The properties of a Navier-Stokes solution of a shock-separated turbulent flow over a flat wall are investigated. Refinements of an algebraic relaxation turbulence model previously shown to be of value for the simulation of separated flows are presented. A simplified analysis applicable near an adiabatic wall is developed and used to help verify the accuracy of the numerical solution. Features of the time-dependent response of a turbulent boundary layer to shock impingement are presented

Baldwin, B.

Hung, C. M.

English

NASA Technical Reports Server

CALCULATION OF A SEPARATED  TURBULENT - 
BOUNDARY LAYER 
Barrett  Baldwin 
Ames  Research  Center 
Ching Ma0  Hung 
DCW Industries  and  Ames  Research  Center 
SUMMARY 
The  properties  of  a  Navier-Stokes  solution  of  a  shock-separated  turbulent 
flow  over  a  flat  wall  are  investigated.  Refinements  of an algebraic  relaxation 
turbulence  model  previously  shown  to  be  of  value  for  the  simulation  of  separated 
flows  are  presented.  A  simplified  analysis  applicable  near  an  adiabatic  wall 
is  developed  and  used  to  help  verify  the  accuracy  of  the  numerical  solution. 
Features  of  the  time-dependent  response of a  turbulent  boundary  layer  to  shock 
impingement  are  presented. 
INTRODUCTION 
Computers now available  are  capable of practical  calculations  of  complex 
flow  fields,  including  separated  turbulent  boundary  layers.  However,  as 
discussed  in  reference 1, development of adequate  turbulence  models  is  a  pacing 
item  that  impedes  progress  toward  that  goal.  Recent  improvements  in  numerical 
methods,  such  as  those  described  in  reference 2, have  made  it  feasible  to  test 
a  variety  of  modifications of existing  turbulence  models  (see,  e.g.,  refer- 
ences 3-5 and  the  more  comprehensive  reference  lists  therein).  For  engineering 
purposes,  it  would  be  practical  to  use  simplified  models  calibrated  from 
experiments  conducted  at  nearby  flow  conditions.  The  shortage  of  experimental 
information on separated  flows  prevents  calibration  with  precision  at  this  time. 
Nevertheless,  it  seems  worthwhile  to  proceed  with  the  development of computer 
codes  for  complex  flows  based  on  simplified  turbulence  models  that  can  be 
.adjusted to accommodate  the.  existing  experiments.  With  presently  available 
information, it may  be  possible  to  accomplish  this  for  high  Reynolds  number 
flows  at  Mach  numbers  up  to 3 using  the  boundary-layer  approximation  for 
viscous  and  Reynolds  stress  terms  in  the  layers  near  solid  surfaces.  More 
complicated  procedures  can be incorporated  when  they  are  justified  or  can  be 
used  to  aid in the  calibration  of  the  simplified  models. 
In reference 5, it  was  shown  that an inner  layer  algebraic  eddy  viscosity 
model  used  by  Clauser  (ref. 6)  provides  better  agreement  with  two  separated flow 
experiments  than  more  conventional  models.  In  this  paper,  the  properties  of  a 
Navier-Stokes  solution  of  a  shock-separated flow based on  a  variation  of  that 
1483 
https://ntrs.nasa.gov/search.jsp?R=19770003419 2020-03-20T07:07:38+00:00Z
model  are  investigated. A simplified  analysis  applicable  near  an  adiabatic 
wall  is  presented.  Favorable  comparisons  of  results  from  this  analysis  with 
the  Navier-Stokes'solution  show  that  the  boundary-layer  approximation  is  valid 
near  the  wall  for  this  case  and  that  the  Navier-Stokes  solution  is  accurate. 
As an  example  of  the  type  of  information  that  can  result  from  such  numerical 
solutions,  features of the  time-dependent  response of a boundary  layer  to 
impingement  of a shock  wave  are  presented. 
