# High-Order Large-Eddy Simulations of a Wind Turbine in Ducted and Open-Rotor
Configurations

Chi Ding<EMAIL_ADDRESS>Bin Zhang  Chunlei Liang  Kenneth Visser
Guangming Yao  Department of Mechanical and Aerospace Engineering, Clarkson
University, Potsdam, New York 13699  Department of Mathematics, Clarkson
University, Potsdam, New York 13699

###### Abstract

High-order large-eddy simulations are performed to study the performance and
flow field of a ducted wind turbine operating at different tip speed ratios.
To evaluate the effects of the duct, simulations with the same tip speed
ratios are also performed on the corresponding open-rotor turbine. It is found
that the ducted turbine consistently obtains higher power outputs than the
open-rotor counterpart, and the duct itself enhances flow turbulence and blade
trailing-edge vortices but weakens tip and hub vortices. Flow bifurcation is
observed at the largest tip speed ratio and is identified to be caused by
blade blockage effects. Comparative simulations are also performed on both
turbines under different yaw angles. It is noticed that the ducted
configuration is insensitive to small yaw angles and maintains higher power
outputs than the open-rotor configuration at all yaw angles. Moreover, it is
observed that the wakes of both configurations recover more quickly as the yaw
angle increases.

##  1 Introduction

Wind power is an important energy source and has a long history of being
exploited [1]. In the 1970s, the interest in developing wind power was boosted
by the oil crisis. At that time, a lot of state-funded projects were launched
to develop wind power. In 1973, the U.S. government approved about 200 million
dollars to support studies on wind turbines [2]. Between 1975 and 1987, the
MOD series of wind turbines were erected in the U.S. [3]. During the same
period, Denmark built two large experimental turbines in the vicinity of
Aalborg [2]. In Sweden, two large wind turbines: WTS-75 and WTS-3, were
installed in cooperation with a German company and a U.S. company,
respectively [2].

One major design task is to extract more energy from wind via, for example,
increasing its blade size. However, the increasing size leads to larger
aerodynamic loads and makes turbine blades more vulnerable to aeroelastic
issues like flutters [4]. In addition, large wind turbines are more difficult
to install in urban areas than small ones [5]. Another solution to improve
turbine efficiency is to place the rotor into a diffuser, called a ducted wind
turbine (DWT) or a diffuser augmented wind turbine (DAWT). By doing so, the
mass flow rate across the rotor plane increases, producing more power output
[6]. The DWT design avoids the problems caused by increasing blade size and is
more suitable for installation in urban areas. DWTs have even more advantages,
including insensitivity to yawed flow and less tip loss [6]. Therefore, DWTs
are a promising way to efficiently harvest wind energy.

To study the performance of DWTs, a lot of theoretical and experimental
research has been carried out. In 1956, Lilley and Rainbird [7] performed a
one-dimensional theoretical analysis on DWTs. They concluded that adding a
duct could improve the power performance, and higher power outputs can be
achieved via larger duct expansion ratios. However, a large expansion ratio
may cause boundary-layer separation. During the 1970s and 1980s, a series of
experiments verified the concept of DWT. Some experiments employing techniques
like ring-shaped flaps and multi-slotted diffusers were also performed to
prevent flow separations [8, 9, 10].

Many computational studies on DWTs have also been reported. In 1981, Fletcher
[11] analyzed a DWT using the blade element momentum (BEM) theory, which took
into account the effects of Reynolds number and wake rotation. Good agreement
with experimental measures was achieved. Vaz and Wood [12] improved this BEM
method by including a high rotor thrust correction and a new formulation for
the far-wake velocity. Koras and Georgalas [13] modeled the rotor of a DWT by
a lifting line and the duct by a combination of vortex rings and source rings.
They used this potential flow method to study the influence of several
geometrical parameters on the power output. However, their method was limited
to DWTs with large tip clearance. Politis and Koras [14] later made progress
by using a lifting-surface approach for duct modeling, and their method was
able to handle DWTs with any tip clearance.

The rapid development of computational fluid dynamics (CFD) technologies has
also boosted the computational studies on DWTs. One of the most popular
approaches is to combine a CFD solver with an actuator-disk (AD) model that
represents a rotor to study DWTs. This approach is usually referred to as the
CFD-AD approach and can provide more details of a DWT flow field at low
computational costs. Phillips et al. [15] applied a CFD-AD approach to
investigate the Vortec 7 turbine — a full-scale DWT design. In the same way,
Hansen et al. [16] analyzed the performance of a DWT and verified that adding
a duct increases the mass flow rate. Abe and Ohya [17] employed a CFD-AD
approach to study a turbine with a flanged diffuser. Their focus was on how
the loading coefficient and the diffuser’s opening angle affect the turbine
performance. Venters et al. [18] used a CFD-AD approach to find an optimized
design for a DWT. They employed two objective functions for the optimization:
one based on rotor power coefficient and the other on total power coefficient.
That study was continued by Sadeghi et al. [19] using new optimization
algorithms.

These computational studies gave a lot of helpful guidance on the design of
DWTs. However, the aforementioned methods inevitably bear significant
simplifications on the turbine geometries. Nowadays, it is viable to simulate
fluid flows about real turbines using more advanced CFD techniques.
Furthermore, the development of high-order methods made it possible to
simulate a flow field at higher spatial accuracies than traditional finite
volume methods. The most popular high-order methods include the discontinuous
Galerkin (DG) method [20, 21], the spectral element (SE) method [22, 23], the
spectral difference (SD) method [24, 25, 26, 27], and the flux reconstruction
(FR) method [28, 29, 30]. The SD and the FR methods are based on the
differential-form governing equations and are two of the most efficient high-
order methods. The FR method is a unified framework that can recover many
existing high-order schemes (e.g., DG and SD schemes) and produce new schemes
that were never reported before. To deal with rotating objects, Zhang and
Liang [31, 32] introduced the curved dynamic mortar concept and applied it to
developing high-order sliding-mesh SD and FR methods. These methods were later
extended to sliding-deforming meshes [33], 3D geometries [34], and general
nonuniform sliding interfaces [35]. Zhang et al. [36] further introduced the
transfinite mortar concept that has no geometric error and makes a sliding-
mesh method arbitrarily high-order accurate in space and high-order in time.
This method has been applied to simulate flows around rotating cylinders of
different cross-sectional shapes [37], flapping wings for energy harvesting
[38], and, more recently, the first high-order eddy-resolving simulation of
flow over a marine propeller [39].

The authors of the present work also applied the above techniques to a
preliminary study of a DWT (designed at Clarkson University by Dr. Kenneth
Visser) at its design condition [40]. In this work, we further the study by
comparing the DWT with the corresponding open-rotor wind turbine (OWT) at
different working conditions. Given that DWTs can be installed in urban areas
where the directions of winds may be affected by buildings, simulations for
the DWT and OWT under yawed inflows are also performed.

The rest of this paper is organized as follows. Section 2 briefly introduces
the numerical methods. Section 3 gives details on the simulation setup and the
numerical validation. In Section 4, computations for turbines under axial
flows are carried out, and results on the aerodynamic loads, pressure, and
velocities are presented and analyzed. Section 5 reports simulation results
for the yawed cases. Finally, Section 6 concludes this study.

##  2 Numerical Methods

###  2.1 The Governing Equations

The three-dimensional unsteady Navier-Stokes equations in the following
conservative form are numerically solved,

$\partialderivative{\mathbf{Q}}{t}+\partialderivative{\mathbf{F}}{x}+\partialderivative{\mathbf{G}}{y}+\partialderivative{\mathbf{H}}{z}=\mathbf{0},$
(1)

where $\mathbf{Q}$ is the vector of conservative variables, $\mathbf{F}$,
$\mathbf{G}$, and $\mathbf{H}$ are the flux vectors in each coordinate
direction. These terms have the following expressions,

$\displaystyle\mathbf{Q}$ $\displaystyle=[\rho~{}\rho u~{}\rho v~{}\rho
w~{}E]^{\intercal},$ (2) $\displaystyle\mathbf{F}$
$\displaystyle=\mathbf{F}_{\text{inv}}(\mathbf{Q})+\mathbf{F}_{\text{vis}}(\mathbf{Q},\gradient{\mathbf{Q}}),$
(3) $\displaystyle\mathbf{G}$
$\displaystyle=\mathbf{G}_{\text{inv}}(\mathbf{Q})+\mathbf{G}_{\text{vis}}(\mathbf{Q},\gradient{\mathbf{Q}}),$
(4) $\displaystyle\mathbf{H}$
$\displaystyle=\mathbf{H}_{\text{inv}}(\mathbf{Q})+\mathbf{H}_{\text{vis}}(\mathbf{Q},\gradient{\mathbf{Q}}),$
(5)

where $\rho$ is fluid density, $u$, $v$, and $w$ are the velocity components,
$E$ is the total energy per unit volume defined as
$E=p/(\gamma-1)+\frac{1}{2}\rho(u^{2}+v^{2}+w^{2})$, $p$ is pressure, and
$\gamma$ is the ratio of specific heats which is set to 1.4 in this work. The
fluxes have been split into inviscid and viscous parts. The inviscid fluxes
are only functions of the conservative variables and have the following
expressions,

$\displaystyle\mathbf{F}_{\text{inv}}$ $\displaystyle=[\rho u~{}~{}\rho
u^{2}\\!+\\!p~{}~{}\rho uv~{}~{}\rho uw~{}~{}u(E\\!+\\!p)]^{\intercal},$ (6)
$\displaystyle\mathbf{G}_{\text{inv}}$ $\displaystyle=[\rho v~{}~{}\rho
uv~{}~{}\rho v^{2}\\!+\\!p~{}~{}\rho vw~{}~{}v(E\\!+\\!p)]^{\intercal},$ (7)
$\displaystyle\mathbf{H}_{\text{inv}}$ $\displaystyle=[\rho w~{}~{}\rho
uw~{}~{}\rho vw~{}~{}\rho w^{2}\\!+\\!p~{}~{}w(E\\!+\\!p)]^{\intercal}.$ (8)

