As the mesh is reﬁned the free surface elevation is seen to be converging towards a mesh independent position. CHAPTER 8. VISCOUS FREE SURFACE FLOW 202

Figure 8.3: The streamlines for a driven cavity ﬂow, with a conventional ﬂoor driven cavity on the

left, and for a ﬂoor driven cavity with a free surface on the right. The ﬂow is calculated using ULTRA

§ §

%z&H& c & &¢% &|{X}~{& %

differencing on a %zy¦% mesh, for Re and Fr . Contours are at intervals for .

0.025 -0.1

0.02

0.015 -0.01 0.01

0.005 0 -0.001

-0.005 Free Surface Elevation Free Surface Elevation -0.01 -0.0001 Fr 0.1 -0.015 Fr 0.2 Fr 0.5 -0.02 Fr 1.0 Fr 2.0 -0.025 -0.00001 0 0.2 0.4 0.6 0.8 1 0.1 1

x Fr

§ %a&b&

Figure 8.4: The free surface elevation for the driven cavity, calculated for a ﬂow with £¥¤ on

¦¡

& %|{ {c

a %ay¦% mesh, with Froude numbers in the range . The free surface proﬁle is shown on the left. On the right is a plot of the depth of the trough on the left hand side of the cavity plotted as a function of Froude number.

Conclusions

The free surface driven cavity flow allows an easy test of a viscous free surface solver. The free surface modelling scheme is seen to converge to a grid independent solution as the mesh is refined, suggesting that the method is consistent and is correctly modelling the flow. The presence of the free

surface is seen to affect the ﬂow in the cavity, increasing the volume ﬂow, and theK movement of the

¢¡ P free surface from it’s rest position is seen to depend on a relationship of the form n+o , which can be explained by an energy argument.

8.3.2 Two-Dimensional Channel Flow

The second problem modelled was a quasi-two-dimensional open channel ﬂow, the geometry of the

problem being shown in Figure 8.6. A free surface ﬂow enters the domain at the left boundary with

contraction occurs in the channel, with the width of the channel being reduced by %a& . Whilst this CHAPTER 8. VISCOUS FREE SURFACE FLOW 203

0.025 -0.005 15x15 0.02 19x19 27x27 35x35 -0.006 0.015 51x51 67x67 0.01 99x99 -0.007 131x131

0.005 0 -0.008

-0.005

Free Surface Elevation Free Surface Elevation -0.009 -0.01

-0.015 -0.01

-0.02

-0.025 -0.011 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 x dx

Figure 8.5: Convergence of the calculated free surface elevation with the reﬁnement of the mesh. Free

¢¡§ §

£"¤ %z&H& %

surface proﬁles shown for a ﬂow with c for a range of meshes in the range to

P K

%zy¦% . Shown on the left are the proﬁles of the calculated free surfaces, on the right the depth of the

trough on the left side of the cavity, plotted as a function of mesh size . n contraction occurs in the n axis, the ﬂow is modelled as two-dimensional with motion in the axis

being ignored, and so the problem is referred to as quasi-two-dimensional. The ﬂow then continues

17 for a distance of H before exiting the domain. The contraction has a smooth proﬁle, of length

and is centred at a station b downstream of the inlet. The upper surface of the domain is modelled as

a free surface, whilst the ﬂoor of the channel is a slip surface. e

Inlet d Outlet

llllllllllllllllllll

g b f g H f

llllllll

lllllllllll

f g

llllllllllll

l lllllll

Figure 8.6: The geometry for the open channel ﬂow problem.

For this ﬂow the Froude number is deﬁned as

¦¡§ ©

(8.14)

whilst the Reynolds number of the ﬂow is

£"¤ (8.15)

By modelling the problem as an inviscid one dimensional ﬂow, the elevation of the free surface can be calculated using the Energy Head method of Bakhmeteff (for example, see Henderson[59]). The

energy head at a location is given by

: (8.16) c

CHAPTER 8. VISCOUS FREE SURFACE FLOW 204

r where : is the depth of the ﬂow, the velocity of the stream, and the acceleration due to gravity. For an open channel ﬂow with no viscous losses the energy head remains constant along the channel.

