Wing profile
evolution driven by computational fluid dynamics
Evolución de perfil alar conducida por dinámica de
fluidos computacional
Cristian C. Rendon 1a,
José L. Hernandez 1b, Oscar Ruiz – Salguero 1c, Carlos A. Alvarez 2, Mauricio Toro 3
1Laboratory of CAD CAM CAE, Universidad EAFIT, Colombia.
Email: a crendo11@eafit.edu.co , b jlhernande@eafit.edu.co , c oruiz@eafit.edu.co
2 Fluid Dynamics Laboratory, Universidad EAFIT, Colombia. Orcid: 0000-0003-0463-330X. Email: calvar52@eafit.edu.co
3 GIDITIC Research Group, Universidad EAFIT, Colombia. Orcid: 0000-0002-7280-8231.
Email: mtorobe@eafit.edu.co
Abstract
In the domain of fluid dynamics, the problem of shape optimization is relevant because is essential to increase lift and reduce drag forces on a body immersed in a fluid. The current state of the art in this aspect consists of two variants: (1) evolution from an initial guess, using optimization to achieve a very specific effect, (2) creation and genetic breeding of random individuals. These approaches achieve optimal shapes and evidence of response under parameter variation. Their disadvantages are the need of an approximated solution and / or the trial - and - error generation of individuals. In response to this situation, this manuscript presents a method which uses Fluid Mechanics indicators (e.g. streamline curvature, pressure difference, zero velocity neighborhoods) to directly drive the evolution of the individual (in this case a wing profile). This pragmatic strategy mimics what an artisan (knowledgeable in a specific technical domain) effects to improve the shape. Our approach is not general, and it is not fully automated. However, it shows to efficiently reach wing profiles with the desired performance. Our approach shows the advantage of application domain - specific rules to drive the optimization, in contrast with generic administration of the evolution.
Keywords: fluid mechanics; shape evolution; wing profile.
Resumen
En el
dominio de mecánica de fluidos, el problema de optimización de forma es
relevante porque es esencial incrementar la fuerza de elevación y reducir la de
arrastre en un cuerpo inmerso en un fluido. El estado del arte actual consiste
en dos variantes: (1) evolución a partir de una estimación inicial usando
optimización para lograr un efecto específico, (2) creación y crianza genética
de individuos aleatorios. Estos enfoques logran formas óptimas y evidencian la
respuesta bajo la variación de parámetros. Sus desventajas son la necesidad de
una solución aproximada y / o la generación de individuos por ensayo - y -
error. En respuesta a esta situación, este manuscrito presenta un método que
usa indicadores de Mecánica de Fluidos (e.g.
curvatura en líneas de corriente, diferencia de presión, zonas de velocidad
cero) para dirigir la evolución de un individuo (en este caso un perfil de
ala). Se presenta una estrategia pragmática que imita las acciones de un
artesano (conocedor de un dominio técnico en específico) para mejorar la forma.
Nuestra aproximación no es general y no está completamente automatizada. Sin
embargo, presenta eficiencia al alcanzar perfiles de alas con el desempeño
deseado. Nuestra aproximación presenta la ventaja de tener un dominio y reglas
de aplicación específicas para realizar la optimización, en contraste con la
administración genérica de la evolución.
In nature, constant perturbations of a fluid in objects make
to change their shape in order to develop their dynamic behavior and evolve.
Examples are eolic erosion or abrasion of rocks by
streams. Similarly, engineering applies shape evolution techniques to develop
devices or tools with optimal performance. Aeronautics focuses in the
optimization of aerodynamic performance in aircraft with CFD. Due to current computational power and mathematical models,
this optimization can be partially conducted in silico,
saving in costly wind tunnel and other experiments. The present work presents a
methodology of experimentation with computational fluid dynamics (CFD) observing
flow characteristics of an individual to evolve its shape achieving a required
lift- and minimize drag- force. The optimization process of a wing profile can be carried
out in two ways, (1) evolution from an initial guess, using optimization, (2)
creation and genetic breeding of random individuals. Optimization methods use an objective function to be
satisfied (e.g. gradient-based method [1, 2]). These methods are successful
under one or two criteria to achieve a specific effect (e.g. lift production
and / or drag reduction). The disadvantage is the need of an initial guess. Ref. [3] determines Multivariable Polynomial Response
Surfaces (MPRS) that express aero-dynamic performance measures (e.g. drag,
lift) as functions of 2D control point sets. The point cloud of the MPRS is
obtained by running Computational Fluid Dynamic simulations. After the MPRS are
obtained, they are used to find the airfoil cross section control points which
achieve the desired drag and / or lift. The 2D control points are constrained,
in order to respect design conditions (e.g. space allowance for fuel
compartment). The training of MPRS makes this method resemble Genetic
Algorithms or Neural Networks. Ref. [4] describes a method to use multi-level
constrains for the design of helicopter rotor blades. Since these blades suffer
considerable challenge from conflicting design conditions, the constraints are
organized in hierarchical manner. A genetic algorithm is used to administer the
constraints, and dimensionality reduction (Principal Component Analysis) and
Multi-Layer Hierarchical Constraint (MLC) methods are used to impose priorities
on the design constraints. A large portion of the effort is devoted to find
reduced representative constraint method out of a large hierarchical constraint
set. Ref. [5] discusses the optimization of the airfoil NACA 2411
