Seminar on

Low Reynolds Number Aerodynamics

Syed Mohsin
(22ETRP700001)

07-01-2023

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 The presentation includes the following details:
 
• Motivation
• Introduction
• Low Reynolds Number
(i) Aerodynamics
(ii) Laminar separation Bubble
(iii) Computational study
• Outcomes and Future Scope
• References
 
 

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 Motivation
Due to advances in energy, actuator and sensor technologies, it is possible to build
small flying devices with mean chords of 5 cm flying at about 10 m/s capable of
performing missions such as environmental monitoring, surveillance and assessment in
hostile environments.

Planes similar to these have been built and flown for nearly a decade, However, these
flying vehicles still have small endurance and range compared to other flying vehicles,
are quite inefficient, and have low payload to airframe weight ratios.

Figure 1: flight Speeds v/s Reynolds-Number of Aircrafts

Low Re number flows are seen on mini, micro and unmanned air vehicles, wind
turbine blades, model aircrafts, birds and little creatures like bees or flies.
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 Introduction

Reynolds Number (Re) is the most important dimensionless number in fluid

dynamics. It is the ratio of inertial forces to viscous forces and is given by the
formula:
VD
Re 


where ρ = density of the fluid, V = velocity, D = pipe diameter, and μ = fluid viscosity.

The Reynolds number (Re) helps predict flow patterns in different fluid flow

situations. At low Reynolds numbers, flows tend to be dominated by laminar flow,
while at high Reynolds numbers flows tend to be turbulent.

[for flow through pipe] [over a flat plate]

Re < 2000, the flow is Laminar Re < 5x10^5, the flow is laminar
2000 < Re < 2900, the flow is Transitional 5x10^5 < Re < 10^7, the flow is
Transitional
Re > 2900, the flow is Turbulent Re > 10^7, the flow is Turbulent

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 (i) Aerodynamics of Low Reynolds number

Title: Aerodynamik des Flugmodells

Year: 1942
Author: Franz Wilhelm Shmitz
Conclusion:
The earliest rigorous study of aerodynamics within this range of Re was conducted by
Schmitz during the 1940’s.

Using a wind tunnel he measured the forces generated by airfoils in the range
2x10^4<Re<2x10^5. He focused his study on three airfoil shapes: a thin flat plate, a
thin cambered plate and a thick cambered airfoil (N60).

His studies showed that the thick cambered airfoil has a critical Re where the
performance changes drastically. Above the critical Re range, the lift to drag ratio is
much higher than below.

When an airfoil is below its critical Re range, the flow is dominated by viscous forces
and remains laminar over the entire airfoil.
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 Figure a. Plot for the N60 airfoil across a range of Re. In this plot, the lift coefficient and drag coefficient are
called ca and cw, respectively (the German convention).

Above the critical Re range, the lift-to-drag ratio is much higher than below. When an
airfoil is below its critical Re range, the flow is dominated by viscous forces and
remains laminar over the entire airfoil.
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 Title: Low Speed Single Element Airfoil Synthesis
Year: 1979
Author: Mc Masters, J.H and Henderson, M.L
Conclusion:

Figure 2. Maximum Lift-to-Drag vs Reynolds number

McMasters plot shows that, smooth airfoils have a higher lift to drag ratio than rough
airfoils at high Re, thus Roughness causes a decrease in performance at Re above 10^5.
Below this Re, roughness is beneficial. Smooth airfoils have a large decrease in lift to drag
ratio, while rough airfoils perform nearly as well as they did at higher Re.

