Panacm 2015 paper

1. 1st Pan-American Congress on Computational Mechanics - PANACM 2015
XI Argentine Congress on Computational Mechanics - MECOM 2015
S. Idelsohn, V. Sonzogni, A. Coutinho, M. Cruchaga, A. Lew & M. Cerrolaza (Eds)
STEADY AND UNSTEADY ANALYSIS OF
AERODYNAMICS WING SECTIONS AT ULTRA-LOW
REYNOLDS NUMBERS (RE < 10000)
DINO P. ANTONELLI1
, CARLOS G. SACCO2
AND JOSE P. TAMAGNO3
1Universidad Nacional de Cordoba y CONICET
Av. Velez Sarfield 1611, PC 5000 Cordoba, Argentina
dinoantonelli@hotmail.com
2Instituto Universitario Aeronautico
Av. Fuerza Aerea S/n, PC 5000 Cordoba, Argentina
csacco@iua.edu.ar
3Universidad Nacional de Cordoba
Av. Velez Sarfield 1611, PC 5000 Cordoba, Argentina
jtamagno@efn.uncor.edu
Key words: aerodynamic wing sections, ultra-low Reynolds, CFD, steady and unsteady
flows.
Abstract. The purpose of this study is to describe phenomena that manifest them-
selves in flows where Reynolds numbers are ultra-low (Re < 10000). To accomplish
this study, mathematical techniques capable of solving the Navier-Stokes equations for
laminar-incompressible flows are used. It is noted that a solver based on the Finite
Element Method provides an appropriate resolution procedure, however, it must also be
noted that because of the incompressible assumption the character of the continuity equa-
tion goes from hyperbolic to elliptic. Because of this, a Fractional Step method which
evolves toward a semi-implicit temporal integrator is used, and to handle the convective
and pressure terms the so called Orthogonal Sub-grid Scale(OSS) algorithm is applied. In
addition, the motion of the finite elements computational mesh through solving the Pois-
son equation and optimizing each element metric, is implemented. Basic useful results
describing the behavior of several 2D geometries at steady ultra-low Reynolds flows, are
presented. Different geometric parameters like thickness ratio, mean lines camber, shape
of leading edge, etc. are changed and its effects evaluated. Flow detachment features
and their impact on main aerodynamic properties are assessed. Wing sections performing
typical unsteady flights like heaving, pitching, flapping and hovering are also analyzed,
and its aerodynamic properties in terms of Strouhal numbers, reduced frequencies and
Reynolds numbers determined.
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2. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
1 INTRODUCTION
The importance of ultra-low Reynolds flows lies in technology applications like MAVs
(Micro Air Vehicles). Numerous cases require a deep understanding of the present phe-
nomena in both steady and unsteady flights, to obtain maximum propulsive and handling
efficiencies. Basically due to Reynolds number effects, aerodynamic characteristics such
as lift, drag and thrust of a flight vehicle change considerably between MAVs and con-
ventional manned air vehicles. In fact, in the nature, birds or insects flap their wings
interacting with the surrounding air to generate lift to stay aloft or producing thrust to
fly forward. The main powered flights are: flapping (flight with free stream) and hovering
(flight without free stream).
Much research in this broad area have been made. The most significant that can
be named are: Kunz [8] in his thesis studied the behavior of different geometries in
steady flows at ultra-low Reynolds number; Guerrero [7] carried out unsteady aerodynamic
studies at ultra-low Reynolds in 2D and 3D configurations built using the NACA 0012
wing section; Pedro et al [10] with the purpose of studying the propulsive efficiency of
flapping hydro-foil NACA 0012 at Re = 1100, (flow density ρ = 1kg/m3
and dynamic
viscosity µ = 0.01kg(m.s) ), also carried out numerical simulations. A Finite Volume
Technique with an additional equation for the pressure, an explicit temporal scheme and
structured grid, were used.
Ranges of non-dimensional numbers found relevant to unsteady flights of biological
”flappers”, are also considered valid for MAVs. A characteristic one for flapping motions
is the Strouhal number St = 2fhha/U, where fh is the frequency and ha the amplitude.
Therefore, the Strouhal number expresses de ratio between the flapping wing velocity
and the reference velocity U. The reduced frequency given bay k = πfc/U is another
parameter that can be interpreted as a measure of unsteadiness comparing the wave length
of the flow disturbance to the chord c.
2D unsteady flow sinusoidal kinematics are characterized by the equations:
h(t) = hasin(2πfht + φh) (1)
α(t) = αasin(2πfαt + φα) (2)
where φh and φα are the phases angles.