The viscous fluxes are functions of the conservative variables and the
gradients. Their expressions are

$\displaystyle\mathbf{F}_{\text{vis}}$
$\displaystyle=-[0~{}~{}\tau_{xx}~{}~{}\tau_{yx}~{}~{}\tau_{zx}~{}~{}u\tau_{xx}\\!+\\!v\tau_{yx}\\!+\\!w\tau_{zx}\\!+\\!\kappa
T_{x}]^{\intercal},$ (9) $\displaystyle\mathbf{G}_{\text{vis}}$
$\displaystyle=-[0~{}~{}\tau_{xy}~{}~{}\tau_{yy}~{}~{}\tau_{zy}~{}~{}u\tau_{xy}\\!+\\!v\tau_{yy}\\!+\\!w\tau_{zy}\\!+\\!\kappa
T_{y}]^{\intercal},$ (10) $\displaystyle\mathbf{H}_{\text{vis}}$
$\displaystyle=-[0~{}~{}\tau_{xz}~{}~{}\tau_{yz}~{}~{}\tau_{zz}~{}~{}u\tau_{xz}\\!+\\!v\tau_{yz}\\!+\\!w\tau_{zz}\\!+\\!\kappa
T_{z}]^{\intercal},$ (11)

where $\tau_{ij}$ is viscous stress tensor which is related to velocity
gradients as $\tau_{ij}=\mu(u_{i,j}+u_{j,i})+\lambda\delta_{ij}u_{k,k}$, $\mu$
is the dynamic viscosity, $\lambda=-\frac{2}{3}\mu$ based on Stokes’
hypothesis, $\delta_{ij}$ is the Kronecker delta, $\kappa$ is the thermal
conductivity, $T$ is temperature which is related to density and pressure
through the ideal gas law $p=\rho RT$, and $R$ is the gas constant.

###  2.2 The Computational Equations

Each mesh element in the physical space is mapped to a standard element in a
computational space. Assume the mapping is: $t=\tau$,
$x=x(\tau,\xi,\eta,\zeta)$, $y=y(\tau,\xi,\eta,\zeta)$, and
$z=z(\tau,\xi,\eta,\zeta)$, where $(\tau,\xi,\eta,\zeta)$ are the
computational time and coordinates. It can be shown that the governing
equations will take the following conservative form in the computational
space,

$\partialderivative{\widetilde{\mathbf{Q}}}{t}+\partialderivative{\widetilde{\mathbf{F}}}{\xi}+\partialderivative{\widetilde{\mathbf{G}}}{\eta}+\partialderivative{\widetilde{\mathbf{H}}}{\zeta}=\mathbf{0}.$
(12)

The computational variables and fluxes are related to the physical ones as

$\begin{bmatrix}\widetilde{\mathbf{Q}}\\\\[2.84526pt]
\widetilde{\mathbf{F}}\\\\[2.84526pt] \widetilde{\mathbf{G}}\\\\[2.84526pt]
\widetilde{\mathbf{H}}\end{bmatrix}=\absolutevalue{\mathcal{J}}\mathcal{J}^{-1}\begin{bmatrix}{\mathbf{Q}}\vphantom{\widetilde{\mathbf{Q}}}\\\\[2.84526pt]
{\mathbf{F}}\vphantom{\widetilde{\mathbf{F}}}\\\\[2.84526pt]
{\mathbf{G}}\vphantom{\widetilde{\mathbf{G}}}\\\\[2.84526pt]
{\mathbf{H}}\vphantom{\widetilde{\mathbf{H}}}\end{bmatrix},$ (13)

where $\mathcal{J}$ is the Jacobian matrix, $\absolutevalue{\mathcal{J}}$ is
the determinant, and $\mathcal{J}^{-1}$ is the inverse Jacobian matrix. Their
expressions are

$\displaystyle\mathcal{J}=\partialderivative{(t,x,y,z)}{(\tau,\xi,\eta,\zeta)}=\begin{bmatrix}1&0&0&0\\\
x_{\tau}&x_{\xi}&x_{\eta}&x_{\zeta}\\\ y_{\tau}&y_{\xi}&y_{\eta}&y_{\zeta}\\\
z_{\tau}&z_{\xi}&z_{\eta}&z_{\zeta}\end{bmatrix},$ (14)
$\displaystyle\mathcal{J}^{-1}=\partialderivative{(\tau,\xi,\eta,\zeta)}{(t,x,y,z)}=\begin{bmatrix}1&0&0&0\\\
\xi_{t}&\xi_{x}&\xi_{y}&\xi_{z}\\\ \eta_{t}&\eta_{x}&\eta_{y}&\eta_{z}\\\
\zeta_{t}&\zeta_{x}&\zeta_{y}&\zeta_{z}\end{bmatrix}.$ (15)

Besides the flow equations, the geometric conservation law (GCL) [41] also
needs to be numerically satisfied to ensure free-stream preservation on moving
grids. The GCL equations and the steps for solving them are described in,
e.g., [33].

###  2.3 The Flux Reconstruction Method

Only hexahedral elements are used in this work, and each element is mapped to
a unit computational element $0\leq\xi,\eta,\zeta\leq 1$ via the following
iso-parametric mapping,

$\begin{bmatrix}x\\\\[2.84526pt] y\\\\[2.84526pt]
z\end{bmatrix}=\sum^{K}_{i=1}M_{i}(\xi,\eta,\zeta)\begin{bmatrix}x_{i}(t)\\\\[2.84526pt]
y_{i}(t)\\\\[2.84526pt] z_{i}(t)\end{bmatrix},$ (16)

where $K$ is the number of nodes that define a physical element, $M_{i}$ is
the shape function (detailed expressions can be found in, e.g., [42]), and
$(x_{i},y_{i},z_{i})$ are the coordinates, of the $i$-th node.

Solution points (SPs, denoted by $X_{s}$) are defined inside each
computational element, and flux points (FPs, denoted by $X_{f}$) are defined
on the boundaries. Figure 1 shows a schematic of the distribution of the SPs
and FPs in the $\xi$-$\eta$ plane for a fourth-order FR scheme. For an $N$-th
order FR scheme, there are $N$ SPs and $N$ FPs in each direction. The SPs and
FPs are chosen as the Legendre points in this study.

Figure 1: Schematic of SPs (round dots) and FPs (square dots) in the
$\xi$-$\eta$ plane for a fourth-order FR scheme.

At the SPs, the following Lagrange interpolation bases can be defined (where
$X_{i}$ is the coordinate of the $i$-th SP),

$h_{i}(X)=\prod^{N}_{s=1,s\neq
i}(\frac{X-X_{s}}{X_{i}-X_{s}}),~{}~{}i=1,2,\cdots,N.$ (17)

The above polynomials also form a basis for polynomials of degrees less than
or equal to $N-1$, i.e., $\boldsymbol{\mathsf{P}}_{N-1}$. The solution and
fluxes within each element can be approximated via tensor products of the
interpolation bases, e.g.,

$\displaystyle\widetilde{\mathbf{Q}}(\xi,\eta,\zeta)$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\widetilde{\mathbf{Q}}_{ijk}h_{i}(\xi)h_{j}(\eta)h_{k}(\zeta),$
(18) $\displaystyle\widetilde{\mathbf{F}}(\xi,\eta,\zeta)$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\widetilde{\mathbf{F}}_{ijk}h_{i}(\xi)h_{j}(\eta)h_{k}(\zeta),$
(19)

where the subscript $ijk$ denotes the discrete value at the $(i,j,k)$-th SP.
Obviously, the solution and flux polynomials are in
$\boldsymbol{\mathsf{P}}_{N-1,N-1,N-1}$ and are continuous within each element
but discontinuous across cell boundaries. Therefore, common values need to be
defined at cell boundaries. In this work, the common solution is calculated as
the average of the discontinuous values from the two sides of a boundary; the
common inviscid fluxes are computed using the Rusanov solver [43]; the common
viscous fluxes are computed from the common solutions and common gradients.

