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NASA-CR-176853 19860022058
A Reproduced Copy OF
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Final Technical Report
Grant No, NAGW-816 //.'1 - ?_.~_. ,. ~ .::, I ''':;' .. -' I
Title of the Grant:
Theoretical and Experimental Investigation o~ Vortex Breakdown
Principal Investiqator:
Prof. Egon Krause, Ph.D.
Periocl covered by report:
JlJnf. 1,1985 - May 31, 1986
Address:
Aerodynamisches Institut
FWTH Aachen
Wullnerstr. zwischen 5 und 7
5100 Aachen
West Germany
(NAS;"-C[\-176353) TIlEORET rUL AND EXPERl~EN1AL INVES11GA1ION CF VCRTEX BFEAKCOWN Final Technical ~efo~t, 1 Jun. 1505 - 31 Ma.y 15E6 ('I€chnis·.;he lIochschule) 117 p CSCL 01A G3/02
Na6-31530
Unclas 43281
I I
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f r [
( "'-, r
I I I I
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t
l
" Introduction
The phenomenon of vortex breakdown is of considerable importance for flows under
critical conditions about delta wings. The leading edge vortices on the leeward side
of the wing can be termed slender until breakdown is about to occur. The
slenderness of the vortices is characterized by the ratio of the core radius Rand
the length L of the undisturbed vortex, i.e. R/L« 1. As long as the sl.:mderness
condition is satisfied, the equations of motion can be simplified considerably for
the description of the flow in the vor~ex. The integration of the simplified
equations is then possible for lorge Reynolds numbers. The breakdown itself must
be analysed by solving the full Navier-Stokes equation. With the slender vortex
approximation the overall computation time can be very much reduced, since the
largest portion of the flow can be determined from the simplified equations, and
extremely fine resolution is only required for the breakdown region, where the full
Navier-Stokes equations must be employed.
The work carried out under this contract is concerned with the formulation and
solution of the slender-vortex approximation and its numerical solution for
incompressible and compressible flow; with the initiation of experimental work and
the validation of the solution; and finally the prescription and determination of
permissable inflow conditions for the slender-vortex approximation. The results of
this analyses are presented in several publications (1), (2), (3), [4J; copies
of the preprints and manuscripts are enclosed.
Formulation of the Problem
The slender-vortex approximation for compressible viscous flow was first formu
J3ted in Refs. (1) and 12). Therein it was shown that the flow is basicly
governed by the local axial pressure gradient, which can be expressed through an
integro-differential equation. According to this analysis the pressure gradient is
mr:inii' delermined by the radial distributions of the circumferential velocity
component, and the temperature. It c:;uld be shown that a hot core tends to
stimulate breakdown and that a cold core delays it. The direct influence of the
compressibility is that the variable density tries to counteract the influence of
positive axial pressure gradients, thereby delaying breakdown.
I · , · i
1.
f ' , · , :', ~
f ,
I ' ~ ; ; ! , I
: 1 : j
.1 · I 1 i q
....
. -
Perrnissnhle Inflow Conditions
The slenderness condition imposed on the flow does not permit one to prescribe
arbitrary inflow conditions, which are required for the integration of the simplified
equations of motion. The inflow conditions are given by a set of radial profilas. for
the axial and circumferential velocity profiles, and the temperature. It Vias shown
in (11 and (2) that the inclination of the traces of the stream lines v/u in the
axial plane satisfiet a second-order nonlinear ordinary differential equation, to be
integrated in the radial direction. The coefficients of that equation are solely
determined by the inflow conditions. Since the inclination of the traces of the
streamlines must obey the slenderness condition, the solution of the equation for
v/u decides as to whether or not the prescribed inflow conditions for u, w, and T
can he admitted. This is described in detail in Ref. [2] •
Numerical Solution
Finite-difference solutions of the slender vorlex approximation were developed for
incompressible flows, and compressible flows. The differential equations were
casted into locally linearized implicit difference equations, which can be inverted
by recursion. Details of the solution for incompressible flow are described in 14);
the solution for compressible flow is described in 12J, and (3]. n.B most
important results computed with the solution so far are also reported in the
references just mentioned: In Ref. (4), it is shown, for example, that the solution
reacts sensibly to axial pressure gradients; comparison calculation were carried for
the experimental data of Faler and Leibovich. Th3 comparison, discussed in
[4} shows that for vorticities with large swirl, the breakdown point c£.nnot be
predicted with the slender vortex approoximation, since it does not take into
acocunt the upstream influence of the breakdown region.
In Refs. (2) and 13) the results t'btained so far for compressible flows are
discussed. The influence of the freestream Mach number and of the temperature
distribution prescribed for the inflow cross section on the flow development in the
downstream direction is clearly demonstrated.
Conclusions
The slender-vortex approximation was analysed for incompressible and com-
. i I .~
l: .. ;~
-
pressible flow. First the equations of motion were reduced inan order of magnitude
analysis. Then compatiblity conditions were formulated for the inflow conditions.
-Thereafter finite-difference-solutions were constructed for incompressible and
compressible flow. Finally it was shown that these solutions can be used to describe
the flow in slender vortices. The analysis of the breakdown process must, however,
be excluded, since its upstream influel1ce cannot be predicted with the slender
vortex approximation. The investigation of this problem is left for future work.
Literature
[ 1 J E. K:-ause: "Schlanke Wirbel in kompressibler Stromung". ZAMM, Band 67,
Heft 4/5. To be published.
[21 C. 1-1. Liu, E. Krause, S. Menne: "Admissable Upstream Conditions for
Slender Compre3sible Vorti::es". AIAA I ASME 4th Fluid Mechanics,
Plasma Dynamics and Lasers Conference, May 12 - 14, 1986, Atlanta, GA,
AIAA-86-1093.
( 31 E. Krause, S. Menne, C. H. Liu: "Initiation of Breakdown in Slender
Co~pressible Vortices". 10 International Conference on Numerical
Methods in Fluid Dynamics, 23 - 27 June, 1986, Beijing, China. To be
published.
[4] L. Reyna, S. Menne: ''Numerical Prediction of Flow in Slender Vortices".
Accepted for publication in Computers and Fluids.
r
- ,
• !
•
SCHLANKE WIRBEL IN KOMPRESSIBLER STRtiMUNG
E. Krause
Aerodynamisches Institut der RWTH Aachen
Willlnerstr. zw. 5 und 7, 5100 Aachen
Der EinfluG der Kompressibilitat auf schlanke Wirbel in stationnrt'7", reibungsbe
haftcter Stromung wird dargestellt. Eine entsprechende AbleitL ng fUr rei bungs
frt:ie Stromung ist in (1) und fUr instationare Stromungen in [2 J gegeben •
. Fur die Untersuchung wird angenommen, daG die Achse des Wirt.cls parallel zur
ankommenden Stromung 1st und daG die Stromung im Wirbel r(Jtationssymme- .
trisch ist. Bezeichnet R den Kernradius in dem Querschnitt senkrecht zur
Anstromrichtung, in dem der Wirbel erzeugt wird, und L die unbekannte Uinge
des Wirbels, gemessen in Stromungsrichtung,
aufplatzt, so erfordert die Schlankheitsbedingung
v R· ; -: 01-1« 1 u L
Gber die de:" Wirbel nicht
(1) .
Dabei ist v die rndiale und u die axiale Geschwindigkeitskomponente. Die mit
Gleichung (1) vereinfachten Erhaltungsgleichungen fUr Masse, Impuls und
Energie entsprechen der Grenzschichtapproximation:
Masse: L 1 9 u 1 + ..!. l 1 9 v r) = 0 ax r or
(2)
Impuls, x-, r- und G -Richtung:
au au an ~ 0 1 au 9U -+9v - =-~ +- - rll-) ax ar ax r ar ar
131
9w2 _ ap r ~"ar 1"
(5) ... _
Energie aT aT ap ap 1 a aT a r 2 au 2 cp l9u -+9v -l:u-+v-+--IrX-)+ll[Ir-I-ll +1-)) ax ar ax ar r ar ar ar r2 ar
(6) Das System wird mit der Zustandsgleichung 9 = P/(RT) geschlossen, Vlobci R die Gaskonstante ist. Zur Uisung des Gleichungssystems (2)-(6) mUsscn Ein
strombedingungen
v:y_. n$r: u:f.lrl w:' .. lrl. T:f .. lrl 17\
i .1
" "j
• !
