1. Introduction
Over-exploitation of natural resources has caused serious ecological and environmental problems globally, threatening human civilization and the world economy [
1]. A popular trend in resolving this issue is to increase the use of renewable energy. Ocean tidal energy, as one kind of renewable energy, has many advantages, such as lower pollution and low exploitation costs. It is also more renewable and predictable compared with fossil energy. Therefore, many countries, including China, are increasing their investment in developing tidal energy [
2]. The development and utilization of tidal energy begins with transforming the kinetic energy of the tide into electric energy using an energy transducer. Energy transducer devices can be divided into two types: Impeller and non-impeller. The turbine is widely used as an impeller conversion device for ocean tidal energy. Turbines can be divided into the horizontal axis and vertical axis according to the relationship between the flow direction and the spindle. For horizontal axis turbines, the spindle is parallel to the flow direction. In contrast, the spindle of a vertical axis turbine is perpendicular to the flow direction. The vertical axis turbine is not affected by flow direction and has the advantages of a simple blade structure, low working speed ratio and low noise. Moreover, the power systems can be arranged above the water. As a result, vertical axis tidal turbines have recently attracted particular attention [
3].
At present, prediction methods for the hydrodynamic performance of vertical axis tidal turbines can broadly be divided into the following four types: Stream-tube model methods [
4,
5,
6], vortex model methods [
7,
8,
9], computational fluid dynamic (CFD) methods [
2,
10,
11] and model test methods [
12,
13,
14,
15]. Stream-tube model methods are based on the momentum theorem and have difficulties predicting the flow field characteristics of turbines when the speed ratio increases to a point past which the momentum equation may diverge, leading to no solution. The classic vortex model uses a bound vortex filament to replace the rotor blade and changes its strength as a function of azimuthal position. However, the model cannot fulfill the Kutta condition, resulting in an inaccurate calculation of the forces on blades. The free vortex method combined with finite element analysis (FEVDTM) of the flow surrounding the blades was developed by Ponta [
8]. This improves the accuracy of vortex models, but the computational cost is expensive. In recent years, the application of CFD methods to analyze turbine performance has become more popular, but too many factors can affect the results of CFD, thus the results need to be verified by experimental tests. Further, the computational cost of CFD is very high and becomes increasingly prohibitive when used for a full-scale turbine or multiple turbines. Compared with actual conditions in engineering applications, the computational results from a small-scale model may have scaling effects, which are difficult to evaluate with accuracy. Model tests are comparatively more reliable but take a longer time and greater effort and thus have a higher cost.
In this paper, a numerical model is developed based on a combination of the boundary element method (BEM) and vortex theory to predict the hydrodynamic performance of vertical axis turbines. The dimension-reducing technique of the boundary element method is used to reduce the computation cost and vortex theory is applied to wake vortices, which are simulated by a system of discrete vortex cores. The Kutta condition is implemented by forcing zero pressure jumping at the trailing edge between the upper and lower surfaces. The time-stepping method is used to solve the unknown strength of sources and vortices and obtain the velocity distribution along the blades at any time. A model test is also carried out to verify our numerical methods.
2. Methods
The blade of a vertical axis turbine is a slender body; it can be treated simply as the two-dimensional cross section of the blade. Here, we consider unsteady, two-dimensional, inviscid, incompressible flows around a multi-blade vertical turbine, as shown in
Figure 1. A moving coordinate system (OXY) fixed on one of the blades is established, as shown in
Figure 2. The motion of the blade at time
t can be represented by two components, the rotating angular velocity
around the origin
o and the translational velocity
U of the origin
o. The azimuthal angle
θ indicates the position of the blade in the track circle,
θ =
θ0 +
ωt, in which
ω is the angular velocity around the rotation center
O and
θ0 is the initial azimuthal angle of the blade.
The velocity of uniform incoming flow is
, which at infinity is
is the velocity potential at point
p and time
t, which meets the Laplace equation in the fluid domain
:
and the non-penetration boundary condition on the blade surface:
where
nb is the normal of the blade surface
Sb; and
r is the vector from the origin
o to any surface point in the coordinate system
xoy.