SYMBOLS 
A 
B 
BF3 
C 
K 
P 
'RT 
s2 
T 
U 
U 
T2 
U 
T3 
V 
vF 
x, Y 
Y+ 
constant  (eq. ( 2 2 ) )  
constant  (eq. (23 ) )  
constant  in  turbulence  model  (eq. ( 6 ) )  
constant  (eq. (17)) 
specific  heat  at  constant  pressure 
universal  function  (en. (18)) 
universal  function  (eq. (19)) 
Karman  constant (0.4) 
Clauser  constant (0.016) 
pressure,  N/m2(lb/ft2) 
turbulent  Prandtl  number (0 .9 )  
constants  in  Sutherland  viscosity  law  (eq. (20)) 
temperature, K (OR) 
velocity  in  x-direction,  m/sec  (ft/sec) 
friction  velocity  (eq. (4)), m/sec  (ft/sec) 
friction  velocity  (eq. (5)) , m/sec  (ft/sec) 
constant  in  Van  Driest  damping  factor  (eq. (8)) 
Van  Driest  damping  factor (eq. (8)) 
Cartesian  coordinates, m (ft) 
law  of  the  wall  variable  (eq. ( 9 ) )  
1484 
6 boundary-layer thickness, m(ft) 
6* kinematic displacement thickness (eq. (lo)), m (ft) 
A relaxation  parameter  (eq. (5)) 
u molecular coefficient of viscosity, kg/m-sec (slugs/ft-sec) 
Ut 
P gas  density,  kg/m3  (slugs/ft3) 
T shear stress (eq. (ll)), N/m2 (lb/ft2) 
turbulent  eddy  viscosity  coefficient  (eqs. (1),(2)) kg/m-sec 
(slugs/ft-sec) 
Subscripts: 
0 initial  profile 
2 value  of yat
3 value  of y at 
at  station  ahead  of  interaction 
which u is evaluated (eqs. (31 ,  ( 4 ) )  
T2 
which 3eq 
is  evaluated  (eq. (6)) 
i position of inviscid shock impingement on wall 
max maximum velocity in profile 
W value  at wall 
METHOD 
Numerical  Method  for  Navier-Stokes  Solutions 
The  basic  numerical  method  used  in  this  investigation  is  described  in 
reference 7. Recently,  MacCormack  has  improved  the  method  such  that  the  calcu- 
lations  require  an  order of magnitude  less  computation  time  than  formerly  (see 
ref. 2). The  compressible  Navier-Stokes  equations  to  be  solved  are  also 
listed  in  reference 7. 
Experiment  Used  for  Comparison 
The  experimental  flow  field  (ref. 8 )  is  depicted  in  figure 1. A shock  wave 
generated  by  a  plate  set  at 13' to  the  free  stream  impinges  on  the  boundary 
layer on the  upper  wind  tunnel  wall. A separation  bubble  containing  reversed 
flow  forms  and  extends  upstream of the  inviscid  shock  impingement  point. A 
pattern  containing  induced  and  reflected  shocks forms. The  free-stream  Mach 
number  is 3 and  the  Reynolds  number  based  on  initial  boundary-layer  thickness 
ahead  of  the  interaction is lo6. 
1485 
I 
Turbulence  Model 
The  two-layer  algebraic  eddy  viscosity  model  used  in  the  present  calcula- 
tions  is  defined  by  the  following  equations: 
P t  )
- KPumaxG* K = 0.0168 
outer 
where p, and p2 are evaluated at y = y, with y2 determined from the 
relations 
The  quantity  u  is  determined  from  the  relaxation  formula 
T3 
3 
U - u  
T 3eq T3 - =  
dx y A = 5  A6 0 
where 
a 
U ~ 3 e q  = BF3 (ky $) 
Y'Y 3 
with y3 determined  from  the  relation 
au 
PkY a 
-+ = ( / y, = 2000 y3 
Y=Y 3 
The  quantity  VF  is  the  Van  Driest  damping  factor 
BF3 = 1.18 
VF = 1 - exp (-y+/V) V = 18 
where 
1486 
The  transition  from  the  inner  to  the  outer  formula  takes  place  at  the  minimum 
value  of  y  at  which  (ut)inner = (ut)outero The  kinematic  displacement  thick- 
ness 6" is  given  by 
max 
The  foregoing  model  differs  from  that  developed  in  reference 5 in  several 
respects.  Replacement  of p with p2p/u2 at  y < y2  in  equation (1) has 
the  effect  of  removing  the  dependence  of  pt/p  on  variations  of  temperature 
and density.in  the  viscous  sublayer.  Although  future  experimental  data  may 
show  such  a  dependence,  it  seems  preferable  at  this  time  to  make  pt/p  dependent 
on  y+  alone,  and  thus  to  maintain  a  close  correspondence  with  incompressible 
flows.  Additionally,  the  factor BFg in  equation ( 6 )  allows  a  close  corre- 
spondence  of  the  present  inner  layer  Clauser  model  with  more  conventional  models 
for  flows  with  zero  pressure  gradient.  Finally,  us.e  of  y+  as  defined  in 
equation (9) in  the  Van  Driest  damping  factor  instead  of  the  definition  used  in 
reference 5 requires  rescaling of V to V = 18. 