After taking the spatial derivatives in Eq. (12), the three flux terms are
reduced to elements of $\boldsymbol{\mathsf{P}}_{N-2,N-1,N-1}$,
$\boldsymbol{\mathsf{P}}_{N-1,N-2,N-1}$, and
$\boldsymbol{\mathsf{P}}_{N-1,N-1,N-2}$, respectively, which are inconsistent
with the solution term. To fix this issue, the degrees of the original flux
polynomial need to be raised, which can be achieved using higher-degree
correction functions [28]. For example, the corrected/reconstructed flux in
the $\xi$ direction is

$\widehat{\mathbf{F}}=\widetilde{\mathbf{F}}(\xi,\eta,\zeta)+\left[\widetilde{\mathbf{F}}^{\text{com}}(0,\eta,\zeta)-\widetilde{\mathbf{F}}(0,\eta,\zeta)\right]\cdot
g_{\text{\tiny
L}}(\xi)+\left[\widetilde{\mathbf{F}}^{\text{com}}(1,\eta,\zeta)-\widetilde{\mathbf{F}}(1,\eta,\zeta)\right]\cdot
g_{\text{\tiny R}}(\xi),$ (20)

where $\widetilde{\mathbf{F}}$ is from (19),
$\widetilde{\mathbf{F}}^{\text{com}}$ is the common flux on a cell boundary,
and $g_{L}$ and $g_{R}$ are the left and right correction functions that are
required to satisfy

$\displaystyle g_{\text{\tiny L}}(0)=1,\quad g_{\text{\tiny L}}(1)=0,$ (21)
$\displaystyle g_{\text{\tiny R}}(0)=0,\quad g_{\text{\tiny R}}(1)=1.$

These conditions ensure that

$\widehat{\mathbf{F}}(0,\eta,\zeta)=\widetilde{\mathbf{F}}^{\text{com}}(0,\eta,\zeta),\quad\widehat{\mathbf{F}}(1,\eta,\zeta)=\widetilde{\mathbf{F}}^{\text{com}}(1,\eta,\zeta),$
(22)

i.e., the reconstructed flux still takes the common values on cell boundaries.
In this work, the $g_{\text{DG}}$ function [28] is chosen as the correction
function. The other two fluxes are reconstructed in the same way. Finally, the
governing equations can be written in the following residual form,

$\evaluated{\partialderivative{\widetilde{\mathbf{Q}}}{t}}_{ijk}=-\left[\partialderivative{\widehat{\mathbf{F}}}{\xi}+\partialderivative{\widehat{\mathbf{G}}}{\eta}+\partialderivative{\widehat{\mathbf{H}}}{\zeta}\right]_{ijk}=\mathbf{R}_{ijk},\quad
i,j,k=1,2,\cdots,N,$ (23)

where $\mathbf{R}_{ijk}$ is the residual at the $(i,j,k)$-th SP. This system
can be time marched either explicitly or implicitly. In the present work, a
four-stage third-order explicit Runge-Kutta method [44, 45] is employed.

###  2.4 The Sliding-mesh SD/FR Method

There are two fundamental types of sliding interfaces in 3D as shown in Fig.
2. For simplicity, assume that the mesh points do not match in the azimuthal
direction but match in the other direction. Further assume that the azimuthal
direction is uniformly meshed. We take the second type to briefly explain how
the method works. More detailed explanation can be found in previous papers,
e.g., [34, 36, 39].

Figure 2: Two types of sliding interfaces.

Curved dynamic mortar elements [32] are employed to communicate between the
two sides of a sliding interface. As shown in Fig. 3, at each time instant, a
cell face $\Omega$ is connected to two mortar elements $\Xi_{1}$ and
$\Xi_{2}$. These curved geometries are mapped to straight ones first. For
$\Omega$, assume that the azimuthal direction is mapped to $\xi$ and the other
direction is mapped to $\eta$ in the computational space. Similarly, for each
mortar, assume these two directions are mapped to $\xi^{\prime}$ and
$\eta^{\prime}$, respectively, in the mortar space. Then these two spaces are
related as: $\xi=o+s\cdot\xi^{\prime},~{}\eta=\eta^{\prime}$, where
$0\leq\xi,\eta,\xi^{\prime},\eta^{\prime}\leq 1$, and $o$ and $s$ are the
offset and scaling of a mortar with respect to a cell face.

Figure 3: Map curved cell face and mortar elements to straight ones.

Any variable $\phi$ on a cell face $\Omega$ and on the left side of a mortar
$\Xi$ can be represented as

$\displaystyle\phi^{\Omega}(\xi,\eta)$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\phi^{\Omega}_{ij}h_{i}(\xi)h_{j}(\eta),$
(24) $\displaystyle\phi^{\Xi,L}(\xi^{\prime},\eta^{\prime})$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\phi^{\Xi,L}_{ij}h_{i}(\xi^{\prime})h_{j}(\eta^{\prime}),$
(25)

where the $(\phi^{\Xi,L}_{ij})^{,}$s are unknown and can be obtained through
the following projection (refer to Fig. 4(a)),

$\int_{0}^{1}\int_{0}^{1}(\phi^{\Xi,L}(\xi^{\prime},\eta^{\prime})-\phi^{\Omega}(\xi,\eta))h_{\alpha}(\xi^{\prime})h_{\beta}(\eta^{\prime})\mathrm{d}\xi^{\prime}\mathrm{d}\eta^{\prime}=0,\quad\forall\alpha,\beta=1,2,...,N.$
(26)

Figure 4: Projections between cell face and mortars.

The same process is repeated on the right side of the mortar. Following that,
a common value, denoted as $\Phi$, are computed on mortars and then projected
back to cell faces (see Fig. 4(b)) according to

$\sum_{k=1}^{2}\int_{\xi=o_{k}}^{\xi=o_{k}+s_{k}}\int_{\eta=0}^{\eta=1}(\Phi^{\Omega}(\xi,\eta)-\Phi^{\Xi_{k}}(\xi^{\prime},\eta^{\prime}))h_{\alpha}(\xi)h_{\beta}(\eta)\differential{\xi}\differential{\eta}=0,~{}~{}\forall\,\alpha,\beta=1,...,N.$
(27)

It can be shown that the two projections, (26) and (27), are equivalent to a
series of 1D projections [46], which make the process very efficient.

##  3 Simulation Setup

###  3.1 Geometry

The DWT considered here was designed at Clarkson University by Dr. Kenneth
Visser based on his previous experimental study [47]. Figure 5 shows two views
of the geometry and a photo of the real product. The diffuser duct has a
profile of the E423 high lift airfoil, with an inlet radius of $R_{i}=1.546$m,
an exit radius of $R_{o}=1.832$m, a width of $W=0.612$m, and an angle of
attack of $\alpha=25^{\circ}$. The rotor has three 1.5m-long blades (i.e.,
$R_{b}=1.5$m) that are $0.388$m downstream from the inlet. More information
about the blade shape can be found in [47]. The cylindrical hub has a diameter
of $D_{h}=0.456$m and is closed by two hemispherical ends of the same
diameter. This configuration results in a small gap of approximately 0.113m
between the blade tips and the duct’s inner surface.

(a)

(b)

(c)

Figure 5: The ducted wind turbine geometry: (a) upstream view, (b) lateral
view, (c) photo of the real product.

###  3.2 Meshes and Boundary Conditions

Figure 6(a) shows the surface meshes of the DWT’s wall boundaries. Along these
boundaries, the first layer of volume mesh has a height of approximately
$3\times 10^{-3}D_{o}$, and the first layer of solution points is about
$1.4\times 10^{-4}D_{o}$ off the walls (for a fifth-order scheme), where
$D_{o}=2R_{o}$ is the diameter of the duct’s exit. Figure 6(b) shows the
meshes on the upstream sliding interface and the duct’s inner surface. In
fact, there are two sliding interfaces, and the other one is downstream at the
duct’s exit. Because the gap between the blade tips and the duct’s inner
surface is too small to allow a third sliding interface there, the duct’s
inner surface is set to rotate with the two sliding interfaces. The speed on
this inner surface is then overwritten to zero. The surface mesh of the OWT’s
rotor is identical to that of the DWT. The OWT allows three sliding interfaces
that form a disk region, as shown in Fig. 6(c). This region has a thickness of
$W$ and diameter of $1.5D_{o}$ and encloses the blades. For both turbines, the
hubs and the hemispherical ends rotate at the same angular speed as the
blades.

(a)

(b)

(c)

Figure 6: Surface meshes: (a) for DWT, (b) for DWT’s sliding interfaces, (c)
for OWT’s sliding interfaces.

The overall computational domain for the DWT has a cylindrical shape, as shown
in Fig. 7(a). The length and diameter of the domain are both $12D_{o}$. The
domain for the OWT has the same shape and sizes. The resulting blockage ratio
is $0.69\%$ for the DWT and $0.47\%$ for the OWT. The overall mesh with a 1/4
cutout is shown in Fig. 7(b), where the red area represents a sliding
subdomain. For the DWT, the sliding subdomain has 29,469 elements, and the
outer subdomain has 266,969 elements, resulting in 296,438 total elements (or
37.1 million degrees of freedom (DOFs) for a fifth-order scheme). For the OWT,
the sliding region has 54,432 elements, and the outer subdomain has 240,905
elements, adding up to 295,337 elements in total (or 37.0 DOFs for a fifth-
order scheme). The inlet is treated as a Dirichlet boundary. The outlet and
outer cylindrical surface are treated as characteristic farfields. All solid
walls are treated as no-slip adiabatic walls. The Mach number of the incoming
freestream flow is set to $Ma_{\infty}=0.08$ to ensure negligible
compressibility effect.

(a)

(b)

Figure 7: Overall computational domain and mesh: (a) domain, (b) mesh.

###  3.3 Nondimensional Parameters

Parameter | Value
---|---
Freestream velocity ($U_{\infty}$) | 10 m/s
Kinematic viscosity ($\nu$) |  $1.4616\times 10^{-5}$ m2/s
Rotation axis |  $x$-direction
Rotation speed ($\omega$) |  $-26.18$ rad/s
Table 1: Physical operating conditions.

The most common physical operating conditions for the present DWT are listed
in Table 1. Once the geometry of the turbine is given, there are two
nondimensional parameters that govern the flow. One is the Reynolds number
$Re$, and the other is the tip speed ratio $\lambda$. In this study, we take
the following definitions

$Re=\frac{U_{\infty}D_{o}}{\nu}\quad\text{and}\quad\lambda=\frac{\omega
R_{b}}{U_{\infty}}.$ (28)

The values in Table 1 give $Re=2.5\times 10^{6}$ and $\lambda=3.93$.
Hereinafter, the Reynolds number is kept constant for all the cases. Two more
values, $\lambda=3.11$ and $\lambda=4.75$, are studied for some cases to
examine the effects of the tip speed ratio.

In what follows, all the simulations use a small nondimensional time step size
of $\Delta t^{*}=\Delta tU_{\infty}/D_{o}=1.25\times 10^{-5}$ for stability
consideration. This time step size corresponds to a blade rotation between
0.0054 degree (for $\lambda=3.11$) and 0.0083 degree (for $\lambda=4.75$),
which is small enough to provide sufficient sampling resolution. Each
simulation was ran for 25 nondimensional time units, and data sets from the
last 12.5 time units were used to calculate the statistics.

