A 2-D Immersed Boundary Method on Low Reynolds Moving Body Jonas Alexandre Silva Pereira Thesis to obtain the Master of Science Degree in Aerospace Engineering Supervisors: Professor José Carlos Fernandes Pereira Doctor Duarte Manuel Salvador Freire Silva de Albuquerque Examination Commitee Chairperson: Professor Fernando José Parracho Lau Supervisor: Professor José Carlos Fernandes Pereira Member of the Committee: Professor André Calado Marta June 2014
Transcript
A 2-D Immersed Boundary Method on Low ReynoldsMoving Body
Jonas Alexandre Silva Pereira
Thesis to obtain the Master of Science Degree in
Aerospace Engineering
Supervisors: Professor José Carlos Fernandes PereiraDoctor Duarte Manuel Salvador Freire Silva de Albuquerque
Examination Commitee
Chairperson: Professor Fernando José Parracho LauSupervisor: Professor José Carlos Fernandes PereiraMember of the Committee: Professor André Calado Marta
June 2014
“There are things known and there are things unknown,
and in between are the doors of perception.”
Aldous Huxley
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Acknowledgments
Comeco por agradecer a minha familia por toda a educacao que me deram e por me darem a pos-
sibilidade de expandir os meus horizontes, nunca me prendendo e sempre apoiando, mesmo quando a
distancia dificulta a relacao.
Guardo um agradecimento especial para a minha namorada, Sofia Pascoal, por toda a paciencia que
me dispensou nos ultimos cinco anos, desde o momento em que nos conhecemos ate ao momento em
que me encontro. Foram muitas provacoes, muitos dias de sacrifıcio e muitos dias de mau humor para
poder chegar aqui.
Agradeco tambem a Fundacao Caridade Pestana pelo apoio financeiro que me ofereceram ao longo
do meu percurso academico, possibilitando com este apoio que eu tivesse uma condicao economica
mais favoravel e que me pudesse concentrar a 100% nas minhas tarefas estudantis.
Tenho tambem de agradecer pelo apoio no projecto de investigacao ’EXTREME – PTDC/EME-
MFE/114343/2009 da Fundacao para a Ciencia e Tecnologia (FCT)’ sem o qual o desenvolvimento
desta tese nao tinha sido possivel.
Um sincero obrigado tambem ao Dr. Duarte Albuquerque pela incansavel ajuda em todos os passos
deste projecto, desde a primeira linha de codigo ate a ultima linha da conclusao.
Um agradecimento tambem ao Prof. Dr. Jose Carlos Pereira pela sabedoria que me foi passando
ao longo dos ultimos 12 meses, e a toda a equipa do LASEF que nunca me negou ajuda em momento
algum.
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Resumo
Neste trabalho um Metodo de Fronteira Imersa numa malha estruturada bidimensional foi desen-
volvido, implementado e os resultados obtidos foram interpretados. Tres tipos de interpolacao para a
correccao da fronteira imersa foram implementados e testados para o escoamento dentro de uma cavi-
dade com solucao analitica. Foi obtida a segunda ordem de convergencia caracteristica da discretizacao
de Volumes Finitos com uma regressao polinomial que usava valores dos pontos materiais e valores
das celulas de fluido vizinhas baseada no Metodo dos Minimos Quadrados.
A verificacao do modelo foi efectuada por comparacao dos resultados obtidos com dados computa-
cionais presentes na bibliografia para o domınio laminar e constante. Foram consideradas caracteris-
ticas do escoamento quantitativas e qualitativas e em ambas semelhanca com as fontes bibliograficas
foi observada.
No capıtulo de fronteiras imersas em movimento obteve-se validacao para o modelo por comparacao
com resultados experimentais e computacionais.
A ferramenta desenvolvida permitiu a analise aerodinamica computational de corpos em movi-
mento e imersos no fluido, representando com precisao o comportamento do fluido e a sua respectiva
interaccao com a estrutura. Foi realizado um estudo aerodinamico a seccao de um tubo de casca de
Eucalipto enrolado, uma das principais causas de propagacao de fogos florestais, onde se apresentam
os resultados das forcas aerodinamicas.
A comparacao bem sucedida dos resultados obtidos com dados empiricos e computacionais confir-
mou que o presente metodo e fiavel para a resolucao de problemas de corpos imersos e que o custo
computational deste tipo de simulacoes pode ser melhorado.
Palavras-chave: Metodo da Fronteira Imersa, Problemas de Fronteiras em movimento, Es-
coamento incompressıvel, Metodo dos Volumes Finitos, Metodo dos minimos quadrados, Interaccao
Fluido-Estrutura, Fogos Florestais
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Abstract
In this work the development, the implementation and the results interpretation of an Immersed
Boundary Method for the incompressible Navier-Stokes equations, in a bi-dimensional Cartesian grid
are shown. Three interpolations for the immersed boundary correction were implemented and tested in
a cavity flow with analytical solution for the case of static boundaries and the model was validated with
comparison of computational results with experimental data for moving boundaries. The second order of
convergence from the FV discretization was obtained with a polynomial regression using material points
and neighbor fluid cells based on the Least Square Method.
The verification of the model was performed by comparison of the obtained results with computational
data from the literature in steady laminar flows. Both qualitative and quantitative features of the flow were
considered and showed considerable resemblance with the literature.
In the moving boundary chapter verification was achieved by comparison with both experimental and
computational data.
The developed code allowed applications for computational modeling of moving boundaries im-
mersed in fluid flows, accurately representing the behaviour of the fluid and the interactions between
the fluid and the body’s structure. The study of a portion from a winding Eucalyptus peel tube, one of the
most common reasons for the propagation of forest wild fires, is performed and the aerodynamic drag
force is shown.
The comparison of the results with both experimental and computational data confirmed that the
method is reliable to solve problems of immersed boundaries and that the computational cost can be
decreased for this type of calculations.
Keywords: Immersed Boundary Method, Moving Boundary Problem , Incompressible Flows,
Finite Volume Method, Least Squares Method, Fluid-structure Interaction, Forest Fires
SIMPLE Semi-Implicit Method for Pressure-Linked Equations
SVD Singular Value Decomposition
TC Tangential Correction
TRI Triangular interpolation scheme
TSTE Taylor Series Truncation Error
UDS Upwind Differencing Scheme
IBM Immersed Boundary Method
KC Keulegen Carpenter Number
NS Navier-Stokes
LSM Least Squares Method
BC Boundary Condition
List of symbols
Please note that some symbols may have different meanings, depending of the context they refer to:
θsep Separation Angle
A,B analytic constants
D, L reference diameter, length
d distance vector between cells P and Pn
ε numerical error
least squares residual vector (WLS)
F(P ) the set of faces of cell P
f face
f coordinates of the face centroid f
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h hydraulic diameter
P cell or volume control (mesh)
P coordinates of the cell centroid P
p pressure
p′ pressure correction
Re Reynolds number
Sf face normal or surface vector (mesh)
S source magnitude
Uf face velocity
u velocity vector
u, v w velocity components
VP cell volume
v vertex
wP weighted function of cell P
xp cell P centroid
x, y cartesian coordinates
Ω arbitrary domain or a cell
Γ boundary of the respective Ω
αu, αp momentum and pressure relaxation factors
∆t time step
φ transported variable or computational variable
ρ density
µ dynamic viscosity
ν kinematic viscosity
Operators
∇φ gradient
∇ · u divergence
∇2φ Laplacian
‖u‖ Euclidean norm
A′ transpose
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Chapter 1
Introduction
1.1 Motivation
The subject of this thesis is an application of numerical finite volume method to moving bodies using
the immersed boundary method.
This work comes from the author’s interest to increase his knowledge after an academic cycle
wherein the fluid dynamic and energy transfer topics have been catching the author’s attention.
In the 21st century the aerospace industry is still relying in static analysis to study time variant phe-
nomena leading to extrapolative and inductive reasoning. This enhanced the author’s motivation to build
a tool that could be able to provide the desired stability, efficiency and accuracy not only for computa-
tional analyses of complex or arbitrary geometries but also to study the behavior of immersed bodies
with user prescribed motion and studies where the dynamic and mechanical behavior of an immersed
body is ruled by the interaction between the fluid and the body structure.
It was also important to provide a study of convergence for the second order immersed boundary
method with verification of 2-D results with literature data for fluid flows and to study the effects of the
history of the computational cells in the boundary’s motion and the consequent integration of aerody-
namic forces in this boundary.
1.2 Aerospace Industry Applications
1.2.1 Static Boundary Analyses
The developments in the aerospace industry in the last two decades has brought an immense range
of new practicabilities and new technologies that made the aerospace vehicles much lighter, economi-
cally viable and eco-friendly than their predecessors from the last decades.
The knowledge to accurately project these more efficient vehicles made accurate static and dynamic
aeroelastic predictions more critical than ever. The traditional methods that assume small deflections
and linear inviscid aerodynamic behaviour are no longer adequate for the precision needed in nowadays
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aircraft project, thus more advanced tools must be deployed in the design cycle.
Figure 1.1: Figure from Fidkowski [2011] representing the CFD mesh on a rocket, showing the positiveimplications of CFD calculations regarding the aerospace industry
In figures 1.1 and 1.2 it is possible to observe the application of these tools in an octree grid for the
aerodynamic analysis of a rocket and to a 2-D leading edge of an airfoil with a triangular mesh.
Figure 1.2: Figure from Fidkowski [2011] representing the cut-cell method applied to an airfoil
A simple implementation of an IBM, like the one present in this work, for laminar flows, can have pos-
itive implications in the study of airfoils, and combining this type of method with optimization algorithms
it is possible to obtain new and more efficient airfoils for low Reynolds and laminar steady applications.
1.2.2 Moving Boundary Analyses
Following the thoughts of the previous paragraphs we can deduce that using a static boundary is a
narrow approach to compute these complex phenomena.
As stated in Engel and Griebel [2004], the small deformations present in linear-elastic solid mechan-
ics can be easily handled by finite-element code, but that does not happen to fluid flow problems. The
difference that allows solid mechanics problem to be handled with the finite-element method is that these
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kind of problems are mostly formulated in a Lagrangian setting. For the case of the fluid flow behaviour
the Lagrangian formulation of the Navier-Stokes equations “is however not suitable for mesh-based dis-
cretization methods because of the complicated structure the “deformations” of a fluid can exhibit. ”
Engel and Griebel [2004].
Figure 1.3: Figure from David L. Rodriguez and Smith [2014] representing an aircraft using the VariableCamber Continuous Trailing Edge Flap
The non-linear behaviour of the fluid structure interactions and the possibility to computationally solve
problems like airfoil oscillations or wing fluttering using moving boundaries is a step forward regarding
the increasing accuracy that is needed in nowadays aerospace vehicles design, figure 1.3 represents
an example.
The full implementation of the IBM, with a tri-dimensional domain, high Reynolds number flows and a
moving boundaries can be a major development for the aerospace industry since it can help deepen the
knowledge of the fluid behaviour and the FSI for a wide number of specific components of aerospace
vehicles, from the turboprop fans to the aerodynamics of the landing gear in the take off and landing and
even for the launching acceleration behaviour of rockets.
1.3 Historical Context
Over the last decades of the 20th century and the beginning of the 21st, the industrial fluid flow studies
verified a substantial development and evolution. There are several applications derived from the precise
knowledge of fluid flows behavior from the study of biological fluids to combustion chambers or airplane
aerodynamics. As the analytical solution of the Navier-Stokes equations still remains unknown, the
numerical methods have been the baseline of complex engineering fluid flows studies.
The default approach regarding space discretization has been Finite Volume or Finite Elements un-
structured mesh methods. Eventough these methods have a real effectiveness in stationary grids, this
process leads to the obvious loss in the computational time when the subject is a moving body or a de-
formable boundary a remeshing process is required, due to the complexity of unstructured remeshing,
and the numerical errors that follows this type of methods have a significant computational cost.
In order to avoid remeshing in every time-step, a fixed-grid method has to be implemented. In this
kind of algorithms the fluid flow is computed in a fixed mesh and the boundaries of the solid body are
moved across the computational domain.