Simplified  Analysis  Applicable  Near  an  Adiabatic  Wall 
The  following  approximate  relationships,  obtained  from  the  compressible 
Navier-Stokes  equations,  are  useful  for  checking  the  adequacy  of  the  mesh  used 
for  the  numerical  solution.  They  may  also  be  useful  for  deducing  values  of 
parameters  in  the  turbulence  model  from  experimental  data  obtained  from  sepa- 
rated  and  attached  boundary-layer  flows  with  pressure  gradients.  Upon  neglecting 
the  convection  and  inertia  terms,  the  steady-state  x-momentum  and  energy  equa- 
tions  can  bs  approximated  by 
Replacement  of 
of  state  for  a 
P + - , 2  C T  1 =P T2 1 
2 + - u2  (adiabatic  wall) 'RT. 'RT 2 2  
the  y-momentum  equation  with  ap/ay = 0 and  use  of  the  equation 
perfect  gas  yields 
T2 P=- 
p 2  T 
With  the  definition 
I 
and  substitution  of  the  foregoing  inner-layer  eddy  viscosity  model  into  equa- 
tion (11) , the  following  expression  for  u+  can  be  derived: 
where 
u+ = (1 - P+)f  (Y+) + €2 f (Y+) 
Y$ 
.,+ 
To obtain  the  relation  between  u+  and  u,  we  use  the  Sutherland  viscosity 
law  in  the  form 
Linearization of the  Sutherland  relation  and  substitution f equations  (12) 
and  (13)  into (14) leads to 
where 
2 + 3S2/Tw 
A =  1 + S2/TW 
B =  
1488 
RESULTS AND DISCUSSION 
A time-dependent  solution  of  the  compressible  Navier-Stokes  equations  based 
on  the  foregoing  eddy  viscosity  model  has  been  carried  out.  The  initial  flow 
field  is  uniform  in  the  x-direction  with a boundary-layer  profile  corresponding 
to  an  upstream  station  in  the  experiment  of  reference 8 .  At the  lower  boundary 
of  the  computational  field,  boundary  conditions  are  imposed  corresponding  to 
the  shock  wave  in  the  experiment  (see  fig. 1). During  the  calculation,  the  shock 
wave  grows  toward  the  upper  nozzle  wall.  Eventually, a steady  state is reached 
corresponding  to  the  conditions  at  which  experimental  measurements  were  made. 
As  an  illustration  of  the  type  of  information  that  can  result  from  such  calcula- 
tions,  features  of  the  time-dependent  response of the  boundary  layer  to  growth 
of.the incident,  induced,  and  reflected  shock  pattern  will  be  presented.  How- 
ever,  the  simplified  analysis  in  the  preceding  section  will  first  be  used  to 
check  the  adequacy  of  the  computational  mesh. 
The  steady-state  Navier-Stokes  solution  was  used  to  compute  dimensionless 
profiles of u+  versus  y+  according  to  equations (9) and  (21).  Three  such 
profiles  are  contained  in  figure 2. The  circles  represent  the  Navier-Stokes 
solution  and  are  at  the  computational  mesh  points.  The  dashed  lines  are  obtained 
from  the  simplified  analysis  (eq.  15)),  using  values  of p+ and  y+  evaluated 
from  the  Navier-Stokes  solution  according to equations (3) , ( 4 ) ,  f 9 )  and  (16). 
