Ik1UNDARY EFFECT ON CAVLTATING FlOW PASE A CYLINDER K. K. Siraf riev
Lhurn.sl Prikladnoi Mekfsanikj i Tekniciseshoi liziki, No. . PP.
03- i0 1965
rise widespread method of redu ing data obtained in modelex-periments ors a flow of finite width to the conditions of in infinite flow lias beers verified experimentally. The experiments were conducted on
cylinders from to 20 mm in diameter ,st a relative constriction of the
working (flamber Cd/b) varying from 0.0i to 1.40. The parameters characterizing the flow, namely, resistance, pressure coefficient, and
ca-vitation number, were reduced to the conditions .55 infinity using extra-poI ation curves as a function of the rei ative constriction ofthe flow to zero
value of the consticsion. It was isown that sise of an equivalent velo-city is not always justified and may lead to errors. Tile cause of these
errors resides SIS the fact that the increase in equivalent velocity at
the profile of the model is nonuniform and is affected not only by tise flow boundaries but also by the stage of cavitation, i.e., tise shape .snd size of the cavitation zone. Tise true value of the correction can be determined only be a series of experiments at different flow
constric-tions but the same Reynolds number. NOTATION
a height of working chamber; b widtis of working chamber; b
maximum width of cavitation zone; C., drag coefficient; d diameter of cylinder; h its height; g acceleration of gravity; k equivalent velo-city coefficient; l length of cavitation zone; l length of separated part of zone; N number of separating cavities; p Isydrornechanical
pres-sure; Pc pressure in cavitation zone; p water vapor prespres-sure; P pressure
coefficient; q velocity head; r radius of cylinders; R Reynolds number;
S Strouhal number; u velocity at cylinder profile; y flow velocity
up-stream from model; v flow velocity with allowance br constiction due to model-equivalent velocity; X drag of model; d relative width of cavitation zone; v velocity increment; y weight of unit volume of water; x cavitation number; X relative length of cavitation zone: X5 relative length of separated part of cavitation zone; u kinematic visco-sity; u angle between axis of piezo aperture in model and direction of
flow.
Subscripts: -for flow of infinite width; h-for flow of finite width:
Prro - Pr
P -
Pr,1, s2 dviq
q=--'
---i;-'
Nd
Cr,
(I) 1. OwIng to the presence of the walls opposite the end faces of the model, the uniform velocity curve at the nseasuring section is
distorted by rise boundary layer as i result of which the drag distribu-tion along die model ilse becomes nonuniform. The effect of these walls can be eliminated, if tire drag is determined from the pressure dIstrIbution at tise mid-section of the model and tire measured prcrure values are related to tise velocity at tire axis of the working (flamber.
however, tise presence of the other two walls must also have air
effect on tise flow past tise middle element of the model as ompared
with a flow of infinite width.
There have been more theoretical than experimental studies of the effect of flow boundaries [I - 15). The theoretical studies relate prlmarïly to tise influence of chamber width on drag, lift force, and
the dlnsenslons of tire cavitation zone, generally for a very smaLl range
of the ratio d/b and small cavitation numbers _
It itas been recommended that the measured values of the hydro-dynamic pressures he referred to a velocity equivalesst to the velocity
of a flow of infinite width and computed, in our investigations, from
tise formula [16-18]
vr,, rk = y - (2)
In studies of cavitation erosion [19, 20] it is often necessary to
know the effect of the boundaries over a quite wide range of values of the ratio d/b on other characteristics of the flow, for example the
Reynolds number R. the Strosthal number S, and the cavitation number
s for different stages of cavitation À..
2. The experiments were conducted in a water tunnel, cresssac: ion of working chamber a X b 20 X 50 mm. Six cylindrical models u-ere
tested at ratios d/d = 0.1 - 0.4. The test conditions are shown sri
ra-bIc 1. The Reynolds numbers R in Table trecre calculated using valtses
of k obtained as a result of tise present islvestiation with extrapolation
of C. As Table
I sisoses, in these experiments the stsfìuencc of the ley-folds number ori tise drag Cx was relatively small.
Lab
y. Schepoukunde
Technische Hog3school
73Deift
Table 1 d, mm d/bv,msec'
k 5 0.05 12 1.03 0.62 8 0.16 7.5 1.09 0.115 lOa 0.2(1 12.5 1.1 1.38 10h 0.20 9.2 1.1 1.02 12 0.24 10 1.18 1.46 15 0.30 4.4 1.21 0.80 20 0.40 5.86 1.44 1.68RCHIEF.
J. APPL. MECI1. AND TECH. 1HYS., NUMBEH 3
b
The experiments consisted of I) observations of the development
of a cavitation zone in relation to the cavitation number, accompanied
by measurements o! tire dlnsenslorrs of the cavitation ;qne; 2) measure-ments of sire pressure distribution at the mid-section of the models, the water temperature, and the barometric pressure. In order to measure the pressure at different points of the cross section. using the same. piezo aperture the model, connected with a vernier scale, could be adjusted
at an angle O between the flow axis and the axis of the piezo aperture.
lise flow axis was found as the axis of symmetry of tise pressure coef-ficient curves P = P (0).
