Moment method for solving nonsimilar laminar boundary layer equations for three dimensional cross flo

Fillo, J. A.

English

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DEVELOPMENT OF A MOMENT METHOD TO SOLVE 
THE THREE-DIMENSIONAL BOUNDARY LAYER EQUATIONS 
REPORT NO. NGR-22-010-019/3/68 
3. A. FILL0 
NASA GRANT NO. NGR-22-010-019 
JUNE, 1968 
DEPARTMENT OF MECHANICAL AND AERO-SPACE ENGINEERING 
8 UNIVERSITY OF MASSACHUSETTS 
AMHERST, MASSACHUSETTS 01002 
https://ntrs.nasa.gov/search.jsp?R=19680019442 2020-03-12T09:53:36+00:00Z
INTRODUCTION 
.Problems i n  t h e  theory of laminar incompressible and compressible three- 
dimensional boundary l a y e r s  at tract  much i n t e r e s t .  
subj.ect appear i n  l i t e r a t u r e  re ferences .  The main ob jec t ives  of t h e  theory are 
t o  c a l c u l a t e  t h e  s k i n  f r i c t i o n  d i s t r i b u t i o n ,  and t o  p red ic t  t he  loca t ion  of t he  
boundary l aye r  s epa ra t ion  su r face  on three-dimensional bodies ,  as w e l l  as t o  
determine hea t  t r a n s f e r  e f f e c t s  during,  say,  t h e  re-entry of space vehic les .  
Comprehensive reviews of the  
1,293 
8 
The three-dimensional boundary l aye r  equat ions,  e i t h e r  f o r  "small" o r  "large" 
croes  flows, are not  e a s i l y  amenable t o  mathematical ana lys i s  owing t o  t h e  complexity 
of t he  set of p a r t i a l  d i f f e r e n t i a l  equat ions governing t h e  phenomena. Exact 
so lu t ions ,  i n  genera1,are  not  found, and r e s o r t  must b e  made t o  numerical procedures,  
or  t o  approximate methods. With regards t o  t h e  na ture  of so lu t ions ,  t he  problem 
may be  reformulated as follows: 
and it i s  known t h a t  numerous and severe  r e s t r i c t i o n s  are imposed as flow condi t ions 
f o r  s i m i l a r  so lu t ions  t o  ex is t .  Considering t h e  d ive r se  app l i ca t ions  of compressible 
and incompressible laminar boundary l aye r  theory,  i t  is seldom t h a t  a l l  of t he  
s i m i l a r i t y  condi t ions are m e t .  Hence, the  question: What approach should be used 
t o  p red ic t  t he  behaviour of t he  nonsimilar laminarboundary l aye r?  
review, Dewey and Gross consider t h i s  quest ion but  with emphasis on two-dimensional 
flows. I n  essence,  
i t  was shown t h a t  t he re  are four  b a s i c  types of approach: 
such as i n t e g r a l  and series s o l u t i o n s  containing f r e e  parameters; s t r i c t l y  numerical 
approach; l o c a l l y  similar methods; and, l a s t l y ,  expansians absut  similar so lu t ions .  
by exact  so lu t ions  i t  is  meant similar so lu t ions  
In  a recent  
Their f ind ings  could a l s o  be appl ied t o  three-dimensional flows. 
approximate techniques 
Two of t h e  above approaches are discussed i n  t h e  present  r epor t  extended t o  
three-dimensional flows: expansions about similar so lu t ions  and an approxiniate 
i n t e g r a l  technique. 
- 2- 
Statement af the Problem 
We consider a body (prevailingly. canvex) 'of'flnite'dimeosions, placed in, a 
uniform air stream. Furthermore, we assume that the inviscid flow around the 
boQy is known and allows an extrapolation towards y = 0 (surface of the body). 
These extrapolated values at the wall are considered as the outer boundary values 
of the boundary layer which we suppose laminar. over the whole surface. 