###  3.4 Resolution Verification

For the method used, the total number of DOFs is
$N_{\text{DOF}}=N_{\text{elem}}\cdot N^{3}$, where $N_{\text{elem}}$ is the
total number of mesh elements, and $N$ is the scheme order. Once a mesh is
generated, we can vary the scheme order $N$ to identify the resolution
requirement for a simulation. The Reynolds number in this study is extremely
high, which makes it prohibitively expensive to get fully resolved solutions
(i.e., to resolve turbulent eddies of all sizes). Thus, we need to identify a
proper scheme order that gives a good balance between computational cost and
solution qualities.

To do this, we employ the thrust coefficient $C_{T}$ and power coefficient
$C_{P}$ as two measurement criteria. These two terms are defined as below

$\displaystyle C_{T}=\frac{T}{\frac{1}{2}\rho U_{\infty}^{2}A_{\text{rot}}},$
(29) $\displaystyle C_{P}=\frac{Q\omega}{\frac{1}{2}\rho
U_{\infty}^{3}A_{\text{rot}}},$ (30)

where $T$ and $Q$, respectively, are the thrust (in fact drag for a turbine)
and the torque (only the $x$-component) acted on turbine blades, and
$A_{\text{rot}}=\pi R_{b}^{2}$ is the swept area of rotor blades.

Three scheme orders, $N=4$, $5$, and $6$, are applied to the OWT at the
operating conditions listed in Table 1. The time-averaged results are
summarized in Table 2. It is observed that the $\overline{C}_{T}$’s from $N=4$
and $5$ are $16\%$ and $5\%$, respectively, smaller than that from $N=6$.
Meanwhile, the $\overline{C}_{P}$’s from $N=4$ and $5$ are $32\%$ and $7\%$,
respectively, smaller than that from $N=6$. The data obviously shows a
converging trend as $N$ increases. Going from $N=5$ to $N=6$ increases the
computational cost by $73\%$, but the changes in the coefficients are
relatively small. Thus $N=5$ is chosen as the scheme order to perform the
simulations.

| $N=4$ | $N=5$ | $N=6$
---|---|---|---
$\overline{C}_{T}$ | 5.31E-01 | 6.02E-01 | 6.34E-01
$\overline{C}_{P}$ | 2.74E-01 | 3.74E-01 | 4.02E-01
Table 2: Results from different scheme orders.

##  4 Simulation of Turbines in Axial Flows

In this section, the results and analysis for the axial flow cases are
presented, in which the free stream moves along the turbines’ axial direction.
Besides the designed tip speed ratio $\lambda=3.93$ ($\omega^{*}=\omega
D_{o}/U_{\infty}=-9.60$), two other tip speed ratios: $\lambda=3.11$
($\omega^{*}=-7.60$) and $\lambda=4.75$ ($\omega^{*}=-11.60$), are considered,
and the loads, vortex fields and velocity fields are analyzed.

###  4.1 Load Analysis

Figure 8 demonstrates the time histories of $C_{T}$ and $C_{P}$ for the two
configurations at the designed tip speed ratio. It is seen that the thrust and
power outputs for both configurations are well converged, and the DWT’s loads
have larger fluctuations than those of the OWT, which indicates that the flow
over the DWT is more turbulent.

(a) Thrust

(b) Power

Figure 8: Time histories of the loads for the two configurations at $\lambda=3.93$. |  | $C_{T}$ | $C_{T,p}$ | $C_{T,v}$ | $C_{P}$ | $C_{P,p}$ | $C_{P,v}$
---|---|---|---|---|---|---|---
DWT | mean | 9.28E-01 | 9.28E-01 | 1.97E-04 | 5.47E-01 | 5.49E-01 | -1.20E-03
r.m.s | 9.79E-03 | 9.79E-03 | 5.79E-07 | 4.87E-03 | 4.87E-03 | 8.27E-06
OWT | mean | 6.02E-01 | 6.01E-01 | 1.91E-04 | 3.74E-01 | 3.75E-01 | -1.44E-03
r.m.s | 1.55E-03 | 1.55E-03 | 5.41E-07 | 7.38E-04 | 7.39E-04 | 1.45E-06
Table 3: Loads and their components of the two turbines at $\lambda=3.93$.

The mean (time-averaged) values, r.m.s (root-mean-square) deviations, and the
corresponding contributions from pressure (with subscript ‘$p$’) and viscosity
(with subscript ‘$v$’) of the loads are listed in Table 3. Overall, the r.m.s
values are about two orders of magnitude smaller than the mean values,
suggesting that the loads are relatively steady. Meanwhile, viscosity
contributions are over three orders of magnitude smaller than pressure
contributions for $C_{T}$, and about two orders of magnitude smaller for
$C_{P}$, revealing that pressure plays a much dominant role on the loads.
Moreover, viscosity has positive contributions to the mean thrust but negative
contributions to the power output. Most interestingly, the DWT’s mean thrust
coefficient is about $54\%$ higher, and the mean power coefficient is about
$46\%$ higher than the OWT’s.

The time-averaged load coefficients (denoted by $\overline{C}_{T}$ and
$\overline{C}_{P}$) at the three tip speed ratios are shown in Fig. 9. For
both turbines, $\overline{C}_{T}$ and $\overline{C}_{P}$ increase with
$\lambda$ in this range. The $\overline{C}_{T}$’s of the two turbines grow at
almost the same pace (with a difference of approximately 0.3 between the two
curves), while the $\overline{C}_{P}$ of the DWT grows faster than that of the
OWT.

(a) Thrust

(b) Power

Figure 9: Mean loads for different tip-speed ratios.

The phase-averaged pressure on the blade surfaces is plotted to show more
details of the load distributions on the blades. As shown in Figs. 10 and 11,
for the DWT, the upstream surface experiences a higher pressure than the
downstream surface. Thus the rotor experiences a drag force pointed towards
the x-direction. The outboard part of the blades, especially the area around
the leading edge, has a higher pressure difference, which means the load
strength is higher in this region. The OWT’s phase-averaged pressure contours
are given in Figs. 12 and 13. A similar pressure distribution as that of the
DWT is observed. One major difference is that the downstream surface has a
smaller area of low pressure than that of the DWT, which is responsible for
the smaller loads than the DWT’s.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 10: The DWT’s phase-averaged upstream surface pressure.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 11: The DWT’s phase-averaged downstream surface pressure.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 12: The OWT’s phase-averaged upstream surface pressure.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 13: The OWT’s phase-averaged downstream surface pressure.

For the DWT, the phase-averaged pressure at $x/D=0.02$ is shown in Fig. 14.
Due to the pressure difference between the two sides of each blade, all three
blades experience a torque pointed towards the negative x-axis. This torque
direction is the same as the rotation direction, which means positive work is
done on the rotor, and the wind energy is transferred to the rotor’s
mechanical energy. Fig. 15 plots the pressure contours for the OWT, and a
similar pressure distribution is observed. The major difference between the
two turbines is still that the DWT has larger low-pressure regions.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 14: The DWT’s phase-averaged pressure at $x/D=0.02$.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 15: The OWT’s phase-averaged pressure at $x/D=0.02$.

###  4.2 Vortex Fields

Isosurfaces of instantaneous Q-criterion, colored by the streamwise velocity,
are plotted in Fig. 16 for both turbines. A view through the central plane of
the DWT at $\lambda=3.93$ is shown in Fig. 17. Compared with the OWT’s flow
fields, there are dramatic changes in the DWT’s due to the addition of the
duct. Several important observations can be made. For example, 1) The DWT’s
flow fields are much wider in the radial direction. 2) The flow fields are
more turbulent with a much broader range of flow structures. 3) Tip vortices
have been dramatically weakened and can hardly be observed. 4) Hub vortices
have almost been entirely suppressed. 5) The blades’ trailing-edge vortices
have been enhanced (see Fig. 17).

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 16: Isosurface of instantaneous Q-criterion
$Q_{cr}D^{2}/U_{\infty}^{2}=60$ in both configurations. Figure 17: Flow
structures in the central x-y plane of the DWT at $\lambda=3.93$.

###  4.3 Velocity Fields

Contours of the mean streamwise velocity, $\overline{u}$, in the central x-y
plane are shown in Fig. 18 for the three tip speed ratios. It is seen that the
flows downstream of both turbines are at reduced speeds compared with the
free-stream flow, which represents a fluid kinetic energy loss. The DWT’s
wakes have much larger low-speed regions than those of the OWT, indicating
more fluid kinetic energy loss, which is consistent with the higher power
coefficients of the DWT.

At $\lambda=3.11$ and $3.93$, the DWT’s low-speed regions have similar
diverging-converging shapes. However, at $\lambda=4.75$, the DWT’s wake
becomes bifurcated. It is conjectured that the bifurcation is caused by the
blockage effects of the blades and the duct. From Fig. 5, it is observed that
each blade is almost perpendicular to the incoming flow around the tip but
more skewed (in other words, more aligned with the flow) towards the root.
When a blade rotates faster (i.e., as $\lambda$ increases), the tip blockage
effect increases more substantially, which, when combined with the presence of
the duct, dramatically reduces the flow speed downstream of the duct. It is
worth mentioning that the gap between the tips and the duct’s inner surface is
too small (about $0.03D$) to allow much fluid to go through. Meanwhile, the
enhanced blockage effect forces more fluid to go through the root region of
the blades, resulting in larger root flow speeds at $\lambda=4.75$ than at
$3.11$ and $3.93$. The combination of the flows in these two regions (i.e.,
tips and roots) results in a bifurcation in the wake. The tip blockage effects
actually also exist in the OWT’s flow fields. As can be seen from Fig. 18, the
flow speeds downstream of the OWT’s tips also decrease as $\lambda$ increases.