However, the influence of drag on parameter variation, boundary layer momentum

thickness, transition point were still to be examined thoroughly

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 Title: Experimental studies of separation on a two-dimensional airfoil at low
Reynolds numbers.
Year: 1982
Author: Mueller, T.J and Batill, S.M
Conclusion:

(a) At Re 4x10^4 ( (b) At Re 4x10^5

Figure 3. Lift curves for NACA 663-018 airfoil (Mueller)

The sudden change in lift at

α ≈ 8◦ and Re = 4 × 10^4 might be related to the change in performance indicated
in
Figure 2. The same airfoil at much higher Re has much larger lift coefficients, 8 8
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 Mueller conducted both visualization and force measurement experiments to
understand the cause for this performance decrease in low Re. His studies focused on
measuring the lift and drag forces on the NACA 663-018 airfoil at
4x10^4<Re<4x10^5.

The lift measurements made at Re =4x10^4 show a dramatic change at α=8 deg. At
Re>10^5 however, the lift coefficient was found to increase linearly with α.

Using smoke visualization, Mueller showed that the drastic change in lift coefficient
at Re=4x10^4 and α=8 degree is due to the formation of a Laminar separation bubble.

Figure a. Smoke flow visualization at Re-40,000 Figure b. Smoke flow visualization at Re-
400,000
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 It seems likely that the decrease in performance as Re decreases explained by
McMasters is related in some way to the formation of the discontinuity in the lift vs
angle of attack plot shown by Mueller.

Due to the limitations of the force balance, the drag force measurements were only
made at Re > 10^5.

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 Title: Airfoils at Low speeds
Year: 1989
Author: Selig M.S and Donovan, J.S
Conclusion:
Another series of extensive airfoil studies at low Re was conducted by Selig starting in
1986 at Princeton University. In this study, lift and drag were measured for 60 airfoils
primarily at 6 × 10^4 < Re < 3 × 10^5.

The lift was measured directly using a strain guage force balance, while the drag was
estimated by the wake measured by a pitot tube traversed vertically through the wake.

The experiments were primarily concerned with improving the endurance and range of
sail planes (non powered airplanes) that operate in this range of Re

The data obtained in this study showed that at Re > 1 × 10^5, each airfoil has a drag
polar that is similar to drag polars at all higher Re. That is, the drag coefficient is low for
a large range of lift coefficients.
However, as Re decreases below this number, many of the airfoils have a significant
increase in drag coefficient at moderate lift coefficients.
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 This is seen in figure 4, where the drag polar for the Eppler 387 (E387) airfoil, as
measured by various facilities, is plotted across a range of Re.
The E387 airfoil is chosen for comparisons because it has been studied by more
researchers at these Re than other airfoils. It is also a championship airfoil at sail plane
competitions. It was designed by Richard Eppler to have a very high lift-to-drag ratio at
Re ≈ 5×10^5.

Figure 4. Drag polar of Eppler 387 airfoil at various Re.

The drag polar of the E387 airfoil at 6 × 10^4 < Re < 3 × 10^5 as measured by different
facilities. The shape of the E387 airfoil is also shown. 1212
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 (ii) Laminar Separation Bubble

The most common explanation for the unusual behavior of airfoils and wings at
these low Re (10^4 < Re < 10^5) is the existence of a laminar separation bubble
(LSB) at certain α.

Selig claims that the LSB causes the peculiar drag increase at moderate lift
coefficients.

Figure 5. Laminar separation bubble

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 A typical LSB is demonstrated in figure 5.

It typically begins with a laminar boundary layer that encounters an adverse pressure
gradient, which causes the boundary layer to separate.

The laminar separated shear flow is unstable and transitions to a turbulent separated
shear flow. The turbulence then transports momentum from the free-stream, across
the shear layer, and down towards the surface.

When the momentum transport is sufficient, the turbulent boundary layer is

considered to be reattached to
the surface, thus closing the separation bubble.

The LSB can be classified as short and long.

Both short and long bubbles have negative effects on aerodynamic performance.
These negative effects may increase drag and decrease lift owing to the altered
pressure distribution caused by the presence of the LSB.

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 In order to improve the aerodynamic performance, there are new methods being
developed
to eliminate the effects of the LSB, besides the high lift devices.

