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## NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

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Title: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

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
•

2
Overview
• INTRODUCTION
• NUMERICAL MODELING AND GOVERNING EQUATIONS
• TURBULENCE MODELING
• VERIFICATION OF THE NUMERICAL RESULTS USING
EXPERIMENTAL DATA

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
2. Numerical Modeling and Governing Equations
• Continuity and Momentum Equations
• Compressible Flows
• Time-Dependent Simulations

5
2.1. Continuity and Momentum Equations
• The Mass Conservation Equation
• Momentum Conservation Equations

i-direction in a internal (non-accelerating)
reference frame
6
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

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
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
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
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
4.1. Simulation of Projectile Flight Dynamics
12
4.1. Simulation of Projectile Flight Dynamics (2)
13
4.1. Simulation of Projectile Flight Dynamics (3)
14
4.2. Hydrodynamic and Cavitation Performances
of Modified NACA Hydrofoil
• Modified NACA 4418 Hydrofoil

15
4.2. Lift Coefficient for Different Turbulence
Models
16
4.2. Pressure Coefficient Around the Blade With
and Without Cavitation
17
4.2. Lift Coefficient of the Blade With and
Without Cavitation
18
4.2. Cavitation at ?80 (Numerical Solution and
Experiment)
19
4.2. Cavitation Cloud Length(Numerical Solution
and Experiment)
20
4.2. Cavitation Inception at ?80 (Numerical
Solution)
21
4.2. Cavitation Development at ?80 (Experiment
and Numerical Solution)
22
4.2. Cavitation Development at ?160 (Experiment
and Numerical Solution)
23
4.3. CFD model of the Calculated Pump-Turbine
24
4.3. Meshing
a) b)
c)
d) e)
a) Spiral case b) Stator c) Wicket gate d)
Impeller e) Draft tube
25
4.3. Number of Mesh Elements
26
4.3. Visualization of the Vapor Development on
the Impeller (Pump Mode)
27
4.3. Results of the Cavitation Caused Efficiency
Drop (Pump Mode)
28
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

29
CONCLUSIONS
• NECESSARRY IMPROVEMENTS IN THE
• NUMERICAL MODELING INCLUDE
• Geometry description
• Flow modeling
• Boundary layer modeling
• Boundary conditions
• Secondary flow effect