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Figure 1.4: Representation of the two paths of spatial discretization available, on the left the body-conformal, on the right a cartesian mesh with complete independence of the boundary’s geometry
The immersed boundary method (IBM) was proposed for the first time by Peskin [1982] to study
the blood flow in the human heart, which was a moving boundary problem. The first simulation was
performed using a simple cartesian mesh, which didn’t conform to the boundary’s geometry. In this
alternative technique, the immersed boundary is represented as a cut trough the Cartesian grid. In
order to maintain the physical nature of the problem a modification in the governing equations at the
vicinity of the immersed boundary is needed. In the end those alterations are the baseline of the IBM
studies.
In the original formulation, the deformable valve wall was represented as a set of nodal forces which
were incorporated in the fluid momentum equations as line forces. A smeared delta function was used to
represent them, resulting in a fuzzy interface spread over several cells as the works of Sanjay R. Mathur
and Murthy [2010] have shown.
In account of the information above, it is possible to assert that the process of imposition of the
boundary conditions in this type of schemes is not trivial, which results in different problems when
compared with body-fitted schemes. It is also important to note that in the case of body-fitted methods
the alignment of the grid lines with the boundary requires a lesser effort in the grid resolution control,
this fact is emphasized specially for higher Reynolds numbers.
To illustrate this, consider the case of the two-dimensional (2D) body of characteristic length L in
Figure 1.4, immersed in a linear steady flow with a boundary layer of thickness δ where it is required to
provide an average grid spacing of ∆n and ∆t in the directions normal and tangential to the body surface.
In the end we can conclude that for moderately high Reynolds numbers for which δ << L , the size of a
body-conformal grid scales as (L/∆t) · (δ/∆n), and a Cartesian grid scales as (L2/∆2n ). Assuming that
∆n ∝ δ and , ∆t ∝ L we deduce that the ratio of sizes for a Cartesian to a body-conformal grid will scale
as (L/ δ)2 .Furthermore for a laminar boundary layer,(L / δ) ∝ Re0.5 Schlichting and Gersten [2000],
implies that the grid-size ratio will scale with (Re)1.0 for 2D bodies. Similarly for a three-dimensional (3D)
body, the grid-size ratio would scale with (Re)1.5. Thus, as the Reynolds number increases, the size of a
Cartesian grid increases faster than the corresponding body-conformal grid.Mittal and Iaccarino [2005].
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Finally it is relevant to verify that a lower grid size may not imply a higher computational cost, because
a substantial percentage of the grid points in immersed boundary treatments is located inside the solid
body region where the equations are not solved.
In account of the simulation of a fluid flow past a solid body, as shown in Figure 1.4 the default ap-
proach, implies the creation and implementation of a structured or unstructured body-conformal grid. In
order to generate this kind of grids, two steps must be taken. First a surface grid covering the boundaries
Γb should be generated, then this boundary grid is used to generate a grid in the volume Ωf occupied by
the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the
governing equations will be transformed to a curvilinear coordinate system aligned with the grid lines as
said in Ferziger and Peric [1996]. As the grid conforms with the surface of the body, the transformed
equations can be discretized in the computational domain with relative ease. If a finite-volume technique
is employed, the integral form of the governing equations will be discretized and the geometrical informa-
tion concerning the grid will be incorporated directly into the discretization. In case an unstructured grid
is employed, then either a finite-volume or a finite-element methodology can be used. Both approaches
incorporate the local cell geometry into the discretization and do not resort to grid transformations.Mittal
and Iaccarino [2005]
The generation of a body-conformal structured or unstructured grid is usually very costly. The main
goal is to construct a grid that provides adequate local resolution with the minimum number of total
grid points. Thus, except for the simplest geometries, these conflicting requirements can lead to a
deterioration in grid quality, which can negatively impact the accuracy and the convergence properties
of the solver, see Ferziger and Peric [1996] for details . In structured grids, the usual path is to divide
the domain in multiple sub-domains, but it is known that the interface of sub-domains will introduce
complexity in the solution of the algorithm.
The advantage of the Cartesian grid-based IB methods also becomes eminently evident for flows
with moving boundaries. The simulation of such flows on body-conformal grids requires the generation
of a new grid in each time step and a procedure to project the solution onto this new grid [Tezduyar 2001].
Obviously these processes will have a higher computational cost and consequently a lower accuracy.
The family of fixed-grid methods has also a large advantage in computational problems where the
boundaries are nearly touching each other, as an example we have the flow of a body inside a cavity.
This type of problems cannot be solved with a moving grid method.
Despite the fact that a large numerical complexity(for static boundaries) and a higher grid resolution
is needed in an immersed boundary approach, the main point is that regarding moving boundaries this
method is faster and accurate. Since 1981 a series of spinoffs of the immersed boundary method have
been developed, and the algorithms are way more complex than the original one. In this work you
will find the explanation and literature review of the most widely spread families of Immersed Boundary
Methods and a few analysis using one of the algorithms exposed.
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1.4 Immersed Boundary Method Literature Survey
The differences among the wide variety of immersed boundary methods are found in how these
boundaries are handled. This boundary treatment is the most important factor in the method’s accuracy
and robustness. Taking the example of the incompressible flow past a arbitrary body, the behaviour of
the fluid is governed by the following equations:
δu
δt+ u · ∇u +
1
ρ∇p− µ
ρ∇2u = 0 and ∇ · u = 0 in Ωf (1.1)
and
u = uΓ on Γb (1.2)
being u the fluid velocity, p the pressure, ρ and µ the density and the viscosity respectively. The whole
domain is represented by Ωt, the domain occupied by the solid body is refered as Ωs, and the fluid
surrounding the body, which is the rest of the domain, is represent by Ωf , where Ωt = Ωf + Ωs. The
study of the outer boundaries is not important, and the immersed boundary is represented as Γb.
In the representation of the coupled system of momentum and continuity from equation 1.1 as a
linear system, we have:
L(U) = 0 in Ωf (1.3)
with
U = UΓ on Γb (1.4)
where U = (u, p) and L is the operator representing the Navier-Stokes equations system as in 1.1.
First it is relevant to explain that in the context of the incompressible Navier-Stokes equations, pressure
is determined by the continuity constraint and, consequently, the continuity equation is considered an
implicit equation for pressure. There are various numerical integration schemes that can explicitly derive
and solve a Poisson equation for pressure, this type of information as we will analyze in chapter 2.
In order to introduce the physical representation of the solid boundaries in the mathematical scheme
there must be a change in equation 1.3. The physical change in the nature of the equations is performed
by the addition of a source term (or forcing functions). The IB methods’ first differences are noticed in
the implementation of this forcing functions. These methods can be included in the continuous form
of the equations, adding a source term, fb leading to L(U) = fb and then applied to the total domain
Ωt. Note that fb = (fm, fp) where fm and fp are the forcing functions applied to the momentum and
pressure, respectively. This equation is subsequently discretized on a Cartesian grid, leading to the
following system of discrete equations that are solved in the entire domain:
[L] U = fb (1.5)
In the second approach, the governing equations are first discretized on a Cartesian grid regardless the
immersed boundary, resulting in a set of discretized equations [L] U = 0 . Following this, the cells
the discretization near the IB is adjusted to account for the boundary’s presence, resulting in a modified
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system of equations [L′] U = r, that is solved in the Cartesian grid. In this equation, [L] is the
modified discrete operator and r represents known terms associated with the boundary conditions on
the immersed surface. The above system of equations can be rewritten as:
[L] U = f ′b (1.6)
where f ′b = r+ [L]U − [L′]U.
The first type of algorithms is often called “continuous forcing approach”, since the force is added to
the system in it’s continuous form, before the Navier-Stokes equations are discretized. In the second
type of approach the force is incorporated in the system after the discretization of the Navier-Stokes,
often called “discrete forcing approach”.
An attractive feature of the continuous forcing approach is that its formulation is independent of the
underlying spatial discretization. In opposition the discrete forcing approach is highly dependent on the
discretization scheme used. This discrete forcing approach feature can be seen as an advantage, since
it enables a direct control over the numerical accuracy, stability and discrete conservation properties of
the solver, see Mittal and Iaccarino [2005] for more detailed information.
Over the next sections, an inside view of these methods as well as some particularities of these
algorithms will be presented.
1.4.1 Continuous Forcing Approach
The continuous forcing approach has two different categories. The first one is based on the method
presented in Peskin [1982] and is mainly used in elastic moving boundaries, while the second is used in
the representation of rigid boundaries.
Elastic Boundaries Treatment
The original IB method was created in order to study the complex fluid-structure coupling in the heart
valves. This problem involves a viscid fluid in a contracting-expanding domain ruled by an elastic be-
havior. The step forward given in Peskin [1982] was the resolution of the fluid flow in moving boundaries
with a Cartesian grid, where the immersed boundaries were represented by a set of elastic fibers and
the location of these fibers was tracked in a Lagrangian fashion by a collection of massless points that
move with the local fluid velocity, Mittal and Iaccarino [2005]. This implies that the properties of the kth
Lagrangian points are ruled by:
δXk
δt= u(Xk, t) (1.7)
The consequent stress and deformation are then related with a law similar to Hooke’s law, and
this stress is then passed to the fluid flow employing a forcing term in the Navier-Stokes momentum
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equations analogous to the stress term in the boundary point:
fm(x, t) =∑k
Fk(t)δ(|x−Xk|), (1.8)
being δ the Dirac delta function that will only be applied in the boundary points. Since the computational
boundaries are not body-conformal, the nodal points usually do not match the boundary points, which is
solved by distributing the force in a small range of cells at the vicinity of the Lagrangian boundary point.
This leads to a minor difference in the forcing term, the Dirac delta function is substituted by a
smoother distribution function,d.
fm(xi,j , t) =∑k
Fk(t)δ(|xi,j −Xk|), (1.9)
Rigid Boundaries Treatment
The method above is important to treat elastic boundaries but for rigid boundaries it’s not adequate.
Although it is trivial to think the rigid body is an elastic body with a very high stiffness the comportment
of the elastic boundary would be considered as a structure attached to an equilibrium location (Beyer
Leveque 1992, Lai Peskin 2000) by a spring with a restoring force F given by:
Fk(t) = −κ(Xk −Xek(t)) (1.10)
where κ is a positive spring constant and Xek(t) is the equilibrium location of the kth Lagrangian
point. In order to represent a rigid body the numerical value of κ would have to be very high, resulting
in a stiff system of equations susceptible to severe stability constraints (Lai & Peskin 2000, Stockie &
Wetton 1998). On the other hand, as noted in the low Reynolds cylinder wake simulations of Lai &
Peskin (2000), lower values of κ can lead to spurious elastic effects such as excessive deviation from
the equilibrium location.
An overview over Continuous Forcing
This approach to the IB method has very attractive features regarding elastic boundaries, since the
physical nature of the problem is very similar to its numerical simulation. Also, the implementation is
simple. Therefore, most of the applications of these kind of methods are in biological fluid flows (Beyer
The last convective scheme that will be explained in this work is the Linear Interpolation scheme.
Since the values of φ from both P0 and P1 are taken into account, the accuracy problems revealed by
the UDS are not present.
In order to solve the accuracy problem in the case of non-uniform meshes, a linear interpolation is
needed. This interpolation takes into account the distance between the surface plane that contains the
cell face and the next equation:
zf = P0 + ηLIN (P1 − P0) = P0 + ηLINd (2.17)
being d the distance vector between the P0 and P1 centroids and ηLIN the blending factor of this scheme,
that can be calculated with the projection of (f − P1) and d to the face normal space that results in:
ηLIN =
(f−P0)·Sf
Sf ·Sf
(P1−P0)·Sf
Sf ·Sf
=(f − P1) · Sf(P1 − P0) · Sf
(2.18)
This scheme was chosen for all 2nd order of accuracy simulations in this work.
2.2.2 Diffusive Schemes
This section will discuss the diffusive schemes, which are the methods applied in order to calculate
the value of the gradient of the dependent variable ∇φ at the face centroid, using the information of the
neighbor cells.
The central difference scheme (CDS) has second-order accuracy and it is a part of the family of Finite
Volume Methods, as the schemes presented in subsection 2.2.1. In this work this type of schemes was
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only used in the discretization of the diffusive term of equation 2.11, it will only be mentioned in this
section.
