Some significant differences between fully developed cavity flows and their corresponding potential flow models are described and an attempt is made to interpret the results in terms of the real fluid properties.  The phenomenon of cavity separation from a smooth surface and the nature and appearance of the cavity interface are given particular attention

Brennen, C.

English

Caltech Authors - Main

Some Viscous 
in Fully 
and Otber Real Fluid Effects 
Developed Cavity Flows 
Discussion by C. Brennen 
Cal i fornia Inst i tute o f  Technology ' 
Pasadena, Cal i fornia 
Some significant differences between fully developed cavity flows and their corres- 
ponding potential flow models are described and an attempt is made to interpret the 
results in terms of the real fluid properties. The phenomenon of cavity separation from a 
smooth surface and the nature and appearance of the cavity interface are given particular 
attention. 
Introduction 
Studies of real fluid effects upon partially or fully developed cavity flow have been 
receiving more attention of late due to the many interesting comparisons of potential 
and inviscid flow theory with experiment. The intention of this short paper is to  briefly 
describe some of the effects observed during a series of water tunnel experiments with 
axisymmetric headforms (some of which have been reported in detail elsewhere) and to 
surmise qualitatively on some of the general implications and effects. The following 
section looks at the phenomenon of separation; the Section on "Cavity Surface Appearance 
and the Nature of the Interfacial Boundary Layer" reviews the appearance of the cavity 
surface and the closure region. Finally the last Section "Effects of 'Drag-Reducing' Polymer 
Added to the Water" introduces some of the rather surprising effects observed when small 
quantities of "drag-reducing" polymer were dissolved in the water of one of the tunnels 
used. 
Cavitation Separation from a Smooth Surface 
When separation occurs at a sharp projecting corner on a headform, potential flow theory 
adequately predicts both the wetted surface pressure distribution and the shape of the 
cavity for some distance downstream in the majority of flows of practical interest (see 
for example Ref. [6] ). When there is no such corner potential flow solutions necessarily 
employ the condition of "smooth separation" in order to locate the position of separation. 
I f  the headform is a regular "streamlined" shape, then the wetted surface pressure gradient 
is usually favorable everywhere except at separation where it approaches zero. But in a 
real flow the wetted surface boundary layer would be expected to separate only in the 
presence of a finite adverse pressure gradient and at the point of zero wall shear stress. 
How in practice this inner boundary layer flow is "matched" with the outer potential 
flow is best illustrated by an example. 
The experimentally observed position of separation from a sphere which is fully 
cavitating lies downstream of that predicted by inviscid theory (see plate I ) ,  the differ- 
ence increasing as the Reynolds number is reduced [8,2 and 31 . The results of these 
references in which spheres of diameters ranging from Yi in. to 3 in. were used indicate 
that the position depends primarily upon cavitation number, U, and Reynolds number, 
Re (Fig. 1 was compiled using those results). That the dependence upon Weber number is 
at most secondary, despite the appearance of a significant "separation meniscus" at low 
Re (see plate 7), is supported by some results with a solution having a much reduced 
- 
surface tension [2] . 
On the other hand measurements by Rouse and McNown [ l o ]  and Konstantinov [9] 
indicate that the pressure on the wetted surface, p, does in fact reach its minimum around 
the theoretical separation position. This suggests that the pressure in the region between 
the theoretical and observed separation position is roughly constant at the cavity value,P, 
(This interpretation is also consistent with the fact that the delayed separation in ex- 
periment does not lead to higher values of the total drag on the headform; i f  anything 
u =0.5 
----- 
THEORETICAL POTENTIAL 
60 - FLOW VALUES 
I oa0.1 
I 04 I 05 
Re 
1 o6 
FIG. 1 VARIATION I N  THE POSITION OF SEPARATION FROM 
A SPHERE WITH CAVITATION NUMBER, a, AND 
REYNOLDS NUMBER, Re, BASED ON UPSTREAM 
VELOCITY AND SPHERE DIAMETER. 
PLATE 1. A N  ENLARGEMENT OF A MICROFLASH PHOTO- 
GRAPH OF THE FLOW I N  THE NEIGHBOURHOOD 
OF SEPARATION FROM A 3in. SPHERE WHEN UP- 
STREAM VELOCITY IS 45 ftlsec AND o = .15. 
SUPERIMPOSED UPON THIS ISTHE CORRES- 
PONDING CAVITY PROFILE FROM POTENTIAL 
FLOW THEORY IN  WHICH THE "SMOOTH 
SEPARATION" POSITION IS AS INDICATED. 
the reverse appears to be the case). Viscous separation may then be effected by the 
existence in this region of pressures slightly below that of the cavity. Provided that the 
residence time of an element of fluid in this region of p< p is small cavitation nuclei 
will not appear. C 
Conversely there will be other cavity flows in which this residence time is sufficient 
for significant growth of cavitation nuclei; these may then grow and combine to initiate 
the  large scale cavity. A photograph of a cavitating hydrofoil (plate 2, reproduced, as is 
plate 3, by permission of Professor A. J. Acosta and V. Arakeri of the California Institute 
of Technology, under Office of Naval Res. Contract), typifies this type qf "nucleate 
separation" as opposed to the "smooth viscous cavitation separation" from the sphere 
(plate 1). Transition of the boundary layer prior to "separation" would also effect the 
issue, delaying possible viscous separation whilst enhancing the chances of nuclei growth. 
On the other hand a quite small amount of cavity ventilation may raisep, sufficiently t o  
eliminate the existence of a possible nuclei growth region. Finally plate 3 is included to 
illustrate an interesting and presumably intermediate form of cavity separation. 
Cavity Surface Appearance and the Nature of the Interfacial Boundary Layer 
During the water tunnel experiments with spheres and other axisymmetric headforms 
(Brennen [2 and 41 ) a study was also made of the appearance of the cavity surface 
PLATE 2. U g 4 O  ftlsec, u S O  19 
CAVITATION OF A THIN BI-CONVEX HYDROFOIL NEAR 
ZERO ANGLE OF ATTACK. U IS THE UPSTREAM VELOCITY, 
following separation. Immediately after that point the surface invariably appeared smooth 
and glassy, suggesting a laminar boundary layer at separation even at the highest Reynolds 
