Investigating Dead Water

Eric William HesterDr Geoffrey Vasil

University of Sydney

February 28, 2016

1

1 Introduction

The term ‘dead water’ refers to water experiencing density stratification, such as occurs with warmfresh water lying on top of cold, salty sea water. The ‘dead’ moniker comes from the increasedresistance that boats experience in such waters. Previous research (Ekman, 1906; Mercier et al.,2011) has concluded that this drag stems from the generation of internal waves at the boundary ofthe fresh and salty water.

The aim of this AMSI summer scholarship has been to investigate the mechanisms behind thiseffect computationally. By modelling the fluid using incompressible Boussinesq Hydrodynamics(Kundu et al., 2012), and using Immersed Boundary Methods (Mittal and Iaccarino, 2005) toincorporate the boat, the flexible spectral code dedalus (Lecoanet et al., 2016) was used to simulatethe dead water phenomenon. A method for calculating the force on the boat was developed andused to both confirm the existence of the effect, and infer its cause, being the generation of intenselow pressure regions behind the boat by the aforementioned internal waves.

2 Background

While there are possible accounts of dead water going back millenia, with Tacitus (98) mentioningseas north of Germany in which one could not row a boat, this effect was first properly observedduring the famous 1893-96 polar expedition of the great 19th century explorer Dr Fridtjof Nansen(Nansen and Sverdrup, 1897). Nansen’s plan was to freeze his ship, Fram, in the ice north of Russiaand ‘ride’ his way to the pole. Fram did not end up heading directly north however, leading himto set out on foot with a single companion, Hjalmar Johansen, a year and a half into the voyage.Unfortunately he was forced to turn back before the pole, reaching a latitude of 86, but he did notreturn empty handed, bringing back a wealth of scientific observations.

Among them was the perplexing phenomenon of dead water, which he noted ‘occurs only wherea surface layer of fresh water rests upon the salt water of the sea’ – a frequent happening nearthe glaciers and river mouths of the far north. Upon returning to Norway in 1896, he relayed thisinformation to the scientist Dr Vagn Walfrid Ekman.

Ekman investigated the dead water phenomenon thoroughly (Ekman, 1906), performing modelexperiments, formulating the basis of a mathematical theory, and even interviewing sailors on first-hand experiences with such waters. His research indicated that the key to the increase in boatdrag, plotted in Fig. 1, was the generation of internal waves at the boundary of the salty and freshwater, shown in Fig. 2.

Research on dead water since Ekman has progressed at a slow pace however. Theoreticalanalysis of the phenomenon has been quite restricted, generally being restricted to some form oflinear analysis, owing to the complexities of the full, non-linear problem (Miloh et al., 1993; Yeungand Nguyen, 1997).

Experimental investigations of the increased drag have fared slightly better, with one of the primeexamples since, a five year old experimental investigation by Mercier et al. (2011), confirming andextending Ekman’s findings to more general stratifications. Despite the aid of modern experimentaltechniques, a deeper understanding of the phenomenon still remains lacking however, with theextant analysis of the problem deemed insufficient by the authors.

2

Figure 1: Ekman’s resistance vs boat speed curves. Thecrosses are experimental data, while the remaining curvesare theoretical. The data is taken as average resistances atthe approximate speeds shown, while the dead water resis-tance model is from linear theory, which Ekman has fittedto the data. Ekman’s experiments used a constant pullingforce, requiring the averaging of velocities (Ekman, 1906).

Figure 2: Time series ofEkman’s experiments, clearlyshowing the generation of in-terfacial waves between thefresh and salt water. Themodel boat is 20 cm long,(Ekman, 1906).

Figure 3: Picture of experimental setup used by Mercier et al. (2011), which similarly applied con-stant pulling forces to the boat. Three layer and linear density stratifications were also investigated,with the authors noting that the drag peaked when the boundary wave crest would break on therear of the boat, which is 20 cm long.

3 Project Motivation and Outline

Previous research on the dead water phenomenon has largely been either experimental (Ekman,1906; Mercier et al., 2011), or theoretical (Ekman, 1906; Yeung and Nguyen, 1997). Consequently,such research has been constrained either by physical restrictions, or simply through the intenseanalytical difficulties associated with fluid mechanics.

This project seeks to circumvent these constraints and attain a deeper understanding of thephenomenon through computational analysis, only recently made possible with the advent of suffi-ciently powerful computers and software.

We utilise the flexible dedalus code, a partial differential equation (PDE) solver, to simu-late incompressible Boussinesq hydrodynamics, a theoretical approximation to the full set of fluidequations valid for small density variations. By exploiting Immersed Boundary Methods, we simplyincorporate the boat into the system of equations, and thereby simulate the dead water phenomenon.

