A model is suggested in which some of the important features of the circulating flow inside the two-dimensional near wake are derived by assuming a slow viscous flow. The theory considers the flow away from the body base. It is found that there is a region of constant speed merging, as we go downstream, into a region of stagnation-apex flow. The velocity returning from the rear stagnation point along the center streamline is shown to be a slowly varying function of the 'wedge-angle,' of the wake and to be roughly one half the velocity at the edge of the shear layers driving the wake-cavity flow. These results seem to be in agreement with experimental data

Abarbanel, S.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19770003544 2020-03-22T12:08:33+00:00Z
I( J//
"1
M
Q0
I C A S E
	 REPORT
THE SLOW RECIRCULATING FLOW NEAR THE REAR
STAGNATION POI.J OF A WAKE
Saul Abarbanel
Report Number 15-18
September 29, 1975
Approved for public release=
distribution unl-inited.
INSTITUTE FOR COMPUTER APPLICATIONS
J
IN SCIENCE AND ENGINEERING 	 r	 `^/^
Operated by the
UNIVERSITIES SPACE RESEARCH ASSOCIATION
ai r^1lrt
^r
NASA' S LANf,LEY RESEARCH CENTER	 ^ MAy
 ll^ i9le
9!	 ,-Hampton, Virginia	 ^LID
f ^9
Auts: 1 r41 
tot _ _THE SLOW RECIRCULATING FLOW NEAP THE REAR
Milt	 Ytnue Sewall
O.g
	
Ball SWIM q 	 STAGNATION POINT OF A WAKE
ua, ':" roL	 q
Ills"PICATION I--
By	
I
	
es • rlecnon xvuuaulrY caoxt	 Saul Abarbanel
^.,	 fY, d. e:tJ 'n ^LUL
ABSTRACT
A model is suggested in which some of the important features of the
circulatingiflow inside the two-dimensional near wake are derived by
assuming a slow viscous flow. The theory considers the flow away from
the body base. It is found that there is a region of constaltt speed
merging, as we go downstream, into a region of stagnation-apex flow.
The velocity returning from the rear stagnation point along the center
stoeamline is shown to be a slowly varying function of the "wedge-angle"
of the wake and to be roughly one half the velocity at the edge of the
shear layers driving the wake-cavity flow. These results seem to be in
agreement with experimental data.
s
This research was sponsored in part by the Air Force Office of Scientific
esearc	 roug	 a uropean uttice Of e ospace KesearcM, AhSL,	 z,
m e	 a es Air grce, under grant	 - -c_,an in part by NASA
rz nl	 - 2-001 while the author was in residence at ICASE, NASA 	 I;-
La,yley Research Center, Hampton, VA 23665.
1
	
