APPLICATION OF THE TRANSPIRATION METHOD
FOR EFFICIENT AEROELASTIC ANALYSIS
USING AN EULER SOLVER
By
CLINT COLEMAN FISHER
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
1993
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 1996
APPLICAnON OF THE TRANSPIRATION METHOD
FOR EFFICIENT AEROELASTIC ANALYSIS
USING AN EULER SOLVER
Thesis Approved:
Dean of the Graduate College
ii
ACKNOWLEDGMENfS
I would foremost like to express my deep gratitude to my major advisor, Dr.
Andrew Arena for his guidance and for the tremendous opportunity that he gave me. His
assistance and direction were invaluable. I would also like to thank my committee
members, Dr. David Lilley and Dr. Peter Moretti.
Funds for the support of this study have been allocated through the NASAAmes
University Consortium Office, under Interchange Number NCC2S10S, and Oklahoma
State University. I wish to express my sincere appreciation to these funding organizations
and to the technical director of the study, Dr. Kajal K. Gupta of the Dryden Flight
Research Center.
Twould also like to give my appreciation to the Department ofMechanical and
Aerospace Engineering for granting me teaching assistantships that helped support me
during my graduate studies.
Finally, I would like to thank my parents for their support, motivation, and
continuous encouragement.
iii
Section
TABLE OF CONTENTS
Page
1. INTRODUCTION 1
1.1. Background 1
1.2. Problem Statement 2
1.3. Motivation for the Study 3
1.4. Rationale 6
1.5. Flutter Analysis 6
1.6. Literature Review 7
1.6.1. Transpiration Concept 7
1.6.2. Transpiration in Full Potential Studies 8
1.6.3. Transpiration in Euler Studies 11
1.7. Objectives of the Present Study 14
2. I\1ETHODOLOGY 16
2.1. Transpiration Development , 16
2.2. Steady Panel Methods 18
2.3. STARS Code 19
2.3.1. Flow Solver 21
2.3.2. Aeroelastic and Aeroservoelastic Solver 22
2.3.3. Code Modifications 24
2.3.3.1. FROMOD 26
2.3.3.2. SOLMOn 27
3. RESULTS 28
3.1. Transpiration Limitations 29
3.2. Steady Flap Case 30
3.3. Plate Case 34
3.4. AGARD Wing 54
3.5. Mesh Sensitivity 66
3.6. Plate with 0.5 Generalized Displacements 70
4. CONCLUSIONS AND RECOMMENDATIONS 74
IV
4.1. Conclusions 74
4.2. Recommendations 75
BIBLIOGRAPHY 77
APPENDIXADDITIONAL DATA 79
v
LIST OF TABLES
Table Page
3.1. Generalized Forces for Mach 0.3 42
3.2. Generalized Forces for Mach 0.95 48
3.3. Generalized Forces for Mach 3.0 52
3.4. Generalized Forces for Old and New Meshes 69
3.5. Summary ofResults for 0.5 Generalized Displacement Plate 71
VI
LIST OF FIGURES
~~ p.
2.1. Transpiration Implementation 18
2.2. Method of Surface Deflection Flowchart 25
3.1. NACA 0012 Airfoil with Deflected Flap 31
3.2. NACA 0012 Surface Pressure Results From Panel Code 32
3.3. NACA 0012 Surface Pressures from Panel Code and from Raj 33
3.4. Surface Mesh for Plate Case 35
3.5. Six Vibrational Modes for Plate Case 37
3.6. Deflected Plate Surface 38
3.7. Actual Pressure Contours, Mach 0.3 39
3.8. Pressure Contours Using Transpiration, Mach 0.3 39
3.9. Pressure Cut Stations 40
3. 10. Pressure Profiles, Mach 0.3 41
3.11. Actual Pressure Contours, Mach 0.95 .43
3.12. Pressure Contours Using Transpiration, Mach 0.95 43
3.13. Pressure Profiles, Mach 0.95 45
3.14. Mach Profiles, Mach 0.95 47
3.15. Actual Pressure Contours, Mach 3.0 49
3.16. Pressure Contours Using Transpiration, Mach 3.0 49
VIi
~re p.
3.17. Pressure Profiles, Mach 3.0 '" ." 51
3. 18. Residual Convergence 53
3.19. AGARD Wing 54
3.20. Deflected AGARD Wing 55
3.21. Actual Pressure Contours, Mach 0.99 S6
3.22. Pressure Contours Using Transpiration, Mach 0.99 56
3.23. Pressure Profiles, Mach 0.99 58
3.24. Actual Pressure Contours, Mach 1.141 S9
3.25. Pressure Contours Using Transpiration, Mach 1.141 60
3.26. Pressure Profiles, Mach 1.141 61
3.27. Actual Pressure Contours, Mach 2.0 62
3.28. Pressure Contours Using Transpiration, Mach 2.0 63
3.29. Pressure Profiles, Mach 2.0 , 64
3.30. Original Plate Mesh , 66
3.31. Pressure Profiles for New and Old Plate Cases 68
3.32. Deflected Plate, 0.5 Generalized Displacernents 70
3.33. Time History of Generalized Forces 72
3.34. Average Percent Error in Generalized Forces 73
A.l. AGARD Wing 80
A.2. Deflected AGARD Wing, 2.63 Units Bending and 0.33 Units Torsion 80
A.3. Actual Pressure Contours, Mach 0.678 81
A.4. Pressure Contours Using Transpiration, Mach 0.678 81
Vlll
Figure Page
AS. Pressure Profiles, Mach 0.678 82
A6. Actual Pressure Contours, Mach 0.99 83
A7. Pressure Contours Using Transpiration, Mach 0.99 83
A 8. Pressure Profiles, Ma,ch 0.99 " ,. 84
A9. Actual Pressure Contours, Mach 2.0 85
AIO. Pressure Contours Using Transpiration, Mach 2.0 85
All. Pressure Profiles, Mach 2.0 86
IX
CHAPTER I
INTRODUCTION
1.1. Background
Ever increasing technological advancements have lead to the development of
aircraft that operate at greater and greater velocities. Today, aircraft that are designed and
manufactured to perform near and above sonic speeds are much more common. For
example, it is typical for aircraft, from military fighter planes to modern airliners, to have
cruise speeds in the transonic range. In the design and performance analysis of these
vehicles, it is important to study their aeroelastic characteristics. Even for vehicles
traveling at subsonic speeds, the flow can cause a body deformation in the structure,
which in turn will affect the flow solution. The coupling between fluid and structural
forces can induce unstable oscillations, Therefore, it is important to accurately predict a
vehicle's flutter boundaries, since inaccurate predictions could result in vehicle failure.
Often in modern threedimensional aeroel,astic analysis, a flow solver is employed
in conjunction with a structural solver. The solutions of the flow field and the surface
oscillation are performed simultaneously. Many flow solvers used today employ a grid
and the surrounding flowfield. Movement ofthe body under investigation due to the
surface loads must be accounted for. This requires that either the surface grid be modified
to match the body deflection, or that the deflection be simulated by some means. One
possibility for simulating the deflection is to modify the surface normals at their original
locations. Thus, to the flow field, the surface appears to have an altered form. This
concept is known as the transpiration boundary condition and is the focus of this study.
1.2. Problem Statement
In unsteady aeroelastic analysis, the position and orientation ofthe surface under
investigation are a function of time. In order for the solution to be determined accurately,
the surface deformation must be represented. The most direct representation would
appear to be regenerating the computational grid. However, in a time stepping approach,
this would require that the grid be regenerated at each time step, since the surface will
change with every time step. Using this procedure with present computer capabilities, a
solution for a single set of parameters using grid regeneration could be on the order of
weeks.
The present study uses an integrated computer code called STARS (STructural
Analysis RoutineS) that is capable of performing the steady and unsteady flu~d and
structural analysis of flight vehicles that are required for determining flutter boundaries.
The Euler based flow solver is a recent addition to the code and is capable of simulating
2
threedimensional compressible inviscid flows. It uses finite element techniques with
unstructured adapted meshes of tetrahedral elements. Flow solutions using this code are
performed in a timemarching fashion. Thus, in unsteady aeroelastic applications, the
surface deformation must be represented at each time step. This is accomplished using the
transpiration boundary condition in which the original computational domain remains
unaltered. The surface deflections are simulated by applying the deflected body normals at
the undeflected body location, thus reducing the time required for determining flutter
boundaries.
1.3. Motivation for the Study
In any field of study it is important to search for ways of improving
solution characteristics, whether it be solution accuracy, expediency, or cost efficiency. A
coupled fluidstructure time marching solution can be highly time consuming. For
example, on present highspeed workstations, the calculation of a single fluidstructure
transient on a threedimensional aircraft configuration using the Euler equations may
require over 100 cpu hours. Therefore, it is highly desirable to develop means of reducing
the required computational time while maintaining solution accuracy. Advancements in
computer technology are continually assisting in this task. However, faster computers are
generally more expensive, so what is gained in speed may be lost in cost. By introducing
3
new concepts in lieu of or in conjunction with existing methods vast improvements in
solution characteristics can often be made without significantly adverse effects.
