An approximation which reduces the computation of three dimensional, laminar, compressible, boundary layer equations to the problem of solving two dimensional type boundary layer equations is presented. A comparison of this method with a fully three dimensional boundary layer calculation is provided

Burbank, R. A.

Fillo, J. A.

English

NASA Technical Reports Server

\J(
ON THE C/i.LCUU\TIQt.i OF
THRE[-DlIIjENSIONAL LAIHNP,R BOUNDARY LAYER FLO\'JS
.L
J. A. Fillo*, R. Burbank '
Uni versity of 11assachusctts, ft,mherst, ~·~(l.ssachusetts
N Nm·1ENCLATURE
pressure
x-component of inviscid flO\'l
z-component of invisc~d rim>!
(THRU)
4\J
(CODE)
4-
(CATEGORY)
12 '1"
(ACCESSION NUMBER)
U -------,,~/l...,j~-----
N (P'AGES)
o
>0 C R - /,2 Keg 10
~ (NASA CR OR TMX OR AD NUMBER)
f '15......
s trear::ful1cti ons
= tranSfOrIi12G coordinates
:: Cartesian coordinates
dens i t.y
GtH'\iii':)' vati able of inte0ration .
distance from cylinder axis to the leading edge of the flat
plate (::~5.7cl11)
:: velocity components in the x,y,z directions
velocity upstream of the flat plate
cyl i nde r radi usf: F(~,A,l;;)dA
u/U
fll G(c;,A,Z;)d>.o .
I;IIU
p
A ::
x,y ,z
a ::
f ::
F ::
9 --
G ::
p ::
U ::
W ::
I' ::v
00
u,v,w
x
a
o
N
.........
....:to .....
_o::llM
OZrl
..... O.r-! r-'
o::>HPt-<N
:, I!"i V) t-
.; N Z • (J"I
..... l:il...:t: .....
I:;;::: '1.-'
lXiHt";)
UP
I
~r:4V) •
V)r:4:;l:>
.:t ~ O·r-!
Z::z:::~~
1-8"'" ::::J
~
Pt-<ro::lUl~
O>o<4-lU
o::ll4-lU1
Z.,.;l(J)U
o Ul
H>o<::::l
8lXi..d
o::lloo:t:U
.,.;loro
pZUl
UOUl
~oro
...:t:o:l:':
U -~
l:ilo::llrl
::I::zro
8H
~+l
··Zoo:t:(J)Q.,
,O....:t
*Associ ate Professol', CepcTtrnent ot j·iechan i C~1 and l\erospc.ce Engi ncc:f'ing
t Graduate p.(~se(lrch 1\55 i s tent, Department of t'1ech';:.ni cal anc J\erospace
Engln201"ing
. N:t~~~NA~ TE~HNI~~~ --I
JNF~~~~~~~'t~~~~ICE
Springfield VA 22151
https://ntrs.nasa.gov/search.jsp?R=19720011664 2020-03-11T19:04:19+00:00Z
2s = 2~/U U~
8* = 2~/U l'J
. z;
K = 2~/UI.UZ;
~
K* . = 2~(H VJ~
'xy"zy = shear stress components
« = dynamic viscosity
v = kinematic viscosity
Subscripts·
x, y, z, n, = di fferenti ati on (except on .)
INTRODUCTI ON
In a recent paper, Wang l introduces an approximati9n which reduces
the computation of three-dimensional, laminar, compressible, boundary-
layer equations to the problem of solving two-dimensional type boundary
layer equations. Assuming familiarity "'lith l~(lng's work, it is to be
noted that the test case chosen by Wang had been previously studied by
Farlllelop2 who had introduced a sma11 cross-flo'>'l perturbati on methoG.
The perturbation method itself reduced the computation of the three-
dimensional boundary layer equations to that of blo-din:ensional type
boundary layer equations. VJhile on the one hand Hang's results compare
favorably with Fannelop's, it would seem that a more stringent test of
l~ang's method vlould be the comparison with a fully three-dimensional
boundary layer calculation. In this note we provide such a comparison
\'lith the results fl~om our three-dimensional calculations. In addition
\'!e consider another aspect of ~'I()n91 s approxi mati on for the cal cul ati on
of th ree- di mens i on a1 fl Oi'lS •.
3The test case which we have chosen is incompressible, laminar
flo\'! past a fl at pl ate with attached cyl i nder (see Fi g. 1). The boundary