Introduction
A complete description! of the structure of the two dimensional near wake
behind a body in motion depends on knowledge of the whole flow field. in
particular one has to know, or derive, the way the boundary layer separates
at the base and becomes d shear layer. This shear layer is driven by the
inviscid free stream and in turn drives the recirculating flow confined inside
the near-wake cavity.
Experiments at low to moderate (Re<200) Reynolds numbers [1] indicate
that the recirculating flow field in the cavity is composed of three parts:
The flow very near the base, the central region, and the flow field near the
rear stagnation point where the cavity closes on itself and where Cie flow,
driven by the shear layers, turns around towards the base. The data show
that in the central region the velocity on the stagnation streamline (connecting
the !ear stagnation point to the center of the base)	 is constant,
independent o f
 the Reynolds number and is about 1/5 the free stream value.
tluc:erical studies for supersonic wake and Re=500 [2] indicato that in the
near wake the viscous and pressure terms dominate while a short distance down-
st-eam of the ,ear stagnation point the two merged shear layers have boundary-
layer-like sl a t!re.
The abn • e observations suggest that one may be able to mudel the flow in
the :.ritral and rear stagnation regions of the recirculating flow field by
consirlering a slow viscous fluid bounded by converging streamlines which
"represent" the bounding shear layers. Away from the apex, i.e. the rear
stagnation point, the velocities along the outer and center streamlines are to
be constant. Mearer the dpex the velocities will depend linearly on the
distance from the stagnation point and very close to it the variation will be
parabolic - as suggestr_-d by Heimenz (See for example Batchlor [3]).
U0
FIGURE 1
rti
J
i
!`
In section 2 the Stokes' equation will be solved for the ventral region,
and it will be found that the returning velocity alcng the center stream line
cannot be assigned arbitrarily but chat it is dictated by the constraint that
on the center line the pressure gradient normal to it must vanish.
In section 3 the linear stagnation region is considered. It is found
that in it the pressure gradient must be radial. Therefore, the solution
in the constant velocity region found in section 2 and the ;elution for this
stagnation region can be matched without changing the wedge angle.
In section 4 we examine the final parabolic approach to stagnation. It
is found that,as in the central (constant velocity) region,there is a con-
straint on the returning velocity along the centerline. This condition,
however matches the one found in section 2 only for limiting small apex
angles. Thus in general this region will produce a discontinuity in the
slopes of the streamlines. However, because our model problem possesses no
natural reference length : this F inal stagnation region can be made as small
a , desired and will not affect the solution elsewhere. An interesting result
;n thi ,; in n•diatY W jhuorrood of the stagnation apex is that the boundary
1 • naWues cannot form an apex angl e of 260 = n radians Kt i i,e the returning
,,eiucinv d ung the t saerline (6 n D) is proportional to see 0
-U 	
it
:i Vt. 10, 
as:; o 
0, 0 .oachen	 n/e.
2, the constant (radial) velocity region
In this region the geometry is as follows;
I
i
t
i
-2-
A_-Mvml.
The boundary conditions are u  = 
- lip on the streamlines 	 h = 0, 0 = 2e 
and u6 = 0 on the same streamlines. Since we assumed the inertia forces
to be nugatory the equation to be solved ^s Stokes', which is bi-harmonic
for the streamfunction,
041P = 0.	 (2.1)
Wi *.h the above, boundary conditions, hrwevrr, Gve shall get a sink- type flow.
We mutt stipulate that along the center ine the fluid is moving away from
the apex (which is not in cu r region of consideration now). Therefore we
c(,nsider the following ,))-obleii:
0
FiGUkF 2
8
i t is clee- that in polar cuordinates the solution to (2.1) .nth its attendant
;^o:. -!a y coi,di tions must ae of the form
0 = rfN)
fran -mh ich
I 	 = ,° , = f' (9),
r	 r 36
u0 _ _ 
4ar
 = -f (o)
IM
- 3 -
and equation (2.1) for the stream function becomes
r3 
V 
4 
ip  = 
f (i v ) + 2f" + f = 0. 	 (2.2)
The boundary conditions for the half-wedge (see figure 2) are
f(0) = 0
	
f(a0) = 0
(2.3)
f 1 (0) = -U0	fI(a0) = V0
The solution of (2.2) subject to (2.3) is
f(a) = A0 (sine-ecose) + B0Bsine - U  cosh
where
A =	 U0
	 1(—%)E) sine -02 
1o	 62-sin260 + U0 0	 0 0
I 1 CY,
(2.4)
(2.5)
and
g	 U0	 (V-2)(6 Cosa -sine ) + 6 - sine Cosa	 (2.6)
0	 02-sin26	
( U0	 0	 0	 0	 0	 0	 o f
0	 0
If we wish the center line a = 6 0 to be a true line of symmetry for the
complete wedge it is not sufficient to have u 6 ( a 0 ) = -f(eo ) = 0. We must
also require that the pressure gradient normal to that streamline vanish;
i.e. (ap/2e)
6=6
 = 0. Now
0
38	 r
- 4 -
P I
YrcYdi., 3!?gk^?'^3xea... ^`.- 	 ..., .., ^x5^fatrri,<1cw'^..,.,.	 ....-,.......^,. ;
where u is the coefficient of viscosity. The symmetry requirement leads
therefore to
(Ao+U0 )sine 0 + B0cose, = 0
	