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4 Dedalus

Dedalus is an open source framework that uses spectral methods to solve Partial DifferentialEquations (PDE’s) (Lecoanet et al., 2016), developed in part by the supervisor of this project,Dr Geoffrey Vasil. It is eminently flexible and easy to use, being written in Python and supportingalmost arbitrary equation sets, yet also very powerful, utilising high performance scientific Pythonlibraries, and supporting MPI parallelisation. A recent paper using dedalus to simulate cold waterconvection is shown in Fig. 4.

The principal behind the spectral methods underlying dedalus involves expressing quantitiesas sums of particular classes of functions, such as Fourier series, or Chebyshev polynomials. Byunderstanding how the derivatives of these functions couple different spectral modes, it is possibleto then reduce the set of PDE’s down to coupled ordinary differential equations (ODE’s), a markeddecrease in complexity. Dedalus then reduces the problem to a matrix problem involving thecoefficients of the spectral functions. By expressing them in a basis returning a sparse matrix(incorporating any boundary conditions), dedalus can then evaluate the system very quickly.

Dedalus can be used to solve both linear and nonlinear cases of initial value problems, boundaryvalue problems, and eigenvalue problems, though our project focuses on initial value problems.

NUMERICAL SIMULATIONS OF INTERNAL WAVE . . . PHYSICAL REVIEW E 91, 063016 (2015)

TABLE II. Dimensionless numbers characterizing the convectionin the lower part of the domain. Lconv is taken to be 0.22 m. We use⟨·⟩x,... to denote an average with respect to the variables listed in thesubscript. The z average in the calculation of Nu is only within theconvection zone.

Ra = gαT 20 L3

conv

νκRe = urmsLconv

νNu = −⟨wT ⟩x,z,t

κT0/Lconv

5.8×107 100 6

to Ttop at z = 0.35 m at the top of the domain. We alsoadd low-amplitude random noise to the temperature field toinitiate convection. In this paper, we analyze the times between35 287 and 39 122 s. This corresponds to a period of aboutten convective turnover times, starting about 70 convectiveturnover times after the beginning of the simulation. We foundthat ten convective turnover times is long enough to buildsufficient statistics to describe both the convection and waves.Although we simulate many convective turnover times, thesimulation up to 40 000 s corresponds to only 4% of a thermaltime, so the thermal structure continues to evolve on longtime scales throughout the simulation. Several dimensionlessnumbers describing the convection in this simulation aregiven in Table II. Because our equations include a Newtoniancooling term, we do not have a constant heat flux through theconvection zone. Thus, we calculate the Nusselt number usingthe average value of wT in the convection zone.

B. Characteristics of convectively generated waves

The convective excitation of waves is depicted in Fig. 1.Panel (a) shows the typical state before a major excitationevent. The convective region contains two counterrotatingconvective cells, with up flows (represented by red cold,buoyant fluid) along the sides of the domain, and a down flow

(represented by blue hot, dense fluid) in the center. Becausewater has a Pr sufficiently higher than 1, the thermal boundarylayer at the bottom of the domain is unstable to the formation ofbuoyant plumes. The turnover frequency of the convective cells(∼2×10−3 Hz) and the plume ejection frequency (∼10−2 Hz)are the important time scales in the convection zone. Theyellow curve in Fig. 2(c) shows the kinetic energy spectrum atthe top of the convection zone (z ≈ 0.19 m). Although thereis a peak at the turnover frequency, the spectrum is fairlyflat between the turnover frequency and the plume ejectionfrequency. At frequencies higher than the plume ejectionfrequency, the spectrum falls off rapidly.

Panel (a) of Fig. 1 shows a particularly vigorous plumeon the right side of the domain at about z = 0.16 m. Thisplume rises into the stably stratified region (water above 4 C)in panel (b), moving the interface between the convectiveand stably stratified regions upwards and generating strong,localized IGWs. The plume is deflected by the stratified fluidabove it, and it deflects leftward. This allows the interface tolower [panels (c) and (d)].

Figure 1 shows the IGWs generated by the plume in thelower right corner of the stably stratified region propagatetoward the upper-left. However, the phase velocity is towardthe lower-left—the upper part of the wave packet starts withpositive vorticity [panel (b)], but shifts to having negative vor-ticity [panel (c)], and then positive vorticity again [panel (d)].This is consistent with the IGW dispersion relation, whichimplies that the group velocity is perpendicular to the phasevelocity. The Supplemental Materials include a movie showingthe flow evolution from 34 630 s to 39 332 s [27].

