Model equations for studying compressible turbulen boundary layer

Donaldson, C. D.

Gilligan, B. D.

Yates, J. E.

English

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A.R.A.P. WORKING PAPER NO. 71-4
A STUDY OF COMPRESSIBLE TURBULENT
BOUNDARY LAYERS USING THE
METHOD OF INVARIANT MODELING
John E. Yates
Coleman duP. Donaldson
Barry D. Gilligan
This work was supported by
the National Aeronautics and Space Administration
Washington, D.C. 20546
under
Contract NASW -1777
Aeronautical Research Associates of Princeton Inc.
50 Washington Road, Princeton, New Jersey 06540
January 1971
i
1.
I . INTRODUCTION
The present report summarizes the current status of a program
to analyze a compressible turbulent boundary layer. It is part of
a continuing effort at A.R.A.P. to develop equations and investigate
general turbulent shear flows by the method of invariant modeling.
This method has been highly successful in the treatment of
incompressible turbulent flows (Refs. 1, 2 1 3). As this report
will show, we are now on the verge of being able to make calcula-
tions for the fully compressible case.
This document is in the nature of a status report. No pretense
is made of giving a fully detailed explanation of the modeling
procedure or of the computer routines that were developed to program
the model equation. A more detailed account will be forthcoming
upon completion of this work.
a [—I
.^ r
(2)
aTq^ _ - k 
axe
2
J = C 
p 
T + -7
Heat Flux
Total Enthalpy
3
1
f
2.
II. MODEL EQUATIONS
The model equations that are appropriate for the study of
compressible turbulent boundary layers are given in the appendix.
Here we outline briefly the key steps and assumptions that are
made to derive the model equations. The compressible Navier-Stokes
equations are the fundamental equations from which the model
equations are derived. In tensor notation, they are
at + ( Pu`^) , = 0
P 
at 
+ 
ujui,J 	 a^ + Ti,iax
( 1 )
3
a	 + (P` UJ ) , ^	
at 
+ ( u ` -r ) , k	 qj
p = pRT
	
State Equation
where
i
F	 Irij = µu1
	+ u^ i - 2 ij ukk Viscous Stress
To derive turbulent equations, we assume each variable to be
the sum of a mean and fluctuating part, e.g.,
P = P + P^ 	 (3)
Substitute these expressions into the Navier-Stokes equations and
compute statistical moments to arrive at differential equations for
statistical quantities. We have derived equations for the
following variables:
No. of Equations
Mean .Density	 p	 1
Mean Velocity	
u	
3
Mean Temperature 	 T	 1
Density Fluctuation
	 p	 1
t.
3.
Temperature Fluctuation	
T	
1
Decisit,y-Temperature- Correlation 	 p' T'	 1
Density-Velocity Correlation
	 p'ui	 3
Temperature-Velocity Correlation 	 T u1	 3
Velocity Correlation	 uiuk	 6
20
Each of these equations (twenty in total) involves higher corre-
ia +.ions, both third and fourth order. To close the equation, we
must give the higher order correlations in terms of variables that
we want to solve for. The method of invariant modeling permits us
to accomplish this task in a rational way. This method has been
described in detail and used successfully in previous work with
the incompressible equations(Ref.1,2). Here we give three typical
model terms that illustrate how the two turbulent length scales
enter into the formulation. The models are as follows:
Uiu^ uk = - n^ I ` uiuk ^^ k + C u i uk ), j + \ UN
, ( ,	 ' _ ^
	
K Tr	 ( )P l ui' ^ + 111, i^ - n
	
6ik3 - uiuk	4
u' u'gmn ,	 ,	 _ i k
ui,muk,n
	
>12
where K = u ' umt is the turbulent kinetic energy. The length
parameters m n and X are related to the macroscale and micro-
scale of the turbulence. In our work, they are assumed to be
related by the formula
n
a + bReA
(5)
Re n = K
µ
where a and b are constants to be determined by comparison with
experimental data. Equation (5) has been used by Glush ko (Ref, 4)
and Bee-'--With and Bushnell (Ref. 5) in their models of the turbulent
Kinetic energy equation.
One final assumption that is used to tierive the turbulent
Equations in the appendix requires some diGcussion. It has been
customary, in previous work, to assume that the fluctuating velocity
field is incompressible, i.e.,
u ^ = 0
	
