Ben Wahls
CNU Advisor: Professor Mark Croom
NASA Langley Advisors: Michelle Lynde and Richard Campbell
Capstone – Final Report
Design of Low Fidelity Transition Analysis Code for Tollmien-‐Schlichting
Abstract:
Aircraft that utilize laminar flow as opposed to turbulent flow achieve an
increase in efficiency through a reduction in drag. Therefore it is important to
develop tools that can effectively analyze a given configuration for its extent of
laminar flow so that designs can be accurately evaluated. Currently a code called
LASTRAC uses a numerical approximation method of the boundary layer theory
equation to accomplish this; however it comes with a long runtime. Therefore a low
fidelity code, called MATTC, has been developed to achieve the same results as
LASTRAC using an empirical relationship between the shape of a configuration’s
pressure distribution and its resulting Tollmien-‐Schlichting (TS) wave growth.
These results are accurate after configuration specific calibration of scaling
coefficients has been completed. TS is the predominant cause for transition from
laminar to turbulent flow for streamwise flow over configurations. The fact that
MATTC only looks at streamwise flow limits its utility; however it is a valuable tool
for use in analyzing certain aspects of configurations.
By creating a global calibration file of various deliberately built pressure
distributions to verify equations developed by using empirical relationships
between TS wave amplitude growth (N-‐factor growth) and local Mach number,
Reynolds number, and pressure gradient, MATTC has been expanded to a new
version that can effectively predict the same transition location as LASTRAC with an
N-‐factor value within a 10% error of the local wing chord without the need for
configuration specific calibration. This is demonstrated in this report by comparing
the MATTC and LASTRAC predicted transition location on the Common Research
Model with Natural Laminar Flow (CRM-‐NLF) wing that was designed at NASA
Langley Research Center to utilize laminar flow. Next, MATTC will be integrated into
the full laminar flow design process already used in the Configuration Aerodynamics
Branch at NASA Langley to further improve the quality of their laminar flow designs.
Introduction:
A key goal of research in aeronautics is to increase efficiency of aircraft in
order to save fuel, money, and produce fewer emissions. One method studied by
researchers in the field at places such as NASA Langley Research Center and Boeing
is the design and implementation of aircraft components that utilize natural laminar
flow instead of turbulent flow within the boundary layer. Boundary layer refers to
the thin region of flow surrounding an aircraft’s surface where viscous effects are
present. Laminar flow is smooth airflow as opposed to turbulent, which is more
chaotic by nature. More specifically the layers of airflow in laminar flow do not
intersect each other; instead they slide along the same path next to each other in a
uniform pattern allowing the internal flow and overall flow to travel in the same
direction. However in turbulent flow the internal flow layers travel randomly with
countless intersections. Although the bulk turbulent flow still travels in a specific
direction, the uniformity of laminar flow results in much less drag when moving
across the surface of an aircraft. Therefore the key goal of laminar flow research is
to develop aircraft geometries that delay laminar-‐to-‐turbulent flow transition. To
successfully complete this research, tools must be developed to accurately and
efficiently determine the point of transition from laminar flow to turbulent flow
along the surface of an aircraft in order to effectively evaluate design concepts.
There are three types of transition from laminar to turbulent flow:
attachment line, cross flow (CF), and Tollmien-‐Schlichting (TS). The focus of this
project will be on TS, which deals with transition occurring streamwise as the flow
moves straight back along the aircraft surface. Currently the most frequently used
transition prediction software is the high fidelity tool called LASTRAC (written in
Fortran); however, analyzing a single two-‐dimensional cross section with this tool
can take many minutes of computation time on the user’s local machine. Even
simple geometries require upwards of 20 cross sections for a design to be thorough.
Each cross section needs to be analyzed for the two modal triggers of transition, TS
and CF, resulting in an inefficient process with a runtime easily exceeding an hour.