1 
I 
WI 
The Navier-Stokes equations, written in a system of curvilinear coordinates 
and based on the following three-families of surfaws: 
y = constant; surfaces parallel to that of the body (y = 0 ) ;  
z = constant; normal surfaces along the skreamlines of the inviscid flow 
at the wall of the body; 
s = constant; normal surfaces along the orthogonal trajectories of the 
streamlines of the inviscid flow at the wall; 
assume after the introduction of Prandtl's boundary layer hypothesis, the form: 
in the directions of (s, y, z); el(s, z) and e2(s, z) are the metric 
3' 
K~ and K are Gaud 
the inviscid flow streamlines and their orthogonal trajectories (e 
normal of the body, being unity by definition). 2 
elements along 
along the outer 
curvature terms, 
-3- 
given by 
6 
and 
''vas ./" /v cpf+ UL+ w2 
2 H=CpT+ (7) 
is t h e  t o t a l  e n e r m ' o f  t he  flow. 
are : 
a t  t h e  wall 
(T is the  Prandt l  number. The boundary condi t ions 
and at  t he  outer  edge of t h e  boundary l aye r  
We assume now, as a f i r s t  approyimation, t h a t  w < u ,  a and - a are de r iva t ives  
a5 a 8  
p a r a l l e l  t o  t h e  w a l l ,  and therefore ,  may be considered as opergtors  which conserve th'e: 
order of a t e r m .  The system becomes, after app l i ca t ion  of Euler ' s  equations t o  
where Eqs. 9,  10, and 1 2  do not  conta in  w, and depend on z only as a simple parameter. 
Furthermore, i f  we agree t h a t  a l l  de r iva t ives  and a l l  i n t eg ra t ions  are executed f o r  
z 5 constant ,  w e  may write 
la,a where dx = elds e, as 8% 
5 
is t h e  length element of an inv i sc id  flow streamline.  
has been recent ly  employed t o  der ive  E q s .  9-12, a l so ) .  
(A formal per turba t ion  procedure 
-4- 
1 
Analysis 
Applying the Levy-Lees transformation 
1L 
9 ( 7 ~ ) p S ~  f w p w  Ue e," dx 
z = constant 
J 
w a=&(* 
ue 
(14) 
to Eqs. (10-12), we obtain 
where 
The boundary conditions are: 
w- 0 (19a) 3- %w 
'L-"" : 3-  .w- 0 (19h) 
Equations (15)-(17) subject to boundary conditions (19a) and (19b) are the equations 
on which we shall focus our attention. 
-5- 
Asymptotic Expansion of the Boundary Layer Equations 
Of the four basic types of approach to the solution of the nonsimilar 
laminar boundary layer equations, it seems that expansions about similar solutions 
has received little, if any, attention for three-dimensional flows. The motivation 
for such an investigation arose out of the results reported in Refs. (5) and ( 6 ) .  
1 In the former exact numerical ca$cylations of the complete nonsimilar, small-cross 
flow, boundary layer equations were compared with locally similar solutions. While 
the results obtained from the two different methods for the main flow were in 
excellent: agreement, the cross flow results, while not drastically different, did 
reveal that further analytical work is necessary. 
In the latter reference only the method of utilizing locally similar solutions 
was used. As pointed out in this work, the application of locally similar methods to 
the secondary flow has no quantitative confirmatioq. With this in mind it was felt that 
some light could be shed on lacally similar methods by the expansion methods. In 
addition, the use of integral or improved integral methods could be placed on 
surer footing if there are more reliable results to compare the integral results with. 
W 
7 
rr" The basis of the expansion method may be traced to the work of Mekayn. The 
I I' 
underlying premise is as follow: if the boundary layer is at all times very nearly 
described by a similar solution, then the direct effects of the nonsimilar terms may 
be calculated by asymptotically expanding the full boundary layer equations in terms 
of small parameters which measure the departure of the solutions from similarity. In 
this way, the accuracy of locally similar methods is explicitly determined by using 
the full nonsimilar equations. 