The effects of the LSB has been investigated by means of various experimental methods,
such as force measurement, velocity measurement by using hot-wire anemometry and
particle image velocimetry (PIV), pressure measurement with pressure transducers, flow
visualization with smoke & oil.

These systems are useful and accurate but also expensive. Therefore investigating all
kind of aerodynamic phenomena via Computational Fluid Dynamics (CFD ) is more
suitable and appropriate.

By using CFD, the flow characteristics of a wing profile or the device (UAV, MAV, wind
turbine) can be easily analyzed.

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 Title: Airfoils at Low speeds
Year: 1989
Author: Selig M.S and Donovan, J.S
Conclusion:
Selig et al mentions that the LSB is responsible for the drag coefficient increase at
moderate lift coefficients when the Eppler 387 airfoil is at Re ≈ 6 × 10^4 (Figure.4)

He states that the effects of a laminar separation bubble are apparent only at Re 6 ×
10^4, where the drag coefficient reaches a maximum of 0.032. He called this drag
increase the “bubble drag” because it is caused by the LSB.

However none of his studies at that time had investigated the boundary layer of the
airfoil tested, and thus no direct evidence of his conclusion could be found.

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 Title: Wind tunnel aerodynamic tests of six airfoils for use on small wind turbines

Year: 1998
Author: Selig, M.S and McGrahanahan, B.D
Content
Later tests by Selig et al used surface oil flow visualization techniques to discern the
existence and size of the laminar separation bubble.

This technique is used to visualize the time-averaged flow properties near the surface.
With it, one can distinguish steady-state laminar and turbulent flow regions,
separation/reattachment points, and the transition regions of a boundary layer.

The tests conducted clearly show that there is a laminar separation point, and that there
is a turbulent reattachment point. However, data were only shown for the relatively
high Re of 2 × 10^5 and 3 × 10^5.

The ”bubble drag” is only discernable for Re ≤ 1×10^5, Re at which surface oil flow
visualization is not available.
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 Selig et al claim that the LSB is responsible for the drag coefficient increase at
moderate lift coefficients.

However, there were no flow field measurements of the boundary layer to support such
findings, only surface oil flow visualizations. Further, there were no flow
measurements at all for the airfoil at Re ≤ 1 × 10^5, where the drag increase is more
apparent.

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 Title: Flow field measurements over an airfoil during natural low-frequency
oscillations.
Year: 2000
Author: Broeren, A.P and Bragg, M.
Conclusion:
Broeren and Bragg conducted detailed studies of the LSB at low Re and related the LSB
to the stall type.
At Re=3 × 10^5, they measured the time -dependent growth and decay of the LSB on
the upper surface of the LRN(1)-1007 airfoil at α = 15◦.Measurements were made using
2-component Laser Doppler Anemometry with measurements made within 0.2mm of
the surface.

By measuring the flow field around the wing: the LSB started small, then grew until it
reached the trailing edge causing full separation, then suddenly became small again.

They also classified airfoils at this Re as having 5 common types of stall : Trailing Edge,
Leading Edge, Trailing Edge/Leading Edge, Thin Airfoil, and Thin Airfoil/Trailing
Edge.

The studies carried out were performed at angles close to stall to study the behavior of 1919
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 Title: Comparison of laminar separation bubble measurements on a low Reynolds
number airfoil in three facilities.
Year: 2005
Author: Ol, M. V., Mcauliffe, B.R., Hanff, E.S and
Conclusion:
A detailed comparison between time-averaged measurements of an LSB was done by Ol
et al.

Experiments were performed at 3 facilities: a tow tank at the Institute for Aerospace
Research, a wind tunnel at the Technical University of Braunschweig, and a water
channel at the Air Force Research Lab. Each facility used PIV to measure the flow field
around the SD7003 airfoil at α = 4 deg and Re = 6×10^4.

