NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

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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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Overview

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

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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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2. Numerical Modeling and Governing Equations

• Continuity and Momentum Equations
• Compressible Flows
• Time-Dependent Simulations

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2.1. Continuity and Momentum Equations

• The Mass Conservation Equation

• Momentum Conservation Equations

i-direction in a internal (non-accelerating)
reference frame
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2.2. Compressible Flows

• When to Use the Compressible Flow Model?
• Mlt0.1 - subsonic, compressibility effects are
  negligible
• M?1- transonic, compressibility effects become
  important
• Mgt1- 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

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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

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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

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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
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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

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4.1. Simulation of Projectile Flight Dynamics
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4.1. Simulation of Projectile Flight Dynamics (2)
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4.1. Simulation of Projectile Flight Dynamics (3)
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4.2. Hydrodynamic and Cavitation Performances
of Modified NACA Hydrofoil

• Modified NACA 4418 Hydrofoil

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4.2. Lift Coefficient for Different Turbulence
Models
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4.2. Pressure Coefficient Around the Blade With
and Without Cavitation
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4.2. Lift Coefficient of the Blade With and
Without Cavitation
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4.2. Cavitation at ?80 (Numerical Solution and
Experiment)
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4.2. Cavitation Cloud Length(Numerical Solution
and Experiment)
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4.2. Cavitation Inception at ?80 (Numerical
Solution)
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4.2. Cavitation Development at ?80 (Experiment
and Numerical Solution)
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4.2. Cavitation Development at ?160 (Experiment
and Numerical Solution)
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4.3. CFD model of the Calculated Pump-Turbine
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4.3. Meshing
a) b)
c)
d) e)
a) Spiral case b) Stator c) Wicket gate d)
Impeller e) Draft tube
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4.3. Number of Mesh Elements
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4.3. Visualization of the Vapor Development on
the Impeller (Pump Mode)
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4.3. Results of the Cavitation Caused Efficiency
Drop (Pump Mode)
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4.3. Analyses of the Flow in the Draft Tube-
Stream Lines Distribution (Turbine Mode)

1. Minimal flow discharge
2. Mode between minimal and optimal mode
3. Optimal mode
4. Maximal flow discharge

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CONCLUSIONS

• NECESSARRY IMPROVEMENTS IN THE
• NUMERICAL MODELING INCLUDE
• Geometry description
• Flow modeling
• Boundary layer modeling
• Boundary conditions
• Secondary flow effect