fluent manual

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TECHNICAL UNIVERSITY OSTRAVA

Fluid Flow Modelling FLUENT, CFX

Milada Kozubkov

OSTRAVA 2014

Content

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Content

LIST OF USED DESIGNATION ....................................................................................................... VI1. INTRODUCTION TO NUMERICAL MODELING OF FLUID FLOW ................................. 1

1.1. INTRODUCTION TO FLUID FLOW ............................................................................................... 11.2. TRANSFER OF MASS, MOMENTUM, HEAT IN NON ISOTHERMAL FLOW OF INCOMPRESSIBLE FLUID 11.3. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS........................................ 31.4. BOUNDARY CONDITIONS OF INCOMPRESSIBLE FLUID FLOW ..................................................... 5

1.4.1. Types of boundary conditions ............................................................................................ 51.5. SOLUTION OF THE FLOW IN SUDDEN EXPANSION SECTION .................................................... 81.6. APPLICATION FVM ON WATER FLOW BETWEEN THE PLATES ................................................. 11

2. GEOMETRY AND COMPUTATIONAL GRID GENERATION ...................................... 362.1. THE TERM "GRID" AND ITS IMPORTANCE FOR THE MATHEMATICAL MODELING .................... 362.2. WORKBANCH, GRID ELEMENTS ............................................................................................. 362.3. CRITERIA FOR GRID QUALITY ASSESSMENT ............................................................................ 382.4. PKLAD VYTVOEN ST ...................................................................................................... 40

2.4.1. Vytvoen st ................................................................................................................. 412.4.2. Definition of the boundary conditions in Gambit ............................................................ 43

3. FLUENT SOFTWARE SYSTEM ........................................................................................... 443.1. OVERVIEW OF METHODS FOR SOLVING OF PARTIAL DIFFERENTIAL EQUATIONS .................. 443.2. INTEGRATION BY FINITE VOLUME METHOD ............................................................................ 443.3. CHOICE OF INTERPOLATION SCHEME .................................................................................... 473.4. CONVERGENCE. ..................................................................................................................... 48

3.4.1. Residuals ........................................................................................................................ 483.4.2. Convergence acceleration ........................................................................................... 503.4.3. Relaxation ...................................................................................................................... 51

4. TURBULENT FLOW OF REAL FLUID ............................................................................... 534.1. CLASSIFICATION OF REAL FLUID FLOW ................................................................................. 534.2. TURBULENT FLOW .................................................................................................................. 55

4.2.1. Theory of turbulence ..................................................................................................... 564.2.2. Methds of mathematical modelingof turbulent flow ................................................. 594.2.3. Reynolds equations ...................................................................................................... 614.2.4. Boussinesq's hypothesis about eddy (turbulent) viscosity ..................................... 64

5. STATISTICAL MODELS OF TURBULENCE ..................................................................... 665.1. MODEL OF MIXING LENGTH (ZERO - EQUATION MODEL) - PRANDTL .................................... 67

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5.2. ONE - EQUATION MODEL ........................................................................................................ 675.3. TWO - EQUATION K- MODEL ................................................................................................. 685.4. RNG K- MODEL ..................................................................................................................... 695.5. REYNOLDS STRESS MODEL (RSM) ....................................................................................... 695.6. MODELING OF FLOW IN NEAR WALLS, WALL FUNCTIONS ....................................................... 70

5.6.1. Theory of wall functions by Launder and Spalding .................................................. 715.6.2. Modeling of flow near the wall in Fluent .................................................................... 725.6.3. Non-equilibrium wall function ...................................................................................... 755.6.4. Using of wall functions and their limitations .............................................................. 765.6.5. Two-layer model (near-wall modelling) ...................................................................... 765.6.6. The influence of grid quality on the choice of wall functions for various models of turbulence ....................................................................................................................................... 775.6.7. The influence of roughness on the wall function ............................................................. 78