Considering the time required to generate a single domain mesh for a flow
solution, it seems highly impractical to rediscretize the computational domain at each time
step. One alternative to this rediscretization was presented by Batina (1989) where he
used a dynamic mesh algorithm that models a triangulated mesh as a spring network.
Each edge of each triangle in the mesh is modeled by a spring whose stiffness for any edge
ij is inversely proportional to the length of the edge.
In this algorithm, the grid points on the outer boundary of the mesh are fixed and a
predictorcorrector procedure that iteratively solves the static equilibrium equations in the
x and ydirections at each time step is used to determine the displacements &xi and Oyi at
each interior node i. The method predicts displacements according to
8>. =2& >"  &~1
, , I
It then corrects the displacements using several Jacobi iterations ofthe static equilibrium
equations using
4
where km is the spring stiffness, and the summations are over all edges of the triangles that
have node i as an endpoint. Finally, the new nodal coordinates are given by
This algorithm was found to produce good results when used in unsteady pitching and
plunging studies, and also in the prediction of flutter boundaries. However, it is necessary
to perform the displacement calculations at each node for every time step in both
coordinate directions. Additional computations will be required if the algorithm is
extended to threedimensions.
This number of calculations can be significantly reduced if the surface deflection
can be simulated without having to alter the existing grid. One way to simulate body
deflection is to apply a transpiration boundary condition at the surface. This is essentially
done by rotating the body normals so that the new normals are in the same directions they
would be in if the body had actually deflected. Thus the original body grid remains
unaffected throughout the flutter investigation.
Transpiration has been used effectively in simulating surface deformations in full
potential solutions and steady and unsteady rigidbody applications in Euler equations. It
5
is the purpose of this study to show the extent to which the transpiration boundary
condition may be used in unsteady aeroelastic problems.
14. Rationale
When the transpiration boundary condition is employed, there is potential for great
savings in time requirements. However, an expedient solution is useless if its accuracy is
questionable. Therefore, showing the extent to which the transpiration boundary
condition is valid will allow flutter investigations to be made much more quickly and with
confidence in results when inside the transpiration limits.
1.5. Flutter Analysis
Pertaining to aerodynamics, flutter is the divergent oscillation of a surface resulting
from a coupling between structural and fluid forces. The prediction of flutter boundaries
has been the subject of a great deal of studies. Methods that have been used in many
recent studies include full potential and Euler methods. Both methods have proven
effective in many flutter investigations
Full potential methods offer accurate solutions in a wide range of applications with
relatively low computational costs. As with Euler methods, .there is the assumption of
6
inviscid flow. However, since these methods use an approximation of the Euler equations,
they require further assumptions. One assumption in the development of the full potential
equation is irrotational flow, which in most cases is a reasonable solution. These
equations also do not allow for entropy changes across shocks. Thus, the existence of
shocks, even in subsonic flow, will introduce inaccuracy to the solution.
Euler methods produce a higher order, more accurate solution than the full
potential methods. They are capable of accounting for viscous and entropy effects,
therefore they can be used for studies of a much wider variety than the full potential
methods, such as at high Mach numbers and with strong shocks. One drawback
with these methods is that they are more computationally intensive. However, with ever
increasing advancements in computer technology, these methods are becoming much more
computationally affordable and widespread.
1.6. Literature Review
1.6.1. Transpiration Concept
In 1958, M. 1. Lighthill presented four alternatives for the treatment of
displacement thickness. One of these alternatives was termed "method of equivalent
sources". The idea used in this method has today developed into what is known as
transpiration. Rather than thickening an airfoil to account for the boundary layer, an
7
equivalent surface distribution of sources is used to • simulate' a thicker airfoil. This is
done by modifying the normal velocity just outside the boundary layer to include
additional outflow due to the boundary layer.
1.6.2. Transpiration in Full Potential Studies
One early study in which transpiration was used in conjunction with a flutter
solution was performed by Sankar, Malone, & Tassa [1981]. The study was performed
using a full potential method. Although not explicitly stated, transpiration was used in
"simulating" the first order bending of an oscillating rectangular wing in subsonic flow.
This was done by applying the zero normal velocity boundary condition for the deflected
surface at the undeflected wing position. Computations were also performed by applying
the boundary condition at the actual surface. The authors reported making both
computations to ensure that the differences were small, however, results were only
presented for the transpiration solutions.
Lift, moment, and phase results were presented for the transpiration computations
at a Mach number of 0.24 and were compared to experiment and Kernel function
solutions. It was concluded that the results compared reasonably well. It was noted that
the simulated results more closely resembled the Kernel function than the experiment.
This was attributed to neglecting viscous effects and the fact that the experiment used a
8
5% thick circular arc airfoil. Furthermore it was concluded that the simulation accurately
and reliably predicts unsteady subsonic potential flow.
This transpiration boundary condition was further used in a study of a fighter wing
in transonic flow [Malone, Sankar, & Sotomayer, 1984]. The study incorporated
transpiration with the full potential equations to estimate the 1st harmonic, real and
imaginary components of unsteady surface pressures at eight different stations on an F5
fighter wing. Computed and experimental results were presented for three transonic Mach
numbers (0.8,0.9, and 0.95). It was concluded for this study that the results correlated
reasonably well. It was further concluded that this method could be used in studying
flutter behavior of fighter type wings at transonic speeds,
Based on these validating results, Malone & Sankar [1985] used transpiration with
full potential equations to study the unsteady pitching oscillation for the RAE wingbody
model in transonic flow (Mach = 0.8). Unsteady surface pressure results were presented.
however, no experimental data was available for comparison,
Then, in 1986, a study was performed exclusively to compare results from a full
potential method using the exact boundary condition and the transpiration boundary
condition [Sankar, Ruo, & Malone]. Results were presented for a NACA 64A006 airfoil
with an oscillating trailing edge flap, a large aspect ratio wing in independent pitching and
plunging, a plunging fighter wing, and a steady rectangular wing.
The NACA 64A006 results were computed at a freestream Mach number of 0,875,
The results showed that pressure distributions were within plottable accuracy of one
another, and that integrated loads were within 10% of each other, The first harmonic out
9
of phase component of the surface pressure distribution was also determined for the flap
case and compared to experiment. Results for both exact and transpiration approaches
and experimental data were found to be in close agreement, the only appreciable difference
being near the sonic line.
The next configuration in the study was a large aspect ratio wing. Results were
presented for three plunge velocities (corresponding to 1, 5, and 10 degrees steady angle
of attack) at Mach = 0.77. At 1 degree, the results proved to be nearly identical. At the
larger plunge values, the results were not as close but still very good (within 10%).
Pitching results using the transpiration approach were also computed at Mach = 0.66 and
compared to results from a vortex lattice method (for pitching rates of 1, 5, and 10
degrees/second). In each case, the transpiration prediction was larger than that of the
vortex lattice method, the worst case being over 30%. However, the vortex lattice
method does not consider airfoil thickness effects and therefore underpredicts the airloads.
Surface pressure distributions at four span locations were present for a Mach 0.95
fighter wing undergoing constant plunging motion (I. 5 degrees effective plunge velocity)
Transpiration and exact approaches were compared with experiment. Both approaches
were found to closely match experiment, except for slightly smeared shocks predicted by
transpiration. This was suspected to be due to the use of a somewhat coarser grid in the
transpiration case.
The final study used transpiration to simulate steady viscous effects of a
rectangular wing in transonic flow. The wing was analyzed at an angle of attack of2.0
degrees and a Mach number of 0.8. Surface pressure distributions were calculated, with
10
and without viscous corrections, and compared with experiment. Solution accuracy
showed improvement when transpiration was used for viscous corrections.
It was concluded from this study that for small amplitude motions, as in aeroelastic
applications, the transpiration boundary condition provides accurate results. Furthermore.
the authors cited a considerable savings in coding effort and memory requirements when
using the transpiration boundary condition.
1.6.3. Transpiration in Euler Studies
The transpiration boundary condition was used with the unsteady Euler equations
in a study of transonic flow past a fighter wing by Sankar, Malone, & Schuster [1987].
The pitch oscillation of the fighter wing was accounted for by changing the boundary
condition and leaving the original surface unmodified. Inphase and outofphase
components of the surface pressure distribution were calculated at four span locations and
compared with experiment. The fighter wing Mach number was 0.8, and the pitching
amplitude was 0.113 degre,es at 40 Hz (zero mean angle of attack), Slightly higher
suction levels were predicted by the Euler solver for the inphase component near the
leading edge. However, overall experimental and calculated results were found to be in
very good agreement. It was concluded by the authors that the unsteady Euler solver with
transpiration was robust enough and accurate enough to be used in aeroelastic studies.
