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NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES. Zoran Markov Faculty of Mechanical Engineering University in Skopje, Macedonia Joint research Predrag Popovski University in Skopje, Macedonia Andrej Lipej Turboinstitut, Slovenia. INTRODUCTION

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NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

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  1. NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Zoran Markov Faculty of Mechanical Engineering University in Skopje, Macedonia Joint research Predrag Popovski University in Skopje, Macedonia Andrej Lipej Turboinstitut, Slovenia Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  2. INTRODUCTION NUMERICAL MODELING AND GOVERNING EQUATIONS TURBULENCE MODELING VERIFICATION OF THE NUMERICAL RESULTS USING EXPERIMENTAL DATA Overview Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  3. 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 1. Introduction Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  4. Continuity and Momentum Equations Compressible Flows Time-Dependent Simulations 2. Numerical Modeling and Governing Equations 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 Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 • i-direction in a internal (non-accelerating) reference frame:

  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 : Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 The Compressible Form of Gas Law • ideal gas law:

  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 Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 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. 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 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: Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 The "eddy" or turbulent viscosity, t, is computed by combining k and  as follows:

  10. Simulation of Projectile Flight Dynamics Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil Cavitation Performances of Pump-turbine 4. Verification Of The Numerical Results Using Experimental Data Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  11. 4.1. Simulation of Projectile Flight Dynamics Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  12. 4.1. Simulation of Projectile Flight Dynamics (2) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  13. 4.1. Simulation of Projectile Flight Dynamics (3) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  14. Modified NACA 4418 Hydrofoil cevka 4.2. Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  15. 4.2. Lift Coefficient for Different Turbulence Models 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 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 Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  18. 4.2. Cavitation at =80(Numerical Solution and Experiment) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  19. 4.2. Cavitation Cloud Length(Numerical Solution and Experiment) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  20. 4.2. Cavitation Inception at =80(Numerical Solution) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  21. 4.2. Cavitation Development at =80(Experiment and Numerical Solution) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  22. 4.2. Cavitation Development at =160(Experiment and Numerical Solution) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  23. 4.3. CFD model of the Calculated Pump-Turbine Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

  24. 4.3. Meshing a) b) c) Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004 d) e) a) Spiral case b) Stator c) Wicket gate d) Impeller e) Draft tube

  25. No. of elements Pcs. Total Spiral case 110.328 1 110.328 Stator channel 16.626 16 266.016 Wicket gate channel 17.918 16 286.668 Impeller 33.138 7 231.966 Draft tube 77.350 1 77.350 Total: 1.008.305 4.3. Number of Mesh Elements 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) 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) 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) Minimal flow discharge Mode between minimal and optimal mode Optimal mode Maximal flow discharge 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 Conference on PDE Methods in Applied Mathematics and Image Processing, Sunny Beach, Bulgaria, 2004

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