Analytic representations of dissipation in partially-submerged reciprocating pipe flow

Abstract

A series of analytic models are presented of a partially-submerged, straight circular cross-section pipe subjected to reciprocating flow, incorporating viscous and turbulent boundary-layer dissipation. This pipe is the most elementary paradigm for a fixed-type Oscillating Water Column (OWC), and its analysis more generally addresses one aspect of reciprocating pipe flow. The derivation of the dissipation terms in the momentum conservation equations from their origin in the Navier–Stokes equation are explicitly identified. The contribution of different damping sources to the overall energy loss is compared. The flow inside the pipe is assumed fully developed along the entire length, neglecting the effect of the flow development region. For relevance to engineering applications, a power take-off system is modelled assuming that the air compresses and expands isentropically in an air chamber at the top of the pipe. Existing theory is adapted into the current models for computing the hydrodynamic coefficients related to the scattered wave and radiation wave. A novel contribution of this paper is the inclusion of damping due to the wall shear stress, modelled for the reciprocating flow system inside the pipe. Comparison between the damping factors in relatively short OWCs confirms the findings in the literature that the major part of the damping is due to the radiation wave. However, it is found that with the increase of the draft length, the wall shear stress damping may also become an important factor in the resulting dynamics.

Introduction

Several phenomena must be dealt with when modelling dissipation in partially-submerged reciprocating pipe flow, andamongst them are the issues of conditional turbulence in the wall shear stress and energy losses by wave radiation. A reciprocating flow cycle consists of an acceleration and a deceleration phase in the positive direction, and the same sequence in the reverse direction, giving two acceleration and two deceleration phases. Like steady flow, the reciprocating flow might be laminar, weakly turbulent or fully turbulent throughout the cycle, depending on the Reynolds number. However, Hino in [1] introduced a new flow regime in reciprocating flow, called the conditional turbulence regime. In this regime, a portion of the flow cycle is laminar and the other portion is turbulent. As ${Re}_{\delta }=U_b\delta ∕\nu$ goes above $550$ ($U_b$ is the cross sectional mean velocity, $\delta =\sqrt{2\nu ∕\omega }$ is the Stokes layer thickness and $\nu$ is the kinematic viscosity), the flow enters into this regime. Above the critical value but at relatively low ${Re}_{\delta }$, most of the cycle is laminar, and turbulence appears in the final portion of the deceleration period. At higher ${Re}_{\delta }$ values, the flow is laminar during the acceleration period and becomes turbulent during most of the deceleration period, as illustrated schematically in Fig. 1. However, the violent turbulence generated in the deceleration period suddenly disappears as the flow reverses direction and starts to accelerate again. As ${Re}_{\delta }$ increases further, turbulence begins to appear in the acceleration period as well. Moreover, [2] shows that the entire flow cycle can become turbulent if the ${Re}_{\delta }$ is large enough.

The aim of the present paper is to introduce wall shear stress into analytic models of a partially-submerged pipe in which the flow is reciprocating, which could be considered the most elementary paradigm of an Oscillating Water Column (OWC). Wave radiation losses are dealt with by adapting existing theory [3].

There are many inventions for extracting renewable energy from ocean waves [4], [5], [6], [7], [8], [9], [10], [11], [12]. They may be classified in two major types. The first type consists of devices that convert the wave energy by means of oscillations of floating bodies such as buoys, cams and floats. The second type is the OWC, which consists of a vertical hollow cylinder or duct, with one end submerged under the water surface, and the other end open to the atmosphere via a turbine. The underwater portion of the duct may be straight or bent. Waves in the ocean cause the water-column in the pipe to push and pull the air above through a bi-directional air turbine (generally the Wells turbine), which generates electricity.

The maximum power is extracted when the device natural frequency coincides with the incoming wave frequency; i.e. the system resonates. Dissipation limits the power the device can extract at resonance. Therefore, the damping factors in the device play a major role in controlling the overall efficiency of the system. In principle, the optimum power is extracted by a fixed-type OWC with linear dynamics when it resonates and the rate of useful power extraction is equal to the rate of damping [8].