11
Transpiration was again used in the unsteady Euler investigation oftwo pitching
wings; a transonic (Mach 0.82) transporttype wing and a subsonic (Mach 0.7) rectangular
wing [Ruo & Sankar, 1988]. The transporttype wing was the LockheedAir ForceNASA
NLR (LANN) wing. Its oscillating pitch amplitude was 0.6 degrees at a frequency
of 24 Hz. The rectangular wing's oscillating pitch amplitude was 2 degrees at a frequency
of 10 Hz.
Inphase and outofphase components of the surface pressures were presented for
calculated and experimental results at four span locations for each wing. For the LANN
wing, there are significant differences in results at some locations. However, according to
the authors, the experimental data at many locations is not considered reliable for this
wing. When the unreliable data is ignored, results for both components show good
agreement. For the rectangular wing, calculated and experimental results agree
everywhere except near the wing tip for the inphase component, which may be due to
viscous effects.
Midspan unsteady pressures were compared for both exact and transpiration
methods. For the LANN wing, a steeper variation of the pressure near shock waves was
predicted by transpiration, but away from shocks, the results were very similar. For the
rectangular wing, almost identical results were predicted by both methods.
For this study, it was concluded that the overall agreement of results was good.
However, the issue of which boundary condition was more favorable for smallamplitude
motions was inconclusive.
12
More recently, a study was performed using transpiration with an Euler method to
simulate control surface deflections [Raj & Harris, 1993]. The study included the trailingedge
flap deflections for a NACA 00 12 airfoil and for an arrowwing body configuration.
Actual and simulated surface pressures were presented for the NACA 0012 airfoil with a
10 degree flap deflection at Mach numbers 0.6 and 0.9. For both Mach numbers, the
transpiration method produced results that were in very good agreement with actual
results.
The arrowwing body configuration was analyzed using both a coarse and a fine
grid, the fine grid having higher resolution around the wing. Simulated results were
computed and compared with experimental data. Surface pressure distributions were
presented at three span locations for 0, 4, and 8 degrees angle of attack with and without a
flap deflection of 8.3 degrees. No experimental data was presented for the zero degree
case with no flap deflection, but results for the fine and the coarse grids were in close
agreement. Results from both grids matched well with experimental data for flap
deflection at zero angle of attack.
Some noticeable differences between computed and experimental results can be
seen for the outermost station in the 4degree angle of attack case, while there is better
agreement at the inner stations. However, these differences are most likely due to
limitations of Euler equations, such as neglecting viscous effects, rather than transpiration.
Also, the finer grid appears to produce data near the leading edge that is slightly closer to
the experimental.
13
In the 8degree angle of attack case, results for the innermost station are very
close. Differences are seen at the middle station, but they are not a result of transpiration.
The differences are a result of Euler computations producing attached flow where
experimental data suggests flow separation at the leading edge. The outennost station
results are not as good as the innermost, but results there are comparable.
Experimental and simulated lift and pitchingmoment coefficients were also
compared for the arrowwing body case. Results were presented for 0, 8.3, and 17.7
degree trailingedge flap deflections at Mach 0.85. For each flap deflection, the simulated
lift and pitching moment results were in very close agreement.
From this study, the authors concluded that transpiration boundary condition was
effective in estimating the changes in aerodynamic forces, moments and surface loading
due to controlsurface deflections, assuming that the Euler equations are capable of
modeling the flow field. It was noted that transpiration was not suitable for simulating
configurations that may produce geometric gaps.
1.7. Objectives of the Present Study
The STARS group, for which this study was performed, has recently added an
unsteady Euler flow solver that uses transpiration to simulate surface deflections and
deformations. The primary objective of this research is to employ this new code over a
14
wide Mach number range in the analysis of unsteady aeroelastic problems, and to
document the extent to which the transpiration boundary condition is effective.
It is evident that the transpiration boundary condition can be used effectively to a
large extent in applications ofrelatively small displacements. It is also apparent that this
boundary condition can be an effective tool in aeroelastic investigations. However, it is
important to be aware of what circumstances will cause the boundary condition to
introduce appreciable inaccuracies. Therefore, this study will present the rationale for
when and why the transpiration boundary condition will give inaccurate results and to
perform. investigations to support the rationale.
A secondary objective of this study is to determine how much influence grid
resolution has on any given solution. This is a direct result of the primary objective
because it is important to distinguish between inaccuracies due to transpiration and mere
differences due to mesh sensitivity.
15
CHAPTER 2
METHODOLOGY
In this research effort, the transpiration concept was applied to various steady and
unsteady cases. The preliminary results were obtain using a steady panel code that was
modified to use transpiration for steady deflections. The majority of the results obtained
from this study used an Euler based code that employs the transpiration boundary
condition in steady and unsteady surface deflections and deformations. Computer codes
were also developed in this study to produce actual surface deflections for use with the
Euler code.
2.1. Transpiration Development
The idea of transpiration was first developed by Lighthill [1958] as a method of
equivalent sources. Lighthill modified the normal velocity just outside the boundary layer
of an airfoil through an equivalent surface distribution of sources to "simulate" a thicker
airfoil. In this way, the effect of a boundary layer is present in the solution without
16

physical representation of a true boundary layer. Applied mathematically, the nonnal
velocity w is
i7. Ow IZ au . dU alZ dU d lot w = ~z = ~z =z+ (u  u):iz =z+ (u  u):lz
o oz 0 ox. dx Ox 0 dx dx 0 .
where z is the distance from the surface and u is the x component of velocity, which takes
the value U just outside the boundary layer. The first term is the original, unmodified
nonnal velocity. The second term represents the normal velocity contribution from the
boundary layer. Thus, the flow field "sees" the effects of a boundary layer that is not
physically present
In this study, transpiration is applied as a boundary condition such that geometric
changes and motions are simulated through surface transpiration. The unsteady boundary
condition states that the velocity component normal to the body surface must equal the
velocity of the body surface,
thus there can be no flow through the surface. With transpiration, this done by adding a
velocity component normal to the tangential velocity so that the resultant velocity is at
some angle to the original surface, as shown in Figure 2.1.
17
°old
Vtranspiration
Figure 2.1. Transpiration Implementation
This velocity is then taken as the new tangential velocity. Thus the solution is performed
on a seemingly different surface.
2.2. Steady Panel Methods
In general, panel methods use a distribution of singularity elements (e.g. sources
and vortices) over a solution boundary to satisfy the solution boundary conditions. This is
accomplished by using discrete singularity "panels" over the body surface (and possibly
other areas, such as the wake). Combining the velocity potential of each singularity
element with that of the freestream, the continuity equation is solved with the proper
boundary conditions to give a unique solution for the velocity potential.
18
One of the conditions that is imposed in a panel method (as in other methods) is
flow tangency~ that is, there can be no flow through the body surface. Thus, in the
solution ofa steady panel problem, the boundary condition is
V·n =0
must be satisfied on each panel, where n is the panel normal. To apply transpiration to a
panel method, essentially all that needs to be done is to rotate either the panel angle or the
panel normal.
The first investigation that was performed used transpiration with a 2D, steady
panel code. This code, written by Arena [1993], employs the SmithHess panel method.
Results were computed for an NACA 0012 airfoil with an actual and a simulated trailing
edge flap deflection. In the actual solution, the coordinates of the original airfoil were
changed at the flap location to create a 10 degree flap deflection. For the transpiration
solution, the angle of each panel in the region of the flap is modified with respect to the
free stream in the calculation of the influence coefficients.
2.3. STARS Code
The computer code system used to generate the majority of the comparison data
for this study was an extension of the original STARS (STructural Analysis RoutineS)
19
computer program. STARS is an integrated FORTRAN code for the multidisciplinary
analysis of flight vehicles (STARS users manual, 1995). Features ofthe system include
structural analysis, computational fluid dynamics, heat transfer, and aeroservoelastic
modules. The most recent version was written under the direction ofK. K. Gupta (1995)
at the NASA Dryden Flight Research Center primarily to support NASA flight operation,
and research and development projects. Until recently, STARS used linearized
aerodynamic theory for the prediction ofthe unsteady flowfields interacting with the
elastic motion of an aircraft. This technique produces adequate results for a wide range of
problems but has some significant limitations, such as only producing valid results for
small perturbation flows. The technique also assumes simple harmonic motion which
hinders studies of arbitrary or transient motions.
Due to these limitations, an unsteady Euler CFD module was added which uses
transpiration to simulate body surface motion. However, this addition was limited to
simulations of singledegreeoffreedom, rigid body, simple harmonic motions. Since a
primary responsibility ofthe STARS group was to calculate aeroelastic effects in support
of fight test operations, the program was modified by the STARS group (Gupta, 1995) at
NASA Dryden to allow calculation of arbitrary motion and aeroelastic effects.
The code uses finite element analysis to derive the frequencies and mode shapes of
the structure. The modal superposition method is then used for the dynamic structural
response analysis. The recent addition to the code is a module that enables the
computation of unsteady aerodynamic forces employing the finite elementbased structural
20
and computational fluid dynamics computations. This code was validated with theoretical
and experimental results.
