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  

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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-­‐

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5. Vassberg,  John  C.;  Rivers,  S.  Melissa;  and  Wahls,  A.  Richard:  Development  of  a  

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