Computational Heat Transfer and Fluid Dynamics

Bud. Univ. of Techn. and Economics Dep. of Aeronautics, Naval Architecture and Railway Vehicles

Budapest University of Technology and Economics

Department of Aeronautics, Naval Architecture and Railway Vehicles

Budapest, H-1111, Sztoczek street 6. building J. 4th floor, Hungary

Tel.: +36-1-463-1922, Fax: +36-1-463-30-80

Computational Heat Transfer and Fluid

Dynamics

BMEKOVRM606

Dr. Árpád Veress

associate professor

Budapest, 07-09-2020

1


Introduction

2


Actuality

3

The world fastest supercomputersSupercomputersMicroprocessors (parallel)Intel x86Alpha

Year of introduction

Bud. Univ. of Techn. and Economics Dep. of Aeronautics, Naval Architecture and Railway Vehicles4

Description of physical processes - basic approaches of mathematical models in space

1. Concentrated parameter type 2. Distributed parameter type

Thermodynamics, Heat Transfer and Fluid Dynamics → Possibilities of the application

of the governing equations

Approaches for Modelling and Conditions for Applications


Description of physical processes - basic approaches for mathematical modelling:

1. Statistical Physics (based on statistical mechanics, kinetic theory of gases)

2. Continuum Mechanics

Definition of Length scale

5



Robert Brown – 1827

Brownian motion

Swept volume of a molecule by unit time:

A c

AcV = ]/[ 3 sm

Then, the number of collisions with other molecule by unit time:

Vnn = ]/[]][/[ 33 sdbsmmdb =

The average time between two collisions: ( ) ( )AcnVnnt 111' === ][s

The average path length between two collisions (mean free path of the molecule):

( )nA'tc 1== ][m For air in case of standard conditions:

=

3197,2

cm

dben

2151 cmeA −=

cme, 573 −=

In 160 Km far from the ground: m80=

In shock waves: cmem 411 −==

6


Length scale - mean free path of

the molecule


1. Statistical Physics (based on statistical mechanics, kinetic theory of gases)

2. Continuum Mechanics

Categorization of the Flow by Local Knudsen Number

Kn=0.0001 0.001 0.01 0.1 1. 10. 100.

Continuum Slip Transition Free

MolecularNavier-Stokes

Kn=0: EulerBurnett

Boltzmann Collosionless

Boltzmann

01,0]m[03,0

]m[7e7,3

LKn

−==

The importance of the local Knudsen number in case of decision about using

continuum mechanics based approaches.

7



Source: Xiao-Jun Gu and

David R. Emerson:

Application of the Moment

Method in the Slip

and Transition Regime for

Microfluidic Flows, RTO-

EN-AVT-194,

http://ftp.rta.nato.int/public//P

ubFullText/RTO/EN/RTO-

EN-AVT-194///EN-AVT-194-

11.pdf (2013.09.01.)


Categorization of the Flow by Local Knudsen Number


Mathematical Models of

Flow – Governing

Equations

9


http://aero-comlab.stanford.edu/Papers/SEVILLE.pdf (2013.09.01.)

Source: Antony Jameson: A perspective on computational algorithms for aerodynamic analysis and design, Progress in Aerospace

Sciences, Volume 37, Issue 2, February 2001, Pages 197–243

10



Flow Modelling by Means of Continuum Mechanics;

Navier-Stokes Equations

Lagrangian and Eulerian specifications

=

==8

1

,

i

ixx Fdt

dumma

dxdydzm =

uVt

u

dt

du+

=

( ) ( ) ( ) ( )dzdydx

zyxx

p

z

uw

y

uv

x

u

t

udzdydx zxyxxx

+

+

+

−=

+

+

+

2

in x direction:

=

==8

1

,

i

ixFdt

dudzdydx

dt

dum

Newton II. law in x direction:

i,xF

dt

du

=

8

1i

i,xF

Source: VKI-LS, Introduction to CFD


uVt

u

dt

du+

=

( ) ( ) ( ) ( )dzdydx

zyxx

p

z

uw

y

uv

x

u

t

udzdydx zxyxxx

+

+

+

−=

+

+

+

2

i,xF

dt

du

=

8

1i

i,xF=

==8

1

,

i

ixx Fdt

dumma

( )t

ut

u

t

u

−

=

As:

( ) ( )VuVuuV −=and:

( ) ( ) ( )VuVt

ut

u

dt

du

+

+

−

=

Hence: 0 (from mass cons.

law)


Navier-Stokes Equations Source: VKI-LS, Introduction to CFD

in x direction:


( ) ( ) ( ) ( ) ( )