6. A MATHEMATICAL MODEL OF TURBULENCE FOR COMPRESSIBLE NON-ISOTHERMAL FLOW ...................................................................................................................... 79

6.1. K- TWO-EQUATION MODEL OF TURBULENCE ....................................................................... 796.2. BOUNDARY CONDITIONS FOR K- TURBULENT MODEL ......................................................... 80

6.2.1. Mass flow rate ............................................................................................................... 806.2.2. Turbulent variables ....................................................................................................... 806.2.3. Pressure at inlet ............................................................................................................ 816.2.4. Pressure at outlet .......................................................................................................... 836.2.5. Outflow ............................................................................................................................ 83

7. EEN PENOSU TEPLA (KONVEKCE, KONDUKCE) ...... CHYBA! ZLOKA NEN DEFINOVNA.

7.1. VOD DO PROBLEMATIKY MODELOVN PENOSU TEPLA ............... CHYBA! ZLOKA NEN DEFINOVNA.

7.1.1. Kondukce veden tepla ............................................................................................. 847.1.2. Konvekce ........................................................................................................................ 86

7.2. MATEMATICK MODEL PENOSU TEPLA ............................................................................... 877.2.1. Penos tepla v tekutinch ............................................................................................ 877.2.2. Penos tepla ve vodivch stnch .............................................................................. 88

7.3. HUSTOTA ZVISL NA TEPLOT A TLAKU, BOUSSINESQOVA APROXIMACE ......................... 897.3.1. Vyjden hustoty pro stlaiteln mdium ................................................................. 897.3.2. Vyjden hustoty pro nestlaiteln mdium ............................................................. 897.3.3. Boussinesquova aproximace ...................................................................................... 90

7.4. OKRAJOV PODMNKY PRO NEIZOTERMN PROUDN .......................................................... 927.4.1. Okrajov podmnky na tenk stn............................................................................ 94

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7.4.2. Okrajov podmnky na tenk dvoustrann stn .................................................... 947.4.3. Vpoet teploty a hustoty tepelnho toku na stn ................................................. 95

8. MODELOVN PROUDN PMS ............................................................................... 1008.1. TRANSPORTN ROVNICE PRO PENOS PMS .................................................................. 1008.2. DEFINICE ZDROJE PMSI .................................................................................................. 101

8.2.1. Zdroj-inlet ..................................................................................................................... 1028.2.2. Objemov zdroj ........................................................................................................... 102

8.3. FYZIKLN VLASTNOSTI PLYN A JEJICH SMS .................................................................. 1038.3.1. Hustota ......................................................................................................................... 1038.3.2. Viskozita ....................................................................................................................... 1048.3.3. Mrn tepeln kapacita .............................................................................................. 1058.3.4. Tepeln vodivost ......................................................................................................... 1058.3.5. Standardn sluovac entalpie a entropie ................................................................ 105

8.4. OKRAJOV PODMNKY PRO PMSI NA VSTUPU, VSTUPU A STN ................................ 1069. VCEFZOV MODELY ...................................................................................................... 109

9.1. VCEFZOV MODELY OBECN ............................................................................................ 1099.1.1. Euler-Lagrangev pstup .......................................................................................... 1109.1.2. Euler-Eulerv pstup .................................................................................................. 110

9.2. VCEFZOV MATEMATICK MODELY .................................................................................. 1119.2.1. Vcefzov model smsi (mixture model) ............................................................... 1139.2.2. Trajektorie pevnch stic v plynu ........................................................................... 116

10. ASOV ZVISL EEN ............................................................................................... 12410.1. DISKRETIZACE ASOV ZVISL ROVNICE ...................................................................... 12510.2. OKRAJOV PODMNKY ...................................................................................................... 127

10.2.1. Tabulka pro asovou okrajovou podmnku ......................................................... 12710.2.2. UDF pro okrajovou podmnku ............................................................................... 129