Symmetriebedingungen auf der Wirbelachse -----_._" .- ..... --- -- -.. -- - - - - .. - -..
r = 0 ; x < x . au - v - w - aT - 0 0- . ar - - - ar - . 18)
und Randbedingungen fUr r- CO
19)
vorgegeben werden. Die Funktionen f1
(r) - f 3(r) und gl(x) und g2(x) mussen als
bekannt vorausgesetzt werden.
Wegen der Schlankheitsbedingung (1) kannen f 1 (r) - f 3(r) jedoch nicht willkurlich
vorgegeben werden. Dies geht aus folgender Betrachtung hervnr: Aus der
Kontinuitats-, Energie- und Zustandsgleichung laOt sich ~ ~ durch folgende
Beziehung ausdrucken: - -.. --. - ... ------- --. --_._.- ... _-- --_. -
_au_ = __ u_..£l?. _ ~ _a_T + 1_1<_-1_) -5 v __ 1 __ o-19 vr )+ _H_ ax Kp ox T ar r a 9r ar 9CpT
(101
Die GraGe H steht zur Abkurzung fUr die Dissipationsfunktion und den Warme
leitungsterm in der Energiegleichung. Nach Einsetzen von G!eichung (10) in die
x-Impu!sgleichung erhalt man nach Integration:
- .::L = exp (- II [
- ! Nach Gleichung (13) ergibt sich stets ein nichtverschwindender Druckgradient
~ ~ (x,r), da in reibungsbehaftetcr Stromung v/u stets von Null verschieden ist.
Ein heiner Kern kann dabei. den Druckgradienten vergroOern, wahrend die
Umfangskomponente den Druckgradienten verkleinert. Dieser andert· seinen
Einflur3 mit der axialen Machzahl. Das geht aus der x-Impulsgleichung .hervor,
die sich in folgende Form bringen laOt: -----._-_.-._-_ .... __ .. _-- .. - .. ---_. __ .... _-
_a (_v) __ r1- ~u2 I ~1 ap '1 w 2 ) v , 1 a { au I (x-11 -- .--.,- --~-- rTJ- .-~H ar u a pu ax 0" ur QU~ r or ar po"u
( 14)
GJeichung (14) zeigt, daO bei Uberschalldurchstromung des Wirbels, d.h. uta > 1
der crste Term sein Vorzeichen andert.
Sollen nur die Funktionen f 1 (r) - f 3(r) auf ihre Kompatibilit5t mit der Schlank
heitsbedingung, Gleichung (1), im Einstrom~uerschnitt GberprGft werden, kann
dies durch Elimination der Druckgradienten aus den Impulsgleichungen gesche
hen: Durch entsprechende Differentation nach x bzw. r und Subtraktion der
resultierenden Ausdrucke voneinander, erhalt rnan nach Zusammenfassen
a 22 ( ~) • G, (x, r ) -~ { ~ I • G2 (x, r)( ~). G3( x ,r I = a ar u ar u u
(151
Die Funktionen G 1 (x,r) - G3{x,r) lassen sich unmittelbar aus den Einstrombedin
gungen ermitteln. Integration von' Gleichung (IS). ergibt dann die radiale
Verteilung v/u, so daO nbgescrl:itzt werden kann, ob die Schlankheitsbedingung,
Gleichung (l), erfUllt ist.
Literatur
1. Krause, E., e'er CnfluG der Kompre!'sibiliUit auf schlanke Wirbel, Aerodyn. lnstitut, RWTH Aachen, H:lft 27, S. 19-23, 1985.
2. Kraus'e, Eo, On Slender V\)rtices, in: Flow of Real Fluids, G.E.A. Meier, F. Obermeier (Eds.), l ectur~ Notes in Phy~ics, Vol. 235, Springer Verlag, S. 211-218, 1985.
.. •
I "
NUMERICAL PREDICTION OF FLOW IN SLENDER VORTICES
* •• Luis Reyna and Stefan Menne
Aerodynamisches institut, RWTH Aachen, West Germany
Abstract
We study the slender vortex approximation with attention put on high Reynolds
number behaviour. It is shown that the breakdown of the approximation coincides
, with the criticality' condition as introduced by Benjamin [12J • We study free
vortices with and without an adverse pressure gradient for viscous and inviscid
flows. Finally we compare to experimental results from Faler and Leibovich [B] ..
*
*.
This research was conducted while the first co-author was at the Aero
dynamisches Institut as an Alexander von Humboldt-scholar.
Correspondence and proofs for correction should be transmitted to Stefan
Menne, Aerodynamisches Institul, Wlillnerstr. zw. Nr. 5/7, 5100 Aachen, West
Germany.
';.'
:;
:',
;.~
. ... "
, "
· .... -.... ·~r / ., .... : :', ",' I
: .. 'j
" .
f'
; . .
:~
. , ,..-* ," .. . . .. .." ..
""; f . .j
*"'t
~~f1
NUMERICAL PREDICTION OF FLOW IN SLENDER VORTICES
. --Luis Reyna and Stefan Menne
Aerodynamisches Institut, RWTH Aachen, West Germany
1. Introduction
Vortex breakdown was initially observed by Peckham and Atkinson [11 for leading
edge vorth::es formed on delta wings at large angle of attack and with large tip
angles. The phenomenon has a drastic influence on the aerodynamical behaviour of
the flow. In flows around wings its presence strongly decreases the Ii ft [2,3] and·
in combustion chambers it can be used to design flame holders [4] • Despite of the
large amount of research on this subject, the problem of vortex bre3kdown can not
be yet considered as being fully understood. The presertt state of the art can be
found in the review articles by Ludwieg [5], Hall [6] and Leibovich [7] .
In experimental investigations Faler and Leibovich [8] found six different cases
of breakdown for a swirl flow in a divergent tube. Two of thf: n are nearly
axisymmetric and called bL.bble type, the remaining have either a spiral or heli
coidal shape. The spiral form is marked by a kink followed by a cork-screw shaped
twisting of the vortex filament. In this case non-axisymmetric effects are im-
".~ portant and have to be included in the analysis of the flow [ 9] •
*
**
This research was conducted while the first co-author was at the Aero-
dynamisqhes /nstitut as an Alexander von Humbolclt-scholar.
Correspondence and prol)fs for correction should be transmit ted to Stefan
Menne, Aerodynamisches Institut, WUllnerstr. zw. Nr. 5/7, 5100 Aachen, West
Germany •
~------------~--------~.r---------
n ~ .j ;J,
.i i
0·' , t
i , I
i (
~
, l ~ : . ,0, .. -, ;·1
1 , ~ t I
•
,,- ~ , . :. ~- ; ic..f .. >i "'I n ""! ' ..
.!
