COMSOL MULTIPHYSICS VERSUS FLUENT 2D NUMERICAL SIMULATION OF THE

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					           Scientific Bulletin of the                                            3rd Workshop on
    “Politehnica” University of Timisoara                                    Vortex Dominated Flows
         Transactions on Mechanics                                             Timisoara, Romania
       Tom 52(66), Fascicola 3, 2007                                             June 1 - 2, 2007




          COMSOL MULTIPHYSICS VERSUS FLUENT:
    2D NUMERICAL SIMULATION OF THE STATIONARY FLOW
         AROUND A BLADE OF THE ACHARD TURBINE
   Andrei-Mugur GEORGESCU, Assoc. Prof.*                 Sanda-Carmen GEORGESCU, Assoc. Prof.
  Hydraulics and Environmental Protection Department      Hydraulics and Hydraulic Machinery Department
   Technical Civil Engineering University Bucharest            University “Politehnica” of Bucharest
      Sandor BERNAD, Senior Researcher                             Costin Ioan COŞOIU, Assist. Prof.
Center of Advanced Research in Engineering Sciences      Hydraulics and Environmental Protection Department
       Romanian Academy - Timisoara Branch                Technical Civil Engineering University Bucharest
              *Corresponding author: Bd Lacul Tei 24, Sector 2, 020396, Bucharest, Romania
               Tel.: (+40) 212433660, Fax: (+40) 212433660, Email: andreig@mail.utcb.ro


ABSTRACT                                                 x, y, z         [m]     Cartesian coordinates
                                                         α               [grd] angle of attack
    Two-dimensional numerical modelling of the sta-
tionary flow around a blade of the Achard turbine, a     λ               [−]     tip speed ratio
new water-current turbine concept, is performed both     ν               [m2/s] water cinematic viscosity
with COMSOL Multiphysics 3.3 and with Fluent 6.0.1       ω               [rad/s] rotational velocity
software, in order to compare the results and the sof-   ρ               [kg/m3] water density
tware capabilities. The k − ε turbulence model has       θ               [grd] azimuth angle of the blade
been selected and same geometry and boundary con-
ditions were considered within computations.             Subscripts and Superscripts
                                                         D          drag (along flow direction)
KEYWORDS
                                                         L          lift (cross flow direction)
    Achard turbine, cross-flow marine turbine, NACA      *          dimensionless variable
airfoil
                                                         1. INTRODUCTION
NOMENCLATURE
                                                             In 2001, the Geophysical and Industrial Fluid Flows
c            [m]       airfoil chord length              Laboratory (LEGI) of Grenoble, France, launched the
cD           [-]       drag coefficient                  HARVEST Project (Hydroliennes à Axe de Rotation
cL           [-]       lift coefficient                  VErtical STabilisé), to develop a suitable technology
                                                         for marine and river hydro-power farms using cross-
c0           [m]       airfoil mean camber line length   flow current energy converters piled up in towers [1,
d            [m]       airfoil maximum thickness         2 & 8]. The hydro-dynamic of these systems is studied
F            [N]       hydrodynamic force                at LEGI with the support of the R&D Division of the
FD           [N]       drag force                        EDF Group, while other laboratories of the Rhône-
FL           [N]       lift force                        Alpes Region are charged with mechanical aspects
g            [m/s2]    gravity                           (3S-INP of Grenoble and LDMS-INSA of Lyon), as
H            [m]       turbine height                    well as with electrical aspects (LEG-INP Grenoble).
k            [m2/s2]   turbulent kinetic energy              In 2006, the Technical Civil Engineering Univer-
p            [Pa]      pressure                          sity Bucharest, in collaboration with the University
R            [m]       turbine radius                    “Politehnica” of Bucharest and the Romanian Academy
                                                         - Timisoara Branch, started the THARVEST Project,
Rec          [−]       chord based Reynolds number
                                                         within the CEEX Program sustained by the Roma-
u = ωR       [m/s]     transport velocity                nian Ministry of Education and Research [5]. The
V0           [m/s]     upstream velocity                 THARVEST Project aims to study experimentally and
w            [m/s]     relative velocity                 numerically the hydro-dynamics of a new concept
14             Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007