(b)
even though the mean flow field may be highly compressible. An
alternate procedure is followed in the present work. Based on exper-
imental and theoretical estimates, we assume that the temperature
and density fluctuations are large compared to the pressure fluctu-
ations, i.e.,
p,2 	 0	
T^2 >> --y—	 (7)
P	 T	 p
r'rom the equation of state, we then obtain a simple proportionality
between p' and T' , i.e.,
P
Finally. Eq. (8) is combined with the equations for density and
temperature fluctuations to actually solve for the turbulent
compressibility. The result is
where
0' = _j Tk^ + uj I-k
,k ,,j	 ,k 
H' = q 	
(10)
J, 
This result is subsequently used to eliminate all equations for
variables involving T'	 The final number of equations is reduced
from twenty to fifteen. These equations are summarized in the
appendix.
III. METHOD OF SOLUTION
The model equations given in the appendix are valid for
fairly arbitrary turbulent compressible flow. To obtain equations
for a two-dimensional turbuleni^ boundary layer, we first eliminate
the mean velocity component ujj 0: 1
 w . Also, in all diffusion-
type terms, we retain only second derivatives with respect to y
the coordinate normal to the wall. The final result is a set of
fourteen nonlinear,coupled, parabolic, differential equations.
They are to be solved subject to boundary conditions at the wall
and in the free stream. Initial data must also be specified at a
given x location or, the wall.
Because of the large number of equations involved, it was
decided that the derivation of the finite difference equations and
the selection of the difference grid must be fully automated. The
firs'. step was necessary to eliminate needless errors in performing
Lq
 very large amoLint of algebraic detail. The second step is needed
to minimize machine computation time and to control the accuracy of
the calculations.
To illustrate the method of solution and show how the two
automatic routines fit into the scheme, we consider a single
nonlinear diffusion equation
au - F(u) 62uTX1
— — — - +1
h ,j +1
1^x
n	 n+1
Referring to the sketch above, we can write Eq. (11) as an implicit
finite difference equation as follows:
n+1	 n	 n+1
	
n+1
	 n+1	 n+lu^j	- u1 _
	 1	 u +1 - u	 u	 - u -1
(12)
(11)
6.
I
where
CY	 ' CtFU+ = F(u^1+
nnU, ,	 1 . •i
un , - --^
n	 n
n _ --- uJ -1
	
U
	 ^	 2
Rearranging terms in (12), we obtain
-Au n+1 + Bu n+1 _ Cun+l = D
	
J-1	 ,j	 j+1
where
OxFi7-
	
( LX) ^
+TFA	
i+ h ,j+	 C	 i+ J+1 J+1
B= hj +h	 + +'	 D= un
,j	 ,j + 1	,j	 ,j + 1
The last result is a linear difference equation for the unknown u
at station n+l . Given data at station n , we can solve this
equation by a familiar standard algorithm(Ref. 6) for u at n+1
The difference equations that result from the boundary layer
equations in the appendix are exactly of the same form as Eq. (13).
The only difference is that u is a vector with fourteen
components. Thus, A, B, and C are 14x14 matrices and D is a 14-
componentvector. Each element of A, B, C, and D requires a
defining equation. In all there are 602 such equations that must be
written out. It is precisely this number that dictates the need
for an automatic routine to write the equations. Furthermore, an
estimate of the time required to make a single step, Ax , is of the
order of 4 seconds on a CDC 6600 computer or 15 minutes on an IBM
1130. This means that each step must be made efficiently. Thus,
the routine to automatically adjust the grid spacing in the x and
y directiors was developed. Each of these routines is described in
the following section.
(13)
(14)
^r
7.
IV. AUTOMATIC DIFFERENCING AND GRID SPACING
A. Automatic Differencing
The routine that automatically differences the tensorial
equations can be reduced to the following basic steps:
1. Code tensor equations (input);
2. Expand into additive terms and set metric tensor;
3. Drop terms that are zero or of higher order and .redefine
coefficients;
4. Compute difference formula (output).
The actual computer program consists of 9000 cards! A detailed
description would require a small book in itself. Here we can only
give a general description of the four basic steps.
1. Code tensor equations (input)
The tensor equations are coded directly in a form that the
computer program will accept. All tensor notations and operations
are given special symbols, e.g.,
Ai
 + A' I	 gij ♦ Q'IJ
Ai ♦ At I	 61 ♦ & "I'J	 (15)
A i ♦ ( A),I
The equation for the mean density, for example, is coded as follows:
U"J*(D),J=-D*(U"J),J-(DU"J),J
	