Vortex breakdown theories assume axial symmetry except Lud,,· ... ey's [10] whic.h
regards it as a hydrodynamical instability of nearly axisymmetric flow. The
important but unanswered question is the dependence of the breakdown type on the
approaching flow. In order to explain the phenomenon Squire [11 J ' Benjamin
[12,13J and Bossel [1 l l,15 J introduced the concept of critical state. The flow is
called supercritical when perturbations can onlt propagate downstream and subcritical
when there is an upstream influence. Squire [l1J suggested upstream perturba
tions would accumulate and produce breakdown at the critical state where the flow
proceeds from super to subcritical and the phase velocity of the perturbations is
zero. Benjamin [ 12] showp.d the existence of two equivalent solutions to a
conjugate flow, one being supercritical and the other subcritical. He then regards
breakdown as a sudden transition from super to subcritical states, similar to the
hydraulic jump. It is not possible to predict vortex· breakdown with this theory, but
it allows a classification of flows. The significance of the theory can be seen in an
experimental observation of Faler and Leibovi"ch [ 8] : "All flows that exhibit
vortex breakdown of the "axisymmetric" form (which we classify as types 0 and 1
disturbance form!:) or "spiral" form (our type 2) are supercritical upstre<lm, In the
sense of Benjamin."
The details of breakdown can be studied by numerical integration of the full
Ni'lvier-Stokes equations. Lavan, Nielsen and Fejer [16] , Kopecky and Torrance
[ 17], Grabowski and Berger [ 18] computed axisymmetric, laminar, incom
pressible and stationary swirling flows. -Grabowski and Berger [ 18] found a
backward flow for subcritical initial profiles in contrast to the classification of
Benjamin [ 12J . Since the double r·ing stucture was not present inside the bubble,
Faler and Leibovich concluded due to their experiments [19] that the numerical .-,
~ solution shculd take into account time dependent periodic asymmmetric motions
[19] . The time dependent calculation .done by Shi [20] showed correctly this
structure. The computed flow was stationary upstream o~ the breakdown point but
instationary and nearly time periodic downstream of it. This and similar
computations ~re restricted to low Reynolds numbers, much lower than the ones
present in technical applications.
Plow pictures show nearly cylindric.'11 stre<lm surfaces upstream and this observa
tion can be used in order to derive ar. approximation of the Navier-Stokes equations
usually called the slender vortex approximation. The assumption behind this
approximation i:: analogous to the one frorn boundary layer theory. The slender
.. _-----,
vortelC approximation has been used by Gartshore [21,22] , Hall [23-25] , Bossel
[ 26], Mager [ 27], Nakamura and Uchida [28] and Shi [20]. The
approximation based on small gradients and small radial velocities fails at the
breakdown point but is vaiid upstream I\f it and generaily for stable vortices.
The purpose of this paper is to study the high Reynolds number behaviour of the
slender vortex approximation. In chapter 2 we prove that the breakdown of the
approximation occurs at the critical state as remarked by Ludwieg [5]. In chapter} we present numerical solutions for different Reynolds numbers. In the
limit of no viscosity the external pressure gradient determines the breakdown. First
preliminary results are shown in [29] . Finally we compare to experimental data
and discuss the advantages and limitations of the approximation.
2. Slender vortex approximation
Consider now the Navier-Stokes equations written in cylindrical coordinates
(x, r, G) with correspondir.~ velocities (u,v,w) in a nondimensional form. The axial
and 'radial velocity compo:-aents are normalized with the axial velocity and the
pressure with its value at the initial static..n for r~ co. The lengths are normalized
with the vortex core radius which marks the region of viscous flow. The
circumferential velocity is normalized with its value at the initial station at the
edge of the vortex core.
Including the slenderness condition
and assuming steady axial symmetric flow leads to a system of equations
A.L.4x t 'IT.(..(.r -t t>x =:: R: f (r,ul")r "D wl. rr =r-
M. WI( "; 'IT-i:-(rW)r '0. ~ [~(rw)r]r \
(r..u.)x + (r'\T)r =- 0
(1.0..)
(2.1)
(2c)
(2d..) called the slender vortex approximation. Here the Reynolds number is based on the
vortex core dimension and the undisturbed axial velocity and p is the pressure.
-_ .. \'
i ,
Notice that the type of this syst'!m is n·o.-./ parabolic for viscous flow and hyperbolic
for invic;cid flow cO:Jplad to two ordinary differential equations.' The first such
•• equation is the momentum equation in the radial direction and the sc':ond
t p
(rtr)rr - ~ (r'I.T}r t f,u~r3 (rlw%)r - ; (~M.r)r] (ru-) c
== ~~ [-~ (~ (r,ur)r)r + ~~ (f(rw)r)r ]
can bc easily derived from (2).
Due to the slenderness condition the approximation based on small gradients and on
a small radial velocity fails in the neigho~urhood of the breakdown point. ,l\ccording
to the theories of Sq:Jire [11] and Benjumin [12] the flow is supercritical
upstream of this position and perturbations can only propagate downstre:lm. As a
consequence the influence of the t-.reakdown bubble is not contained :n the slender
vortex approximation.
Symmetry conditons are imposed along the r = 0 axi:;:
A..t.t-= 0 I '\T:=O J W= 0 (Lfa.)
At the outer radiu~ thr. type of the system allows three boundary conditions for
viscous flow and one for the pressure for inviscid flow; the physical boundary
conditions for free vortices are
(-p't' i,u.:z.)(r, X) --> C J (lfb)
r~co
where C is a constant taken from the initial conditions and p. (x) is the axternal
prcssur'!. These boundary conditions are also valid for the inviscid s),stem when the
radial velocity has a negative sign at infini~y, otherwise only the pressure
can be given
(Ire)
The slender ·vortex approximation in vorticity, circulation, stream-function
formulation becomes
,u f)C + 1T fr -;c. (~T'r)r
.Q = (~"fr)r
• '---~.-----. "'-~-~--."""""--:--.-.'
-"~"""
(5"1»
(5c)
r I t
i i(
.. ..
with the vortic:ity I2. = -ur ' local circulation rr = rw, slrenm-functiiJn 'If nr-d
v~locity compor ~nt.s
.., I..J.. ::: C - r ·\t'r
'lr; 1: "fx In the vorticity, circulation, stream-function formulation the swirl influence is
given explicitly in the vorticity tran~p')rt. eq,·~·ion by the term I, 3 r / in ::ontra~t r
to the form~lation in primitive variables where this coupling fol!ows impll~itly)v';,
the pressure. For an isoltlted vortex it is advantageous to consider the pe .. tu~b?t.ion
of the stream-function from paralh:1 flow. In this case the parameter 8 in eq.(ba)
is equal to onej elo;e £ i: equal to Z!!rt),
The boundary conditions. (4) corresponding to the vorticity-stream-function for
mulation are at the axis of symmetry
J (- ) +0..,
At the outer radius t!.e boundary conditior.s (l;b) t.anslo~e into
)
r~oo r_co
where C =,(Pi
+ ~u~) is a constant ~a!{en from the initial conditions .,nd p .... (x) is
the external pressure. The parameter p-> specifies the rate of the circumferential
velocity lit the edge of the vortex core to the axi;:d velocity at infinity in the initial
station. "l'hese boundary condit;ol"s are .;:so valid for the inviscid sy"tem w"en the
radial velocity has a negative sign at infinity, otherwise only the pressure
-) can be given
1'-~ l'co (x) t-+co
Finally for viscous flows in a pipe the no-slip conditior.
,IJ.. =. 0,.', "tf~ 0 J W i= 0 holds at the outer radius.
For free vortices the corresponding bou:·.dary conditions for (3) are
'\T d...J.J..oo tr=O at:. r==o and. 'lTr+--r+~== 0 a.t r~co
where v..,.. ex) = -{ 2(C-Pc.o (x»'is a given function.
,. .... .. ~-.
(t cl.)
(7-e.)
t
'I ". ~
t ~
~,
~. -:. , i ;..
J . .... 1 -, "1'. ., " ,-'=.-,,' ~-. :e ,. i ~ . ... [..--
~: ~.
I :/ ;l: •
.
.(
f ): r * ~, ~ , I .,\ ,.,
'.t .. . ,4
.-,~.-h "
r-J.~_ . 't
'~> tt ~ ,'A #"; .,1!t-.~-. 1\' , t!~,.J
: \
..