by using genetic (PANEL) algorithms. The point set of the polygonal form of the
airfoil is replaced by the PARSEC parameters for the purposes of lowering the
size of the tuning variable set. The PARSEC parameterization is an airfoil -
dedicated dictionary that translates fewer airfoil design parameters into full
geometric profiles that are needed for the fluid dynamics simulation. This
reference emphasizes the articulation of PANEL, PARSEC and Genetic Algorithms
for the sake of getting a coarse optimization, which effectively lowers the
computational expenses. Ref. [6] presents an optimization of the landing for a
morphing airfoil, conducted via iso-geometric
analysis of potential flow. The iso-geometric
analysis is a low - fidelity 2D one, that addresses both the fluid and the
stress / strain of the profile (seen as Timoshenko beam). This reference makes
emphasis on the direct usage of the beam B-Rep for the (i.e. iso-geometric) analysis of profile and fluid. Ref. [7] focuses on the optimized design of super-critical
wings. The manuscript uses 2D supercritical airfoil optimization (vis-a-vis
pressure distribution). This optimization is the mapped to each cross section
of the wing in the span direction via a so called 2.75D transformation. This
transformation translates, back and forth, the pressure distribution between
the wing and the 2d cross sections (i.s. airfoils).
The 2.75D transformation is a fitted function, that maps the wing parameters
onto the pressure distribution along the wing. The method is a heuristic /
empiric one, natural in an area in which the staggering computational and
experimental expenses make reasonable such approximations. Ref. [8] develops a fluid-structure interaction model for a
wind turbine. The authors implement an iterative procedure to optimize the
geometry of the blade through performance theories and then compare the results
obtained with a standard blade profile. They conclude with the obtaining of
greater torques for the turbine in the optimized model but, at the same time,
with greater stresses and structural deformations. The creation and genetic breeding of random individuals
modifies its flow conditions and / or the geometry, searching to improve the
aerodynamic performance of the individual. Refs. [9, 10] change the flow
direction on the individuals. Refs. [11, 12, 13] modify surface geometry of the
individuals. These experimentations can be conducted in wind tunnels and / or
CFD. The disadvantage of these methods is the trial - and - error way to
achieve the desired performance. Ref. [14] presents the fitting of parametric B-Spline curves
to large sets of points originated in the cross section of an airfoil. The
manuscript optimizes different curve parameters (stages, knot sequences, stage
degree, control polygon, continuity, etc.) to obtain a reasonable curve
fit with a minimum of computational effort (given the large point set). This
manuscript does not seek to design or re-design the airfoil profile, as it
takes already existing ones. Therefore, it does not make the connection between
wing profile against hydro- or aero-dynamic flow conditions. Ref. [15] implements CFD simulations for different radius of
curvature of a tracheal carina. The manuscript performs the parameterization of
the carina based on a simple bifurcation model and variates the radius of
curvature. Although the methodology discusses relations between the radius of
curvature and flow behaviour, it does not apply any optimization
over the carina shape. Optimization methods need of an initial guess to be carried
out. Creation of random individuals present a trial - and - error methodology. Experimental
approaches concentrate on the variation of geometrical and/or flow conditions
and do not seek optimal conditions. This work intends to evolve, gradually, an
initial rectangular profile into a wing profile using Fluid Mechanics
indicators. Our approach is a pragmatic strategy to drive the optimization.
However, it is not general and it is not fully automated. Table 1 presents an
overview of the literature review: The experiment is carried out in the software ANSYS Academic
Research Fluent, Release 17.2. The initial model consists in a 2D profile ( 1.
2.
3.
Steady state flow (i.e. the derivative of the fluid properties with
respect to time is equal to zero). 4.
Transition Shear Stress Transport model (SST) for CFD solution. SST
model is highly accurate in the predictions of flow separation. Captures eddies
phenomena and reaches convergence. Notice that, due to the finite element size and differential
equation modeling, the phenomenon of eddies is not really modelled here. At
this modeling level, we only make use of the fact that zero velocity boundaries
in the interior of the fluid domain (i.e. not related to material walls) mark
the existence of regions in which phenomena such as eddies occur. Our
(admittedly draconian) approach is to deny such regions to the fluid by moving
the wing profile to those limits. Since there is a zero velocity in such new
profile boundaries, we do not violate continuity laws, and in coarse manner
simply avoid the problematic eddy regions, without modeling them. At this time, we are conscious of the fact that the finite
element mesh used to model the flow must be optimized. Such an optimization
includes both topological (i.e. interpolation degree, number of nodes, etc.) as
well as geometrical (sensitive element size) aspects. We have used generic and
possibly non – efficient mesh topology and geometry. Future endeavors shall
include such considerations. Fig. 2 shows sizing and inflation methods used for
the first stage of the process. Todas Shape evolution process is
carried out in a pragmatic and intentional way, evolving the shape from a
rectangular profile into a wing profile adding or removing material Fig. 3
illustrates the evolution process. 1.