numbers (9 X 10' in the case of the 3 in. sphere). With most headforms (eg. plate 4) a 
system of waves appeared a short distance from separation, and growth in amplitude of 
these waves during convection downstream normally led to wave break up and a turbulent 
interfacial boundary layer along the rest of the length of the cavity. As the tunnel speed 
was reduced both the wavelength and the distance from separation to the point of wave 
break up increased. By employing ventilation so as to produce fully developed cavities at 
further reduced .tunnel velocities a point was reached atwhich break up ceased to occur 
and the waves persisted along the length of the cavity (plate 5). These wave patterns can 
be fairly satisfactorily ascribed to the growth of an instability in the separated laminar 
boundary layer, the observed frequency being close to that which has the maximum spatial 
144 
PLATE 6. U 35 ftlsec 
WAVE PATTERNS O N  T H E  SURFACE O F  CAVIT IES BEHIND 
A PART-SPHERICAL H E A D F O R M  OF BASE DIAMETER 2.34 in. 
A N D  A 3 in. D IAMETER DISC. (U IS T H E  UPSTREAM VELOCITY) .  
amplification rate in the layer at or shortly after separation [41 . Among the variously 
shaped headforms tested only the 3 in. disc produced cavities which were completely 
clear and glassy at all possible speeds (plate 6). 
The nature of the interfacial boundary layer can be seen to have at least one interesting 
implication. I f  that layer i s  turbulent then for most practical flows the rate of diffusion 
of dissolved air from the water into the cavity will be much greater than that given by 
diffusion within a purely "potential" flow, See Ref. [ I ] .  (The same will be true of the 
diffusion of heat required to produce vaporization at the interface). However, in the 
PLATE 7. PLATE 8. 
VENTILATED CAVITIES BEHIND A '/4 in. DIAMETER SPHERE IN 
WATER AND I N  A 50 PPM AQUEOUS SOLUTION OF UNION 
CARBIDE WSR 301 "POLYOX". UPSTREAM VELOCITY 12 ftlsec. 
steady state, this mass rate of diffusion of air into the cavity must be matched by the rate 
at which air is entrained away from the cavity in the wake of the highly turbulent closure 
region. In a recent paper the author compared experimentally measured entrainment rates 
with theoretically estimated inward rates of diffusion of dissolved air for cavities with 
a turbulent interface behind a particular headform (Brennen (1968a)). A "semi-theore- 
tical" linear relationship between the partial pressure of air in the cavity,pcA and the air 
content of the water, P, was thus obtained and this compared favorably with the experi- 
mental graph of these quantities. Gadd and Grant 171 found a very much lower value 
;f pcp/P in their experiments with a 5 in. diameter disc, presumably because the 
lamtnar" interface produced by that headform gave rise to much smaller values of the 
inward diffusion rate. 
Effects of "Drag-Reducing" Polymer Added t o  the Water 
Plates 7 and 8 have been included to highlight the principal effect which the additon 
of 50 PPM. of the long chain molecular polymer, Polyox (Union Carbide WSR 301) 
has upon fulty developed cavity flows. The cavities in these cases had to be produced by 
ventilation due to the low tunnel velocity (12 ftlsec). In the case of the water flow an 
increase to around 16 ftlsec in the tunnel velocity was required before the instability 
waves described in the last section appeared, It seems clear from plate 8 that the effect 
of the polymer is to produce an instability in the flow around the headform which is then 
reflected in the surprisingly regular distortion of the separation line. Other similar 
additives such as Dow Separan AP 30 and guar gum produced similar effects (Brennen 
P I  1. 
Concluding Remarks 
In conclusion it may be useful to suggest a few particular areas in which further 
research is indicated: 
(i) Since much of the latter part of Section 2 is conjectural further experimental 
studies of separation would clarify issues. Headform surface roughness and the gas 
nuclei content of the "free stream" may also be significant in this respect. 
(ii) An analysis of the flow in the immediate neighborhood of a viscous cavitation 
separation point, perhaps along the lines of the treatment given to noncavitating 
separation by Catherall and Mangler (1966) would be interesting, though the problem 
may be complicated by the liquid contact angle. 
(iii) The real flow in the region of cavity closure and in the cavity wake is normally 
marked by considerable turbulence and at least small scale unsteadiness. Further studies 
would increase understanding of the turbulent mixing processes (for example the work 
of Young and Holl (1966) i s  a welcome contribution) and entrainment mechanism 
(see Section 3) in this region. 
Acknowledgement 
This work has been prepared partly under the support of the Office of Naval Research. 
Plates 1,4,5,6,7 and 8 are reproduced by permission of the Director, Ship Division, 
National Physical Laboratory. 
References 
[ I  I Brennen. C. 1968a. The dynamic balances of dissolved air and heat in natural cavity flows. 
N.P.L. Ship Div. Rep. No. 115. 
[2] Brennen, C. 1968b. Some cavitation experiments with dilute polymer solution. N.P.L. Ship 
Div. Rep. No. 123. 
[3] Brennen. C. 1969a. A numerical solution of axisymmetric cavity flows (to appear shortly 
in J. Fluid Mech.). 
[4] Brennen, C. 1969b. Wave patterns on the surface of hydrodynamic cavities. (Manuscript 
in preparation.) 
[5] Catherall, D. and Mangler, K. W. (1966). The integration of the two dimensional laminar 
boundary layer equations past the point of vanishing skin friction. J. Fluid Mech., 26, 1. 
[61 Fage, A. and Johansen, F. C. 1927. On the flow of air behind an inclined flat plate of infinite 
space. Proc. Roy. Soc. (London) A1  16, 170-197. 
[7] Gadd, G. E. and Grant, S. 1965. Some experiments on cavities behind disks. J. Fluid Mech. 
23.4. 
[81 Hsu, E-Y. and Perry, B. 1954. Water tunnel experiments on spheres in cavity flow. Calif. Inst. 
of Tech. Rep. No. E-24.9. 
[9] Konstantmov, V. A. 1950. Influence of Reynolds number on the separation (cavitation) 
flow. David Taylor Model Basin Translation, No. 233. 
[ l o ]  Rouse, H. and McNown, J. M. 1948. Cavitation and pressure distribution: headformsat 
zero angles of yaw. State Univ. of Iowa, Studies in Engineering, Bull. 32. 
[ I l l  Young, J. 0. and Holl, J. W. (1966). Effects of cavitation on periodic wakes behind sym- 
metric wedges. ASME Trans. J. Basic Eng.88 D, 163-176. 