-. , 4
'" :'\
Initial conditions ha\~ to be provided for the axial nnd circumferential velocity
M.. (r, Xo) = Mo (Y')
W Cr, '1<'0) ::= \\10 er)
The pressure, the radial velocity, the vorticity, the circulation and the stream-
. function can be obtained from them.
Solutions of the slender vortey approximation can cease to exist due to two
different mechanisms: first, when the axial velocity vanishes somewhere in the
flow and second, when the radial velocity become~ unbounded. The first case is
similar to the boundary layer separation while the second is related to the
criticality condition introduced by Benjamin [12]. The se("'ond situation arises
since there is no viscosity present along the axial direction and therefore
unbounded gradients along it can appear. Since the viscosity present in (2) only
controls tt)e radial gradients of the axial and circumferential velocity components
axial gradients and also the radial velocity can grow infinitely. Moreover, in this
case the solution tends to that of the homogeneous problem (3).
Indeed numerical solutions of the slender vortex approxir:'ation sh,w that the
viscous forces are really small compared to pressure and inertia forces near
breakdown of the slender vortex equations.
As in boundary layer theory t'1e behaviour of the approximated .md the full Navier
Stokes s)'slem are identical shortly before breakdown when the proper boundary
conditions are imposed. We return to the point when comparing predictions of the
slender vortex approximation and expcrimental results.
We now comtJare this breakdown' condition to the criticality condition ~s introduced
by Benjamin [12] (see a!so Hall [6 J ). Benjamin co'nsiders steady, inviscid and axisymm~tric flow. Intl'oducing the
streamfunctio~ 'f' with u = ~ 'rr and v = - ~~, the tot31 head h = P + ~u2 +v2 +\,,.2)
and the circulation r = rw the equation of motion for these flows can be
summarized in a single eQuo::tion fur the stream-function.
~ • F".. - -Ali
;:;:r~ .. -. \t.J~~. !~~~!t:'::Z;·'::*'~1i~::t:tm~t~,;·~~m~~~:ffl':\i!fjte,t·5·~£~~~rg ... f 4~~JJ
..
I . 'I .1 -, d i J
~( ;1
1 :1
~f ..i
~ tl l
l
We now consider a perturbation of 0 qu;;sicylindrical flow
"/(>:,1')= 1'c>:,r) + c f(x,r) eYx (10)
where £.« 1 and ~ ~« 3 ~. This perturbation introduced in (9) and neglecting
higher order terms in e yields an equation for F. The critical state F corresponds . c
to 'f = ° and in this case
fct"r- ~fer + [-~(~).tr)r+ riM.%. (r1 w1)r]Fc ==0 ('11)
If the solution with initial data F = ° and F = 1 at r=O vanishes away from the c cr axis the flow is subcriticalj when this solution vanishes at both the axis and outer
radius R it is said to be critical and otherwise supercritical •
In comparison to eq.(ll) the equation for the radial velocity (8)
(r'lT")rr - i (rv-)r + [-.:;. (~.ur)r+ r3~:' (r2. w2.)r] (r1Y) = 0 ('12)
shows an identical form with the boundary conditons
tT d. .u. cO 0 a:C r 1&0' 'R.. V'~ 0 a.-C Y'-= 0 a.\'\ 0" tTr + r 1- c:l x = where the outer radius R has a finite value with R» 1.
The general solution of eq.(l2) reads
(r1T) = C H (r) where the boundary condition (rv)(r=O) = ° is already used.
With the constant C the solution function H(r) can be adapted to the boundary
condition· at r=R. Excluding the trivial solution v; 0, i.e. v(r=R)'*O, the demand
r-7'R.: H~ 0
leads to a critical case, since necessarily the constant C has to go to infinity and
consequent! y
0< r < "'R.: (rv) ~ 00
This charact~.ristical behaviour of ~he radial velocity is present in the slender
vortex approxi~ation in the vicinity of the breakdovm region. A comparison of this
critical case with the previously described critical flow state according to the
theory of Benjamin [121 leads to a complete agreement: eq.(ll) and (12) are
equivalent to e:Jch other just as the boundary and the criticality conditions since
the condition F = 1 excludes only the trivial solution. cr
\ I I !
I I t
, . Now :.ve can see that the breakdown condition for the slender vortex approximation
corresponds to the condition from Benjamin [12] and breakdown can be seen as a
transition from supercritical to subcritical flows as remarked by Ludwieg [5] . Hall [6] reached also the sarne conclusion. He explained the equivalence using
the existence of two solutions for the slender vortex a small distance apart from
each other which do not coincide as the distance tends to zero. He then shows that
"the condition for the appearence of arbitrarily large axial gradients turn out to be
identical to the condition for the critical" [6] .
3. Solution and results
We used both the primitive variables and the vorticity-stream function formulation
for the numerical solution of the slender vortex Clpprcximation. In a first approach
we used primitive variables with centered discretization. The radial velocity is
evaluated only at the midpoints of axial intervals (see fig. 1). The discretization
only use two "time" levels making it convenient for variable axial spacing. The·
function value distribution is shown in fig. 1.
Here foo denotes the numerical solution at the (i,i> node corre:;ponding to IJ
(x,r) = (i6x,jDr), where 6x and 6r are the axial and radial spacing. The equations
are solved marching along the axial direction as the type of the system indicates it.
At each new axial station a s)'stem of nonlinear equations must be solved. This is
done using Ne~ton's approach and a linear band sol ver for the Jacobian invei'sion.
In order to decrease the band width, the symmetry condition for the axial velocity
is applied as
/
. .
~g'!fi:":·:t@·~·t¢~15i:;:W;t:¥J;.s"g;] ~'4.,;"~.l~mtftaOM#pnk~;':::iwk'Z· 'i ·:;Uh,":·,,·;;;:::= t : ::T1- .. ;Zlb' 'i$, '?':fr'TJ~'
instead of three point extrapolation formula. (This boundary condition also holds in
the inviscid case). The discretization is second order accurate in botl1 radial end
axial direction and unconditionally stable with respect to the size of Ax.
Nevertheless during the calculation the size of t::.. x is decreased when the Courantv. ~
Friedrichs-lewy number CFl = max . ...!.. A2 exceeds a predetermined constant I u. u r .
CFl. • t::.. x is also decreased when the Newton's procedure fails to reduce the max
residual below the tolerance limit TOl after NEW iterations. max
Normally we use CFl =2, NEW =3 and TOl=10-4, being the solution not max max sensitive to these values.
The second numerical approach was applied to the vorticity stream-function
formulation (6). We use again centered discretization with function values
determined at node points except the radial velocity which is evaluated £It inter
mediate axial points 0+ !,i> (Crank-Nicolson formulation). The resulting discreti
zation is unconditionally stable and second order accurate. At each new axial
station a tridiagonal system must be solved for the stream,;.function, vorticity and
circulation. For a free vortex the equations are not coupled through the boundary
conditions and.therefore can be solved iteratively as follows. We initialize
(0)
M..iot.{.j = )J..<',j (0)
U--\+"I~,j::: tJi-"l1tl (1{,)
and using these values compute r (1) and then.Q.(l).
Next the stream-function "p (1) is computed and a new iteration can be started~ The
procedure is repeated until
where iterative procedure has a low storage
requirement, .?nly two axial stations has to be kept. It has the drawback to be only
convergent for,supercritical profiles. This is not the case for the Newton's-i,eration
approach that allows subcritical initial profiles. On the other hand initially
subcritical profi les are of little physical inte rest since these f lows can be regarded
as already broken down according to the theories of Squire [ 11] and Benjamin
[12] •
I
. .~
i
i I
The converger~ce problem translates 'into increasing number of' iterations as the
flow reaches criticality conditions. In these situations the axial step is decreased in
order to keep the amount of work low; this also gives a better axial location for the
breakdown point. Each line the axial step is decreased, the' iteration is restarted
with the last converged values.