IGWs are continually excited as the plumes detaching fromthe bottom boundary layer approach the interface. Viscositytypically damps the waves before they can propagate tothe top of the domain. This is depicted in Fig. 2(a), aspectrogram of the kinetic energy density, i.e., a plot ofω⟨K⟩x ≡ 0.5 ω⟨|u(ω,x,z)|2⟩x as a function of ω and z,

FIG. 1. (Color online) Four simulation snapshots near t = 37 400 s. The bottom part of the domain (below the thick black line) shows thetemperature field. Recall that below 4 C, cold water (red) is less dense than hot water (blue). The thick black line is the 5 C isotherm andshows the boundary between the convective region (below) and the stably stratified region (above). The top part of the domain (above the thickblack line) shows the vorticity field associated with IGWs.

063016-3

Figure 4: Dedalus simulation of convection in cold water by Lecoanet et al. (2015). Waterexperiences a minimum in density at 4 C, leading to convection when colder water is heated frombelow. The top part of the plot shows the fluid vorticity, highlighting the wave generation takingplace, while the bottom half represents the fluid temperature. The paper successfully determinedthe mechanism behind the wave excitation, attributing it to bulk excitation from Reynolds stresses,as opposed to interface forcing.

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5 Mathematical system

5.1 Fluid Dynamics Operators

Before detailing the Initial Value Problem considered in our project, we first explain the materialderivative, an important differential operator in the field of fluid dynamics. The material derivativecalculates the change in a quantity that is following the fluid, thus requiring the incorporation ofboth change in time and position. Provided the time t, spatial coordinates (x, y, z), and the fluidspeed u = (u, v, w), the convective derivative takes the following form,

D

Dt=

∂

∂t+ u · ∇ =

∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z, (1)

which stems simply from applying the chain rule to the total time derivative operator.

The key point is that this operator is nonlinear, invalidating many analytical techniques com-monly applied to linear differential equations, and hence necessitating numerical analysis of theequations. We now move on to the system of equations used in this project.

5.2 Governing Equations

The particular initial value problem considered in our project is a simplification of the full equationsof Fluid Mechanics known as incompressible Boussinesq hydrodynamics. Incompressible Boussinesqhydrodynamics rests upon two assumptions; that the fluid is incompressible (an appropriate ap-proximation for water), and that the density ρ is close to constant (water varies in density from1000 kg m−3 to 1030 kg m−3 for very fresh and very salty water respectively). Exploiting theseassumptions, and our assumption of two-dimensionality, we now specify the system of equations.

The first equation details the condition of incompressibility, telling us that the velocity vectorfield u has zero divergence,

∇ · u =∂u

∂x+∂w

∂z= 0. (2)

The assumption of nonuniform density ρ(x, t) is accounted for in the second equation, wherethe material derivative tells us that it advects with the fluid; while the Laplacian operator,

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2, (3)

informs us that it diffuses as it advects. The rate of diffusion is governed by κ, the diffusivity, whichis assumed to be a constant, leaving us with the expression,

Dρ

Dt= κ∇2ρ. (4)

The final equation, the momentum equation, tells us what causes the fluid to accelerate, beingpressure gradients ∇p, viscous effects ν∇2u, and gravity gz. It is this equation that requires thecondition of small variations in density, as it ignores the effect of changing density on the pressuregradient or viscosity terms, accounting for it solely through the final gravitational term,

Du

Dt= −∇p

ρ0+ ν∇2u− ρg

ρ0z. (5)

A summary of the equation terms is provided in Table 1.

Having developed the governing equations of the system, we now specify the domain and bound-ary conditions (B.C.’s) of the problem.

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Table 1: The following quantities are accounted for in Boussinesq Hydrodynamics.

Symbol Quantity Symbol Quantityt time (x, y, z) spatial coordinatesu velocity (u, v, w) velocity componentsρ density κ density diffusivityp pressure ν kinematic viscosity

5.3 Domain, Boundary Conditions and Initial Conditions

Dedalus is capable of solving differential equations defined on rectangular, annular, and recentlyspherical domains (Lecoanet et al., 2016). Currently, we restrict ourselves to a two dimensionalrectangular domain x ∈ [0, L], z ∈ [−H2, H1], with initial density stratification in two horizontallayers [−H2, 0] and [0, H1], as shown in Fig. 5.

Note that we are required to smooth the density gradient to prevent Gibbs phenomena; numer-ical instabilities resulting from using a finite number of spectral functions to approximate rapidlychanging quantities. All other quantities are initially set to zero.