( ) ( )z

zTkwvu

y

yTkwvu

x

xTkwvu

z

wH

y

vH

x

uH

t

E

zzzyzxyzyyyx

xzxyxx

++++

++++

+

+++=

+

+

+

2

2

VTcE v

+=

2

2

VTcH p

+=

RTp =

FvP =( )T

( ) ( ) ( )0=

+

+

+

z

w

y

v

x

u

t

( ) ( ) ( ) ( )zyxz

uw

y

uv

x

pu

t

u zxyxxx

+

+

=

+

+

++

2

( ) ( ) ( ) ( )zyxz

vw

y

pv

x

vu

t

v zyyyxy

+

+

=

+

++

+

2

( ) ( ) ( ) ( )zyxz

pw

y

wv

x

wu

t

w zzyzxz

+

+

=

++

+

+

2

( )ttanConsuAudA

x

u

A

V ==⎯⎯→⎯

=

0

maF =

13




• Compressible, ideal gas in a relative stationary system,

• Newtonian continuum fluid, which can be either laminar or turbulent,

• Homogeneous (one material), isotropic material,

• Taking account of transient processes,

• Flow, in which the friction is included between fluid layers sliding on each other,

• Free from any force field (e.g.: gravity and electromagnetic field),

• Free from any sources and sink,

• Conservation form → discontinuities (contact discontinuities, slip lines and shock

waves) are handled.

The conditions in physical point of view for the application of the nonlinear partial

differential system of equations introduced in the previous slide:

;v;

;p;vvv

t

n,n,n

00

0021

=

===−

;v;

;p;v

t

n

00

00

==

;v;

;p;v

t

n

00

00

=




CFD

Computational Fluid

Dynamics

15


▪The CFD stands for Computational Fluid Dynamics, which is a way

of modelling flows with the help of a computer using principles

found in math and physics.

▪ It is one of the most important key elements of modern development

processes today.

▪With the help of CFD one can develop more cost effective, more

environmental friendly and safer vehicles, products and processes

with higher performances and higher efficiencies.

▪ It can be effective tool for simulating processes, which are expensive

or cannot be implemented.

▪Beside verification and plausibility check, the validity of simulations

can be checked by experiments or others (e.g.: benchmarks).

What is the CFD (Computational Fluid Dynamics)?

16


▪ The results of the simulations give back the results of the

measurements within less than 5-10 % in the 80 % of the

industrial applications.

▪ It can be used in the full life-time cycle of the products.

▪ Significant amount of cost, time and capacity can be saved by using

CFD.

▪The physical processes and so the root cause of the problems can be

recognized more easily and faster due to the possibilities of wide

visualization techniques (e.g.: parameter distributions, streamlines,

velocity vectors in any arbitrary cross sections).

▪More physics (e.g.: fluid dynamics, heat transfer, structural

mechanics, electromagnetics) can be coupled together at reasonable

computational costs.

The Advantages of CFD

17


The Advantages of CFD

▪ It can be used where the measurement is difficult, would influence

the investigated process or it is impossible (e.g.: Mars Mission).

▪ The numerical simulation can be parameterized, easily reproduced

and automated.

▪ The simulations can be combined with optimization algorithms.

▪The whole development process cannot be based only on

calculations, validation is required.


Main Application Area

• Vehicle industry (aerodynamics (internal, external), engine

operation, air conditioning),

• Aerospace engineering,

• Safety (prediction of fire- and smoke-spreading, modelling of

detonation and other hazard phenomena),

• Turbomachinery,

• Environmental protection,

• Production and operational process in heavy, light, chemistry and

food industry

• Building industry (heating, cooling, air conditioning, drain and piped

water)

• Weather forecast, climate models

• Astronomy

19


Forrás: Introduction to ANSYS CFX, Lecture 02 – Introduction to CFD, CFX-Intro_14.0_L02_IntroCFD_CFX.pdf (2013.09.01.)

Summary of Basic Operation of CFD


Mathematical Models of

Flow - Turbulence

21



Navier-Stokes Equations; Turbulent Flows


Forrás: Introduction to ANSYS CFX, Lecture 07 - Turbulence, CFX-Intro_14.0_L07_Turbulence .pdf (2013.09.01.)


Navier-Stokes Equations; Turbulent Flows




Navier-Stokes Equations; Turbulent Flows -Theory










Forrás:

Jurij SODJA : Turbulence models in CFD

http://www-f1.ijs.si/~rudi/sola/Turbulence-models-in-CFD.pdf (2013.09.01)


NS Equations; Simulation Techniques for Handling Turbulence












RANS; Modelling of Turbulence -Theory










t

u’

u

uuuu += vvv +=

www += ppp +=

dtut

utt

t+

=

0

0

1

+=

( )dtut

utt

t+

=

0

0

11~

Reynolds Átlagolás Favre ÁtlagolásNagy sebességű,

összenyomható áramlás esetén.

uuu += ~vvv += ~ www += ~ ppp += +=

hhh +=~

eee += ~TTT +=

~jjj qqq += jLj qq =

L: laminar transport of the heat


RANS; Reynolds and Favre Averaging -Theory - DASFLOW


0~~~=

+

+

+

z

w

y

v

x

u

t

+

+

+

−=

+

+

+

zyxx

p

z

uw

y

uv

x

uu

t

uF

xz

F

xyF

xx ~~~~~~~

+

+

+

−=

+

+

+

zyxy

p

z

vw

y

vv

x

vu

t

vF

yz

F

yy

F

yx ~~~~~~~

+

+

+

−=

+

+

+

zyxz

p

z

ww

y

wv

x

wu

t

wF

zz

F

zyF

zx ~~~~~~~




uuVx

u TF

xx−−

=

~

3

2~2

vvV

y

v TF

yy−−

=

~

3

2~2

wwVz

w TF

zz−−

=

~

3

2~2

vux

v

y

uF

yx

F

xy−

+

==

~~

wux

w

z

uF

zx

F

xz−

+

==

~~wv

y

w

z

vF

zy

F

yz−

+

==

~~

( )