(2.8)
Since Ao and Bc depend on Bo , Uo and V0 
/U
0 
we find that for a given
driving velocity U 	 and a given semi-apex angle 0o equation (2.8) yields
a value for Vo/Uo . The variation of go = Vo/Uo with 0o is shown in
figure 3. Note that for o<Bo^y/2, $o varies monotonically between 0.5
and 2/n = .637. Thus for most values of 0 o that one expects to encounter,
in experiments for example, the value of S o = Vo/U0 will be confined to
fairly narrow bounds. This seems to be in agreement with experiments [1]
where at different (low) Reynolds number the same constant value of ur/Um
was measured along the stagnation streamline. The measured value of
lu r/Um ( = 0.2 would give our results the right order of magnitude if Uo,
the velocity characteristic of the "bottom" edge of the driving shear-layer,
is taken to be about 1/3 the local free stream value.
3. Thelinear) stagnation region
We assume that from a certain distance, r o , the velocity along the
bout.Jir.g streamlines begins to decelerate linearly in r. This is not an
unreasonable assumption and is in line with Heimenz` results [3] for stagnation
point flow. Thus we set in this region (r < ro)
>U = r2f(6).
Consequently
u  = rf`(0),
U0 = -2rf(6),
t;
	
'^^+^',^e'Aa"`7-wry.^-'wr4ij2U*`ii.Yi^.; +ri^7!£'P'il''^ aS[+Z+3L°s.G =^"1E^5^`3.^+'.::a"CP'3="
•
^4'^wYTSTa'N:1•xwaw-wyes^.rvva. ...-:'-^.. ^_ .. ___.	 ..
v
and the bi-harmonic equation reduces to
r204ip = f(iv) + 4f" = 0
	 (3.1)
The boundary conditions are:
f(0) = 0
	 f(eo) = 0
(3.2)
1r ( 0 ) = -U 1	f'(eo) = VI .
Since r has not been scaled yet we shall say that this stagnation region
is defined by r < r o
 = 1. Then in order for the velocity along the bounding
streamline to be continuous in the two regions r > 1 and r < 1 we must
set U I = Uo and V I
 = Vo.
The solution to the system (3.1), (3.2) is
	
f = A I (sin2e-26) + B I (cos2e-1) - U o6	 (3.3)
where
U  (1+e,)taneo-260
AI 4	 . ao -tan6o
(3.4)
Uo (sineocosao)(1
+BI
+200tan6o)- (),+BI)aoB
1
_
4
(3.5 )
ao-taneo
	 sineoceseo
,I VVooB 1 = U - Uo (3.6)
It is easily established that for the present case 3p/;o = 0 everywhere
and hence there is no constraint on the value of B 1 • It follows that it was
legitimate to require continuity of the radial velocity on the bounding stream-
lines at r = 1 ; i.e., to set, as we did, BI = Bo.
kkisz
It is interesting to note that the choice of	 = r2f(6) is the only one
Ileading to ap/ae = 0 everywhere. This is seen by examining the expression
for the pressure gradient:
au	 u
v ae = °2 " +e r aor 
r2	
(3.;)
If 
U  
and u 	 are calculated from ^ = rmf(e) we find
1 ae 
= ,'n-3(2-m)(f"+m2f)
	
(3.8)
ur a6
It is seen from (3.8) that ap/ae vanishes everywhere for m = 2. This
-`
	