The greatest difference between the diffusive term of equation 2.11 and the convective term is that
the first one represents a derivative of the variable φ and the second represents the value of the same
variable. In order to calculate this derivative on the face of the cell a finite central difference is applied
using two points, the cell centers of the cells adjacent to the face.
(∇φ)f = (φP1− φP0
)d
d · d(2.19)
As the central differencing scheme uses consistent expressions to evaluate the fluxes at the faces of the
control volume the conservativeness of the scheme is guaranteed. On the other hand the transportive-
ness of the method is not assured, as the coefficient that takes into account the downstream value of
the variable is the same as the upstream value, causing the scheme to not recognize the flow direction
for high Peclet numbers (Pe), where the Pe is defined as:
Pe =ρu
Γ/δx(2.20)
Lastly, the accuracy of this method is defined by the Taylor series truncation error which despite
being second-order, it is important to state that the cell Peclet number has significant part to play in this
evaluation. If Pe < 2 the boundedness of the scheme is not granted, and as obvious nor the accuracy as
said in Versteeg and Malalasekera [2007], so in order to obtain these assessments a relation between
the fluid properties (ρ and Γ), the flow properties (u) and the grid properties (δx) must be calculated.
2.2.3 Cell Centered Gradient Schemes
In order to improve the accuracy of some Finite Volume schemes or to have the values of the gradient
terms of the governing equations,for example the pressure gradient, the gradient of the variable φ must
be computed in the center of each cell.
As only orthogonal structured grids were used these calculations do not rise a wide number of prob-
lems, so it was chosen to use the simplest method in order to obtain these quantities.
The finite volume method applies the Divergence Theorem (equation 2.8) in the gradient of the
variable ∇φ, transforming it in a summation of the fluxes flowing in each cell’s face. The Gauss method
can be derived from equation 2.21:
∫P
∇φdV =
∫∂P
φ · dS → (∇φ)P =1
VP
∑f∈F(P )
φfSf (2.21)
It’s important to note that the second order assumption inside the cell was considered as well as the
Gauss-Legendre quadrature.
The quantity φf can be computed with any of the convective schemes presented in subsection 2.2.1
except with the Upwind based schemes. In case the convective scheme is dependent of the value of
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∇φ a deferred correction approach can be implemented considering an initial guess.
2.2.4 Time Schemes
This work baseline is a moving body which implies the existence of governing equations that rep-
resent the temporal evolution of the flow variables. As observed in equation 2.7, the temporal term is
represented by a temporal derivative of ρφ, which turns the computation of this term, as the diffusive
or the convective scheme, dependent of a numerical scheme capable of handling with consistency and
accuracy the time progression.
In order to apply the computational methods it is convenient to represent the equations 2.1 and 2.2
as :δφ
δt= F (φ, t) (2.22)
where F represents the sum of all the other terms in equation 2.7, the convective term, diffusive term,
pressure gradient term and source terms.
F (φ, t) = C (φ, t) +D (φ, t) + PG (φ, t) + S (φ, t) (2.23)
It is also necessary to define an initial temporal stage, initial condition, that is represented as:
φ (t0) = φ0 (2.24)
By combining equations 2.22 and 2.23 we are able to conclude that a wide variety of time-schemes
can be created based on these premisses. This type of schemes can be divided in two different groups,
the first with the so-called explicit time-schemes and the second with the implicit time-schemes. In the
explicit time-scheme the value of δφδt is calculated based on a finite difference as the following expression:
φn+1 − φn
∆t= F (φn, tn) (2.25)
The equation 2.25 represents one of the simplest modes to calculate this term, the first order accurate
explicit Euler time scheme.
Within this group a lot of methodologies can be applied, for example the Runge-Kutta time schemes,
that presents intermediate time-steps and follows the expression stated in equation 2.26 and 2.27 .
φn+1/2∗ = φn +
∆t
2F (φn, tn) (2.26)
φn+1 = φn+1/2∗ +
∆t
2F(φn+1/2∗ , tn+1/2
)(2.27)
The case represented in equations 2.26 and 2.27 is the 2nd order Runge-Kutta time scheme.
As already said above there is a wide variety of time schemes, since the goal of this study is to
acknowledge the interaction between a fluid flow and a structural element, the immersed body, an implicit
20
method for the time-scheme had to be utilized. This other category is based on the following equation:
φn+1 − φn
∆t= F
(φn+1, tn+1
)(2.28)
Where the F (φ, t) term is calculated in a future time stage, n + 1, therefore the coupling of the
pressure gradient and the velocity must be solved using SIMPLE or PISO, which will be explained in
section 2.3.
Since the equation 2.2 is non-linear it is mandatory to perform spatial iterations between every time-
step, because the face velocity, Uf is necessary and only appears at the time stage n. In every spatial
iteration the matrix with the terms representing the momentum equation will change contributing to the
convergence of Uf that is close to the value of the same variable in the time step n + 1. Naturally as
the velocity in the faces change the pressure gradient term will have an analogous evolution, Mittal and
Iaccarino [2005].
2.3 Pressure Velocity Coupling
The Navier-Stokes equations are coupled between two different variables, pressure and velocity,
so it becomes relevant to state how this coupling is handled by our numerical methods. This type
of algorithms has an extreme importance in the overall solution since the compatibility between the
numerical schemes can affect the accuracy, stability or the conservation properties of the problem.
Similarly to the problems in other sections, there are different ways of handling the pressure-velocity
coupling depending on the type of time-scheme that is chosen. For instance, when explicit time-schemes
are implemented it is usual to apply a fractional-step algorithm to solve the Navier-Stokes equations and,
for steady or implicit computations, the conventional approach is to implement the SIMPLE or the PISO
algorithms.
Since all computations in this work are either steady or implicit only the coupling for this type of flows
will be analyzed, being the formulation of PISO out of the purpose of this work.
The Semi-Implicit-Method for Pressure-Linked Equations (SIMPLE) was first introduced in Patankar
and Spalding [1972] and it is the most widely spread algorithm to solve the pressure-velocity coupling in
steady or implicitly discretized cases. In order to simplify the presentation only the steady case will be
shown.
The first step of the algorithm is the computation of an estimation for the velocity field u∗, that must
satisfy the momentum equations, taking in account the values from previous iteration . The steady
equation represented as 2.29 is solved in an implicit form and the convection terms are linearized.
F∑f=1
Unf u∗f − ν
F∑f=1
(∇u∗)f · Sf = −VPρ
(∇pn)P (2.29)
being Unf the face velocity defined as ufn · Sf , and Sf the surface normal vector Sf · nf . As a result a
21
linear system of equations of the following form is obtained:
1
αuapu
∗p +
F∑l=1
= alu∗l = −VP
ρ(∇pn)P +
1− αuαu
apunP (2.30)
Where αu is the under relaxation factor for the momentum equations, ap represents the main diagonal
matrix values from the momentum system, and al represents the other non-zero terms of the system
matrix. The αu term, the under relaxation term, is essential to solve non-linear systems. After the
prediction of the velocities, u∗, an interpolation is applied in order to get the face velocity values, U∗f .
Once we used a collocated grid during the whole work it is relevant to emphasize that there can be some
incompatibilities in the pressure at the face’s centroids and at the cell’s centroids. This incompatibilities
can be explained by the impossibility to force the value of the pressure, as the pressure term is a
gradient, a differential term. In order to contradict this problem a Rhie and Chow. [1983] interpolation
is performed to obtain the U∗f values. In this interpolation the cell centered pressure gradient from the
momentum equations is needed and the face velocity values are defined as:
U∗f = u∗f · Sf −[αuVpρaP
]f
[(∇pn)f −
(∇pn
)f
]· Sf (2.31)
The value u∗f represents the interpolated face centered velocity, which is calculated with a convective
scheme, note that it cannot be an upwind based scheme. The over-lined terms are computed with a
linear interpolation of the values stored at the cell’s centers.
There are some points to explain in equation 2.31. The first one regards the term αuVP
ρapthat is
the relation factor between the pressure gradient and the velocity field that comes out of the momentum
equation. The pressure gradient (∇p)f is calculated with a diffusive scheme and the(∇pn
)f
is computed
from the values of the cell centered pressure gradient in P0 and P1.
Having satisfied the momentum equation, in the previous steps, the velocity field must be updated
in order to satisfy the continuity equation, through a computation of a pressure correction p′ to solve the
Poisson equation,
F∑f=1
[αuVpρaP
]f
(∇p′)f · Sf =
F∑f=1
U∗f (2.32)
After the computation of the pressure correction, the velocity values are calculated with p′ satisfying
the continuity equation, and the face velocities are defined as:
Un+1f = U∗f −
[αuVpρaP
]f
(∇p′)f · Sf (2.33)
and the cell centered velocities as:
un+1P = u∗P −
[αuVpρaP
](∇p′)f (2.34)
The Rhie-Chow interpolation ensures that (∇p′)f and (∇p′)P are compatible with each other, and
22
the pressure at the new time-step is then obtained with:
pn+1 = pn + αP p′ (2.35)
where αP is the pressure under relaxation factor, needed in the SIMPLE algorithm. The under relaxation
factors αu and αP have typical values of 0.8 and 0.2 respectively.
This algorithm, as most of the algorithms in computational mechanics, is iterative and the computa-
tion of the residual is one of the steps of the method. In this case the momentum residuals RM can be
obtained by:
RM =apu
n+1P +
∑Fl=1 alu
n+1l + VP
ρ ∇paPmax (u)
(2.36)
Since the volumes of the cells are not all the same it is important to adimentionalize the residual by
the Volume. Then the continuity residuals can be calculated by:
RC =
F∑f=1
U∗f (2.37)
As usual the 2nd norm of the residuals is computed in order to control the stopping criteria of the
algorithm, if this second norm is lower than the tolerance asked by the user, normally between 10−6 and
10−10, the algorithm stops, otherwise it starts again from equation 2.29.
SIMPLE Summary
In order to simplify the implementation and the comprehension of this algorithm here is a summary
of the method.
1. Define the pressure, p, the velocity field,u and the face’s velocity Uf for the iteration n
2. Assemble the matrix system of the momentum equations. The face velocity can change in every
spatial iterations so the matrix values can change too, store the main diagonal values ap and the
cell centered pressure gradient (∇p)P
3. Solve the linear system of equations to obtain the prediction of the velocity, u∗.
4. Compute the prediction of the face velocity, U∗f with the Rhie-Chow interpolation, equation 2.31.
This interpolation is dependent of the cell center velocity u∗P , the pressure, the cell centered pres-
sure gradient and the matrix main diagonal values, ap.
5. Compute the continuity for each cell and assemble the matrix for the pressure correction. As the
values of the first matrix change in every outer iteration this pressure correction matrix needs to be
assembled in every iteration.
6. Solve the pressure correction system to obtain p’.
7. Correct the face velocities, Uf and the cell center velocities, u for the iteration, n + 1 applying
equations 2.33 and 2.34 respectively.
23
8. With the updated values for the face and centroid velocities update the value of the pressure for
n+ 1 with the equation 2.35.
9. At last compute the momentum and continuity residuals, Rm and RC and check if the stopping
criteria have been met, if not repeat from step 1.
2.4 Boundary Conditions
The CFD calculations purpose is to numerically simulate the behavior of the fluid transport proper-
ties, of energy, mass and momentum, in a prescribed domain and in order to accurately calculate all
these properties from the fluid and from the flow the implementation of the boundary conditions to the
computational domain is mandatory. There are no calculations without boundary conditions and in the
case of unsteady flows initial conditions.
The boundary conditions are the connection between the physical reality of the flow and the mathe-
matical meaning of the numerical resolution, which adds a physical significance to the numerical code.
The comprehension of the accurate implementation of the boundary conditions is vital for every compu-
tational mechanics calculations.
2.4.1 Mathematical Formulation of the Boundary Conditions
As all other features in computational mechanics the boundary conditions are no more than a sim-
ulation of the reality by means of ma thematic interpolation or differentiation and so all the boundary
conditions can be divided in three major categories.