Since few grid points are required outside the vortex core we use a transformation
in the radial direction
r: (OJrMCl)()~ (5': (O'()mo,. ... )
r _ taft (.~ (5) (18)
rt?tQ.,)( - -cak (~6",,1I.)() The uniform spacing in the r; -direction gives almost a uniform distribution in the
radial direction when ~ «1 and an accumulation of grid points near the axis max when G' ~ 1. The pressure is obtained integrating the radial momentum max equation
where
The initial value is the pressure at r:5 • 9 is an even function of C5 • In order to max decrease the truncation error the pressure is integrated analytically outside the
vorticity core where the circulation is constant.
The initial values are taken as
M.. ::: -1 + oc f (r) with o
Wo: f.>~(r) wi~'" ~cr)::-{~C2,-r2.) Jr~11 ,r I r":? If
, (20)
These polynon:ial distributions are alroady used by Mager (27) , Grabowski and
Berger [18] 'and Shi [20] and " ••• \.vere chosen to approximate the experimen
tally-measured velocity in vortex cores such as those of trailing vortices ••• " [10] .
The parameter 0<. controls the shape of the axial profile, uni form flow for OC = 0,
jetlike for 0<. >' 0 and wakelike for c:« O. The circulation corresponds to solid body
rotation for r« 1 and to potential flow outside the core of vorticity. The
polynomials fulfill symmetry conditions and providr; a smooth transition form the
, ... !!O.s:"MB
......
/ J
:;:;. g =
solidly rotating core into the outer flowfield. The parameter r.:, speci ties the rate
of the circumferential velocity at the.edge of the vortex core to the axial velocity
at infinty in the initial station.
We now return to the boundary conditions for free vortices. We assume constant
pressure gradient
~"Poo clx
==7: Then the slender vortex equations can be scaled using
.., X=X,'t
" and therefore the pressure gradient for the transformed sy.>tem is equal to one. The
flow is relevant for values x < ~ since the outer continuitly desacceleration of the
outer flow makes the outer axial velocity vanish at ';(' = i. When no pressure gradient is present in the outer flow the scaling
';::f. X X== ~ J
(23)
makes the equations and the initial conditions Reynolds number independent.
Notice that the axial coordinate is stretched but the radial coordinate is left
unchanged in contraJt to boundary layer type of scaling (e.g. Hall [25J). The
result is an increasin~ breakdown length for increasing Reynolds number in contrast
to experimental observations [30J for which the location reaches a limit. Due to
this behaviour at high Reynolds number we do not expect this to be the physical
mechanism behind breakdown which we believe to be pressure induced.
For tube flows the numerical procedure for the stream-function formulation w?s
slightly changed due to coupling of the vorticity and stream-function through
boundary conditions. The stream-function was computed from the fourth order
differential equation obtained when (6c) is introduced in (6a) and using a five point
centered di:;crctization. For the primitive variable formulation these new boundary
conditions int~oduced no new complications.
\
In order to resolve the boundary layer on the tube wall, the following stretching
function was used
with R(x) being ti,e tube radius ::md G'm = 0.0.
,/
., j
,/
,/
Faler and Leibovich [0] measured several velocity distributions which were used
as initial condition for the flow calculations.
We study our breakdown criterion described in chapter 2 by applying it to free
flows driven by a pressure gradient which reach critical state already in the initial
section. We used simplified profiles that allow us to do analytical work:
. {-1+~) ,«.-: o 1 )
. {2.~r ) Wo = ~
r
) r~ r'ff r ~~Yi'
These profiles are an approximation to the profiles used in the numerical examples.
The axial velocity distribution is discontinous and the circumferential velocity can
be described by solid body rotation inside the core and potential flow outside of it.
Introducing these distributions in (12) we obtain
and (2b)
where
Since the axial velocity is discontinous jump conditions have to be prescribed
[31J : 'tJ"~
[:u:-J= 0 * where [r] denotes the jump in the values of ~ across r.
The solution of eq.(l6) reads
v- == C" ;.1 Cl) cl
tr::: cz'r + r for r ~ ill (.or r > ~ yz
(2~)
(1.8)
where J1
is t~~ Bessel function of the first kind of first order. The cor';tant c2 is
determined by the pre~sure gradient at infinity. Since the condition v = 0 at r = 0 is
already used, the constants c1
and c3
are determined by the jump conditi,)ns (27).
It follows
(lQ.o.)
"
.'
"
,;
:i ,i
',i
I n "
,I 'I , . !
I : I
;. ,
I
y. . I
..
.... ':J
l C:t
at
Criticality is already reached when
that is
OCG. = J 0,0 with.
[ J..A%.(r...r J'*' - -) =0 r .v.. r
(31)
(j is the first non trivial zero of the zeroth Bessel function J ). This critical 0,0 0
curve is drawn in figure 2 as a dotted line. The curve marked by triangles shows the
numerical profiles which are initially critical. Due to the undershoot in the axial
velocit)· for c« 0 of the discontinuous analytical distribution when compared to the
numerically used profiles, the limiting curve lies too high producing a destabilizing
effect (oc> 0 produces the inverse effect). The overshoot of the analytically used
profiles for the circulation shifts the cu~ve C>C G(~) upw:::rds. The numerical results
can be summarized as follows: Or the right side of the limiting curve in fig. :2 no
solution of the slender vortex approximation can be found (subcritical region). At
the left there exists a region where vortic~s have a limited breakdown length. In
the following region at the left no vortex breakdown appears at all. The vortices
are stable and dissipate themselves away •
3.1. Isolated vortex for inviscid and viscous flow
Figs. 3-6 shoW results for the free vortex with e;: = 0, {:> = .8 and ~ = O. This
vortex breaks down at ~e = 0.015. In fig. 3 we show the relative str'tncgth of the
forces present in the axial momentum balance. Here F p is the pressure force, FI
the inertia and F V the viscous force. Initially pressure and inertia forces are almost
equal while viscous forces are much smaller. Near breakdown both pressure and
inertia forces show a dramatic inc· ease in their strength. This increase is not
.r~ v .rt"\ f-I 'f-I .
j u,w,p,.. v u,w,p
"\ v ,.."\ .... .-1 " ''''
j -1 I t"\ _v. 11"\ 'f-I '" 'I-"
i -1 . 1 1+-
2
1.5
ex
1.0
:7) 0.5
stable unstable
o
-0.5
-1. 0 ~~--,-----.-----,.-----,..--...,......., o O.l. 0.8 1.2 1.6 _ 2.0
- .----------.
" . I' , . , ; ..
..
1
70
t 50
40
30
20
10
0
a=O , B=.8
( RXe )8.0.= 1.3 x 10-2
-----
0 .2 .4 .6 .8
/
I I I I I . /
-Fp / . / --
-FI
Fv ---I
1.0 1.2 x10-2
x ... -Re
. • l·· t ~.
..
.~~ , 1 .• J
present in the viscous forces. Near brp.akdown the flow is essentially inviscid,
viscosity plays a secondary role. This is in agreement with the corresponding
theories. of Squire [111 and Benjamin [12J and· the critical condition which calls
for a non trivial solution to the homogeneous version of equation 0).
Fig. 4 shows the radial dependence of the three components of the velocit y vector
at different axial stations. The circumferential velocity decreases in magnitude and
the axial velocity develops a wakelike profile as the flow reaches breakdown
conditions. The radial velocity shows a dramatic increase just before breakdown,
similar to that of the pressure and inertia forces. Fig. S presents the radial distribu
tion of the pressure. The behaviour of the pressure is quite similar to that of the
axial and circumferential velocity component. Fig. 6 shows the axial distribution of
the pressure and of the radial velocity at di fferent .ldial distances from the sym
metry axis. Notice an almost linear change in the pressure until shortly before
breakdown. In fig. 7 and 8 the velocity components and the pressure are shown at
different axial stations for inviscid flow. Fig. 9 demonstrates the in:ial dependence
of the pressure for inviscid flow. The good agreement of the breakdown process
compared to the viscous case indicates that breakdown is an inviscid phenomenon.