2.3.1. Flow Solver
The flow solver portion of STARS is an Euler based code capable of simulating
three dimensional compressible inviscid flows. It uses finite element techniques with
unstructured adapted meshes oftetrahedral elements. The mesh is generated using an
advancing front technique, which has the advantages ofapplication to arbitrary shapes,
varying grid density in the domain, and the ability ofadaptive mesh generation in
accordance with solution trend.
The flow solver is comprised of different modules that perform mesh generation
and flow solution. The five main modules are as follows:
• SURFACE: generates the two dimensional front
• VOLUME: generates the threedimensional tetrahedral mesh
• SETBND: sets the boundary conditions on the domain
• EULER_STEADY: performs the steady Euler flow solution
• EULER UNSTEADY: performs the unsteady Euler flow solution
21
There is also mesh geometry and flow visualization through the XPLOT and ZPLOT
modules of the STARS system. Adaptive mesh techniques are also possible through the
module REMESH.
To begin a flow solution study, the user must define the curve components, surface
components, curve segments, and surface regions that are necessary to describe the
geometry of the problem. The user must also define the background mesh and the nodal
spacing parameters. Taking these definitions as inputs, the code automatically generates
the surface and volume meshes using the advancing front technique.
The user must then define the boundary condition types for the curves and surfaces
to be used by the preprocessor. Finally, the user must create a namelist file that assigns
flow conditions and coefficients. The code then takes all of the user input files, plus the
files it generates in constructing the mesh and preprocessing the flow infonnation, and
performs the steady flow solution. The unsteady flow solution is then perfonned using the
steady flow solution as the initial condition.
2.3.2. Aeroelastic and Aeroservoelastic Solver
The structural solver performs nonlinear, CFDbased aeroelastic and
aeroservoelastic analysis. The structural modeling uses a finite element method, thus
creating a unified approach using finite element analysis for both the flow and the
structural solution. The natural frequencies (co) and modes (4)) are computed by solving
22
Mii+Ku =0
where M and K are the inertial and stiffness matrices, respectively, and u is the
displacement vector. The steadystate Euler solution is then performed using either an
explicit or a quasiimplicit, local time stepping solution procedure that employs a residual
smoothing strategy. The equation of motion in the frequency domain is
Mil + Cq + Kq + f. (t) + fJ (t) =0
where
~
M = inertial matrix (= <t>TM<1», and similarly
K,C = stiffness and damping matrices
q = displacement vector (= <t>TU )
~
f. (t) = aerodynamic (CFD) load vector (= <t> ~ pA), where p is the Euler
pressure, A the appropriate surface area, and <t>. the modal vector
pertaining to aerodynamic grid points interpolated from relevant
structural nodes
and
~
fl(t) = impulse force vector (= <1>Tf, )
where fi is a number of modes input by the user. The statespace form of the equation is
23
where
x [:J
A ~ [:'K d,cJ
b.(I) ~ [M~f. (I)J
b,(I) ~ [M~f,(I)J
and a time response solution of the statespace equation in an interval nt (= tnI  tn) is
obtained as
The structural deformations u and velocities u are then computed and used by the CFD
code to change the velocity boundary conditions at the solid boundary. Then a onestep
Euler solution using a global timestepping scheme is performed and the process is
repeated for the required number of steps.
2.3.3. Code Modifications
The original unsteady Euler code that was used in this investigation performs
surface deflections using transpiration only. If comparisons of these transpiration resuhs
24
were to be made against actual deflection results, it would be necessary to produce a
method of generating a deformed surface mesh. Essentially, the problem reduces to
assembling a code that modifies a given surface as desired and a code for calculating the
correct normals. Once the deflected surface is obtained with the proper normals, the
steady Euler code can be used to perform the flow solution.
The two codes generated to complete the process for obtaining the actual solution
were FROMOD and SOLMOD. A flow chart depicting the process is presented in Figure
2.3.
Mode
Shapes
Figure 2.2. Method of Surface Deflection Flowchan
The flow chart shows that the original surface mesh does not need to be modified to
deform the surface. A discussion of each code is presented in the foHow sections.
25
2.3.3.1. FROMOD
The three basic components required for performing surface deflection are the
original, unmodified mesh, the mode shapes of the different modes, and scaling factors
specifying the generalized displacements. The unmodified mesh can be obtained by
generating the mesh on the original surface. The mode shapes are obtained from the finite
element structural solver. The final requirement is a set generalized displacements, which
is provided by the user.
The original surface is altered by displacing each node point in each coordinate
direction, x, y, and z, on the surface by an amount that is determined from the mode
shapes and their generalized displacements. For example,
xncw(i) = Xold(i) + .1x(i)
where ~x(i) is determined by
n
~x(i) =L <t> a (i, j)* f] (j)
J=1
where <1>.(i,j) is an array of mode shapes
1;0) is an array of generalized displacements, input by the user
and n is the number of modes
26
Essentially, FROMOD is used to generate a deformed surface mesh. It reads in
the surface definition and the generalized forces and prompts the user for the modal
scaling factors. Using this data, the code generates an array of nodal displacements and
adds these displacements to the original nodes, as previously described. Then the code
writes the new surface file. One advantage of this procedure is that the connectivity of the
surface is unchanged, so it can simply be transferred from the old surface file to the new
one. The final requirement is a code to calculate the normals for the deflected surface.
This code is discussed in the following section.
2.3.3.2. SOLMOD
A second modification code was required to complete the process for determining
actual deflection solutions. This code, called SOLMOD, was needed to calculate the
surface normals for the new mesh. SOLMOD reads in the file containing the normals,
corrects the normals, and then overwrites the old normals file with the new one.
The computational process for modifying the normals already existed in the Euler
code because it is used in the transpiration solution. In this process, the normals at each
node in the mesh are calculated using an area weighted average of surface triangulations
that contain that node. Since this coding was already present, it was simply combined
with the proper read and write statements to create SOLMOD.
27
CHAPTER 3
RESULTS
The main objective of this study was to determine the effectiveness of the
transpiration boundary condition in unsteady aeroelastic applications. This was to be done
by performing investigations of various geometries over a wide range of Mach numbers.
By doing so, this boundary condition could be employed in the prediction offlutter
boundaries. The rationale is to discuss the instances in which transpiration will introduce
error into the solution, and to generate comparison data to show the effectiveness of
transpiration in various applications. The following section contains a discussion of the
limitations. Subsequent sections present results demonstrating the capability of
transpiration.
The first set of results that is presented is for a steady, trailing edge flap deflection,
as a preliminary investigation. The next two sets of results comprise the core of the
research, the first being a 2 by I flat plate and the second being the AGARD 445.6 wing.
In these cases, solutions were performed for actual deflections and compared to simulated
deflections using transpiration. The final cases of the study address the issue of mesh
sensitivity in computational fluid dynamic solutions
28
3.1. Transpiration Limitations
The idea of transpiration is essentially to alter the boundary condition when
performing a flow solution. In using transpiration, if the surface under investigation
deflects, the surface normals are modified to account for the deflection. Therefore, if the
deflection is small, transpiration can very accurately predict the solution. This is because
the relative position of one part of the surface to another will have little effect. However,
as the deflection increases, the accuracy of the transpiration solution will decrease. Since
transpiration only accounts for the orientation of the normals, the effect of translation
between two points on the surface will not be accounted for in the solution.
As an example of this translation problem, consider a wing that has first mode
bending such that the free end is no longer in the same plane as the fixed end. When
transpiration is used to simulate the bend, the normals along the wing will be altered in the
wing's original position. Thus the effect that a point on the displaced end has on points
elsewhere will not be completely accurate, and increasing the displacement will add to the
maccuracy.
One consequence of the translation effect is error introduced by intersecting
shocks and surfaces. Consider a shock wave originating at the nose of an aircraft whose
wing is oscillating in first mode bending. In the transpiration solution, the shock will
29
intersect the wing in its original position. However, in the actual solution the wing is
displaced, thus the shock intersection will not be in the same location.
Other problems such as this could result when there is internal surface translation
of the body under investigation. As mentioned, however, in cases of small deflection, such
as in flutter problems, transpiration can produce very accurate results.
3.2. Steady Flap Case
In order to gain a better understanding of the transpiration concept and to perform
some preliminary investigations, a panel code was modified to employ transpiration.
Actual and simulated surface pressures were computed for a NOO 12 airfoil at Mach 0.6
and zero degree angle of attack with a steady 10 degree flap deflection. The results were
obtained using a FORTRAN code written by Arena that employs the SmithHess panel
method. Slight modifications to the code were made to obtain the transpiration results.
Figure 3.1 shows the deflected flap, located at 80% of the chord.
30
O. 5 rrr"~,r.,...rr.,..,
0.4 3+++tt++++;
0.3 ++++tt++++;
0.2 4I+++lI+++t
0O.iE"1tt±±a~~
0. 1++++++++t+I
0.2 ++++++++++l
0.3 ++++++++++I
0.4 ++++++++t+I
0.5 4.",~:+rm+:miImr+'l"I'TT'i~'!'TThrr+:rrM
O ..... N(T')~I,{)C'Ol'roo> .....