Dirichlet Boundary Conditions
The Dirichlet Boundary condition type is the most basic type of boundary. This type simply states the
imposition of a value to the φb variable in the boundary face.
Convective Term As mentioned above the treatment of the convective term of the Navier-Stokes equa-
tions has the finality of computing the value of φ in a face. The Dirichlet treatment for this term is the
simplest, the value of the variable φb in the boundary is the value of φf in the face of the boundary cell.
φb = φf (2.38)
Diffusive Term In contrast to the convective term, that intends to simulate the variable value, the
diffusive term has the purpose of simulating the variable differential, which results in a more complex
solution. It is important to state that the diffusive term in the presence of a boundary must respect the
following equation:
(∇φ)f Sf =
(∂φ
∂n
)b
‖Sf‖ (2.39)
24
while in the case of a Dirichlet Boundary condition the handling of the boundary shall be done using a
finite difference formula for the gradient such as equation 2.40
(∂φ
∂n
)b
=φb − φPdn
(2.40)
where dn represents the distance between the face centroid and the cell centroid P . This can be ob-
tained with the following expression:
dn = (f −P) · Sf‖Sf‖
(2.41)
Neummann Boundary Conditions
The Neumman boundary conditions is the type of boundaries where not the value of the variable
is fixed but its gradient along the boundary normal vector,(∂φ∂n
)b
= gb and so the treatment of the
convective and diffusive term is a little different than that of the Dirichlet type.
Convective Term In the case of the convective term and since the prescribed value of the boundary
is the gradient along the normal direction and the convective term represents the variable and not a
differential of it, an interpolation is necessary in order to get the value of φb. This interpolation is usually
linear and of the form:
φf = φp +
(∂φ
∂n
)b
(fb −P) · Sf‖Sf‖
(2.42)
Where fb represents the centroid coordinates of the boundary face and P is the index of the cell contain-
ing the boundary face. In the case of an implicit method of resolution the contribution of φP have to be
added to the solution matrix and the other term must be added to the right hand side of the linear system
of equations. The value of Uf can be computed from the vectorial product of the boundary velocity and
the face area vector, Sf
Diffusive Term In the case of Neummann boundary conditions the diffusive term is easier to handle,
as the term is a differential and the type of boundary represents a differential we can take the equation
2.39 and replace the value of(∂φ∂n
)b
with the value of gb
Robin Boundary Conditions
The Robin Boundary conditions type is the least used and it consist in a linear combination of the
two previously mentioned types, imposing a boundary condition ruled by the following form :
φb + c
(∂φ
∂n
)b
= d (2.43)
Where c and d are constants. This type of boundary conditions wasn’t implemented in the SOL code
and it wasn’t used in this work.
25
2.4.2 Physical Boundary Conditions
The physical boundary is a set of generic conditions types that simulate specific boundary behav-
iors. Every specific type of physical boundary conditions has a combination of mathematical boundary
conditions that creates the conditions of velocity and pressure within a physical behavior similar to what
is wanted in the prescription of that boundary. In this section the types of physical boundary conditions
implemented in the SOL code will be explained, using the same nomenclature that is used along the
work.
Wall
In order to respect the no slippery condition a Dirichlet boundary condition is implemented in the
velocity of the face, where the tangent component of the the boundary face velocity is equal to the
velocity of the wall. The impermeability condition states that in the normal direction, the velocity has
a zero value, thus there’s another Dirichlet boundary condition for the velocity. For the pressure a
Neummann BC is imposed with a null gradient.
Inlet
The Inlet BC is simple and it’s basically the imposition of a fixed value to the velocity vector. A
Dirichlet BC for velocity, and a Neummann null gradient BC for the pressure.
Outlet
In the outlet BC a Neummann BC with a null gradient is applied for the velocity, at every time-step
or iteration a new extrapolation of the velocity value in the boundary, ub is required. The easiest way
of doing this is by a first order approximation considering the velocity in the boundary is equal to the
velocity on the boundary containing the cell.
ub = uP (2.44)
A linear extrapolation can be executed taking into account the cell centered gradient of the velocity
components where:
Uf = uf · Sf (2.45)
By knowing that an extrapolation was made it is obvious that the continuity of mass may not be totally
respected. So to guarantee the extrapolation a computation of the mass flux through all the inlets and
outlets is needed.
Qin =∑
f∈∂Ωinlet
Uf (2.46)
Qout =∑
f∈∂Ωoutlet
Uf (2.47)
26
When convergence is acquired the values of Qin and Qout cancel each other, or in the other hand a
forcing can be implemented with a flux correction or using the ratio between Qin and Qout.
Qcorr = −(Qin +Qout) (2.48)
Rflux =−QinQout
(2.49)
These correction terms are applied to the face velocity updated by its value, as on equations 2.50
and 2.52. Proving that both methods are accurate and have close convergence rates.
U∗f = Uf +Qcorr‖Sf‖Soutlet
(2.50)
where the Soutlet represents the total outlet area, and can be obtained by:
Soutlet =∑
f∈∂Ωoutlet
‖Sf‖ (2.51)
or in the case of the Ratio:
U∗f = UfRflux (2.52)
Outlet - Pressure Outlet
The pressure outlet is an alternative way to build a domain outlet. It is created with a Dirichlet BC for
the pressure and a Neummann null gradient condition for the velocity field.
The most important feature of this boundary is the fact that, contrary to the outlet BC (that have a
free distribution) the pressure is forced to have a constant distribution on the boundary.
Moreover this type of BC presents another unique feature, if SIMPLE is used the computation of
a velocity correction will be needed once the pressure gradient isn’t null as in other boundaries,which
makes the pressure correction value, for the velocity-pressure coupling, be equal to zero. This leads to a
corrected value of the face velocity, Uf on the boundaries. This treatment already ensures the continuity
of mass in the domain so no more treatments are required.
Symmetry Plane
This type of BC is implemented with the imposition of Neummann BC with zero gradient for all the
variables. The only detail is that in vectorial variables, such as velocity, the normal component of the
variable shall have a different treatment to afixed zero value Dirichlet BC. This is due to the fact that the
vector quantities cannot have a normal component in the symmetry plane.
27
2.5 Immersed Boundary Interpolation Schemes
The specificity of this work is the integration of an immersed boundary on a fluid flow, which as
mentioned in chapter 1.4, has a different treatment from the other types of boundaries. The features
that distinguishes the interpretation of this type of conditions it the fact that the computational boundary
is not geometrically coincident with the physical boundary. So in order to maintain the accuracy of the
results an interpolation of the velocity field must be made and this value will then be the fixed value of a
Dirichlet BC on the computational boundary.
After the identification of the boundary cells and the discretization of the continuity and momentum
equations with a finite volume scheme, the interpolation mentioned above is used to calculate the face
center values of the velocity, which do not exist as in the finite volume schemes the velocity is stored in
the cell center. In this work three types of interpolations, with different truncation error orders, were used
and in this section we’ll deepen the implications of each one.
In the interpolations algorithms some material points were used and so a routine that can retrieve
the closest solid points to the boundary faces, and their coordinates was implemented and executed.
2.5.1 First Order Interpolation - Interpolation 0
This interpolation scheme is the simplest and is based on the collocation of the closest solid point
value in the face value of the immersed boundary. So for every immersed boundary face the above
mentioned algorithm selects the closest solid point and uses its value on the boundary face, as in
equation 2.53.
φf = φs (2.53)
2.5.2 Linear Interpolation with Distance Weighting Factor - Interpolation 1
The linear interpolation with the distance weighting is made by the use of two points, one solid and
one fluid and is of the form:
φf = φc (1− η) + ηφs (2.54)
where φf represents the interpolated value for the face center variable, φc the cell center value for the
variable in question for the cell containing the boundary face, φs represents the value of the variable in
the solid material point closer to the immersed boundary face in question, in the end η represents the
weighting factor of the interpolation and can be defined as :
η =d1
d1 + d2(2.55)
where d1 represents the distance from the cell center to the face center and d2 is defined as the distance
between the computational boundary face center and the closest material point in solid region. In figure
2.2 we can observe the graphical representation of the two values.
28
Figure 2.2: Representation of the distances used to calculate the geometrical factor η in order to use thefirst order interpolation scheme
2.5.3 Second Order Interpolation with a Quadratic Polynomial obtained with the
Least Square Method - Interpolation 2
The two previous subsections described two different approaches to obtain the face value of velocity
in the boundary face, but they have at the most a first order of accuracy. The first (subsection 2.5.1 )
is really outdated as it was the interpolation scheme used in the earlier computational fluid dynamics
calculations and the second one even tough is better still has only a first order of accuracy. In order
to obtain a breakthrough a second order interpolation was needed and based on the work of Sanjay
R. Mathur and Murthy [2010] this interpolation was implemented in the SOL software.
In order to obtain a second order of accuracy interpolation scheme a quadratic polynomial of the
following form has been created:
φf = β0 + β1x+ β2y + β3x2 + β4y
2 + β5xy (2.56)
where the βn coefficients are determined by the least square method. This method was chosen because
it gives the user the possibility to obtain an approximated solution for a problem with more equations than
unknowns.
The polynomial in equation 2.56 was formed using a stencil with some of the velocity values of the
nearest neighbor cells, with shared vertex with the boundary face, and with the closer material points
(points in geometrical boundary of the solid region). These points have a prescribed velocity vector, the
velocity of the moving boundary, and in case a neighbor cell is an IB cell, the nearest material point to
that IB face is taken in account to the construction of the stencil.
As there is variation in the number of IB faces near the interpolated face a selection algorithm was
implemented in order to choose the number of solid or material points in the stencil. The dimension of
the stencil varies for each IB face.
As verified in the figure 2.3 the stencil for the interpolation of the velocity value for face f has 10
points which is clearly shown in figure 2.3.
After choosing the points of the stencil the construction of the matrices and vectors for the least
29
Figure 2.3: Schematic representation of the stencil for the generic face f . The cell centers of theneighbor cells are represented in red, the face centers of the IB cells in blue and the solid materialpoints in black
square interpolation was performed. For the polynomial construction the velocity of every point can be
approximated in a matrix form of the equation 2.56 by M~β where ~β = [β0, β1, β2, β3, β4, β5]T and M :
M =
1 x1 y1 x2
1 y21 y1x1
1 x2 y2 x22 y2
2 y2x2
......
......
......
1 xn yn x2n y2
n ynxn
(2.57)
where n is the number of points of the stencil, that must be at least 6. In the least square method the βi
values are calculated in order to minimize the square of the difference between the real variable φ and
the interpolated value, so the objective is to get the minimization of ‖φ−M~β‖2. The works of Kariya and
Kurata [2004] showed that this difference is minimized when the β coefficients are obtained as:
β =(MT ·M
)−1MTφ (2.58)
As stated in chapter 1.4 correction is necessary in order to maintain the continuity equation and to
prevent the interpolation to inject mass in to the system. This correction is a multiplication of the negative
fluxes by a ratio(R =
∑U+∑U−
). This correction is applied before the momentum equations are solved, in
order to maintain mass conservation in the domain.
The algorithm of implementation of the second order polynomial using the least square method can
be summarized as:
1. Identify the stencil’s points, using the method explained above.
2. Obtain the coordinates of the selected cell’s centroids as well as the material points coordinates in
IB face centered referential.
3. Construct the M matrix obtaining the values of x and y of the velocity
4. Computation of the β coefficients using equation 2.58 with the M matrix and the velocity values of
the previous outer iteration.
30
5. Apply the mass flux correction in order to maintain the mass conservation.
As the referential used was an IB face centered the value of φf = β0 and no further calculations were
needed.
2.6 Algebraic System of Equations Solvers
After all the steps presented in chapter 2 for the implementation and application to the flow rises
the problem of solving the matrix to get the solution. The problems in computational fluid dynamics
are solved in a way that turns the problem in an algebraic problem of the form Ax = b, being A a
sparse matrix where only the non-zero values are stored. Because of the size of the total matrix, it
would be computationally ineffective to store the thousands of zero terms in the matrix. This algebraic
problem is then solved iteratively. In this work the linear systems were solved using the bi-conjugated
gradient stabilized method (BIGCSTAB), and the incomplete LU decomposition. The AZTEC library of
FORTRAN was used and these features were already implemented in the SOL-code.