of water-current turbine, called Achard turbine, in                    radial supports are shaped with NACA 4518 airfoils,
collaboration with the LEGI partners involved in the                   while the circular rims are shaped with lens type airfoil.
HARVEST Project. The Achard turbine, a cross-flow                         The turbine radius is R = 0.5 m, and the turbine
marine or river turbine with vertical axis and delta                   height is H = 1 m. In Figure 2 we present the Achard
blades is studied in France mainly with regard to marine               turbine geometry generated in MATLAB (the upper
applications, to extract energy from tidal currents in                 and lower rims are not represented here). Along each
costal locations. But the Achard turbines are also                     delta blade, the airfoil mean camber line length c0
suitable to be placed in big rivers, as the Danube, and
                                                                       varies from 0.18 m at z = 0 , to 0.12 m at the ex-
to produce the desired power by summing elementary
                                                                       tremities, where z = ±0.5 m. Between the leading
power provided by small turbine modules [6 & 8].
                                                                       edge of the blade’s extremity and the leading edge
                                                                       of the blade at mid-height of the turbine, there is a
                                                                       30º azimuth angle.




     Figure 1. The Achard turbine [LEGI courtesy]
                                                                                   Figure 2. The Achard turbine geometry
    In Figure 1 we present the Achard turbine that is
studied now in Romania. The turbine description and                       At z = ± H 4 = ±0.25 m, the horizontal cross-
the generation of its geometry are explained within                    section of the runner gives airfoils with c0 = 0.15 m
the section 2 of this paper.                                           (the mean value of the mean camber line length
    In this paper we focus on the 2D numerical mod-                    along the delta wing), and with the chord length
elling of the stationary flow around a blade of the                    c = 0.1494 m (see section 3). Within this paper, the
Achard turbine. The blades are shaped with NACA                        2D computations correspond to the cross-plane
airfoils. The 2D modelling corresponds to the flow                     placed at z = 0.25 m level (see Figure 3), where the
around an airfoil in a horizontal cross-section of the
runner, situated at one quarter of the turbine height,                       0.5


starting from the top. Different azimuth angles of the
                                                                                                                                     o
                                                                             0.4
                                                                                                                                 θ=0

blade are analysed. Two types of airfoils are considered
                                                                             0.3
within the section 3: a curved one (NACA 4518) and
a straight one (NACA 0018). The simulations are                              0.2

performed both with COMSOL Multiphysics 3.3                                  0.1

software and with Fluent 6.01 software. The results                      y [m] 0
and the software capabilities are compared within
the section 4 with experimental data. The paper con-                        −0.1

clusions are summarized in section 5.                                       −0.2                         o
                                                                                              θ = 120                                                  o
                                                                                                                                               θ = 240
2. ACHARD TURBINE DESCRIPTION                                               −0.3



   The vertical axis Achard turbine from Figure 1
                                                                            −0.4


consists of a runner with three vertical delta blades,                      −0.5

sustained by radial supports at the mid-height of the                               −0.5   −0.4   −0.3       −0.2   −0.1     0
                                                                                                                           x [m]
                                                                                                                                   0.1   0.2     0.3     0.4   0.5

turbine, and stiffened with circular rims at the upper
and lower part of the turbine. The blades and their                         Figure 3. Computational runner cross-section
Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007     15

three blade profiles have a mean camber line length                             Its dimensionless value is m ∗ = m c = 0.03758 . As
of c0 = 0.15 m. The values of the azimuth angle of                              percentage of the chord, m = 3.758% ≅ 4% , so the
                                                      {
the blades in Figure 3 are θ = 0 o ; 120 o ; 240 o , in               }         first digit of the NACA airfoil is 4.
counter clockwise direction.                                                                                 (a) NACA 4518
                                                                                    0.02


3. AIRFOIL SELECTION
                                                                                    0.01

                                                                                y [m]
                                                                                        0

    The runner blades are shaped with NACA airfoils                                −0.01

of four-digit series [13], where the first digit is the                            −0.02
maximum upper camber m (as percentage of the chord),
                                                                                            0         0.05                   0.1    0.15
                                                                                                                 x [m]

the second digit is the distance p of the maximum
upper camber from the airfoil leading edge (in tens of                              0.02
                                                                                                             (b) NACA 0018

percents of the chord), and the last two digits describe                            0.01
the maximum thickness of the airfoil, d, as percent                             y [m]
                                                                                        0
of the chord length. For the Achard turbine blades,                                −0.01
we consider d = 18 %.
    In a xOy plane, the airfoil coordinates {x, y} are
                                                                                   −0.02
                                                                                            0         0.05                   0.1    0.15
                                                                                                                 x [m]