The resolution of the PIV system was adequate to resolve the boundary layer, and hence
the LSB that is present
at the required angle of incidence.

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 While there is an LSB for this airfoil at these conditions, and while it is at an Re < 10^5,
the drag polar of the SD7003 has no characteristic ”bubble drag” such as the E387 shown
in figure 4.

The drag polar of this airfoil, as found is shown in figure 6 at various Re, including the
one investigated by Ol et al.

Figure 6. Drag polar of a SD7003 airfoil at various Re.

It seems that the LSB measured by Ol et al does not cause a drag increase at moderate lift
coefficients. Thus the separation bubble is having very little effect on the shape of the
drag polar of the airfoil, and may not be the explanation for the drag increase at moderate
lift coefficients found for the E387 at Re = 6 × 10^4. 2121
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 This paper provides limited information on the bubble growth and how it effects the
forces generated by the wing.

Also, no force measurements were made, and it is not clear how the LSB effects the
forces generated by the airfoil

Comparison of nominally identical experiments in three very different facilities - a

water tow tank, a wind tunnel and a water tunnel – shows encouraging qualitative
similarity in the bubble shape and velocity fields, as well as Reynolds stress
distributions. However, discrepancies in the measured location and flow structure of the
bubble remain.

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 (iii) Computational Study

Title: Design Optimization and Investigation of Aerodynamic Characteristics of Low

Reynolds Number Airfoils
Year: 2021
Author: Ali Arshad., Brando Rodrigues and Inigo Martinez.
Conclusion
This paper aims to develop a simple and efficient design optimization methodology for the
low Reynolds number airfoils.

XFOIL is used as an aerodynamic solver while modeFRONTIER workflow is employed

for the design optimization purpose.

The airfoil SG6043 is used as the reference airfoil for optimization due to its common
applications when long-endurance characteristics are desired.

A simple design optimization methodology with the integration of XFOIL in the

modeFRONTIER workflow environment is proposed in this paper.
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 The optimization results are further compared with the results of the numerical
simulations.

The development of new airfoils in this work is composed of the following steps (4
steps)

1. The first step is the selection of typical low Re airfoil for a specific application such
as long endurance.
The airfoil SG6043 is chosen as the reference airfoil based on its common application in
wings when a long-endurance flight is desired.

2. The second step is the validation of the use of XFOIL as a reliable software for
aerodynamics characteristics prediction.
This is achieved by comparing the results of the data generated by XFOIL for two
airfoils. Airfoils SG6043 and E387 are considered for the investigation because of their
widely available wind tunnel results data.

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 3. In the third step, the optimization of the airfoil geometry will be performed.
Since optimization is done on the basis of Cl, Cd and Cm, XFOIL is used as the
aerodynamic solver. modeFRONTIER is used as the optimization software which is
coupled with XFOIL.

4. The fourth and final step is the validation of the newly generated airfoil using
RANS based simulations in ANSYS Fluent with the transition-sensitive turbulence
model γ-Reθ, which allows the assessment of the aerodynamic performance of the
airfoil and evaluation of laminar separation bubbles with higher accuracy.

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 Work Flow
1. XFOIL Validation

Since XFOIL is utilized for design optimization purposes, as a first step, a short study
is conducted to validate the effectiveness of this software. For this, the airfoils SG6043
and E387 were selected because of their widely available wind tunnel results data.

Figure 7. presents a comparison of the experimental and XFOIL results. The results
suggest that the aerodynamics coefficients predicted by XFOIL are quite consistent
with that obtained from the wind tunnel.

Figure 7. Comparison of the results of XFOIL predictions with the experimental wind tunnel results. a Lift
and moment coefficients versus angle of attack for SG6043 at Re= 300,000; b drag polar for SG6043 at
Re= 300,000;
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 2. Design Optimization

For the optimization of the airfoil geometry, XFOIL was integrated into the
modeFRONTIER work environment as the aerodynamic solver. This integration
methodology of two software automated the iterations for several geometries of the base
airfoil i.e. SG6043.