Recently my NASA mentors in the Configuration Aerodynamics Branch at
NASA Langley developed another code, called MATTC (also written in Fortran),
which uses an approximation technique based on observations of LASTRAC results
that can analyze a two-‐dimensional cross section for TS in a fraction of a second. A
drawback to this code is that it currently only works for predicting one of the types
of transition, TS, and therefore it cannot be utilized alone when predicting the
transition front on an aircraft component that has sweep. Another limitation is
MATTC only yields comparable results for TS with LASTRAC for Mach numbers up
to about 0.75. This means it cannot be used when designing supersonic
configurations and is unreliable for transonic conditions. Finally, as it will be shown
in the theory section, MATTC must be calibrated with a LASTRAC run for each
specific configuration analyzed. The calibration process decreases the efficiency of
the tool, since LASTRAC must also be run anyways.
This project will be focused on creating a tool for TS flow stability analysis
that runs as quickly as MATTC, yields accurate results when compared to
LASTRAC’s high fidelity results, works effectively for Mach numbers up to 1.6, and
will not need to be calibrated before running a new configuration. This project will
be used to determine the transition location and envelope frequency growth for
two-‐dimensional cross sections cut streamwise from different components of
aircrafts.
Theory:
The mathematical approach to the concept of a low fidelity transition
prediction tool involves the approximation of results found from boundary layer
theory. The boundary layer of a flow describes the fluid nearest the surface in the
region where viscous effects are most significant. Levin and van Ingen indicate that
the most essential relationship for boundary layer theory is derived from the Navier
Stokes equations, and is called the boundary layer equation, describing boundary
layer behavior for a flat plate configuration:
𝑢 𝜕𝑢𝜕𝑥 + 𝑣
𝜕𝑣𝜕𝑦 = − (1/𝜌)
𝜕𝑝𝜕𝑥 + 𝜈
𝜕!𝑢𝜕𝑦!
Where 𝑢 is the tangential boundary layer velocity, 𝑣 is the normal velocity to the
surface, 𝑥 is the distance traveled streamwise along the surface, 𝑦 is the distance
normal to the surface at a given 𝑥 value, 𝜌 is the density of the flowing fluid (varies
with 𝑥 and 𝑦 values of position), and 𝑝 is pressure (also varies with 𝑥 and 𝑦
position). The continuity equation for this system is:
!!!! + !!
!! = 0
Where, again, 𝑣 is the flow velocity normal to the surface at a given location and 𝑢 is
the tangential velocity. The value 𝑣 is negative if suction is present on the surface,
with boundary conditions:
(1)
(2)
For y=0, u=0 and v=v0
For yà∞, uàU(x)
Where U(x) is the velocity of the edge of the boundary layer. These equations show
that the flow inside the boundary layer will travel more slowly across the surface as
it is more affected by the friction of the surface than flow further away in the y
direction. It is important to note from these equations that a relationship between
flow velocity (Mach number), density of flow (implying compressible flow), and
pressure gradient exists. This is where the idea for approximating these
relationships for MATTC originated. LASTRAC uses a method developed that
simplifies a system into a set of linear differential equations called the eN method,
where N is called the N-‐factor and is a non-‐dimensional coefficient describing the
amplitude of waves traveling through the flow. This process essentially breaks a
curved configuration into a large series of flat plates placed next to each other. This
method is used to calculate the growth of TS waves streamwise on a surface to
determine where transition occurs. The experimentally accepted value for the
critical N-‐factor (amplitude of wave that causes transition and is dependent on the
environment) is 9 for a free atmospheric environment; however, this number has
been up for debate since the 1950s with some reports claiming a value as high as 13
(J.L. van Ingen).
As stated before, the version of MATTC used prior to this capstone project is
based on the fact that TS growth is dependent on Mach number, Mach gradient, and
Reynolds number. For Mach number effects, the term used is derived from a
(3)
Prandtl-‐Glauert scaling method, which is an approximation developed to account for
the effects of compressible flow up to transonic conditions:
𝐴 = √(1−𝑀!)
Where M is the free stream Mach number, but this again only allows for an accurate
frequency growth curve up to M=0.75.