For a channel with a varying cross section, as shown in Figure 8.6, Equation (8.14) can be rewritten in

terms of the volume ﬂow rate ,

: (8.17) where is the breadth of the channel. Substituting the volume ﬂow rate into the deﬁnition for the

energy head gives

:x (8.18)

P P

and are both constant along the channel, so Equation (8.18) can be used as a non-linear expression

for : , and solved for the depth of the flow along a channel of varying breadth . For any section there can be two real solutions, one with a Froude number greater than 1 (a supercritical flow), the other with a Froude number less than 1 (a subcritical flow). As the energy head is reduced the two solutions converge, and there is a point where both solutions converge at the critical flow where the Froude number equals 1. For the test of the CFD code only subcritical flows were modelled.

The problem was modelled using the numerical free surface code, with the solutions being compared

with the Energy Head model. The ﬂow was modelled using both the FOU and QUICK differencing

% & c & y & % %z& %a&H& %z&H&H&

schemes, at Froude numbers of & and for Reynolds numbers of and , on

yH&y b&U

meshes of and & cells.

In Figure 8.7 the free surface elevation and the sectional energy head are shown for ﬂows calculated

using the Energy Head method, and using the numerical scheme with FOU and QUICK differencing. %a&b&

The numerical solutions are calculated on a &^ mesh for a Reynolds number of for Froude

% &

numbers in the range of & to .

Since the surface elevation given by the energy head method depends only on the cross section of the ﬂow, the method predicts a smooth transition between the two channel widths. This is in contrast to the solutions obtained with the viscous numerical scheme, with both differencing schemes exhibiting

some waviness in free surface elevation downstream of the contraction. For the solutions at Froude

y &

numbers of & and both methods also predict a small rise in the free surface level immediately

upstream of the channel contraction.

The difference between the solutions calculated with the FOU and QUICK differencing schemes is most visible in the waves generated downstream of the contraction. For the FOU calculations the

waves quickly die away for all Froude numbers. However, for the QUICK solutions the waves dampen

¢¡§

at a much slower rate, and for the & solution the waves extend to the downstream boundary and the solver could not converge to a steady solution. In addition, for this calculation the ﬂow upstream of the contraction has failed to converge, and is seen to exhibit grid scale oscillations.

The effect of the Reynolds number upon the free surface elevation is shown in Figures 8.8 and 8.9, for

% & y

open channel flow with Froude numbers of & and respectively. In both figures flows calculated

%a& %z&H& %a&H&b& using FOU and QUICK differencing at Reynolds numbers of % and are compared with the analytic Energy Head solutions.

Low Reynolds numbers are seen to perturb the ﬂow from the inviscid analytic solution, but as is seen in Figure 8.10, as the Reynolds number is increased the free surface elevation at the downstream boundary converges to the analytic solution.

The effect of viscosity on the free surface and energy head is most apparent at low Reynolds numbers

when the viscous forces are greatest. At Reynolds numbers of % , the viscous forces cause a large standing wave in the contraction at both Froude numbers. In addition, the loss of energy due to viscous forces causes a decrease in the energy head downstream of the contraction, lowering the free surface elevation at the outlet. CHAPTER 8. VISCOUS FREE SURFACE FLOW 205

Contraction Flow, Analytic Solution Contraction Flow, Analytic Solution 0.01 1.2 Fr 0.1 0.005 Fr 0.2 Fr 0.3 Fr 0.4 0 1.15

-0.005

-0.01

1.1 -0.015

-0.02 Free Surface Elevation Velocity at Free Surface 1.05 -0.025

-0.03 Fr 0.1 Fr 0.2 -0.035 Fr 0.3 1 Fr 0.4 -0.04 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Contraction Flow, Re 100, FOU Contraction Flow, Re 100, FOU 0.01 1.2 Fr 0.1 0.005 Fr 0.2 Fr 0.3 Fr 0.4 0 1.15

-0.005

-0.01

1.1 -0.015

-0.02 Free Surface Elevation Velocity at Free Surface 1.05 -0.025

-0.03 Fr 0.1 Fr 0.2 -0.035 Fr 0.3 1 Fr 0.4 -0.04 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Contraction Flow, Re 100, QUICK Contraction Flow, Re 100, QUICK 0.01 1.2 Fr 0.1 0.005 Fr 0.2 Fr 0.3 Fr 0.4 0 1.15

-0.005

-0.01

1.1 -0.015

-0.02 Free Surface Elevation Velocity at Free Surface 1.05 -0.025

-0.03 Fr 0.1 Fr 0.2 -0.035 Fr 0.3 1 Fr 0.4 -0.04 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Figure 8.7: Free surface elevation (left) and surface velocity (right) for a two dimensional open channel with a contraction, using the Energy–Head method (top), and the CFD code using FOU differencing (middle) and QUICK differencing (bottom). CHAPTER 8. VISCOUS FREE SURFACE FLOW 206

As would be expected, the two differencing schemes give similar solutions at low Reynolds numbers.

