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  • View in gallery

    (top) Schematic diagram of the topography at TTP. It is shaped like a ridge on a sloping sidewall and therefore generates both internal waves and eddies, which create regions of relatively high and low pressure. To simplify the problem and more easily identify flow mechanisms contributing to form drag, the system was split into two cases: (bottom left) the internal wave–generating ridge case and (bottom right) the eddy-generating headland case. In this paper, only the headland case is discussed. The coordinate axes shown on the headland diagram align with the axes in the numerical model.

  • View in gallery

    (left) Plan views of the potential flow SSH calculated from the potential flow pressure field, such that SSH = p/ρ0g. The black contours show the potential flow streamlines. (a) At maximum flood and ebb tides, the SSH has the shape shown, with a region of low pressure at the tip of the headland. (b) At slack tide, when the acceleration is greatest, the SSH is tilted as shown. This tilt is constant throughout the channel except (c) near the headland, where the slope deviates slightly from the background slope. This extra tilt increases the magnitude of the inertial drag.

  • View in gallery

    (a) The tidal velocity with respect to time for two tidal cycles. (b) The total form drag (black), the inertial drag (gray), and the separation drag (dashed) are shown, as well as (c) the corresponding power. The power is simply the product of the velocity and the drag. (d) The cumulative average of the power is shown. Over time, the cumulative—or running—average of the power is approaching an asymptote. Although the amplitude of the inertial drag is nearly as large as the amplitude of the total drag, the separation drag accounts for all of the cumulative power losses after a complete tidal cycle. These curves are from the base run. The relationship between the velocity, drag, and power is similar for the other runs.

  • View in gallery

    Each model run is represented by a bar. The gray part of the bar is the amplitude of the inertial drag, and the black part of the bar is the amplitude of the separation drag; both are calculated with a sinusoidal fit to the data. Their sum is close to the amplitude of the total drag, but not exact because of phase differences between the separation and inertial drags. The white triangles point to the magnitude of the bluff body drag, assuming a drag coefficient of 1. The runs are organized (left)–(right) with the base, the runs where the bluff body form drag parameter DBB was changed, the runs where the tidal excursion distance divided by the along-channel headland length λ was changed, and the runs where the headland aspect ratio α was changed. The magnitude of the inertial drag is generally much larger than that of the bluff body drag; in most cases, the magnitude of the separation drag is close to the bluff body drag.

  • View in gallery

    The relationship between the actual inertial drag and the estimated inertial drag is shown. The inertial drag is estimated by Dinertial = (1 + Δ/L)ρ0VU0ω, where the volume V = πΔLH. All of the runs, except the 0.25λ run, fall just above the 1:1 line, so their actual inertial drag is slightly less than the estimated inertial drag. This could be because the scaling relationship was derived for flow around an ellipse, not flow around a Gaussian-shaped headland.

  • View in gallery

    Each column shows how the separation drag changed as the three experimental parameters—(left) DBB, (middle) λ, and (right) α—were scaled. Each row shows a property of the separation drag: (top) the amplitude (in Newtons), (middle) the phase difference between the separation drag and the velocity (in degrees), and (bottom) the tidally averaged power that the drag can remove from the flow (in watts). The gray lines in (a),(b),(c) in the bluff body drag with a drag coefficient of 1 and in (g),(h),(i) show the tidally averaged power loss resulting from the bluff body drag, 〈P〉 = ½U0DBB.

  • View in gallery

    Histogram of drag coefficients for the separation drag for all of the runs. Most of the runs fall on or just above a typical drag coefficient of 1. The exceptions are the cases where α was changed to create a sharper or more streamlined headland. The last exception is the anomalous 0.25λ run.

  • View in gallery

    (top to bottom) Top view snapshots of the headland during the last six time steps of a tidal cycle for the base case, showing the evolution of the residual SSH and associated vorticity field throughout an ebb tide.

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Dissecting the Pressure Field in Tidal Flow past a Headland: When Is Form Drag “Real”?

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  • 1 School of Oceanography, University of Washington, Seattle, Washington
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Abstract

In the few previous measurements of topographic form drag in the ocean, drag that is much larger than a typical bluff body drag estimate has been consistently found. In this work, theory combined with a numerical model of tidal flow around a headland in a channel gives insight into the mechanisms that create form drag in oscillating flow situations. The total form drag is divided into two parts: the inertial drag, which is derived from a local potential flow solution, and the separation drag, which accounts for flow features such as eddies. The inertial drag can have a large magnitude, yet it cannot do work on the flow because its phase is in quadrature with the velocity. The separation drag has a magnitude that is nearly equal to the bluff body drag and accounts for all of the energy removed from the flow by the topography. In addition, the dependence of the form drag on the tidal excursion distance and the aspect ratio of the headlands were determined with a series of numerical experiments. This theory explains why form drag can be so large in the ocean, and it provides a method for separating the pressure field into the parts that can and cannot extract energy from the flow.

Corresponding author address: Sally J. Warner, School of Oceanography, University of Washington, Box 357940, Seattle, WA 98195-7940. Email: sally2@u.washington.edu

Abstract

In the few previous measurements of topographic form drag in the ocean, drag that is much larger than a typical bluff body drag estimate has been consistently found. In this work, theory combined with a numerical model of tidal flow around a headland in a channel gives insight into the mechanisms that create form drag in oscillating flow situations. The total form drag is divided into two parts: the inertial drag, which is derived from a local potential flow solution, and the separation drag, which accounts for flow features such as eddies. The inertial drag can have a large magnitude, yet it cannot do work on the flow because its phase is in quadrature with the velocity. The separation drag has a magnitude that is nearly equal to the bluff body drag and accounts for all of the energy removed from the flow by the topography. In addition, the dependence of the form drag on the tidal excursion distance and the aspect ratio of the headlands were determined with a series of numerical experiments. This theory explains why form drag can be so large in the ocean, and it provides a method for separating the pressure field into the parts that can and cannot extract energy from the flow.

