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Singularity method applied to the classical Helmholtz flow coupling procedure with boundary layer calculation
Ph. Legallais, J. Hureau
To cite this version:
Ph. Legallais, J. Hureau. Singularity method applied to the classical Helmholtz flow coupling proce- dure with boundary layer calculation. Journal de Physique III, EDP Sciences, 1994, 4 (6), pp.1053- 1068. �10.1051/jp3:1994186�. �jpa-00249166�
Classification Physics Abstracts
02.30 47.90
Singularity method applied to the classical Helmholtz flow
coupling procedure with boundary layer calculation
Ph. Legallais and J. Hureau
Laboratoire de Mdcanique et d'Energdtique, E-S-E-M-, rue Ldonard de Vinci, 45072 Orldans Cedex 2. France
(Received16 December 1993, accepted 24 Marc-h 1994)
Rksum4. Nous mettons en muvre une mdthode de singularitds pour calculer l'dcoulement autour d'un obstacle h parai courbe quelconque en prdsence d'un sillage de Helmholtz. La rdpartition de
densitd tourbillonnaire sur la paroi baignde de l'obstacle est calculde par l'application de la condition de glissement. La condition d'dvolution stationnaire est dcrite sur la premidre partie des lignes de glissement afin de ddterrniner leur position, la gdomdtrie de la seconde partie provenant
d'une Etude asymptotique, Nous jugeons de la validitd de la mdthode en comparant les rdsultats
avec ceux obtenus par une mdthode dtalon utilisant la transformation conforme, et qui est une
adaptation de la mdthode de Levi-Civita. Le bon accord entre les deux nous permet d'envisager l'extension de la mdthode au cas multi-obstacles. Nous proposons ensuite une procddure de
couplage avec un calcul de couche limite appliqude au cas du cylindre circulaire. Nous retrouvons l'existence des premier et deuxidme rdgimes, ce dernier dtant bien prddit pour ce qui est de l'angle
de ddcollement et du coefficient de train6e. La prddiction du premier rdgime est plus approxima- tive, avec pour cons6quence immddiate une surestimation du nombre de Reynolds critique.
Abstract. A free streamline wake model based on singularity distribution is proposed in order to
treat the flow past an arbitrary curved obstacle with Helmholtz's wake. The slipping condition
gives the vortex distribution on the obstacle and the steady evolution condition is written on the first part of the free streamlines in order to find their locations, the geometry of the second part being fixed by an asymptotic study. The validity of the method is judged by comparing results with those obtained by a formulation, to be used as a standard, which encloses conformal mapping and is an adaptation of Levi-Civita's method. Good agreement leads us to envisage extending the method to multi-element systems. Correlatively, we show a coupling procedure with a boundary layer calculation. Applied to the circular cylinder, it allows to bring out the existence of sub-and
supercritical ranges, Although the latter is well predicted for the separation angle and the drag coefficient, the former is only approximately approached, with an overestimate of the critical Reynolds number as an immediate consequence.
1. Introduction.
In a potential theory framework, treatments of wake flows past arbitrary curved obstacles have been discussed by numerous authors since the early days of fluid mechanics, one century ago.
One of the ideas was to represent the fluid region close behind the body which physically
contains one or more vortices and where viscous effects dominate, by a motionless area, then necessary at constant pressure. Extending the area to infinity, the model becomes still more convenient as d'Alembert's paradox is removed it's the classical case studied by Helmholtz and Kirchhoff that we are concerned with here.
When one experimentally investigates the separated flow over a blunt body, it can be observed, for sufficiently high Reynolds numbers, which is the most frequent case, that the free shear layers, limiting the wake near the body, are very thin and rather steady, that the
mean values of the velocity vanish, and that the wake underpressure may be taken as constant, at least as a first approximation.
After basic developments due to Helmholtz (1868), Kirchhoff (1869) and Rayleigh (1876) who both treated the case of the plate, fundamental works of Levi-Civita (1907) and Villat (1911) helped to build up the theory, the former reducing the original problem to a Dirichlet
one, the latter emphasizing the need to deal with an integral functional system. Mathematicians like Birkhoff and Leray contributed to the maturation of the model, the latter particularly looking into existence and unicity conditions.
The latest exhaustive accounts are found in Jacob's [10] and Wu's [23] writings, in the early
sixties. The complexity of solving the problem added to the limits of the model may account for the fact that no earlier than the eighties did some authors, taking peculiar developments of
computers into consideration, decide to have a close look at the theory.
Elcrat and Trefethen [5] propose a computation method, which consists in determining the conformal mapping of the potential slit plane directly onto the physical plane rather than onto the log-hodograph plane, as in Kirchhoff's method. They use a modified Schwarz-Christoffel
integral, based on Monakhov results [16], and then are able to treat any polygonal obstacle.