−

+

−+−−

=

=

+

+

+

+

+

=====

=====

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

~

2

1

2

1~~~

2

1~~

2

1~~

2

1~

i

jijii

j ji

iij

i

ijijj

j j

i

iij

i

iij

j ji

ii

i

ii

uuux

uuuuhuqx

uuuuuhux

uuuuet






















Forrás: www.tech.plym.ac.uk/sme/dsgn313/CFDNotes06.ppt (2013.09.01.)


RANS; Turbulence Modelling -Theory






( )

+

+−

−=

+

j

t

jj

iji

j

jx

k

xk

x

uuu

x

ku

t

k **

~~

( )

+

+−

−=

+

j

t

jj

iji

j

jxxx

u

kuu

xu

t

2

~~

kt =

25

13=

2

1* =2

1= ( ) tMFf **

0

* 1*

+=

( )tMFff **

00 * −=

100

9*

0 =125

90 =

+

+

=0

4001

6801

01

2

2*

k

k

k

k

if

if

f

jj

kxx

k

=

3

1

801

701

+

+=f

( )3*

0

kijkij S= 0=

2

3* =4

10 =tM

( ) ( )0

2

0

2

ttttt MMMMMF −−=

( )

=

01

00

xif

xifx

2

2 2

a

kM t =

k *=

2/1kl =

+

=

i

j

j

iij

x

u

x

uS

~~

2

1

−

=

i

j

j

iij

x

u

x

u~~

2

1

kS

S




( ) ( ) ( ) +=+

A

vnn

A

dAUSdUHdUHUdAt

=

k

E

v

u

U ~

~

~

( ) ( )

+

+

+

=

n

n

n

yn

xn

n

n

V

kV

VpE

npVv

npVu

V

UH*

*

*

~

~

~

( )

=

S

S

US

k

0

0

0

0


RANS; Turbulence Modelling - DASFLOW - Equations in

Conservative Form


( ) ( ) ( ) +=+

A

vnn

A

dAUSdUHdUHUdAt


RANS; Turbulence Modelling - DASFLOW - Equations in

Conservative Form


( ) m,1k

UU

UtUU

UU

m1n

1k

k

0k

n0

=

=

+=

=

+

−• Runge-Kutta módszer

m=4

( ) ( ) ( ) =+

−−=

==

ij

k

kijkijvn

k

kijkijn

ij

ij USHHA

Ut

4

1

,,

4

1

,,

1

( ) ( )=

=4

1

,,k

kijkijvnvn HdUH

ij

( ) ( ) R

vn

L

vnvn UHUHH +=2

1

( ) ( ) ijij

A

AUSdAUS

ij

=

i,j

i,j+1

i+1,j

i,j-1

i-1,ji+1/2,ji-1/2,j

LR L

RL

R

L

Ri,j+1/2

i,j-1/2n

ij,2

ij

ij,1

ij,2

ij,3

ij,4

nij,1

nij,3

nij,4

x

y

ey

ex


RANS; Turbulence Modelling - DASFLOW - Discretization










Forrás: www.tech.plym.ac.uk/sme/dsgn313/CFDNotes06.ppt (2013.09.01.)






RANS; Modelling of Turbulence - Models
















RANS; Modelling of Turbulence - Handling Flow at Solid Walls










yUy =+

wU =

+= uU

U

Cyu += ++ ln1

( ) 5.5ln1 += yUUU

++ = yu

yU

y lnln =+

Law of the wall by Prandtl

y+ < 0.1δ

dy

dUw =

?

30=+y




































Forrás: Introduction to ANSYS CFX








ANSYS CFX






ANSYS CFX





ANSYS CFX




Forrás: ANSYS, Inc., ANSYS CFX-Solver Theory Guide, Release 14.5, ANSYS, Inc. Southpointe, 275 Technology Derive

Canonsburg, PA 15317, [email protected], http://www.ansys.com, USA, 2012

ANSYS CFX






RANS; Modelling of Turbulence - Inlet Boundary Condition















Re-Normalisation Group


RANS; Modelling of Turbulence - Summary










Numerical Modelling of

Flow

Discretization of the

Governing Equations

80


FINITE DIFFERENCE, 1. Introduced by Euler in the 18th century.

2. Governing equations in differential form→ domain with grid→ replacing the

partial derivatives by approximations in terms of node values of the functions→

one algebraic equation per grid node→ linear algebraic equation system. 3. Applied

to structured grids.