(16)
The asterisk denotes multiplication as in FORTRAN programming.
2. Expand into additive terms and set metric tensor
After the input is carefully checked, the program proceeds to
expand tensor sums and parenthesized expressions into a string of
additive terms. Then the zero elements of the metric tensor ke
eliminated and unit multipliers are dropped. The reduced--timme
then printed and punched for the purpose of editin If it ft -deem!
necessary. For the present problem, there are 1549 t-drms 1- t&3
at this point in the operation. The a uation for lid seams
has 10 terms that are displayed below (of. Eq. 4 ^^6 __
ai
i
i
(17)
y	 + g
7.
2
3
4 =
6 -D*(U"2),2
7
-D*(U"3),3
8 -(DU"1),1
9 -(DU"2), 2
10
-(DU"3),3
Drop terms that are zero or hi gher order and predefine coeffi-
cients
The next step is to hand-edit the terms for the particular
problem at hand. For the two-dimensional boundary layer, we set
U'3 = 0, ( ),3 = 0 and drop all but second y derivatives in the
diffusive terms. The number of terms then reduces to 226. For
example, the mean density equation becomes
1	 U "1*(D),1
2	 +U'' 2* (D), 2
3	 =	 (18)
5	 -D*(U"2),2
6	 -(DU"2),2
Another part of the editing process is to gather all coefficients of
differential expressiors into a single term and give it a name for
the purpose of precalculation. This is an economy measure to avoid
multiple evaluation of the same expression in the final program.
The mean density equation does not happen to have any such
coefficients.
4. Compute difference formula (output
The final step is to give the edited equations to the
for differencing. The output at this stage is an expli
formula on card for each nonzero element of the matrix=
and D discussed in Section III. We omit the actin
the mean density equation here to avoid explanati
changes. Suffice it to say that we get the _
elements of the A. B. and C matrices-and 	 _	 'W
In total, there are approximately 70 elemen --^-=
matrices that are nonzero. With the" fir
routine to write the subroutine to d-^ --
tion algorithm.
9.
B. Autome 1;ic Grid Spacin
The routine that adjusts the grid spacing is composed of two
parts. The first part chooses the step size for integration along
the wall (x-direction) and decides whether to back up or go forward
at each step. The second part redistributes the y points normal
to the wall. A brief description of the logic involved in each
part follows.
After arriving at station n by integration, we have the
dependent variables available at stations n and n-1 . Thus we
cancompizte the maximum percentage change of each variable in the
last step.	 If that change exceeds a given limit, the program
backs up and takes a smaller bx increment. If the change is
within its bound, the next step is taken. The new step size is
computed from the maximum gradient of the most rapidly changing
dependent variable.
Once the decision is made to go forward, the second part of the
program examines the distribution of y points and adds and removes
points as needed. The main criterion for adding or removing points