Hureau, Mudry and Nieto [8] suggest a new formulation of the problem, inspired by Levi- Civita's method, in which they map the potential slit plane onto the interior of the upper half unit circle, seeking a log-hodograph variable in this plane by means of Schwarz-Villat's
formula. A relaxed numerical algorithm gives the solution to the problem. This method allows to deal with arbitrary curved obstacles. For several bodies, like airfoils with inclined flap, good Cz(a) curves agreement with experimental datas can be observed.
But, as it was not possible to modify the wake pressure, fixed at P~, often leading to unrealistic calculation of drag, many authors, hydrodynamicists among them, devised new models underlain by experimental facts, to provide a variable wake underpressure. In order to
improve modelling, some kind of artifice had to be introduced in the flow field, so the simple
vision of a flow around an obstacle with its ensuing motionless wake extending to infinity, had to be left.
The reentrant jet model (Fig. la), introduced by Prandtl and Wagner, and further developed
in the forties by Kreisel [14], Efros [4], Gilbarg and Rock [7], although preferred by some theorists, as it roughly looks like real flow and agrees quite well with it in drag for small
C~ rear, overestimates the backward jet phenomenon, particularly jet width, physically
significantly weakened by turbulent mixing.
Models with finite motionless wake allow wake pressure to change. The model considered
by Cisotti [3] and Kolscher [13] (Fig. lb) is unsatisfactory as d'Alembert's paradox reappears, while Riabouchinsky's image model [21] (Fig. lc), though it gives non vanishing drag, does not propose a definite rule in the choice of the fictitious body image.
The dissipative wake model of Wu [24] (Fig. ld), whose drag calculation results agree with
experimental ones in cavitational flows, is one of the most attractive models. The wake is divided into two parts, the first being a constant pressure near-wake, and the second, being
described by a potential flow where pressure recovers free stream value at infinity.
a) b)
c) d)
Fig. I. a) Reentrant jet model, b) Finite wake model, cl Riabouchinsky's image model. d) Wu's
dissipative wake model.
A first class gathers all these models which apply mostly to cavitational flows, accurately predict body drag, and in which wake underpressure is fixed by experimental data. They use
methods involving conformal mapping, and then consider single bodies, the largest part of the results conceming those of simple geometry like plates or wedges.
More recently, new models have appeared, concerned with lifting airfoils, often near stall, sometimes after it. As a matter of fact, experiments confirm that maximum lift is obtained with
a rear separation of moderate size, whose simulation has to be inserted in a global computation
method. They are mostly based on singularity distributions, then adapted to the treatment of multi element systems and, in addition, they allow extension to three-dimensional flows.
Distributed vortices simulate presence of body boundary, and sources (singular or distributed)
account for a separated region.
Bhateley and Bradley [I have imagined a simplified model consisting of a source embedded
inside each obstacle. The wall opens according to the separation points, determined by
experimental observations. Calculations were applied to multiple airfoils including leading edge slats and trailing edge flaps.
Jacob's famous model ill, 12] (see also [25] for application to flows in 2D cascades) (Fig. 2), concerning high lift devices, produces an outflow region by using a source
distribution on the rear part of the airfoil, sink addition downstream the trailing edge rapidly becoming the rule. In these papers, potential flow and boundary layer calculations are
combined to locate the computed separation points. In order to estimate the unknowns of the
problem, and particularly source distribution, conditions of equal pressure at three points and conditions of constant pressure at the boundary of the wake in the vicinity of the airfoil have to be verified.
Parkinson and Yeung [20] have lately proposed a variante using conformal mapping applied
to a separated flow around a single airfoil with a spoiler or a split flap. Considering the rear pressure coefficient as a data, given by physical measurements, the pattem adds two sources to the potential flow over the circular cylinder on the portion of the cylinder corresponding to the
separated flow region of the airfoil, its separation points being stagnation pojnts in the cylinder plane.
Wvi%<id outer flow
smwdatod
bound«i dead air refo~
ldi%@««men modwl
»
zlz
Fig. 2. Jacob's model.
The purpose of this paper is to propose a method based on singularity distribution applied to the classical Helmholtz and Kirchhoff case. The method using conformal mapping plays the part of the standard one and has been chosen for its rigorous theoretic basis. In spite of its crude
modelling of the wake region, we saw above that it can be a good tool for the treatment of the lift configuration of an airfoil fitted with a flap or a spoiler (we do not consider the limited case of cavitating flows with cavitation number equal to zero). After a brief survey of the standard
method (Sect. 2), we shall develop the theory of the direct approach, notably aspects bound to the asymptotic behavior of free streamlines which is not usually taken into account (Sect. 3).
