## Abstract

This paper explores the influence of blockage and free-surface deformation on the hydrodynamic performance of a generic marine cross-flow turbine. Flows through a three-bladed turbine with solidity 0.125 are simulated at field-test blade Reynolds numbers, *O*(10^{5}–10^{6}), for three different cross-stream blockages: 12.5, 25 and 50 per cent. Two representations of the free-surface boundary are considered: rigid lid and deformable free surface. Increasing the blockage is observed to lead to substantial increases in the power coefficient; the highest power coefficient computed is 1.23. Only small differences are observed between the two free-surface representations, with the deforming free-surface turbine out-performing the rigid lid turbine by 6.7 per cent in power at the highest blockage considered. This difference is attributed to the increase in effective blockage owing to the deformation of the free surface. Hydrodynamic efficiency, the ratio of useful power generated to overall power removed from the flow, is found to increase with blockage, which is consistent with the presence of a higher flow velocity through the core of the turbine at higher blockage ratios. Froude number is found to have little effect on thrust and power coefficients, but significant influence on surface elevation drop across the turbine.

## 1. Introduction

The urgent need to establish a clean, safe and affordable energy supply has placed an increased emphasis on the exploitation of new renewable energy sources. The ocean offers immense potential for clean energy extraction and, besides wave and tidal barrage technologies, tidal stream turbines have been identified as prospective marine energy converters. The extractable energy potential owing to tidal stream turbines around the British Isles is estimated to be 18 TWh per year [1], which equates to 5 per cent of the UK’s energy consumption in 2008 [2]. However, this resource has not yet been exploited and as the underlying flow physics of tidal turbines in confined flow conditions is not fully understood, fundamental research is still required to ensure tidal energy is harnessed as cost effectively as possible.

To this end, this paper presents numerical investigations of the hydrodynamic performance of generic marine cross-flow turbines with the objective of furthering the understanding of flows through such devices. Analytical and small-scale laboratory investigations [3] have revealed that the flow through a cross-flow turbine is extremely complex with simultaneous attached and separated flow regions as well as reattachment, dynamic stall and blade–vortex interaction events. Hence, proper numerical modelling of the flow through such turbines necessitates the use of an incompressible viscous flow model that can properly simulate separated flows and account for turbulence effects. Hence, in this study, we choose to use a mesh-based computational fluid dynamics (CFD) model solving the Reynolds-averaged Navier–Stokes (RANS) equations. In addition to providing the turbine’s torque and power outputs, the CFD model renders a flowfield that can be interrogated to determine vortex structures and blade pressure distributions, and hence permits a detailed analysis of the turbine’s flow physics.

For the case of horizontal axis cross-flow turbines, we examine the effect of free-surface proximity on the turbine. Firstly, we consider a blockage ratio of 50 per cent, where we define blockage ratio as the ratio of the frontal area of the turbine to the area of the flow passage. A volume of fluid (VOF) model is used to model changes in the free surface associated with the energy extraction of the turbine at three Froude numbers, *Fr*, of 0.082, 0.097 and 0.131.

Moreover, at a fixed *Fr* of 0.082, the effect of changes in blockage ratio from 50 to 25 and to 12.5 per cent on free-surface deformation and turbine performance are examined. We also investigate the influence of free-surface modelling approximation and compare the deforming free-surface simulations with those in which the free surface is a fixed rigid lid.

## 2. Numerical method

The simulations presented in this paper have been conducted with the commercial CFD package Fluent [4], which solves the RANS equations using a finite-volume approach. For this study, Fluent is used as a two-dimensional, incompressible flow solver. As the flow problem is unsteady, the evolution of the flowfield is solved on a time-marching basis, in which the flow equations are solved implicitly.

The computational domain, a two-dimensional slice orthogonal to the turbine’s axis of rotation, is made up of three sub-domains: (i) a far-field domain, (ii) a turbine domain consisting of a circular rotating mesh, and (iii) discrete circular domains around each blade. To simulate the rotation of the rotor, the circular turbine mesh with embedded blades is allowed to move relative to the outer inertially fixed domain.

