Title: SHAPE OPTIMIZATION OF AXISYMMETRIC CAVITATOR IN PARTIALY CAVITATING FLOW

1. Title: SHAPE OPTIMIZATION OF AXISYMMETRIC CAVITATOR IN PARTIALY CAVITATING FLOW Department of Mechanical Engineering Ferdowsi University of Mashhad Presented by: Mahmoud Pasandideh fard

2. Preface: • Introduction • Governing Equation • Modeling the Cavitator • Selection Process of the Optimum Design Cavitator • Results

3. Introduction • 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. Cavitation regimes: During flight when maneuvering of the vehicle is necessary, the partial cavitation may also occur. 1- Partial Cavitation 2- Supercavitation

4. Introduction The cavitation phenomenon has been studied extensively in the literature: • 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.

5. 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. • 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. There are only few studies that considered the optimization of the cavitation phenomenon on axisymmetric bodies.

6. Governing Equation 1) Boundary Element Method Green third identity: φ : Thedisturbance velocity potential G: The source ring of unit strength : The dipole ring of unit strength

7. Governing Equation • Boundary Conditions 1) The dynamic boundary condition The dynamic boundary condition derived based on the Bernoulli equation is imposed on the cavity interface. 2) The kinematic boundary condition The kinematic boundary condition is based on the fact that there is no flow crossing the body surface or the cavity interface.

8. 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.

9. Modeling the Cavitator A parabolic relation as of Choi et al. is used for the curve fitting:

10. Selection Process of the Optimum Design Cavitator • Important parameters to optimize the cavitator at a constant cavitation number are: • The total drag coefficient of the projectile, CD. • The geometric parameters of the cavitator, b1 , b2. • The weighing parameter w2. For modeling actual cavitators:

11. 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: 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.

12. 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.

13. 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.

14. Results 2) The Effect of the Projectile Radius Table 1. The optimized cavitator results for various projectile dimensionless radii at different cavitation numbers for a dimensionless conical length of 5. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. Thanks