	
is fortunate since otherwise we would have Sl # S° and would not have been
able to match the velocities on the center streamline of the regions at
r = r° = 1, or at any other station. We are now is a position to combine
the results of this section with the previous section to specify a flow-field
of a slow viscous recirculating fluid in a wedge-like region (with the possible
exception of the very near neighborhood of the apex):
'f
r>1
r[Ao (siO -ecose) + B°esine - Uoecosel
o<e<e
p =	 — °	 (3.9)
r2 [Al (sin2e-2e)	 B l (cosU-1) - Uo 1	 0<r<l
with A° , Bo , Al and B1 given by (215), (2.6), (3.4) and (3.5) respectively.
There remains the question of the continuity of * and its derivatives, at
r = 1, at all 8 . and not just along the bounding streamlines, 6 = 0 and
0 = eo , where the boundary conditions were chosen to assure us of continuity.
Clearly there is no analytic matching uniformly valid in 6 . Numerica:
calculations of f and f' as functions of a for various e o are shown
in figures 4 and 5.	 It is seen that the matching is quite good.
4. Final approach to the stagnation apex
For 0<r<r 2<<1 we might have a parabolic appraoch to stagnancy, i.e.,
i
0 = r•3 f( 6 )
	 (4.1)
This small r 2 is likely to be of the order of the diffusion length proper
to this problem, L  = o(u/Pu o ), where we assumed tacitly that L  << 1 and
hence	 U	 U L
1>>R = o » a d
From (4.1) we have
u  = r2f'(a)
u 9 = -3r2f(e)
and the bi-harmonic equation for this case becomes
r3V40 = f (iv) + lof'^ + 9f = 0
	
(4.2)
with boundary conditions
f(0) = 0
	 fNo) = 0
(4.3)
f°(0) = -U2	 f '( 6 o ) = V2
where continuity of velocity along the bounding streamline (6=0) demands
that U1 = r 2 
U 2 
or U2 = Uo/r 2. 
Solving (4.2) and (4.3) for f(e) we find
U
f = A2 (sine - 3 WO) + 82 (cose-cos3e) - 3 si06	 (4.4)
1
a 2 = 2cosB0
(4.9)
EMI
where
I]
2
A -
	 -3sin2e0+0 cose0-1
(4.5)2	 4 2	
sin2e0
U2	 2coseo-S2
62
 - T	
sine
(4.6}
o
V
62
 = ^?	 (4.7)
2
From (3.3) it follows that the symmetry requirement (ap /ae = 0 at 8 = e0)
takes the form
f"(e 0 ) + 9f(e o ) = 0
	