The turbine is of radius *R*=3.82*c*, where *c* is the blade chord length. The domain extends 16*R* upstream and 24*R* downstream of the centre of the turbine and 2*R*, 4*R* and 8*R* laterally to either side of the turbine, resulting in blockages of 50 per cent, 25 per cent and 12.5 per cent, respectively. As the computational domain is two dimensional, the turbine blades are implicitly assumed to be infinitely long. A consequence of this assumption is that drag owing to blade support struts or end plates is neglected. The turbine blades are chosen to be National Advisory Committee for Aeronautics (NACA) 0015 blade sections.

The following boundary conditions were used in the simulations: (i) no slip conditions on the blade surfaces, (ii) unperturbed streamwise flow conditions across the upstream (inflow) boundary, (iii) symmetry conditions on the lateral boundaries (or VOF on the upper fluid boundary for deforming free-surface simulations), and (iv) either a constant or hydrostatic pressure condition along the downstream (outflow) boundary for rigid lid and free-surface simulations, respectively. The RANS equations were closed, using the shear stress transport (SST) *k*−*ω* turbulence model. For proper boundary-layer resolution with this model, the wall normal resolution is set such that *y*^{+}=0(1) for the wall adjacent elements [4], where the dimensionless wall unit , and *τ*_{w}, *ρ*, *ν* and *y* are the wall shear stress, fluid density, fluid kinematic viscosity and wall normal distance to the wall adjacent cell centre.

For the simulations in which the free surface is permitted to deform, the domain extends above the expected air–water interface region to encompass an additional region of air. The interface region straddling the expected air–water interface is more highly resolved in order to permit surface tracking, using the VOF method. The use of this method necessitates the free-surface elevation at both the inflow and outflow boundaries to be set in advance through Dirichlet-type boundary conditions. However, setting both levels implicitly prescribes the energy removal from the flow within the simulated domain. As the level of energy removal by the turbine cannot be specified *a priori*, the elevation at the outflow boundary must be iteratively adjusted so as to achieve a solution that satisfies the imposed upstream Froude number while simultaneously not constraining the energy removed by the turbine. In this work, this is achieved through a feedback loop implemented in a user-defined function that samples the upstream flow height and adjusts the downstream water height recursively so as to achieve the target upstream Froude number.

Temporal discretization is achieved using the semi-implicit method for pressure-linked equations scheme for pressure–velocity coupling, and second-order upwind for the momentum and turbulence transport equations.

## 3. Validation

The hydrofoils of a cross-flow turbine undergo changes in their angle of attack as the turbine rotates. It is therefore important that the numerical model is validated, not just for static foil considerations, but also for dynamically pitching foils. We first compute lift and drag coefficients, *C*_{L} and *C*_{D}, for a two-dimensional NACA 0015 aerofoil section over a range of angles of incidence, *α*, and compare results with experimental wind-tunnel data obtained from the Sandia National Laboratories [5]. The force coefficients are defined as
3.1
where *L* and *D* are the blade lift and drag forces per unit span, and the incident flow velocity. Secondly, we simulate a dynamically pitching two-dimensional NACA 0015 aerofoil oscillating by ±4^{°} around mean pitch angles of 4^{°} and 13^{°} and compare computed lift and drag data with experimental wind-tunnel data [6].

The validation stage enabled the influence of numerical parameters, such as mesh resolution, time step and turbulence model, to be investigated. In this study, we are concerned with field-test Reynolds numbers and hence the static blade validation stage was carried out at a Reynolds number based on blade chord *Re*_{c}=6.8×10^{5} and the dynamic oscillatory blade validation stage at *Re*_{c}=2×10^{6}, where
3.2
The grid used was a hybrid mesh comprising a very fine-structured grid for the near-wall region, a fine-structured grid for the region within one chord length of the hydrofoil and an unstructured grid for the remainder of the domain. Each hydrofoil surface was discretized with a total of 320 cells. The first grid spacing from the surface in the wall normal direction was 5×10^{−5}*c*, which resulted in 0.2≤*y*^{+}≤3 depending on blade incidence.