¦¡§ %

In addition, the solutions are almost identical for the low Froude number ﬂow ( & ) where the

ﬂow is nearly one-dimensional.

y %a&b&

However, at a Froude number of & and Reynolds numbers of and greater, the solutions exhibit a marked dependence on the differencing scheme used. The FOU schemes show few waves downstream of the contraction, at most there being two waves in the calculated energy head (including that in the contraction). In contrast the QUICK solutions have many more waves, with the energy head having six full waves extending to the downstream boundary. Clearly the numerical diffusion of the FOU scheme is having a marked effect on the solution, with the excess diffusivity suppressing the formation of waves downstream of the contraction. This excessive viscous dissipation in the FOU solutions also exhibits itself as a lowering of the free surface elevation at the ﬂow outlet, as shown in Figure 8.10, the FOU solutions having a lower free surface than those calculated with the QUICK differencing scheme.

Contraction Flow, Fr 0.1, FOU Contraction Flow, Fr 0.1, QUICK 0.0015 0.0015 Analytic Analytic Re 1 Re 1 0.001 Re 10 0.001 Re 10 Re 100 Re 100 Re 1000 Re 1000 0.0005 0.0005

0 0

-0.0005 -0.0005 Free Surface Elevation Free Surface Elevation -0.001 -0.001

-0.0015 -0.0015

-0.002 -0.002 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Contraction Flow, Fr 0.1, FOU Contraction Flow, Fr 0.1, QUICK 1.007 1.007 Analytic Analytic Re 1 Re 1 Re 10 Re 10 Re 100 Re 100 1.0065 1.0065 Re 1000 Re 1000

1.006 1.006

Energy Head 1.0055 Energy Head 1.0055

1.005 1.005

1.0045 1.0045 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Figure 8.8: Free surface elevation (top) and Energy–Head (bottom) for a two dimensional open chan-

nel with a contraction, using FOU differencing (left) and QUICK differencing (right). Solutions for a

¦¡§

% %£¥¤%a&b&H&

Froude number of & and Reynolds numbers in the range .

Finally, the convergence of the method with decreasing grid size is shown in Figure 8.11. The free

surface elevation, and energy head are calculated using the QUICK differencing scheme at a Reynolds

& % yb&^

number of %z&H& and a Froude number of . Solutions were calculated upon three meshes, of

b&U &|

y and cells respectively.

For the low Froude number problem, the perturbation in the calculated energy head reduces as the

mesh is reﬁned, suggesting that this is an artifact of the numerical error, and that there is negligible

¢¡x§ y

perturbation in the fully converged solution. However, at the higher Froude number ( & ), the perturbation in the energy head does not seem to be converging to zero, and the waves downstream of the contraction are persistent, although the variation in wavelength suggests that the correct wavelength of the ﬂow has not been captured. CHAPTER 8. VISCOUS FREE SURFACE FLOW 207

Contraction Flow, Fr 0.3, FOU Contraction Flow, Fr 0.3, QUICK 0.01 0.01 Analytic Analytic Re 1 Re 1 Re 10 Re 10 0.005 Re 100 0.005 Re 100 Re 1000 Re 1000

0 0 -0.005 -0.005

Free Surface Elevation -0.01 Free Surface Elevation -0.01

-0.015 -0.015

-0.02 -0.02 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Contraction Flow, Fr 0.3, FOU Contraction Flow, Fr 0.3, QUICK 1.06 1.06 Analytic Analytic Re 1 Re 1 1.058 1.058 Re 10 Re 10 Re 100 Re 100 1.056 Re 1000 1.056 Re 1000

1.054 1.054

1.052 1.052

1.05 1.05 Energy Head Energy Head

1.048 1.048

1.046 1.046

1.044 1.044

1.042 1.042 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Figure 8.9: Free surface elevation (top) and Energy–Head (bottom) for a two dimensional open chan-

nel with a contraction, using FOU differencing (left) and QUICK. differencing (right). Solutions for

¦¡§

y %x£¥¤%z&H&b&

a Froude number of & and Reynolds numbers in the range .