(a) $\lambda=3.11$

(b) $\lambda=3.93$

(c) $\lambda=4.75$

Figure 18: Contours of mean streamwise velocity $\overline{u}$ in the central
x-y plane: left, DWT; right, OWT.

To quantitatively see how the wake flows develop, we plot the mean streamwise
velocity profiles in Figs. 19-21. Overall, the wakes show a consistent
recovery trend as the flows travel downstream. At almost every location and
for every $\lambda$, the DWT’s wake has lower speeds than the OWT’s. The
notable exceptions are at $x/D=5$ and for the $\lambda=3.93$ and $4.75$ cases,
where the OWT’s wakes have lower speeds around $y/D=0$ because of the presence
of strong hub vortices.

The effects of the blades and the duct on the wakes are most evident on the
profiles at $x/D=0.5$. Since the mean flow is symmetric about $y/D=0$, we thus
only focus on the $y/D>0$ part to examine the effects. For the DWT, there is
always a local minimum of $\overline{u}$ around $y/D=0.5$, which is caused by
the blockage effect of the duct. The value of this local minimum decreases as
$\lambda$ increases (the values are approximately 0.41, 0.32, and 0.19 for
$\lambda=3.11$, 3.93, and 4.75, respectively), suggesting an increasing
blockage effect. Right below this minima is a local speed maximum. As
$\lambda$ increases, the maximum’s location moves downward (toward the root),
and its value increases. More specifically, the locations are $y/D=0.33$,
0.28, and 0.14, and the corresponding values are 0.50, 0.52, and 0.66,
respectively, for $\lambda=3.11$, 3.93, and 4.75. These speed distributions
agree with our conjecture that increasing the rotational speed will cause
stronger tip and duct blockage effects and enhanced root flows, which are
responsible for the bifurcation.

(a) $x/D=0.5$

(b) $x/D=1$

(c) $x/D=3$

(d) $x/D=5$

Figure 19: Profiles of mean streamwise velocity for $\lambda=3.11$ at
different streamwise locations in the central x-y plane.

(a) $x/D=0.5$

(b) $x/D=1$

(c) $x/D=3$

(d) $x/D=5$

Figure 20: Profiles of mean streamwise velocity for $\lambda=3.93$ at
different streamwise locations in the central x-y plane.

(a) $x/D=0.5$

(b) $x/D=1$

(c) $x/D=3$

(d) $x/D=5$

Figure 21: Profiles of mean streamwise velocity for $\lambda=4.75$ at
different streamwise locations in the central x-y plane.

##  5 Simulation of Turbines in Yawed Flows

This section focuses on several yawed flows at $\lambda=3.93$. To facilitate
the discussion, besides the original physical coordinate system, we introduce
a flow coordinate system x′-y′ as shown in Fig. 22, where x′ is along the
freestream flow direction, and y′ is perpendicular to the freestream flow
direction. The yaw angle $\gamma$ is defined as the angle between the flow
direction and the axial direction of the turbine, i.e., the angle between x′
and x. Four yaw angles: $\gamma=0^{\circ}$, $10^{\circ}$, $20^{\circ}$, and
$30^{\circ}$, are considered for both the DWT and the OWT. It is worth noting
that because of axial symmetry, we do not consider the sign of the yaw angle
here.

Figure 22: Definitions of coordinates and yaw angle.

###  5.1 Load Analysis

The two turbines’ mean thrust and power coefficients at the four yaw angles
are plotted in Fig. 23. It is seen that the DWT experiences more drag and
extracts more energy than the OWT at all yaw angles. The DWT’s performance is
found to be insensitive to small yaw angles (e.g., for $\gamma<10^{\circ}$).
Other than that, the two coefficients of both turbines decrease as the yaw
angle increases, which is a direct result of the flow rate decrease across the
turbines’ swept areas.

(a) Thrust

(b) Power

Figure 23: Mean thrust and power coefficients for yawed flows at
$\lambda=3.93$.

To see the relative performance change at different yaw angles, we normalize
both coefficients with their values at $\gamma=0^{\circ}$. The normalized
coefficients are plotted in Fig. 24. Except for small yaw angles for the DWT,
the yaw angle is found to have more substantial effects on the power
coefficients than on the thrust coefficients. For example, from
$\gamma=0^{\circ}$ to $\gamma=20^{\circ}$, the OWT sees a $6\%$ relative drop
in the thrust coefficient but a $12\%$ relative drop in the power coefficient.
From $\gamma=0^{\circ}$ to $\gamma=30^{\circ}$, the OWT experiences a $13\%$
and $26\%$ relative drop on the $\overline{C}_{T}$ and $\overline{C}_{P}$,
respectively, while the DWT sees a $20\%$ and $29\%$ relative drop in the two
coefficients.

(a) Normalized $\overline{C}_{T}$

(b) Normalized $\overline{C}_{P}$

Figure 24: Normalized mean thrust and power coefficients for yawed flows at
$\lambda=3.93$.

###  5.2 Vortex Fields

Isosurfaces of the instantaneous Q-criterion for the two turbines are shown in
Fig. 25. For simplicity and fair comparison, the plots are presented in the
flow coordinates (i.e., the x′-y′ coordinates). Obviously, nonzero yaw angles
have made the wakes asymmetric about the flow direction. These asymmetries can
be qualitatively explained as follows. In the extreme situation of infinitely
large blade rotation speed, the flows are completely blocked in the turbines’
cross-sections, and the turbines behave like two disks. At a finite rotational
speed, the turbines still roughly behave like disks with some cross-sectional
flow rates. The upper leading edges of the “disks” cause much larger flow
separations than the lower trailing edges, resulting in the present asymmetric
shapes of the wakes. Furthermore, as the yaw angle increases, the relative
angle of attack of the “disks” decreases, resulting in weaker wake vortices.
Of course, the blades’ rotation motions add more complexities to the flow
fields. For example, for the OWT, the spiral tip vortices are still present
even for large yaw angles, but the hub vortices gradually disappear as the yaw
angle increases.

(a) $\gamma=10^{\circ}$

(b) $\gamma=20^{\circ}$

(c) $\gamma=30^{\circ}$

Figure 25: Isosurfaces of the instantaneous Q-criterion
$Q_{cr}D^{2}/U_{\infty}^{2}=60$ for yawed flows.

###  5.3 Velocity Fields

Contours of the mean streamwise flow velocity in the central x′-y′ plane at
different yaw angles are plotted in Fig. 26. It is seen that the wakes are
still at lower speeds compared with the freestream. However, the sizes of the
low-speed regions decrease as the yaw angle increases, which agrees with the
decreasing energy extraction performances revealed in Fig. 23. Although the
results are presented in the flow coordinate, deflections of the wakes are
still observed. Overall, the OWT shows more considerable wake deflections than
the DWT. For the DWT, as the yaw angle increases, the angle of attack of the
duct’s upper portion increases, which has led to larger flow separations and
blockage effects around the duct’s upper trailing edge. Strong interactions
are seen between the duct’s upper wake and the DWT’s main wake. For the duct’s
lower portion, the situation is just the opposite. The angle of attack
decreases as the yaw angle increases, so no severe flow separation is observed
in this region.

(a) $\gamma=10^{\circ}$

(b) $\gamma=20^{\circ}$

(c) $\gamma=30^{\circ}$

Figure 26: Mean streamwise velocity contours at $\lambda=3.93$ in the central
x′-y′ plane.

The mean streamwise velocity profiles at several locations in the central
x′-y′ plane are shown in Figs. 27-29 for the three nonzero yaw angles. To
facilitate the discussion, we define the centerline of a wake as the line that
connects points with minimum local streamwise flow speeds. This centerline
divides a wake into an upper part and a lower part. From the profiles, it is
seen that in the near field (e.g., $x^{\prime}/D=0.5$ and $1.0$), the upper
wakes overall travel slower than the lower wakes, indicating that the turbines
may have extracted more energy via their upper parts. A comparison of the
profiles further downstream, e.g., at $x^{\prime}/D=3.0$ (Figs. 27(c), 28(c),
and 29(c)), and for different yaw angles reveals that the wakes of both
turbines recover faster for larger yaw angles. Finally, the wake deflection is
better judged from flows sufficiently downstream of the turbines, e.g., at
$x^{\prime}/D=5.0$. For $\gamma=10^{\circ}$ (see Fig. 27(d)), the DWT’s wake
is slightly more deflected than the OWT’s. For $\gamma=20^{\circ}$ (see Fig.
28(d)), the OWT’s wake is slightly more deflected. For $\gamma=30^{\circ}$
(see Fig. 29(d)), the OWT’s wake is obviously more deflected.

(a) $x^{\prime}/D=0.5$

(b) $x^{\prime}/D=1$

(c) $x^{\prime}/D=3$

(d) $x^{\prime}/D=5$

Figure 27: Mean streamwise velocity profiles for $\gamma=10^{\circ}$.

(a) $x^{\prime}/D=0.5$

(b) $x^{\prime}/D=1$

(c) $x^{\prime}/D=3$

(d) $x^{\prime}/D=5$

Figure 28: Mean streamwise velocity profiles for $\gamma=20^{\circ}$.

(a) $x^{\prime}/D=0.5$

(b) $x^{\prime}/D=1$

(c) $x^{\prime}/D=3$

(d) $x^{\prime}/D=5$

Figure 29: Mean streamwise velocity profiles for $\gamma=30^{\circ}$.