layer of interest is formed on the plate upstream of the cylinder. Th1S
problem has been stud1ed by Sowerby3 wich a Blasius-type series solution ~
and by [Nvyer4 "lith a fi nite di fference method. Our method di ffers from
Dwyer's in that \'/e introduce boJO stream functions such that the continui-
ty equation is satisfied.
ANALYSIS
. The governing boundary layer equations, \'Jritten in Cartesian
coordi nates are
u+v+w =0x y z
uux + vUy + \'JU z = - IIp Px -I- VUyy
U\'J + V1;1 + \'I'IJ = -lIp Pz + \)'1/
X Y z .yy
subject to boundary conditions
(1 )
(2)
(3)
y = 0: u ::: V ::: W = 0, y
where -lIp Px = UUx + l'JU z ' -l/p Pz =
and
= 00 : u = U, w = ~J
2U z(x-x)H 00 _
{1/a2)[(x-x)2+z2J2
( 4)
(5 )
(6)
for the present problem under investigation.
Introducing a Blasius-type transfomation, n =
and ~ ::: Z as well as stream functions
k(U/2vx)2y, ~ = x
u = 1[; , v = -'1[; - ¢ w:: ¢y X z' y (7)
4where
1jJ(X,y,z) t::(2vxU) 2f(~,n,r;); cp(x,y,z) ( 8)
Eqs. (2) (lnd (3) becon;e
F + (1+S/2)fF + S(1-F2) + K(l-FG) + (S*-K/2)gF
nn n n
+ 2~(f F -FF ) + 2~lul(g.F -GF) = as n s i;; n I:;
G + (1+S/2)fG - S(1-G2) + K*(l-GF) + (s*-K/2)gG
nn n. . n
+ 2F;,(f~G -FG~) + 2~tUJ(g G -GG) -, a
'" n '" i;; n I:;
(9 )
(10)
Consequently the equations for condition (b) are
with the boundary conditions at n = 0 f = F = G = g = a and at
n = ~ : F = 1, G = 1.
To integrate Eqs. (9) and (10) two initial conditions are needed, and
these are deduced by considering two limiting cases of Eqs. (9) arid (10):
(a) ~ = 0,1:; t 0; (b) ~ t 0, i;; = O. Eqs. (9) and (lO), for condition (a),
reduce to Blasius equations. Referring to Fig. (1) condition (b) is the.
Illine of symmetryll condition. Jl.long this line \'1 = 0 as \'Jell as H. In
other words w/\~ = G is indeterminate. On the other hand, \'/z and VIz are
no~ zero along z = 0 so that by applying L'Hopital's rule to w/W,
G = w /W along I:; = O.
I:; I:;
determined from Eqs. (9) and (10) by simply letting z:; go to zero.
To solve Eqs. (9). and (1O) as well as the equations resulting from
the initial conditions, an implicit finite difference procedure of
Crank-Nojcholson type is applied. The resulting non:-linear equations are
replaced initially by linear difference equations. An iteration procedure·
is used until the non-linear differe~ce equations corresponding to the
grid points employed have been solved. Calculations proceeded in the
downstream direction, for a given z-station, until flO'.v reversal appeared
"5
in the u-component of velocity. Step sizes used in the calculations
are: 6~ = 6s = .61, 6n = .20.
In order to check ~'lang's approximation against the full three-
dimensional calculation, \'1e must sirilplify Eqs. (9) and (10) accordingly; ~
We replace derivatives in the crossflow direction, i.e., the s deri-
vatives, by their values at the edge of the inviscid flow, thereby re-
ducing Eqs. (9) and (10) to quasi-u'JO dimensional equations. Therefore,
the following relations will be used:
GF
1;
= G(l/U u - l/U U F) - G{l-F) l/U U ,
1; s s
- G(l-G) -w1 W ~ g = 0
s s
GG
s
(11 )
Insofar as shear stress data and flow reversal are of importance,
Tables 1 and 2 are a comparison between the full and approximate three-
dimensional calculations of F (=f ) and G (=9 ~ ) at n = o. These values
n nn n nn
are proportional to the shear stress at the surface since
L
xy
au i
= ~_I