Corresponding author address: Sally J. Warner, School of Oceanography, University of Washington, Box 357940, Seattle, WA 98195-7940. Email: sally2@u.washington.edu

1. Introduction

Form drag is a force that results from pressure differences across obstacles in a flow field. In the ocean, form drag has been measured in a coastal setting on a ridge with hydraulically controlled flow (Moum and Nash 2000; Nash and Moum 2001) and in an estuarine setting on a headland with sloping sidewalls subjected to tidal currents (Edwards et al. 2004; McCabe et al. 2006). In these studies, the form drag was found to be at least 2 times (Nash and Moum 2001) and up to 50 times (Edwards et al. 2004) larger than the frictional drag. The form drag was also assumed to be responsible for the creation of flow features such as internal lee waves and eddies and to contribute to increased mixing rates observed near these sites. The importance of form drag in the ocean is not just confined to coastal regions; on a global scale, the performance of a hydrodynamic model of the ocean’s tides was improved by using a drag parameterization that accounted for the creation of internal waves over rough topography in the deep ocean (Jayne and St. Laurent 2001). On much smaller scales, form drag resulting from surface waves has been shown to be a dominant force on benthic organisms in shallow regions (Lowe et al. 2005).

The motivation for this study comes primarily from observations of the tidal flow near Three Tree Point (TTP) in Puget Sound, Washington, by Edwards et al. (2004) and McCabe et al. (2006). Puget Sound is a tidal estuary with an average depth of 200 m and a width of 5 km near TTP. From above, TTP looks like a sharp triangular headland, about 1 km wide (across channel) and 1 km long (along channel) at its base. Beneath the surface, it is situated on a sloping sidewall; hence, it behaves both like a headland that requires water to travel around it and a ridge that requires water to travel over it, as depicted in Fig. 1. Eddies and internal waves are both observed on the downstream side of the topography. A numerical model of the site (Edwards et al. 2004) matched the observations reasonably well. The average amplitude of the total form drag in the model, integrated over a 10-km segment of the tidal channel, was about 2 × 107 N (Edwards et al. 2004, their Fig. 16). To get a better sense of the size of this measurement, we can compare it to a theoretical estimate of the steady bluff body drag,
i1520-0485-39-11-2971-e1
assuming the following values from Edwards et al. (2004, their Figs. 4 and 9): the average density ρ0 = 1023.5 kg m−3, the projected frontal area of the topography S = 200 m × 1000 m, and the amplitude of the tidal velocity U0 = 0.15 m s−1. We find that in order to get the bluff body drag to match the measured drag, a drag coefficient CD of 8.7 must be used, where a typical value is O(1). The largest drag coefficient found in steady flow situations with Reynolds numbers greater than 103 is 2.3, which comes from flow around a concave C-shaped object (Hoerner 1965). In a theoretical and numerical model study, MacCready and Pawlak (2001) found that drag coefficients for stratified, steady flow over ridges on sloping sidewalls were between 1 and 1.2. What creates the increased form drag at TTP? The results of this research suggest that the increased form drag is a result of the oscillating nature of the flow.

To simplify the problem of oscillating flow over TTP-like topography, we decided to look at eddy-generating headlands and internal wave generating ridges separately (Fig. 1). This paper will focus just on the results from the headland study. The dynamics of eddy formation behind headlands, capes, and islands have been studied in the laboratory (Boyer and Tao 1987; Klinger 1994; Cenedese and Whitehead 2000), in numerical models (Black and Gay 1987; Signell and Geyer 1991), and in field experiments (Wolanski et al. 1984; Geyer 1993). In this work, a numerical model was used to explore headlands of different sizes and shapes in a range of flow situations, with the goal of isolating the mechanisms that create form drag. The total form drag was broken up into two parts: the inertial drag and the separation drag. The inertial drag is based on theoretical calculations of potential flow, and the separation drag is the residual that remains after the inertial drag is subtracted from the total drag. It accounts for flow features, such as eddies. In this paper, we will first develop a theoretical basis for the inertial drag by looking at the case of oscillating potential flow around a cylinder (section 2a). We will then expand this theory to include the case of a headland in a channel (section 2b). The numerical model is explained in section 3. In section 4, the relative amounts of inertial and separation drag from each model run are examined and compared to the bluff body drag. The abundance and strength of flow mechanisms that make up the separation drag are explained in the latter part of section 4. The findings of this study are summarized in section 5.

2. Theory

a. An introductory example: Potential flow around a cylinder

A lot of insight into tidal flow around headlands can be gained by looking at oscillating flow around circular cylinders. This is not a new problem. Engineers were motivated to study the details of this flow field because they wanted to calculate the hydrodynamic forces resulting from waves on offshore cylindrical structures (Faltinsen 1990). An analytical theory for oscillating flow around cylinders at low Reynolds numbers was developed by Wang (1968). Laboratory (Bearman et al. 1985; Obasaju et al. 1988) and numerical (Justesen 1991) studies have been conducted with regard to this problem over a range of Keulegan–Carpenter (the ratio of drag forces to inertia) and Reynolds numbers. However, when it comes to flow around topography in the ocean, the Reynolds number can be on the order of 109 because the length scales are so large. None of these studies explored flow at such high Reynolds numbers except Jones et al. (1969), who studied flow around oscillating cylinders at high Mach numbers. An important difference between the motivation of these studies and the research discussed in this paper should be noted. Engineers care about the total drag force exerted on their structures, whereas, in an oceanographic context, we are more interested in the energy that topography can remove from the tidal currents. The different objective has led us to focus on different aspects of the problem.