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Setting up the model

1 1000 kg/m3

periodicperiodic

z

x2 1020 kg/m3

w =@u

@z=

@

@z= 0

w = u =@

@z= 0

Figure 5: Diagram of the two-dimensional domain and boundary conditions used in thededalus simulations.

Dedalus only allows for non-periodic boundary conditions in a single dimension, requiring thatwe apply periodic boundary conditions on the left and right boundaries. We model the bottomboundary as an unmoving ground surface by setting the fluid velocity to zero,

u(x,−H2, t) = 0. (6)

The top boundary is modelled as being a free surface by instead setting the fluid stress to bezero, which reduces to specifying that the fluid vorticity ζ is zero,

ζ(x,H1, t) =

[∂w

∂x− ∂u

∂z

]z=H1

= 0 (7)

We further simplify this condition by additionally specifying that the vertical velocity is zero onthe top boundary,

w(x,H1, t) = 0. (8)

We justify this simplification by appealing to both experimental investigation (Ekman, 1906; Mercieret al., 2011) and theoretical arguments which show that the amplitude of any surface wave is much

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smaller than that of any interfacial wave, for sufficiently low speeds. The reason behind this reducedamplitude stems from the much greater density jump from air to fresh water, compared to fresh andsalt water. Essentially, a fresh water wave in air possesses much more potential energy than a saltwater wave in fresh water, as the energy required to ‘lift up’ the heavier salt water is compensatedby the consequent ‘lowering’ of the similarly dense salt water. As such, we can safely apply thisapproximation to our system, as our boat moves sufficiently slowly that the above conditions hold.

Finally, we specify that the vertical derivative of the density is zero along the top and bottomboundaries, corresponding to a constraint that there is no flux of the density through the boundaries,ensuring that we do not ‘leak’ salinity from our domain,

∂ρ

∂z

∣∣∣∣z=H1,−H2

= 0. (9)

Only one (important) ingredient is now missing from our current system – the boat.

5.4 Immersed Boundary Methods

Intuitively, one may incorporate a boat in the system by applying the boundary condition that thefluid velocity u must match the boat velocity U along the boat surface. However, as previouslymentioned, dedalus can only simulate regular domain shapes, unlike the irregular boat case asshown in Fig. 6.

To circumvent this seemingly intractable problem, we exploit Immersed Boundary Methods.Broadly speaking, an Immersed Boundary Method replaces a boundary condition defined over anirregular geometry by incorporating an added term in the equations, now defined over a regulardomain (for more information, see Mittal and Iaccarino (2005)).

More specifically, we approximate the boundary condition of matching velocities along the boatsurface by adding a final term to the momentum equation, now defined over a rectangular domainΩ,

Du

Dt= −∇p

ρ0+ ν∇2u− ρg

ρ0z − γ B(x, t) (u−U). (10)

Essentially, this term acts to damp the fluid speed u to the boat speed U when within the boat. Theboat, denoted by B(x, t) : Ω→ [0, 1], is unity within the boat, and tapers off to zero elsewhere, asshown in Fig. 7. This corresponds to modelling the boat as a porous medium, albeit of sufficientlylow porosity γ−1 that it essentially no fluid flows through it. It is important to note that B(x, t)must again be smooth, greatly complicating the calculation of the force on the boat.

21

z

x

u = U

Figure 6: Dedalus is incapable of simulat-ing irregular domains resulting from includ-ing a matching velocity boundary conditionon the boat boundary.

22

z

x

U

B = 0

B = 1

Figure 7: We use an immersed boundarymethod to approximate the problem over anregularly shaped domain, by incorporating adamping force term over the boat B.

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5.5 Boat Drag Calculation

For a sharp boundary corresponding to a boat, it is fairly simple to calculate the net force. Onesimply integrates the forces due to pressure, as well as viscosity, over the considered boundary, i.e.integrate the fluid stress along the boundary. However, the incorporation of a diffuse boundary onthe boat, resulting from the Immersed Boundary Method, introduces considerable complexity.

In the course of this project I derived a means by which to calculate the net force on the boatwhich is reproduced below.