nondimensionalised with respect to the chord length                             Figure 4. Blade profiles for c0 = 0.15 m: (a) curved
c, with x ∗ = 0 at the leading edge and with x ∗ = 1                            airfoil NACA 4518; (b) straight airfoil NACA 0018
at the trailing edge (the dimensionless variables are
denoted with an asterisk).                                                          Thus, the airfoil type corresponding to the Achard
                                          ∗
                                                      {
    The dimensionless coordinates x ∗ , y s of the airfoil   }                  turbine blades is NACA 4518, an airfoil with the
                                                                                mean camber line along the runner circumference
mean camber line are defined as:                                                (Figure 4a). The computations from section 4, are
                                                                                performed for the profile NACA 4518 and also for the
   ∗
  ys =
          m∗
              ∗2
                   (2 p   ∗
                                   )   2
                              − 1 x ∗ for 0 ≤ x ∗ < m ∗ , and                   straight profile NACA 0018, which can be generated
          p                                                                                      ∗
                                                                                from (2) for y s = 0 , since m * = 0 in (1). For the
    ∗
   ys =
               m∗
                               (           )
                              ⎡ 1 − 2 p ∗ + 2 p ∗ x ∗ - x ∗ 2 ⎤ for       (1)   NACA 0018 airfoil, the chord length is c = c0 = 0.15 m
          (1 − p )  ∗ 2       ⎢
                              ⎣                               ⎥
                                                              ⎦                 (Figure 4b). That last choice is due to the fact that
                                                                                experimental and numerical data are available for such
                                                   m∗ ≤ x∗ ≤ 1 ,                a straight profile, corresponding to a Darrieus marine
where p ∗ = 0.5 , since the mean camber line of the                             turbine, a vertical axis cross-flow turbine with two
                                                                                straight blades of 0.15 m chord length [3, 4, 7 & 9].
airfoil is along the circle of radius R = 0.5 m, as in
Figure 3. So, the second digit of the NACA airfoil is                           4. NUMERICAL RESULTS
5, because the maximum upper camber is placed at a
half-distance between the leading edge and the                                      The numerical simulations are performed with
trailing edge.                                                                  COMSOL Multiphysics 3.3 [11] (a Finite Element
                                   {           }
    The coordinates x ∗ , y ∗ of the upper and lower                            Method based software), and with Fluent 6.01 [12]
                                                                                (a Finite Volume Method based software), using the
surfaces of the NACA airfoil are defined by:                                     k − ε turbulence model. The stationary 2D flow
                   d ⎛                                                          around a blade of the Achard turbine is analysed both
        ∗
  y∗ = ys ±           ⎜ 0.29690 x ∗ − 0.12600 x ∗ −
                   20 ⎝                                                         for the profile NACA 4518 (Figure 4a), for which
                                                                          (2)   the mean camber line is along the circle of radius
       − 0.35160x ∗ + 0.28430x ∗ − 0.10150x ∗ ⎞ .
                   2            3            4
                                               ⎟                                 R = 0.5 m, and for the straight profile NACA 0018
                                               ⎠
                                                                                (Figure 4b), for which other results are available in
   The chord length c can be expressed upon the                                 literature.
runner radius and the mean camber line length c0 :
                                                                                4.1. Geometry of the models
  c = 2 R sin (c0 2 R ) .                                                 (3)       The 3D blades geometry is realised in MATLAB
                                                                                as in Figure 2. At a certain z-level, each blade profile
For R = 0.5 m and c0 = 0.15 m, we obtain the mean
                                                                                is generated in 101 nodes on the upper airfoil surface,
chord length c = 0.1494 m at z = 0.25 m.                                        and 101 nodes on the lower airfoil surface, using (2).
   The maximum upper camber is defined as:                                      The dimensional (x, y) coordinates of those 202 nodes
  m = R (1 − cos(c0 2 R )) .                                              (4)   are then imported within the two software used for
16             Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007