Spatial convergence tests for the static, dynamic and full turbine simulations identified a mesh size of 68 400 for the circular domains containing each blade as being optimal with regard to a balance between computational cost and solution accuracy. This resulted in around 200 000 elements across the entire domain depending on blockage ratio.

Figures 1 and 2 show comparisons between the experimental data from static blade tests and the CFD computations using the SST *k*−*ω* turbulence model; figures 3 and 4 show the corresponding comparisons for dynamic/oscillating blade tests.

Comparing the numerical and physical data in figures 1 and 2, it is apparent that aside from a broadly constant offset in *C*_{D}, the lift and drag forces of the static blade are well predicted by the numerical model until stall. The numerical boundary layer is stipulated to be fully turbulent leading to an over-prediction of the skin friction drag upstream of the physical transition point, and an over-prediction of the form drag owing to the increased width of the turbulent boundary layer above that in the real flow. This results in the over-prediction of *C*_{D} for the attached flow region, as shown in figure 2.

The observed differences in the stalling behaviour are due to the CFD model’s inability to simulate laminar to turbulent transition. As the numerical boundary layer is stipulated to be fully turbulent, blade stall is characterized by a trailing edge event that renders a benign change in lift and drag. By contrast, the physical experiments exhibit stall characterized by laminar separation and, the formation and spreading of a separation bubble that results in a sharper drop in lift and increase in drag.

The dynamic oscillatory blade tests were based on the physical experiments conducted by Piziali [6]. The blades were dynamically pitched by ±4^{°} about mean pitch angles of 4^{°} and 13^{°}. For both tests, blade incidence was first increased from the mean pitch angle to the maximum angle of attack, *α*_{max}=8^{°} and 17^{°}, respectively, before being reduced to the minimum angle of attack, *α*_{min}=0^{°} and 9^{°}, respectively, and then back to the mean pitch angle. The reduced frequency, *k*, for both test cases was 0.10, where
3.3
and *ω* is the angular frequency of oscillation.

Similar observations may be drawn for the dynamic blade tests as were drawn for the static blade tests. The lift and drag forces for the 4±4^{°} oscillating blade tests are well predicted, as shown in figures 3 and 4. Because the experimental drag force was obtained from pressure tap measurements, the skin friction component of the computationally determined drag force should be neglected in order to make proper comparison with the experiments. In figure 4, the indicated datasets plot computed pressure drag only. It is seen that the form and width, i.e. difference in value between increasing and decreasing incidence, of the lift and drag hysteresis curves are well reproduced in the simulations, indicating that the vorticity shed from the blade is being well represented numerically.

For the dynamic blade tests about a higher mean pitch angle, 13±4^{°}, significant differences arise between the experimentally and computationally derived forces, as shown in figures 3 and 4. The numerically derived lift force is over-estimated as *α* is increased from its initial position at 13^{°}. This is due to a more benign numerical stall mechanism than appears to have occurred in the physical experiments, which itself is a result of stipulating the numerical boundary layer to be fully turbulent. The less acute stall mechanism is also responsible for the significant underestimation of *C*_{D} in the simulations and the significantly less distinct hysteresis loops in both computed lift and drag forces.

## 4. Results and discussion

Having established the required numerical resolution for cross-flow turbine simulations at *Re*_{c}=*O*(10^{6}), we now investigate the performance of a two-dimensional Darrieus-type cross-flow turbine of the same geometric configuration as that tested experimentally by McAdam *et al*. [7]. The three-bladed turbine is of radius *R*=3.82*c*, yielding a solidity *σ*=0.125, where
4.1
and *N* is the number of blades. The mean blade Reynolds number for the simulations tests was calculated to be 4×10^{5} to 3×10^{6} depending on the tip–speed ratio, *λ*, defined as
4.2
where *ω* is the turbine’s angular velocity. The simulations were run until statistically converged, which required up to 80 rotor revolutions depending on the tip–speed ratio, *λ*.