000000000
Figure 3.1. NACA 0012 Airfoil with Deflected Flap
Actual and simulated surface pressures were generated using the SmithHess code. These
pressures were corrected for compressibility effects using PrandtlGlauert correction. The
corrected results are presented in Figure 3.2.
31

~
~Sf~
=>X'1<Q(
11 it ;Jl
c~1(
)Olel(' 0
X~Q( ~SlR oix)( 0
0
0
0
~:()'t>' :>"0<0< <
°c~~
IS ~ ~  d<....
I "'0<00;
~·5 O"'d!
X .",l~~
0 . x O)() ~olio(po'" '"II It
x
0x
0 S.H. actual
x S. H. transpired
.. . / .. J II I I "1 • r I
1.5
1
0.5
Cp 0
0.5
1
1.5
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
xJc
Figure 3.2. NACA 0012 Surface Pressure Results from Panel Code
It can be seen from Figure 3.2 that transpiration is very effective in simulating the flap
deflection with only slight differences occurring near the flapairfoil intersection.
This study was also performed by Raj using a steady Euler method that also
employs transpiration. In that study, transpiration results were also compared against
actual flap deflection results. Figure 3.3 shows the actual and transpiration simulated
results obtained by both the SmithHess code and by Raj.
32
~.
0:':; "<j1 ~iC
C {l~O
"', ~~ 0
(;! ~~~~ ~"
i, i>,.! x~cfi} Q211i ~,.j'lCI!l" 0
o III
° x 0
°c x:
~ llD~1Ij1 fae....
2 wI' 'h.C\!l :ra•• ~~ II be,. ~"lJ> ~: l>. P.R. actual 't
!(f'o
• 0 P.R. transpired
~
jI 0 S.H. actual
It " SH transpired
, . . I'" 'I' .. , I'" I
1.5
1
0.5
Cp 0
0.5
1
1.5
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
xJc
1
Figure 3.3. NACA 0012 Surface Pressures from Panel Code and from Raj
The results from the two studies are in very close agreement. This not only shows the
effectiveness of transpiration, but it also serves as a validation of the SmithHess results.
Considering the magnitude of the deflection. the transpiration boundary condition
appears to be very effective. This preliminary investigation suggests that transpiration has
strong potential in simulating significant deflections.
33
3.3. Plate Case
One ofthe primary cases in this. investigation involved a flat plate 2 units long and
1 unit wide. This type of plate is representative ofconfigurations that have been used in
well documented studies on panel flutter. As defined by Dixon, panel flutter is a selfexcited
oscillation ofthe external surface skin of a flight vehicle which results from the
dynamic instability of the aerodynamic, inertia, and elastic forces of the system.
Information gained from studies involving panel flutter has assisted in the pursuit for flight
vehicles with increasing speeds. This investigation of panel flutter is gives a new approach
to this well known problem, as it allows for flutter investigations in which the surface
under consideration is never required to actually deform.
The plate used in this study was centered on a surface 4 units long by 3 units wide.
This plate and the surface mesh are shown in Figure 3.4.
34
Figure 3.4. Surface Mesh for Plate Case
This figure shows that the grid resolution on and immediately around the plate is very
high. This was necessary to generate a smooth flow solution over the plate. The
freestream flows over the top of the plate, with both sides of the plate being at
atmospheric pressure. The plate edges are pinned to the bottom surface. NI surfaces are
defined as walls and all edge boundaries are defined as far field. The plate is flat, but can
be made to defonn, as described by Section 2.3.3, or defonnation can be simulated using
transpiration. The deflection can be any combination of six bending modes of the plate.
In plate flutter, the magnitude of deflection in each mode would not necessarily be the
same. However in this study, all modal deflections were of equal magnitude for an
arbitrary deflection.
35
The first case compares actual and transpiration results for a generalized
displacement of 0.1 in all modes. As an illustration, each mode shape is shown separately
in Figure 3.5.
36
Figure 3.5. Six Vibrational Modes for Plate Case
37
The superposition of modes that produced the surface used in this case is presented in
Figure 3.6.
Figure 3.6. Deflected Plate Surface
The resulting deflection produces a maximum deflection that is over 7% of the width of
the plate. This case was selected as an ini.tial attempt to detennine limitations of
transpiration. Arbitrary deflection was used to maintain generality; 0.1 amplitudes were
used to give a significant deflection. Surface pressures and generalized forces were
compared for rreestream Mach numbers 0.3,0.95, and 3. In all cases, the actual and
simulated surface pressures were very similar. The generalized forces were comparable,
but did show some appreciable differences.
Figures 3.7 and 3.8 show actual and transpiration surface pressure contours,
respectively, for the Mach 0.3 case.
38
Figure 3.7. Actual Pressure Contours, Mach 0.3
Figure 3.8. Pressure Contours Using Transpiration, Mach 0.3
39

It is evident that these results are very similar. For a more detailed comparison of pressure
results, crosssectional surface pressures are presented at three stations across the plate.
These stations run lengthwise along the plate and are located at onequarter, onehalf, and
threequarters of the width. Figure 3.9 shows the stations across the deflected plate.
Figure 3.9. Pressure Cut Stations
The pressure profiles at these stations are presented in Figure 3.10.
40
"
J~ 1/ ' ) '
\. J \ ,/
V ~
0< "
,  ..J ~ J1'~. . 
\ J 'i,. I
\I \ ;  .~.
1"\
~ I I\...
V 1\ I
\ I


0.45
0.30
0.15
Cp 0.00
0.15
0.30
0.45
1 0.5 0 0.5 1 1.5 2 2.5 3
X
0.45
0.30
0.15
Cp 0.00
0.15
0.30
0.45
1 0.5 0 0.5 1 1.5 2 2.5 3
X
0.45
0.30
0.15
Cp 0.00
0.15
0.30
0.45 .
1 0.5 0 0.5 1 1.5 2 2.5 3
X
v Transpiration
o Actual
"T1
~.
...,
(1l
w
......
0
'"0
I ~
;
Y~,!ll'.J\'I , I
(1l
Ji,c:::,W
Vl
\ " IXI ';1 v.> I I (
~.~
C
~
+ ...,
(1l  .".".,C
0
~
(1l
~a: ~
(") :r
0w
These pressure profiles show that there is very good agreement between the two results.
The largest differences appear to be in regions where the plate is furthest displaced from
its original position, which would be expected. Table 3.1 shows the generalized forces for
the six modes of each case.
Table 3.1. Generalized Forces for Mach 0.3
Mode Actual Transpiration
1 22.14 41.66
2 87.48 91.38
3 54.55 57.12
4 4.649 2.229
5 17.65 18.56
6 57.16 58.33
The generalized forces are very comparable except in modes 1 and 4. Large differences in
generalized forces will be discussed at the end of this section.
The surface pressure contours for the Mach 0.95 case are shown in Figures 3.11
and 3.12.
42
Figure 3.11. Actual Pressure Contours, Mach °95
Figure 3.12. Pressure Contours Using Transpiration, Mach 0,95
43
Again, the pressure contours in each case are very similar. The pressure profiles for this
Mach number are presented in Figure 3.13.
44

C")
It')
N
('\j
~
T"" X
It')
ci
0
It')
c?
...,.
~'"' I
"<Ill
" ... ~ 1
~I'"
~ ".,
4.......
0 .~
~..... ~
I. " I ,
C")
It')
('\j
('\j
It')
T"" X
It')
ci
0
It')
c?
T"" ,
L.. I
~
c ... .1'\..
:"'l'lIl""
~~ rP
( hi. ....1:10
L•• m" ....~""" .
~
,
'1 I
C")
It')
('\j
('\j
It')
x
It')
ci
0
It')
q
T""
I
I ....
" 0' '"boa
~
.",
L.",.oo
("
.....,..0 I..;.;
I~'"r\
•
I I I
~ ~ ('\j 0 ('\j ~ to (XJ to '<t ('\j 0 ('\j ~ ~ (XJ <D '<t ('\j 0 N '<t <D (XJ
0 ci ci ci c? 0 , q c? ci ci ci ci c? c? 0I c? ci 0 ci ci 9 cIi c? c?
0 0 co
0 0
t I c / .Q
i.i.i.
ro '0.
.2 'c"
/ I III () ... « IFigure
3. 13, Pressure Profiles, Mach 0.95
45
These pressure profiles show excellent agreement between the two data sets with only
slight differences being seen near some pressure peaks and in some regions ofsharp
pressure changes. The transpiration solution also seems quite capable of handling the
formation ofthe two shocks. This can be seen again in the Mach profiles at the same
locations in Figure 3.14.
46
•i
"
.""
.oj
~
~
~ .....
)
..~r. . .... .0
J
:,,,•
LO"tMN ..... OO'lCOrlD
,;~~~~~oc:icici
JaqwnN Lpe~
M
U"l
N
N
~
X
U"l
6
0
U"l q
..