the flow simulation. The 2D flow domain extent in the                  on a PC with 2 GB of memory and 3 GHz Intel proc-
xOy plane is scaled with respect to the NACA profiles                  essor, while in Fluent it is less than 25 minutes, on a
mean camber line length that is c0 = 0.15 m. The                       PC with 4 GB RAM and 3.2 GHz dual-core Intel
origin of the system is placed on the profile chord, at                Xeon processor.
one quarter from the leading edge. The rectangular                         Within both software, the following boundary con-
flow domain extents ranges from − 15c0 to + 25c0 on                    ditions are considered: At the left side of the domain,
                                                                       on the water inflow boundary, a constant upstream
x-direction, and from − 3.5c0 to + 3.5c0 on y-direction.
                                                                       velocity V0 = 4.71 m/s and a turbulent intensity of
    Both COMSOL Multiphysics software and Fluent
                                                                       2% are imposed. At the right side of the domain, on
software use the same geometry of the flow domain.
                                                                       the water outflow boundary, a zero relative pressure
The discretization has the same number of nodes on
                                                                       is considered. On the upper wall, as well as on the
the profiles, but the total number of cells is slightly
                                                                       lower wall of the domain, a slip symmetry condition
different from one software to the other (see Figure 5).
                                                                       is selected. On the profile surface, a logarithmic wall
The meshing consists of triangular cells in COMSOL
                                                                       function is selected. In Fluent, a discretization of sec-
and quadrilateral cells in Fluent. For about 33000 cells,
                                                                       ond order is used both for the velocity and pressure.
the resulting run-time in COMSOL exceeds 10 hours,




      Figure 5. Zoom of the domain discretization at α = 0 o , in COMSOL Multiphysics (upper image)
                                       and in Fluent (lower image)
Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007                      17


   The chord based Reynolds number is defined as                                               within the range V0 ≤ w ≤ 3V0 . The velocity ratio
 Rec = w c ν , where w is the relative velocity. The tip                                        w V0 upon the azimuth angle, that is w V0 = f (θ ) ,
speed ratio λ = ωR V0 has the imposed value λ = 2 .                                            as well as the angle of attack variation α = α (θ ) are
The relative velocity w on the blade at the leading                                            presented in Figure 6.
edge is obtained by composing the upstream velocity                                                In this paper, most of the results are computed
V0 and the transport velocity u = ωR = λ V0 :                                                  for discrete values of the pair of angles {α ;θ } . Due
                                                                                               to the symmetry, for the straight profile NACA 0018,
  w = V0 1 + 2λ cosθ + λ2 = V0 1 + 4(1 + cosθ ) ,                                        (5)
                                                                                               the flow behaviour obtained at a certain positive α
where the azimuth angle θ defines the position of the                                          value, is identical to the one obtained at the corre-
blade around the circle, in counter clockwise direction.                                       sponding negative α value.
    For our simulations, the Reynolds numbers exceed                                           4.2. COMSOL Multiphysics versus Fluent
7 ⋅ 10 5 for the whole range of the azimuth angle,                                                We present the results, namely the pressure field
         [               ]
θ ∈ 0 o ; 360 o , placing the phenomenon behaviour                                             and/or the velocity field around the blades, obtained
within the self modelling region with respect to the                                           with both software, for the two types of profiles,
Reynolds number. In this paper, 2D numerical com-                                              NACA 4518 and NACA 0018, at different azimuth
putations are performed on a fixed blade, the value of                                         angles θ (see Figures 7÷9).
the upstream velocity being taken so that the Reynolds                                            The results show that the flow behaviour descrip-
number on the fixed blade exceeds 10 5 , thus the flow                                         tion is similar in both COMSOL Multi-physics and
may be assumed to have the same characteristics as                                             Fluent software. Unfortunately, it is difficult to fit the
in the real rotating case.                                                                     same colour scheme on both software post-processors.
    The angle of attack α , considered between the                                                Fluent seems to give more accurate results. For
chord and the w-direction at the leading edge, is also                                         MATLAB users, the advantage of using COMSOL
related to the azimuth angle θ through the following                                           Multiphysics is obvious in pre-processing step, where
relations:                                                                                     the geometry can be generated directly with MATLAB
                                                                                               sources.
  α = arccos
                         (λ   2
                                       )
                                  − 1 V02 + w 2
                                                − arcsin
                                                          c
                                                            , for                              4.3. Influence of the number of elements
                                  2λ V0 w                2R
                                                                                                   Within this paragraph we will discuss the influence
                                                     [
                                               θ ∈ 0 o ; 180 o , and  )                  (6)   of the number of computational cells. The simulations
  α = arccos
             (λ               2
                                       )
                                  − 1 V02+w     2
                                            + arcsin
                                                      c
                                                        , for
                                                                                               realised in COMSOL Multiphysics need less than 10
                                  2λ V0 w            2R                                        minutes when the grid consists of about 17000 cells,
                                                     [
                                               θ ∈ 180 o ; 360 o .        )                    while the run-time is 60 times greater for twice as
                                                                                               many cells (e.g. about 10 h for near 33000 cells).
                                                                                                   In Figure 10 we present the results obtained using
             4                                                                      40         COMSOL for the velocity field around the blade
                                                     velocity ratio                            profiles NACA 4518 and NACA 0018, at α = 30 o
                                                                                               (          )
         3.5                                                                        30
                                                     angle of attack
             3                                                                      20
                                                                                                θ = 120 o , for about 16500 cells, and 32800 cells
                                                                                               respectively. It seems that the flow behaviour (e.g. the
         2.5                                                                        10
                                                                                               vortices development) doesn’t change substantially
                                                                                               when multiplying by two the number of elements.
  w/V0