The Reynolds number effect is related to boundary layer thickness, which led
to an exponential term of the form:
B = (Rex)b
Where Rex is the local Reynolds number at a location in millions (example: 30 for a
Reynolds number of 30,000,000), and b is a constant. For the current MATTC, b =
0.5 has proven effective at lower Mach numbers.
For Mach gradient effects it has been shown that certain favorable gradients
over a length of the surface are strong enough to dampen TS growth, so the term for
this effect looks at the difference in Mach gradient values at each point on the
surface in comparison the point where growth first starts occurring:
C = c * (!"!"− 𝑑)
Where !"!" is the Mach gradient along the surface at a given point, d is the Mach
gradient value where TS growth first starts occurring chordwise on the surface, and
c is a scaling constant. The variable d must be determined from looking at the
pressure distribution along the surface of the aircraft, and little c would be
optimized for each speed regime (subsonic, transonic, and supersonic) to give the
best results.
(4)
(5)
(6)
Putting all of these terms together gives the current MATTC relationship with
N-‐factor growth:
𝑁! = 𝑁!!! + (𝐴 ∗ 𝐵 ∗ 𝐶 ∗ 𝑑𝑠)
This equation means the present N-‐factor level is equal to the previously calculated
N-‐factor level plus the product of the three terms discussed above with the distance
between the current point and previous point where the initial N-‐factor is set to 0 at
the leading edge. This equation gives the basis for my work on expanding this tool to
the transonic and supersonic regimes.
The essential differences between MATTC and LASTRAC come with the level
of approximation techniques used to simplify the problem and the specifics of
conditions. While MATTC solves the one simple equation shown above for each
point, LASTRAC solves its eN method equations for every possible stream wise
energy wave within a set of frequencies (given as input) at every point measured.
The end result from LASTRAC shows the amplitude growth (plotted as N-‐factor, NF,
(7)
Figure A: LASTRAC Output Figure B: MATTC Output
which is a non-‐dimensionalized unit) shape of the specific waves from the set of
given frequencies with the largest amplitude growth (Figure A); whereas MATTC
gives the location of the largest amplitude of a wave present at a point without
knowing which wave was amplified (Figure B). The curve plotted by MATTC is
called the envelope amplitude growth.
Through the procedure detailed in the following section, the updated version
of MATTC was created; this is the goal of this capstone project. In this version of
MATTC empirical relationships between local Mach number, Reynolds number, and
pressure gradient (instead of Mach gradient) and N-‐factor are used to develop a new
set of equations to predict N-‐factor growth and transition point of various
configurations. Just like the old version, each of the relationships listed above is
represented by a unique equation in the code, and each equation has a scaling
coefficient attached to it. Once the shape of the function being used was determined
to accurately illustrate the specific effect (Mach number, Reynolds number, pressure
gradient), the coefficient was optimized using a program called MTCCAL. First is the
new equation to account for compressible flow effects present on TS growth
(especially at high Mach numbers). In the original MATTC, this term, (equation (4)),
was a Prandtl-‐Glauert approximation relationship, which accurately describes
compressibility effects through subsonic speed conditions before falling off in the
transonic regime. This new term was found to accurately account for
compressibility effects in amplitude growth through both transonic and supersonic
speeds as well as the continued accuracy in subsonic conditions:
𝐵 = !!!!(!!!!!)
(8)
Where m is local Mach number (Mach number of flow at a specific point on the
surface instead of the overall speed of the aircraft itself). The first version of
equation (8) that was tested used (𝑚 +𝑚!), however it was not sensitive enough to
change in N-‐factor growth rate. The term was increased to the power shown in
equation (8) to be more sensitive to curves in N-‐factor growth. As before, Reynolds
number effects on TS growth are determined by a simple function:
𝐶 = (𝑅𝑒)!