Goal: To satisfy a lift force s.t. 2.
Criteria: Reduction of pressure on the upper surface by
increasing there the stream velocity in order to produce pressure difference
(i.e. lift force). Reduction of drag by producing laminar flow (avoid streamlines
divergence from Avoid zero velocity neighborhoods. The Fluid Mechanical indicators to conduct the shape
evolution are three. Velocity scalar map, pressure scalar map and streamlines
curvature. These indicators are analyzed in each stage of the evolution.
Velocity- and pressure- scalar map are taken directly from the ANSYS
postprocessor as a result of the solution of the Navier
- Stokes equations. Curvature of the streamlines are obtained as follow. A
function interrogates ANSYS database. Then, curvature is calculated from Eq. 3
as a discrete curve how it is indicated in [19]. Where Four iterations were carried out, observing the fluid
mechanics indicators (mentioned in section 3.3) for each stage of the process.
The results are illustrated in this section. Figs. 4 and 5 show shape- and
force- evolution respectively, as follows.
Stage 1. Fig 4 (a), (b), (c) presents symmetry between the upper and lower
surfaces, resulting in null lift. High pressure in front produces a drag
significantly greater than lift. Streamlines Figure 4. Evolution scalar maps of velocity, pressure and
streamlines. Source: the authors.
5(b)). The back is rounded reducing the zero velocity
neighborhoods.
Stage 4. Fig 4 (j), (k), (l). The lift reaches 13000 N >10000 N (see Fig. 5
(b)). The zero velocity zones are filled by the object. The streamlines fit
completely to the profile. Velocity at lower surface is largely equal to the
flow velocity∞.
Three algorithms are implemented for the stages analysis. To
calculate the complexity of these algorithms the measure variable is the number
of elements in the mesh\ 𝑁𝑒.
Being the number of elements in a horizontal line in [−𝑤,𝑤] or vertical line in . Table 3 shows a brief description of the algorithms. Source: the authors. Figs. 4 (b) and 5 (a) show that perpendicular surfaces to
the flow increase drag by high pressure zone in front. Streamlines show the
response of the corner rounding favoring both reduction of drag (Fig. 5 (a)
shows higher reduction of drag) and laminar flow (see Fig. 4 (f)). Streamlines
along the evolution validate the reduction of drag by making the flow more
closed to laminar [16]. It occurs when there is not separation between
streamlines and the profile. Production of lift seems favored by an asymmetric
shape respect flow direction where the inclination is a determinant aspect. Zero velocity combined with low pressure zones suggest
presence of eddies and this zones can be filled by the object improving the
aerodynamic behavior. In this sense, mathematical models based into reducing
zero velocity and low pressure zones can be developed taking into account that
there is no transfer of momentum at their boundary. Both, the experimental
method presented, and a hypothetical mathematical model could be automated in a
future work. This methodology can be applied for the development of devices and
the understanding of fluid dynamics with submerged bodies. Ω Rectangular orthogonal simulation domain ∈ 𝑅2 with center in
(0,0). 𝑥 ∈ [−𝑤,𝑤] and 𝑦 ∈ [−ℎ,ℎ]. Γ Wing profile represented as a simple closed curve ∈ 𝑅2 immersed in Ω. 𝑉∞
Flow velocity at 𝑥
= −𝑤. 𝑉 Velocity magnitude at a point ∈ Ω. 𝑃𝑟𝑒𝑓
Magnitude of reference pressure. 𝑃 Pressure magnitude at a point ∈ Ω. 𝐹𝐿 Lift force acting on Γ. 𝐹𝐷 Drag force acting on Γ. 𝐶 Streamlines curvature. 𝑁𝑒 Number of mesh elements. [1]
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1. Introduction
2. Literature
Review
2.1. Conclusions of
the literature review
3. Methodology
3.1.1. Computer
Experimental Setup
) immersed in a fluid (Ω) moving at a
certain velocity (
) such that as seen in Fig. 1 
is bounded for parameters w
and h. 
is defined at
the first stage by the parameter a and b in Table 2. Assumptions
is a Newtonian
fluid region 
with constant
density and viscosity. This is because the Mach number for 
is less than 0.3 being an incompressible
flow [18]. 
rigid with no
slip condition. Therefore, Velocity (V) in body boundary is 0. 3.2. Shape
Evolution Process
≥10000 N and to reduce drag force 
with respect to 
. Eq. 1 and Eq. 2 show how the forces
are computed with their discrete form [18]. 
). 3.3. Fluid
Mechanics Indicators
is the
i-th vertex of the streamline is, 
is the vector going from 
to 
and
is the curvature
at . The calculation of 
in all the streamlines allows to
draw the curvature scalar map. 4. Results
4.1. Algorithms
Complexity
5. Conclusions and
Future Work
Glossary
References