Some Viscous and Other Real Fluid Effects in Fully Developed Cavity Flows

A numerical solution of axisymmetric cavity flows (to appear shortly in J. Fluid Mech.).

Brennen, C. 1968b. Some cavitation experiments with dilute polymer solution. N.P.L. Ship Div. Rep. No. 123.

Brennen, C. 1969b. Wave patterns on the surface of hydrodynamic cavities. (Manuscript in preparation.)

Brennen. C. 1969a. A numerical solution of axisymmetric cavity flows (to appear shortly in J. Fluid Mech.).

Cavitation and pressure distribution: headformsat zero angles of yaw.

Gadd, G. E. and Grant, S. 1965. Some experiments on cavities behind disks. J. Fluid Mech. 23.4. [81 Hsu, E-Y. and Perry, B. 1954. Water tunnel experiments on spheres in cavity flow. Calif. Inst. of Tech. Rep. No. E-24.9.

I I Brennen. C. 1968a. The dynamic balances of dissolved air and heat in natural cavity flows. N.P.L. Ship Div. Rep. No. 115.

Influence of Reynolds number on the separation (cavitation) flow. David Taylor Model Basin Translation,

Konstantmov, V. A. 1950. Influence of Reynolds number on the separation (cavitation) flow. David Taylor Model Basin Translation, No. 233.

Some cavitation experiments with dilute polymer solution.

Some experiments on cavities behind disks.

The dynamic balances of dissolved air and heat in natural cavity flows.

The integration of the two dimensional laminar boundary layer equations past the point of vanishing skin friction.

Wave patterns on the surface of hydrodynamic cavities.

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Some Viscous and Other Real Fluid Effects in Fully Developed Cavity Flows

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