I
i,~
#,.0 ... r
"q Ic~ ....., ....
,.~
f:'" .
, ~ ""'" ... "'00
I"'" ;•
...I
10 "tMN..OO'lCOrlD
~~~~~~oooo
JaqwnN Lpe~
M
10
N
N
~..
X
10
6
0
U"l
6I
....I
......•
1
~.....
l 0 0 • 51
~ ~ ~ ~Itj'lij
~
~ . •
"" ......
I
!••
,. ,. ..
U"l"tMN ..... OO'lCOrlD
,....:r~~~,....:OOoo
JaqwnN 4:le~
C")
10
N
N
~..
X
10
6
0
U"l
9
..
I
1
I c
.Q
iii / '
iii
'0..
VI
.2 c
I co
I u '
~ I
\ /
Figure 3.14, Mach Profiles, Mach 0.95
47
Notice how there are only minor difference near the shocks, and seemingly no differences
at the shocks. The generalized forces for this case are presented in Table 3.2.
Table 3.2. Generalized Forces for Mach 0.95
Mode Actual Transpiration
1 283.2 475.1
2 1457 ~ 1483
3 1500 1495
4 61.35 34.72
5 233.5 236.3
6 869.1 886.9
These forces are also comparable, with some noticeable differences again being seen in
modes 1 and 4.
Figures 3.15 and 3.16 show the surface pressure contours for the Mach 3.0 case.
48
Figure 3. 15. Actual Pressure Contours, Mach 3.0
Figure 3.16. Pressure Contours Using Transpiration, Mach 3.0
49

As in the two previous cases, the pressures appear to be very much alike. Figure 3. 17
shows the pressure profiles.
50
,.
,
",
"'I
:,".~
"
'<
I: ,1
I,
"
""
'.,:,
A'
~ ';
• l.
1 ~ .... : 1.
'\. I , y'"
"II"'" ,...",. .
11.
~""l
~ "'\
\ I \  \ J , ,, ~
IW"'" 

L I I I I I I ,
f,l
11
I 
I I ... il J I 
H
,
+I
1
 I
I
L \ 7
 
7_ o· ~g
"\l,.,I
~...........
......... 
1
"T ....rrJ
05 1 1.5 2 2.5 3
X
0.30
0.25
020
0.15
C
0.10
P 0.05
000
0.05
0.10
0.15 .
1 0.5 0
0.30
0.25
0.20
0.15
C
0.10
P 0.05
0.00
0.05
0.10
0.15
1 0.5 a 0.5 1 1.5 2 2.5 3
X
0.30
0.25
0.20
0.15
C
0.10
P 0.05
0.00
0.05
0.10
0.15 .
1 0.5 a 0.5 1 1.5 2 2.5 3
X
o Actual
• Transpiration
'Tj
~.
'"1
(t)
\.oJ .....
l
'i:1
'""1
(t)
CJl
CJl
VI c::  '""1
(t)
'i:1
'""1
0::n
~
"CJl
~
I:»
('") ::r
w
0
Most regions show good results, however, some interesting differences can be seen in the
region near the end ofthe plate. These differences could be due to the abrupt change in
slope where the plate ends, since the edges of the plate are pinned and not clamped. Table
3.3 gives the generalized forces for this case.
Table 3.3. Generalized Forces for Mach 3.0
Mode Actual Transpiration
1 2593 2041
2 534.7 253.1
3 3708 3826
4 848.3 809.4
5 574.8 373.4
6 2248 2194
Here, the differences in generalized forces are seen in modes 2 and 5.
As an illustration of one of the criterion for convergence, the residuals for this case
are presented in Figure 3.18.
52
.,
,,
..
2 3 4 5 6 7 B 9 10
Time Step
,\
\I
\ ,
\ \ \
\ \
\
\ "\
\. \\
r.~ \ '::
',......"'I:::_~ ~
I , I OE+O
o
1E6
2E6
4E6
5E6
6E6
(ij
::J
~ 3E6
tJ
Cl::
'.
Figure 3.18. Residual Convergence
This shows a well converged solution with tinal residual values being on the order of 1O'll'.
It is evident from these plots that transpiration very accurately simulates the
deflection. However, this is not readily assumed when studying the generalized forces.
Since the surface pressures match so closely, it can be deduced that generalized forces
may be very sensitive to ·small differences in surface pressures. However, differences may
be negligible if the structural spring force is large relative to the generalized forces. Also,
since in one case there is actual plate defonnation while in the other case the deflection is
only simulated, the internal mesh of each domain will be different. Subsequently, as will
be shown in later sections, the solution can be strongly dependent on the mesh. Therefore,
53
not all of the differences seen can necessarily be attributed to the transpiration boundary
condition.
3.4. AGARD 445.6 Wing
The next case that was investigated was the AGARD 445.6 wing. This wing is a
standard aeroelastic test configuration which has been investigated experimentally in the
Langley Transonic Dynamics tunnel. Using this wing allows for transpiration comparison
in practical application. Also, in this case, the surface is surrounded by the flow unlike the
plate case where the flow is only on one side, thus adding to the complexity of the
problem. The original, undeflected wing is shown in Figure 3. 19.
Figure 3.19. AGARD Wing
54
i'""
The wing was given a generalized displacement of 2 units in first mode bending and 2
units in first mode torsion. The deflected wing is shown in Figure 3.20.
Figure 3.20. Deflected AGARD Wing
The severity of the deflection is apparent from this figure. It is so significant that an
approximate 9 degree angle of attack is created at the tip ofthe wing where there should
be none. In addition, the end of the wing has translated below its original position.
Results were obtained for Mach numbers of 0.99, 1.141, and 2.0, using this deflection.
Even with this unrealistic deformation at these Mach numbers, transpiration produces a
very accurate simulation.
The surface pressure contours on the lower surface of the wing for Mach 0.99 are
shown in Figures 3.21 and 3.22.
55

Figure 3.21. Actual Pressure Contours, Mach 0.99
Figure 3.22. Pressure Contours Using Transpiration, Mach 0.99
56
Both results show the same regions of strong pressure gradients and both predict the
fonnation ofa shock near the middle ofthe tip of the wing.
Actual and simulated surface pressure profiles at three stations for the Mach 0.99
are shown in Figure 3.23.
57
1.0 .
0.8 .,
\ 0.6 ~~RR~
0.4 o ~ . 'b Cp0.2 . '"
0.0 ~ 0.2 .
0.4 .,
0.6 'Tj
~. 20 25 30 35 40 45 50
.., ~ ~ X
~
w 1.0
N 0.8 . w
0.6 '"0
.O4~ , .~
..,
~
Cp0.2 k: . [J)
[J) c:
Vo 0.0 ..,
00 ~ • •
'.".,0 0.2
0 0.4
t::!I
06 ~
~ 10 15 20 25 30 35 40 .[J)
3:: X ~
()
::r 1.0
0 0.8 \C)
\C) 0.6
0.4 ., CP.O.2j bc <\ 0.0
0.2 .
o Actual I
0.4
Transpiration I 0.6
0 5 10 15 20 25 30
X
The two data sets show very good agreement near the root of the wing. There is also
good agreement near the tip of the wing except very close to the trailing edge, where the
transpiration solution predicts a larger pressure peak. The largest differences are seen at
the middle ofthe wing where the data is somewhat shifted near the trailing edge.
Surface pressure contours are presented for the Mach 1. 141 case in Figures 3.24
and 3.25, again for the lower surface.
Figure 3.24. Actual Pressure Contours, Mach ).] 41
59
Figure 3.25. Pressure Contours Using Transpiration, Mach 1.141
Overall, the results appear to be very good from studying the pressure contours. The
pressure profiles for this case are presented in Figure 3.26.
60
0 0 0
L() "'T M
L()
L() III
"'T M N ..... r"'. 0 0 0
"'T M N
L() >< L() >< III X
M N ...... J.. 0 0 0
(") N ....
II
L()
L() L() "
N
0 0
N
I I 0
0 N "'T <0 ~ <"! 0 N "'T <0 CO ~ <"! 0 N "'T <0 CO ~ <"!
0 0 0 0 .... 0 0 0 0 0 0 0 0 0 0 ....
c.. c.. c..
U U U
c0
i.i...i. I,
'Q. f
en III
I
:J C
13 I1l
« ~
Figure 3.26. Pressure ProfIles, Mach 1. 141
61
Results near the root of the wing are extremely good, except for a slight difference in
pressure peaks at the trailing edge. The agreement decreases with increasing distance
from the root. This demonstrates the limitation described in Section 3. 1 of transpiration
not accounting for actual surface translation, since the wing is increasingly displaced from
its unmodified position moving from wing root to tip. However, despite the very large
deflection and rotation at the tip, the results there are quite good, with the largest percent
error being less than 15%.
The surface pressure contours on the lower surface for the Mach 2.0 case are
shown in Figures 3.27 and 3.28.