2 NUMERICAL SIMULATION
2.1 Governing equations
The two-dimensional time-dependent Navier-Stokes equations are solved using the fi-
nite element method, assuming incompressible-laminar flow which is justified since the
Mach number of MAV flight is M << 0.3 and the Reynolds number Re < 10000. Con-
servation of mass and momentum are described by:
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3. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
.u = 0 in Ω × (t0, tf ) (3)
∂(u)
∂t
+ (u. )u = −
1
ρ
p + ν 2
u + fe in Ω × (t0, tf ) (4)
where Ω represents the analisys domain with boundaries Γu Γσ, whereas (t0, tf ) is time
interval of analisys. The u is the two-dimensional flow velocity vector, ρ the constant
density, ν the kinematic viscosity and p the pressure.
To represent the unsteady flow, eqs. 3 and 4 are solved in a fixed inertial reference
frame incorporating a moving mesh following the Arbitrary Lagrangian Eulerian (ALE)
formulation [6]. This method combines the advantages of both the Lagrangian and Eu-
lerian approaches. In the Lagrangian approach the computational mesh is moved such
that the nodes follow material particles during motion. The computational mesh is fixed
and the fluid moves with respect to the mesh. The ALE method incorporates a moving
mesh using the Lagrangian method, where the mesh follows the motion of the geometry
boundary, whereas the equations are solved using the Eulerian approach.
In order to obtain the ALE equations the velocity u in the convective term of the
momentum equation needs to account for the mesh motion. Therefore the velocity of the
mesh um is subtracted from the flow velocity in the convective term. Then the Navier
-Stokes equations in ALE formulation are obtained by:
.u = 0 (5)
∂(u)
∂t
+ (c. )u +
1
ρ
p − ν 2
u − fe = 0 (6)
where c = u − um is the convective velocity that represent the difference between fluid
velocity and mesh velocity.
In the present work, the algoritm of mesh movement is based in operations of opti-
mization of smoothing, developed to the R ANSYS software package [3].
2.2 Fractional Step algorithm
The equations previously presented can’t be solved by a numerical standard form be-
cause incompressibility gives raise to a flow field restriction. There are several algorithms
to deal with this difficulty and the Fractional Step method is one of them. The method
meets the LBB condition through the use of same order of approximation for velocity and
pressure.
To apply the Fractional Step algorithm the momentum equation is divided in two parts:
ˆun+1
= un
+ δt un+θ
. un+θ
+ γ
1
ρ
pn
− ν n
un+θ
+ fn+θ
(7)
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4. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
un+1
= ˆun+1
−
δt
ρ
( pn+1
− γ pn
) (8)
and in the last two equations a new variable ˆu known as fractionary momentum, is
introduced. If in eq. 8 the divergence is taken and the continuity equation is applied,
results:
2
(pn+1
− γpn
) =
ρ
δt
.ˆun+1
(9)
Through this equation the pressure is calculated. In addition, γ is a numerical parameter
such that its values of interest are 0 and 1. The θ parameter determine the kind of
temporal approximation.
2.3 Discret form of equations
The Finite Element Method is used to discretize the govern equations and provides an
appropriate resolution procedure [9]. The resultant scheme is of first order (γ = 0) and
the temporal discretization (θ = 0) results in Euler forward. The test functions (ψh, φh)
∈ Ψh × Φh are used such as 1
:
1
δt
(ˆun+1
h , ψh) =
1
δt
(un
h, ψh) − (un
h. un
h, ψh) − ν( un
h, ψh) − (fn
e , ψh) (10)
( pn+1
h , φh) =
ρ
δt
(ˆun+1
h − un
h, φh) − ( un
h, φh) (11)
(un+1
h , ψh) = (ˆun
h, ψh) −
δt
ρ
( pn+1
h , φh) (12)
The last equations system is semi-implicit because eqs. 10 and 12 are explicit (lumped
mass matrix) and eq. 11 for the pressure computation is implicit.
2.4 Stabilized scheme
The discretization of convective terms yields numerical instabilities, therefore stabiliza-
tion methods must be used. In this work the Orthogonal Subgrid Scale (OSS) algorithm
is applied [4],[5],[11]. The expresion for the convective stabilization term is:
STBu = τ1(un
h. un
h − πn
h, un
h ψh) (13)
where πn
h is the convective term proyection and it is defined in eq. 18. This equation
add to momentum eq. 10 and it is evaluated in tn
, therefore it remains explicit.