Section 4 will be devoted to comparisons between the two approaches and will describe a coupling procedure of the Helmholtz method with boundary layer calculation applied to the
case of the circular cylinder.
2. Standard method.
Let us consider the ideal incompressible and irrotational flow past bodies with Helmholtz's wake (Fig. 3). Boundary conditions for complex velocity w(z) hold three conditions ; the first (I) is the infinity condition, the second (2) expresses that the solid is a streamline and the third (3) fixes the velocity modulus on free streamlines :
iirfl W(Zj=v~ (ij
-w
Im (w dz
= 0 on AB (2)
(w( =V~ on ~U~. (3)
, ~
Outstanding author Levi-Civita proposed to solve the initially difficult mixed problem for the complex velocity in the physical z-plane by mapping the potential plane vi onto the upper half unit circle plane (<) (Fig. 4).
In the (-plane, he determines the function Q
= &+ iT, related to the complex velocity W(2) by
w(z)
= v~ e~'~ (4)
Es
B
v=o
P=P~
A
Z~
Fig. 3. Helmholtz's flow pattern.
(o
~~~
~ oq
B ~~ ~~~~
,
A
,
, ,
, ,
, ,
, ,
,', , ,
,, ,
Fig. 4. The potential plane and the upper half unit circle plane.
bringing out the velocity argument in real part and a combination of its modulus in imaginary part.
Transforming onto the half unit circle is made intentionally in order to apply Schwarz's reflexion principle, which allows extension of the primary domain of the upper half unit disk to
the lower one, thereby opportunely removing free streamline conditions. Determining the
Q (f function proceeds from the application of Schwarz-Villat's formula solving Dirichlet's
problem, but is generally resisted, as pointed out by Villat [22], by the ignorance of the function
6(~r = &(e'" (Villat's function (5)
The real flow is obtained with dz
=
df/w, so that
~ _
~~~_~~=ke' e ~
where r~ is the rgument of the image of the stagnation
point on U+ We
constant k is In order to emove the velocity ingularity at the
stagnation point,
function 12~ = 6~ + ir~ (which is the solution of the
I
=
6 Ho + = T To T (~r)
= T(e'"). (7)
JOURhAL DE PHYS<QUE «i -T 4 N'fi JUNE19Q4
The 6~ function takes the values -I and +I
as ~r belongs to [0, ~r~[ or to
2 2
]~r~, gr] respectively, and the T~ function is that associated to 6~.
Laboratory contribution (see [8] and [9]) consists in emphasizing the 6 function in writing it 6 (" )
~ p ° E(" ) (8)
It is a compound of the p function (p is the tangent angle to the wall) and of the
~ function defined by
s « e jo, gr j
- s ~ s (« ) e jo, Lj (9)
where curvilinear abscissa s is the parameter on AB, of length L. So, we bring out the
s function, one-to-one correspondence applying U+ on the wetted wall AB. We then deal with
a functional system whose unknowns are the three functions
W - ~, f, E (10)
and angle ~ro.
The four equation system reads
G(~r, ~r~) jr ? sin ~r'sin~((~r + ~r~)/2)
. e(~r
= with G (~r, ~r~) = d~r' (I I)
G(gr, ~ro)
o exp(f(~r'))
. f(m)= lim 31j~i(«') ~~ «'~ (12)
ST,_~,,, o 1-2(cos~r'+(~
. I(")
" p °E(">-) (131
. ~ro =
"
+ i(~r) d~r (14)
2 gr
j
We work out a relaxed iterative algorithm solving the previous functional system. Beginning
with an arbitrary e correspondence, it is then possible to determine the values of the other unknowns by formulas (12), (13) and (14). The iterative procedure starts by a new calculation of correspondence s by (I I). It stops as soon as a normalized error associated to one or more
unknowns is reached. Knowing complex velocity in all points of the wetted wall, we can then get total force acting on the body by integration.
3. Method based on singularity distribution.
3.I THEORETICAL FORMULATION. We refer to the flow in figure 5, boundary conditions
being exactly the same as the preceding ones. The direction of the arrows indicates increasing
curvilinear abscissa, the positive side lying to the left by convention. Streamlines
~ (BQ~) and ~ (AQ<) are velocity tangential discontinuity lines, commonly named free
, ,
streamlines. The method here developed consists in simulating both the wetted wall and the free streamlines by vortex distribution (for further developments see [15]).
On the obstacle, the piecewise constant distribution is determined by a slipping condition, which requires the normal component of the velocity relative to the wall to vanish. In order to
calculate free streamline locations, the primary idea is to apply a steady evolution condition all
~ Q~
/~
B
V=0
P=P~
~
Fig. 5. Description of the Helmholtz flow approached by singularity method.
over the lines. On each one, we express a tangential jump of the fluid velocity between a zero value inside the wake and a V~ value outside it.