Figure 5 outlines the fluid mechanical principles of a cross-flow turbine. The incident flow velocity, , once modified owing to the presence of the rotor by induction factors to yield a local velocity vector (*U*_{x},*U*_{y})^{t}, can be superimposed on the blade rotational velocity, *U*_{θ}, to yield the resultant flow velocity relative to the blade, *U*_{r}, which forms an angle of incidence, *α*, with the blade chord line. The circumferentially resolved components of the resulting lift and drag forces are the drivers behind a Darrieus-type turbine, as they combine to give the resultant torque per unit length of blade, *Q*, about the axis of turbine rotation,
4.3
where *L* and *D* are the sectional lift and drag forces. In the investigations to follow, we present the turbine’s performance in terms of its power, *C*_{P}, and thrust, *C*_{T}, coefficients defined by
4.4
and
4.5
where *A* is the projected frontal area of the turbine per unit span of rotor, i.e. *A*=2*R*. We present simulations for blockage ratios from 12.5 to 50 per cent, where
4.6
is the blockage ratio and *h* the depth of the flow. The free surface is modelled using both deforming VOF and rigid lid boundaries, where in the case of the deforming free-surface boundary simulations, the Froude number, *Fr*, is defined as
4.7

### (a) Turbine performance

Figures 6 and 7 show the variation of power coefficient, *C*_{P}, and thrust coefficient, *C*_{T}, with tip-speed ratio for blockage ratios *b*=50, 25 and 12.5 per cent operating at *Fr*=0.082 in the case of the VOF simulations.

The power curves exhibit the expected bell-type shape with low power at low *λ* owing to high angles of attack and blade stall, low power at high *λ* owing to low angles of flow incidence and hence low blade lift, and peak power occurring at intermediate tip–speed ratios. The flow blockage is observed to have a significant influence on device power in several different ways. Firstly, peak power, , is highly dependent on the blockage ratio, achieving, in the case of the VOF simulations, a value of 1.23 in the case of 50 per cent blockage and as little as 0.52 for 12.5 per cent blockage. The width of the positive power curve and the tip–speed ratio at which peak power occurs are also observed to increase with flow blockage.

The influence of blockage can be examined by considering the time-mean flow through the turbine and the resulting time-mean induction factor. Figure 8 shows the phase-averaged (or long time-mean) flowfields at the tip–speed ratio at which occurred for both the lowest and highest blockage cases simulated; both cases shown use the rigid lid free-surface approximation. Superimposed on these flowfields are streamlines defining a streamtube of width 6*c* at 12*c* upstream of the centre of the turbine. The relative expansion of the streamtubes indicates the relative degree of flow induction between the two blockage extremes. The lowest blockage case, despite exhibiting a significantly lower thrust coefficient, is seen to decelerate the flow through the turbine more significantly than the high blockage case. This results in the more blocked turbine experiencing a higher incident flow velocity, which in turn presents the turbine blades with higher angles of flow incidence.

The change in the range of blade incidence owing to the change in the streamwise velocity underlies the shift of the power curve to a higher range of *λ* at higher blockage ratios. At low *λ*, the higher incidence associated with higher blockage results in a decrease in the power take-off owing to blade stall, while at high *λ*, it results in an increase in *C*_{P} owing to increased blade incidence, and hence an overall shift of the power curve to higher *λ*.

Also observed in figure 7 is a significant increase in turbine thrust with increased blockage ratio. This is consistent with the analytic models of Whelan *et al*. [8] and Houlsby *et al*. [9] in which increased blockage acts to constrict the bypass flow, accelerating it and creating a pressure drop in the bypass streamtube and hence a greater pressure drop across, and therefore thrust on, the turbine than would have otherwise been achievable in unbounded flow.

The increase in time-mean mass flux through the turbine at maximum power is calculated to be 1.7 times between the low- and high-blockage cases. The increase in thrust, and therefore pressure change across the turbine, for higher blockage ratios, together with the increased mass flux, results in the higher power coefficients observed for the more blocked turbines, and their ability to exceed the Betz limit for unconfined flows. In terms of the blade forces, the increase in power generated derives from an increase in both flow incidence angle, *α*, and resultant velocity, *U*_{r}, for a given tip–speed ratio.