The  upper  two  profiles  are  at  stations  aft  of  reattachment  where  relatively 
small  pressure  gradients  are  present.  The  bottom  profile  is  in  the  region of 
constant  pressure  ahead  of  the  separated  region,  which  extends  from 
-2.6 < (x - xi/60)'2 0.5. The  close  correspondence  between  the  approximate 
and  numerical  solutions  leads  to  two  conclusions: (1) Use  of  the  boundary- 
layer  approximation  with  additional  neglect  of  convection  and  inertia  terms 
is a valid  approximation  near  the  wall  for  small  pressure  gradients;  and 
(2) the  computational  mesh  used  for  the  Navier-Stokes  solution  provides  adequate 
resolution.  However,  are  the  same  conclusions  valid  at  stations  where  strong 
pressure  gradients  exist? 
Profiles  near  separation  and  in  the  middle of the  reversed  flow  bubble 
are  shown  in  figure 3. Again  the  Navier-Stokes  solution  is  represented  by 
symbols  and  the  simplified  boundary-layer  approximation  by  dashed  lines. 
Near  separation,  the  approximate  results  are  invalid  for  values  of y+ greater 
than  about 300 because  of  neglect  of  the  convection  and  inertia  terms.  In  the 
middle  of  the  separation  (lower  curve),  the  simplified  analysis  retains  validity 
to  large  values  of  y+.  The  resolution  of  the  Navier-Stokes  solution  is  again 
shown  to  be  adequate.  Profiles  near  reattachment  are  shown  in  figure 4 .  The 
two  solutions  agree  closely  over a large  range  of'  y+  in  this  region. 
With  confidence  in  the  resolution  of  the  Navier-Stokes  solution  established, 
it  is  of  interest  to  observe  the  time-dependent  response  of  the  boundary  layer 
to  shock  impingement.  Figure 5 contains  plots  of  wall  pressure  distribution 
at a series  of  time  intervals  after  the  start  of  the  calculation.  Shortly 
after  the  shock  reaches  the  boundary  layer  and a reflected  shock  has  formed,  the 
wall  pressure  rise  is  steep  and  extends  about  one boundary-1ayer.thickness 
upstream  of  the  inviscid  shock  impingement  point,  which  occurs  at 0 on  the 
1489 
abscissa  scale.  During  succeeding  time  intervals,  the  pressure  gradient 
decreases  and  the  pressure  rise  moves  upstream.  Eventually, a steady  state  is 
reached  in  which  the  initial  pressure  rise  occurs  about  three  boundary-layer 
thicknesses  ahead  of  the  inviscid  shock  impingement  point.  The  plot  at  the  top 
includes  the  corresponding  experimental  steady-state  pressure  distribution  from 
reference 8 for  comparison. 
Figure 6 contains  plots  of  the  skin-friction  distributions  after  the  same 
series of time  intervals.  In  the  plot  at  the  top,  the  calculations  are  in  close 
agreement  with a Preston  tube  measurement  of  the  initial  skin-friction  coeffi- 
cient  Cf  and  oil-flow  observations  of  separation  and  reattachment  points 
(ref. 8). Skin-friction  measurements  were  not  made  at  other  stations  in  this 
experiment. 
REFERENCES 
1. Chapman,  D.  R.;  Mark,  H.;  and  Pirtle,  M.  W.:  Computers  vs  Wind  Tunnels  for 
Aerodynamic  Flow  Simulation.  Astronautics  and  Aeronautics,  vol.  13,  no. 4, 
April  1975, pp. 22-30. 
2.  MacCormack,  R.  W.: A Rapid  Solver  for  Hyperbolic  Systems of Equations. 
Paper  presented  at  5th  International  Conference  on  Numerical  Methods  in 
Fluid  Dynamics,  The  Netherlands,  June  28  to  July  3,  1976. 