To begin our theoretical analysis, we start by looking at the two-dimensional, inviscid, irrotational potential flow around a cylinder, as detailed by Lamb (1930, chapter 4, section 68). This approach of looking at the potential flow solution as a way to gain insight into the different parts of the form drag is similar to that explained by Dean and Dalrymple (1984, chapter 8). The velocity potential ϕ for a cylinder is defined as
i1520-0485-39-11-2971-e2
where U(t) is the free-stream velocity, a is the radius of the cylinder, and r and θ are the polar coordinates. By definition, the gradient of the velocity potential is equal to the velocity around the cylinder, ϕ = u = (u, υ). To calculate the pressure p on the surface of the cylinder, start with the inviscid, Boussinesq, irrotational momentum equation:
i1520-0485-39-11-2971-e3
Replace ∂u/∂t with (∂ϕ/∂t) and solve for the pressure:
i1520-0485-39-11-2971-e4
i1520-0485-39-11-2971-e5
This is the unsteady, irrotational Bernoulli’s equation, and C(t) appears as a time-dependent constant of integration. Making use of the relations (∂ϕ/∂t)|r=a = Ut2a cosθ and u · u|r=a = 4U2 sin2θ, which can be calculated from Eq. (2), we can solve for the pressure on the surface of the cylinder:
i1520-0485-39-11-2971-e6
The first term on the right-hand side of Eq. (6) represents a constant pressure offset around the whole cylinder that is a function of time. The middle term defines low pressure regions on the sides of the cylinder that lie tangent to the background velocity. Its magnitude is greatest when the velocity is at its maximum. The third term represents a horizontal pressure gradient across the cylinder. This gradient is strongest when the velocity is zero and the acceleration Ut is at its maximum.
The drag force per unit length of cylinder δz, D/δz, resulting from this pressure field can be calculated by integrating the pressure [Eq. (6)] over the surface of the cylinder:
i1520-0485-39-11-2971-e7
i1520-0485-39-11-2971-e8
i1520-0485-39-11-2971-e9
It is interesting to compare the size of this drag force [Eq. (9)] to the bluff body drag [Eq. (1)]. To do this, it is assumed that the drag coefficient in Eq. (1) is 1 and that the projected frontal area of the cylinder is 2aδz. Taking the flow to be oscillatory with the form
i1520-0485-39-11-2971-e10
where ω is the frequency of oscillation, it can then be assumed that Ut scales like ωU0. Hence, the ratio of this drag force to the bluff body drag is
i1520-0485-39-11-2971-e11
where LT = 2U0/ω is the tidal excursion distance and we have assumed CD = 1. For a short tidal excursion distance, this ratio is large, so the drag associated with the potential flow is expected to be much larger than the bluff body drag. When the tidal excursion distance is long, the drag from the potential flow will be small compared the bluff body drag. This makes sense because a long tidal excursion distance is approaching the steady limit; by definition, steady potential flow does not exert a force on a body (Kundu and Cohen 2004, 166–167).
This potential flow example can be taken one step farther by calculating the work that the drag can do on the flow. The work, W, is the time integral of the power P over an oscillation period, and the power is the product of drag [Eq. (9)] and the velocity [Eq. (10)]:
i1520-0485-39-11-2971-e12
Hence, the drag associated with potential flow around a cylinder cannot do work on the flow. Although this is a seemingly obvious result that has been shown in places like Batchelor (1967, p. 355) and Dean and Dalrymple (1984), using the same theoretical basis for flow around a headland will help us dissect the pressure field into the parts that can and cannot do work on the flow.

b. Potential flow around a headland: The inertial drag

The framework that was developed in section 2a will now be applied to the case of potential flow around a headland in a channel. By doing this, the part of the pressure field that cannot do work on the flow can be isolated. This will be referred to as the potential flow pressure field and the corresponding form drag will be called the inertial drag.

The first step to calculating the inertial drag is to find the two-dimensional, incompressible, irrotational streamfunction around a headland in a channel. Conveniently, the streamfunction ψ for potential flow satisfies the Laplace equation,
i1520-0485-39-11-2971-e13
The boundary conditions are defined such that the velocity far from the headland is equal to the free-stream velocity from Eq. (10), ∂ψ/∂y = U(t), and the velocity normal to a solid surface is zero, ∂ψ/∂s = 0, where s defines the along-stream coordinates of the boundary. Equation (13) can be solved numerically by using the method described by Kundu and Cohen (2004, 182–187). A typical streamfunction around a headland looks like the black contours in Fig. 2 (left). Throughout the tidal cycle, only the magnitude of the streamfunction changes, not the shape.
Similar to the case of flow around a cylinder, to get the pressure field, integrate the inviscid, irrotational, Boussinesq momentum Eq. (3) along streamlines from a point xA = (x0, yA) at the entrance of the domain to any point on the same streamline within the domain xB = (xB, yB):
i1520-0485-39-11-2971-e14
where s is the streamwise direction and us is the streamwise velocity whose magnitude is u2 + υ2. To get the potential flow pressure field, we solve Eq. (14) for p:
i1520-0485-39-11-2971-e15

The potential flow pressure field is made up of three parts, each of which has different physical impacts on the flow. The “offset” is a change in the pressure field over the whole domain with respect to time. Theoretically, it should be the pressure at the entrance to the channel; however, the choice of its value does not affect the inertial form drag because it is constant throughout the domain.