An OWC basically works on the principle of a forced mass–spring–damping system, where the water column represents the mass, and the body force on it (due to gravity) plays the role of the spring restoring force. The damping, which is the primary focus of this study, consists of different terms representing different mechanisms of energy loss as the OWC operates.

A number of previous theoretical studies of fixed-type OWCs have been conducted. These models typically result in a damped harmonic oscillator. Extensions to this basic model include modifications to model the free surface effects in the OWC [13], more sophisticated models of the air chamber and Power Take-Off (PTO) system [14], and the calculation of hydrodynamic coefficients for radiation and diffraction waves [3], [15]. Parameter studies using these models have investigated the effect of water depth, air compressibility, and turbine characteristics [14], the optimization of the turbine parameters [16], [17], [18], and the performance of OWCs in specific circumstances, such as at the tip of a breakwater [19]. However, all the above mentioned works considered the flow inside the device as a potential flow, neglecting the impact of the dissipation from viscosity and turbulence.

A number of dissipation mechanisms exist in the OWC. The first dissipation mechanism, as mentioned above, is the wall shear stress in which boundary layers introduce viscous and turbulent dissipation causing a loss of energy. A second mechanism is the radiation of waves. Pressure fluctuations in the air chamber cause a secondary wave to radiate away from the OWC, carrying energy with it. In the potential-flow assumption, which is used to model radiation, energy cannot be lost, but is radiated to infinity, and thus in practical terms represents energy lost from the oscillating flow in the pipe. A significant amount of damping must come from the PTO system, which is, in general, a turbine-generator arrangement and this, of course, represents the useful power extracted from the waves. Vortex formation at the submerged end of the pipe is also likely to play a role.

The impact of these different dissipation mechanisms on the overall power output has not been rigorously studied. However, damping is one of the most important design parameters, required to estimate the total amount of power loss. Laboratory experiments have reported on power losses in scale models of actual devices [20]. Theories on calculating the radiation damping and hence estimating the efficiency of an OWC have been thoroughly derived in [13], [14]. However, very few theoretical studies have incorporated the dissipation from wall shear stresses into their models [21], [22]. Recent numerical studies on the flow dynamics of an OWC using standard computational fluid dynamics packages which used turbulence models, e.g. $k$-$ϵ$, $k$-$\omega$ etc., successfully validated the numerical models against the experimental results by evaluating the overall performance of OWCs [23], [24]. However, rigorous studies on estimating different damping factors for the high Reynolds number reciprocating flow in OWCs are yet to be done.

In this paper, a fluid-dynamical approach is taken to derive the mass–spring–damping model of a partially-submerged pipe from the Navier–Stokes equations, in order to examine the essential physics of the OWC without undue complexity. The radiation damping derived in [3] is adapted to the present model after transforming to a rigid body model as explained in [25]. Air in the air-chamber is considered compressible, and it is assumed that the change of air pressure maintains a linear relationship with the mass flow rate through the turbine (as in a Wells turbine model [26]).

The first-order flow created inside the OWC by ocean waves is reciprocating, i.e. it is periodic and completely reverses direction. A key novel contribution of the present work is the inclusion of the resulting reciprocating shear stress, modelled based on the theory developed in [27] and the Blasius correlation for steady turbulent flows in smooth pipes, as justified in [28]. For OWCs with a short draft, the amount of energy loss due to vortex formation at the submerged free-end is quite significant [29], however, for OWCs with a long draft as proposed in [30], the vortex loss might not be as significant as the radiation and shear stress losses. Estimating the energy loss due to the vortex formation at the entrance requires a substantial amount of work to understand the unsteady and nonlinear formation of vortices, and the subsequent viscous and turbulent energy dissipation process. A study of this problem does not exist at present. In this paper, the flow inside the device is assumed fully-developed, neglecting the effect of the developing region and vortices at the entrance which is a reasonable assumption for OWCs with long draft. However, results for short-drafted OWCs are also presented to at least estimate the contribution of shear stress damping to the overall damping. Future work will estimate the loss due to entrance vortices and include it in the present model to create a comprehensive model for OWCs with short draft. A comparison of overall power output is presented for different wall shear stress models.