FINITE VOLUME, 1. Governing equations in integral form→ solution domain is subdivided into a finite number of contiguous control volumes→ conservation equation applied to each CV.

2. Computational node locates at the centroid of each CV.

3. Applied to any type of grids, especially complex geometries

4. Compared to FD, FV with methods higher than 2nd order will be difficult, especially for 3D.

FINITE ELEMENT,

1. Similar to FV

2. Equations are multiplied by a weight function before integrated over the entire domain.


RANS; Discretization - CFX


Forrás: ANSYS CFX-Solver Theory Guide, ANSYS Inc., Southpointe, 275 Technology Drive, Canonsburg, PA 15317, October

2012





2012





2012





2012

° : parameters of the next iteration step, : parameters of the actual iteration step,

First or Second order Implicit Backward Euler method has been used for solving the

system of algebraic equations.





2012


RANS; Solution of System of Algebraic Equations - CFX



2012

Pressure-Velocity Coupling: ANSYS CFX uses a co-located (non-staggered) grid

layout such that the control volumes are identical for all transport equations. As

discussed by Patankar, however, naïve co-located methods lead to a decoupled

(checkerboard) pressure field. Rhie and Chow [2] proposed an alternative discretization

for the mass flows to avoid the decoupling, and this discretization was modified by

Majumdar to remove the dependence of the steady-state solution on the time step.

Similar strategy is adopted in ANSYS CFX.

Solver: ANSYS CFX uses a coupled solver, which solves the hydrodynamic equations

(for u, v, w, p) as a single system by means of a fully implicit discretization of the

equations at any given time step. For steady state problems, the time-step behaves like an

‘acceleration parameter’, to guide the approximate solutions in a physically based

manner to a steady-state solution. This reduces the number of iterations required for

convergence to a steady state, or to calculate the solution for each time step in a time-

dependent analysis. ANSYS CFX uses first or second order (better for transient due to

the higher accuracy) Backward Euler scheme for time discretization (see slide 85).


RANS; Solution of System of Algebraic Equations - CFX


System of partial

differential equations,

L(Ť)

Exact solution, Ť

Discretization

Consistency

System of

algebraic

equations

Stability

Approximate

solution, T

• Initial and boundary conditions

Convergence

if Δx, Δt 0

Mathematical properties of the numerical discretization


RANS; Numerical Methods and Their Characteristics


The Main Steps and

Rules of Completing a

CFD Task

89


The Main Steps of a CFD TaskP

re-p

roce

ssin

g

• Goal, definition and review of the task which has to be solved and

mapped that into the modelling space, making time and cost plan,

• Creating geometrical (flow) model,

• Making of the numerical mesh in the flow domain,

• Definition of the material properties,

• Setting up related physical models and their parameters,

• Imposing boundary conditions and assigning them to the geometry,

• Specify initial conditions,

• Setting solver parameters,

• Start calculation and evaluate convergences,

• View, analyze, and evaluate the results,

• Verification, plausibility check and validation,

• Mesh and parameter sensitivity analyses,

• Documentation preparation with especial care for the suggestions

for improvements in case of relevancies. Post

-

pro

cess

ing

90


Geometry – Flow field

91

The Main Steps of a CFD Task


The Main Steps of a CFD Task – Discretization of the Flow

Field


Forrás: Introduction to ANSYS CFX, Lecture 10 - Best Practice Guidelines - CFX-Intro_14.0_L10_BestPractices (2013.09.01.)


Field




Field




Field



Részletesen lásd: Mesh Metrix, Mesh_metrix_in_ANSYS_WB_13_v11.ppt


Field




Field




Field




Field




Field




Field




Field




Field




Field




Field




Field



Field

Rotational flow domain of a

centrifugal compressor. The

quality and the resolution of the

mesh at the high gradient

regions should be improved

until it has influence for the

results. Minimum 20 cells are

necessary in case of the smallest

gap at compressible flow.


Typical boundary conditions in case of CFD simulations: inlet, outlet,

solid wall, symmetry, opening (far field) and periodic. The boundaries

should be placed on such a far distance from the object under

investigation that the disturbances should not be propagated till the

boundaries (e.g.: the streamlines should not cross the opening

boundaries, the flow should not enter at the outlet section) together with

the minimal number of cells with the mesh independent results.

108

The Main Steps of a CFD Task – Material Properties and

Boundary Conditions

periodic

boundary B

outlet

frictionless

wall

opening

boundary or

symmetric

periodic

boundary B

inlet

inlet

outletperiodic

boundary A

periodic

boundary A

solid

wall

solid

wall


Convergence

=

=

pN

1i

2

i

i

p

10N

1log

109

The Main Steps of a CFD

Task – Solution and

Convergence


Forrás: Introduction to ANSYS CFX, Lecture 05 - Solver Settings and Output File - CFX-Intro_14.0_L05_SolverSettings_OutFile

(2013.09.01.)

The Main Steps of a CFD Task – Solution and Convergence



(2013.09.01.)




(2013.09.01.)



