markov - Conference on PDE Methods in Applied Mathematics...

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  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 Faculty of Mechanical Engineering  University in Skopje, Macedonia Joint research Predrag Popovski University in Skopje, Macedonia Andrej Lipej Turboinstitut, Slovenia  
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Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES 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 
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Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES 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
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Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES and Image Processing, Sunny Beach, Bulgaria, 2004 4 2. Numerical Modeling and      Governing Equations Continuity and Momentum Equations Compressible Flows Time-Dependent Simulations
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Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES and Image Processing, Sunny Beach, Bulgaria, 2004 5 2.1. Continuity and Momentum Equations The Mass Conservation Equation Momentum Conservation Equations ( ) ( ) ij i i j i i j i j p u u u g F t x x x τ ρ + = - + + + 2 3 j i l ij ij j i l u u u x x x μ δ = + - i-direction in a internal (non-accelerating) reference frame: ( 29 m i i S u x t = +
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Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES 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 : 1 2 0 1 1 2 s p M p γ - - = + 2 1 1 2 o s T M T γ- = + The Compressible Form of Gas Law ideal gas law: ( ) / op s p p RT ρ= +
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Z. Markov: NUMERICAL APPROACH IN SOLVING
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markov - Conference on PDE Methods in Applied Mathematics...

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