Where Re is Reynolds number and c is approximately 0.5. Finally the last equation
accounts for slopes in the pressure distribution of a configuration:
𝐷 = 𝑑 ∗ [ 𝑐𝑝𝑠 + 𝑢𝑑𝑓𝑠 ]
Where “cps” is the slope of the pressure distribution at a specific point, and “udfs” is
the slope of a function called the universal damping function at that same location
on the surface of the aircraft. The universal damping function is a curve with the
shape close to a square root function that, when present in a pressure distribution,
results in zero TS growth, making it an ideal pressure distribution for delaying
transition due to TS growth. The relationship in equation (10) compares the
difference in slopes of the actual pressure distribution being looked at with the
slope of the universal damping function at that same x-‐value location in the function.
Once again, as in the original version of MATTC, the four coefficients: a (one final
overall scaling coefficient), b, c, and d, are optimized before the following equation is
used to determine the new N-‐factor value for each location on the surface of the
aircraft based on the value calculated for the previous surface location:
𝑁! = 𝑁!!! + 𝑎 ∗ 𝐵 ∗ 𝐶 ∗ 𝐷 ∗ 𝑑𝑠
(9)
(10)
(11)
Where ds, as in the original version of MATTC, represents the step size along the
airfoil. The process followed to verify and calibrate equations (8) through (11) is
detailed in the following section.
Methods:
This new low fidelity stability analysis tool was developed in a similar way
that the first version of MATTC was created. It was based on patterns recorded from
LASTRAC results on the envelope N factor growth due to changes in Mach number,
Reynolds number, and pressure gradient along the chord of a cross section. The new
analysis tool was implemented as a subroutine in the original version of MATTC to
keep the data input and output formatting consistent, and the original equations for
N-‐factor were simply commented out.
As stated before, the equations accounting for each of the three effects listed
above were developed one at a time through isolating the specific parameter
associated with them. The process for developing equations (8) through (10) was
the same. Using another Fortran code, called MAKECP, pressure distributions can be
created with the user’s desired characteristics. This program allows the user to
input a desired free stream Mach number, Reynolds number, angle of sweep, and Cp
(non-‐dimensional pressure coefficient). Free stream Mach number refers to the
speed that the aircraft is traveling and has to do with compressible flow effects,
Reynolds number is a value representing viscous flow effects, angle of sweep is the
angle the leading edge of the component being analyzed forms with the oncoming
flow, and Cp level is the amount of pressure desired after a rapid initial increase. In
addition, the user inputs values for variables x1, x2, x3, x4, and UDF.
The UDF value describes a scalar multiple of the universal damping function
discussed previously and is used to influence pressure gradients in the distributions
(UDF = 0 for flat plate). The non-‐dimensionalized abscissa values called x/c, as
shown in Figure 1, describe location along the upper surface of the cross section
where 0 is the leading edge and 1.0 is the trailing edge. The stretch from the initial
point to x1 is a rapid decrease in pressure level up to Cp level at x1, x1 to x2 is a
region of zero pressure gradient, x2 to x3 is the region with the pressure gradient
specified by the user’s UDF value, and x3 to x4 is the region where a slight shock is
introduced into the distribution to force transition no further on the airfoil than x3,
or 60% local chord. For all cases in this project: x1 = x2 = 0.01, x3 = 0.6, and x4 = 0.8.
The variable x3 is given this value since sustaining laminar flow across 60% of the
chord is a common design goal when working to obtain natural laminar flow.
Using the MAKECP program, sets of pressure distributions were created
x1=x2=0.01 x3=0.6 x4=0.8
Figure 1: Example Pressure Distribution
Mach=0.1, Reynolds=20 million, UDF=-‐0.4
x/c
specifically to isolate one effect at a time. These sets of distributions were run
through LASTRAC and MATTC to develop the proper equation shape for the effect.
Each distribution run through LASTRAC was added to a calibration file using a
program called MTCADD. This calibration file was then used by MTCCAL to
determine the best set of coefficients (a, b, c, and d) for the new version of MATTC
up to that point. After developing the last equation, every run used throughout the
project had been added to the same calibration file using MTCADD and the final set
of ideal coefficients for the updated version of MATTC was produced by MTCCAL.