I To demonstrate the method for three-dimensional flows, we conslder the 
(obtained from Eqs. 15 and 16 by setting incompressible momentum equations 
-6- 
subjec t  t o  t h e  boundary condi t ions 
5 
It is t o  be noted t h a t  Eq. 20 is t h e  transformed two-dimensional boundary l aye r  
equation. 
t o  t h e  appropr ia te  expansion w a s  t h e  invers ion  of independent va r i ab le s ;  i.e., t h e  
change of va r i ab le s  from 9 
t h e  crossflow equation, Eq. 21, w e  f i nd  t h a t  Eqs. 20 and 2 1  transform t o  
8 Merk was t h e  f i r s t  t o  expand t h i s  complete nonsimilar equation. The key 
, t o  8 ,  '1,. Adopting t h i s  change of va r i ab le s  f o r  
where 
For small w e  perform an  asymptotic expansion of 5 and Wof t h e  form 
(27) 
Subs t i t u t ing  Eqs. 26 and 27 i n t o  Eqs. 23 and 24, as w e l l  as 25a and 25b, y i e lds ,  on 
equating l i k e  powers o f 6  , t he  following sets of equations : 

I 
-8- 
Equations 28 and 29 represent the similar solutions of Falkner and Skan while 
Equations 32 and 33 represent the first correction f, to the main flow velocity profile 
which arises from nonsimilar terms. Equations 30 and 31 are the Corresponding similar 
solutions for the corss flow and Equations 34 and 35 represent the first correctionUI, 
to the secondary flow velocity profile because of nonsimilar terms. 
that 6 is not too large so that the series solution in powers off will, converge 
satisfactorily. 
c. 
It is assumed 
Before proceeding with a discussion on the integration of the equations, a few 
comments on the question of similarity and local similarity of the ciposs flow question 
are pertinent. 
two velocity components that are different from zero and parallel to the wall are present 
in the external flow, the pertaining velocity Uor W of the exfernal flow must necesgarily 
be chosen as the scale factor for XI and w. If, an the other hand, no W-component 
is present in the external flow which corresponda to the use of streamline coordinates, 
an arbitrary function of the dimension of a velocity as scale factor may be chosen for 
w. While this discussdon has been with respect to similar solytions, it is nevertheless 
valid for locally similar soultions as well, In the two previ0usJ.y cited references 
( 5 and 6), where sfreamline coordinates were used, the question of arbitrariness in 
the scale factor is what lead to different small crms flow, locally similar equations. 
Two different scale factors were introduced. The present w w k  is concerned with the 
dimensionless cross flow velocity component, w/w, and, hence, the reason why the leading 
terms, in e o ,  check with the incompressible equations of Reference [ 5 I and not 
with those in Reference [ 6 3 .  
9 In fhe work of Geis it was pointed out that for similar solutions if 
Returning to the problem of integrating Equations 28-35, one finds that methods 
have been discussed [4,10] 
subsequent higher order terms). 
to integrate the equations for Eo and fl (as well as 
The problem before us is to findu), and W,. 
-9- 
As in the case of computing fl, the primary difficulty arises in computing the 
inhomogeneous term %( $,%) which contains derivatives of w, with respect to both 
s a n d  t. This is a difficult term to calculate numerically because a number of 
similarity solutions in the neighborhood of 
problem is not insurmountable in light of the numerical methods'discassed in 
must be known with high precision. The B 
v 
[ 101 for 
the calculation of fl, 
eliminating the parameter 
Some simplification of the equations may be achieved by 
by defining new dependent variables 
At present work is being carried on in the investigaCion of Equations 30, 31, 34 and 35. 
Moment Integral Conditions 
Returning to the compressible forms of the equations, Equations 15 -19b, a 
dfscussion of the moment equations will follow. 
has been presented in Reference [11] for two-dimensional flows and, in which, 
The basic formulation of the method 
Equations15 and16 have been discussed in detail, What we shall be primarily 
interested in is Equation 17. 
at whichW=O we obtain a series of integral I= "2. Introducing a finite "edge", 
conditions by multiplying Equation 17 by (m = 0, I, 2 , , . . , )  and by integrating 
termwise from 0 to . It is assumed that the streamwise and cross flow boundary 
layers are described by the same thickness, a usual 
5 %  
9 
assumption for the three- 
dimensional case. The first term on the left hand side of Equation -..- 17 baoomes 
L 
m 0 (36a) 
m-1 (46b) 
-10- 
For mal the influence of the transport properties throughout the boundary layer is 
present. 
introduce a mean value of C, for m7-1, denoted C., and varying with 9 and to consider 
the remainder to be a function of 9 and 5 , known from a previous seep-' dnL'the 
integration of the ordinary differential equations. 