2.7 Immersed Boundary Method Mesh Treatments
The only thing thas was missing for the calculations conclusion was the insertion of the immersed
boundary in the mesh. In this work only cartesian and structured grids were used but all the mesh
treatments are applicable to unstructured grids. Two different treatments were executed in the cartesian
mesh, and a small explanation of them will be given in this section.
The first path that was taken to insert the immersed boundary in the mesh, was the simplest one and
was only used in the fixed IB boundary analysis, chapter 3. In this method an algorithm was implemented
in order to label all the cells with one of three features. These three features were, solid if all vertex of
the cell in question were inside the solid region, fluid if the cell center and all the vertex of the cell were
outside the solid region or IB if at least one vertex of the cell was inside the solid region and the others
were in the fluid region.
After this selection, the cells marked as solid were taken away from the mesh, and to the IB cell’s
faces that were neighbors of the fluid cells one of the interpolations from section 2.5 was chosen thus
providing us with the immersed boundary to begin our calculations.
An example of the result of this method is given in figure 2.4 for a 40x40 grid with an Immersed
Boundary, this mesh was used in the case of the analytical cavity, section 3.1.
This method would be effective if the main goal of this work was to study steady objects with the IBM.
However for the case of moving boundaries it was as computationally ineffective as a body-fitted mesh,
since the solid cells are obliterated from the grid. Thus in the case of a moving body a new mesh would
have to be created and new cell numbers would have to be found what would really interfere with the
speed of the computations. It’s important to remember that the main goal of this method was to prevent
the re-mesh at every time-step in case of a moving grid. Also, changing the total number of cells in every
31
Figure 2.4: 40x40 grid after the cut of an immersed 2-D cylinder with r=0.1
time-step implies changes in the memory allocation for the matrix of the system of equations in every
time iteration, which is inconvenient and computationally ineffective.
In order to ease the transition from one time-step to another a constant mesh would have to be
saved, let’s set the example of the 80x80 mesh present in the figure 2.5, the selection of the cells’ types
was the same as in the first method, but instead of cutting off the solid cells a new approach was taken.
Figure 2.5: Appearance of the 80x80 grid on the left, and the cloned faces on the right
This new approach was based in the cloning of the interface faces, instead of cut of the solid cells.
The faces of the IB cells that were neighbors to fluid cells were cloned and duplicated disconnecting the
fluid and solid cells. In other words the single block of mesh that existed was divided in two separate
blocks, one of fluid cells and one of solid cells. In the fluid cells block, the Navier-Stokes were solve
and all boundaries corrections were performed, while in the solid cells block the equations didn’t require
solving as the pressure and velocity fields are null inside of the solid region.
32
Figure 2.6: Schematic of the fresh fluid cells resultant from the boundary motion from one instance oftime to the following. Image from Mittal and Iaccarino [2005] figure 5 pp 252
As already said one of the main goals of this project was to obtain a procedure to compute the results
for a moving boundary. When this process was implemented we faced some new problems.
The main problem we had to solve comes from the motion of the body and the obvious variation of
the cells contained in the solid block of mesh and the cells contained in the fluid zone. What is implied
by these changes in the nature of the cells is that the cells that were in the solid region and change
to the fluid region, represented in yellow in figure 2.6, were not provided with and estimation to enter
the iterative algorithm to solve the Navier-Stokes equation. This happens because the cell’s numeration
is constant alongside all the computations. This results in the nonexistence of an initial estimation
for the values leading to a decrease in the method’s stability and preventing the iterative method from
convergence.
un0 i,j = Un−1IBcg + ω ⊗ (ri,j − rIBcg) (2.59)
The solution to this problem was obtained by indenting all the cells that changed from the solid
zone to the fluid block of mesh and including an initial guess for the computations equal to the value
of the immersed body center of mass velocity for the cases of non-rotating immersed boundaries and
applying the angular term for the cases when rotation was present, this led to equation 2.59. Where un0 i,j
represents the initial guess for the i, j freshly cleared cell, Un−1IBcg the velocity vector of the immersed
body center, ω the angular velocity vector of the boundary and ri,j − rIBcg the distance from the cell
center to the center of rotation of the immersed body, thus providing a valid initial guess for the whole
computations that insured convergence of the computed cases.
2.8 Fluid-Structure Interaction (FSI)
One of the goals of this work was the successful implementation of an algorithm that could provide
accurate results for the free motion of a body immersed in fluid, where its motion is caused by the
interaction of the fluid behaviour with structure.
For this purpose it’s important to explain a few steps of the implementation in this work. Knowing that
in this type of methods the geometrical boundary is not coincident with the computational boundary, the
33
boundary forces calculated with this method are obviously computed at the computational boundary.
This leads to some errors in the values of the forces in the physical boundary, so we had to ensure
a first order extrapolation using the normal component of the gradient values of the pressure and the
shear stress and computed it from the IB faces, where the forces were calculated, to the material points
in the geometrical boundary using the distance between them.
Having the local extrapolated values stored in the material points we proceeded to an integration of
the pressure and the friction drag at the boundary. Since the boundary was represented by a set of
points this integration was not trivial. This integration was performed applying equation 2.60 and 2.61,
to the full set of the material points and adding the discrete values to obtain the value of the pressure
and the friction in the boundary.
p(θ)k/2 = (θk+1 − θk) ||rk+1/2||︸ ︷︷ ︸||dl||
[p (θk) + p (θk+1)
2
](2.60)
Fpressure =
k=np−1∑k=0
−pk+1/2
rk+1/2
||rk+1/2||(2.61)
Where p(θ)k/2 represent the average value of pressure between two material points and ||dl|| the
discrete arc of circumference between θk+1 and θk.
The calculation of the shear stress in the computational boundary is similar but for this case in the
summation we substitute p with the respective contribution ν ∂vt∂t .
After the integration the values of the friction drag and the pressure a the boundary were applied in
the center of mass of the immersed body (∑
F = mbodya) and its dynamical behaviour was computed
with:
un+1 = un + a∆t (2.62)
un+1/2 =un+1 + un
2(2.63)
xn+1center = xncenter + un+1/2∆t+
a (∆t)2
2(2.64)
The interpolation for the value of u is computed using a scheme similar to the Leapfrog temporal scheme
as can be seen in Ferziger and Peric [1996]. Thus providing us with a viable solution to compute the
dynamic motion of the IB.
34
Chapter 3
Static Immersed Boundary Results
After the presentation of the principlesof computational fluid dynamics we can now start interpreting
some numerical results from our algorithms.
In this chapter, the results will be presented in order to obtain validation of the immersed boundary
method. The first section, section 3.1 will show the analytic results for a specific cavity, these results are
then compared with a numerical solution obtained with the Finite Volume method.
After the presentation of this results we have section 3.2 where the same case is tested but now with an
immersed boundary in the domain.
The last sections of this chapter, sections 3.3 present the results of the immersed boundary method for a
static 2-D cylinder in a cross flow with various Reynolds numbers. Tests were performed for a spectrum
of Re numbers that in our predictions would have a laminar behaviour.
3.1 Analytic Cavity
The first case in study was the flow on a cavity, an analytical solution for this problem was already
implemented in the software that was used throughout this work, and it was the result of the works of
T. M. Shih and Hwang [1989], M. Kobayashi and Pereira [1999] and Pereira and Pereira [2001]. The
comparison between the results of an analytical solution and a numerical solution will make possible the
verification of the model in order to proceed to more complex flows and geometries.
In the studied case the dynamic viscosity, ν = 1.0, which corresponds to Re = UL/ν = 1.0. These
values were chosen in order to have convective and diffusive schemes with the same load on the solu-
tion.
An additional source term was added to the 2-D momentum equations represented as:
B(x, y) =8
Re
[24
∫ζ1(x)dx+ 2ζ ′1(x)ζ ′′2 (y) + ζ ′′′1 (x)ζ2(y)
]−64 [Φ2(x)Ψ(y)− ζ2(y)ζ ′2(y)Φ1(x)]
(3.1)
35
Where the ′ represents a differentiation and the functions ζ1(x), ζ2(y) Φ1(x), Φ2(y) and Ψ(y) are:
ζ1(x) = x4 − 2x3 + x2 (3.2)
ζ2(y) = y4 − y2 (3.3)
Φ1(x) = ζ1(x)ζ ′′′1 (x)− ζ ′1(x)ζ ′1(x) (3.4)
Φ2(y) =
∫ζ1(x)ζ ′1(x)dx (3.5)
Ψ2(y) = ζ2(y)ζ ′′′2 (y)− ζ ′2(y)ζ ′′2 (y) (3.6)
This computation his performed in a 2-D cavity in a square domain of [0; 1]2 and all the boundaries are
Dirichlet boundary conditions, three of them with a zero velocity, and the fourth, the upper boundary
(y=1.0), has an imposed velocity of :
u(x, 1) = 16ζ1(x) (3.7)
The analytical velocity field, that has no dependence of the Reynolds number is:
u(x, y) = 8ζ1(x)ζ ′2(y) (3.8)
v(x, y) = −8ζ ′1(x)ζ2(y) (3.9)
and the pressure field is:
p(x, y) =8
Re
[24
∫ζ1(x)dxζ ′′′2 (y) + 2ζ ′1(x)ζ ′2(y)
]+64Φ2(x) [ζ2(y)ζ ′′2 (y)− ζ2(y)ζ2(y)]
(3.10)
The numerical solutions were obtained for four cartesian grids, being the coarser a 20x20=800 cells,
being the other successive refinements of 40x40 = 1600 cells, 80x80=6400 cells and 160x160=25600
cells. The CDS scheme was used for the diffusive term as the linear interpolation was for the convective
and the Gauss method for the gradient schemes of the Finite Volume discretization of the domain.
In order to analyze the deviation of solutions between the numerical and analytical results the mean
and maximum values of the error were plotted in relation with the hydraulic diameter of the grid in
figures 3.3. The maximum value of the error was the maximum deviation between the analytical and the
numerical result in a cell, the mean error was calculated as equation 3.11 states.
εu =1
Vt
NC∑i=0
‖uanal − usol‖Vi (3.11)
where Vt is the total volume of fluid cells and Vi the single cell volume.
So in figure 3.1, we can see on the first frame the velocity u component field, then on the second the
v velocity component of the analytal solution.
By using the analytic solution and the numerical solution it was possible to compute the error field
36
Figure 3.1: Analytic velocity fields
for both components of the velocity, that are plotted in figure 3.2. This part of the analysis is of vital
importance because after this analysis an immersed boundary would be placed in the domain in order
to validate the model.
Figure 3.2: Error between the numerical and analytical solutions of the NS equations in a cavity
The evolution of the results of the analytical cavity can be observed in table 3.1 and it can be evalu-
ated by its graphical representation in figure 3.3. With these data and knowing that the power law from
the regression of the error with regard to the hydraulic diameter implies the order of convergence of the
method it is possible to state for this case that p ≈ 2 which implies that the accuracy order for this Finite
Volume discretization is equal to 2.
3.2 Analytic Solution with Immersed Boundary
In order to test the algorithm of the IBM an analytic cavity similar the one previously shown was
implemented, but in this case, instead of only solving the NS equations an immersed boundary was set
on the domain, the Immersed body was a solid 2-D cylinder with a radius of 0.2, and in the solid material
Table 3.1: Values the maximum and mean error for both components of velocity in the Analytical Cavitycase
Figure 3.3: Decay of the mean error with the hydraulic diameter of the grid for the analytic cavity, on theleft for the u component of velocity and on the right for the v component
points the velocity of the analytical solution was implemented.
As said in section 2.5, three types of interpolation schemes were tested for the immersed boundary
and, in the next subsections, the differences in the results of these schemes will be analyzed.
The theoretical formulation of these interpolations was already explained, but in the case of the ana-
lytic cavity a slight difference is implemented, in order to mask the presence of the immersed boundary,
to have an analytical case to validate the model, instead of the collocation of the values of φf using the
solid points the same was implemented but using the analogous solution of the analytical cavity.