These coefficients will be reported later in the data section. Essentially, MATTC was
updated/optimized for one effect, then two, and finally all three.
The first effect examined was the influence of Mach number on amplitude
growth. To do this, every parameter in MAKECP was held constant except free
stream Mach number, so Reynolds number = 30 million and pressure gradient
(UDF) = 0. Free stream Mach number was varied from 0.1 to 1.6 at intervals of 0.1.
Figure 2 shows the pressure distributions for Mach number of 0.1 and 1.6, clearly
showing they are identical except for a higher initial pressure level for the lower
Mach number.
Mach = 0.1 Mach = 1.6
Figure 2: Pressure distribution dependence on Mach number
By the very nature of a Prandtl-‐Glauret approximation, it becomes inaccurate
at predicting compressible flow effects in the supersonic speed regime; equation (8)
was developed from the need for a larger growth at higher Mach numbers without
affecting the already accurately predicted growth at lower speeds. The quadratic
term in the denominator accomplished this quite well, acting essentially as a higher
order approximation term. In addition, the old version of MATTC used free stream
Mach number in its compressibility term, but as seen in equation (8) local Mach
number is now being used. The local Mach number provided more accurate
calibration results because it better represents the compressibility of the local flow
near the boundary layer as opposed to the free stream Mach number. As stated
earlier, every run in the set of pressure distributions for isolating compressibility
effects was then put through MTCADD to begin building the overall calibration file.
Next the effects of Reynolds number were examined. This time a set of
pressure distributions were made with Mach numbers of 0.1, pressure gradients
(UDF) of 0, and varying Reynolds number. The Reynolds numbers (in millions)
evaluated were: 1, 5, 10, 20, and 30. A Mach number of 0.1 was used to remove all
compressibility from the flow, and therefore allowing us to say confidently that
Reynolds number effects cause the changes in TS growth seen with this set. This
effect proved to be the simplest to capture as the pressure distributions themselves
showed no differences depending on the Reynolds number used. In fact, the same
equation from the original MATTC, equation (5), showed accurate results for all
configurations. Again, the LASTRAC results from the runs were added to MTCADD to
further build the calibration file.
An important thing to note was that the smaller the Reynolds number, the
farther the start of TS growth was along the surface of the cross section. LASTRAC
picked up on this effect, while MATTC always incorrectly predicted growth to begin
right at the leading edge of the cross section being analyzed. Since Reynolds number
was isolated, this is clearly an effect that could be controlled using the same
Reynolds number variable as in equation (9), C. An “if-‐statement” was added just
before equation (11) in the code comparing the current location on the surface
(from 0 to 1.0) to the inverse of the now calculated value of “C” in equation (9). If the
x/c location was smaller than the inverse of “C” then no N-‐factor growth would
occur. This improved the quality of the results from MATTC in all future runs.
Finally, the effects of pressure gradients were studied. The idea of comparing
the actual pressure distribution slope at each x/c location to the universal damping
function arose after discovering pressures with the form of the UDF showed no
additional TS growth. Before, MATTC had to calculate multiple values at each point
within the actual N-‐factor growth subroutine being updated in this project; however
the slope of the UDF and actual pressure distribution at each is already calculated
elsewhere in the code. This allows convenient access to these values while cutting
back on runtime even further by removing some calculations.
The set of pressure distributions for this case all had Mach numbers of 0.1
(again to remove compressible flow effects) and Reynolds numbers of 20 million,
which doesn’t affect results since the effect has already been accounted for by
equation (9). The UDF scaling values used were -‐0.4, -‐0.2, 0, 0.2, and 0.4. The shapes
of these are shown Figure 3.
