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

Z. Markov: NUMERICAL APPROACH IN SOLVING THE PDE FOR PARTICULAR FLUID DYNAMICS CASES

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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

( ) ( ) iji i j i i

j i j

pu u u g F

t x x x

2

3ji l

ij ijj i l

uu u

x x x

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

mii

Suxt


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2.2. Compressible Flows

When to Use the Compressible Flow Model? M<0.1 - subsonic, compressibility effects are negligible M1- 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 :

120 1

12s

pM

p

211

2o

s

TM

T

The Compressible Form of Gas Law ideal gas law: ( ) /op sp p RT


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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

( )Ft

1

( )n n

Ft

1 13 4( )

2

n n n

Ft

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

tk b M

i k i

μDk kρ μ+ G G ρε-Y

Dt x σ x

The turbulent kinetic energy, k, and its rate of dissipation, , are obtained from the following transport equations:

2

t1ε k 3ε b 2ε

i ε i

μDε ε ε ερ μ+ C G C G C ρ

Dt x σ x k k

2

t μ

kμ =ρC

ε

The "eddy" or turbulent viscosity, t, is computed by combining k and as follows:

C1=1.44; C2=1.92; C=0.09; k=1.0 and =1.3


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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

Part/Subassembly: Multiple

Mass: 14.968Kg

Volume: 3140185.9077mm̂3

Centroid: X: 307.1229

Y: 0

Z: 0

Moments of inertia: X: 22273.0654

Y: 1641718.8812

Z: 1641715.2464

Products of inertia: XY: 0

XZ: 0

YZ: 0

Radii of Gyration: X: 38.5752

Y: 331.1828

Z: 331.1824

Principal moments and X-Y-Z directions about centroid:

I: 22273.0654 along [1,0,0]

J : 229871.3222 along [0,1,0]

K: 229867.6875 along [0,0,1]

Projectile 105 mm M1 - mass analysis


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4.1. Simulation of Projectile Flight Dynamics (2)

Comparison of the projectile trajectory elements obtained as a result of thesimulation with those given in the range tables

Initial conditions:

Elevation angle 0

Muzzle velocity V0

Method forobtaining of the

results

Max

imum

ran

ge[m

]

Tim

e of

fli

ght

[sec

]

Max

imum

heig

ht[m

]

Ang

le o

f im

pact

on t

he g

roun

d[]

Impa

ct v

eloc

ity

[m/s

]

Simulation 3069 12.8 201 15.4 231

Range tables 3100 12.9 203 15.6 233

0 = 14

V0 = 267 m/sError [] 0.1 0.77 0.98 1.28 0.85

Simulation 3568 17.6 381 24.3 205

Range tables 3600 17.7 383 24.6 207

0 = 22

V0 = 238 m/sError [] 0.88 0.56 0.52 1.21 0.96

Simulation 7794 31 1188 34.7 253

Range tables 8100 31.5 1229 35 259

0 = 28

V0 = 376 m/sError [] 1.3 1.58 3.33 0.85 2.31


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Maximum range comparison

0

2000

4000

6000

8000

10000

10 15 20 25 30

Elevation angle [deg]

Ma

x.

ran

ge

[m

]

Simulation Range tables

4.1. Simulation of Projectile Flight Dynamics (3)

Comparison of the projectile's maximum range


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4.2. Hydrodynamic and Cavitation Performances of Modified NACA Hydrofoil

cevka

• Modified NACA 4418 Hydrofoil


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4.2. Lift Coefficient for Different Turbulence Models

NACA 4418

0.435

0.44

0.445

0.45

0.455

0.46

0.465

0.47

0.475

0.48

experiment k-e standard(20000)

k-e standard(90600)

k-e RNG k-e K-L k-e K-Lscalable

k-omega default k-omega defaultKL

RSM default

Cl


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4.2. Pressure Coefficient Around the Blade With and Without Cavitation

NACA 4418

-5

-4

-3

-2

-1

0

1

2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x/L

Cp

exp s2 exp s7 Fluent-s7 Fluent-s2


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4.2. Lift Coefficient of the Blade With and Without Cavitation

NACA 4418

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16

alfa

Cl

experiment sigma 7 Tasc Flow sigma 7 experiment sigma 2 TascFlow sigma 2


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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) Spiral caseb) Stator c) Wicket gated) Impellere) Draft tube

a) b) c)

d) e)


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4.3. Number of Mesh Elements

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


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4.3. Visualization of the Vapor Development on the Impeller (Pump Mode)


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0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

max

TascFlow

Model test

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)

a) Minimal flow discharge

b) Mode between minimal and optimal mode

c) Optimal mode

d) 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

numerical approach in solving the pde for particular fluid dynamics cases

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