The second part of the potential flow pressure field is the “dip.” This part results from the fact that water must travel faster as it goes through a constriction; by Bernoulli’s formula, faster moving fluid must be at a lower pressure. The dip is most prominent when the tidal velocities are maximized. The dip part of the potential flow pressure field can be seen in Fig. 2a, where the sea surface height (SSH) is obtained by dividing the pressure in Eq. (15) by ρ0g. The dip is symmetric across the axis of the headland, and hence the inertial form drag associated with the dip is zero; any pressure changes created by the dip on the left-hand side of the topography are balanced by equal pressure changes on the right-hand side. Although this term does not contribute to the inertial form drag, by removing it from the total pressure field we are better able to visualize the residual pressure field.

The third part of the potential flow pressure field is the “tilt/jump,” as seen in plan view in Fig. 2b and in side view in Fig. 2c. The sea surface height has a constant slope near the beginning and ends of the domain in regions where the channel’s cross-sectional area is constant. We refer to this slope as the “tilt,” but it is often simply called the pressure gradient. Close to the headland, the actual slope deviates from the background value, and we refer to this part as the “jump.” This is the pressure perturbation classically associated with the “added” or “apparent” mass (Lamb 1930). In cases where the across-channel headland width is equal to the along-channel, e-folding length of the headland, the relative contributions to the form drag from the jump and the tilt are nearly equal. As the across-channel width of the headland increases, the relative size of the jump also increases. For a more gentle constriction, the tilt term would dominate. Of the three terms in Eq. (15), the tilt/jump term is the only one that can contribute to the inertial form drag because it is the only one that is not symmetric across the axis of the headland. At slack tide, velocities are zero but accelerations are at a maximum and the pressure difference between the left and right sides of the headland has its greatest magnitude.

The most important thing to realize about the inertial drag is that it cannot do work on the flow. The only part of the potential flow pressure field that can contribute to the inertial drag is the tilt/jump term. Because this term has a phase that scales with the time derivative of the streamline velocities, by construction, the phase of the inertial drag will always be in quadrature with the phase of the velocity. This means that, although the amplitude of the inertial drag may be significant, the tidally averaged power of the inertial drag is always zero.

Additionally, we must make note of the limitations and errors associated with this calculation. The streamfunction and its associated pressure field are calculated assuming that the channel has a rigid lid and that flow in the channel is 2D, with no z variation. In both the real ocean and in the numerical model, the sea surface height changes and hence pressure fluctuations at the entrance of the channel will take time to propagate down the channel. In the above pressure field calculation, in contrast, fluctuations at the entrance of the channel are immediately “felt” throughout the length of the channel. In this study’s numerical model, the surface gravity wave speed is approximately 44 m s−1, which means that a tidal wave will take about a minute and a half to travel the length of a 4-km headland. This is much faster than other variations in the sea surface height field that occur at the tidal frequency ω. Hence, the sea surface height change associated with the wave does not create a significant amount of form drag. Another limitation that should be noted is that the potential flow pressure field found with the above method is not the exact same pressure field that would result from isolating the irrotational parts of the flow in the model. The basic characteristics would be the same; however, certainly the presence of eddies changes the shape of the irrotational flow field in the model. Despite these limitations, we decided to use the described method to calculate the potential flow pressure field, because it has a theoretical basis that is very useful in explaining the different parts of the flow field and is found to account for much of the form drag created by the headland that previous studies such as Edwards et al. (2004) could not explain.

c. Total and residual form drags

The method for computing the inertial drag was explained in section 2b, but this only accounts for the irrotational, inviscid part of the flow. The total form drag is commonly thought of as the area integral of the bottom pressure times the bottom slope. It can be derived from the three-dimensional, Boussinesq momentum equation (e.g., McCabe et al. 2006). In our case, there is no bottom slope, so we rederived the form drag equation to account for the flat bottom and curving sidewalls. The resulting expression for form drag around a headland is
i1520-0485-39-11-2971-e16
where Ap is the area of the wall containing the headland projected onto the xz plane, p is the pressure on that wall, ξ is the headland extent in the y direction, and ξx is the slope of the headland in the x direction. Figure 1 shows the directions of the axes. The negative sign appears in this equation because the form drag removes momentum from the flow.
The separation drag is simply the difference between the total form drag and the inertial drag at each time step,
i1520-0485-39-11-2971-e17
This is known as the Morison equation (Dean and Dalrymple 1984), and it has been used in previous studies as a useful way to divide up the form drag (Lowe et al. 2005). Similarly, the residual pressure field can be obtained by subtracting the potential flow pressure field from the total pressure field. The separation drag can either be calculated with Eq. (17) or directly from the residual pressure field using Eq. (16). The most important thing to note about the separation drag is that, because the inertial drag is nondissipative, the tidally averaged power removed by the total form drag and the separation drag is equal.

3. Methods

a. Specifics of the ROMS model

To investigate the form drag resulting from oscillating flow around headlands, a series of numerical experiments were performed using the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005). ROMS is a three-dimensional, free-surface, hydrostatic, terrain-following-coordinate model that solves the baroclinic and barotropic momentum equations separately. Although our model was not intended to accurately model the dynamics near Three Tree Point, the physical dimensions and stratification of our idealized setup were chosen to roughly resemble the main basin of the Puget Sound, where Three Tree Point is located. The channel in the model was 100 km long, 10 km wide, and 200 m deep, with closed sides and open ends. The salinity at the surface was 30 and increased linearly to 32 at depth, which resulted in a buoyancy frequency of 0.01 s−1. Although the model was stratified, there were vertical sidewalls and barotropic tides, so most of the flow outside the frictional boundary layer remained barotropic. Compared to a model run in a channel without stratification, the separation drag changed by only 1.7%.