We begin by ignoring the varying density of the fluid and treat it as homogeneous, ρ = ρ0.We shall see that this approximation remains valid, as the boat remains entirely within the toplayer throughout the simulation. Additionally, it simplifies the derivation, allowing us to rearrangeequation 10 as follows,

Du

Dt= −∇p

ρ0+ ν∇2u− gz − γB(u−U),

= ∇(−[p

ρ0+ gz

])+ ν(∇ · ∇)u− γB(u−U),

= ∇ ·(−[p

ρ0+ gz

]I + ν∇u

)− γB(u−U),

= ∇ ·T− γB(u−U), (11)

where I is the identity matrix (or the Kronecker delta), a second rank tensor. We can subsequentlyexpress the second rank tensor T as,

T = −(p

ρ0+ gz

)(1 00 1

)+ ν

∂u

∂x

∂w

∂x∂u

∂z

∂w

∂z

.

Assuming that the boat is neutrally buoyant we conclude that it possesses a density equal to thefluid, ρb = ρ0. Hence, we define the boat volume |B|,

|B| =∫

Ω

B dV, (12)

the boat momentum Pb,

Pb = ρ0

∫Ω

BU dV, (13)

and the fluid momentum Pf ,

Pf = ρ0

∫Ω

(1−B)u dV. (14)

Having defined the fluid momentum, we now take its time derivative,

dPfdt

= ρ0d

dt

[∫Ω

(1−B)u dV

],

= ρ0

∫Ω

∂

∂t[(1−B)u] dV, (Liebniz rule)

Pf = ρ0

(∫Ω

(1−B)∂u

∂tdV −

∫Ω

∂B

∂tu dV

). (15)

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We then find an expression for the time derivative of the fluid momentum by integrating equation11, multiplied by (1−B),∫

Ω

(1−B)Du

DtdV =

∫Ω

(1−B)∇ ·T dV − γ∫

Ω

(1−B)B(u−U) dV. (16)

Firstly, we deal with the left hand side,∫Ω

(1−B)Du

DtdV =

∫Ω

(1−B)∂u

∂tdV +

∫Ω

(1−B)u · ∇u dV,

=Pfρ0

+

∫Ω

u∂B

∂tdV +

∫Ω

(1−B)u · ∇u dV. (17)

Secondly, we use the product rule, as well as Gauss’ theorem, to show that the first right hand termequates to ∫

Ω

(1−B)∇ ·T dV =

∫Ω

∇ · ((1−B)T) dV −∫

Ω

T · ∇ (1−B) dV,

=

∫∂Ω

(1−B)T dA+

∫Ω

T · ∇B dV. (18)

Thus, we find that,

Pfρ0

=

∫∂Ω

(1−B)T · dA+

∫Ω

T · ∇B dV − γ∫

Ω

(1−B)B(u−U) dV . . .

−∫

Ω

u∂B

∂tdV −

∫Ω

(1−B)u · ∇u dV. (19)

Now, we use Newton’s Second Law and Eq. 19, which tells that the net change in systemmomentum is equal to the external forces on the system (where Fb is an external force on theboat),

Pb + Pf = ρ0

∫∂Ω

T · dA + Fb,

ρ0 |B| Ub = ρ0

∫∂Ω

T · dA− ρ0

[∫∂Ω

(1−B)T dA+

∫Ω

T · ∇B dV − γ∫

Ω

(1−B)B(u−U) dV . . .

−∫

Ω

u∂B

∂tdV −

∫Ω

(1−B)u · ∇u dV]

+ Fb, (20)

leaving us with the following expression for the boat acceleration,

Ub =1

|B|

[∫∂Ω

BT · dA−∫

Ω

T · ∇B dV + γ

∫Ω

B(1−B)(u−U) . . .

+

∫Ω

u∂B

∂tdV +

∫Ω

(1−B)u · ∇u dV]

+Fb

ρ0 |B|. (21)

Now, we simplify this expression.

Firstly, we have that the fluid vorticity and pressure are defined to be zero along the topboundary, the only region where B intersects with ∂Ω. Since we additionally have hydrostaticequilibrium, letting us cancel the gravitational contribution, we therefore conclude that BT = 0 onall of ∂Ω.

To simplify the partial time derivative of B, we analyse the motion of the boat. Currently,we only allow horizontal translation of the boat, which prevents the boat volume from changing

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(getting larger or smaller as it dips and rotates in and out of the water). We justify this assumptionby noting that the boat does not move fast enough for any appreciable torque or vertical forces todevelop, verified experimentally by Mercier et al. (2011). Hence, we express the boat function as,

B(x, t) = B(x− xb(t)), (22)

where xb(t) is some reference point on the boat. Using the chain rule, we find that the partialderivative is given by,

∂

∂tB(x, t) =

∂

∂tB(x− xb(t)),

= −∂B(x− xb(t))

∂(x− xb(t))· ∂xb(t)

∂t,

= −∇B ·U . (23)

Now, we exploit tensor calculus manipulations to simplify the convective derivative term. Usingabbreviated tensor notation (where doubled indices imply summation), we find that,

u · ∇u = ui∂iuj ,

= ∂i(uiuj)− uj∂iui, (product rule)

= ∂i(uiuj), (zero divergence)

= ∇ · (u⊗ u) ,

= ∇ · (uu) , (24)

where ⊗ corresponds to the outer/direct product.