##  6 Summary

High-order implicit large-eddy simulations of a ducted wind turbine and its
open-rotor counterpart under different tip-speed ratios and yaw angles have
been successfully performed. The simulations employed a fifth-order spatial
scheme, a third-order temporal scheme, and about thirty million degrees of
freedom. This is the first time that a high-order method being applied to a
comprehensive study of a complete ducted wind turbine without any geometric
simplification.

The simulation results reveal that the ducted turbine has much higher power
outputs than its open-rotor counterpart for all the given tip-speed ratios.
From pressure contours, it is identified that the loads concentrate more on
the blades’ outboard part, especially around the leading edges. From the
isosurfaces of instantaneous Q-criterion, it is noticed that the presence of
the duct has made the flow field more turbulent, weakened the tip and hub
vortices, and enhanced the trailing edge vortices of the ducted turbine. As
the tip-speed ratio increases, flow bifurcation is observed in the flow field
of the ducted turbine. The contours and profiles of the mean streamwise
velocity reveal that the bifurcation is caused by the increasing blockage
effects of blade tips with increasing rotation speed.

Under yawed flow conditions, the ducted turbine is still found to have larger
power outputs than the open configuration. It is confirmed that the ducted
turbine’s performance is insensitive to small yaw angles. Other than that, as
the yaw angle increases, the thrust and the power coefficients both decrease,
with the latter decreasing more than the former. At nonzero yaw angles, the
vortex fields are highly asymmetric for both turbines, and an interpretation
using disk analog and angle of attack has been established. The velocity
fields and profiles show that a large yaw angle may cause obvious flow
deflections in the wakes, especially in the open turbine’s wakes. Finally, the
wake flows are found to recover more quickly at larger yaw angles.

## Acknowledgment

The authors would like to express our acknowledgments to Clarkson University
for financial support through the IGNITE Fellowship Program. C. Liang would
also like to acknowledge the support by an Office of Naval Research grant (No.
N00014-20-1-2007) monitored by Dr. Ki-Han Kim. The computing hours were
granted through the DoD HPC Modernization Program.

## References

  * Shepherd and Zhang [2017] W. Shepherd, L. Zhang, Electricity generation using wind power, World Scientific, 2017.
  * Hau [2013] E. Hau, Wind turbines: fundamentals, technologies, application, economics, Springer Science & Business Media, 2013.
  * Burton et al. [2011] T. Burton, N. Jenkins, D. Sharpe, E. Bossanyi, Wind Energy Handbook, John Wiley & Sons, 2011.
  * Ochieng et al. [2018] F. X. Ochieng, C. M. Hancock, G. W. Roberts, J. Le Kernec, A review of ground-based radar as a noncontact sensor for structural health monitoring of in-field wind turbines blades, Wind Energy 21 (12) (2018) 1435–1449.
  * Dighe et al. [2020] V. Dighe, F. Avallone, G. Bussel, Effects of yawed inflow on the aerodynamic and aeroacoustic performance of ducted wind turbines, Journal of Wind Engineering and Industrial Aerodynamics 201 (2020) 104174.
  * Bontempo and Manna [2020] R. Bontempo, M. Manna, On the potential of the ideal diffuser augmented wind turbine: an investigation by means of a momentum theory approach and of a free-wake ring-vortex actuator disk model, Energy Conversion and Management 213 (2020) 112794.
  * Lilley and Rainbird [1956] G. M. Lilley, W. J. Rainbird, A preliminary report on the design and performance of ducted windmills, College of Aeronautics, Cranfield, 1956\.
  * Foreman et al. [1978] K. M. Foreman, B. Gilbert, R. A. Oman, Diffuser augmentation of wind turbines, Solar Energy 20 (4) (1978) 305–311.
  * Gilbert and Foreman [1979] B. L. Gilbert, K. M. Foreman, Experimental demonstration of the diffuser-augmented wind turbine concept, Journal of Energy 3 (4) (1979) 235–240.
  * Igra [1981] O. Igra, Research and development for shrouded wind turbines, Energy Conversion and Management 21 (1) (1981) 13–48.
  * Fletcher [1981] C. Fletcher, Computational analysis of diffuser-augmented wind turbines, Energy Conversion and Management 21 (3) (1981) 175–183, ISSN 0196-8904.
  * Vaz and Wood [2018] J. R. Vaz, D. H. Wood, Effect of the diffuser efficiency on wind turbine performance, Renewable Energy 126 (2018) 969–977.
  * Koras and Georgalas [1988] A. D. Koras, C. G. Georgalas, Calculation of the Influence of Annular Augmentors on the Performance of a Wind Rotor, Wind Engineering (1988) 257–267.
  * Politis and Koras [1995] G. K. Politis, A. D. Koras, A performance prediction method for ducted medium loaded horizontal axis windturbines, Wind Engineering (1995) 273–288.
  * Phillips et al. [2002] D. G. Phillips, P. J. Richards, R. Flay, CFD modelling and the development of the diffuser augmented wind turbine, Wind and Structures 5 (2_3_4) (2002) 267–276.
  * Hansen et al. [2000] M. Hansen, N. Sørensen, R. Flay, Effect of placing a diffuser around a wind turbine, Wind Energy 3 (4) (2000) 207–213.
  * Abe and Ohya [2004] K. Abe, Y. Ohya, An investigation of flow fields around flanged diffusers using CFD, Journal of Wind Engineering and Industrial Aerodynamics 92 (3-4) (2004) 315–330.
  * Venters et al. [2018] R. Venters, B. Helenbrook, K. Visser, Ducted wind turbine optimization, Journal of Solar Energy Engineering 140 (1).
  * Sadeghi et al. [2018] N. B. Sadeghi, B. Helenbrook, K. Visser, Ducted wind turbine optimization and sensitivity to rotor position, Wind Energy Science 3 (1) (2018) 221–229.
  * Hill [1973] W. R. T. Hill, Triangular mesh methods for the neutron transport equation, Tech. Rep., Los Alamos Scientific Lab., N. Mex.(USA), 1973\.
  * Cockburn et al. [2012] B. Cockburn, G. Karniadakis, C. W. Shu, Discontinuous Galerkin methods: theory, computation and applications, vol. 11, Springer Science & Business Media, 2012\.
  * Patera [1984] A. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, Journal of Computational Physics 54 (3) (1984) 468–488.
  * Karniadakis and Sherwin [2013] G. Karniadakis, S. Sherwin, Spectral/hp element methods for computational fluid dynamics, Oxford University Press, 2013\.
  * Kopriva and Kolias [1996] D. Kopriva, J. Kolias, A conservative staggered-grid Chebyshev multidomain method for compressible flows, Journal of Computational Physics 125 (1) (1996) 244–261.
  * Kopriva [1996] D. Kopriva, A conservative staggered-grid Chebyshev multidomain method for compressible flows. II. A semi-structured method, Journal of Computational Physics 128 (2) (1996) 475–488.
  * Kopriva [1998] D. Kopriva, A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations, Journal of Computational Physics 143 (1) (1998) 125–158.
  * Liu et al. [2006] Y. Liu, M. Vinokur, Z. Wang, Spectral difference method for unstructured grids I: Basic formulation, Journal of Computational Physics 216 (2) (2006) 780–801.
  * Huynh [2007] H. T. Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA paper 2007-4079, 2007.
  * Huynh [2009] H. T. Huynh, A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion, AIAA paper 2009-403, 2009.
  * Wang and Gao [2009] Z. J. Wang, H. Gao, A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, Journal of Computational Physics 228 (2009) 8161–8186.
  * Zhang and Liang [2015a] B. Zhang, C. Liang, A simple, efficient, high-order accurate sliding-mesh interface approach to FR/CPR method on coupled rotating and stationary domains, AIAA paper 2015-1742, 2015a.
  * Zhang and Liang [2015b] B. Zhang, C. Liang, A simple, efficient, and high-order accurate curved sliding-mesh interface approach to spectral difference method on coupled rotating and stationary domains, Journal of Computational Physics 295 (2015b) 147–160.
  * Zhang et al. [2016] B. Zhang, C. Liang, J. Yang, Y. Rong, A 2D parallel high-order sliding and deforming spectral difference method, Computers & Fluids 139 (2016) 184–196.
  * Zhang and Liang [2016] B. Zhang, C. Liang, A high-order sliding-mesh spectral difference solver for simulating unsteady flows around rotating objects, in: 31st Symposium on Naval Hydrodynamics, Monterey, CA, 2016.
  * Zhang et al. [2018] B. Zhang, Z. Qiu, C. Liang, A flux reconstruction method with nonuniform sliding-mesh interfaces for simulating rotating flows, AIAA paper 2018-1094, 2018.
  * Zhang and Liang [2021] B. Zhang, C. Liang, A conservative high-order method utilizing dynamic transfinite mortar elements for flow simulation on curved sliding meshes, Journal of Computational Physics (2021) 110522.
  * Zhang and Liang [2019a] B. Zhang, C. Liang, High-order numerical simulation of flows over rotating cylinders of various cross-sectional shapes, AIAA paper 2019-3430, 2019a.
  * Zhang and Liang [2019b] B. Zhang, C. Liang, High-order numerical simulation of flapping wing for energy harvesting, AIAA paper 2019-3338, 2019b.
  * Zhang et al. [2021] B. Zhang, C. Ding, C. Liang, High-Order Implicit Large-Eddy Simulation of Flow over a Marine Propeller, Computers & Fluids (2021) 104967.
  * Ding et al. [2022] C. Ding, B. Zhang, C. Liang, K. D. Visser, G. Yao, High-order large-eddy simulations of a ducted wind turbine, AIAA paper 2022-1147, 2022\.
  * Thomas and Lombard [1979] P. D. Thomas, C. K. Lombard, Geometric conservation law and its application to flow computations on moving grids, AIAA Journal 17 (1979) 1030–1037.
  * Bathe [2006] K. J. Bathe, Finite Element Procedures, Klaus-Jurgen Bathe, 2006.
  * Rusanov [1961] V. V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, Journal of Computational and Mathematical Physics USSR 1 (1961) 267–279.
  * Spiteri and Ruuth [2002] R. J. Spiteri, S. J. Ruuth, A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM Journal on Numerical Analysis 40 (2002) 469–491.
  * Ruuth [2006] S. Ruuth, Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Mathematics of Computation 75 (253) (2006) 183–207.
  * Zhang [2016] B. Zhang, A high-order computational framework for simulating flows around rotating and moving objects, Ph.D. thesis, The George Washington University, 2016\.
  * Kanya and Visser [2018] B. Kanya, K. Visser, Experimental validation of a ducted wind turbine design strategy, Wind Energy Science 3 (2) (2018) 919–928.