Fig. 10 shows the dependence of the breakdown length on the initial profiles for
free vortices without adverse pressure gradient. Here only supercritical profiles are.
considered. We also show the corresponding results obtained by Shi [20] by
solving the time dependent Navier-S~okes equations. The shnder vortex approxima
tion is in good agreement with his numerical results.
For a given f.>, there exists a lower bound for ~ leading to initially supercritical
profiles. It is possible nevertheless to numerically solve system (2) starting with
subcritical profiles when using primitive variables in conjunction with a direct
solver. We found that all such flows exhibit breakdown ilt some axial distance from
the initial profile.
The increase of the jet type profile provides a stabilizing effect on the flow. For OC
large enough ~nd passed a critical v;111Je the vortex does not break down at all and
viscosity only flattens the profile·s (~his is usually referred as aging of the vortex)
(see. fig. 11). The effects of ~ on the flow are still important near its critical
value. For ~ passed this value the axial velocity at the axis decreases initially but
after reaching 0 minimum value increases again with no breakdown of the flew.
There is no unbounded growth in either the pressure gradient or the radial velocity
any longer •
\ \ \ \
. \ ~.
: \ ---+_. 1
i
I
.1 L. i
\ I. : .
. ·.i
0 . ~
?: . CO t to
x N 0 I
0 0 .-- .---...j'
X
C"'1 . ~
II N 0 CO -
xl&! 0 - N 0 ~ Q) . X 0::: .-- N
II X ~ > 0 CO
t " X 8 0 0.. X 0 co co .-- ~
en en .--.-- t'-... .--
CO 0 1/
. ~
~ 0 ~
. ' .. 0 .-- '-\+ " ,
I :J
0
t II CO tS 1/
Xl& x to
0 0 .--
~ . 0 N co ~ 0 L- __ . . . . -.... ,
.:
I ' I
! I I , . I
ex = 0,0 = .8 - \
-2.0
r
t 1.6
1.2
.8
.4
100 x ~e = 1.3 O~~--~~~--~--~I----~I -~~
-.1 0 .2 .4 .. 6 .. 8 1.0 --a- p
1='t~. _ s .
•
'-I ~ .
(~) = 1.3 x 10-2 Re B.D.
u
r = .84 1.0 ~~;::=-_~..:...:..:::.=-_____ __ f·8
.6
.6 r = .84
P
t·4
"~1
.2
.. a .:-.1 ~---""-------'-__ ""' __ -"-__ --'--__ "'--I _
o .2 .4 .6 .8 1.0 1.2x10-2 X --t~ __
Re 'f(~. 6 ..
~ __ 0_- ____ ~_
----.. - ---~- .. -- -_._----- --- (
~ ~ ---------y-~ . - -
: , .
2.0
r
1.5
1.0
. 0.5 x·1"=0.0373
0.036 o . L-~~L-__ ~ __ L-~ __ ~
o 0'.6 0.7 0.8 0.9 u 1.0
r
~~ ..•
2.0
1.5
0.5
o
o
0.024
0.036
~ .---.----.. --~ " ... f . I, ", ~ :'~'.' .. ::-., "
-.-.--~ ~~- ......... -...,.)/-, .... ~. ~ .... ~-" ",., . -.... -~ ., ... ;.;. ... ~ ...
'2.0
r .
1.5
1.0
O.S
a o 0.2 0.4 0.6 0.8 1.0
w
2.0
r
1.5 ~
.. )
. 1.0 Tt~.8 ..
. ,
0.5
o
-0.2 P
-0.4
-0.6
- 0.8 L-----:--.
-1.0
o 1 2 ,~ ...
I :,' ! I
, ;'
/
3 x·"(
i " I " I -: !
• I' ~ " ,
I
/ '.: . .-'
.1
" N If)
~O N
......t I
M 00 0 ...- "
~ --- N , ...-1/ (j) X
~ I 0 l-O 0(1)
I (1)...- Net: .-> II ...- "-
I 0 QJ 0 CO
Zo:: -- COX a
I .
a ~I 8 ~t I
..c - 3= :J (f) :J
I r---o II II . to <J tS ~ ...-~
I I .....:t ...-
I I
If) (J) N . 0
...- "
I ...-
I 0 0 ...-...-
I I
CO
) I I
0 ~
. I ~ ....
I ......t H-
I I
N M tD
I' 0 0
0
0 to N CO ......t 0 ....s . . N ...- ...-
If)
m 0 ~ 0 II II
C 8 0. X co co
co ~
~
J ~
~.
II ~,
-. Ln 0 Ln . N N ~
0
:J II L
0 ~
. ,)
(-;'
Lf)
0 0
co ~
N I
0 ~
x
Xl~ .....:t ~
N co-
0 ~
co
to
. "-
....:t
N
0
,~' l'
--.
'.
!:
~ ~
~ . ... ~
i I
, " ~L ' . . ;.:
,\;: "-
11 ~i .. ~ r
G ~J . '. . , t \ rl 'j
;~
~, \ 'J " ., ;, , I '..f
II I I
.! .J I": ,.1
~i "
:--i '; 'i
'"j -1 ;1 ~ 'j
1 J "
:1
I (:'j I' ......
¥
.---
Pig. 12 shows results for vortices with constant external pressure gradient * = 7: and fig. 120 an increase in the breakdown length for increasing
Rly'n~lds number with its value attaining a limit for Re .... CXI. This is the expected
behaviour for vortices that exhibit breakdown without external pressure gradient.
Vortices that are stable in absence of pressure gradient (here c( = 1.1) can break
down for Re''t' large enough {in this case Re" it 0.1) but are stable' otherwise. as
once more expected.
Pig. 12b shows the dependence of the breakdown length for small pressure gradient.
Por 'l: small we can see xSO/Re reading the expected values obtained from the
vortex without pressure gradient shown in fig. 10. Por increasing Reynolds numbers
the ratio xSO/Re tends to zero since xsotends to the limit ,shown in the left
picture. Pig. 13a and Db show similar results obtained from different swirls. We
can see again the disappearence of breakdown for stable vortices for small enough
pressure gradients.
In order to stress the influence of the pressure gradient on the flow we sketch in
fig. 14 the results from figs. 10. 12 and 13 combined for Re·'t' = 10. The flow is
shown to be extremely sensitive to small pressure fluctuations at large Reynolds
numbers. A change in 1% in the pressure. normalized with the dynamic head. over . one vortex core length translates into a 100% change in the breakdown length for a
Reynolds number of 1000.
The behaviour of the flow' shown in figs. 12 and 13 can also be found in
experimental investigations. Werl~ [29] found that the breakdown length of a
leading edge vortex formed on a delta wing does not change for high Reynolds 4 numbers (Re
L> 10 ) for an angle of attack of 20 degrees. If w~ assume that the
adverse pressure gradient of the vortex lies approximately in the range of 0.5 to 2%
of the dynamic head over the vortex core radius, then the value of Rr·7: lies
between 50 and 200.
Par these va',ues the breakdown length has approximately reached its ultimate
value acccpteq for Re·T-t CO (Pig. 12a and 12b). The extreme sensitivity against
adverse pressure gradients shown in fig. 14 can be observed also in fig. 15 (from
[30J ). Although the obstacle is positioned far downstream of the brc'lkdown
point, there is a signi ficant change in the breakdown length.
, ..... ./
.. ,!
"
"
,-
I
" " E
,'~
1:; ,'1 "
" f~~ I
f:~ f:
" " ./ ,,'
: .- " ., .-'-~: ....