is based on a local curvature parameter b^ that is computed from
three successive y points.
If b, < E l (given limit), the center point is removed.
If E l < 6  < E 2 (given limit), no change is made.
If b^	 E 2 , two points are added.
Further limitations are imposed on addition and removal of points
to ensure that the ratio of two successive step al .-es does not
become too large.
The automatic grid spacer has been written and debugged on an
existing program for incompressible flow. It has been found to
work satisfactorily.
IF
t
G
e
c
e
v
V. SUMMARY AND RECOMMENDATIONS
The accomplishments of the present program are summarized as
follows:
1. The model equations appropriate for the investigation of
compressible turbulent shear flow were derived.
2. The model equations were written in finite difference form
by an automatic differencing program that was developed
under this contract.
3. An automatic grid spacing program was developed to ensure
the economic solution of the difference equations.
All main parts of the full computer program have been written.
It remains to debug the final program and proceed to the solution
of actual problems.
It is recommended that the following steps be carried out in
the sequel to the present work:
1. Complete the final debugging of the turbulent boundary
layer program. Perform check runs to compare with
previous incompressible results.
2. Make complete transition runs for the following four cases:
a. M = 0	 adiabatic wall
b. M = 3	 adiabatic wall
c. M = 3	 Twall/ e = 0.5
d. M = 3
	 Twall/Te = 0.1
Exact values of M and T_ 'i^`' should be chosen for comparison
with available experimental dita.
REFERENCES
1. Donaldson, Coleman duP.: Calculation of Turbulent Shear Flows
for Atmospheric and Vortex Motion. Dryden research Lecture,
9th AIAA Aerospace Sciences Meeting, New York, January 1971.
2. Donaldson, Coleman duP. and Sullivan, Roger D.: Decay of an
Isolated Vortex. Boeing/AFOSR Aircraft Wake Turbulence
Symposium, Seattle, 1-3 September 1970.
3. Donaldson, Coleman duP., Sullivan, Roger D. and Rosenbaum,
Harold: Theoretical Study of the Generation of Atmospheric
Clear Air Turbulence. 8th AIAA Aerospace Sciences Meeting,
AIAA Paper No. 70-55, January 1970.
4. Glushko, G.S.: Turbulent Boundary Layer on a Flat Plate in
an Incompressible Fluid. Bull. Acad. Sci. USSR, Mech. Ser.
No. 4, 1965.
5. Beckwith, Ivan E. and Bushnell, D.M.: Detailed Description
and Results of a Method for Computing Mean and Fluctuating
Quantities in Turbulent Boundary Layers. NASA TN D-4815,
1968.
6. ' Keller, P_.B.: The Numerical Solution of Parabolic Partial
Differential Equations. Mathematical Methods for Digital
Computers (A. Ralston and A.S. Wil , eds., John Wiley an
Sons, Inc., New York, 1960.
11.
F-
D
f^
s
12
APPENDIX
TURBULENT MODEL EQUATIONS
Mean Density (p)
Mean Velocity (ui)
puj u	 = - P',-, ' u	 - ( pu' u`^' )
	 - ( p' u' uJ )i,,j	 i,^	 i 	 i	 ,i
+ g j k (A VT cp'ui) ,k + (P^uk),i
+ µ g'(ui ^^+ u^^ i }	 3 bi u'I
,J
Mean Temperature (T)
paj	 j	 p u	 li	
p	 )IJ
+ gjk TAf .) k	 -	 uj^
P	 v
+	 uJk r gki (u^^^ + 
u^,^ )	2 bk uIx^ + g,jk
	
