1. Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Zoran Markov M.Sc. Faculty of Mechanical Engineering University in Skopje, Macedonia mzoran@mf.ukim.edu.mk Predrag Popovski Ph.D. Faculty of Mechanical Engineering University in Skopje, Macedonia predrag@mf.ukim.edu.mk

2. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 2 Overview INTRODUCTION NUMERICAL MODELING AND GOVERNING EQUATIONS TURBULENCE MODELING VERIFICATION OF THE NUMERICAL RESULTS USING EXPERIMENTAL DATA

3. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 3 1. Introduction Solving the PDE equations in fluid dynamics has proved difficult, even impossible in some cases Development of numerical approach was necessary in the design of hydraulic machinery Greater speed of the computers and development of reliable software Calibration and verification of all numerical models is an iterative process

4. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 4 2. Numerical Modeling and Governing Equations Continuity and Momentum Equations Compressible Flows Time-Dependent Simulations

5. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 5 2.1. Continuity and Momentum Equations The Mass Conservation Equation Momentum Conservation Equations i-direction in a internal (non-accelerating) reference frame:

6. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 6 2.2. Compressible Flows When to Use the Compressible Flow Model? M<0.1 - subsonic, compressibility effects are negligible M1- transonic, compressibility effects become important M>1- supersonic, may contain shocks and expansion fans, which can impact the flow pattern significantly Physics of Compressible Flows total pressure and total temperature : The Compressible Form of Gas Law ideal gas law:

7. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 7 2.3. Time-Dependent Simulations Temporal Discretization Time-dependent equations must be discretized in both space and time A generic expressions for the time evolution of a variable is given by where the function F incorporates any spatial discretization If the time derivative is discretized using backward differences, the first-order accurate temporal discretization is given by second-order discretization is given by

8. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 8 3. Turbulence Modeling Standard CFD codes usually provide the following choices of turbulence models: Spalart-Allmaras model Standard k-  model Renormalization-group (RNG) k-  model Realizable k-  model Reynolds stress model (RSM) Large eddy simulation (LES) model

9. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 9 Transport Equations for Standard k- model The turbulent kinetic energy, k, and its rate of dissipation, , are obtained from the following transport equations: The "eddy" or turbulent viscosity,  t, is computed by combining k and  as follows:

10. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 10 4. Verification Of The Numerical Results Using Experimental Data Simulation of Projectile Flight Dynamics Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil Cavitation Performances of Pump-turbine

11. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 11 4.1. Simulation of Projectile Flight Dynamics

12. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 12 4.1. Simulation of Projectile Flight Dynamics (2)

13. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 13 4.1. Simulation of Projectile Flight Dynamics (3)

14. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 14 4.2. Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil cevka Modified NACA 4418 Hydrofoil

15. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 15 4.2. Lift Coefficient for Different Turbulence Models

16. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 16 4.2. Pressure Coefficient Around the Blade With and Without Cavitation

17. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 17 4.2. Lift Coefficient of the Blade With and Without Cavitation

18. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 18 4.2. Cavitation at =8 0 (Numerical Solution and Experiment)

19. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 19 4.2. Cavitation Cloud Length (Numerical Solution and Experiment)

20. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 20 4.2. Cavitation Inception at =8 0 (Numerical Solution)

21. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 21 4.2. Cavitation Development at =8 0 (Experiment and Numerical Solution)

22. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 22 4.2. Cavitation Development at =16 0 (Experiment and Numerical Solution)

23. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 23 4.3. CFD model of the Calculated Pump-Turbine

24. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 24 4.3. Meshing a) Spiral case b) Stator c) Wicket gate d) Impeller e) Draft tube a) b) c) d) e)

25. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 25 4.3. Number of Mesh Elements No. of elementsPcs.Total Spiral case110.3281 Stator channel16.62616266.016 Wicket gate channel 17.91816286.668 Impeller33.1387231.966 Draft tube77.3501 Total:1.008.305

26. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 26 4.3. Visualization of the Vapor Development on the Impeller (Pump Mode)

27. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 27 4.3. Results of the Cavitation Caused Efficiency Drop (Pump Mode)

28. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 28 4.3. Analyses of the Flow in the Draft Tube - Stream Lines Distribution (Turbine Mode) a) Minimal flow discharge b) Mode between minimal and optimal mode c) Optimal mode d) Maximal flow discharge

29. Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 29 CONCLUSIONS NECESSARRY IMPROVEMENTS IN THE NUMERICAL MODELING INCLUDE: Geometry description Flow modeling Boundary layer modeling Boundary conditions Secondary flow effect

30. Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Zoran Markov M.Sc. Faculty of Mechanical Engineering University in Skopje, Macedonia mzoran@mf.ukim.edu.mk Predrag Popovski Ph.D. Faculty of Mechanical Engineering University in Skopje, Macedonia predrag@mf.ukim.edu.mk