We then use Gauss’ theorem and the product rule again to simplify the final term in Eq. 21,∫Ω

(1−B)u · ∇u dV, =

∫Ω

(1−B)∇ · uu dV,

=

∫Ω

∇ · [(1−B)uu] dV −∫

Ω

uu · ∇ (1−B) dV, (25)

=

∫∂Ω

(1−B)uu · dA +

∫Ω

u(u · ∇B)dV. (26)

From the boundary conditions, we have that there is no velocity normal to the edge, hence, thefirst term must disappear. Putting all this together, we find that Eq. 21 reduces to,

Ub =1

|B|

[−∫

Ω

T · ∇B dV + γ

∫Ω

B(1−B)(u−U) +

∫Ω

[(u−U) · ∇B]u dV

]+

Fbρ |B| . (27)

Put in this form, the origin of each of the terms is apparent. The first term arises as a naturalextension to integrating the fluid stresses over a diffuse boundary. The second term comes aboutas a result of the mismatch between velocities, and importantly acts only on the boundaries ofthe boat, as would be expected of an impermeable material. The next term also depends on thismismatch between fluid and boat velocities, acting only on the boundary of the boat.

Currently, we are only interested in the drag on the boat due to the fluid, meaning that weexclude the final external forcing Fb from the calculation.

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6 Results and Discussion

After incorporating the mathematical problem into dedalus, several simulations were run, bothwith and without density stratification. Two examples of each, sharing otherwise identical param-eters, are reproduced for this report.

In each simulation, the velocity of the boat is defined to gradually increase from zero to a constantmaximum velocity Umax, with the resultant force on the boat is calculated using equation 27. Inreality (and experiment), it is typically the motive force on the boat that is specified to be constant,with the resultant velocity used to infer the existence of increased drag. However, this increaseddrag leads to lower, oscillating velocities in the stratified case, preventing accurate comparison ofdrag between the simulations at a particular velocity. By specifying the boat velocity manually, wemore accurately capture the difference in boat drag between the stratified and unstratified case.

6.1 Simulation parameters

The simulation parameters used in the two simulations are summarised in Table 2, with the onlydifference being the presence or lack of density variations.

The depth of the top layer H1 = 1 m, the bottom layer H2 = 2 m, and the dimensions of the boat`bx = 4 m, `bz = 0.5 m were chosen so as to reasonably reflect those of a small boat moving throughshallow water. In the second simulation, the density of the top layer ρ1 = 1000 kg m−3 correspondsto that of fresh meltwater, while the lower density ρ2 = 1030 kg m−3 is that of very salty seawater.Again, the density gradient is chosen to have a smooth transition region of size δz = 0.1 m, bothto ensure computational stability, and accurately approximate the imperfect stratification of realwaters.

By linearising the problem and assuming irrotationality, one can use these parameters, and theacceleration due to gravity g = 9.8 m s−2, to determine the theoretical maximum interfacial wavephase speed cφmax,

cφmax =

√(ρ2 − ρ1)g

/(ρ2

H2+ρ1

H1

). (28)

The maximum boat speed Umax is then chosen to be 0.7cφmax, as Ekman observed that this pro-portion (approximately) lead to the greatest drag. To ensure a gradual acceleration, the speed ofthe boat Ub at time t was chosen as

Ub(t) =Umax

2

[tanh

(t− ττ

)+ 1

], (29)

where τ = 1.5 s is the characteristic acceleration time of the boat. Consequently the boat approxi-mately reaches maximum speed at t = 6 s.

The boat B(x, t), with centre xb(t) = (xb(t), H1), has the shape of the bottom half of a hy-perellipse, with semi-major axis `bx, and semi-minor axis 2`bz. Again, the boat is smoothed overa thickness δ = 0.1. This was achieved, as with all other smoothed quantities, by using the errorfunction, resulting in the following expression,

B(x, t) =1

2

[1− erf

(∣∣∣∣x− xb(t)`bx/2

∣∣∣∣6 +

∣∣∣∣z −H1

`bz

∣∣∣∣− 1

)/δ

](30)

The inverse porosity (‘stickiness’) of the boat γ = 50 was chosen to be high enough that noappreciable diffusing of water through the boat could be observed, while the kinematic viscosity ofthe water, ν = 5× 10−5 m2 s−1 was chosen to be as low as feasible without observing any Gibbs

11

phenomena in the fluid. The true kinematic viscosity of water is roughly one fiftieth of this value,but the Reynolds number Re of the given simulation is of the same order as Mercier and Ekman,

Re =H1Umax

ν≈ 6000, (31)

allowing us to view this as a reasonably accurate simulation.