I'
II',E
L
dO 18F'-w i-.
-ömm /5,a.
SN-..r.kia
a.ar wu0m
ru ,1a u..
i ULV
IU!A
!1
a. rAi
a..,"
200. "yI...
2.'.U
,
kI
:
,Omrn 2004I
7.1-ß -U
.pi ,uiu
.ii: &!IU
:80' d5mm Fig. 2 -o.The drag of the cylinders was found by planimetry from the pres-sure graphs P = f(r) without allowance for friction drag, which forms
only a small fraction of the total drag (l-21v).
The results of the experiments were analyzed using extrapolation curves, on which the value of the parameter was plotted as a function
of the ratio d/b. The value of the parameter obtained by extrapola-tion to the value d/b = O was taken as equal to that for a flow of
mho-rute width.
5. lire results of visual measurements of the cavitation zone are
presented in tire graphs showing the lengths X = l./d and Ao = 15/d and the maximum width of the zone 8c b/d as a function of the ratio d/b (Fig. l the graph for d = 10 mm was constructed for R 1.38 1O.
Some idea of the structure of the zone is given by X, the length of the separated part of the zone. The length l is defined aS the
dis-tance from tise axis of the model to tire point at which rire jeu bound-ing the cavitation zone join, while the length 1 is defined as the
dia-tance from the axis of she model to the point of transition from the part of the cavity adjacent to rire model to tire part with a bubbly
structure at the end of the cavity. As found prevIously [21, 22], the pressure ira the hollow part is equal to tire water vapor pressure Pv in
the bubbly part of the cavity PcPv An examination of Fig. i leads to
the following conclusion.
In spite of the similarity of the Reynolds numbers for the models in each of tIre two groups (diameters 5, 8, arid 15 mm and diameters IO,
12, and 20 mm), there is no similarity in the development of the cavita
tion zones. In all stages, with increase in the ratio d/b the cavitation
zone develops closer to the model as a result of an increase in positive pressure gradient [23]. Thus, for models 5 and 8 mm in diameter un tise initial stage we have = I and for a model 15 mm in diameter =
At the same time, with increase of d/b the maximum width of the
ca-vitation zone moves closer to tire model and decreases from S = 1.5 ro S = 1.25. For the separated stage, at X = , the number x = 0.5,
where-as for zero pressure gradient it must be equal so 0.
The pressure distribution of six cylinders in the absence of cavita-tion is shown in Fig. 2, the pressure coefficient relacavita-tion P(d/b) in Figs.
3,4, the drag of these cylinders C(d/b) and the angles corresponding
to the characteristic points on the pressure distribution curves (dl h), O', (d ¡b), in Fig. 5. Applying the extrapolation method to
C (d/b), we can express it by means of the two formulas
C 0.81 + 5 (ci/ 5)2, C, = 0.74 + 5(d / 5)2. (3) These differ with respect to the constant, possibly as a result of the difference in Reynolds numbers (Table 1), except for the C5 for tire cylinder d = 15 mm. Tire drag of this cylinder is considerably dif-ferent from tirar which mf girt have been expected from (3). This
dis-crepancy may be attril,uted to the fact that the pressure mjn is less than what it should be according to the relation min (d/b) for rire
cylinders with d = 5.8 and 10 mm (Fig.4). In its turn, the deviation of mín frorrr tise general law is probably due to the unique character of the dependence of tire angle 0mln (dÍb). There is a certain value d/b 0.20, at which 0min assumes a minimal value and after wisich
the point with the value mIn begins to move in tise direction of flow
(Fig. 5).
-.2
-24
5
-08
-4
74
ZU. PRIKL. MEKH. I TEKIU'. FIZ., JULY-SEPTEMBER
19135o 02!
Fig. 4
Accepting this explanation of the deviation of Cx and minfor d = = 15 mm from the general laws in terms of the simultaneous opposing influence of the flow boundaries and tise Reynolds number, in the sub-sequent analysis of the pressure distributioms over the cylinders we used the extrapolation method (Figs. 3, 4).
Employing tIsis method, we shall consider the question of tire value of the equivalent velocity for reducing tise pressure to the conditions
for a flosi of infinite width. In accrodance with the definition of the
pressure coefficient P (1), the increase in velocity at different points of the mid-section of cylinders in a flow of irrfinite width and a flow of finite width may be found from the formulas
=
U
i
-co
(4)
wheie tise subscript indtcats that the given qisaistity relates to a
rnsxlci with J/b O, and the subscript b tos model with J/b > O.