The tides were modeled as propagating waves in the shape of simple sinusoids, oscillating at the M2 frequency. The maximum tidal velocity and amplitude varied for each run (explained in section 3b). To force the tides, both the sea surface height and the barotropic velocity oscillated at the same period and phase on the left open boundary of the channel. The tidal wave was allowed to freely propagate downstream at a shallow water wave speed of 44.3 m s−1. The open boundary conditions were set so that mass, momentum, and waves are allowed to leave the domain. Specifically, this meant that the free surface had Chapman boundary conditions, the barotropic momentum had Flather boundary conditions, and both the baroclinic momentum and the tracers had radiation boundary conditions (for an overview of boundary conditions in ROMS, see Marchesiello et al. 2001). Nudging the tracers toward background values and the baroclinic momentum equation toward zero was necessary in 15-km-long regions at each end of the domain. This prevented any possible numerical wave reflection off of the open boundaries. The time scale of the nudging was set to be about 2 h for the tracers and 12 h for the baroclinic momentum equation. Analysis was done only in the region 30 km upstream and downstream from the headland crest, which ensured that the results were not influenced by the nudging regions at either end.

The horizontal grid of the model was stretched in the along-channel direction; however, this stretching did not affect the analysis region, where the grid size was 100 m in both the along- and across-channel directions. Exceptions to this were made for two of the cases with particularly pointy headlands, where the grid size had to be reduced to 50 m. With this resolution, there were always at least 10 grid points in the along-channel direction on each headland. There were 20 vertical grid boxes, spaced to provide more resolution near the top and bottom of the domain.

The headlands were Gaussian shaped. Their across-channel extent ξ was defined at every x coordinate by the equation
i1520-0485-39-11-2971-e18
where Δ is the maximum across-channel headland width; L is the along-channel, e-folding length of the headland; and xmid is the coordinate in the middle of the domain, where the headland crest is located. Numerically, the headlands were simulated using a masking region in the shape of Eq. (18).

The model used a generic length-scale mixing scheme within the domain. The bottom boundary used a quadratic drag law with a drag coefficient of 3 × 10−3. The side boundaries were set to have free-slip conditions. The effect of no-slip versus free-slip boundaries was tested. Only very small changes were detected when no-slip side boundaries were used instead of free-slip side boundaries. The lack of change is thought to be due to the fact that the headland itself does not have a perfectly smooth shape, because it is defined only to the resolution of the grid. This somewhat jagged nature to the shape of the headland makes it act as if there were no-slip sidewalls, even though they were set to be free slip.

The Coriolis force within the model was set to zero. The decision not to include the Coriolis force was justified by the fact that the Rossby number at Three Tree Point is about 10. Because the effect of the Coriolis force would be minimal, it was decided to leave it out in order to more clearly focus on the dominant mechanisms in the flow. In cases with larger headlands, such as in Freeland (1990) and Magaldi et al. (2008), the Coriolis force is an essential part of flow dynamics.

The model was run for 10 full tidal cycles in order to get the domain “spun up” to a tidally steady state. The analysis was performed on the data from the eleventh and twelfth tidal cycles. Throughout this paper, we use the time scale lunar hours (3726 s) for consistency with the forcing frequency. To avoid breaking the Courant–Friedrichs–Lewy (CFL) condition, the model’s time step was 29.8 s, except in the two cases with smaller grids where Δt = 7.5 s was used. Data were saved once per lunar hour.

b. Nondimensional parameters and model runs

The goal of this series of numerical experiments was to test how different headland shapes and flow conditions affected the form drag. To do this, a “base run” was created that had dimensions somewhat close to those found at TTP. We then identified two nondimensional and one dimensional parameter that we thought would affect the form drag. Each parameter was systematically varied to cover a range of parameter space, with the other two held constant at their base values. The first nondimensional parameter was the tidal excursion distance divided by the along-channel headland length,
i1520-0485-39-11-2971-e19
which is equal to the Keulegan–Carpenter number (mentioned briefly in section 2a) divided by a factor of π. The second parameter was the aspect ratio of the headland,
i1520-0485-39-11-2971-e20
which defined how sharp or streamlined each headland was. The final parameter was the expected bluff body drag DBB from Eq. (1), with CD = 1. These three parameters form a system of equations that can be solved for the maximum tidal velocity U0, the headland e-folding length L, and the headland width Δ. All other model variables such as the forcing frequency ω, the depth H, the channel width, and the stratification remained constant. The final parameter that had to be determined was the tidal amplitude, which is related to the tidal velocity in the following way:
i1520-0485-39-11-2971-e21
where c = gH is the surface gravity wave speed. All of the experiments, including the relative size of the nondimensional parameters and headland dimensions, are listed in Table 1. Each experiment has been given a name based on the scaling of the parameters. For instance, run 2α has an aspect ratio that is twice as steep as the base run.

4. Results

a. Time series of inertial and separation drags

Before delving into the results from all of the model runs, it is important to understand how the different parts of the drag behave throughout the tidal cycle. In Fig. 3, the tidal velocity, drag, and power from the base run are all plotted with respect to time over two tidal cycles. The velocity shown here and used in the power calculations is a spatial average of the velocity throughout the analysis region of the domain. The velocity and the three parts of the drag all have the same frequency, but their relative phases are offset from one another. At any point in time, the sum of the separation drag and the inertial drag equals the total drag. The same is true for the power and cumulative averaged power.