             2                                                                      0
                                                                                        α




         1.5                                                                        -10
                                                                                               So, to get a general view of the flow, computations
                                                                                               with about 17000 cells can be acceptable due to the
             1                                                                      -20        significant run-time economy.
         0.5                                                                        -30            Of course, as long as pressure coefficients are
                                                                                               concerned, the values obtained for 33000 cells are
             0                                                                      -40
                                                                                               closer to the experimental existing results.
                 0                90           180           270              360
                                                θ
                                                                                                   Highly accurate computations are reported by
                                                                                               Ervin et al [4] for the flow within a Darrieus marine
   Figure 6. Velocity ratio w V0 [−] and angle of                                              turbine with two-blades of NACA 0018 profile. They
  attack α [grd] versus the azimuth angle θ [grd]                                              used the Turb’Flow software, which is less dissipative
                                                                                               than COMSOL and Fluent. For more than 140000 cells,
   So, the resulting angle of attack varies within the
                                                                                               the computations are really time consuming (about
                     (                     )
range α ∈ − 30o ; 30o , more precisely, from − 29.85 o                                         50 days) on a workstation with 8 GB of memory and
to + 29.85 o . For λ = 2 , the relative velocity (5) varies                                    3.2 GHz dual-core Intel Xeon processor.
18            Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007




 Figure 7. NACA 4518 profile (left images) and NACA 0018 profile (right images), at θ = 0 o , in COMSOL
 Multiphysics (1st & 3rd row) and in Fluent (2nd & 4th row): pressure (first 2 rows) and velocity (last 2 rows)
Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007   19




     Figure 8. Velocity field for NACA 4518 profile (left images) and NACA 0018 profile (right images), at
                   θ = 60 o , in COMSOL Multiphysics (upper row) and in Fluent (lower row)




      Figure 9. Pressure field (left images) and velocity field (right images) for NACA 4518, at θ = 120 o ,
                       in COMSOL Multiphysics (upper row) and in Fluent (lower row)
20            Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007




  Figure 10. Velocity field around the NACA 4518 (left images) and NACA 0018 (right images), at α = 30 o
     (        )
    θ = 120 o , in COMSOL Multiphysics, for about 16500 cells (upper row) and 32800 cells (lower row)




 Figure 11. Pressure field (left image) and velocity field (right image) around the NACA 4518, at α = −30 o
                             (            )
                          θ = 240 o , in COMSOL Multiphysics (about 33000 cells)
    In Figure 11 we present the results obtained using                the flow around the NACA 4518 and NACA 0018
COMSOL for the pressure field and the velocity field,                 profiles, at different values of the angle of attack α .
around the blade profile NACA 4518, at the angle of                        The drag coefficient c D and the lift coefficient
attack α = −30 o and azimuth angle θ = 240 o , for                     c L are computed for each profile, based on the ele-
about 33000 cells. The flow behaviour can be com-                     mentary drag force (dFD ) distribution, and on the
pared with the one obtained for the blade placed in
mirror position (with respect to the flow direction) in               elementary lift force (dFL ) distribution around the
the Figure 10 (see left image, lower row).                            profile boundary, using the following relations:
4.4. Results at different angle of attacks                                          2                               2
   Within this paragraph we present some results
                                                                         cD =
                                                                                 ρ c w2   ∫ dF  D   and c L =
                                                                                                                 ρ c w2   ∫ dF  L   .    (7)
obtained with COMSOL Multiphysics software for
Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007                               21