, .,
~·a""...L-~' .. '~>5~El·"""7,,:-:r·-:-··~:':_~~~~~, .:.. .•. _~,:·". __ ~M._".~_..:....~_· ' .... ''', ... "", . - :.. ___ ~_: _0 ;,:,;::,:;. __ -,:,-:,~_. -~ .. _.~. ____ ~~"."..i11
.24
3DXl
t ,20
.16
.12
~08
~04
o
-. .".
"
•
~ = 1.095
U = u'(o',O) -1
0= Wk Uco
.-.-'
((=1.1
.5
10-2 10-1 10° 101 102 103 10t.·
--Ja- Re Xl
fl~. 'fl(a.).
.. ' .
6 ((=1.1 \
x 10-2 . B = 1.095
5 u=UIO,O)-l x B.D. . ~ = Wk Re 4 Uro
t 3
.8 \\ :.J......
2
1
o 1 1.5~9 10-2 10-' 10° 10' 102 103 lOt.
-- Re x l
F~d' 12, (b).
\ , " \. " • "t ,
\ \
, , , , : ..... , , , ; "
.. -"
J
'.
•
./
\.
.ia;,'
, ,
J ., I
J
,
&i~?eoJ.jt":.""" ".i'j(::!,_:!~!,:::":t,·:::".=t.,:::,,i. 1 )P ... ...:-: '" :V",.,, .. _~,~, nL-YU',~_ ...
.24
'~
• ;:~,:-, tsJ
). X 1" n =.8
\\,
.20 . - 1 a ~ ,u (o.O)
.16 Wk o = U
co
.12 a. =.2
.08 .08
.04 ~ a 0
-.1
o y y --I I I
10-2 10-1 10° 10' 102 103 10' ~ Rex,1"
, '
'.
Fi~. tf3(lA).
-,
.. '. , .
:24 ,
xs.o. x 1" ~ = .8944
-1 t .20 ex = U(O,O)
ex =.5 Wk n = Uco .16
/ . 4 -0
9 p-12 ~
. 3 I ,...
.08
.04 . 1
o 10-2 10-1 10° 10' 102 103 10t.
-~ Rex1'
f(~. 13 (b)~
" -",:
" . ".::.,
." \ . , , .•
• J i ,.
/ /' / h
I , '
i I ; ,.
"-
.~
.~:-;-~
"
1.2
-to
t . 6
.4
.2
o
-.2
0
\ \ ,
!
I \
--0- Re x1"= 0
--o--ReX1"=10 ~ /
........ c:r'
........ ........
/ /
........ ~ ................ B = 1095
/ ",-
/
/: /' /, .
...-,.c ...-
!.8 __ -_____ ~L~----~
...
.2 .4 .6 .8 1.0 1.2 1.4 1.6 , ,
x10-2 ( X
'!
B.D. .. Re
fl~ . -1tr.
--.
I i • . . ,
-.j.' . r
, .
.~' ~::\ .. g~::t: '\> "~iJ ,::> ::<t " .. \:" ,1" .1\\,\. y!: t.~t .;.~ , '" .',r-4:"'~'. '-, ,
'.
')
\ r'"
'J
OF POOR QUALiTY
/ l
/
/
> ,
'.
j'
3.2. Comrmrison to experimental results
Faler and Leibo\'ich [0] performed several experiments on vortices in divergent
tubes. The swirl component was generated by swirl \' ",nes and afterwards the flow
was led through the tube. Profiles for the axial and circumferential velocities were
measured 1/} diameter upstream of the divergent part of the tube. The opening
angle for the tube was 1.4}0 and various mass fluxes and swirl vane angles of
attack investigated. They found six di fferent breakdown types for the flow, the
corresponding breakdown lengths and their dependence on the Reynolds number and
swirl parameter are shown in fig. 16. The Reynolds number is based here on the
averaged axial velocity rmd the diameter of the tube.
Since the slender vortex approximation only deals with axisymmetri~ flows, only
bubble-type brenkdown is considered in this comparison (types "0" and "1"ar.cording
to tne terminology used in [ 8J). Stable bubble structures are only present for
certain ranges of circulation GJ and Reynolds number Re.
One more inconvenience causes from the lack of detniled information on the
boundary layer structure for high Reynolds number!'. Fig. 16 shows the numerical
and computational results combined. The initial distributions are constant along
each solution curve a'nd coincide' with' the experimental value,s only at nodes marked
(*). For lc) = 1.07 we can see a good agreement, the trend of decreasing breakdown
length for increasing Reynolds numbe~s is present and the n~merical values are'
similar. For Re around 2500 and below no breakdown occur in the numerical
experi ments.
For W = 1.54 the . _merical prediction of the ,breadown length decreases for
increasing Reynolds number Inrger than 3000. This behaviour reverses itself for
1000 .!: Re f 3000. For Re ~ 1000 ther~ is no breakdown present in the numerical
experiments. The prediction of the breakdown length is now far off from its
experimental counterpart. The phys:cnl reason behind this phenomenon is the
sensitive behaviour of the flow on prc:;:;urc variations. The pressure prediction
obtained from the slender vortex approximation does not take into 'account the
influence of the bubble presence on th;: pressure field acting on the vortex core. We
propose that a good prediction of the breakdown location can be obtained from an
inviscid calculation for the vortex core with an e.xternal pressuie gradient obtained
from the numerical solution of the full Navier-Stokes equations without the
.... _-------
.... ,
'. , '
,
I
I I
! i
,"
/ j/
I I •
/"
"
r. ,
, "
"
.,-r
~."'" ~'
~ 6
-..-........ --~,---~-~-~-~~,""-'~~-~-
. ,
,W =:=J.51.· ~
-0-\-1--0'0 -1-
, :'1 -s
w =1.07 W :: 1. 28 '?i' '2-
O'..l -~-1- ''t!\ ~2" ~1) b:~ 1 -2,
, . 'n
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.. ··0.1 -2~ ,~, ¥-2-. " 1 -2 .Q _ 2
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-187 -~1-W - , - ~5 _ __ 2 ' " ".
-'0- _ ~O ~1- -3~2-~ - -Ol~~?,~_ ~';;:O- -O':l~ ~2:-4
---"3- /6 .t:f
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L. L. -==--S-/ W = 1. 2
o -3 o
Fi~. 16
.0
a~·i:J· = 1.54
, 'u_-~:----o ,-- n,'l
3 6 9 x/r tube 12
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-~--
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/ ... /
--
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slenderness ap;>.oximation. This is an explanation for the fact that the numerical
prediction strongly deviates from the experimental results. Moreover, the assump
tion of sel fsimilar inflow profi les is not fulfi lied in this case (Re'= 6000). To
compare with experimental data we had to assume selfsimilarity to transfer to
other Reynolds numbers because only a few inflow profiles are given in [8].
The lack of upsteam influence in the approximation leads to predictions for the
breakdown length which are longer than the observed values. Nevertheless, for all
. calculations of Wand Re the inflow profiles are supercritical and upproach critical
state near breakdown.
Faler and Leibovich make the following remark:
"All flows that exhibit vortex breakdown of the "axisymmetric" form (which we
classify 8S types 0 and 1 disturbance forms) or "spiral" form (our type 2) arc
supercritical upstream, in the ~ense of Benjamin" [8] '.
4. Conclusions
The slender vortex approximation was studied in particular for high Reynolds
numbers. For free vortices without external pressure influences the breakdown
length is proportional to the Reynolds number. For free vortices v:ith adverse
pressure gr.adients, the breakdown length is inversely proportional to the .. alue of
its gradient. Flows with small pressure gradients take a long distance t~ breakdown.
For low Reynolds number the prediction of the simplified system agree quite well
with the. ones obtained from solutions of the full Navier-Stokes equations. It was
fOlJnd that the flow becomes quite sensitive on pressure fluctuations for high
Reynolds numbers and that the failing of the slender vortex equations corresponds
to the critical condition from Benjamin [12J for inviscid flows. The last comment
holds since viscous forces arc negligible near breakdown compared to inertia and
pressure forces. The viscous forces do playa role for low pressure gradients by
controlling the aging process. The predictions from the approximating system were
compared to experimental results and for low swirl a good agreement · .... as obtained.