Table 1 shows the comparison between geometric parameters of the base SG6043 airfoil
and the new optimized
airfoil.

Table 1. Geometric parameters

comparison

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 The effect of these geometric parameters can be visualized in terms of lift, drag and
pressure coefficients and boundary layer behavior as represented in Figure 8.

Figure 8. Cl versus angle of attack for

SG6043 and SG6043mod airfoils

The most notable differences between the airfoils are the SG6043mod greater values of
Cl max and stall angle i.e. 1.7968 at 17.5° against 1.6326 at 16° of the SG6043.

This increase can be explained by the increase in thickness, camber, and leading-edge
radius. Thicker airfoils also have thicker boundary layers (i.e. more turbulent), which
makes it more resistant to the adverse pressure gradient, allowing the airfoil to reach
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 3. Numerical Simulations

Figure 9 shows the plot of lift coefficient versus angle of the attack obtained by the
numerical simulations in ANSYS Fluent and compared to the results obtained by
XFOIL. Even though Fluent predicted an earlier stall in comparison to XFOIL, the
overall comparison is highly accurate.

Figure9. Lift coefficient of SG6043mod obtained by Fluent

and XFOIL

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 Table 2 summarizes the comparison between Fluent and XFOIL results

Table2. Fluent and XFOIL results comparison

The drag polar obtained from XFOIL and Fluent simulations is plotted in Figure
10.

Figure 10. Drag polar for SG6043mod, obtained by

XFOIL and Fluent
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 Summary:

In this work, design optimization of a low Reynolds number airfoil was performed by
the integration of two
software i.e. modeFRONTIER and XFOIL.

The newly generated airfoil demonstrated improved aerodynamic characteristics in

comparison to the reference airfoil.

Numerical simulations in ANSYS reconfirmed the improved results obtained from the
proposed optimization methodology.

For the low Reynolds number airfoils, XFOIL provided good agreement with the
experimental results
and proved to be a reliable tool. However, future investigations on the high lift airfoils
using XFOIL can be carried out

Further evaluation of the airfoil using wind tunnel testing can also be done.

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 Title: Basic Understanding of Airfoil Characteristics at Low Reynolds Numbers
(10^4–10^5)
Year: 2018
Author: Justin Winslow, Hikaru Otsuka, Bharath Govindarajan, and Inderjit
Chopra
Conclusion:
A computational study was conducted on various airfoils to simulate flows at Reynolds
numbers (Re) primarily between 10^4 and 10^5 to provide understanding and guidance
for MAV and other low-Reynolds-number designs.

The computational fluid dynamics tool used in this study is a Reynolds-averaged Navier–
Stokes solver with a Spalart–Allmaras turbulence model and a correlation-based laminar–
turbulent boundary-layer transition model.

The airfoils investigated in this study include NACA 0009, NACA 0012, flat plate airfoils
(1, 3, and 5% thickness), and thin cambered plates (3, 6, and 9% camber).

Airfoils were examined for lift and drag performance as well as surface pressure and flow
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 Computational method
The CFD solver used in this study to investigate low-Reynolds number aerodynamics
is the Transonic Unsteady Rotor Navier–Stokes 2D (TURNS2D)

A key feature in TURNS2D is the inclusion of a laminar–turbulent transition model

coupled with the Spalart–Allmaras (SA) turbulence model, which improves the
numerical solution by better predicting the flow transition physics.

TURNS2D Validation

To demonstrate the capability of TURNS2D, validation cases were carried out by

comparing previous experimental results to current computational results for
NACA0009 airfoil by comparing the pressure distribution on the upper surface of the
airfoil at various angles of attack at a Reynolds number of 5 × 10^4. Results for lift and
drag were also compared

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 Summary
The results from TURNS2D show the extreme sensitivity of airfoil performance to
changing the Reynolds number below 10^5, particularly for conventional airfoils such
as the NACA 0012.