At infinity,
satisfies the velocity potential for uniform incoming flow,
Introducing
as the perturbation velocity potential,
can be decomposed into two parts:
Assuming zero initial perturbation,
should satisfy the following equation and boundary conditions:
The boundary element method is used to solve the time-discrete Laplace equations in combination with the Neumann boundary conditions as described by Equations (6)–(9). As shown in
Figure 3 for the
hth blade (
h = 1,…,
Z;
Z is the number of turbine blades), sources
are distributed on the blade surfaces
, linear vortices
γf(
s) are placed along the arced centerline of the blade profile
to generate lift force. A time-discrete vortex model is used to simulate trailing vortices
, which are shed from the trailing edge and move at the local particle velocity, as shown in
Figure 3. As a result, at an arbitrary point
p in the fluid field, the induced velocity can be expressed as:
in which
To compute Equation (10), we divide each blade surface
Sb into
N elements at which the sources
(
j = 1,
N) are distributed, and surface
Sf into
M elements with a linear distribution. Assuming vorticity intensity
at the leading edge and
at the end of the arced centerline, the vorticity intensity on the surface
Sf can then be described as:
at which
is the length of the arced centerline. The total intensity of the vortices placed on
Sf is:
With the movement of each blade after a time step Δt, a point vortex is generated at the trailing edge. As a result, a discrete vortex street is formed.
According to the surface boundary condition in Equation (7), Equation (10) for each surface element
i can be discretized as
where
Z is the number of blades,
is the unit normal vector of the
ith element on surface
Sb and
k represents the number of time steps at current time
tk (
tk =
k × Δ
t).
In Equation (13), there are N × Z equations with (N + 3)Z unknown quantities, including the source strength of the N × Z element on the blade surfaces at current time tk, the vortex strength of Z blades, the vortex strength of newborn trailing vortices at the current moment and their positions . Thus, 3Z equations must be added to make Equation (13) close.
First, the Kutta condition of equal pressure is used on each blade to obtain
Z equations, which can be expressed as:
That is, the pressure difference between the top and bottom surface of the airfoil trailing edge (T.E.) is zero. Applying the unsteady Bernoulli equation at an arbitrary point in the fluid field, the dynamic pressure
P can be expressed as:
Applying Equation (15) to Equation (14) produces:
where
is the pressure on the upper surface of the airfoil trailing edge,
is the pressure on the lower surface of the airfoil trailing edge,
is the perturbation velocity potential on the upper surface of the airfoil trailing edge,
is the perturbation velocity potential on the lower surface of the airfoil trailing edge,
is the vector of the upper surface of the airfoil trailing edge in the local body-fixed coordinate system and
is the vector of the lower surface of the airfoil trailing edge in the local body-fixed coordinate system.
After obtaining the perturbation velocity on the blade surface unit numerically from Equation (10), the pressure distribution in each element can be calculated using the unsteady Bernoulli equation. Then, the hydrodynamic force on the airfoil blade can be obtained by numerical integration. The time derivative of the perturbation potential can be obtained by the finite difference method:
At any time
, the perturbation velocity potential
is determined by the line integral of the perturbation velocity
. Theoretically, the starting point of the integral should be infinity. However, in actual calculation the starting point is 10 times the distance from the rotation center
O of the water turbine. Integrating from the starting point to the rotation center
O, then from
O to the lower point
of each trailing edge, the perturbation velocity potential of any element center
in the whole airfoil can be derived:
where
is the perturbation velocity potential at
.
Second, the Kelvin condition is applied at each blade to obtain
Z equations as follows:
where
is the total intensity of vortices on
Sf of the
hth blade at time
tk and
is the total intensity at the preceding time step
tk−1.
Third, the position
of newborn trailing vortices of the
hth blade at the current time can be determined as:
where
is a constant coefficient,
.
The blade surface pressure coefficient is defined as:
where
ρ is the density of water. The resulting force on the blade is:
The moment on the blade reference point
is:
The tangential force coefficient of the turbine blade is defined as:
and the normal force coefficient of the turbine blade as:
where
is the turbine blade tangential force,
is the turbine blade normal force,
C is the turbine blade chord and
b is the turbine blade length.