Finally, the diffusivity of the density κ = 1× 10−4 m2 s−1 was taken so as to ensure computa-tional stability while preventing excess diffusion of the interface.

Table 2: The given quantities were used in the simulations below.

Parameters Symbol Value UnitsTop layer depth H1 1 mBottom layer depth H2 2 mDomain length L 12 mBoat length `bx 4 mBoat depth `bz 0.5 mTop layer density ρ1 1000 kg m−3

Bottom layer density ρ2 1030 kg m−3

Density gradient sharpness δz 0.1 mGravitational acceleration g 9.8 m s−2

Maximum boat velocity Umax 0.31 m s−1

Acceleration timescale τ 1.5 sBoat sharpness δ 0.1 mInverse porosity γ 50 s−1

Kinematic viscosity ν 5× 10−5 m2 s−1

Reynolds number Re 6000Density diffusivity κ 1× 10−4 m2 s−1

The homogeneous simulation was run an a quad core late 2015 iMac Air 5K with 4 GHz pro-cessors, while the stratified simulation was run on a 12 core Mac Pro with 2.4 GHz processors.Each simulation has a spatial resolution of 96 grid points per metre in each direction, and tookapproximately 24 hours to run.

6.2 Simulation results

Reproduced in Fig. 8, are plots of the vorticity ζ at times t = 0 − 50 s in the homogeneous andstratified simulation respectively. From these plots we can easily observe the formation of vortices,which we see take on vastly different qualities in the two simulations. A plot of the fluid density inthe second simulation is also provided in Fig. 8, clearly showing the generation of internal waves.

In the unstratified case, little changes can be observed in the fluid vorticity, where a sizeablerear vortex generated immediately behind the boat persists throughout the entire simulation. Thisis in strong contrast to the stratified simulation, in which several new phenomena are observed.

While initially (before 10 s), the simulations are broadly similar, we notice the generation of aninternal wave after this time in the second simulation. The left side of the wave peak subsequentlydevelops a Kelvin-Helmholtz instability which progressively spreads during the simulation. Impor-tantly, we see that this internal wave ‘traps’ vorticity shed by the boat, leading to the generation ofa trailing vortex behind the wave crest at t = 30 s. On close observation, an additional rear vortexcan be seen at the trailing edge of the boat, which corresponds to a greatly diminished form of therear vortex from the first simulation. From the exaggerated drag at these times, we can conclude

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that the existence of this trailing vortex, and the diminished size of the rear vortex, must play akey role in the increased resistance of the dead water phenomenon.

Reproduced in Fig. 9 is a plot of the boat drag for each simulation, calculated according toEq. 27. The drag forces has been normalised by the mass of the boat for each simulation, meaningthat the plots show (the negative of) the acceleration that the boat would experience if only thefluid acted on it clearly demonstrating an increased drag in the stratified case, in accordance withexperiment.

As would be expected of the linear theory, the generated internal wave moves faster than theboat, leading to the merging of the trailing and rear vortices at t = 45 s. Importantly, we see thatthere is a corresponding drop in the drag on the boat at this time. This is somewhat contrastedwith observations by Mercier et al. (2011), who notes that ‘as the boat evolves, the amplitude ofthe interfacial waves grows up to a maximum value where the waves catch up with the boat andbreak on its hull ... This corresponds to the moment where the drag force is maximal’.

This apparent discrepancy is not as definite as it may immediately appear however. Firstly,Mercier’s view of ‘maximal drag’ was not determined by force measurements, but rather by notingthat the ‘the boat is almost standing still’. This minimum speed would actually occur after thepeak drag is attained, which would happen during the period of deceleration preceding the boatstopping. Additionally, the simulations are slightly confounded by the fact that the interface hasbeen disturbed, which doesn’t occur to the same extent in the experimental investigations of Mercieror Ekman. And it is mixing of the interface that was confirmed to be a key part of the disappearanceof the effect. It is believed that this instability may stem from the acceleration of the boat takingplace too quickly. A smaller boat may also lead to smaller disturbances.