Convergance of Free Surface Location, Fr 0.1 Convergance of Free Surface Location, Fr 0.3 -0.00118 -0.0118

-0.0012 -0.012

-0.0122 -0.00122

-0.0124 -0.00124 -0.0126

-0.00126 -0.0128 -0.00128 -0.013

Free Surface Elevation -0.0013 Free Surface Elevation -0.0132

-0.00132 -0.0134

Analytic Analytic -0.00134 FOU -0.0136 FOU QUICK QUICK -0.00136 -0.0138 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000 Reynolds number Reynolds number

Figure 8.10: Convergence of downstream free surface elevation with increasing Reynolds number, for

% & y &|

Froude numbers of & (left) and (right), on a mesh.

CHAPTER 8. VISCOUS FREE SURFACE FLOW 208

This is also suggested by the plots of free surface elevation (at the downstream boundary) plotted as a function of mesh size in Figure 8.12. As the mesh is reﬁned the calculated free surface elevation converges towards the analytic solution, with the QUICK solution having a lower error than that calculated with the FOU differencing scheme. However, there is still a substantial error for the solution calculated on the ﬁnest mesh tested.

Contraction Flow, Re1000, Fr 0.1, QUICK Contraction Flow, Re1000, Fr 0.3, QUICK 0.0002 0.002 Analytic Analytic 32 x 5 32 x 5 0 0 62 x 8 62 x 8 92 x 11 92 x 11 -0.002 -0.0002

-0.004 -0.0004 -0.006

-0.0006 -0.008 -0.0008 -0.01

Free Surface Elevation -0.001 Free Surface Elevation -0.012

-0.0012 -0.014

-0.0014 -0.016

-0.0016 -0.018 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Contraction Flow, Re 1000, Fr 0.1, QUICK Contraction Flow, Re 1000, Fr 0.3, QUICK 1.0055 1.05 Analytic Analytic 32 x 5 32 x 5 1.049 1.0054 62 x 8 62 x 8 92 x 11 92 x 11 1.048 1.0053

1.047 1.0052

1.046 1.0051 1.045 Energy Head Energy Head 1.005 1.044

1.0049 1.043

1.0048 1.042

1.0047 1.041 -4 -2 0 2 4 6 -4 -2 0 2 4 6 x x

Figure 8.11: Convergence of Free surface elevation (top) and Energy–Head (bottom) for a two dimen- %

sional open channel with a contraction, using QUICK differencing, for a Froude number of & (left)

y yH&y H& &

and & (right). Calculations performed on three meshes of , and cells, for a ﬂow

§ %a&b&H& with a Reynolds number of £¥¤ .

Conclusions

Subcritical free surface channel ﬂow was modelled for a channel with a contraction, with inlet Reynolds

%z&H&H& & % &

numbers in the range % to and inlet Froude numbers in the range to . Solutions were

calculated using the FOU and QUICK differencing schemes, and the results were compared with predictions of the Energy Head method.

The effect of viscosity on the ﬂow is seen, with viscosity damping wave generation, and energy losses occurring due to viscous dissipation lowering the free surface elevation at the ﬂow outlet. As the Reynolds number increased the free surface at the outlet rose, converging towards the Energy Head solution, and waves appeared downstream of the contraction.

The excessive dissipation of the FOU differencing scheme led to a reduction in the downstream free surface elevation, and damped the waves generated in the contraction, making the ﬂow appear much like one with a lower Reynolds number. As the mesh was reﬁned the methods converged to a grid independent solution. CHAPTER 8. VISCOUS FREE SURFACE FLOW 209

Convergance of Free Surface Location, Re 100 Fr 0.1 Convergance of Free Surface Location, Re 100 Fr 0.3 -0.001185 -0.0119

-0.01195 -0.00119

-0.012 -0.001195

-0.01205

-0.0012 -0.0121

-0.001205 -0.01215 Free Surface Elevation Free Surface Elevation -0.00121 -0.0122

-0.001215 Analytic -0.01225 Analytic FOU FOU QUICK QUICK -0.00122 -0.0123 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 dx dx

Figure 8.12: Convergence of downstream boundary free surface elevation with respect to mesh size,

& % & y

at a Reynolds number of %a&H& , and Froude numbers of (left) and (right).

CHAPTER 8. VISCOUS FREE SURFACE FLOW 210

8.3.3 Free Surface Ship Flow

As a final application of the method, the free surface solver was used to model the flow about a Wigley ship hull. The Wigley hull is a standard ship shape, defined with a rectangular section in side elevation, and with a parabolic profile for the beam both in length and draft. The hull section was originally designed by Wigley for providing experimental data for comparison with Mitchell’s thin ship theory, and over the years a considerable body of experimental data has been built up for these

hulls.