T,J1L	 3	 (,C-v 	 /k
13
Density Fluctuation ()
u^ pp _ -2pp u'j ^j
 - 2p^u 
rP , ,j + g'jk(AA p 
+ 2 _- 1 µ ^^ 9 k uj u u I _	 K
'YP	 A^	 'k	 I 9JI 3
T
+ Ps P;'^ 51k 9ki (u^j. ^ + ui,i)	 2 dk al3	 2p
+ Ili/ p- ,^' o r a l _	 K	 km -
	
-	
- 2 k -m
2 A	 g	 $ k g.3 g (u^, m + um, ^ ) 3 b^ 
am
_ 
2_ - 1 
g k	
T P + T
-Y5
	
,k ^ ^	 -
	
P ,I	 P
	
t	 77 _ 77
+k (
T ) I kp	 + T
	
2k+ T	 +TP2
	
P , ,	 p /, 2	 } k
	
P ,	 p
_ gki 71+ ks T p r 2 T	 + T
	
ksT p'^ + ks T p k
P ,k	 P
+ µ(y - 1)p^ 2	 2gki u ^j	 urur _ 6
	
K
	
-Y5 A^	 , k 	 ^^ 3
li)+ gJ' ( u;uk g1k 3 [gkm(aj,. + %,J )	 3 dk a  I'M])
t14.
Density-Velocity Correlation (p'ui)
-
p
7—	 77Pu ^ui ,J + 2plui (pIu I )	 - p'ui' PuJ
p u	 ( dui, ) - u i u	 -PP.,
+ pgJ Af ( P 'ui) ok + (P'ug),i
+ 2Af 9jk (P ,ui)
,k + (P^uk)^i P,i +AAR gJk 7 ui
+ 2A f p : i uJ + u i (A f p
, 
i)
+ p bA2 EstESt + AA)
 (PIuki) k
p83estESt + r Wu")
_2 (_y
 - l 1119ki u^ C u u _	 K
	
'YP	 ' k` ' 	9 k 3 ^, i
µ8T
u
J k p^ui gki ( u^^^ + u^ ^) - 3 
^
dk
p	 U,3	 ,
2 gJ
 
I( u^uk 
_ g^ k 3 `	 gkm(u^ 
m + um, J ) 2 a ,^ umm	 -_
	
``	
,i	 3	 _
52(-y- 1
	 ki
	 -
+	 g	 k	 T	 T	 j
	'YP	 'k	 p ^ I P ui +	 (P ui ) ,I	 t
_	 P
(u^u^ -	 K	 + T P'ui glk	
--
2p _A A	 P 	 3	 =°
!ST
[(ksT
	 kT
+ p 
P rui T,ip k + T p	 -	 P^ui + 
8 
(P,ui)^k
_	 P , k	 P__
+ k8T P C u' u' - g, K	 _	 p 2 g j u' u' - g	 K 1 µ	 ==
2pAf	 i k	 i^ 3^	 A^	 \ i Z 1^ 
- µ8T P6ki (u	 + u ) - 2 bk ul
P	 i'^	 ^' i 	3 i '^ k
	
==
ANN
- -	 -
15.
Velocity Correlation (uiuk)
	
PuJ(uiuk),J = - p'uk uJui,J - p'ui u Juk 	- puku	 ui, J 	 J
P 
uiu ' uk ,J	 u-i k(PuJ) ,J + gJ^ PA 3 (ul ) k,
• ( uUk)	 + ( uiuk) i	 + A3K uJ (Plug)
I) I J	 i ,k
• (P 7uk)^ i 	+ Af g I j [( P ,U;) ^ k + (P7uk)^^ ui,J
,J
• A3K gil (p'u^)^ i
 + ( 1ui
	
uk,J
p bA ES s + Af (u^u
	
i	 )4),k
p/bA e S^+Av/K- (uku'	 -
+pa e Est + VK u'u,' -g K\3 st	 A	 i	 ik 3 1
i T
	 -
u
+ µ gJI(uiuk) ,J,^ - 2 i ulk + µ, J 9Ji(uiuk) E
+ (u ' uJ i )	 + (u ^ uJ l	 + J¢(	 K`
	
k	 k	 ), i
	 g\ uiuk -
 gik 3 J^
T
µs p^uk gJi (ui ^^ + 51,i ) 
- 
3 
ai 51
P	
,J
ST p^ui J1 (5+ u^ k)
	
2 6J u^	
-
p _ L
	
, _	 ^ J
,
	
µST p'u'	
+ SST 
p 	 u^_g
k)	
-(Tj g	 Jk lP	 ,J	 2pA 3R	 3!
	
[gj '( ai'l	 -	 _ 2 aJ u'	 _ [( VsT P
^--	 P	 , J
+ µBS p ( u ^ u ^ _	 K	 J1-	 -	 _ 2 J
2 A f	 J i gJi 5 g (uk^ l + uJ, k)	 dkP	 -


A study of compressible turbulent boundary layers using the method of invariant modeling

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