Regardless, the merging of these vortices can be seen to result in the generation of a large rearvortex immediately behind the boat, similar to that of homogeneous case, which coincides with aminimum in the drag experienced by the boat. To attain a deeper understanding of the behaviourof the boat drag, we separate the force from Eq. 27 into its components, which we label as,

1. Fp: Drag resultant from pressure in the first part of the T tensor.

2. Fv: Viscous drag due to the remaining component of T.

3. Fγ : Drag resulting from the immersed boundary method term∫

Ωγ(1−B)B(u−U).

4. FR: The final term in Eq. 27,∫

Ωu[(u−U) · ∇B].

From the force component plot in Fig. 10, see that pressure drag term dominates, with theporous drag being approximately one third this magnitude, and the remaining components severalorders of magnitude smaller, allowing us to ignore their contributions.

To further examine the dominance of Fp, we investigate the fluid pressure in each simulation,shown in figures 11, 12, and 13. Immediately, it becomes evident that there exists a much greaterregion of lower pressure behind the boat in the stratified case. We can then conclude that it is thegeneration of this intense low pressure at the rear of the boat that ‘pulls’ it backward, accountingfor the increased drag of the dead water effect.

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Figure 8: Time series of the fluid vorticity in the homogeneous simulation (a), and fluid vorticity(b) and density (c) in the stratified simulation. It can easily be seen that the vorticity differs greatlybetween each simulation in figures (a) and (b), with a greatly diminished vortex immediately behindthe boat in the two layer case, and entirely new trailing vortex behind the wave peak. The mergingof these two vortices is seen to correspond to a strong decrease in the boat drag (see Fig. 9). Thedensity plot also clearly demonstrates the generation of internal waves by the boat.

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Figure 9: A time-force plot of the boat drags for each simulation. The drag forces have beennormalised by the boat mass, meaning that the plot expresses the acceleration the boat wouldexperience from the fluid. As can be seen, the drag is much higher for the stratified case.

Figure 10: A plot of the force components Fp and Fγ for each of the boats. The remainingcomponents Fv and FR are omitted, as they are more than three orders of magnitude smaller thanFγ . The majority of the force on the boat is due to Fp, hinting at the origin of the phenomenon.

Figure 11: Homogeneous fluid pressure at t = 25 s. The rear low pressure region does not contactthe boat, preventing it from ‘pulling’ on the boat.

Figure 12: Stratified fluid pressure at t = 25 s, displaying the trailing low pressure region behindthe boat responsible for the increased drag of dead water.

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Figure 13: Time series of the fluid pressure in the homogeneous (a), and stratified (b) simulations. Amarkedly more pronounced low pressure region can be seen at the rear of the boat in the stratifiedsimulation, accounting for the increased drag. The period of maximum resistance, at t = 25 s,coincides with the presence of an additional high pressure region before the boat.

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7 Conclusion

Considerable progress in understanding the dead water phenomenon has been made in this project.By mathematically modelling stratified water using incompressible Boussinesq Hydrodynamics, thepowerful dedalus code was used in conjunction with Immersed Boundary Methods to simulatethe motion of a boat in both a two-layer and homogeneous density system. A method to calculatethe force on the diffuse boat was developed and used to both confirm the existence of the deadwater effect, as well as infer the origin of the phenomenon, being due to generation of low pressuresbehind the boat by interfacial waves.

8 Future Developments

While a considerable amount of work has been done on this project, it is by no means near comple-tion. We soon seek to investigate the dependence of the effect on the many system parameters, suchas the extent of stratification, and depth of each layer. In particular, we aim to explore how therelationship between the boat parameters δ, γ, and the fluid parameters ν, affects the simulation.

Additionally, we will soon be running simulations in which a constant force is applied to theboat, allowing us to more accurately compare our results with previous experimental investigation.Other constraints, such as those of strict horizontal translation or zero normal surface velocity, arealso being gradually lifted.

Of particular help in these future investigations will be access to the Artemis High PerformanceComputer at the University of Sydney, a 1600 core computer, which we are currently workingto install dedalus on. With access to vastly increased computing power, we will be able tosimulate more realistic parameter regimes in far less time than is currently possible. Perhaps mosttantalisingly, it is expected that we will be able to extend our simulations to three dimensions, themain phenomenological deficiency of the current simulations.

9 Acknowledgements

I wish to thank both AMSI and the University of Sydney for having provided this incrediblyrewarding opportunity to develop my research skills through the Vacation Research Scholarship. Inparticular, I also wish to think my supervisor Dr Geoffrey Vasil, whose intellectual, philosophical,and computational advice made the project far more achievable, and enjoyable, than otherwisepossible.

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