In figures 6 and 7, we observed little difference in simulated power and thrust coefficients between the two free-surface models, rigid lid and deformable free surface (VOF), for the lower blockages considered. However, for the 50 per cent blockage, there is a modest increase of up to 6.7 per cent in *C*_{P} and 9.0 per cent in *C*_{T}. This increase in power and thrust is attributed to an increase in the streamwise flow velocity owing to the increase in effective blockage arising from the reduction in flow depth downstream of the rotor, which must occur as energy is removed from the subcritical flow, recalling that *Fr*=0.082. Figure 9 plots the centreline velocity magnitude for *b*=50 per cent and *λ*=3, which was the tip–speed ratio that gave for both rigid lid and VOF boundaries. It may be observed that the flow velocity through the turbine remains up to 5 per cent higher for the free-surface deformation case, leading to the modest increase in power and thrust. It is also of interest to note that the downstream recovery of the two flows is quite different despite exhibiting similar thrust levels; the recovery in wake velocity taking far longer in the case of the deforming free surface than the rigid lid simulations.

### (b) Free-surface deformation

The change in flow depth through energy extraction is explored in figures 10 and 11. The streamwise variation in free-surface elevation from that far upstream, , is illustrated in figure 10 for the case of 50 per cent blockage and various tip–speed ratios (note the expanded vertical scale in this figure). The free surface lifts slightly ahead of the turbine, and falls through the turbine as energy is removed from the flow, reaching a minimum depth between two and three turbine radii downstream of the centre of the turbine. Following this point, the core and bypass flows remix, resulting in further energy loss and a gradual rise in the free-surface elevation. It is evident that although the free-surface elevation appears asymptotic towards the domain outflow, 24*R* downstream of the turbine centre, full wake recovery is not achieved within the computational domain. Figure 11 summarizes the computed height changes, *Δh*, from domain inflow to the fully remixed downstream condition for the different blockage and tip–speed ratios considered. Note that as the fully recovered height is not achieved within the computational domain, we use analytic methods to calculate the necessary change in free-surface height to satisfy momentum conservation given the thrust recorded on the turbine and the upstream flow conditions *Fr* and *b*. Total energy removal, including that lost in wake mixing and therefore the change in free-surface elevation, is correlated with device thrust (in the case of the rigid lid approximation, energy removal is linear in thrust) and hence figure 11 is seen to mirror the thrust variations plotted in figure 7. For the lower two blockage ratios, the maximum depth change is 0.13 per cent of the upstream depth, whereas for the highest blockage case, the maximum depth change is still just 0.51 per cent of .

### (c) Hydrodynamic efficiency

We next consider the efficiency of the energy extraction process and define a hydrodynamic (or basin) efficiency *η* as the ratio of useful power extracted by the turbine to the total power removed from the flowfield,
4.8
The power removed from the flowfield must necessarily include all of the energy lost in wake mixing. As identified in figure 9, the flow does not fully remix, i.e. obtain a uniform velocity profile over the depth, before the simulation domain outflow is reached. In calculating the power removed from the domain, we therefore analytically determine the fully remixed downstream state given the turbine thrust and upstream flow conditions. In the case of the rigid lid approximation, it can be shown that the hydrodynamic efficiency reduces to [10]
4.9
while for the case of a deformable free surface, *η* may be determined from solution of more complex equations given *Fr*, *b*, *C*_{P} and *C*_{T}.

The significance of the hydrodynamic efficiency is that it characterizes how well the turbine interacts with its environment to produce useful power. Inefficient power conversion, i.e. low *η*, results in a greater head drop, and therefore greater environmental impact, for a given level of useful power generation. Although much of the present industry focus is given to maximizing device kinetic efficiency, i.e. *C*_{P}, as tidal turbines are deployed in greater numbers, the importance of overall environmental impact and constraints, and therefore hydrodynamic efficiency, will become more prominent.