Let us write absolute complex velocity at any point of the flow field as w(z)
= wo(z) + w~(z) + M'~(z) (15)
wo non perturbated flow velocity (V
~ w~ wetted wall induced velocity, w~ free streamlines induced velocity.
A median parameter of affix = describes the wake particle motion. It is related to the ambiant flow by the evolution condition
3f
~
l
~ ~j~~
& ~ ~ ~ i ~~ ~ "' Setting
x = t T (17)
we obtain the steady evolution condition
~~
=
w>* (18)
aX
whose solution = only depends on x (for more details on the median description of vortex sheet, see Mudry [18]). According to the wake panicle evolution theory, we introduce vortex density y carried out by the wake particle defined, for each free streamline, as
~ az
~ ~ ~~~ j~~
as
where curvilinear abscissa s is the parameter chosen for the free streamline description. With p, the tangent angle to the line, we write
d=
= ds e~P (20)
from which we deduce median and jump expressions of the complex velocity
w~* = V~ e~'P (21)
w',* =
' V~ e~'P (22)
[w]~ = w) w>/
= V~ e~'P (23)
[M'Is " w/ wj
= V~ e~'P (24)
One infers densities y~ and y~
~
=
Re [w], ~~
=
Re (- V~ e~ ~P e'P)
= V~ (25)
as y, =
Re [w], ~~
=
Re (- V~ e~'P e'P)
= V~ (26)
as
Now, let us look at the induced velocity expressions. In a point ( not belonging to a free streamline, the complex velocity induced by both free streamlines has the ordinary form
~~,~__£ fi
~~ ~~~~
~ ~" Z~Z'~~
which, with (20), (25) and (26) where the units are so chosen that V~
= I, leads to W~~<)
=
i j~ i ~'l'dz
j~ ~i
l'dzj
(28)
,
The complex velocity induced by arc AB is given by the same Cauchy integral for a point ( not belonging to AB, but, for a point belonging to AB, the expression on the positive side differs from the previous one and can be seen as a direct application of the Plemelj-Privalov
theorem (see, for example Muskhelichvili [19])
where denotes the principal value of the Cauchy integral.
y~ distribution on the wetted wall is obtained by applying the slipping condition ( e AB Im (w+ (( ) d()
=
0 (30)
With the above expression for w'( (29), simply noting y and p density y(() and tangent angle p(() respectively, one gets the developed form
i~'P
~-<P<Z> ~-<pj=>
~~
2 gr
~
f z ~~
~
( z ~~ ~~~ ~ ~~
,
3.2 ASYMPTOTIC GEOMETRY. It appeared that the single steady evolution condition was not
sufficient to take a region without fluid motion into account (actually, velocity vanishing on the
side of the wake of each free streamline does not extend to the whole wake). Computations made with increasing free streamline lengths did not allow to record the growth of the
simulated wake (as calculation began with a horizontal wake shape).
To simulate a motionless wake, it was then necessary that the free streamline geometry contain this information. That is why we have been led to separate each line into two parts, the first one respecting the classical steady evolution condition, the geometry of the second part laid down by an asymptotic behavior, analysed by Birkhoff and Zarantonello [2]. Let us note T~ (resp. T~) the point of line ~ resp. ~ separating its first part from its second one
, ,
(Fig. 5).
Let us examine the asymptotic behavior of the flow at infinity. We know that the following
function maps the lower half plane of parameter variable t onto the slit plane of the complex potential ~f (Fig. 6)
f
=
t~ (32)
(fl
(ti o~
~ ~
llllllllllllll/ llllllllllllllllllllllll
B A
Fig. 6. The potential plane and the lower half plane.
Let us expand complex velocity in the vicinity of infinity so that
w> = exp I (a, t~' + a~ t~ ~ + a~ t~ ~ + a~ t~ ~ + (33)
where a
j, a~, a~, are all reals. Flow geometry in the physical plane is then determined by the
quadrature
Idf
z = (34)
w
= I iai t~' 2 a( + ia~) t~ ~ + 2 t dt (35)
which gives, after integrating and only keeping those terms of exponent higher than 2 in w~ ' expansion
z=zo+t~-2ia,t- '+ia~~ Int~-t(+2iajto+ ~~+ia~~ lnt(. (36)
As the two free streamlines correspond to the negative and positive parts of the real axis of the t plane, the shape of the motionless wake is essentially parabolic. The unknowns aj and aj are supplied applying the momentum theorem to the domain of figure 7 bounded by
free streamlines and by a fixed wall. The domain is here considered infinite.