The term stabilization of pressure to be added to the eq. 11 is:
1
The notation used in the equations mean: (a, b) = a.bdΩ
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5. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
STBp = −(τ2( pn
− ξn
h ), φh) (14)
where ξn
h is the gradient pressure term proyection and it is defined in eq. 19. In addition,
it is evaluated in tn
, therefore it remains explicit.
The complete stabilized scheme is obtained:
1
δt
(ˆun+1
h , ψh) =
1
δt
(un
h, ψh) − (un
h · un
h, ψh) − ν( un
h, ψh) − (fn
e , ψh) − (15)
−(τ1(un
h · un
h − πn
h), un
h · ψh)
( pn+1
h , φh) =
ρ
δt + τ2
(ˆun+1
h − un
h, φh) − ( un
h, φh) +
τ2
δt + τ2
( ξn
h , φh) (16)
(un+1
h , ψh) = (ˆun
h, ψh) −
δt
ρ
( pn+1
h , φh) (17)
(πn
h, ψh) = (un
h. un
h, ψh) (18)
(ξn
h , ψh) = ( pn
h, ψh) (19)
where ψh ∈ Ψh. The system of equations of eqs. 15, 17, 18, 19 are solve in explicit
form with lumped mass matrix and the system resultant of eq. 16 is solve in explicit form
through of conjugate gradients with diagonal pre-conditioner.
It is noted that the formulation of the scheme isn’t in the ALE framework. To ac-
count the mesh velocity is necessary introduce in convective and stabilizations terms, the
convective velocity c.
Finally the boundary conditons in viscous tensor and velocity are:
• Imposed velocity: u = uc
• No slip: u = 0
• No traction: n.¯σ.n = 0
3 VERIFICATION OF NUMERICAL CODE
For the cases studies included in Table 1 the following parameters are considered:
pitching and heaving frequencies fα = fh = 0.225[Hz], reducy frequency k = 0.7096,
maximum heaving amplitude ha = 1, phase angle ϕ = 90◦
, Strouhal number St = 0.45
and the variable parameter is the pitching amplitude.
From Table 1 it can be concluded that the results obtained in this work compare well
with those given by [10] and [7], up to αa = 15◦
, but no so much for greater angles
αa = 20◦
and αa = 25◦
. In this cases computed values from this work tend to overpredict
results given by the other autors. Specific investigations about the reason for differences
were not made.
5

6. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
Pedro et al.[10] Guerrero[7] Present Work
αa
¯ct CLmax
¯ct CLmax
¯ct CLmax
5◦
0.4324 8.3333 0.4245 8.0828 0.4311 8.2078
10◦
0.6511 7.4834 0.6576 7.1699 0.6556 7.2400
15◦
0.8226 6.6307 0.8360 6.5435 0.8246 6.3904
20◦
0.9337 5.8176 0.9389 6.1133 0.9960 5.5113
25◦
1.0046 5.0558 0.9601 5.6080 1.0900 4.9910
Table 1: Average thrust ¯ct and maximum lift CLmax coefficients. Comparison for flapping NACA 0012
cases between present work, Refs. [10] and [7].
4 RESULTS AND DISCUSSION
4.1 Steady analisys of aerodynamic airfoils
Figure 1: Lift and drag coefficients for thickness ratios and camber ratios over standart NACA 4 digit
airfoils. (a) and (b) thickness ratio effects. (c) and (d) camber ratio effects. (The inviscid curve is from
[8]).
To analize how the thickness ratio behaves in ultra-low Reynolds flow, comparisons
between four digits symetric NACA airfoils (0002, 0006, 0008) at two Reynolds numbers
(Re = 2000 and Re = 6000) are carried out. Results are presented in Fig. 1.a and 1.b.
Consider first the 0006 and 0008 thickness airfoils. In the quasi-linear ideal range of angles
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7. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
of attack (0◦
− 6◦
) for a symmetric airfoil (Fig.1.a), a very important reduction in lift can
be observed. If the Reynolds number is decreased the lift slope tends to improve, but such
improvement is comparatively less than the decrease due to thickness. However, in NACA
0002 airfoil the lift slope becomes greater when the Reynolds is higher (Re = 6000). It
appear that his behavior is related to a lower leading edge suction peak and subsequent
delayed stall. Note from Fig. 1.a that as the thickness ratio get smaller, the closer
the viscous results get to the ideal values. Fig. ??.b shows the strong increase of drag
coefficients as the Reynolds number decreases.