   We mention that the above curvilinear integrals,                                                         In Figure 13, our computed results for the NACA
which can be computed within the software pre-                                                           4518 profile are plotted together with our results for
processor, give the resulting drag force and lift force                                                  NACA 0018, in order to show that there are no
on the profile, as:                                                                                      significant differences between the drag and lift
                                                                                                         coefficient values, due to the slight curvature of the
                  ∫
        FD = dFD and FL = dFL .               ∫                                                    (8)   4518 airfoil with respect to the straight-one.
    The numerical values of the drag coefficients and                                                    5. CONCLUSIONS
lift coefficients are compared in Figure 12 with the
experimental data obtained at Sandia Laboratories                                                            In this paper, 2D numerical computations are
[10] for the NACA 0018 airfoil, both in c D = c D (α )                                                   performed both with COMSOL Multiphysics 3.3
                                                                                                         software and Fluent 6.0.1 software, in order to depict
and c L = c L (α ) representations, but also in polar                                                    the stationary flow behaviour around a cross-section
diagram c D = c D (c L , α ) . We notice that there are                                                  of a fixed blade of the Achard turbine. The value of
some differences between our computed values and                                                         the upstream velocity is taken so that the Reynolds
the experimental ones, especially at low angles of                                                       number on the fixed profile exceeds 10 5 , thus the
attack (see the polar diagram).                                                                          flow may be assumed to have the same characteris-
                           NACA 0018                                                                     tics as in the real rotating case.
         2                                                     1.5
                                                                               Sandia Labs [10]
                                                                                                             As mentioned above, both software produce simi-
        1.5                                                                    COMSOL results            lar results. Fluent results seem to be more accurate,
    c
     D
         1                                                      1
                                                                                                         but this is probably due to our greater experience in
                                                                                                         using Fluent. The run-time also seems to favour the
        0.5
                                                                                                         use of Fluent for more complicated problems (i.e.
         0
                                                               0.5                                       greater amount of grid cells), but as the two computers
              0       50         100   150     200
                             α [grd]                    c
                                                           L                                             used for the numerical simulations are far from having
        1.5
                                                                                                         the same computational power, this statement must
                                                                                                         be observed with reserves.
                                                                0

         1
                                                                                                             On the other hand, for profiles that can be generated
cL 0.5
                                                            −0.5                                         from mathematical known equations (like the NACA
         0
                                                                                                         profiles), the interoperability between COMSOL and
    −0.5
                                                                                                         MATLAB software gives a plus on the geometry
        −1
              0       50         100   150     200
                                                               −1
                                                                     0   0.5        1        1.5     2   generation aspect for COMSOL.
                             α [grd]                                               cD
                                                                                                             Although global results with respect to the pressure
   Figure 12. Drag and lift coefficients versus the                                                      coefficients on the airfoils agree well with experi-
  angle of attack (left images), and polar diagram                                                       mental static results, it is obvious that such compu-
(right image) for the NACA 0018 airfoil: numerical                                                       tational approaches cannot predict the dynamic stall
  results in COMSOL and experimental data [10]                                                           phenomenon that is reported in the case of vertical
                                         COMSOL Multiphysics results
                                                                                                         axis cross-flow turbines. Moreover, 2D simulations
        0.8                                                                                              can be acceptable for blades with constant cross-
        0.6
                                                       NACA 4518
                                                       NACA 0018
                                                                                                         sectional profiles along the z-axis, but do not permit
    c
     D
                                                                                                         an accurate description of the flow for vertical axis
        0.4
                                                                                                         cross-flow turbines with varying blade cross-section
        0.2                                                                                              along the z-axis like the delta blade of the Achard
          0
                                                                                                         turbine.
         −30               −20         −10             0
                                                     α [grd]
                                                                         10             20          30
                                                                                                             Those last two aspects are to be considered in
                                                                                                         further work performed by our team.
        1.5

         1

        0.5
                                                                                                         ACKNOWLEDGMENTS
c
 L
         0                                                                                                  Authors gratefully acknowledge the CEEX Pro-
    −0.5
                                                                                                         gramme from the Romanian Ministry of Education
        −1
                                                                                                         and Research, for its financial support under the
    −1.5
      −30                  −20         −10             0
                                                     α [grd]
                                                                         10             20          30   THARVEST Project no. 192/2006. Special thanks
                                                                                                         are addressed to Dr Jean-Luc Achard, CNRS Research
    Figure 13. Variations c D = c D (α ) and c L = c L (α )
                                                                                                         Director, and to PhD student Ervin Amet from LEGI
          for the NACA 4518 and 0018 airfoils                                                            Grenoble, France, for consultancy and documentation
                                                                                                         on the Achard turbine.
22              Proceedings of the 3 rd Workshop on Vortex Dominated Flows. Achievements and Open Problems, Timisoara, Romania, June 1 - 2, 2007


                                                                         7. Maître T., Achard J-L., Guittet L., Ploeşteanu C. (2005)
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