As the Reynolds number decreases there is an increase in drag, particularly because of

premature flow separation and failure to reattach, resulting in a large decrease in lift
with the maximum lift coefficient decreases by approximately 46% between the
Reynolds number of 10^5 and 10^4.

As flat plate thickness-to-chord (t∕c) decreases (from 5 to 1%), lift and drag
characteristics improved, with the 1% flat plate being the most efficient at a Reynolds
number of 10^4, a consequence of lower form drag.

Examination of the boundary layer at a low Reynolds number shows significant

differences between thin plate airfoils and thicker airfoils.

The NACA 0012 is more susceptible to trailing edge separation at a low Reynolds
number compared to flat plates where there is reattachment of the boundary layer
downstream. Allowing the flat plate to provide more lift than the NACA 0012 at 3434
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 It is evident that at low angles of attack (α ≤ 5 deg ), the pressure distribution
predicted by TURNS2D agrees well with experimental data and the location and
magnitude of the pressure peak captured quite accurately.

At higher angles of attack the agreement is less satisfactory but still follows the
trend as the angle of attack increases. The incorrect prediction of the pressure curve
could be attributed to modification required in the transition model employed within
the CFD formulation for low-Re flows.

There are discrepancies in the pressure distribution between the experiments and
numerical predictions.

The performance characteristics of various airfoils investigated are in steady flow

conditions.

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 Title: Aerodynamic shape optimization of airfoils at ultra-low Reynolds numbers
Year: 2018
Author: MEEDHU GEOGY UKKEN and M SIVAPRAGASAM
Conclusion
The flow over NACA 0008 airfoil is studied computationally in the ultra-low Reynolds
number regime Re [1000, 10000] for various angles of attack α [ 0 deg, 8 deg]

The laminar flow separation occurs even at low angles of attack in this Reynolds
number regime. Significant increase in the values of drag coefficient is seen with a
decrease in Re. Lift-to-drag ratios are consequently very low.

An adjoint-based aerodynamic shape optimization methodology is employed to obtain

improved aerodynamic characteristics in the ultra-low Re regime., which is a numerical
method for efficiently computing the gradient of a function in a numerical optimization
problem.

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 The main objectives of this study are two-fold. Firstly, to evaluate the aerodynamic
characteristics of NACA 0008 airfoil in the ultra-low range Re [ [1000, 10000].

Secondly, to obtain optimal airfoil shapes resulting from aerodynamic shape

optimization for three different objective functions, namely, (i) minimization of drag
coefficient, Cd, (ii) maximization of lift coefficient, Cl, and (iii) maximization of lift-
to-drag ratio, (Cl/Cd), in this Re range
Aerodynamic optimization procedure

In this paper, a adjoint-based aerodynamic optimization methodology is

employed. With the equation given as:
ψ=…………………(i)

Where, ψ is the lagrangian multiplier introduced which converts a constrained

optimization problem into an unconstrained one

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 Optimal airfoil shapes

(i) Minimizing Cd0

The first problem of interest is to determine the shape of a minimum drag body for Re
[1000, 10000]. Certain constraints were imposed during the design cycle.

The airfoil chord length was maintained at x/c = 1 throughout the design process by
fixing the leading and trailing edges.
Starting from the NACA 0008 airfoil the flow and adjoint computations were carried
out as in figure 11. The convergence history of the objective function, for example, for
Re = 2000 is shown in figure 11.

We see that Cd decreases monotonically with design iterations. The evolution of the
airfoil shapes as the design process progresses is also depicted

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 Figure 11. Convergence history of Cd minimization and evolution of airfoil shapes
with design iterations for Re = 2000.

The optimal airfoil shown in figure 11 at Re = 2000 had Cd = 0.0728 less than the
NACA 0008 airfoil. The
reduction in Cd is brought about by a reduction in pressure contribution to Cd.