At slack tide, the inertial drag is applying its maximum force, and a quarter of a tidal cycle later, when the velocity is at its maximum, the inertial drag is zero. The power associated with the inertial drag has equal positive and negative parts, so its cumulative average is zero when calculated over even half of a tidal cycle. The phase of the separation drag leads the velocity. At slack tide, the separation drag is about three-quarters of a lunar hour past its zero, which is 22° ahead of the velocity. Because the separation drag and the velocity are close to being in phase with each other, throughout the tidal cycle, the power associated with the separation drag is mostly removing energy from the flow. This differs from the power associated with the total and inertial drags, which have both positive and negative parts. The most important thing to notice from Fig. 3 is that although the amplitude of the inertial drag is much larger than the amplitude of the separation drag, the separation drag accounts for all of the tidally averaged power loss.

b. Inertial drag: Amplitude

The data discussed in the previous section from each model run is condensed into Fig. 4, which shows the amplitude of a sinusoidal fit to the separation drag (black bar) and the inertial drag (gray bar). The relative sizes change significantly, especially in the cases where λ was scaled. The estimate of the inertial drag by Dean and Dalrymple (1984) is
i1520-0485-39-11-2971-e22
where CM is the inertial drag coefficient, V is the volume, and U(t) is tidal velocity defined in Eq. 10. For an ellipsoid, the inertial drag coefficient may be shown analytically to be
i1520-0485-39-11-2971-e23
where a and b are the along- and across-stream semimajor axis lengths of the ellipse (Dean and Dalrymple 1984). For the Gaussian headlands used in this experiment, there is no analytical solution for CM, so we instead approximate a = L and b = Δ and the inertial drag coefficient becomes CM = 1 + Δ/L. The exact volume of a Gaussian headland is πΔLH. Using these values for the inertial drag coefficient and the volume gives
i1520-0485-39-11-2971-e24
as a prediction of the inertial drag for a Gaussian headland. The validity of Eq. (24) is seen in Fig. 5, where the actual inertial drag is compared to this estimate, assuming ∂U(t)/∂t scales as U0ω. Equation (24) overestimates the inertial drag slightly, as all but one of the runs fall slightly above the 1:1 line. This could be because Eq. (24) was derived for flow around an elliptical cylinder, not flow around a Gaussian-shaped headland in a channel, two shapes that have different flow patterns. The exception to this trend is the 0.25λ run, which falls below the 1:1 line. This headland was the largest of any of the headlands tested. Its across-channel width Δ was over half the width of the channel and nearly double the size of all of the other headlands. When we reran this model case in a channel that was twice as wide (20 km), we found that the inertial drag amplitude was much more in line with the trend found for the other runs. The opposite wall was having an effect on the flow patterns in this anomalous case.
Just as in Eq. (11), we will calculate the ratio of the inertial drag to the bluff body drag:
i1520-0485-39-11-2971-e25
As before, we assume that the bluff body drag coefficient is 1 and the tidal excursion distance is LT = 2U0/ω. As was found with the cylinder, when the tidal excursion distance is small, the inertial drag dominates over the bluff body drag; as the tidal excursion distance becomes longer, the impact of the inertial drag wanes.

c. Separation drag: Amplitude

To see how the separation drag changed as λ, α, and DBB were scaled, the results from all of the model runs have been organized into Fig. 6. The amplitudes of the separation drag for all the model runs are shown in the top panels. In the three cases where the bluff body drag DBB was scaled, with λ and α held constant, the separation drag roughly followed the predicted value from Eq. (1), as is seen in Fig. 6a.

In the cases where the tidal excursion parameter λ was scaled, the separation drag stayed relatively constant and nearly equal to the bluff body drag, as can be seen in Fig. 6b. Hence, the separation drag does not appear to depend on λ. The exception to this trend is the 0.25λ run. As discussed in the previous section, we found that the flow around this headland was significantly affected by the presence of the wall on the opposite side of the channel, which worked to increase both the separation and inertial drags. In the wider channel, the amplitude of the separation drag for this headland differed by only 6.4% from the bluff body drag.

In the four cases where α was scaled, the separation drag does not equal the bluff body drag, as seen in Fig. 6c. A large α corresponds to a sharper headland, and the separation drag is significantly larger than the bluff body drag. When α is small, the headland is more streamlined and the separation drag is smaller than the bluff body drag. The reason for this discrepancy is because we assumed a drag coefficient of 1 when calculating the bluff body drag. However, it is expected that sharper headlands would have higher drag coefficients than more streamlined headlands. A histogram of the drag coefficients calculated using CD = Dseparation/(½ρ0ΔHU02) is shown in Fig. 7. Eight of the runs have drag coefficients that are close to 1. This corresponds well to Hoerner’s (1965) estimate of a drag coefficient of 1.17 for flow around a cylinder at a high Reynolds number. The exceptional runs are the three cases where α was scaled and the anomalous 0.25λ run. The case with the sharpest headland (run 2α) has the largest drag coefficient, and the case with the most streamlined headland (run 0.5α) has the smallest drag coefficient. A drag coefficient of 4 is surprisingly large. However, it is hard to compare this number to experimentally determined drag coefficients, which have only been found for steady flow conditions around objects of all shapes (Hoerner 1965) and for cylinders in oscillating flow (Obasaju et al. 1988) but not for odd-shaped objects in a channel subjected to oscillating flow. The important point is that, as the headland aspect ratio α increases, so does the drag coefficient. From the trends that are observed in Fig. 6 (top), we can say that the separation drag is proportional to the bluff body drag, provided a reasonable drag coefficient is used.

d. Separation drag: Phase and power

The plots in the middle row of Fig. 6 show the phase of the separation drag in relation to the phase of the tidal velocity. In every case, the drag leads the velocity. Similar phase leads have been found in laboratory experiments of oscillating flow around a cylinder (Obasaju et al. 1988) and at Three Tree Point, where it was found that the phase of the drag led the tidal velocity by 1–2 h (McCabe et al. 2006), which is between 30° and 60°.