Originally a version of equation (10) was used without the scaling coefficient
d, and this showed to have the correct shape of the N-‐factor growth curve. However,
it did not always have the correct amplitude, leading to the final version of equation
(10) that included the d scaling coefficient multiplied by the difference in the
current pressure distribution slope from the slope of the UDF. This was the last
effect studied, so once this set of runs was added to the calibration file with
MTCADD the final set of calibrated coefficients (a, b, c, and d) are calculated with
Mach = 0.1, Reynolds = 20 million
UDF = -‐0.4 UDF = -‐0.2 UDF = 0
UDF = 0.2 UDF = 0.4
Figure 3: Pressure distributions created for gradient
MTCCAL. At this point the new updated version of MATTC theoretically will not have
to be calibrated again, and the next step is to test the code on actual configurations
within any of the conditions included in the full set of calibration parameters: free
stream Mach numbers of 0.1 to 1.6, Reynolds numbers of 1 million to 60 million, and
pressure gradients of UDF = -‐0.4 to UDF = 0.4.
Data:
To test the updated version of MATTC, the program was used to predict the
laminar flow present on a version of the Common Research Model’s (CRM) wing
designed to utilize natural laminar flow (CRM-‐NLF). The original CRM (shown in
Figure 4) is a generic open geometry created by NASA to enable facilities around the
world to test and compare results. It is an effort to advance the aeronautics industry
as a whole and encourage international cooperation. The CRM is designed at a free
stream Mach number of 0.85 and a Reynolds number of 0.1087 per inch.
It is important to record the output scaling coefficients from MTCCAL, and so
these values that are plugged into the new MATTC’s equations are found in Table 1.
Although it has not been fully tested yet, utilizing these values should mean that the
code will not have to be recalibrated every time a configuration is analyzed.
a 6.05381
b 0.42742
c 0.61387
d 0.82798
Table1: Scaling Coefficient Values for MATTC
Figure 4: Common Research Model
Therefore the final set of equations in the new version of MATTC based on
equations (8) through (11) is:
𝐵 =1
1+ (0.42742)(𝑚! +𝑚!)
𝐶 = (𝑅𝑒)!.!"#$%
𝐷 = (0.82798) ∗ [ 𝑐𝑝𝑠 + 𝑢𝑑𝑓𝑠 ]
𝑁! = 𝑁!!! + 6.05381 ∗ 𝐵 ∗ 𝐶 ∗ 𝐷 ∗ 𝑑𝑠
Using the output transition locations predicted by MATTC and LASTRAC
when analyzing the CRM wing the transition front plotted in Figure 5 was
constructed showing the region of laminar flow on the surface. The transition
location was calculated at 14 stations (locations where cross sections are cut to be
Figure 5: Predicted TS transition front (LASTRAC vs MATTC)
Leading Edge Trailing Edge
Chord/Stream Wise Flow
Fuselage
Laminar Flow Region
(12)
(13)
(14)
(15)
analyzed) along the wing, and together provide the predicted region of laminar flow.
In addition, no laminar flow is assumed to exist right where the wing connects to the
fuselage and at the wing tip.
Table 2 shows the percent error of the results given by the new version of
MATTC compared to LASTRAC. The goal is to be within a ±5% local wing chord.
Station 1 (cross section at the point where the wing meets the fuselage) and station
16 (wing tip) are excluded from the table since no laminar flow is found at these
locations as described before.
Station Number MATTC Trans.
Location (x/c)
LASTRAC Trans.
Location (x/c)
Δx/c -‐ (in percent
local chord)
2 0.46 0.42 4%
3 0.45 0.46 -‐1%
4 0.50 0.54 -‐4%
5 0.55 0.56 -‐1%
6 0.62 0.61 1%
7 0.66 0.62 4%
8 0.60 0.64 -‐4%
9 0.67 0.65 2%
10 0.59 0.56 3%
11 0.59 0.54 5%
12 0.63 0.59 4%
13 0.68 0.67 1%
14 0.62 0.60 2%
15 0.49 0.47 2%
Table 2: Δx/c of MATTC transition front points versus LASTRAC
Discussion and Conclusions:
As seen in Table 2, MATTC has Δx/c position error less than or equal to 5% at
every location on the CRM-‐NLF wing for its predicted transition location resulting in
strong confidence of its solutions. This was accomplished without calibrating the
code to this specific configuration, meaning the goal of creating a code without the
need for additional calibration is a success for this configuration. The original
version of MATTC could also produce an accurate transition front for the CRM-‐NLF,
but it required calibration with LASTRAC runs. This defeated the purpose of MATTC,
which, again, is to drastically reduce computation time. Similar testing must be done
on many more actual configurations, especially ones in the supersonic speed regime,
in order to proclaim the program’s total validity for all future configurations.