To analize the camber behavior a comparison between NACA 2302, 4302 and 6302 is
performed. Lift results of numerical simulations plotted in terms of α − α0, being α0 the
zero lift angle, are shown in Fig. 1.c. The 6302 airfoil lift slope, is 30% greater than 2302
airfoil. If CL = 0, note the large increment of CD as the camber increases Fig. 1.d, it can
be attributed to leading edge early detachments.
4.2 Unsteady analisys of aerodynamic airfoils
Figure 2: Average thrust coefficient ¯ct and propulsive efficiency η for NACA airfoils (0004, 0006 and
0012) in heaving motion. Plots (a) and (b) apply to f = 1[Hz]. Plots (c) and (d) apply to f = 2[Hz].
The fundamental parameter of unsteady analysis is the Strouhal number, defined as
St = 2fha/U. Taylor et al. [12] and Triantafyllou et al. [13] performed a study of wing
frequencies and amplitudes, and cruise speeds across a range of birds, insects, fishes and
cetaceans, to determine Strouhal numbers in “cruising” flight. They found 75% of the 42
7

8. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
species considered fall within a narrow range of 0.19 < St < 0.41 [7]. Therefore, a similar
range of Strouhal numbers has in this work been selected.
4.2.1 Heaving
The kinematic of heaving motion is given by eq. 1. The analysis is applied to NACA’s
four digits (0004, 0006 and 0012) symmetrical airfoils. The kinematics parameters are:
two values of heaving frequencies fh and a variable Strouhal number throughout the
heaving amplitude ha and Re = 1100. The average thrust coefficient ¯ct and propulsive
efficiency η = ¯ct/ ¯cp (where ¯cp is the power coefficient input), are in terms of the Strouhal
number presented for fh = 1[Hz] and fh = 2[Hz] in Fig. 2.a and 2.b, and in Fig. 2c and
2d respectively. Note the numerical results obtained by Guerrero [7] in Fig. 2.a and 2.b.
Figure 3: Comparision at differents times of velocity contours between NACA 0004 and NACA 0012
airfoils (fh = 1[Hz] and St = 0.3).Times (a) and (e) t = 0.45[s], (b) and (f) t = 0.86[s], (c) and (g)
t = 1.29[s], (d) and (h) t = 1.64[s].
The flow motion topology expressed by Figure 3, helps to understand the results of
the simulations. Therefore, comparisons at different times of velocity contours between
NACA 0004 (Fig. 3 a,b,c,d) and NACA 0012 (Fig. 3 e,f,g,h) are shown. The formation
of leading edge vortices (LEV) and its convection into the wake, can be observed [1].
4.2.2 Flapping
The name flapping is applicable to a combined motion of heaving and pitching, conse-
quently the kinematics relations given by equations 1 and 2 are simultaneously applied.
In Fig. 4a and 4b are shown average thrust coefficients ¯ct and propulsive efficiencies
η applicable to NACA symetric airfoils 0004 and 0012, as function of the pitching angle
8

9. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
5◦
< αa < 25◦
with fα = fh = 0.3, ha = 0.5[m], ϕ = π/2 and St = 0.3. In Fig. 4c and 4d
are shown average thrust coefficients ¯ct and propulsive efficiencies η applicable to NACA
symmetric airfoils 0004 and 0012, as function of the heaving amplitude 0.025 < ha < 0.5
(0.05 < St < 1) with fh = fα = 1, αa = 15◦
and ϕ = π/2.
Figure 4: Average thrust coefficient ¯ct and propulsive efficiency η in heaving motion for thickness ratio
variation in NACA 0004, 0012 at Re = 1100. (a) and (b) pitching amplitude variable αa. (c) and (d)
heaving amplitude variable ha.
4.2.3 Hovering
The kinematics relations in hovering flight are given by equations 1 and 2 simulta-
neously applied. The kinematic parameters utilized in the simulation of hovering are:
ha = 0.5, fα = 0.75, fh = 0.75 and ϕ = π/2 and the Reynolds number is defined by
Re = 2fhρπhac/µ because the free stream velocity is null. Average lift coefficients ¯cl
and η efficiencies were obtained simulating the hovering of a NACA 0012 airfoil, and are
plotted as function of the Reynolds number in Figures 5a and 5b. The Reynolds number
covers the range 100 < Re < 1000.