The relationship between the tidal excursion distance parameter λ and the phase of the separation drag is shown in Fig. 6e. When the tidal excursion distance is shorter than the headland length—cases of small λ—the drag leads the velocity by about 45°. An analogy may be drawn to the description by Batchelor (1967, 353–358) of a 45° phase lead of the frictional stress on an oscillating flat plate. On the other extreme, when the tidal excursion distance becomes longer than the headland length, the drag becomes nearly in phase with the velocity. This makes sense because as the tidal excursion distance gets longer, the flow is heading toward a quasi-steady state. So, in general, when λ is less than one, the phase lead of the drag will be about 45°; when λ is greater than 1, the drag and velocity will be in phase with each other; and when λ is close to 1, the phase difference will fall in a transitional region between 45° and 0°.

The bottom row of Fig. 6 shows the tidally averaged power that the separation drag can remove from the flow. This is equal to the power loss from the total drag, as explained in section 2a. Recall that power is the product of drag and velocity, so for two functions oscillating at the same frequency such as U(t) = U0 sinωt and D(t) = D0 sin(ωtφ), where φ is the phase difference between the two, the tidally averaged power is 〈P〉 = ½D0U0 cos(φ). Now, looking at Figs. 6b,e,h, the relationship between the phase of the separation drag and the tidally averaged power can be seen. The gray line is the theoretical power of the bluff body drag, 〈P〉 = ½DBBU0. Although the amplitude of the drag changes very little when λ is scaled (except for the 0.25λ run), the power loss more than doubles when λ is increased from 0.5 to 4. This is partly due to the phase change and partly due to the increasing tidal velocity. Although knowing the amplitudes of the drag and the velocity will give a good indication of the expected power loss, the phase can also create changes in the total power loss.

In the cases where the aspect ratio of the headland changes, there is a distinct phase change, as can be seen in Fig. 6f. For streamlined headlands, the phase is approaching the 45° limit that was observed for the small λ cases. However, unlike when the tidal excursion parameter changed and there was an abrupt jump from a 45° to 0° phase lead, here the phase seems to increase in a nearly linear fashion. It is possible that there are limits near 0° and 45° for this parameter too. The experimental range of α is more limited than the range of λ, so possibly the aspect ratio all of our runs fell in the transitional-phase region between 45° and 0°. As expected, because the amplitude and the phase of the separation drag are both increasing as α gets larger, the power in Fig. 6i is also increasing substantially.

For the cases where the bluff body drag was scaled, there is a 10° change in phase as the drag increases (Fig. 6d), but this variation is small compared with the other two cases. Because the phase shift is small, the change in power is due more to the increasing separation drag amplitude and tidal velocity than to the change in phase between the drag and the velocity.

e. Description of flow through a tidal cycle: What mechanisms create the separation drag and its phase?

To gain intuition about the mechanisms that create the separation drag, it is useful to look at the residual pressure field and contours of relative vorticity over half of a tidal cycle. Figure 8 shows an ebb tide for the base case. The residual pressure field is found by simply subtracting the potential flow pressure from the total pressure. The form drag depends on the pressure right next to the headland’s edge, especially in regions where the headland has a steep angle ξx. The greatest form drag occurs when there is a high pressure on one side of the headland and a low pressure on the other.

Looking at the six panels depicting the ebb tide of the base case in Fig. 8, two prominent mechanisms that create form drag can be seen: low pressure regions associated with eddies and high pressure regions resulting from water “piling up” behind the headland and the eddies. In the first panel, at slack tide, just before the tide begins to ebb, there is an eddy left over from the previous flood and there is also a high pressure region just to the right of the eddy. Because the low pressure and high pressure regions are both on the same side of the headland, the form drag is nearly zero at this time step. As the tide begins to ebb (flow to the left) during the next two lunar hours, the strength of the eddy dipole increases. However, its direct contribution to the form drag is weakened as it is pushed toward the tip of the headland. At the same time, the high pressure region on the upstream side of the headland is growing in size and magnitude. Not only are the eddies blocking the ebb tide from getting around the headland, but their circulation pushes more water back upstream. The separation drag reaches its maximum two lunar hours after slack tide, at t = 8, when there is the most water piled up on the upstream side of the headland.

At the following time step, the tidal velocity reaches its maximum and the eddy dipole begins to move downstream, away from the headland. Not all the vorticity travels with the dipole. Some remains on the leeward side of the headland for the next four lunar hours, until the tide changes direction and begins to flood. At the same time, the water that was piled up on the upstream side of the headland is slowly making its way to the left, around the headland. However, even though the high pressure on the upstream side is waning, the low pressure on the downstream side associated with the eddy is increasing in strength, so a substantial amount of form drag is still present all the way through the remainder of the ebb tide. The form drag only takes two lunar hours to go from zero to its maximum, but it then takes four lunar hours to return back to zero. After the ebb tide, the currents switch and the pattern repeats itself in the opposite direction. There is very little difference in the magnitude of the eddies or the separation drag between the flood and ebb tides.

All of the other runs have basically the same mechanisms creating the form drag as the base run. There are some small differences that can be noted. For instance, in cases with sharp headlands, the eddies are larger and stronger, and hence there is more separation drag. The drag is also more in phase with the velocity because, when the dipole eddy is shed downstream, a greater amount of vorticity remains on the headland than in the base case and it takes longer for the piled up water to flow around the headland. Therefore, the maximum form drag and the maximum velocity occur at nearly the same time. In general, for all of the runs, the eddies create low pressure regions in their vicinity and block the flow in nearby areas to create high pressure regions. This creates the separation drag. The position and timing of the eddies’ release determines the phase of the drag.