For higher swirl upstream effects on the pressure produced by the breakdown
bubble deteriorates their agreement. This can not be incorperated into the slender
vortex approximation.
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I.cknowledgemcnt
This research was carried out in the frame of thc. special rcsearch progrom
"Vortical Flows in Aerodynamics (SFB25)." The authors are indebted to Professu.
E. Krause, Ph. D., who suggested a part of this investigation. They olso thank Dr.
D. Hanel for many helpful discussions.
~ .. :.}l ',-1 .'.::t:i;!.._-
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Referenc~
1. D. H. Peckham Ilnd S. A... Atkinson, Preliminary results of low.speed wind
tunnel tests on a Gothic wing of aspect ratio 1.0. Acro. Res. Counc. CP 508
(1957).
2. H. Chevalier, Flight test studies of the formation and dissipation of trailing
vortices. J. Aircraft 10,14-18 (1973).
J. A. J. Bilanin nnd S. E. Widnall. Aircraft wake dissipation by sinusoid;:)1 in
stability and vortex brenk down. AIAA paper 73-107 (1973).
4. J. Swithenbank and N. Chigier, Vortex mixing for supersonic combustion. Proc.
12th Symp. Combust. Inst ... 1153-1162, Pittsburgh: Combust. Inst. (1969).
5. H. Ludwieg, Vor.tex breakdow.1. Dtsch. Luft Raumf:lhrt Rep. 70-40 (1970).
6. M. G. Hall, Vortex breokdown. Ann. Rev. Fluid Mech., Vol. 4, 195-218 (1972).
7. S. Leibovich, The structure of vortex breakdown. Ann. Rev. Fluid Moch., Vol.
10, 221-.246 (1978) •
8. J. H. Faler and S. Lcibovich, Disrupted states· of vortex flow :md vortex
breakdown. Phys. Fluids, Vol. 20, 1385-1400 (1977).
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10. H. Ludwieg, Zur Erklarung der Instabilit&t der liber angestellten Delta-fWgeln
auftretenden freien Wirbelkerne. Z. F1ugwiss. 13, 437-442 (1965) •
11. H. B. SqUire, Analysis of the vortex breakdown phenomenon. P<lrt 1.' Aero.
Dept., Imperial Coli., London, Rep. 102 (1960). Also in Miszdlnnecn der
.A..ngewar:dten Mechanik, Berlin, Akademie Verlag, 306':H2 (1960).
\
12. T. B. Benjamin, Theory of the vortex breakdown phenomenon. J. fluid Mech.
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D. T. B. Benjamin, Some devclopmer.:s in the th(!or}, of \'ortex breaKdown. J. Fluid
Mc:ch. 20,65-04 (1967).
, ..
--" .'
14.
· .... ..:. , " .. ~.,.. -~- .
H. H. nor-sci, Inviscid "nd viscous rnodels or the vortex hrenkc.lown phenornt:/lon.
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flows in circulnr ducts. Ph·)·s. Fluids, Vol. 12, 1747-1757 (1969).
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eddies in a • otnting stream. Comput. Fluids I, 289-300 (1 ?73).
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vortex breakdown. J. Fluid Mech. 75, 525-544 (1976).
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a vortex breakdown. J. Fluid Mech. 86, 313-335 (1978).
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RWTH Anchen, Aachen (1983).
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tional vortex. NRC Can. Aero. Rep. LR-378 (1963).
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Aero. Res. Counc. R. &. M. 3467 (1965).
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Vol. 7, 53-110 (1966).
. 25. M. G. Hall, A new approach to V:lrtex breakdown. Proc. 1967 Heat Transfer
Fluid Me,ch. Inst., Stanford Univ. Press, 319-340 (1967) •.
26. H. H. Bossel, Vortex computatior. by the method of weighted residuals using
exponentials. AIAA J. 9, 2027-2034 (1971).
27. A. Mager, Dissipation and breakdown .of a wing-tip vortex. J. Fluid Mech. 55,
609-628 (1972).
28. Y. Nakamura and S. Uchida, Calculations of axisymmetric swirling flows by
!
~I
r ,.>-,., F·· ..
L I
I [ ! I i
I \.
I . j
I r
. .
. -~. ---_ .. _--:-.--_. "'.-: .,', h •• '
I...;..·.' ~ .' • . ~--.- -: . _, • ___ • __ .... '_ .__ .4.,
I
29. E. I<rau:.c, L. RC}'na and S. Menne, Druckiinderung bei ;txialsymmetrischem
Wirbelaufplatzen. Abh. Acrodyn. Inst. RWTH Aachen 27,1-.10 (1985).
)0. H. Wcrl~, Sur I'eclntcmcnt des tourbillons d'apex d'une aile delta nux faiblcs
vitesscs. Ln Rcch. A~rosp. No. 74, 23-30 (960).
:no S. Menne, Approximation fur schlanke Wirbel mit diskontinuierlichen Anfangs
verteilungcn, pnpur submitted for presentation at the GAMM- Tagung, Dort
mund 1986 •
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F"ig. 1:
Fig. 2:
-{;;:J:-
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Fig. 3:
Fig. 4:
Fig. 5:
Fig. 6:
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Coordinotc systern and rJrid derinition
Stability diDgram for \'l,rtex breakdown
I · / I • .. , I
Transition from super- to subcritical flow according to Benjamin [12J
Breakdown at inflow section for slender vortex approximation, initiol
condition (20)
Region of temporary stable vortices
Breakdown at inflow section for slender vortex approximation, initial
condition (25)
Axial variation of the inertia, pressure, and viscous forces per unit
volume as computed with the viscous slender-vortex approximation
Radial profiles. of the axial, radial, and circumferential velocit},
components computed with viscous slender-vort,ex approximation
Radial profiles of the static pressure as computed with viscous slender
vortex approximation
Axial variDtion of the axial velocity component and of the static pressure
as computed with the viscous slender-vortex approximation
Fig. 7a: Axial velocity profiles computed 'with slender vortex approximation for
i'r)viscid flow
Fig. 7b: Radial velocity profiles computed with slender vortex approximation for
inviscid flow
Fig.7c: Circumferential velocity profiles cOl,nputed with slender vortex approxi-
, i
-.-.... -. ~ .. -
-' ."
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'0
, .
; , ' J
j: \
Fig. 0:
Fig. 9:
, ,.
I· .,.1.,_
.i
Pressure profiles cornrlJtcd with slender vortex approximation for
inviscid flow
, Axial pressure variation computed with slender vortex approximation for
inviscid f'''w
Fig. 10: Shape parameter" as a function of the breakdown length. Computed for
several swirl rates
Fig. 11: Axial variation of the axial velocity component computed with viscous
slender vortex approximation
Fig, 12a: Breakdown length as a function of the Reynolds number for constant
externally imposed pressure gradient 1:" _ Computed with viscous slender
vortex approxiamtion
Fig, 12b: Br~akdown length as a fu~ction of .imposed external pressure gradient -r: for constant Reynolds number. Computed with viscous slender vortex
approximation
-Fig. 13a: Same as Fig- 12a. The ~wirl parameter ~ = 0.8
Fig, Db: Same as Fig. 12a. The .~;wirl parameter ~ = 0.8944
Fig. 14: Influence of an externally imposed axial pressure gradient 7: on the
breakdown length as computed with the viscous slender-vortex approxi
mation
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Fig. 15: Influence of an extcrnnlly impor.l!d axial pressure gradient on the
breakdown length. (From [~O])
Fig. 16: Comparison of experimzntal data (from [8 J ) and numerical results wit"h
slender vortex approximation. The type of disturbance (0-6) and its mean
axial location vs. Reynolds number.
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End of Document