As said previously for every type of interpolation a convergence study was made, but in order to save
space only the results for the 160x160 grid are showed in this section.
The following figures represent, the velocity u component error field (εu) and v component error field
(εv).
3.2.1 Interpolation 0
In the case of Interpolation 0 instead of collocating the value of φf = φsolid in the face of cell P , what
was collocated was the value of φ of P cell of the analytical solution of the cavity problem.
In table 3.2 the results for the error are summarized, and we can view the values for mean and
maximum error in the domain for the case of interpolation 0.
In this case it is observable that the error increases at least one order of magnitude regarding the
analytical case without boundary, this was expected as the Finite Volume discretization for the case
shown in 3.1 was second order of accuracy and the order of accuracy of this interpolation is lower as
will be shown in section 3.2.3.
38
Figure 3.4: Velocity error field for the 160x160 grid with Interpolation 0
Figure 3.5: Decay of the mean error with the hydraulic diameter of the grid with Interpolation 0, on theleft for th u component of velocity and on the right for the v component
3.2.2 Interpolation 1
The case of interpolation 1 is still expected to be a first order interpolation and analogously to inter-
polation 0 the values that enter the interpolation are the values that would have been in the boundary’s
face cell if the boundary did not exist, then the process mentioned in chapter 2 is applied. The results
for the refined mesh can be viewed in figure 3.6. And analogously to the previous subsection the results
for the errors can be viewed in table 3.3. An evolution can be noticed in the results of this table, in
comparison with table 3.2, this was expected and can be explained in section 3.2.3.
Table 3.2: Values of the maximum and the mean error for both components of velocity in the AnalyticalCavity case with immersed boundary and interpolation 0
39
Figure 3.6: Velocity error field for the 160x160 Grid with Interpolation 1
Table 3.3: Values of the maximum and the mean error for both components of veloctiy in the AnalyticalCavity case with immersed boundary and interpolation 1
3.2.3 Interpolation 2
The second order interpolation is a little more complex than the first two. In the case of the analytic
cavity instead of putting the analytic value of the cell in the boundary, as in the other two interpolations,
these values are put in the construction of the matrix that will then be solved by the Least Square Method
in order to get the value that will be put in the boundary’s face.
The error field for the second order interpolation can be observed in figure 3.8. It’s observable in this
figure, in the case of the u component, that the error distribution is similiar to figure 3.2, and we have to
consider the fact that the error peaks in the immersed boundary are negligible, what can be interpreted
as good indicator that the method is indeed valid. In the v component we can see a zone near the
boundary with a high error (red zone), but this can be explained by the fact that even for the analytic
cavity case, in section 3.1, the zone for the highest error is located on the same place as the immersed
boundary is in this case, this occurs because the order of magnitude of the v component is almost one
scale (10−1) lower what leads to a higher dispersion of the error field on the domain, contrary to the
other interpolation schemes.
As the other two interpolations we can see the error summary in table 3.4. It is worth noticing that
in this case the error values are closer to the values from table 3.1, where the maximum deviation in
the error from the analytic cavity is found on the mean error in u component, εu, and has a numerical
value of 27%, for the other errors we have for εumax, εv, εvmax the deviation between the value with
40
Figure 3.7: Decay of the mean error with the hydraulic diameter of the grid with Interpolation 1, on theleft for th u component of velocity and on the right for the v component
Figure 3.8: Velocity error field for the 160x160 Grid with Interpolation 2
interpolation 2 and without IB is 3% , 22% and 1.9% respectively, for the most refined grid. This means
that for the maximum errors the two computations have almost identical errors.
In the following section, using figures 3.9 an analysis is performed in order to validate the premise
that this interpolation is second order accuracy, what would explain why the results for this part are closer
to the analytical results than the results from the other interpolations.
Error Decay
As a matter of interpretation of the results for the analytic cavity and in order to validate the model,
as said previously, the mean and maximum error for both components of the velocity for 4 different grids
were analyzed. These 4 values for each variable made possible an interpolation with a power law of the
Table 3.4: Values the maximum and mean error for both components of velocity in the Analytical Cavitycase with immersed boundary and interpolation 2
41
Figure 3.9: Decay of the mean error with the hydraulic diameter of the grid with Interpolation 2, on theleft for th u component of velocity and on the right for the v component
Error p0 p1 p2εu 0.659 0.904 2.018
εumax 0.359 0.406 1.955εv 0.865 1.114 2.026
εvmax 0.731 0.829 1.937
Table 3.5: Values of the power law interpolation for the decay of the error with the hydraulic diameter ofthe grid
results for maximum and mean error, this interpolation can give an objective criteria to determine the
order of accuracy of the interpolation in question.
In figures 3.5, 3.7 and 3.9 we can observe the decay in the errors with the decrease of the mean
hydraulic diameter of the grid. In table 3.5 the results of the power law for the interpolations are summa-
rized.
It’s observable that the decay of the error for the consecutive refinements of the grid is following a
pattern that can be described with a power interpolation. The power terms present in table 3.5 can guide
us on an interpretation of the results to validate the model.
In the case of the first interpolation, interpolation 0 (section 2.5.1) with the exception of the power
term for the maximum error in the v component of the velocity all the results are in the vicinity of 1 what
can let us deduce that the order of accuracy of this method with this prescribed interpolation scheme is
almost first order.
The second interpolation, interpolation 1, ( section 2.5.2) shows us the result for a supposed first
order interpolation with linear weighting of the distance between the solid point and the face’s centroid,
what means that more information is taken into account thus a higher accuracy order should be obtained.
The results show that eventhough the numerical value of the power law is higher than in interpolation
0, it is not yet enough to achieve second order accuracy, the decrease in the error is closer to 1 than
the other interpolation and in the case of the εv it’s even higher than the unity but still p ≈ 1 being this
interpolation first order accurate.
At last the Interpolation with a Quadratic Polynomial obtained with the Least Square Method, Interpo-
lation 2 (section 2.5.3) shows a real jump in the quality of the results. Not only the absolute value of the
mean and maximum error, in both components of the velocity, have decreased more that one order of
magnitude in this case relatively to the other 2 interpolations, but the power law for every single variable
is near 2, giving us second order accuracy with an Immersed Boundary method. This made possible
42
the inclusion of a boundary in the domain and still maintain the same order of accuracy of the results of
section 3.1 with a cartesian grid.
3.3 Flow Around a 2-D Static Cylinder
After the realization of the analytical cavity case the verification process continued to the next phase,
the analysis of a 2-D static cylinder in a steady cross flow. The results were obtained for three different
Re numbers the first two cases are with low Reynolds number flows, Re = 20 and Re = 40, and have
an expected steady behaviour, the last case is performed with a higher Reynolds number, Re = 100 in
order to obtain verification for a case that is expected to be unsteady.
The results in section 3.2 show that the second order of accuracy was only obtained with the LSM
interpolation. To save computational costs the decision not to perform the calculations for the other
boundary’s faces interpolation schemes was taken.
The data from Subhankar Sen and Biswas [2009], Guilmineau and Queutey [2002] and Cheny and
Botella [2010] was used for validation. So in order to achieve validation for this boundary treatment we
compared five different parameters from the literature: the drag coefficient(cD) associated to the flow; the
distribution of the pressure coefficient ( cp) around the physical domain, obtained with an extrapolation
of the pressure results in the computational boundary; the distribution of the friction coefficient (cf ) what
will specifically give the result for the separation angle (θsep); the last parameter will be the bubble length,
Lsep, that can be defined as the length at which the streamwise velocity measured in the centerline of
the cylinder returns to positive values.
The results and the dimensions of the grid are presented with regard to the diameter of the circular
cylinder, and for all cases shown in this work the chosen diameter was 0.02 m. The dynamic viscosity
was chose as 0.02 in order to have an easier control of the Reynolds number regarding the hydraulic
diameter of the cylinder. As the Reynolds number regarding the flow around a cylinder is described as :
Re =U∞D
ρν(3.12)
where U∞ represents the velocity at the inlet boundary, D the diameter of the cylinder, ρ the volumetric
mass of the fluid, and is prescribed with a unitary value by the software in which all calculations were
performed, and ν represents the dynamic viscosity. With the above mentioned value of ν, we can
conclude that in all calculations ReD = U∞.
Geometry and Grid
Before the presentation of the results it is important to show the chosen geometry and the process of
meshing. The geometry used to perform the calculations was a 2-D rectangle with Xu+Xd on the longer
sides and 2h in the smaller sides, having the cylinder fixed at the origin of the domain. The right-hand
side boundary condition (x = −Xu) was an inlet boundary condition, as explained in section 2.4.2, the
other 3 boundaries (x = Xd, y = −h and y = h) had a pressure outlet boundary condition. The upper
43
heightName total no Cells no Boundary cell faces Xu (D) Xd ( D ) h (D)m1 38048 80 5 35 5m2 152192 160 5 35 5m3 363072 320 5 35 5m3* 363072 320 5 10 5
Table 3.6: Geometry and grid data for the static boundaries computations
and lower boundary were chosen to have outlet pressure instead of wall because of the effect of the
wall’s boundary layers on the flow.
In this analysis, four different grids were tested, three of them in order to guarantee the convergence
of results with the refinement of the mesh and one with a different domain length in order to understand
the effect of blockage described in Subhankar Sen and Biswas [2009]. The blockage B is defined as the
ratio between the cylinder diameter to the domain width. The above mentioned article states that this
blockage parameter is of major importance in the analysis of a 2-D cylinder since there is a wide scatter
of results for B ≥ 0.01. The grids present in this chapter all have a blockage parameter, B = D2h = 0.01.
The features of the grids are shown in table 3.6 and the the meaning of Xu, Xd and h is explained by
figure 3.10.
Figure 3.10: Explanation of the domain in the left and mesh representation on the right
3.3.1 Reynolds 40
The steady laminar flow of Re = 40 around the circular cylinder is the main section of this chapter.
The bibliographic data for this particular case was more extensive than for the previous case, thus
making the comparison of results and its consequent validation easier.
Four different simulations were carried on, one for each of the grids presented in table 3.6. The fourth
simulation was performed only for this case because this flow had more bibliographic data. This flow
was expected to be a steady laminar flow.
With an immediate observation of figure 3.11 we can understand the similarities between the results.
The main difference can be pointed out as a small difference in the cp value over the leading edge is
a little higher than the values shown in the bibliography. This small discrepancy over the results can
be explained with two reasons. The first is the difference in the blockage effect of the computational
44
Figure 3.11: Image taken from Subhankar Sen and Biswas [2009]-pp 97, representing the distributionof the pressure coefficient along θ on the left. On the right distribution of pressure coefficient, cp with θ,obtained with most refined grid
calculations. In the case of Subhankar Sen and Biswas [2009] the figure shown represents a domain
with a blockage effect of B = 0.05 and in our calculations the chosen domain had B = 0.01.
The other source for the error comes from the calculations of the pressure around the physical
boundary. As the computational boundary is wider than the physical boundary and we only have pres-
sure values for the cell centers an extrapolation was needed in order to obtain the pressure around the
cylinder. This interpolation was effectuated using the values of the pressure gradient in the boundary’s
faces and the distance of each face center to the closest material point in the physical boundary of the
cylinder.
The separation angle θsep is defined as the angle at which the the boundary layer separates from
the boundary, which is represented as the angle at which the friction drag is null. In figure 3.12 we can
notice the friction distribution along θ for the more refined grid on the right and the distribution of the
friction coefficient in H. Ding and Xu [2004]. It’s important to state that as in figure 3.11 they started
to account the angle at the leading edge, and in the computations executed in this work the angle that
has a zero value is the trailing edge of the cylinder. Also worth of note, the image of the right has the
representation for the whole cylinder and the case from Droge [2007] only 180o are shown.
Figure 3.12: Distribution of friction coefficient, cf with θ, on the left for the medium refined grid, on theright picture from H. Ding and Xu [2004]-Figure 5.3 D pp 84
In the obtained results the maximum skin friction is higher than the results from Droge [2007], the
divergence in the results is explained by the fact that the computational boundary is not coincident with
Table 3.7: Results from different bibliographic sources and all grids for Re = 40
the physical boundary, besides that our boundary condition is not a true wall with no slip condition
but is almost as a source/sink term in order to correct the distance from the physical boundary to the
computational.