In figure 12, we examine the hydrodynamic efficiency of the turbine for blockage ratios of 25 and 50 per cent for both the rigid lid and VOF free-surface models. In the *η*−*C*_{P} plane, each curve is traversed in a clockwise manner with increasing *λ*. It is evident that for both blockages presented, maximum hydrodynamic efficiency occurs at the tip–speed ratio for maximum kinetic efficiency. Further, the hydrodynamic efficiency is greater for the more blocked case. This is due to the higher flow velocity through the turbine for the more blocked case, and hence the core and bypass flows are closer in velocity, and less energy is lost in remixing them downstream of the turbine. There are only small differences between the hydrodynamic efficiencies computed with the rigid lid and VOF models, with the latter exhibiting slightly greater efficiency, which is consistent with the drop in free surface providing an increase in effective blockage for the VOF simulations.

### (d) Influence of Froude number on turbine performance

For the case of 50 per cent blockage, we now consider the influence of Froude number on turbine performance by considering the various performance metrics through figures 13–16 for turbines operating at Froude numbers typical of likely field values: *Fr*=0.082, 0.097 and 0.131. Froude number is observed to have a relatively slight influence on turbine power and thrust coefficients, with both increasing with *Fr*: in the case of power, increases from 1.22 to 1.25 between the lowest and highest Froude numbers considered. Likewise maximum thrust, , increases by around 3 per cent with *Fr*. The greatest change in performance metrics with Froude number is seen in the change in free-surface elevation between the far upstream and far downstream states. The change in free-surface height is seen to be much larger with higher Froude numbers, reaching 1.36 per cent for the highest thrust and Froude number considered. Despite the greater change in free-surface elevation, the hydrodynamic efficiency remains similar for all Froude numbers and shows a small increase with *Fr*.

## 5. Conclusions

A two-dimensional RANS-based CFD model has been used to examine the influence of blockage on the hydrodynamic performance of a marine cross-flow turbine operating at a mean blade Reynolds number of *O*(10^{5}–10^{6}). Two representations of the free-surface boundary are considered: a rigid lid and a deformable free surface. Free-surface deformation is modelled using a VOF surface tracking approach in which the height of the downstream boundary is recursively adjusted so as to achieve a target upstream Froude number without advance knowledge of the energy extracted by the turbine.

The numerical investigations showed that at *Fr*=0.082, increasing the cross-stream blockage presented by the turbine from 12.5 to 50 per cent resulted in an increase in the maximum kinetic power coefficient, *C*_{P}, from 0.52 to 1.23. Moreover, the increase in blockage resulted in an increase in the streamwise velocity through the turbine, which resulted in the entire power curve being shifted to higher tip–speed ratios.

Rigid lid and deformable free-surface simulations were seen to render very similar turbine performance, particularly at the lower levels of blockage considered, *b*≤25 per cent. For *b*=50 per cent, there were 6.7 and 9.0 per cent increases in maximum power and thrust coefficients when the free surface was allowed to deform. This is attributed to the additional flow confinement inferred by a drop in the free-surface elevation.

Even for the high levels of power and thrust coefficient witnessed for high blockage, the drop in free-surface elevation was small in all cases: for *Fr*=0.082, the maximum drop in free-surface height across the domain was 0.68 per cent for 50 per cent blockage. Hydrodynamic efficiency was seen to be strongly influenced by blockage, and increases with blockage, as the velocity shear between core and bypass flows decreases.

Froude numbers representative of field conditions were explored, 0.082≤*Fr*≤0.131. Turbine performance, kinetic and hydrodynamic efficiencies were only marginally influenced by Froude number, while free-surface elevation change was found to be strongly dependent on Froude number.

## Acknowledgements

The authors acknowledge the support of the Oxford Martin School, the UK Engineering and Physical Sciences Research Council and Research Councils UK.

## Footnotes

One contribution of 14 to a Theme Issue ‘New research in tidal current energy’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.