On the other hand the wake topology is studied in Fig. 6. It can be observed the
LEV and TEV (trailing edge vortex) at Re = 150, αa = 20◦
and ϕ = π/2 , caused
by some stroke and its motions indicated by arrows. Three typical mechanism needed
to understand the behavior of hovering are present: wake capture and diffusion effects,
dynamic stall and roll-up effects.
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10. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
Figure 5: Hovering motion over NACA 0012 airfoil with ha = 0.5, fα = 0.75, fh = 0.75 and ϕ = π/2.
(a) average lift coefficient ¯cl. (b) efficiency η hovering motion for two pitching amplitudes.
Figure 6: Velocity contours in hovering motion to NACA 0012 at Re = 150, αa = 20◦
and ϕ = π/2. (a)
t = 0.6s, (b) t = 0.92s, (c) t = 1.20s, (d) t = 1.64s, (e) t = 2.00s, (f) t = 2.26s. (LEV C and TEV C are
the previous stroke vortexes to capture and LEV N the new vortex generated).
5 CONCLUSIONS AND FUTURE WORK
What has been sought with this work is the confirmation that current computational
knowledge can to the ultra-low Reynolds number applications (Re < 10000), be extended.
The main assumptions made about the flow field are: two dimensional incompressible,
fully laminar steady and unsteady flows. The fully laminar assumption is the most phys-
ically accurate in the range of Reynolds numbers and angles of attack of interest here.
10

11. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
5.1 Steady Analysis
Two dimensional analysis allows a broad spectrum of parameters to be considered and
provides a baseline for future more detailed studies. The geometric parameters investi-
gated included thickness, camber etc, and it was intended to address the fundamental
question of whether section geometry is still important at ultra-low Reynolds number.
The most obvious effect of operating at ultra-low Reynolds numbers is a large increase
in the section drag coefficient, however, the increase in drag is not reciprocated in lift,
resulting in a large reduction in the L/D.
A ultra-low Reynolds numbers flow, is dominated by viscosity and the so called Bound-
ary Layer concept is no longer applicable. Here it is generalized as the lower velocity
viscous flow region adjacent to the body over which the pressure gradient normal to the
surface is almost null. The extended constant pressure from the surface implies that the
wing section effective geometry is significantly altered. As a result the pressure recovery
is reduced and besides impacting on drag, at positive angles of attack the large effect is on
lift. Viscous effects in thin wing sections thickness, significantly reduce the leading edge
suction peak and the associated reduction in slope of the adverse pressure recovery, delays
the onset of the stall. It can be stated that leading edge separation is delayed in thin
wing sections and trailing edge separation is delayed in thicker sections. This behaviour
could be potentially beneficial to lifting performance.
The effects of camber do not differ significantly from those observed at much higher
Reynolds numbers. The fact that as the Reynolds numbers and section maximum thick-
ness are reduced the details of the thickness distribution becomes less relevant, it allows
to conclude that the camber-line is the dominant factor in performances.
5.2 Unsteady analysis
The highlight of the Finite Element software here used, is the ability to create mobile
grids needed to simulate unsteady flights like heaving, flapping and hovering.
Symmetric wing sections are considered in studying the heaving motion. Average thrust
coefficients and propulsion efficiencies are computed for given motion frequencies, and are
plotted in terms of a Strouhal number determined using the amplitude of heaving. As a
help for understanding the simulation results, figures are shown where velocity contours
for two wing sections are compared at different times. The generation and displacement
of vortices as the wing section executes the heaving motion, are well described.
Combinations of pitching and heaving motions (flapping) were simulated for symmetric
wing sections, and thrust coefficients and propulsive efficiencies determined. Maximum
pitch and vertical displacement amplitudes were taken as variables of plotting.
Hovering, is perhaps the type of flight that capture the greatest interest in the develop-
ment of MAV. Average lift coefficients and power efficiencies are obtained simulating the
flight of a symmetric wing section and the results plotted in terms of an ad-hoc Reynolds
number defined taking into account that there is no free-stream. Classical aspects of
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12. Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno
the hovering flight (leading and trailing edge already generated vortices, its capture and
diffusion by the wake, and corresponding new generation), are shown.
5.3 Future area of research
So far, the fluid dynamic studies have been conducted and applied to a rigid 2D model.
It is intended to extend first, the ultra-low Reynolds number area of research to 3D rigid
finite span lifting wings, to the development of the viscous flow region adjacent to the
surface, to describe stall patterns and wakes coming forth. Later, the rigid wing will be
replaced by an elastic model and the coupling fluid-structure accounted for.
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