5. Conclusions

In this study of form drag generated by oscillating flow in a channel around headlands, the drag was divided into two portions. The inertial portion was derived by numerically solving for the pressure field associated with the irrotational, inviscid part of the flow. The inertial drag can have a significant amplitude; however, it cannot do work on the flow because its phase is in quadrature with the velocity. The other part of the form drag is the separation drag associated with flow features such as eddies. The power that can be extracted from the flow by the separation drag is equal to the power associated with the total drag.

Headlands with different sizes and flow characteristics were tested with a numerical model. These experiments confirmed the results of Edwards et al. (2004): in oscillating flow situations when the tidal excursion distance is nearly equal to the topographic length, the total drag will be larger than the bluff body drag because of the added impact of the inertial drag. Furthermore, we showed that in cases where the tidal excursion distance was shorter than the topographic length, the inertial drag was larger than the separation drag and the separation drag led the velocity by about 45°. As the tidal excursion distance becomes longer, the inertial drag becomes smaller and the separation drag and the velocity become more in phase. The magnitude of the separation drag was close to the bluff body drag estimate provided the aspect ratio of the headland was close to 1; otherwise, sharper headlands have higher drag coefficients and hence more separation drag than streamlined headlands.

Acknowledgments

The authors thank David Darr for his help with computers and the ROMS model. Useful feedback and suggestions were given by Matthew Alford, LuAnne Thompson, Danny Grünbaum, and Ryan Lowe. Thank you also to two anonymous reviewers for their insightful comments. This work was funded by NSF Grant OCE-0425095.

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Fig. 1.
Fig. 1.

(top) Schematic diagram of the topography at TTP. It is shaped like a ridge on a sloping sidewall and therefore generates both internal waves and eddies, which create regions of relatively high and low pressure. To simplify the problem and more easily identify flow mechanisms contributing to form drag, the system was split into two cases: (bottom left) the internal wave–generating ridge case and (bottom right) the eddy-generating headland case. In this paper, only the headland case is discussed. The coordinate axes shown on the headland diagram align with the axes in the numerical model.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 2.
Fig. 2.

(left) Plan views of the potential flow SSH calculated from the potential flow pressure field, such that SSH = p/ρ0g. The black contours show the potential flow streamlines. (a) At maximum flood and ebb tides, the SSH has the shape shown, with a region of low pressure at the tip of the headland. (b) At slack tide, when the acceleration is greatest, the SSH is tilted as shown. This tilt is constant throughout the channel except (c) near the headland, where the slope deviates slightly from the background slope. This extra tilt increases the magnitude of the inertial drag.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 3.
Fig. 3.

(a) The tidal velocity with respect to time for two tidal cycles. (b) The total form drag (black), the inertial drag (gray), and the separation drag (dashed) are shown, as well as (c) the corresponding power. The power is simply the product of the velocity and the drag. (d) The cumulative average of the power is shown. Over time, the cumulative—or running—average of the power is approaching an asymptote. Although the amplitude of the inertial drag is nearly as large as the amplitude of the total drag, the separation drag accounts for all of the cumulative power losses after a complete tidal cycle. These curves are from the base run. The relationship between the velocity, drag, and power is similar for the other runs.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 4.
Fig. 4.

Each model run is represented by a bar. The gray part of the bar is the amplitude of the inertial drag, and the black part of the bar is the amplitude of the separation drag; both are calculated with a sinusoidal fit to the data. Their sum is close to the amplitude of the total drag, but not exact because of phase differences between the separation and inertial drags. The white triangles point to the magnitude of the bluff body drag, assuming a drag coefficient of 1. The runs are organized (left)–(right) with the base, the runs where the bluff body form drag parameter DBB was changed, the runs where the tidal excursion distance divided by the along-channel headland length λ was changed, and the runs where the headland aspect ratio α was changed. The magnitude of the inertial drag is generally much larger than that of the bluff body drag; in most cases, the magnitude of the separation drag is close to the bluff body drag.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 5.
Fig. 5.

The relationship between the actual inertial drag and the estimated inertial drag is shown. The inertial drag is estimated by Dinertial = (1 + Δ/L)ρ0VU0ω, where the volume V = πΔLH. All of the runs, except the 0.25λ run, fall just above the 1:1 line, so their actual inertial drag is slightly less than the estimated inertial drag. This could be because the scaling relationship was derived for flow around an ellipse, not flow around a Gaussian-shaped headland.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 6.
Fig. 6.

Each column shows how the separation drag changed as the three experimental parameters—(left) DBB, (middle) λ, and (right) α—were scaled. Each row shows a property of the separation drag: (top) the amplitude (in Newtons), (middle) the phase difference between the separation drag and the velocity (in degrees), and (bottom) the tidally averaged power that the drag can remove from the flow (in watts). The gray lines in (a),(b),(c) in the bluff body drag with a drag coefficient of 1 and in (g),(h),(i) show the tidally averaged power loss resulting from the bluff body drag, 〈P〉 = ½U0DBB.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 7.
Fig. 7.

Histogram of drag coefficients for the separation drag for all of the runs. Most of the runs fall on or just above a typical drag coefficient of 1. The exceptions are the cases where α was changed to create a sharper or more streamlined headland. The last exception is the anomalous 0.25λ run.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Fig. 8.
Fig. 8.

(top to bottom) Top view snapshots of the headland during the last six time steps of a tidal cycle for the base case, showing the evolution of the residual SSH and associated vorticity field throughout an ebb tide.

Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4173.1

Table 1.

The specifics for all of the numerical experiments. The experimental parameters that characterized each model run were the bluff body drag estimate DBB, the aspect ratio of the headland α, and the tidal excursion distance divided by the headland length λ. The model parameters that were then calculated were the across-channel width of the headland Δ; the along-channel, e-folding length of the headland L; the amplitude of the tidal velocity U0; and the tidal height η0.

Table 1.
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