% q

For the numerical calculations, a hull with a length:beam:draft ratio of : q : was used, the beam of

q] q8¡

the hull having the proﬁle

Pa©

§£¢

c n

?¦¥ ¥ ?«ª

¨§ §

% % ¬¨

: (8.19)

where ¢ is the maximum beam, the length of the hull, and the draft. Such a hull is drawn in section in Figure 8.13 with a perspective view being shown in Figure 8.14.

Plan

Bow Elevation Beam Elevation

Figure 8.13: Plan, beam and bow elevations of a Wigley hull.

Figure 8.14: Perspective view of a Wigley hull.

c & c®yby & cHO¯ & y

Flows were calculated with Froude numbers of & and , for Reynolds numbers in the

range of %a&H°£¥¤±%z&H² , the range of Reynolds numbers being limited by the lack of a turbulence model in the ﬂuid modelling code. The ship hull was not allowed to trim, but instead was ﬁxed in space. The mesh was generated using a non-orthogonal algebraic grid generator. CHAPTER 8. VISCOUS FREE SURFACE FLOW 211

Free Surface Elevation at Re 1000 Free Surface Elevation at Re 10000 0.015 0.015 Froude No. Froude No. 0.2 0.2 0.233 0.233 0.267 0.267 0.3 0.3 0.01 0.01

0.005 0.005

0 0 Elevation at centerline/hull Elevation at centerline/hull

-0.005 -0.005

-0.01 -0.01

-1 0 1 2 3 -1 0 1 2 3 %z&®³ Figure 8.15: Wave proﬁles along the centreline of the domain at Reynolds numbers of %a&b° and .

Free Surface Elevation at Fr 0.267 Free Surface Elevation at Fr 0.3 0.012 0.015 Reynolds No. Reynolds No. 1e3 1e3 0.01 1e4 1e4 1e5 1e5 0.008 0.01

0.006

0.004 0.005

0.002

0 0 -0.002 Elevation at centerline/hull Elevation at centerline/hull -0.004

-0.005 -0.006

-0.008

-0.01 -0.01

-1 0 1 2 3 -1 0 1 2 3

c® ¯ & y

Figure 8.16: Wave proﬁles along the centreline of the domain at Froude numbers of & and .

The calculated free surface elevations along the centreline of the domain (or on the surface of the hull where present) are shown in Figures 8.15 and 8.16, the former showing the effect of a variation in

Froude number at a ﬁxed Reynolds number, whilst the latter has a ﬁxed Froude number with a varying

Reynolds number. The bow (or front) of the hull is located at & , and the stern (or rear) is

' §

located at & . As can be seen, for the range of Froude numbers and Reynolds numbers tested

' the Froude number has a strong effect on the wake distribution whilst the Reynolds number has only a minor effect.

Free Surface Elevation at Fr 0.267 Free Surface Elevation at Fr 0.3 0.015 0.015 Reynolds No. Reynolds No. 1e3 1e3 1e4 1e4 1e5 1e5 4.5e6 ITTC 1983 1e7 Shearer 1951 0.01 9e6 Shearer 1951 0.01

0.005 0.005

Elevation at hull 0 Elevation at hull 0

-0.005 -0.005

-0.01 -0.01

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4

cHO¯ & y

Figure 8.17: Wave proﬁles along the ship’s hull at Froude numbers of & and . Numerical

calculations compared with experiment. CHAPTER 8. VISCOUS FREE SURFACE FLOW 212

Free Surface Elevation at Fr 0.233 Re 10000

Free Surface Elevation at Fr 0.267 Re 10000

Free Surface Elevation at Fr 0.3 Re 10000

c®yHy & cHO¯ & y

Figure 8.18: Free surface for ﬂow past a Wigley hull at Froude numbers of & and and

at a Reynolds number of %a&®³ . The vertical axis has been stretched for clarity. CHAPTER 8. VISCOUS FREE SURFACE FLOW 213

Free Surface Elevation at Fr 0.3 Re 1000

Free Surface Elevation at Fr 0.3 Re 10000

Free Surface Elevation at Fr 0.3 Re 100000 y

Figure 8.19: Free surface for ﬂow past a Wigley hull at a Froude number of & and Reynolds numbers