3.  Hung,  C.  M.  and  MacCormack,  R.  W.:  Numerical  Simulation  of  Supersonic  and 
Hypersonic  Turbulent  Compression  Corner  Flows  Using  Relaxation  Models. 
AIAA,Paper 76 - presented  at  AIAA  9th  Fluid  Dynamics  Conference  in 
San  Diego,  July  14-16,  1976. 
4. Horstman,  C.  C.: A Turbulence  Model  for  Nonequilibrium  Adverse  Pressure 
Gradient  Flows. AIM Paper  76-412,  presented  at  AIAA  9th  Fluid  and 
Plasma  Dynamics  Conference  in  San  Diego,  July  14-16,  1976. 
5.  Baldwin,  B. S. and  MacCormack,  R.  W.:  Modifications  of  the  Law  of  the 
Wall  and  Algebraic  Turbulence  Modeling  for  Separated  Boundary  Layers. 
AIAA  Paper  76-350,  presented  at AIAA 9th  Fluid  and  Plasma  Dynamics 
Conference  in  San  Diego,  July 14-16, 1976. 
6.  Clauser,  F.: On Turbulent  Flow  Near a Wall.  Jour.  Aero.  Sci.,  vol.  23, 
no. 11,  1956, pp. 1007-1011, 
7. MacCormack,  R.  W.  and  Baldwin,  B. S.: A Numerical  Method  for  Solving 
the  Navier-Stokes  Equations  with  Application  to  Shock-Boundary  Layer 
Interactions.  AIAA  Paper  76-1,  presented  at  AIAA  13th  Aerospace  Sciences 
Meeting  in  Pasadena,  Jan.  20-22,  1975. 
8. Reda,  D.  C.  and  Murphy, J. D.:  Shock  Wave  Turbulent  Boundary  Layer  Inter- 
action  in  Rectangular  Channels,  Part  11:  The  Influence  of  Sidewall 
Boundary  Layers  on  Incipient  Separation  and  Scale  of  Interaction.  AIAA 
Paper  73-234  presented  at AIM 11th  Aerospace  Sciences  Meeting  in 
Washington,  D.  C.,  Jan.  10-12,  1973. 
1490 
70 
60 
50 
40 
30 
U+ 
20 
IO 
0 
-10 
-20 
-30 
x-X( 
80 
 P+ Y; 
0-2.714 0.733 72.4 
A-2.413 1.22 120 
0-0.603 2.29 464 I 
I I I I I 
o IO 102 103 1 0 4  105 
Y+ 
Figure 3. -  Velocity profiles near 
separation. 
1 
40 
U+ 
30 
20 
Ut 
, 
I I I I 
I IO 102 103 104 105 
Y+ 
0 
Figure 2.- Attached velocity profiles. 
A 1.508 0.501 412 i 
0 0.603 0.951 459 
o 0.301 1.2 I 469 
40 t 
30 
U+ 
20 
IO 
0 
I 
Figure 4.- Velocity profiles near 
reattachment. 
1491 
I 
-m-"*4- 
4.467 x sec 0 EXPERIMENT (Ref. 8 )  
0 EXPERIMENT (Ref. 8 )  -002 - @"- c-"" 
0 ~ " " " " "  
\ 
0 
./ 4.467 x 10-3 sec 
" 
"------- 
-.002 I I I 
'" ' 
I I I I 
1.161 /""-- 
0 
1.161 
/@-"""" 'X"' 
0.876 """ 
"-"-A 
-002 - """ . 0' 0.876 /c- 
\ 
0 Cf  0 ' 
0""""-- 
0.589 /------ 
"""4' -002 - """_ . 0° 0.589 
0 0 .  \, /' \d' 
-.002 ' 1 1 1 1 1 1 " 1  .~"""-- 
.002 /' 0.291 
0 0 
""""' 
""-""\,;/~ 
4 L  2 0.000 sec -.0021 I I I I I I I I -002 r 0.000 sec 
Figure 5.- Response of wall-pressure Figure 6 . -  Response of  skin-friction 
distribution. distribution . 
1492 


Calculation of a separated turbulent boundary layer

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server