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 (ii)Maximizing Cl at α = 2

The next objective function considered is maximizing Cl at α = 2.

The leading and trailing edges of the airfoil were fixed to maintain the desired α.

With these constraints, the optimization was carried out starting from the NACA
0008 airfoil. The optimal Cl values and airfoil shapes obtained are shown in figure
12.

Figure 12. Optimal airfoil shapes for

maximizing Cl objective function

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 Significant increase in Cl values are obtained at all Re considered. The optimal airfoils
have evolved into rather thin profiles with distinct droops near the leading and trailing
edges.

The streamlines over the optimal airfoils are displayed in figure 13a and the contours of
vorticity magnitude in figure 13b

Figure 13a. Streamlines, and (b) contours of vorticity magnitude over the optimal airfoil for maximizing Cl; Re
A recirculation region is seen on the upper = 2000
surface of the optimal airfoil near the mid-
chord. This recirculation enhances the suction on the airfoil upper surface and
consequently its Cl. The recirculation region slowly disappears with increasing Re. The
recirculation region on the lower surface also vanishes with increasing Re.
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 Maximizing (Cl/Cd) at α = 2

The optimal airfoil shapes obtained for maximizing (Cl/Cd) are shown in figure 15.

The optimal airfoils have similar geometric features as obtained with the Cl
maximization objective function. The maximum thickness of the optimal airfoils was
about 6% occurring at about x/c = 0.10 for Re = 1000 and moves slightly rearward to x/c
= 0.14 for Re = 10000.

Figure 15. Optimal airfoil shapes for

maximizing (Cl/Cd) objective
function

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 Considering the Re = 2000 case to show the Cl/Cd improvement.
The streamlines and contours of vorticity magnitude over the optimal airfoil are plotted
in figure 16.
The recirculation region on the airfoil upper surface leads to Cl enhancement. A
recirculation region is
also seen on the lower surface of the airfoil near the leading edge. These recirculation
regions persist till Re = 10000.

Figure 16. (a) Streamlines, and (b) contours of vorticity magnitude over the optimal airfoil for
maximizing (Cl/Cd); Re = 2000
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 Summary

The flow field over NACA 0008 airfoil was studied computationally in the ultra-low
Reynolds number regime Re [ [1000, 10000] for various angles of attack a [ [0, 8].

The flow being laminar it was seen to separate at very small angles of attack. The onset
of flow separation is at α = 6 for Re = 1000, and occurs at lower a with increasing Re.

The slope of the Cl- a curve is lower than inviscid thin airfoil theory value of Cl,α = 2π.
Due to early flow separation the maximum Cl obtainable is drastically reduced as
compared to the high Reynolds number values.

An adjoint-based aerodynamic shape optimization methodology was employed to obtain

improved aerodynamic characteristics in the ultra-low Re regime. Three different
objective functions, namely, (i) minimization of
drag coefficient, Cd, (ii) maximization of lift coefficient, Cl, and (iii) maximization of
lift-to-drag ratio, (Cl/Cd), were considered.

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 References
• Schmitz, F.W. (1942) Aerodynamic des Flugmodells, C. J. E. Volckmann Nachf.
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• Selig, M.S., Donovan, J.F. and Fraser, D.B. (1989) Airfoils at low speeds, H.A
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three facilities`, 35th AIAA Fluid Dynamics Conference and Exhibit, pp. 1-11.
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 • Arshad, A., Lucas, B. and Inigo, M. (2021) Design Optimization and Investigation of
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• Ukken, M. G. and Sivapragasam, M. (2018) Aerodynamic shape optimization of

airfoils at ultra-low Reynolds numbers, Indian academy of sciences, 44(130), pp. 1-14.

• Desert, T., Jardin, T. and Bezard, H. (2019) Numerical predictions of low Reynolds
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