%a&H³ %z&H² of %z&H° and . CHAPTER 8. VISCOUS FREE SURFACE FLOW 214

The calculated free surface profiles along the hull are shown in Figure 8.17 and are compared with some experimental data. Whilst the experimental profiles are at a much higher Reynolds number than the numerical calculations, and are for a turbulent flow as opposed to the laminar calculations, it was hoped that the relative insensitivity of the solutions to Reynolds number would give some agreement between numerical and experiment. However, whilst the order of magnitude of the wave

elevation is approximately correct, the wave proﬁles do not agree. It should be noted however that cHO¯

the experimental wave proﬁles do not agree well for the & Froude number case where two sets of

data are available.

c®yby & cHO¯ & y

Plots of the free surface elevation are given in Figure 8.18 at Froude numbers of & and

and at a Reynolds number of %a&H³ . The hull of the vessel is at the right rear of the domain travelling towards the right. The generated waves look plausible, being conﬁned to a region aft of the ships hull and radiating at an angle to the direction of travel.

Free Surface Elevation at Fr 0.3 Re 100000 y

Figure 8.20: Free surface for ﬂow past a Wigley hull at a Froude number of & and a Reynolds

number of %z&H² . Numerical predictions compared with the Kelvin Wave Angles.

This is better shown in Figure 8.19 where the free surface elevation for ﬂow past a Wigley hull at a

y %a&H° %a&H³ %a&b²

Froude number of & and Reynolds numbers of and are plotted, and in Figure 8.20 where

& y

the ﬂow at a Reynolds number of %z&H² and a Froude number of is compared with the Kelvin Wave

q®¶ q¶

X8´tµ q q % q

Angles, which are 8´tµ for the outer boundary of the wake, and

Pº8» P ¢ 'b¹ ° ·½¼|¸

for the angle of the w° ·¸aves to the centreline (from Lighthill[102]). There is a good agreement 'b'H¹ between the numerical and theoretical wake angles. CHAPTER 8. VISCOUS FREE SURFACE FLOW 215

Maximum Wave Height Minimum Trough Depth 0.013 -0.004 Reynolds No. 1000 10000 0.012 100000 -0.005

0.011 -0.006

0.01 ¾ ¿ -0.007

0.009 Maximum Wave Height Minimum Trough Depth -0.008 0.008

-0.009 0.007 Reynolds No. 1000 10000 100000 0.006 -0.01 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 Froude Number Froude Number

Figure 8.21: Maximum wave height, and minimum trough depth, as functions of Froude number.

Finally the magnitude of the wake elevation is plotted as a function of Froude number in Figure 8.21. ¢¡ As with the driven cavity ﬂow there is a P relationship between surface elevation and Froude number, with a lesser dependency on Reynolds number, the lower Reynolds numbers damping the movement of the free surface.

8.4 Conclusions

A surface tracking method using a surface fitted mesh and a collocated SIMPLE CFD algorithm has been developed to model steady state viscous free surface flows. A pressure boundary condition is imposed at the free surface, with the flow across the free surface being tracked to find the free surface geometry.

The method has been tested on two simple free surface problems, a two-dimensional driven cavity with a free surface, and a two-dimensional open channel ﬂow. For the driven cavity the method displayed mesh convergence, and the solutions revealed a relationship between free surface elevation

and Froude number of the form K

¢¡

P :o (8.20) which can be explained by an energy argument.

For the open channel flow the calculated solutions were compared with the predictions of the Energy Head model. The solutions are in good agreement, converging with the Energy Head solutions as the mesh is refined and as the Reynolds number is increased. Viscosity is shown to have a strong effect on the free surface position, with the free surface waves being damped for the low Reynolds number flows, and the length of the wave train generated by the contraction increasing as the Reynolds number is increased. Viscous dissipation also affected the free surface elevation at the outlet, with the free surface elevation (and calculated energy of the exiting flow) being lower for the low Reynolds number solutions. The excessive diffusivity of the first order differencing schemes is also displayed by the solutions to the open channel flow, with the solutions calculated using the FOU scheme showing more viscous losses and less waves at the surface than the solutions calculated with the higher order QUICK scheme.

The method was finally applied to the modelling of free surface flow about a ships hull, the calculations being performed for a range of Reynolds and Froude numbers, and the solutions compared with experiment. Whilst the Reynolds number for the computed flows is lower than that for the available experimental data, and there was no turbulence model implemented in the numerical model, the results are still in broad agreement with each other. The wake angles are as those predicted by Kelvin’s

ship wave theory, with an outer wake envelope of % , and with the waves at the edge of the wake O¹

aligned at an angle of . 'H'H¹