# One more proof about the spectrum of Transposition graph

Artem Kravchuk<EMAIL_ADDRESS>Sobolev Institute of Mathematics,
Ak. Koptyug av. 4, Novosibirsk 630090, Russia  Novosibirsk State University,
Pirogova str. 2, Novosibirsk, 630090, Russia

###### Abstract

A _Transposition graph_ $T_{n}$ is defined as a Cayley graph over the
symmetric group $Sym_{n}$ generated by all transpositions. This paper shows
how the spectrum of $T_{n}$ can be obtained using the spectral properties of
the Jucys-Murphy elements.

###### keywords:

Transposition graph; integral graph; spectrum;

###### MSC:

[2010] 05C25, 05E10, 05E15

††journal: Siberian Electronic Mathematical Reports

##  1 Introduction

_Transposition graph_ $T_{n}$ is defined as a Cayley graph over the symmetric
group generated by all transpositions. The _spectrum_ of a graph is defined as
a multiset of distinct eigenvalues of its adjacency matrix together with their
multiplicities. The graph $T_{n},n\geqslant 2$, is bipartite, so its spectrum
is symmetric about zero. The largest eigenvalue of $T_{n}$ is equal to
$\binom{n}{2}$ which implies that all eigenvalues of $T_{n}$ lie in the
interval $[-\binom{n}{2},\binom{n}{2}]$. In [7] it was shown that the
eigenvalues of $T_{n}$ correspond to partitions $\lambda\vdash n$, and
formulas for multiplicities of these eigenvalues were given. It was also shown
in [7, Lemma 3] that $T_{n}$ is integral which means that its spectrum
$Spec(T_{n})$ consists of integers. Later and independently in [11, Theorem
2], it was shown that $T_{n}$ is integral. The obtaining of the spectrum in
[7] is based on the Zieschang theorem [15, Theorem 1]. This paper shows how to
obtain the spectrum of $T_{n}$ in an another way, using the spectral
properties of the Jucys-Murphy elements. The Jucys-Murphy elements are
significant in the study of representation theory related to symmetric groups
[13] and have diverse applications. For example, spectral properties of the
Jucys-Murphy elements have previously been used to find the spectrum of the
Star graph [1]. Also in [3, 4], Jucys-Murphy elements are used to study the
spectral properties of this graph.

The paper is organised as follows. First, in Section 2 we give basic facts
from the representation theory of the symmetric group that are needed in
further proofs. After that in Section 3 we describe the Jucys-Murphy elements
and introduce the $J_{\Sigma}^{n}$ operator and study it’s spectrum. Then in
Section 4 we show that the operator $J_{\Sigma}^{n}$ is equal to adjacency
matrix of $T_{n}$.

##  2 Preliminaries

###  2.1 Partitions and Young tableaux

A nonincreasing sequence of positive integers
$\lambda=(n_{1},\dots,n_{k})\vdash n$, for which
$n=\sum\limits_{j=1}^{k}n_{j}$, is called a _partition_ of $n$.

A _Young diagram_ is a finite set of cells arranged in left-justified rows,
with the lengths of the rows in non-increasing order. It is not hard to see
the bijection between Young diagrams and partitions.

We say that a Young diagram has the shape $\lambda$, where
$\lambda=(n_{1},n_{2},\dots,n_{k})$ if the $i$-th row has $n_{i}$ cells. We
use French notation, in which the rows are constructed from bottom to top.
Figure 2.1 shows Young diagrams corresponding to all partitions of $n=4$.

[4] |
---|---
|  [1]
[3] |
|  [2]
[2] |
|  [1]
[1] |
[2] |
|  [1]
[1] |
[1] |
[1] |

Figure 1: Young diagrams corresponding to all partitions of the number 4.

A _standard Young tableau_ of the shape $\lambda\vdash n$ is the filling of a
Young diagram of the shape $\lambda$ with numbers from $1$ to $n$ such that
the numbers in each row and each column are in ascending order. Let
$\mathcal{T}_{\lambda}$ be the set of standard Young tableaux of the shape
$\lambda\vdash n$. Figure 2.1 shows all standard Young tableaux from
$\mathcal{T}_{(2,2,1)}$.

5 |
---|---
3 |  4
1 |  2
|  4
3 |  5
1 |  2
|  5
2 |  4
1 |  3
|  4
2 |  5
1 |  3
|  3
2 |  5
1 |  4

Figure 2: Standard Young tables of the shape $\lambda=(2,2,1)$.

For each cell at position $(i,j)$, the _hook_ $H_{ij}^{\lambda}$ is defined as
the set of cells $(k,l)$ such that $k=i$ and $l\geqslant j$ or $l=j$ and
$k\geqslant i$. The hook length $h_{ij}^{\lambda}$ is the number of cells in
$H_{ij}^{\lambda}$. The number of standard Young tableaux of the shape
$\lambda$ is obtained using the hook-length formula [2, Theorem 1]:

$f_{\lambda}=\frac{n!}{\prod_{(ij)}h_{ij}^{\lambda}}.$ (1)

Consider some standard Young tableau
$t_{\lambda}\in\mathcal{T_{\lambda}},\lambda\vdash n$. For each
$i\in\\{1,\dots,n\\}$ we define $c_{\lambda}^{t}(i)=y-x$, where $x$ and $y$
are the abscissa and ordinate of number $i$, respectively. For example, for
the first table from Figure 2.1 we have
$c(5)=3-1=1,c(4)=0,c(3)=1,c(2)=-1,c(1)=0.$

Note that the sum $c_{\lambda}^{t}(i),i\in\\{0,\dots,n\\}$ does not depend on
how the numbers in the standard Young table $t_{\lambda}$ are filled. Let us
denote by $s_{\lambda}$ the following sum:

$s_{\lambda}=\sum\limits_{i=1}^{n}c_{\lambda}(i).$ (2)

For example, for the first Young diagram in Figure 2.1, $s_{\lambda}=-6$, and
for the Young diagram in Figure 2.1, $s_{\lambda}=3$.

Let $\lambda=(n_{1},n_{2},\dots,n_{k})\vdash n$.

###### Lemma 1

$s_{\lambda}=-\sum_{j=1}^{k}\frac{n_{j}(n_{j}-2j+1)}{2}.$ (3) Proof.

Let us consider the $j$-th row in the Young tableau of the shape $\lambda$.
Elements from this line add the expression
$\sum\limits_{i=1}^{n_{j}}(j-i)=-\frac{n_{j}(n_{j}-2j+1)}{2}$ to
$s_{\lambda}$. $\square$

Note that different partitions can correspond to the same value of
$s_{\lambda}$. For example, $s_{(3,3)}=s_{(4,1,1)}=-3$.

The _conjugate of a partition $\lambda$_, denoted by $\lambda^{\prime}$, is
also a partition of $n$ where its parts are the nonincreasing sequence
$\lambda^{\prime}=(n_{1}^{\prime},n_{2}^{\prime},\ldots,n_{\kappa}^{\prime})$,
where $n_{j}^{\prime}=\sum\limits_{i|n_{i}\geqslant j}1$.

###### Lemma 2

$s_{\lambda}=-s_{\lambda^{\prime}}.$

Proof. The proof is the same as for Lemma 1 if we consider the columns for the
Young table corresponding to $\lambda^{\prime}$.

###  2.2 Regular representation

Let $GL(V)$ represent the collection of all invertible transformations of
vector space $V$ onto itself, known as the general linear group of $V.$ A
representation of a group $G$ on a vector space $V$ over a field $\mathbb{C}$
is a homomorphism $\rho:G\longrightarrow GL(V)$. $V$ is also called _$G$
-module_.

Let $W$ be a vector subspace of $V$. $W$ is considered $\rho$-invariant if
$\rho(g)(w)\in W$ for any $g\in G$ and $w\in W$. A representation $\rho$ is
defined as irreducible if the only invariant subspaces of V are $\\{0\\}$ and
$V$. The irreducible representations of $Sym_{n}$ are indexed by partitions of
$n$. We denote by $V_{\lambda}$ the irreducible module associated with the
partition $\lambda\vdash n$.

Let $G$ be a finite group of order $n$. Consider a vector space $V$ of
dimension $n$, having a basis whose elements correspond to the elements of
$G$. We denote this basis as $\\{e_{g}\\}_{g\in G}$, where $e_{g}$ is the
basis vector corresponding to the element $g$ of the group $G$. The _regular
representation_ of the group $G$ is a representation $\rho$, defined as
follows:

ρ(g)(e_h) = e_gh for all $g,h\in G$.