The Main Steps of a CFD Task – Visualization of the Results












Mach Number Distribution

0

0,3

0,6

0,9

1,2

1,5

1,8

0 2 4 6 8 10

lenght [m]

Mach

nu

mb

er

Exact

Numerical

M1=1.1

Source: Starken, H., 1971, “Untersuchung der Strömung in ebenen Überschallverzöger-ungsgittern,” DLR-Forschungsbericht 71-99.

119

Visualization - Validation – Numerical Solutions of the Euler Equations


s = 0 m

0

0,01

0,02

0,03

-10

0

0 100 200 300 400 500 600

Velocity [m/s]

y_E

XP

[m

]

experiment

k-ω model

A mérés forrásanyaga: Gerolymos, G. A.; Sauret, E. & Vallet, I. (2003). Oblique-Shock-Wave/Boundary-Layer Interaction using Near-Wall Reynolds-Stress Models, Université Pierre-et-Marie-Curie, AIAA 2003-3466, 33rd Fluid Dynamics Conference, 23-26 June 2003 Orlando, Florida, USA

120

Visualization - Validation – Numerical Solution of the Navier-Stokes Equations


Mach szám eloszlás

121

Visualization - DASFLOW Program – Numerical Solutions of the Navier-Stokes Equations in Cascade


Appendices I.

The Effect of Concave

and Convex Curvature

122


The Effect of Concave and Convex Curvature

Fp

Fc

RFp

Fc

Convex Concave R

pc FFmRa ,RH 2 pc FFmRa ,RH 2

vpFa pH

állandóvp

+ 2

2

1

vpFa pH

állandóvp

+ 2

2

1

Real flow (with friction):n v

123


Convex

n v

124

Real flow (with

friction):







cR

v

n

p 2

=

ps

shroud

hub

ss

shroud

hub

ps ss

n

Rc

ss

ps

shroud

hub

Rc

n

Passage Vortex

Blade Vortex

ssps

shroud

hub

127

Secondary Flow in Cascades


Secondary Flow in Cascades

(Passage Vortex)


Secondary Flow Patterns in Rod Bundles in nuclear reactor


Secondary Flow Pattern in a Rectangle Cross Flow Channel


Secondary Flow Pattern in a Knee Pipe


Appendices II.

DASFLOW Software and

Its Industrial

Applications

132


DASFLOW – Tanszéki fejlesztésű CFD és inverz

tervezésre alkalmas program


Áramlásmodellezés – Kontinuum-mechanika alapján

RANS egyenletek – Reynolds és Favre átlagolás - DASFLOW

t

u’

u

uuuu += vvv +=

www += ppp +=

dtut

utt

t+

=

0

0

1

+=

( )dtut

utt

t+

=

0

0

11~

Reynolds Átlagolás Favre ÁtlagolásNagy sebességű,

összenyomható áramlás esetén.

uuu += ~vvv += ~ www += ~ ppp += +=

hhh +=~

eee += ~TTT +=

~jjj qqq += jLj qq =




0~~~=

+

+

+

z

w

y

v

x

u

t

+

+

+

−=

+

+

+

zyxx

p

z

uw

y

uv

x

uu

t

uF

xz

F

xyF

xx ~~~~~~~

+

+

+

−=

+

+

+

zyxy

p

z

vw

y

vv

x

vu

t

vF

yz

F

yy

F

yx ~~~~~~~

+

+

+

−=

+

+

+

zyxz

p

z

ww

y

wv

x

wu

t

wF

zz

F

zyF

zx ~~~~~~~




uuVx

u TF

xx−−

=

~

3

2~2

vvV

y

v TF

yy−−

=

~

3

2~2

wwVz

w TF

zz−−

=

~

3

2~2

vux

v

y

uF

yx

F

xy−

+

==

~~

wux

w

z

uF

zx

F

xz−

+

==

~~wv

y

w

z

vF

zy

F

yz−

+

==

~~

( )

−

+

−+−−

=

=

+

+

+

+

+

=====

=====

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

~

2

1

2

1~~~

2

1~~

2

1~~

2

1~

i

jijii

j ji

iij

i

ijijj

j j

i

iij

i

iij

j ji

ii

i

ii

uuux

uuuuhuqx

uuuuuhux

uuuuet




( )

+

+−

−=

+

j

t

jj

iji

j

jx

k

xk

x

uuu

x

ku

t

k **

~~

( )

+

+−

−=

+

j

t

jj

iji

j

jxxx

u

kuu

xu

t

2

~~

kt =

25

13=

2

1* =2

1= ( ) tMFf **

0

* 1*

+=

( )tMFff **

00 * −=

100

9*

0 =125

90 =

+

+

=0

4001

6801

01

2

2*

k

k

k

k

if

if

f

jj

kxx

k

=

3

1

801

701

+

+=f

( )3*

0

kijkij S= 0=

2

3* =4

10 =tM

( ) ( )0

2

0

2

ttttt MMMMMF −−=

( )

