<commit_before># Validation of a vortex line model for a medium solidity vertical-axis turbine

_<NAME>, <NAME>, <NAME>, <NAME>, <NAME>,
<NAME>_


## Abstract

_TODO: Write abstract._


## Introduction

To accelerate the development of marine hydrokinetic turbine technology, Sandia
National Laboratories (SNL) developed the Code for Axial and Cross-flow TUrbine
Simulation (CACTUS), based on Strickland's VDART model [@Strickland1981],
originally developed for SNL in the 1980s to aid in the design of Darrieus
vertical-axis wind turbines (VAWTs). Upgrades to CACTUS beyond VDART include
ground plane and free surface modeling, a new added mass correction, and a
Leishman--Beddoes (LB) dynamic stall (DS) model, in addition to that from
Boeing--Vertol (BV) [@Murray2011].

CACTUS was previously validated using experimental data from relatively low
solidity $\sigma = Nc/(2 \pi R)$ (or simply chord-to-radius ratio $c/R$)
rotors---the Sandia 5 m Darrieus ($c/R = 0.08$), 34 m Darrieus test bed ($c/R =
0.05$), and the VAWT 850 tapered H-rotor ($c/R = 0.05$). However, when applied
to a high solidity H-rotor, the UNH-RVAT, CACTUS significantly overpredicted
blade loading, and therefore mean performance coefficients [@Michelin2014]. It
was therefore of interest to validate CACTUS with a medium solidity
vertical-axis (a.k.a. cross-flow) turbine: the US Department of Energy (DOE)
Reference Model 2 (RM2).

The RM2 was the subject of an experimental investigation in the University of
New Hampshire (UNH) tow tank, where mechanical power output, overall streamwise
drag or thrust, and near-wake velocity were measured with a 1:6 scale physical
model. A Reynolds number dependence study was performed, which showed strong
$Re$-dependence below and weak $Re$-dependence above a chord-based Reynolds
number $Re_c \sim 10^5$ [@Bachant2016-RM2-paper].


### Objectives

In this study we sought to evaluate the effectiveness of the CACTUS vortex line
model for predicting the experimental performance results acquired for the 1:6
scale RM2 physical model experiments at UNH. Furthermore, we hoped to establish
some best practice guidelines for its application. In case performance
predictions were not adequate, a survey of static foil coefficient input data
was also performed.


## Methods

### Numerical setup

The 1:6 scale RM2 experiment performed in the UNH tow tank, for which the data
is available from [@Bachant2016-RM2-data], was replicated for a tow speed of 1
m/s, which corresponds to a turbine diameter Reynolds number $Re_D = 1.1 \times
10^6$. To match the experimental blockage ratio (10%), wall panel source
elements were added to the model, corresponding to the tank's 3.66 m wide by
2.44 m deep cross-section. Turbine rotor and wall geometry is shown in Figure 1.

![Figure 1: Turbine rotor and wall panel geometry.](figures/walls.png)

Static foil coefficient data for the NACA 0021 profiles was taken from Sheldahl
and Klimas [@Sheldahl1981], as it is the only dataset for the moderate Reynolds
numbers simulated here. However, it is important to note that this dataset is
"semi-empirical" in that extrapolations were made for various profiles and
Reynolds numbers. Recently, Bedon et al. [@Bedon2014] showed with a double
multiple streamtube (DMST) momentum model that this dataset may be unreliable at
lower Reynolds numbers.

The model was run for 8 revolutions, over the latter half of which performance
quantities were averaged.


## Results

### Verification

Sensitivity of the model results to the time step (or number of time steps per
revolution $N_t$) and number of blade elements was assessed, for which the
results are shown in Figure 2. Ultimately, the number of time steps per
revolution and number of elements per blade were chosen as 24 and 16,
respectively. These values may not indicate a typical "converged" configuration,
but were chosen for practicality, since the computational expense increases
about an order of magnitude when doubling $N_t$. For $N_t = 24$, the expense was
approximately 0.1 CPU hours per simulated second.

![Figure 2: Static foil data comparison for $Re_c = 1.5 \times 10^6$](figures/verification.png)


### Mean performance

The performance curve of the RM2 was simulated using both the Boeing--Vertol and
Leishman--Beddoes dynamic stall models, as well as with dynamic stall modeling
deactivated, and is shown in Figure 3. Dynamic stall has a very significant
deleterious effect on $C_P$ at the tip speed ratios of interest, and especially
at lower $\lambda$. The LB and BV DS models produce relatively similar results,
though the BV performance predictions at lower tip speed ratios do not match the
experiments as well as the LB results do.