(4.8)
Using (4.4), (4.5) and (4.6) we find that 6 2 is constrained to take the
value
Evidently 52 ( 0 0 ) # 60 ( 00 ) ; in fact for the range of interest, 0<e0—< Tr /2
62 (00 ) > Bo (e o ) with equality taking place only at 6 0 = 0. Thus in the
present model there is rio continuous matching possible between the linear and
parabolic stagnation regions. Either, if we match the velocities, there is
a discontinuity in the slopes of the bounding and other streamlines, or --
if we leave the wedge shape undisturbed -- there is a discontinuity in the
velocity field. This was to be expected in a relatively crude model as the
one employed here. However, the extent of the parabolic region is very
small and its effect on the rest of the flow field should be small. The main
point is that the two major regions -- that of constant velocity and the linear
stagnation one -- do match well to yield a flow field with properties charac-
teristic of available low Reynolds numbers data.
It is interesting to note that the expression for 82(ao) effectively
excludes the possibility of a blunted wedge shaped cavity with closure by
streamlines perpendicular to the center line, i.e., e o = n/ 2 . In fact,
one would guess that at the apex the closure is cusped.
REFERENCES
I
1. A. S. Grove, F. H. Shair, E. E. Petersen and A. Acrivos, "An Experimental
Investigation of the Steady Separated Flow Past a Circular Cylinder,"
J.F.M., 19, part A, pp. 60-80. (1964)
2. J. S. Allen and S. I. Cheng, "Numerical Solutions of the Compressible
Navier-Stokes Equations for the Laminar McFr Wake," Phys. of Fluids,
13, No. 1, pp. 37-52 (1970).
3. G. P.. Batchelor, An Introduction to Fluid Dynamics, Cambridge University
Press (1967) - pp. 287-288.
- 1.0 -
0
(D
,i
Ln
4
e^
N cr.)	 CD	 QC^	 p	 p
0
4.4.
r
n f,
n ^
Cw^
rr
C
t~
n
I
-4 o
u
J
w	
S
p C
N cl
41
li.
N
tp
Lr
U
y
c
C>
1
r-1
v
fz,
In
CJ CA	 00	 L-	 CO	 Ln	 m	 CV	 r	 d	 N	 m
r--1	
i•	 i•	 i•	 i•
O
r7
4
1
CD
U-3	 cD
1	 i
_ o
a^
Cl.)
C)
cn
to
N
C) a.
N
LO
,4
d
UNCLASSIFIED
SEFU.y:':Y CLASSIF I CATION - OF THIS PAGE (R1t+n Data Fnrrtedl
/ (y)REPORT  DOCUMENTATION PAGE READ INSTRUCTIONS
-^^'	
r'-ar 3f BEFORE COMPLETING FORM3•1 ,, 
	c	 L'N	 ABGF^^	 F,	 2. (knvT ACCESSION NO.
F.td, ) -	 f^
,
 :' J
REf_IPi7.NT'S CATALOG NUMBER
4. TITLE (and Subtitle)  T.Yp,g CIS ore,,01 A 0 01 .
^	
EREO
BITERIMI /rl; t^ 
THESLOW ^LECIRCULATING -FLOW NEAR THE REAR
6. P,:RFORMING ORG. REPORT NUMBERSTAGNATION POINT OF AWAKE a '	 1i
7. AUTHOR(.)	 //,-- B. CONTRACT OR GRANT NUMBER(.)
SAUL ABARBANEL	 j	 F
=
AF tR W 237 - 7i
y
9. Pt RFORMING ORGANIZATION NAME AND ADDRESS 10.	 ROG RAM ELEMENT, PROJECT, TASK
TEL-AVIV UNIVERSITY^
AREA 6 WORK UNIT NUMBERS
681, 07
	 ^
DEPARTMENT OF MATHEMATICAL SCIENCES	 ^()t - 978	 Ol ^^ ry
	
r	 /^
RAMAT-AVIV, TEL-AVIV, ISRAEL 61102E
11. CONTROLLING OFFICE NAME AND ADDRESS IZ. REPORT DATE	 ^.-
AIR FORCE OFFICE OF SCIENTIFIC RESEARCH/NA% ^^ Sept 75
. NUMBER OF^PAG TBUILDING 410	 `
BOLLING AIR FORCE BASE, D C	 20332 14
IA.	 MONITORING AGENCY NAME b ADDRE55(if dlllerant f
r
om Controlling Office) 15, SECURITY CLASS. Poi Clio report)
/ i,	 %`	 / UNCLASSIFIED•f	 '^	 I 15e. DECLASSIFICATION , DOWNGRADING
—ter. SCHEDULE
16.	 DISTRIBUTION STATEMENT (ol this RePart)
Approved for public release; distribution unlimited.
7.	 DISTRIBUTION STATEMENT (ol the eb,+trecf entered In Blnak 20, It different from Report)J
fe
16. SUPPLEMENTARY NOTES
- 9
19,	 KEY WORDS (Continue an revcrae Bide if necessary and identify by black number)
NEAR WAKE
BASE FLOW
RECIRCULATING FLOW
4 .20.	 ABSTRACT (Continue on reverse tide fl necessary and Identify by block number)
A model is suggested in which some of the important features of the cir-ulating
flow inside the two-dimensional near wake are derived by assuming a slo g viscous
flow.	 The theory considers the flow away from the body base.	 It is found that
there is a region of constant speed merging, as we go downstream, into a region
of stagnation-apex flow.	 The velocity returning from the rear stagnation point
along the center streamline is shown to be a slowly varying function of the
4wedge-angle4 of the wake and to be roughly one half the velocity at the edge of
the shear layers driving the wake-cavity flow. 	 These results seem to be in
.avaAmpnt with exnerimental.-data.... _...	 ..	 :..
i	 .


The slow recirculating flow near the rear stagnation point of a wake

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server