=

01

00

xif

xifx

2

2 2

a

kM t =

k *=

2/1kl =

+

=

i

j

j

iij

x

u

x

uS

~~

2

1

−

=

i

j

j

iij

x

u

x

u~~

2

1



RANS egyenletek – Reynolds és Favre átlagolás – Konzervatív

összevont forma - DASFLOW

( ) ( ) ( ) +=+

A

vnn

A

dAUSdUHdUHUdAt

=

k

E

v

u

U ~

~

~

( ) ( )

+

+

+

=

n

n

n

yn

xn

n

n

V

kV

VpE

npVv

npVu

V

UH*

*

*

~

~

~

( )

=

S

S

US

k

0

0

0

0



RANS egyenletek – Konzervatív forma – RARS - DASFLOW

( ) ( ) ( ) ( ) −+== Rn

Ln

RLnn UHUHU,UH

~UH

2

1 ( )( )LRRL

n UUU,UD −−

i

n

i

n

i

i

nn WrˆUD =

=4

1

+−

+

−

=

cˆ

pVn

cˆ

pVn

Vsc

p

Wn

2

−

+=

cnV

cnV

nV

nV

ˆ

( ) Tn vu.,v,u,r 221 501 +=

Txyxyn )nvnu(,nˆ,nˆ,r −−= 02

( ) ( )T

nyxn Vcc

cncv

cncu

ccr

++++= ˆˆ

ˆ

ˆ2

ˆ,ˆˆ

ˆ2

ˆ,ˆˆ

ˆ2

ˆ,

ˆ2

ˆˆ

23

( ) ( )T

nyxn Vcc

cncv

cncu

ccr

−+−−= ˆˆ

ˆ

ˆ2

ˆ,ˆˆ

ˆ2

ˆ,ˆˆ

ˆ2

ˆ,

ˆ2

ˆˆ

24



RANS egyenletek – Konzervatív forma – MUSCL -

DASFLOW

( ) ( ) ( ) ( )32

2

2

2

1xxx

x

Uxx

x

UUxU ixixi ii

+−

+−

+=

( ) ( ) 2/1i2/1ii

L

2/1i 114

1UU −++ −+++=

( ) ( ) 2/1i2/1ii

R

2/1i 114

1UU −+− ++−−=

U(x)

x

U(x)

x

iii UU −= ++ 12/1

3/1=

1,1−=



RANS egyenletek – Konzervatív forma – MUSCL (limiterek) -

DASFLOW

U(x)

x

U(x)

x

( ) ( ) 2/12/12/1 114

−++ −+++= i

i

i

ii

i

L

i rrr

UU

( ) ( ) 2/12/12/1 114

−+− ++−−= i

i

i

ii

i

R

i rrr

UU

22

2/1

2

2/1

2

2/12/1

2

22

++

+=

−+

−+

ii

iiir

710 −

3/1=



RANS egyenletek – Diszkretizáció - DASFLOW

( ) m,1k

UU

UtUU

UU

m1n

1k

k

0k

n0

=

=

+=

=

+

−• Runge-Kutta módszer

m=4

( ) ( ) ( ) =+

−−=

==

ij

k

kijkijvn

k

kijkijn

ij

ij USHHA

Ut

4

1

,,

4

1

,,

1

( ) ( )=

=4

1

,,k

kijkijvnvn HdUH

ij

( ) ( ) R

vn

L

vnvn UHUHH +=2

1

( ) ( ) ijij

A

AUSdAUS

ij

=

i,j

i,j+1

i+1,j

i,j-1

i-1,ji+1/2,ji-1/2,j

LR L

RL

R

L

Ri,j+1/2

i,j-1/2n

ij,2

ij

ij,1

ij,2

ij,3

ij,4

nij,1

nij,3

nij,4

x

y

ey

ex


Mach Number Distribution

0

0,3

0,6

0,9

1,2

1,5

1,8

0 2 4 6 8 10

lenght [m]

Mach

nu

mb

er

Exact

Numerical

M1=1.1

DASFLOW Program – Egyenletek – Validáció

(súrlódásmentes)


s = 0 m

0

0,01

0,02

0,03

-10

0

0 100 200 300 400 500 600

Velocity [m/s]

y_E

XP

[m

]

experiment

k-ω model

DASFLOW Program – Egyenletek – Validáció

(súrlódásos)


DASFLOW Program - Lapátrács numerikus áramlástani vizsgálata

Probléma leírása


Korrekt Inkorrekt_1

Inkorrekt_2 Inkorrekt_3

Probléma leírása




ptoin= 796127 Pa, Tto

in= 530.7 K

alfain=43o

pout= 654600. Pa, ptoout_ave=780525. Pa,

Ttoout_ave=530.5 K, alfa_rout_ave= -43.7o

Korrekt


ptoin= 796127 Pa, Tto

in= 530.7 K

alfain=43o

pout= 654600. Pa, ptoout_ave=780525. Pa,

Ttoout_ave=530.5 K, alfa_rout_ave= -43.7o


Inkorrekt_1






0=

+

+

y

G

x

F

t

U iii ( )Ti vuU ,,0= Ti vup

uuF ),,( 2

+=

Ti pvuvvG ),,( 2

+=

2D-s összenyomhatatlan áramlás Euler

egyenletei konzervatív és dimenziós alakban:

0=

+

+

y

G

x

F

t

U iii ( )Ti vuPU ,,=Ti vuPuuF ),,( 22 +=

Ti PvuvvG ),,( 22 +=

Chorin módszere szerint:

pP =

Tüzelőanyag sugárszivattyú direkt numerikus

optimalizálása - Ipari alkalmazás


0dHUdt

n =+

+

+==

yn

xn

n

n

PnvV

PnuV

V

nHH

2

• Integrál egyenletek

( ) m,1k

UU

UtUU

UU

m1n

1k

k

0k

n0

=

=

+=

=

+

−

DHUt j

k,j

N

kk,jn

j

j

f

11

1

+−=

=

• Runge-Kutta módszer





optimalizálása - Ipari alkalmazás - Validáció


g




e




Modell Alap Letörés Optimalizált 1 Optimalizált 2

mbe [l/h] 129,8 128,9 126 115,9

mki [l/h] 307 326,5 350,33 369,1

Sz.k. 2,36 2,53 2,78 3,18




Szárnyprofilra, NACA 65-4101

GUI view of the DASFLOW in-house 2D CFD analysis and design software

Optimalizáció; inverz tervezőeszköz kidolgozása és alkalmazása






A számítás folyamata:

1. Kezdeti

geometria

2. Adott áramláshoz tartozó

nyomás eloszlás (preq)

előállítása és CFD

beállítások

3. Hálógenerálás

4. CFD számítás

(analízis)

5. Optimalizálási

kritériumnak

megfelel?

Optimális

Nyomáseloszlás

(preq) alkalmazása

7. Falmódosító eljárás

IGEN

STOP

NEM

6. Inverz analízis,

áteresztő fal

1. Kezdeti geometria (NACA 65 410)

2. Adott áramláshoz tartozó nyomás eloszlás (preq)

előállítása és CFD beállítások




Stratford limiting flow at given Reynolds

number

2

00

0p

V21

ppC

−= : Location of the start of the

positive pressure gradient

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1

x/c

Cp

Cp_min=-1.7

Cp_min=-3.5

Cp_min=-2.5

Cp_min=-1

Canonical pressure distribution:

Pressure coefficient:

Max. area with

Stratford limiting

pressure

increment (near

to the separation

limit)

reqp

p req is the output of the optimization

(n=6)

160

0





1. Kezdeti

geometria




beállítások


4. CFD számítás

(analízis)

5. Optimalizálási

kritériumnak

megfelel?

Optimális

Nyomáseloszlás

(preq) alkalmazása


IGEN

STOP

NEM


áteresztő fal


4. CFD számítás (analízis)





1. Kezdeti

geometria




beállítások


4. CFD számítás

(analízis)

5. Optimalizálási

kritériumnak

megfelel?

Optimális

Nyomáseloszlás

(preq) alkalmazása


IGEN

STOP

NEM


áteresztő fal

5. Optimalizálási kritériumnak megfelel?

6. Inverz analízis, áteresztő fal





1. Kezdeti

geometria




beállítások


4. CFD számítás

(analízis)

5. Optimalizálási

kritériumnak

megfelel?

Optimális

Nyomáseloszlás

(preq) alkalmazása


IGEN

STOP

NEM


áteresztő fal





Eredmények 10 inverz iterációt

követően:

Eredeti profil

Optimalizált profil

Belépő torlóponti nyomás:

ptot,in=112799 [Pa];

Belépő torlóponti hőm.:

Ttot,in=293.15 [K];

Kilépő statikus nyomás:

pstatot,out=101325 [Pa].

Hálóméret: 87×30

Iteráció szám: 5000

Konvergencia kritérium

sűrűség NKÉ: 1e-5.6

Peremfeltételek:




Eredmények 10 inverz iterációt

követően:




Eredmények a kezdeti profil esetén és 10 inverz iterációt követően:

Cl-alpha Plot of NACA 65-410 Profile

-0.5

0.0

0.5

1.0

1.5

2.0

-6 -4 -2 0 2 4 6 8 10

angle of attack [deg]

lift

co

eff

icie

nt

[-]

Measurement

Analysis init.

Analysis opt.



Fokozati kompresszió viszony nagyságát befolyásoló tényezők:


),),(( 21 −= aks CnUf

Eredmények lapátrácsra 10 inverz iterációt követően:

A 0,62 Mach-szám (pstat,out=83325 [Pa]) és Cp

= -1,4-hez tartozó nyomáseloszlás

A lapátrácsban kialakuló nyomáseloszlás

0,62 Mach-szám (pstat,out=83325 [Pa]) és Cp

= -1,4 esetén

181,static,ks =




),),(( 21 −= aks CnUf

Eredmények lapátrácsra 10 inverz iterációt követően: 181,static,ks =

0,62

0,3

0,5

Mach-szám:

( )Cp,pf

p

p

out,stat

in,stat

out,stat

=

==




),),(( 21 −= aks CnUf

Eredmények lapátrácsra 10 inverz iterációt követően: 181,static,ks =

( )Cp,pf

p

p

out,stat

in,stat

out,stat

=

==


Appendices III.