Figure 3.27. Actual Pressure Contours, Mach 2.0
62

Figure 3.28. Pressure Contours Using Transpiration, Mach 2.0
Strong pressure gradients can be seen near the leading and trailing edges in both cases.
The pressure change from leading edge to trailing edge appears to be more uniform in the
transpiration case than in the actual deflection. However, in both cases the pressure
gradient in this region is small, therefore small differences are somewhat magnified. The
small differences are more evident in the surface pressure profiles shown in Figure 3.29.
63
0.3
0.2 . 0.1 ~.....:
0.0
Cp 0.1 : ~r~
0.2 ~ 1j
0.3 II
0.4 .
"T1 0.5 '
QQ'
c: 20 25 30 35 40 45 50
...,
(1) X
w 0.3...,
IV
\D 0.2
""d
0.1 ..., 0.0 7~~ (1)
U>
U> Cp 0.1 .
C1\ c: .... ...,
(1) 0.2 . f
.".".,d 0.3 ~
0::n 0.4
(i" 0.5
y' ~
3:
10 15 20 25 30 35 40
~ X
(l ::r 0.3
IV 0.2
0 _01
1
00' l>' .=e ~ Cp 0.1 •
d •
0.2 • ,
6 I
0.3
o Actual I .
0.4 0
Transpiration I 0.5
0 5 10 15 20 25 30
X
Notice that the pressures away from the leading and trailing edges are almost identical
near the root and middle of the wing. In fact, there is very good overall agreement for
these two stations, with slight differences seen near the leading edge at the middle station..
Again, the station near the tip shows the largest differences, as expected.
This wing was also studied with generalized displacements of 2.63 units in first
mode bending and 0.33 units in first mode torsion. Comparisons were made for Mach
numbers of0.678, 0.99, and 2.0. These results are presented in the Appendix.
An ofthese cases clearly show that transpiration is effective at simulating even
relatively large deflections in transonic and supersonic flows. Another point in the
application oftranspiration is in flutter studies of a severly deflected body, such as this
one. Rather than using methods such as deforming meshes or transpiration to deflect the
surface and perform flutter investigations, it would be simpler and possibly more accurate
to perform the steady solution on the actually deflected surface, then perform the unsteady
aeroelastic analysis from this solution. The deflected surface could be obtained using
methods in this study, and the flutter solution could be determined with confidence using
transpiration, since it is known to be accurate in simulating small disturbances.
As previously mentioned, some discussion of the sensitivity of the mesh to the flow
solution is necessary so that all solution differences will not be falsely attributed to
transpiration. This discussion is presented in the following section,
65
3.5. Mesh Sensitivity
The original plate mesh was much less refined than the one that was used to obtain
the results presented in plate section. The original mesh is shown in Figure 3.30.
Figure 3.30, Original Plate Mesh
When observing the solutions from this original mesh, it was apparent that the elements
were too small and too few to make the solution smooth. Therefore, the solution was
probably somewhat inaccurate. A new mesh was constructed to produce a more accurate
solution. This mesh was presented in Figure 3.4.
66
Pressure profiles are presented in Figure 3.3 1 for both the new and old actually
deflected meshes for 0.1 generalized displacements at Mach 3.0.
67
4..,
,.oJ
~ ~ P"
.....c~
~
1<' I4l~1.0'"
~~
0~0~01t)01t)0
I")NN .......... OOO
c:ic:ic:ic:ic:ic:ic:iqq
c..
U
I")
~N
N
~
X
It)
c:i
0
It)
<;i
.....
I
,~
1~
.lJ
~~. 0'''''
~""~ ell
00 00 • r.
&~
rJl(/: r'
4'i!~
0~01t)01t)01t)0
(")NN .......... OOO .....
c:ic:ic:ic:ic:ic:ic:iqq
c..
U
I")
~N
N
~
X
~
c:i
0
It)
<;i
.....
I
It.".
[]
I • [MIl
~""0 i'G""
lb ~. "'Ii ••~
~
~JT
o It) o It) 01t)0 It) 0
(")NN .......... 0 00 .....
c:ic:ic:ic:ic:ic:ic:ic:ic:i I I
c..
U
I")
~
N
N
~.....
><
It)
c:i
0
It)
<;i
..... I
'I. '" .·;1:~ ..
;;;
rJl ;;;
Q) Ul
~ Q)
~
~
Q) '0
Z (5
0
" '.
'I
Ir,
'."
.~
:1 "'.1':
"
Figure 3.31. Pressure Profiles for New and Old Plate Cases
68
Some rather large differences are seen between the two data sets. In many areas, the
differences are larger than differences that were found in the various actual and
transpiration comparisons. Since the deflections and the flow conditions are the same, the
difference must be the mesh.
The generalized forces that were obtained for the plate using the old mesh are
presented in Table 3.4 along with the values from the new mesh (from Table 3.3).
Table 3.4. Generalized Forces for Old and New Meshes
Mode Old Mesh New Mesh
1 2718 2593
2 1274 534.7
3 2564 3708
4 842 848.3
5 2872 574.8
6 713 2248
There are significant differences in the two sets ofvalues. Again, in some cases, the
differences are larger than differences between actual and transpiration results.
Obviously, the more refined the mesh, the more accurate the solution will be.
However, a more refined mesh contains more elements and nodal points, thus requiring
more computational effort. The investigator must determine a balance between accuracy
and practicality. Due to this mesh sensitivity, a final case was investigated where only one
mesh was used to obtain comparison results. This case is presented in the following
section.
69
3.6. Plate with 0.5 Generalized Displacements
Since transpiration produced good results in the first plate case presented, a plate
deflection 5 times greater than that ofthe first was used. As mentioned, the flow solution
can be dependent on the mesh. Therefore, to eliminate error introduced by mesh
differences, actual deflection results were not compared to transpiration results. Instead,
transpiration was used with the deflected plate to simulate a flat plate. Ideally, the
solution to a problem of this type should be equivalent to the flow across a flat plate,
where the surface pressures and the generalized forces are zero, and the Mach number is
equal to the freestream everywhere.
The superposition of modes for this case is shown in Figure 3.32.
Figure 3.32. Deflected Plate, 0.5 Generalized Displacements
70
..
.,
..,,
.j
I'
~
The remarkable amount of deflecti.on is obvious from the figure. The deflection is so
significant that it produces maximum displacement that is greater than 35% of the width.
Solutions were obtained for Mach numbers 0.3, 0.8, 0.95, and 3. In each case, a well
converged deflected solution was used as the starting point for the transpiration solution.
In all cases, transpiration was effective in "removing" the deflection. However, results for
the Mach 0.95 case were not as good as the others after the same number of iterations.
A summary ofthe final results for this case is given in Table 3.5.
Table 3.5. Summary ofResults for 0.5 Generalized Displacement Plate
Mach 0.3 I Mach 0.8 Mach 0.95 Mach 3.0
Mach Number 0.3 ± 0.00001 0.8 ± 0.00005 0.95 ± 0.04 3.0 ± 0.001
Pressure Order of 10<> 10'" 10°" 10<>
Magnitude
Average Percent 2.98e6 1.62e5 8.27e3 5.23e6
Difference in GFs
From the table it can be seen that the overall Mach number becomes approximately the
freestream Mach number in each case. Also, the surface pressures in each case are
essentially zero with respect to their original deflected values, which were on the order of
101 to 10°. The average percent error in generalized forces was calculated by averaging
all ofthe ratios of final force value and initial force value for each mode. As an illustration
of the generalized forces approach to zero, a plot ofthe generalized force time history for
Mach 3 is presented in Figure 3.33.
71
466 10 12
Time Step
2
'i\
\
'" , 1\ , "'.1  , ''~~ ~
I .~/  
v/ I
,_V
I/~
/
1/
I
6000
8000
o
10000
6000
6000
4000
~ 4000
o
u..
"C 2000
Il
,~
(ij 0
Ii c::
Il
(!) 2000
Figure 3,33, Time History of Generalized Forces
"
This plot is representative of the time history of generalized force for each of the other
Mach numbers as welL
A plot of average percent error versus Mach number is presented in Figure 3.34,
72
1E+0
LJ.. <.9 1E1
c
~ 1E2 0
~
~ w 1E3 ~c
Q)
Cl) 1E4 co
~
Q)
«> 1E5
,.
x
x
0.5 1 1.5 2 2.5 3 '.
Mach Number
1E6 ++'~~~~+;'l
o
Figure 3.34 Average Percent Error in Generalized Forces
This plot suggests that transpiration was less effective in removing the deflection in the
transonic range. However, since transpiration did not appear to be less effective in the
other cases in the transonic range, a generalization cannot be made. Overall results for
this case are very good, considering the magnitude of the deflection that is being
"removed" by transpiration.