But in contrast to the result of the separation angle has an undeniable proximity to our obtained
values. This can be viewed in table 3.7. The maximum value of divergence in from the more refined grid
to the data from the bibliography is nearly 1.2% which is negligible.
Regarding the separation angle it’s important to state the grid convergence of results for the different
grids, being the results for the more refined grids more similar to the bibliographic sources’ results.
Another item that has to be compared in order to obtain validation to this model is the drag coefficient
Cd. As shown in table 3.7 the values for this coefficient are very similar too.
Figure 3.13: Representation of streamlines in the vicinity of the immersed boundary at Re = 40. Onthe left the results computed with the IBM, on the right figure taken from Subhankar Sen and Biswas[2009]-figure 6c p 101
In figure 3.13 we can observe a few streamlines in the wake zone of the cylinder. It is also possible
to observe the bubble lenght Lsep.
Although the figure from the bibliography does not have a scale it is possible to conclude that the
behaviour of the streamlines is similar and that even tough the value of Lsep has some small differences
the overall behaviour of the flow does not act different. The difference in the Lsep are justified by the fact
that the domains are different between both cases and the outer boundaries are not symmetry planes
but pressure outlet, that implies a higher blockage effect.
46
3.3.2 Reynolds 20
For the case of Re = 20, three different grids for a similar domain were used. In Droge [2007]
and H. Ding and Xu [2004] we can find the results for computational cases and the data from those
bibliographic sources is summarized in table 3.8. In figure 3.14 we can observe the streamlines obtained
by the performed test case. We can compare this streamlines with figure 3.13 to understand that the
wake zone is as expected smaller in the case of Re = 20. There’s an obvious similarity in both figures
from 3.14 helping to continue the validation of this method. We can observe in both figures the behaviour
of the separation bubble and the two small steady vortex in the wake region.
From the performed simulations we were able to obtain the distribution of pressure coefficient along
θ, present in figures 3.15 for the more refined grids.
Figure 3.14: Representation of the streamlines in the vicinity of the immersed boundary. On the right anedited figure from Xu and Wang [2006]- Figure 26 a) p 481. On the left hand side the results obtainedwith the developed method
Figure 3.15: Distribution of friction coefficient, cf with θ on the most refined grid
Figure 3.15 has the scatter representation of the friction coefficient along the angle of the circular
cylinder, once again we can find θsep where the friction coefficient has a zero value. In this case it
happens for an angle of 44.5o very similar to the cases in the literature, and once again being the angle
the comparison factor with the most confluent results in the bibliographic sources it is valious to the
validation of the model.
Other important feature to observe is the distribution of the pressure coefficient along the angle, that
can be found in figure 3.16. This figure shows the comparison between the values obtained by our
calculations and the calculations performed by Xu and Wang [2006]. We can observe that there’s a
divergence in the maximum value of the pressure coefficient. But even though the results have small
Table 3.8: Results from different bibliographic sources and all grids for Re = 20
differences from the bibliographic source if an analogy is made between figures 3.11 and 3.16 a relation
can be found. That happens because as the Reynolds numbers increases, the multiplication factor to
obtain the adimentionalization has an inverse 2nd order of power with U∞ that is directly proportional
to Re. This implies that for a decreasing Re as, we will have increasing aerodynamic adimensional
coefficients, and that is observed, as in figure 3.11 the maximum value for the pressure coefficient is
near 1 for the case of Re = 20 the maximum value of cp is near 1.5.
The summarized results can be found in table 3.8.
Figure 3.16: Distribution of pressure coefficient, cp with θ. On the left an image edited from Xu and Wang[2006] Fig. 30b) p 483. On the left the results obtained for the most refined grid
Once again it we have a small deviation for the Lsep factor present in table 3.8. The values of this
item have a significant change in the bibliographic sources analyzed for this work and it seems to be
very difficult to have an accurate result for it. We can observe too the proximity of results for the case of
θsep and for the drag coefficient specially for the most refined grid.
48
Chapter 4
Moving Immersed Boundary Results
The main objective of this work was the implementation of a method capable of obtaining the correct
behaviour of a fluid flow with an immersed moving boundary. In chapter 3 we proved with analytical data
and comparison with other computational methods that the chosen algorithm to handle the immersed
boundary was able to obtain results with second order accuracy as it was intended.
In this chapter the results for moving boundaries will be shown in various flows types. The first
section of this chapter, section 4.1, presents the results of the immersed boundary method with a moving
boundary for one of the cases specified in H. Dutsch and Lienhart [1997]. This specific case in the
literature presents computational and experimental data.
In section 4.2, a simulation was carried to obtain results to be compared with Guilmineau and
Queutey [2002] in order to understand if for another case of a moving boundary, the results were still
valid. In this case, the kinematics of the immersed boundary were forced into a sinusoidal movement in
a perpendicular direction of the flow.
A simulation similar to the one present in 4.2 was performed in section 4.3, but in this case the
moving boundary was oscillating in the streamwise direction.
In section 4.4 we studied the fluid-boundary interaction for a free rotating solid boundary in a cross-
stream.
The last section, section 4.5, can be viewed as an application study of a free moving cylinder inside
a chamber that simulates the flow of a 2-D whistle.
4.1 2-D Horizontally Oscillating Cylinder in a Static Fluid
This section presents the results for a simulation of an oscillating moving boundary immersed in a
static fluid, with Re = 100 for the maximum velocity of the body and a Keulegen-Carpenter number
KC = Ae
2πωD = 5 .
The present results were compared with the data available in H. Dutsch and Lienhart [1997]. The
computational experiment performed in this section was carried out to reproduce one of the analysis
performed in the mentioned literature.
49
The diameter of the 2-D static cylinder was defined as D = .02m, and for a square domain of [10D]2.
The center of the body was animated with a sinusoidal motion as described in equation 4.1.
xcenter = 0.159 sin (2π0.05t) (4.1)
The amplitude of the motion was established in order to obtainRe = 100 for a ν = 10−6, the frequency
of the oscillation was implemented to be equal to the literature. The maximum velocity of the body was
a result of the Reynolds number and the Keulegen-Carpenter number and was settled as Umax = 0.005.
As shown in chapter 2, in this type of method the Courant must be set with a value that doesn’t allow
the boundary to move more than one computational cell between time-steps, so the time step chosen
for these computations was dt = 0.1 and the courant number= Umax∆t∆x = 1/20.
The pressure and the vorticity isolines for four different phases of a cycle were captured and com-
pared with the computational results of H. Dutsch and Lienhart [1997], which can be seen in figures
4.1
0o (a) (b)
96o (c) (d)
192o (e) (f)
288o (g) (h)
Figure 4.1: On the left pressure isolines obtained with the proposed method at four different phaseangles, on right the results computationally obtained in H. Dutsch and Lienhart [1997], pp 258 figure 6,for four different phases of the periodic motion
In figure 4.1 we can observe the similarity in the pressure contours in both computational analyses
for the four phases of the motion described in the literature. Furthermore, in figure 4.2 we can make
a similar observation but this time for the vorticity isolines. In the frames from H. Dutsch and Lienhart
[1997] the negative values of vorticity are represented in a dashed line and the positive values with a
complete line. The results obtained by this method represent the negative values of vorticity in a scale
50
of blue and the positive in a scale of red.
0o (a) (b)
96o (c) (d)
192o (e) (f)
288o (g) (h)
Figure 4.2: On the left vorticity isolines obtained with the proposed method and, on right the resultscomputationally obtained in H. Dutsch and Lienhart [1997], pp 258 figure 6., for four different phases ofthe periodic motion
In order to obtain validation for this method, a comparison with experimental results was also nec-
essary. In figure 4.3 we can observe the comparison of the x component of the velocity for three
different phases of the cycle. In each phase of the cycle, the velocity was analyzed at two different
lines:x = −0.6D and x = 1.2D. There’s similar tendency of the experimental data with the results
computed with the immersed boundary.
The data presented in figure 4.3a) shows that there’s a velocity profile caused by the forward motion
of the IB that generates a low pressure zone in its wake thus inducing the velocity in the fluid. This
velocity profile is the most intense in all eight frames because of the proximity of the body to the probe in
this instant. Another interesting feature of this frame is the negative velocity present at ||y|| > r, where
a negative velocity is induced by the continuity equation, to respect the mass balance.
The second frame has a velocity profile with small intensity, that is caused by the fact that in the
moment when those velocities were captured the body was going towards the probe and was still far
from it.
At the second moment of capture, corresponding tp 210o, we can observe that the intense velocity
profile from frame 4.3a) is dissipating as the body moves away from the probe while at x = 1.2D location
the velocity profile is starting to get preponderance as the body moves closer to the probe.
Finally when the IB reaches 330o, the body is in the same place as it was in previous frames but in
this case it’s moving backwards. The velocity profile shown in figure 4.3 f) is caused by the low pressure
zone inherent to the wake of the body. The lower intensity of this frame compared to the first is caused
by the distance of body to the probe in the moment of the capture of the velocities.
51
180o (a) (b)
210o (c) (d)
330o (e) (f)
Figure 4.3: On the left the results for x = −0.6D and on the right x = 1.2D. The red line representsthe computational results obtain with the IBM and the black squares represent the empirical data fromH. Dutsch and Lienhart [1997], pp 258 figure 6. The y-axis represents the y coordinate of the line probeand the x-axis represents the u component of the velocity
Finally in figure 4.3e) we have a similar behavior to frame b), as the body is moving towards the
probe but it’s still far away.
4.2 2-D Oscillating Cylinder In Cross-Flow
In this section we replicated the computational experiment executed by Guilmineau and Queutey
[2002].
This computational experiment is intended to predict the behaviour of a flow with Re = 185 with a
cylinder oscillating in a perpendicular direction to the stream. The amplitudes of the oscillations were
defined as Ae/D = 0.2.
The center of the oscillating cylinder was ruled by equation 4.2 in the y coordinate while the x coor-
dinate is maintained constant.
52
ycenter(t) = Aesin(2πSeS0t) (4.2)
where S0 represents the wake oscillations frequency of the unsteady case at Re = 185 which according
to the bibliographic sources is S0 = 0.201Hz and Se represents the percentage. The value for the
cyllinder oscillation frequency was chosen to be below the natural oscillation frequency with Se = 0.8,
meaning the oscillation frequency was set to 80% of the natural shedding frequency.
In figure 4.4 it’s possible to see this simulation. The first frame (a) shows the stream lines obtained
by the calculations made with the tool developed for this work and the second frame shows the re-
sults obtained by Guilmineau and Queutey [2002], the original computational experience we intended to
replicate.
The distribution of the stream lines in both frames is very similar, in both frames we can observe that
the stream lines bend in the wake region, this bending is obviously caused by the momentum applied by
the immersed body in the fluid surrounding it.
(a) (b)
Figure 4.4: On the left (a) the streamline distribution obtained with the proposed method for Se=0.8, onright the results obtained at Guilmineau and Queutey [2002], figure from 11a, pp 787
In the figure 4.5 we can observe the distribution of the streamwise velocity in the domain and we can
note the presence of an unsteady laminar wake for this type of flow. Cross-referencing figure 4.5 and
figure 4.4 we can state that for this flow, with the body animated with a movement with a frequency that
is 80% of the shedding frequency there’s no formation of vortex, and the subsequent zones with negative
streamwise velocity as was present in the literature, thus providing the method with another successful
comparison.
4.3 2-D Horizontal Oscillating Cylinder
In this section, instead of having the oscillations in the normal direction to the flow we implemented
a sinusoidal movement to the boundary on the streamwise direction of the flow.
With the data presented in figure 4.6 we were able to analyze some quantitative features of this fluid
flow. It was evident that the behaviour of the drag coefficient is sinusoidal as expected.
53
Figure 4.5: Distribution of streamwise velocity with Se=0.8
Figure 4.6: Drag coefficient over time for the inline oscillations, time is adimentionalized by U∞D
54
(a) (b)
(c) (d)
Figure 4.7: In this group of figures we have represented: (a) the vorticity countours; at (b) the pressure;(c) the stream wise component of the velocity; (d) the other component of the velocity. All presented atthe origin when the IB has a positive center velocity (going downstream)
In figures 4.7 to 4.10 we can observe the behaviour of the main variables of the flow at 4 different
phases of the boundary motion.
