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Title:
SHAPE OPTIMIZATION OF AXISYMMETRIC CAVITATOR
IN PARTIALY CAVITATING FLOW
Department of Mechanical Engineering
Ferdowsi University of Mashhad
Presented by:
Mahmoud Pasandideh fard
Preface:
 Introduction
►
Governing Equation
► Modeling the Cavitator
► Selection Process of the Optimum Design
Cavitator
► Results
Introduction
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The drag reduction of submerged projectiles is a problem that has
attracted the attention of many researchers.
The formation of cavitation due to the lower viscosity of vapor phase
compared to the liquid phase, has been taken into consideration as a drag
reduction technique.
1- Partial Cavitation
Cavitation regimes:
2- Supercavitation
During flight when maneuvering of the vehicle is necessary,
the partial cavitation may also occur.
Introduction
The cavitation phenomenon has been studied extensively in the literature:
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Early analytical study of the cavitation was conducted by Efros (1946)
who analyzed the supercavitating flow using the conformal mapping .
Uhlman (1987, 1989) developed the nonlinear boundary element model
(BEM) for cavitating flow and successfully used the method for both
partially cavitating and supercavitating flows about the two dimensional
hydrofoils .
Varghese et al. (2005) used the BEM to investigate the partial cavitation
about the axisymmetric bodies.
Rashidi et al. (2008) solved the partial and supercavitating flows using
both the BEM and the VOF models.
Introduction
Since the cavitator drag plays a significant role in calculating the total
drag of the body, obtaining the optimum cavitator such that at a given
cavitation number the total drag coefficient becomes minimum, is the
main objective in this study.
There are only few studies that considered the optimization of
the cavitation phenomenon on axisymmetric bodies.
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Choi et al. (2005) analyzed the axisymmetric cavitator for
supercavitating flows using the Boundary Value Problem (BVP) based
on the potential flow.
Shafaghat et al. (2008, 2010) investigated both the two-dimensional and
axisymmetric cavitators in supercavitating flow conditions and generated
different cavitators using the same parabolic relation as of Choi et al.
Governing Equation
1) Boundary Element Method
Green third identity:


2 ( x)    [ ( x) G ( x, x)  G ( x, x)  ( x)]dS ( x)
S
n
n
φ : The disturbance velocity potential
G: The source ring of unit strength
G
: The dipole ring of unit strength
n
G( x, x ) 
1
x  x
Governing Equation
 Boundary Conditions
1) The dynamic boundary condition

 1    sx
n
The dynamic boundary condition derived based on the
Bernoulli equation is imposed on the cavity interface.
2) The kinematic boundary condition

 n x
n
The kinematic boundary condition is based on the fact that there
is no flow crossing the body surface or the cavity interface.
Governing Equation
2) Numerical Solution
It is assumed that :
1) The fluid is a mixture of vapor, liquid and non condensable gas.
2) The flow field is also assumed to be homogeneous.
 The Reynolds average Navier stokes equations are applied to
solve the flow field and the Reynolds stress model (RSM) is used
as the turbulence model.
The boundary conditions used in the numerical
simulations.
Modeling the Cavitator
A parabolic relation as of Choi et al. is used for the curve fitting:
(1   ) 2 Z 1  2 (1   ) Z 2 w2   2 Z 3
Z ( ) 
(1   ) 2  2 (1   ) w2   2
Z1  (b1 ,0), Z 2  (b1 , b2 ), Z3  (0,0.5)
Selection Process of the Optimum Design
Cavitator
Important parameters to optimize the cavitator at a constant cavitation number are:
1) The total drag coefficient of the projectile, CD.
2) The geometric parameters of the cavitator, b1 , b2.
3) The weighing parameter w2.
For modeling actual cavitators:
0  b2  Z 3
0  b1  1
0  w2  2
Selection Process of the Optimum Design
Cavitator
For a certain shape of the projectile at each cavitation number of 0.1, 0.12
and 0.15, the optimum cavitator is obtained as follows:
1.
2.
3.
4.
5.
First, by varying the geometric parameters, a large number of cavitators are
generated (For a certain shape of the projectile and a specific value of
cavitation number, close to 10,000 cavitators).
Solving the fluid flow over these cavitators using the BEM method, the total
drag coefficient (CD) is calculated.
The cavitator with a minimum CD is optimal.
In the next step, several cavitators with a total drag coefficient close to that of
the optimized cavitator are also simulated using the CFD code to examine the
optimization results.
The optimum cavitator is finally selected based on both the BEM and CFD
simulations.
Results
1) Model Validation
Figure 1: The variation of the cavity length versus
the cavitation number from the BEM method, CFD
simulations and experiments.
Figure 2: The variation of the total drag coefficient
versus the cavitation number from the BEM method,
CFD simulations and experiments.
Results
2) The Effect of the Projectile Radius
Figure 3: The optimum shapes of the cavitators at
different cavitation numbers for a dimensionless
projectile radius of 0.7 and dimensionless conical
section length of 5.
Figure 4: The optimum shapes of the cavitators at
different cavitation numbers for a dimensionless
projectile radius of 0.9 and dimensionless conical
section length of 5.
Results
2) The Effect of the Projectile Radius
Figure 5: The optimum shapes of the cavitators at
different cavitation numbers for a dimensionless
projectile radius of 1.1 and dimensionless conical
section length of 5.
Table 1. The optimized cavitator results for various
projectile dimensionless radii at different cavitation
numbers for a dimensionless conical length of 5.
Results
3) Optimum Cavitator vs. Disk Cavitator for the Base Case
Figure 6: The comparison of the cavity shape and length of
a) the disk cavitator with that of
b) the optimum cavitator
at a cavitation number of σ=0.12 for a dimensionless body
radius of 0.9 and dimensionless conical length of 5.
Results
3) Optimum Cavitator vs. Disk Cavitator for the Base Case
Figure 7: Drag coefficients versus the cavitation number for the
optimum cavitator as compared with those of a disk cavitator.
Results
3) Optimum Cavitator vs. Disk Cavitator for the Base Case
Figure 8: Variation of the pressure coefficient distributions on the
optimum and disk cavitators from the BEM and CFD methods at a
cavitation number of 0.12 for a dimensionless body radius of 0.9
and dimensionless conical length of 5.
Conclusion:
 The results show that for all cavitation numbers, the cavitator that creates the
cavity covering the conical portion of the projectile with a minimum drag
coefficient is optimal.
 Increasing the cavitation number causes the optimum cavitator to have a nose
and approaches the disk cavitator.
Although the optimum cavitator produce a smaller cavity and more frictional drag
than the disk cavitator, the serious decrease of the pressure drag coefficient caused
by such cavitators, leads to a significant reduction of the total drag coefficient.