73
CHAPTER 4
CONCLUSIONS AND RECOMMENDATIONS
4.1. Conclusions
The primary objective of this research project was to examine the effectiveness of
transpiration for simulation of structural deformations in steady and unsteady aeroelastic
applications. The majority ofthe investigations were performed using a recently modified
version of a highly integrated, finite elementbased code for the multidisciplinary analysis
offlight vehicles. A supplement to this code was developed in this study which allows for
the generation ofdeflected meshes using modal superposition. This research
demonstrated that the transpiration boundary condition has strong potential for
applications in unsteady aeroelastic analysis, such as in the prediction offlutter boundaries.
The following conclusions were reached during this investigation:
1. The transpiration boundary condition is effective in simulating even relatively large
displacements over a wide range of Mach numbers. Some the results support the
rationale presented for when transpiration will lose accuracy. However, there is no
74
strict criteria for when the transpiration boundary condition will breakdown.
These results show that for applications similar to the ones presented, such as
flutter prediction, transpiration can be a very effective tool in simplifYing the
analysis.
2. Solutions involving the application ofa domain mesh can be sensitive to the
refinement of the mesh. The researcher performing studies using domain meshes
should investigate the sensitivity of the mesh to his or her particular application,
and depending on the desired accuracy, employ the mesh that is the most practical.
3. The codes developed in this study, FROMan and SOLMan, can accurately and
simply be used to perform surface deflections. In cases where a surface is
significantly deflected from its original position before it begins to oscillate, it may
be effective to use a surface deflection scheme such as this to deform the body and
then perform the flutter investigation from this initial condition.
4.2. Recommendations
The cases of this study used relatively simple geometries, as compared to full body
configurations. According to the transpiration concept, there should be little or no
accuracy lost in using transpiration in more complex cases. However, in instances such as
75
IIi
III
'~
I~
.~
·il
'"
"I
"
:~ ,
''"..""I
'I
I'~I
intersecting shocks and surfaces, it would be interesting to see how the transpiration
boundary condition perfonns.
Also, considering the findings on mesh sensitivity, it is recommended that any
future studies using the transpiration boundary condition to compare simulated deflections
with actual deflections begin with a documentation ofthe sensitivity ofthe solution to the
mesh.
Finally, since the majority ofthese results compared computer simulations with
one another, it may be advisable to compare with experimental results when available to
confirm the results from the simulations.
76
BffiLIOGRAPHY
Anderson, D. A, Tannehill, 1. C., & Pletcher, R. H. (1984). Computational Fluid
Mechanics and Heat Transfer. New York: Hemisphere Publishing Corporation.
Batina,1. T. (1989). Unsteady Euler Airfoil Solutions Using Unstructured Dynamic
Meshes. AIAA Paper 890115, American Institute of Aeronautics and
Astronautics.
Bharadvaj, B. K. (1990). Computation of Steady and Unsteady Control Surface Loads in
Transonic Flow. AIAA Paper 900935, American Institute of Aeronautics and
Astronautics.
Dixon, S. C. (No date given). Comparison ofPanel Flutter Results from Approximate
Aerodynamic Theory with Results from Exact Inviscid Theory and Experiment,
NASA TN D3649,
Gupta, K. K. (1995). An Integrated. Multidisciplinary Finite Element Structural. Fluids.
Aeroelastic. and Aeroservoelastic Analysis Computer Program, Edwards, CA:
National Aeronautics and Space Administration.
Henne, P. A (1990). Applied Computational Aerodynamics. Washington, DC:
American Institute of Aeronautics and Astronautics, Inc.
Lighthill, M. 1. (1958). On Displacement Thickness. Journal ofFluid Mechanics, 1(4),
383392.
Malone, 1. B. & Sankar, L. N. (1985). Unsteady Full Potential Calculations for Complex
WingBody Configurations. AIAA Paper 854062, American Institute of
Aeronautics and Astronautics.
Malone, J. B., & Sotomayer, W. A (1984). Unsteady Aerodynamic Modeling ofa
Fighter Wing in Transonic Flow. AIAA Paper 841566, American Institute of
Aeronautics and Astronautics.
Raj, P., & Harris, B. (1993). Using Surface Transpiration with an Euler Method for CostEffective
Aerodynamic Analysis. AIAA Paper 933506, American Institute of
Aeronautics and Astronautics.
77
•II
••
·1
'~
:! t,
""""
:.,'
Ruo, S. Y., & Sankar, L. N. (1988). Euler Calculations for WingAlone Configuration.
Journal of Aircraft, 25(5), 436441.
Sankar, L. N., Malone, 1. B., & Schuster, D. (1987). Euler Solutions for Transonic Flow
Past a Fighter Wing. Journal of Aircraft, 24(1), 1016.
Sankar, L. N., Malone, 1. B., & Tassa, Y. (1981). An Implicit Conservative Algorithm for
Steady and Unsteady ThreeDimensional Transonic Potential ,Flows. AIAA Paper
811016, American Institute of Aeronautics and Astronautics.
Sankar, L. N., Ruo, S. Y, & Malone, 1. B. (1986). Application of Surface Transpiration
in computational aerodynamics. AIAA Paper 860511, American Institute of
Aeronautics and Astronautics.
Thomson, W. T. (1988). Theory of Vibration with Applications (3rd ed.). New Jersey:
Prentice Hall.
78
~
.l .'
APPENDIXADDITIONAL DATA
79
Figure A.I. AGARD Wing
Figure A.2. Deflected AGARD Wing, 2.63 Units Bending and 0.33 Units Torsion
80
Figure A.3. Actual Pressure Contours, Mach 0.678
Figure A.4. Pressure Contours Using Transpiration, Mach 0.678
81
o
(")
o
l/)
N
o
N
o
Q.
U
Jr
I
.1J
I , • I
NOCOc.o~NON~ ""':""':cioooooo I I I I I I
0 0
l/) ~
l/) l/)
~ (") r 0 0
~ (")
l/) >< l/) >< (") N
I\ 0 .J 0
(") N
l/) l/)
N
0 0
I N ....
NOCOc.o~NON~ NOCOc.o~NON~
'7~9999000 ""':""':ooooocici I I I I I ,
Q. Q.
U U
"iii
::J
ti «
c:
o
ro
.!:
Q.
U) c:
~
~
Figure A.5. Pressure Profiles, Mach 0..678
82
Figure A.6. Actual Pressure Contours, Mach 0.99
Figure A. 7. Pressure Contours Using Transpiration, Mach 0.99
83
0.8
0.6 ~.
0.4 ~\
0.2 0\
Cp 0 ~...~
0.2
0.4
0.6
'Tj 0.81 , , , . I ' TTI""ll'IIlf'~ ~.
.., 20 25 30 35 40 45 50
(11 X
?> 08
00 0.6
lo'tI O4~ :~ ..,
(11 02 tf.l
tf.l c Cp 0
00 ..,
0.2 ~
(11
.lo.,'tI 0.4
0 0.6 ~
(11 0.8
~tf.l
10 15 20 25 30 35 40
~
I» X
0::r 0.8
0 0.6 ..
\D
\D 0.4
~2j • ".......~ Cp 0
.
0.2 0
• Actual I 0.4
0.6
Transpiration I 0.8
a 5 10 15 20 25 30
X
Figure A.9. Actual Pressure Contours, Mach 2.0
Figure A.IO. Pressure Contours Using Transpiration, Mach 2.0
85
0 0 0
~ .q M
U"l \() r' fooU"l .q C"'l N ",&'' 0 0 fooO :r C"'l N
U"l >< U"l X U"l X
C"'l N
".~ 0 0 0
C"'l N
~_c..
U"l U"l U"l
N
0 0 0
I . I ., N I I I T I
C"'l N 0 N C"'l (") N 0 N C"'l C"'l N 0 N C"'l q ci ci ci ci ci q q ci ci ci ci ci ci q ci ci ci
I I , , ,
0 0 Q.
U U U
c:
0
.~
0 ro en
::l c:
ti til
« ' ~
Figure A.II. Pressure Profiles, Mach 2.0
86
VITA ~
Clint C. Fisher
Candidate for the Degree of
Master of Science
Thesis: APPLICATION OF THE TRANSPIRATION METHOD FOR EFFICIENT
AEROELASTIC ANALYSIS USING AN EULER SOLVER
Major Field: Mechanical Engineering
Biographical:
Personal Data: Born in Oklahoma City, Oklahoma, On September 18, 1970, the
son ofLyndall C. and Barbara S. Fisher.
Education: Graduated from Putnam City West High School, Oklahoma City,
Oklahoma, in May 1988; received Bachelor of Science degree in
Mechanical Engineering with a minor in Mathematics from Oklahoma State
University, Stillwater, Oklahoma in May 1993. Completed the
requirements for the Master of Science degree with a major in Mechanical
Engineering at Oklahoma State University in May 1996.
Experience: Employed by the Oklahoma Department of Transportation in an
internship program; employed by Oklahoma State University as an
undergraduate and as a graduate research assistant; Oklahoma State
University, Department ofMechani.cal and Aerospace Engineering, 1992 to
1993 and 1994 to 1995.
Professional Memberships: American Institute of Aeronautics and Astronautics,
American Society ofMechanical Engineers.