In 4.7 we find the vorticity, pressure and both components of the velocity when the body passes the
origin moving alongside the flow direction.
As predicted, as the IB is moving downstream we can observe in figure 4.7 in the second frame (b)
two stagnation points represented with a local maximum in the pressure field. The y-component of the
velocity is symmetric regarding the y = 0 axis. In the stream-wise component of the velocity we can
observe a small distance of the wake region from the IB, this distance is explained by the momentum
that the immersed boundary is transferring to fluid in that region.
(a) (b)
(c) (d)
Figure 4.8: In this group of figures we have represented: (a) the vorticity countours; at (b) the pressure;(c) the stream wise component of the velocity; (d) the other component of the velocity. All presentedwhen the IB reaches its most downstream position
55
The second phase of the motion represented in figure 4.8 is at the peak of the downstream position of
the immersed body. In this moment the velocity of the IB is zero. The stagnation point at the trailing edge
disappears and we note the presence of a low pressure wake (b) and negative streamwise velocity(c).
Simultaneously the vorticity contours start developing a zone with contrary values of vorticity, which can
be explained by the negative acceleration of the body.
(a) (b)
(c) (d)
Figure 4.9: In this group of figures we have represented: (a) the vorticity countours; at (b) the pressure;(c) the stream wise component of the velocity; (d) the other component of the velocity. All presented atthe origin when the IB has a negative center velocity (going upstream)
In figure 4.9 the representation of the moment when the IB passes back at the origin but now with
an upstream velocity (u < 0 ). This four frames are very different to the situation presented in figure 4.7.
The pressure in the leading edge is higher because the IB is moving towards the fluid, thus influencing
the wake to have a lower pressure and really low velocity.
(a) (b)
(c) (d)
Figure 4.10: In this group of figures we have represented: (a) the vorticity countours; at (b) the pressure;(c) the stream wise component of the velocity; (d) the other component of the velocity. All presentedwhen the IB reaches its most upstream position
56
In figure 4.10 we have the immersed body at its most upstream position, the velocity of the body is
equal to zero. Again we can observe a small formation of vorticity in the wake region.
In this flow we can observe the behaviour of these 4 variables and analyze the influence of the
changes caused by the periodicity of the body’s velocity. Since the body is animated with a periodic
velocity we can assert that the effective Re number of the flow is instantaneous. We can observe real
differences in the stages where the higher and lower velocities are implemented, at this two points we
have really different effective Re numbers for the flow. Besides that, when the body is animated with
downstream motion, a new stagnation point is created in the trailing edge. Obviously this stagnation
point, with its pressure peak, has a major influence on the velocity behaviour in the wake downstream
of the body.
4.4 Rotating Flattened Cylinder
One of the advantages of this type of algorithms in the resolution of flows around immersed bound-
aries is the possibility to perform analysis with arbitrarily defined bodies without the requirement of
creating a new computational mesh, unlike the case of body-fitted grids.
This shape was defined by 3 semi-circles, one with radius equal to r = 0.01 and the other two with
half of the radius, producing the form present in figure 4.11. This geometry was chosen in order to have
a simplified model of a section from winding Eucalyptus peel tube, one of the most common reasons of
the propagation for forest wild fires.
Figure 4.11: Representation of the flattened 2-cylinder form, composed by 3 semi-circles, one withradius r and another two with radius=r/2
This part of work was mainly motivated as part of an ongoing investigation regarding forest fires.
The premisses of this investigation was that the winding Eucalyptus peel was usually animated with an
angular frequency of 10Hz which meant an angular velocity of ω = 20π /s. We computed our solutions
for a Re = 40.
Once again we expected a periodic solution for this flow. Although we were on a steady laminar
domain, as can be seen in section 3.3.1, the asymmetry of the body and the rotation imposed on it
made us expect that the perturbations induced on the fluid would have a sinusoidal behavior.
In figures 4.12 we can see the pressure distribution for eight phases of a period, namely θ = 0, θ = π4
, θ = 1π2 , θ = 3π
4 , θ = π, θ = 5π4 , θ = 3π
2 , θ = 7π4 .
In these 8 frames it is possible to observe that the pressure around the immersed boundary is chang-
ing with the position of the flattened half of the body. From those frames we can remark that in 4.12a)
57
(a) (b) (c) (d)
(e) (f) (g) (h)
Figure 4.12: 8 phases of the distribution of pressure for one cycle of the flattened cylinder at Re = 40and f = 10Hz, and streamlines representation
Figure 4.13: Evolution of drag with time
and 4.12e) the peak of pressure in the stagnation point is lower that in the other phases of the motion.
Another interesting fact present in figure 4.12 is the variation of the streamlines with the rotation of the
body. We can see in 4.12a) that the vortex is really small and in the following 4.12b) and 4.12c) there’s
a development of the vortexes until figure 4.12d) is reached, where the vorticity starts decreasing.
When we reach the second half of the motion we can see that once again there’s formation of vorticity
but in this case the length of recirculation is larger. The vortexes that were formed in the first half of the
cycle are dissipated by the rotation of the body.
Just as the pressure in the leading edge varies with time, the distribution of the pressure in the rest
of the boundary is also changing accordingly which will produce a temporal variation in the aerodynamic
forces represented in figure 4.13 for the case of the force in the streamwise direction. In table 4.1 we
have the minimum, maximum and average values for the aerodynamic forces of this case.
Force Average Maximum MinimumDrag (N) 21.31 25.02 15.97Lift (N) -0.59 4.235 -5.94
Table 4.1: Values for the average, minimum and maximum values of the aerodynamic forces for theflattened cylinder case
58
In figure 4.13 we have the representation of the streamwise aerodynamic force in time. The evolution
of this force is not a pure harmonic as it’s noticeable the presence of a local maximum in each period.
This local maximum is explained by the asymmetry of the body. When the flattened half of the body is
in the trailing edge the surface exposed the wake behaves like a cylinder of radius equal to the radius of
the bigger semi-circle. In the other halve of the period, the behaviour is influenced by the complexity of
the geometry.
4.5 Free Motion of 2-D Cylinder Inside a Cavity
Another example of a possible application for this method, and similarly for the computational tool
implemented in the SOL-code, is to obtain the results of the behaviour of an immersed body inside a
chamber with a circular flow. The most representative flow of this type is the flow inside a whistle, having
the interaction between the flow and the small sphere inside the chamber.
In our literature survey we were able to understand that noise produced by a whistle is a result of
the exit jet break down, shedding creates the acoustic characteristic field of the whistle. The interaction
between the immersed cylinder inside the chamber as explained in Liu [2012] is not the dominant effect
but it modulates and destabilizes the output jet that typically impinges in a blade to create the shedding
mechanism.
The tool was only developed for 2-D problems, so instead of a sphere inside a 3-D chamber we
simplified the problem for a bi-dimensional flow inside a chamber with a 2-D circular cylinder inside.
This domain has one particularity, the mass flow that comes from the inlet is going directly to the
outlet and the flow inside the chamber is created with the shear stress from the stream going to the
outlet. This shear layer is accelerating the steady flow inside the chamber thus influencing the pressure
field around the immersed body, this pressure differential will then force the body to move.
The behaviour of the body is computed with the dynamic laws of acceleration after the integration of
the pressure and shear stress in the whole boundary, that is obtained by the resolution of the Navier-
Stokes equations for the fluid region of the domain, as explained in section 2.8.
The collisions between the IB and the chamber’s walls were computed as elastic collision with e = 1.
A simplification had to be performed because when the boundary was close to the limits of the domain,
when there was less than two cells between boundaries, a pressure peak would be present in these
cells making the method fail to converge.
In order to solve this problem an offset for the fixed boundaries was programmed to the code. Re-
suming every time the distance between the immersed body and the one of the fixed walls was lower
than ε = 0.003, ε being the distance of the immersed boundary to the wall, which corresponds to three
cells, an elastic collision was simulated and the the body’s velocity component that was perpendicular
to the fixed wall switched sign.
With figure 4.14 we are able to understand the behavior of the immersed boundary inside the cham-
ber as these eight frames represent two cycles of the body dynamics.
In the first frame 4.14a) the IB is at center of the chamber and starts moving towards the high
59
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 4.14: Eight frames represent two periods of the quasi-cyclic behavior of the IB inside the whistle,velocity magnitude
60
momentum and low pressure zones, represented in red. When the IB reaches this high momentum
zone , it creates a low pressure zone in the trailing edge, observable in the dark blue bubble, thus
projecting the immersed boundary.
The momentum gained in the projection is high enough for the IB to reach the chamber’s wall and
therefore collide with it, the moment before this collision is represented in c) as the IB is moving towards
the wall, and in d) the IB is moving away from the wall.
After the collision and the dissipation of the momentum gained in frame b) the cylinder repeats the
behavior as it is sucked to the low pressure zones and cycle continues.
61
Chapter 5
Conclusions
The IBM was implemented in an unstructured finite volume code and several verification and valida-
tion tests were undertaken.
Three different interpolation schemes for the boundary correction were tested using an analytical
solution in order to assert if it was possible to maintain a 2nd global discretization order of the Finite
Volume.
From this analysis we were able to conclude that if the polynomial used in the correction of the
immersed boundary is of the same order (in this case second order) of the discretization method it is
possible to obtain the FV global second order accuracy. This polynomial was constructed using solid
material points and the cell centroids from the fluid cells where the correction was supposed to be
implemented. The polynomial is obtained using a regression based on the Least Square Method.
Analysis for flows with Re = 20 and Re = 40 with an immersed 2-D cylinder were performed with an
effective success in both fields, the qualitative and quantitative. For these cases we were able to obtain
results for the separation angle, θsep, drag coefficient, CD, and for length of the wake zone, Lsep. These
results have shown a good correspondence with the results from the literature. The pressure distribution
and the shear stress alongside the cylinder were computed and a comparison with the ones from the
literature was also successful. In order to obtain the pressure distribution and the shear stress along
the immersed boundary we had to implement a tool for the extrapolation of these quantities from the
neighbor fluid cells to the solid material points.
The IBM was applied to a moving boundary for a periodic case experimentally investigated by
H. Dutsch and Lienhart [1997]. The pressure and vorticity contour plots as well as the velocity curves at
different line probes and time instants were compared with the computational and the experimental data
available. The comparison of these values was satisfactory thus asserting the validation of the method.
Finally the IBM was applied for the case of a rotating Eucalyptus peel tube with an incoming flow at
Re = 40 in order to study the aerodynamic behaviour of this complex geometry and to a 2-D cylinder
moving freely in a cavity flow that was the simplification of a whistle with an immersed body inside
its cavity. The dynamic laws of motion were used based in the resultant aerodynamic forces from the
interaction between the fluid and the body. These dynamic laws included the ellastic collisions of the
62
cylinder with the wall of the cavity of the computational domain.
The proposed method is reliable to solve problems that include the immersion of a body in a fluid
flow and it improves the efficiency using parallel computation producing accurate results at a low com-
putational cost, specially for low Re number flows.
In summary this type of method is in an area of computational mechanics that has room for improve-
ment and a promising investment thus providing the aerospace market with powerful tools to analyze
the behaviour of bodies immersed in fluids.
Future Work
An improvement to this method would be the evolution to a 3-D domain, since the principles are the
same and are already well defined. The current method can also be applied to unstructured grids in
application where this feature could be important.
Other future implementations of this method is to obtain the behaviour of a body immersed in a
turbulent flow with a high Re number,by using either RANS (Reynolds Average Navier-Stokes) or LES
(Large Eddy Simulation). These methods are currently being implemented and validated in the SOL-
code.
The last suggestions are related with the order of convergence of the method, where adaptivity
and high order discretization schemes will improve the results. In the case of high order discretization
schemes, it was proven in this work that the only requirement for the immersed boundary correction is
that the least square polynomial degree needs to be equal or higher than the order of the FV discretiza-
tion.
63
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