ay Iy~O
aw !
= ~-'
ay. -0; y-
(12)
Fig. (1) contains the flow reversal line as determined by both rr.-ethods.
Up to a z or s value of 6.10, the full and approximate three-dimen-
sional calculations predict flov." reversal in the u-component, the approxi-
mate method predi cti ng flo\'J revers a1 further downs tream from the 1eadi ng
edge than the full tllree-dinlensional calculations. At s = 7.32, the full
three-dimensional calculations predict flow reversal \'/hile the approxirr.ate
method does not. By the approximate method, F (0) (or f (0)) reaches a
n ' nn
6minimum and then increases with x ors, the minimum point shifting to
smaller s values as l; increases. The calculations of G (0) by both
n
methods, and for all l; values, even in the region \'/here the approximate"
method fails to predict flow reversal, show good agreement.
Hhile in tVJQ-dimensional, incompressible flOV!, the necessary
condition for flow reversal is an adverse pressure gradient, \'!e find
that for three-dimensional flows an adverse ~ressure gradient is not
necessarily a necessary condition for flo\'1 reversal. As a matter of
fact, for the present problem at l; = 9.05 and flm'l reversal, the
pressure gradient is favorable. As DNyer4 noted, the flow reversal of
the u-component of vel,ocity is caused by crossflo\'1 effects near the
bottom of the boundary layer. That is, flO\'I reversal is driven by the
convective term, wU
z
' which is positive and overcomes the negative Px
term. Consequently \t1U is an important term, and in particular, u ,
z z
insofar as fl 0\'1 reversal is concerned so that by approximating U
z
in
certain rGgions of the flow field, wU
z
can take on values which do not
balance properly with the pressure gradient term. This is the case at
l; = 7.32 - the quasi-blo dimensional approximation breaks dO\t1ll as flm<1
reversal is apPl~oached, although for a large range of s values, the
approximate values of Fn(O) al~e in good agreement with the full three-
dimensional calculations. This is also true at l; = 9.15.
At most, then, It/hat one may conclude from these results, is that
Wang1s" approximation may not be valid in some applications and, mOl~e
likely, in some particular region of a flow field, for expmple, as
di scussed above. Whel~e the method does not break dO\'Jn, agreement is
most satisfactory with the full calculations.
7To explore the possible extension of t~angls method for further
research, we considered the follovJing calculation: adopting the
approximat~ method, Eq. (11), along the line ~ = 3.05 = constant, and
using the subsequent results as initial conditions for the full three-
dimensional calculations, we find that at the next station, ~ = 3.66,
flow reversal is predicted at the same ~ value as that obtained by
-'starting the calculations along the line of symmetry, ~ = O. In other
words, the approximate method may be of interest in calculating
three-dimensional floVJS whel~e aline of symmetry is not present. The
major difficulty in such flows are the initial conditions with respect
to the crossflo.'! derivatives and their calculation. The idea is to
apply the approximate method along some initial line and then revert
t6 the full three-dimensional calculations. This idea is being applied
to othe I' problems as we11 .
REFERENCES
lWang, K. K.., IIP.n Effectl ve Approximati on for Computi ng the Three-