The regular representation of $Sym_{n}$ can be decomposed into irreducible
submodules in the following manner [14, Prop. 1.10.1]:

$\mathbb{C}[Sym_{n}]=\underbrace{\left(V_{\lambda_{1}}\oplus\cdots\oplus
V_{\lambda_{1}}\right)}_{\dim(V_{\lambda_{1}})}\oplus\cdots\oplus\underbrace{\left(V_{\lambda_{k}}\oplus\cdots\oplus
V_{\lambda_{k}}\right)}_{\dim(V_{\lambda_{k}})}=\bigoplus_{\lambda\vdash
n}V_{\lambda}^{dim(V_{\lambda})},$ (4)

and $dim(V_{\lambda})$ = $f_{\lambda}.$

##  3 Jucys-Murphy elements and spectrum of $J_{\Sigma}^{n}$

The _Jucys-Murphy elements_ are elements of group algebra
$\mathbb{C}[Sym_{n}]$. These elements were introduced independently by Jucys
in 1966 [5] and Murphy in 1981 [12] and are defined as follows:

$J_{i}^{n}=(1,i)+(2,i)+\dots+(i-1,i)=\sum\limits_{j=1}^{i}(j,i),i=2,\dots,n,$
(5)

where by the summation of transpositions means the summation of the matrices
corresponding to these transpositions in some representation.

The Jucys-Murphy elements can be viewed as a linear operator acting on the
basis vectors of the regular representation as follows:

J_i^n (e_σ_k) = ∑_j=1^i e_(j,i)⋅σ_k.

The spectrum of Jucys-Murphy elements is obtained by the following theorem.

###### Theorem 1

[6] Let $\lambda\vdash n$. Then there exists a basis
$(v_{t})_{t\in\mathcal{T}_{\lambda}}$ of irreducible module $V_{\lambda}$,
indexed by standart Young tables of shape $\lambda$, such that for all
$i\in\\{2,\dots,n\\}$, one has:

$J_{i}^{n}(v_{t})=c^{t}_{\lambda}(i)v_{t}.$ (6)

Let us denote by $J_{\Sigma}^{n}$ the following expression:

$J_{\Sigma}^{n}=\sum\limits_{i=2}^{n}J_{i}^{n}.$ (7)

Consider the basis $(v_{t})_{t\in\mathcal{T}_{\lambda}}$ from Theorem 1. From
the definition of $J_{\Sigma}^{n}$ and $s_{\lambda}$ it follows directly that
$\\{s_{\lambda}\\}_{\lambda\vdash n}$ are eigenvalues of the operator
$J_{\Sigma}^{n}$:

$J_{\Sigma}^{n}(v_{t})=s_{\lambda}v_{t}.$ (8)

The following lemma gives an expression of the multiplicity of the eigenvalue
$s_{\lambda}$ for the operator $J_{\Sigma}^{n}$.

###### Lemma 3

$mul(s_{\lambda})=\sum\limits_{\eta:s_{\alpha}=s_{\lambda}}f_{\alpha}^{2}$.

Proof. By (4), the number of irreducible modules $V_{\lambda}$ is
$f_{\lambda}$. In each of $V_{\lambda}$ the number of eigenvectors $v_{t}$
corresponding to the eigenvalue $s_{\lambda}$ is equal to $f_{\lambda}$ too.
Note also that $mul(s_{\lambda})=mul(s_{\lambda^{\prime}})$. $\square$

We denote by $\eta_{\lambda}=s_{\lambda^{\prime}}=-s_{\lambda}$. Then we are
able to formulate the following theorem.

###### Theorem 2

The spectrum of $J_{\Sigma}^{n}$ consists of the eigenvalues corresponding to
the partitions $\lambda\vdash n$. Any partition $\lambda\vdash n$ corresponds
to an eigenvalue $\eta_{\lambda}$ of $J_{\Sigma}^{n}$ by the following
expression:

$\eta_{\lambda}=\sum_{j=1}^{k}\frac{n_{j}(n_{j}-2j+1)}{2}$ (9)

with multiplicity
$mul(\eta_{\lambda})=\sum\limits_{\alpha:\eta_{\alpha}=\eta_{\lambda}}f_{\alpha}^{2}$.

##  4 Equivalence of $J_{\Sigma}^{n}$ and the adjacency matrix of $T_{n}$

Let us denote by $A_{T_{n}}$ the adjacency matrix (or equal linear operator)
of $T_{n}$. Then we can formulate the following lemma.

###### Lemma 4

$A_{T_{n}}=J_{\Sigma}^{n}.$

Proof. Let us see how the operator $A_{T_{n}}$ acts on an arbitrary basis
element $e_{\sigma_{k}}$:

$A_{T_{n}}(e_{\sigma_{k}})=\sum\limits_{\sigma_{l}:(\sigma_{l},\sigma_{k})\in
E(T_{n})}e_{\sigma_{l}}.$ (10)

An edge $(\sigma_{l},\sigma_{k})\in E(T_{n})$ if and only if
$\sigma_{l}=(i,j)\sigma_{k}$ for any transposition $(i,j)$. Therefore, (10)
can be rewritten in the following form:

$A_{T_{n}}(e_{\sigma_{k}})=\sum\limits_{i<j}e_{(j,i)\sigma_{k}}.$ (11)

Now let us consider the action of $J_{\Sigma}^{n}$ on $e_{\sigma_{k}}$:

$J_{\Sigma}^{n}(e_{\sigma_{k}})=\sum\limits_{i=2}^{n}J_{i}^{n}(e_{\sigma_{k}})=\sum_{i=2}^{n}\sum\limits_{j=1}^{i}e_{(j,i)\sigma_{k}}=\sum\limits_{j<i}e_{(j,i)\sigma_{k}}.$
(12)

Thus, the action of operators $J_{\Sigma}^{n}$ and $A_{T_{n}}$ on an arbitrary
basis element $e_{\sigma_{k}}$ is the same, so operators $J_{\Sigma}^{n}$ and
$A_{T_{n}}$ are equal to each other. $\square$

Lemma 4 shows that the spectrum of $T_{n}$ can be expressed using Theorem 2.

##  5 Discussions and further research

In this paper, it was shown how one can use the spectral properties of Jucys-
Murphy elements to obtain the spectrum of $T_{n}$.

However, by looking at the expression ( 9) it is not possible to answer the
following questions:

  * •

are all integers from the interval $[0,k],k\in\mathbb{N}$, lie in the spectrum
of $T_{n}$?

  * •

what is the asymptotics for the number of unique values in the spectrum of
$T_{n}$?

Ideally, one would like to be able to answer the question whether a given
integer number $k\in[-\binom{n}{2},\binom{n}{2}]$ lies in the spectrum of
$T_{n}$. The papers [8, 9, 10] partially address these issues.

## Acknowledgements

The author was supported by the Mathematical Center in Akademgorodok, under
agreement No. 075-15-2022-281 with the Ministry of Science and High Education
of the Russian Federation.

## References

  * [1] G. Chapuy, V. Feray, A note on a Cayley graph of $\mathfrak{S}_{n}$. https://doi.org/10.48550/arXiv.1202.4976
  * [2] JS. Frame, G. de B. Robinson, RM. Thrall, The Hook Graphs of the Symmetric Group, _Canadian Journal of Mathematics_ 6 (1954) (316-324). https://doi.org/10.4153/CJM-1954-030-1.
  * [3] S. Goryainov, V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich, PI-eigenfunctions of the Star graphs, _Linear Algebra and its Applications_ , 586 (2020) (7-27). https://doi.org/10.1016/j.laa.2019.10.018.
  * [4] V. Kabanov, E. Konstantinova, L. Shalaginov, A. Valyuzhenich, The Star graph eigenfunctions with non-zero eigenvalues, _Linear Algebra and its Applications_ , 610 (2021) (222-226). https://doi.org/10.1016/j.laa.2020.09.042.
  * [5] A. Jucys, On the Young operators of the symmetric groups, _Lithuanian Journal of Physics_ , VI(2) (1966) (180-189).
  * [6] A. Jucys, Symmetric polynomials and the center of the symmetric group ring, _Reports Math. Phys._ 5 (1974) (107-112). https://doi.org/10.1016/0034-4877(74)90019-6 .
  * [7] K. Kalpakis, Y. Yesha, On the bisection Width of the Transposition network, _Networks_ , 29 (1997) 69–76. https://doi.org/10.1002/(SICI)1097-0037(199701)29:1%3C69::AID-NET7%3E3.0.CO;2-A
  * [8] E V. Konstantinova, A. Kravchuk, Spectrum of the Transposition graph, _Linear Algebra and its Applications_ , 654 (2022) (379-389). https://doi.org/10.1016/j.laa.2022.08.033.
  * [9] E. V. Konstantinova, A. Kravchuk, Distinct eigenvalues of the Transposition graph, _Linear Algebra and its Applications_ , 690 (2024) (132-141). https://doi.org/10.1016/j.laa.2024.03.011.
  * [10] A. Kravchuk, Constructing segments of quadratic length in Spec(Tn) through segments of linear length. https://doi.org/10.48550/arXiv.2404.00410.
  * [11] E. V. Konstantinova, D. V. Lytkina, Integral Cayley graphs over finite groups, _Algebra Colloquium_ , 27(1) (2020) (131-136), https://doi.org/10.1142/S1005386720000115.
  * [12] G. Murphy. A new construction of Young’s seminormal representation of the symmetric group. _J. Algebra_ , 69 (1981) (287-291). https://doi.org/10.1016/0021-8693(81)90205-2.
  * [13] A.M. Vershik , A.Y. Okounkov, A New Approach to the Representation Theory of the Symmetric Groups. II. _J Math Sci_ 131 (2005) (5471-5494). https://doi.org/10.1007/s10958-005-0421-7.
  * [14] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer, New York, second edition, 2001. https://doi.org/10.1007/978-1-4757-6804-6.
  * [15] P. Zieschang, Cayley graphs of finite groups. _J. Algebra_ 118 (1988) (447-454). https://doi.org/10.1016/0021-8693(88)90033-6.