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 Predrag Popovski Ph.D. Faculty of Mechanical Engineering University in Skopje, Macedonia

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, Overview INTRODUCTION NUMERICAL MODELING AND GOVERNING EQUATIONS TURBULENCE MODELING VERIFICATION OF THE NUMERICAL RESULTS USING EXPERIMENTAL DATA

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, 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

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, Numerical Modeling and Governing Equations Continuity and Momentum Equations Compressible Flows Time-Dependent Simulations

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, Continuity and Momentum Equations The Mass Conservation Equation Momentum Conservation Equations i-direction in a internal (non-accelerating) reference frame:

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, 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:

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, 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

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, 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

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, 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:

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, 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

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, Simulation of Projectile Flight Dynamics

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, Simulation of Projectile Flight Dynamics (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, Simulation of Projectile Flight Dynamics (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, Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil cevka Modified NACA 4418 Hydrofoil

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, Lift Coefficient for Different Turbulence Models

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, Pressure Coefficient Around the Blade With and Without Cavitation

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, Lift Coefficient of the Blade With and Without Cavitation

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, Cavitation at =8 0 (Numerical Solution and Experiment)

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, Cavitation Cloud Length (Numerical Solution and Experiment)

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, Cavitation Inception at =8 0 (Numerical Solution)

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, Cavitation Development at =8 0 (Experiment and Numerical Solution)

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, Cavitation Development at =16 0 (Experiment and Numerical Solution)

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, CFD model of the Calculated Pump-Turbine

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, Meshing a) Spiral case b) Stator c) Wicket gate d) Impeller e) Draft tube a) b) c) d) e)

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, Number of Mesh Elements No. of elementsPcs.Total Spiral case Stator channel Wicket gate channel Impeller Draft tube Total:

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, Visualization of the Vapor Development on the Impeller (Pump Mode)

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, Results of the Cavitation Caused Efficiency Drop (Pump Mode)

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, 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

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, 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 NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES Zoran Markov M.Sc. Faculty of Mechanical Engineering University in Skopje, Macedonia Predrag Popovski Ph.D. Faculty of Mechanical Engineering University in Skopje, Macedonia