![Figure 3: RM2 performance curves simulated with CACTUS using the Boeing--Vertol, Leishman--Beddoes, and no dynamic stall model.](figures/perf-curves.png)

<commit_msg>Start section on wake results<commit_after># Validation of a vortex line model for a medium solidity vertical-axis turbine

_<NAME>, <NAME>, <NAME>, <NAME>, <NAME>,
<NAME>_


## Abstract

_TODO: Write abstract._


## Introduction

To accelerate the development of marine hydrokinetic turbine technology, Sandia
National Laboratories (SNL) developed the Code for Axial and Cross-flow TUrbine
Simulation (CACTUS), based on Strickland's VDART model [@Strickland1981],
originally developed for SNL in the 1980s to aid in the design of Darrieus
vertical-axis wind turbines (VAWTs). Upgrades to CACTUS beyond VDART include
ground plane and free surface modeling, a new added mass correction, and a
Leishman--Beddoes (LB) dynamic stall (DS) model, in addition to that from
Boeing--Vertol (BV) [@Murray2011].

CACTUS was previously validated using experimental data from relatively low
solidity $\sigma = Nc/(2 \pi R)$ (or simply chord-to-radius ratio $c/R$)
rotors---the Sandia 5 m Darrieus ($c/R = 0.08$), 34 m Darrieus test bed ($c/R =
0.05$), and the VAWT 850 tapered H-rotor ($c/R = 0.05$). However, when applied
to a high solidity H-rotor, the UNH-RVAT, CACTUS significantly overpredicted
blade loading, and therefore mean performance coefficients [@Michelin2014]. It
was therefore of interest to validate CACTUS with a medium solidity
vertical-axis (a.k.a. cross-flow) turbine: the US Department of Energy (DOE)
Reference Model 2 (RM2).

The RM2 was the subject of an experimental investigation in the University of
New Hampshire (UNH) tow tank, where mechanical power output, overall streamwise
drag or thrust, and near-wake velocity were measured with a 1:6 scale physical
model. A Reynolds number dependence study was performed, which showed strong
$Re$-dependence below and weak $Re$-dependence above a chord-based Reynolds
number $Re_c \sim 10^5$ [@Bachant2016-RM2-paper].


### Objectives

In this study we sought to evaluate the effectiveness of the CACTUS vortex line
model for predicting the experimental performance results acquired for the 1:6
scale RM2 physical model experiments at UNH. Furthermore, we hoped to establish
some best practice guidelines for its application. In case performance
predictions were not adequate, a survey of static foil coefficient input data
was also performed.


## Methods

### Numerical setup

The 1:6 scale RM2 experiment performed in the UNH tow tank, for which the data
is available from [@Bachant2016-RM2-data], was replicated for a tow speed of 1
m/s, which corresponds to a turbine diameter Reynolds number $Re_D = 1.1 \times
10^6$. To match the experimental blockage ratio (10%), wall panel source
elements were added to the model, corresponding to the tank's 3.66 m wide by
2.44 m deep cross-section. Turbine rotor and wall geometry is shown in Figure 1.

![Figure 1: Turbine rotor and wall panel geometry.](figures/walls.png)

Static foil coefficient data for the NACA 0021 profiles was taken from Sheldahl
and Klimas [@Sheldahl1981], as it is the only dataset for the moderate Reynolds
numbers simulated here. However, it is important to note that this dataset is
"semi-empirical" in that extrapolations were made for various profiles and
Reynolds numbers. Recently, Bedon et al. [@Bedon2014] showed with a double
multiple streamtube (DMST) momentum model that this dataset may be unreliable at
lower Reynolds numbers.

The model was run for 8 revolutions, over the latter half of which performance
quantities were averaged.


## Results

### Verification

Sensitivity of the model results to the time step (or number of time steps per
revolution $N_t$) and number of blade elements was assessed, for which the
results are shown in Figure 2. Ultimately, the number of time steps per
revolution and number of elements per blade were chosen as 24 and 16,
respectively. These values may not indicate a typical "converged" configuration,
but were chosen for practicality, since the computational expense increases
about an order of magnitude when doubling $N_t$. For $N_t = 24$, the expense was
approximately 0.1 CPU hours per simulated second.

![Figure 2: Static foil data comparison for $Re_c = 1.5 \times 10^6$](figures/verification.png)


### Mean performance

The performance curve of the RM2 was simulated using both the Boeing--Vertol and
Leishman--Beddoes dynamic stall models, as well as with dynamic stall modeling
deactivated, and is shown in Figure 3. Dynamic stall has a very significant
deleterious effect on $C_P$ at the tip speed ratios of interest, and especially
at lower $\lambda$. The LB and BV DS models produce relatively similar results,
though the BV performance predictions at lower tip speed ratios do not match the
experiments as well as the LB results do.

![Figure 3: RM2 performance curves simulated with CACTUS using the Boeing--Vertol, Leishman--Beddoes, and no dynamic stall model.](figures/perf-curves.png)


### Near-wake mean velocity

To predict the interactions of turbines within arrays and with the
environment...

Potential flow methods such as that used here may not be as useful when
predicting the evolution of the wake since they do not model nonlinear advection
or turbulent transport---as higher fidelity Navier--Stokes models would---but
adequate resolution of the near-wake mean velocity field may help determine
optimal layouts for tightly spaced cross-flow turbine arrays.

_TODO: Write section on wake results._


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