As in the case of the streamwise momentum equation, it is convenient to 
* 
Let 
I" 
(37) 
and for m>l, Equation 36b beomee 
Selecting ZV, so that 
Equation 38, assumes the form 
m a  1 
4us to the case of the streamwise momentum equation, Equations 36 - 44b provide 
a8 many Integral conditions for the cross flow velocity profile as I s  wQnted by lettdng 
1 e 8,  2, 2, * * *  
Summarizing, the integral conditions corresponding to m 0,1 in Equations 36 ..I 44b 
taka on thcl following forms: 
( 4 5 )  
m = Q  
-12- 
0 The inqegra l  moment condi t ions  f o r  f and g may be  found-in Reference [ 111. 
* YL’  
I n  order  t o  i n t e g r a t e  t h e  above equat ions appropr ia te  p r o f i l e s  f o r  f 
and ware needed. Since polynwdals  were u t i l i z e d  f o r  f and g, it is  convenient 
t o  express I& i n  terms of a polynomial. 
appropr ia te  parameters must be  i n  t h e  polynomial t o  account f o r  change of s i g n  i n  the  
truss flow p r o f i l e .  
awoun t  of and p r e d i c t  t he  change i n  s i g n  of t h e  secondary flaw. 
z 
Of course,  i n  doing soyone must recognize that: 
This i s  an important p a r t  of t h e  s tudy t o  be  a b l e  t o  t a k e  
To ga in  a b e t t e r  understanding of t he  behaviour of t he  c ross  flow equations with 
regards t o  t h e  moment method, t h e  incompressible equations are being inves t iga ted .  
A t  p resent  the  example chosen i s  a problem with flow reversal. 
REFERENCES 
1. Moore, F. K., "Three-Dimensional Boundary Layer Theory," Advances in Applied Mechanics, 
Vol. 4, Chapeer 4 .  Editors - H. L. Dryden, Th. von Karman, Academic Press, 1956. 
2. Cooke, J. C. and Hall, M. G., "Boundary &ayers in Three-Dhensions," Royal Aricraft 
Establishment Technical Note, Report No. 2635, February, 1960. rp 
3. Schlichting, HI, "Three-Dimensional Boundary Layer Flow," Deutsche Forschungsanstalt ,- 
fur Luft-u. Raumfahrt E., V., Braunschweig Institute fur Aerodynamik, December, 1962. 
Dewey, C. F. and Gross, J. F., "Exact Similar 6olutions of the Laminar Boundary- 
Layer Equations," Advances ip Heat Transfer, Vol. 4 ,  Academic Press, New York, 1967. 
Fannelop, T. IC., "A Method of Solving the Three-Dimension81 Laminar Boundary Layer 
Equatiws with Application to a Lifting Body, AIAA Paper 67-159, 1967. 
4, 
5 .  
6. Kang, S. W., Rae, W. J., and Dunn, M. G., "Effects of Mass Injection on Compressible 
Three-Dimensional Laminar Boundary Layers with Small Secondary Flow," CAL Report 
No.<'AI-2187-A-2, 1966. 
7 .  Meksyn, P., "New Methods in Laminar Boundary-Layer Theory," Pergamon Press, Oxford, 
1966. 
8, Merk, M. J., "Rapid Calculations for Boundary-Layer Transfer Using Wedge Solutions 
and Asymptotic Expansions," J. Fluid Mech., 5, 460-480, 1959. 
9, Geis, T., "Similar Three-Dimensional Boundary Layers," NASA TTF-30, 1961, 
10, Evans, H. L., "Laminar Boundary-Layer Theory," Addlson-Wesley Publishing Company, 
Reading,i1968. 
11, Libby, P. A. and Fox, H., "A Moment Method for Compressible.Laminar Boundary Layers 
and Some Applications,'' Int, J. Heat and Mass. Trans., 8, November, 1965. 


Development of a moment method to solve the three-dimensional boundary layer equations

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