MATTC produces accurate transition locations (within 5% local chord) for all
pressure distributions in the set generated to validate the compressibility equation
(equation (8)) without requiring a configuration-‐specific calibration. However, the
program must be tested on actual supersonic configurations before it can be
proclaimed as valid for free stream Mach numbers up to 1.6.
A general observation of the differences in N-‐factor growth predicted by
MATTC versus LASTRAC is that the curves are less linear in the LASTRAC results.
MATTC consistently shows the same general shape growth as LASTRAC, but the less
linear the growth is in LASTRAC the more the two codes will differ in growth
pattern. This is true even if the two programs end up at about the same amplitude.
Due to the difference in N-‐factor growth curve sensitivity, discussions have
already begun about adding a fourth equation into MATTC that uses local Mach
number, like equation (8) for compressibility, which pays more attention to the Cp
level of the configuration. The idea was that equation (8) would account for this
since a different pressure level would simply mean the flow speeds up/slows down
more over certain areas and therefore should be accounted for by the
compressibility term; however, even at subsonic speeds (where there are no
compressibility effects present) pressure level seems to have an effect on N-‐factor
growth. Therefore the following equation form is currently being studied to see if it
can account for changes in N-‐factor growth due to differing pressure levels:
𝑒𝑒 = (!"!)!
In this equation, lm is local Mach number, M free stream Mach number, and e is
another scaling coefficient to be optimized. The idea behind this equation is that the
rational number will be larger the further the pressure level (and therefore local
Mach number) is from zero. These are the cases where MATTC consistently predicts
too little growth, meaning equation (16) could very well be the solution to this
problem, although it has not been tested thoroughly yet.
Eventually MATTC will be integrated into the Configuration Aerodynamics
Branch’s Automated Laminar Flow Design Optimization process. In this process a
target pressure distribution is entered as input and then the given geometry is
changed slightly from a program called “CDISC” to try and reach the target pressure
provided. This process repeats many times, sometimes pushing 100 full design
cycles to reach a quality laminar flow result. Right now the target pressure
distribution can be generated two ways. The first is through an educated guess
based on the geometry’s initial pressure distribution or a code. The second is by
(16)
running a program similar to the old version of MATTC that requires calibration to
the configuration to be useful. LASTRAC cannot be used to analyze the slightly
changed geometry each time as it could potentially add many hours or days of
runtime to the process. It is simply not efficient. This new version of MATTC will be
integrated into this design process so that the slight changes to the geometry can be
analyzed automatically in less than a second during each cycle of the design process.
This will allow the target pressure distribution to be updated each time to
continuously create the best design possible without adding more than a few
seconds to the total runtime of the process. The addition of MATTC to this full
laminar flow design process will result in higher efficiently and will also lead to even
better results than the branch at NASA Langley is already obtaining.
Bibliography
1. Ori Levin. “Stability analysis and transition prediction of wall-‐bounded
flows.” KTH Mechanics. https://www.mech.kth.se/~ori/Public/lic.pdf
2. J.L. van Ingen. “The eN method for transition prediction. Historical review of
work at TU Delft.” Faculty of Aerospace Engineering, TU Delft, the
Netherlands. https://repository.tudelft.nl/islandora/object/uuid:e2b9ea1f-‐
5fa1-‐47c0-‐82ad-‐64d1c74a5378/datastream/OBJ
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Laminar Flow Analysis and Design,” AIAA 2011-‐3527, 2011.
4. Chang, C.-‐L., “The Langley Stability and Transition Analysis Code (LASTRAC):
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