Titen the influence of the flow boundaries on tise velocity at any of tise mid-section may he estimated from the velocity irscre-ment in accordance with the forsnula
The results of computations of z3.v are presented in the graph of Av(d/b) for six cylinders, two curves being given for the cylinder d =
lO mm: at velocities y 0.2 and 12.5 msec1 (Fig. G). Tise increase
20 o 20 o 40% 40 (8)
To reduce tise values of the cavitation number to the, essn'hitlons corresponding to an iii finitely wide flow, it s necessary to misc die sattle
method is for C,5. iliese relations have been plotted on the graph of (J/b) for three stages of cavitation: for the initial stage = U, for
X
4
2 o
Fig. 7
tise stage Xc = 3, and for the separated stage X = (Fig. 8). The
in-cresnents Div. determined from this graph. depend on the stage of
ca-vitation. Vor each state of cavitation there is a particular equivalent velocity for reducing x to the conditions corresponding to an infinite flow. For cylinders J = IO and 12 mm the least value of Div is that for 'c = The values of Div increase both in the direction of = O
and in the direction of X =
The results of the experiments to elucidate the effect of the boundaries on C5 und P can also be used to clarify the question of the value of the equivalent velocity in computing the Stroshal number S.
0/5
Fig. S
For developed cavitation it is possible to write for the jet bordering the cavity, from the section M to the section of maximum constriction of the jet by tise cavity Mc
P r2 P V2
1g (8)
From Eq. (8) we find the pressure coefficient at the point Mc
PPco_1
2Tq VI
(9)
analogous to the pressure coefficient at some point of the model (see
formula for P (1)).
We can put Pc = i'.,.' and then
(10)
At constant velocity y, the cavitation number will be a function of the velocity v. The number of separating cavities N
Sv/J nsust
also be found in relation to these changes its v. (lo this bassa, forexample. irs experiments to determine S for models with J =10 and 12
mm it X = U the equivalent velocity y may be equal so the velocity obtained from the extrapolation curve for . in sise .sbove-mner.tiorsed stages of cavitation. To reduce tise Reynolds tiutnber to she conditions for an infinite flow, ir is logical to take tise fisc rerssents Div equal so
the velocity increments for computing C5. 0J2 0/b
-
00 02 02 03/1
d/oo4J. AP PL. MEC II. AND TECH. PUYS., NUIBER 3
7%o 0/6 032
Fig. 5
in equivalent velocity over the profile is nonuniform. The changes in
Lsv are most considerable in tise ieading half of the prcstiie and at large
ratios Jib. This means that to correct the pressure coefficient for the effect of the flow boundaries the equivalent velocity must be found for each point of the profile. It is not permissible to use formula (2).
although for such an accurate consideration of the effect of flow width laborious experiments are required. To avoid tisese difficulties in find-ing C, by the pressure distribution method, it is necessary find C from tise ursreduced pressures, determirse the equivalent velocity va,, using
the extrapolation of C to the value J/b U, and find tise equivalent
velocity increment from the formula I C b
Av=() 1.
xoThe results of computing Div from the data of our experiments and from the data of other authors are presented in Fig. 7, in which: I de-notes experimental data obtained in our investigations, 2 the same sc-cording to Fage [2]. and 3 the same acsc-cording to Thoma [1); the broken line represents the theoretical relation of [3]. the solid curve was constructed using points 1. The relation Div(dib) is satisfactorily de-scribed by the equation of a parabola
Av=300(d/b)2. (7)
The experimental values for Div obtained by all the authors differ considerably from the theoretical values.
20 o O so o o DO Av = Ash Am',, (5) 72 /e4 Fig. C
76
By way of example. Table 2 gives the results of determining by the extrariolation method the vaincs of v fit) which must be used to re-duce the values of the parameters (first column) to the coridltlon3 at Infinity forfiow past cylinders with d = 10 and 12 mrt.
Table 2
The last row of Table 2 gives v computed from formula (2).
CONCLUSIONS
Reduction of the experimental data characterizing cavitating flow past a cylinder in the straight working chamber of a water tunnel to the conditions corresponding to a flow of infinite width using the equivalent velocity found from the continuity equation may lead to errors in determining these parameters.
The cause of these errors resides in the fact that the increase in equivalent velocity over the profile of the cylindrical model is
nonuniform and is affected not only by the flow boundaries but also
by the stage of development of cavitation, i.e. the change in the
shape and dimensions of the cavitation zone as a function of the cavita-tion number. The true value of the correccavita-tion can be found only from a series of experiments at different ratios d/b for identical Reynolds numbers and cavitation stages using the extrapolation method.
According to studies of the scale effect by the method described in [2r)1. in experiments on cavitation erosion it is necessary to use the cavItation drag for cylinders determined under identical conditions of comisirictiomi of the working chamber by the model. In comparing tle results of experiments conducted in different chambers it is necessary rake nro account the effect of a possible difference in pressure gra-dieiiis dong tise axis of the chambers on the value of the cavitation pa a octe rs
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5 September 1964 Moscow Para-meter d, mm Para-meter d. mm Is ii) 2 C 12 17.5 33 R 17.5 2-2.5 18 26 19 22 39 52 5ffllfl 17 30