Industrial Applications of

a CFD and Inverse

Design Tool Developed at

VKI

170


Többfokozatú centrifugál kompresszor összekötő-

csatorna lapátozásának tervezése – Ipari alkalmazás

tömegáram: 340 m3/h –ig,

nyomás: 345 bar-ig

A-C többfokozatú kompresszor

lapát belépő él

lapátnélküli diffúzor

fordító könyök

visszatérő csatorna

forgórész bemenet

lapát kilépő él



csatorna lapátozásának tervezése – Ipari alkalmazás


Többfokozatú

centrifugál kompresszor

összekötő-csatorna

lapátozásának tervezése

– lapátkiterjesztés –

Ipari alkalmazás

st

in

to

in

st

in

st

outp

pp

ppC

−

−=

st

in

to

in

to

out

to

in

pp

pp

−

−=



csatorna lapátozásának tervezése – ÁLT – Ipari

alkalmazás

( )flm

bl

th

blssps tg.R.Wdm

d

cosRzcosWW

−=−

2

1. t

m CWR =

dm

dC

zWW

t

ssps

cos

2 1

=−

=

TE

LE

t

t

bl dmcosC

C

2

1

fl

U

dm

WV fl

Wps

th

R

R+R

Wssds

z

R2

m

s

0V =


RQ

bbl


csatorna lapátozásának tervezése – ÁLT – Ipari

alkalmazás

=

TE

LE

t

t

bl dmcosC

C

2

1

=

TE

LE

blbl dm

R

)tan(),m(


Többfokozatú centrifugál

kompresszor összekötő-

csatorna lapátozásának

tervezése – ÁLT – Ipari

alkalmazás

st

in

to

in

st

in

st

outp

pp

ppC

−

−=

st

in

to

in

to

out

to

in

pp

pp

−

−=



csatorna lapátozásának tervezése – Inverz Módszer –

Ipari alkalmazás

A C


B D



Ipari alkalmazás


R



Ipari alkalmazás


Többfokozatú

centrifugál kompresszor

összekötő-csatorna

lapátozásának tervezése

– Inverz Módszer –

Ipari alkalmazás

st

in

to

in

st

in

st

outp

pp

ppC

−

−=

st

in

to

in

to

out

to

in

pp

pp

−

−=




csatorna lapátozásának tervezése – Lapátelhajlítás – Ipari

alkalmazás

cR

v

n

p 2

=

ps

shroud

hub

ss

shroud

hub

ps ss

n

Rc

ss

ps

shroud

hub

Rc

n

Csatorna örvény

Lapát örvény

ssps

shroud

hub



csatorna lapátozásának tervezése – Lapátelhajlítás – Ipari

alkalmazás



csatorna lapátozásának tervezése – Eredmények - Ipari

alkalmazás

Tervezés

Nem

kiterjeszt

ett

ÁLTÁLT + Inverz

Tervezés

+ negatív

lapátelh.

P2o [Pa] 299699.1 299526.6 299696.6 299698.5

P2s [Pa] 159038.5 182298.2 174936.4 169050.8

P3o [Pa] 236665.3 261108.8 264954.2 265275.3

P3s [Pa] 225215.1 258238. 258414.2 258119.0

0.44813 0.3277 0.27847 0.263

Cp 0.4705 0.6478 0.669106 0.681

[kg/s] 4.68 4.5 4.64 4.5


Összefoglalás

• A numerikus áramlástani módszerek segítségével jobban megérthetők a fizikai

folyamatok többek között a vizualizációs eszközöknek köszönhetően.

• Kapcsolt fizikai folyamatok modellezése is lehetséges elfogadható számítógépi

kapacitással.

• A numerikus módszereket optimalizációs algoritmusokkal is lehet csatolni.

• Alkalmazásukkal jelentős költség- és kapacitás-csökkenés érhető el.

• Kivitelezhetetlen, extrém körülmények közötti, illetve nagy költségű mérések

kiváltására is alkalmas.

• Az analízisek paraméterezhetőek, könnyen megismételhetőek minimális

ráfordítással az előírt geometriai változtatásokat követően.

• A számítási eredmények validációjára és paraméter-érzékenységi vizsgálatok

elvégzésére minden esetben szükség van.

185


Thank you for your kind attentions.

BME, Vasúti Járművek, Repülőgépek és Hajók Tanszék

Stoczek u. 6. J. ép. 4. em. 426

H-1111, Budapest

Telefon: +36 1 463-1922

Fax: +36 1 463-3080

e-mail: [email protected]

186

Forrás: ANSYS, Inc., ANSYS CFX-Solver Theory Guide, Release 13, ANSYS, Inc. Southpointe, 275 Technology Derive


Forrás: ANSYS, Inc., ANSYS CFX-Solver Theory Guide, Release 14.5, ANSYS, Inc. Southpointe, 275 Technology Derive


Documents

Computational Heat Transfer and Fluid Dynamics