Dimensional Larrdnar Boundary-Layer FlovJs,1I AI/\A Journal, Vol. 9, No.8,
Aug. 1971, pp. lG49-1651.
2Fannelop, 1. K.; IIA Method of Solving the Three-Dimensional Laminar
Boundary Layer Equations "'Jith Application to a Lifting Reentry Body,1I
AIAA Journal, Vol. 6, No.6, June 1968, pp. 1075-1084.
3 "Sowerby, L., liThe Three-Dimensional Lamin"ar Boundary Layer on a
Flat Plate,1I Journal of Fluid t·1echanics, Vol. 22, Pt. 3, 1965, pp. 587-598.
4Dwyer, H. A., IISo1utlon of a Three-Dimensional Boundary-Layer Flow
with Separation,1I AIAl\ Journal, Vol. 6, No.7, July 1968, pp. 1336-1342.
TABLE 1
cm,1PARlsorJ BETV/EDJ FULL AND APPROXIr'll,L\TE
THREE-DII~ENSIONAL BOUNDflRY LAYER CALCULATIONS OF F (0)
n
8
l; = 3.05 em. l; = 6. 10 em. s = 7.32 em. S = 9.15 em.
E;-em. Full Approx. Fuli Approx. Full- Approx. Full Approx.
0 .4696 .4696 .4696 .4696 .4696 .4696 .4696 .4696
.61 .4677 .4677 .4677 .4677 .4678 .4678 .4678 .4679
3.66 .4631 .4631 .4637 .4637 .4640 .4640 .4646 .46'16
7.93 .4508 .4508 .4528 .4529 .4539 .4540 .4558 .4560
10.37 .4408 .4408 .4442 .4444 .4461 .4464 .4493 .44%
13.42 .4230 .4231 .4296 . 4300 .4331 .4337 .4389 . .4397
17.69 .3814 .3818 .3974 .3989 .4056 .4075 .4188 .4213
20.13 .3417 .3428 .3691 .3722 .3827 .3865 .4039 .4087
23.79 .2340 .2393 .3006 .3120 .3315 .3444 .3765 .3906
.24.40 .2051 .2126 .2839 .2987 .3198 .3361 .3715 .3885
25.01 .1701 . 1811 .2646 .2843 .3067 .3275 .3661 .3869
25.62 . 1250 .1431 .2714 ~~2685 .2916 .3187 .3605 .3861
26.23 .0946 .2126 .2512 .2735 .3097 .3543 .3863
26.84 .1685 .2322 .2495 .3008 .3471 .3876
27.45 .0874 .2113 .2061 .2919 .3378 .3906
28.06 .1880 .1041 .2835 .3209 .3955
28.67 . 1617 .2759 .2547 .4028
29.28 .1310 .2694 .4130
29.89 .0926 .2646 .4269
30.50 .2623 .4452
31. 11 .2632 .4687
TABLE 2
Cor~PARISON BErt-JEUl FULL AND APPROXJt.1.L\TE
THREE-DH!lENSIONAL BOUNDARY LAYER CALCULATIONS OF G (0)
n
9
r; = 3.05 em. r; = 6.10 ern. r; = 7.32 em. r; = 9.15 em.
--
~-em. Full Approx. Full ApPl"OX. Full Approx. Full Approx.
0 .4696 .4696 .4696 .LI696 .4696 .4696 .4696 .4696
.61 .5309 .5307 .5298 .5295 .5292 .5289 .5280 .5277
3.66 .8567 .8555 .8488 .8477 .8443 .8432 .8363 .8353
7.93 1.355 1. 351 1.333 1.330 1.321 1. 318 1.300 1. 297
10.37 1.669 1.664 i.637 1.632 1. 618 1. 614 1.586 1.582
13.42 2.102 2.094 2.050 2.044 2.021 2.015 1.970 1.965
17.69 2.809 2.798 2.713 2.703 2.661 2.650 2.569 2.559
20.13 3.288 3.278 3.152 3.139 3.077 3.064 2.949 2.935
23.79 4.173 4.172 3.928 3.909 3.797 3.774 3.580 3.552
. 24. LW 4.348 4.351 4.075 4.055 3.932 3.904 3.693 3.660
25.01 4.534 4.543 4.230 4.207 4.071 4.039 3.809 3.770
25.62 4.729 4.750 4.392 4.366 4.216 4.178 3.9?.7 3. ;::-1
26.23 4.973 4.561 4.532 4.366 4.321 4.048 3. ~ ..
26.84 4.723 4.707 4.520 4.469 4. 171 4. -,
27.45 4.892 4.891 4.647 4.621 4.296 4. ~
28.06 5.086 4.754 4.777 4.408 4.::· _
28.67 5.293 4.938 4.411 4.4~&
29.28 5.513 5.102 4.545
29.89 5.747 5.270 4.648
30.50 5.441 4.746
31. 11 5.613 4.837
·~ a =6.1 elll
38
33
E /a
() /
,- 28 /,;'
><
-
23 r
/\/
t
.0 _-...JI,-_J'_-JLt> z- enl
~ 5 10 15
Uco = 3050 ern / sec
Fi g. 1
10
Flow geometry and the pl~edietion of flow reversal: --- quasi-
two-dimensional (approximate); __ fully three-dimensional.


On the calculation of three dimensional laminar boundary layer flows

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