10.3. PKLAD VYHODNOCEN ASOV ZVISL LOHY .......................................................... 12911. MEN A VPOET TURBULENCE PI OBTKN VLCE V AERODYNAMICKM TUNELU ................................................................................................. 134

11.1. TEORETICK ROZBOR LOHY OBTKN VLCE ............................................................. 13411.2. FYZIKLN EXPERIMENT .................................................................................................... 13411.3. MATEMATICK MODEL ...................................................................................................... 13911.4. VYHODNOCEN VSLEDK FYZIKLNHO EXPERIMENTU A MATEMATICKCH MODEL .. 142

12. LITERATURA......................................................................................................................... 14913. PLOHA - STAC PRAVIDLA ....................................................................................... 152

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13.1. DEFINICE A PRAVIDLA ....................................................................................................... 15213.2. PKLADY .......................................................................................................................... 15313.3. SROVNN S VEKTOROVM OZNAENM .......................................................................... 15413.4. SUBSTANCILN DERIVACE ............................................................................................... 155

14. PLOHA - ZKLADN STATISTICK METODY ........................................................... 15714.1. STEDN HODNOTA ........................................................................................................... 15714.2. REYNOLDSOVA PRAVIDLA ................................................................................................ 15814.3. ROZPTYL, SMRODATN ODCHYLKA, TURBULENTN INTENZITA, KOVARIANCE, KORELACE 159

15. PLOHA - ROZKLAD V TAYLOROVU ADU .............................................................. 16115.1. DEFINOVN PROBLMU ................................................................................................... 16115.2. TAYLORV ROZVOJ ........................................................................................................... 161

List of used designation

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LIST OF USED DESIGNATION Poznmka: oznaen, u nho nen uveden rozmr, reprezentuje obecnou promnnou.

a okamit hodnota veliiny a fluktuan sloka veliiny a asov stedovan sloka veliiny a obecn vektor

S ,A plocha m2 A Van Driestova konstanta 1

DC konstanta 1 C empirick konstanta 1 C konstanta 1

1C empirick konstanta 1

2C empirick konstanta 1

3C empirick konstanta 1 c rychlost zvuku ms-1

vc mrn tepeln kapacita pi konstantnm objemu Jkg-1K-1 pc mrn tepeln kapacita pi konstantnm tlaku Jkg-1K-1

mnD , difzn koeficient pro pms n ve smsi m2s-1 f frekvence s-1

ii Ff , sla N

cf Coriolisv parametr s-1 E energie Jkg-1 E empirick konstanta 1 JFi i-t sloka sloka vektoru hustoty toku veliiny J

G termick produkce turbulentn kinetick energie m2s-3 Gr Grashofovo slo 1 g thov zrychlen ms-2

fh souinitel pestupu tepla Wm-2K-1


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h statick entalpie Jkg-1 oh celkov entalpie Jkg-1

I intenzita turbulence % kji

, , jednotkov vektor ve smru os x, y, z 1 J bilancovan veliina

wpe JJJ , , hmotnostn tok pes stny konench objem kgm-2s-1 k turbulentn kinetick energie m2s-2

Pk turbulentn kinetick energie v logaritmick vrstv m2s2 I , L dlkov mtko turbulence m ml smovac dlka m

M Machovo slo 1 jn j-t sloka normlovho vektoru 1

n vektor vnj normly k ploe 1 p tlak Pa

opp operan tlak Pa

sp statick tlak Pa P mechanick produkce turbulentn kinetick energie m2s-3 JP hustota produkce veliiny J

Pr molekulov Prandtlovo slo (stnov funkce) 1 h ,Pr t turbulentn Prandtlovo slo 1

Q prtok m3s-1 r mrn plynov konstanta J.kg-1.K-1 q tepeln tok Jm -2s-1 R rezidul

R normalizovan rezidul 1 Ra Rayleighovo slo 1 Re Reynoldsovo slo 1

yRe turbulentn Reynoldsovo slo buky 1

wr polohov vektor na stn m S modul tensoru stedn rychlosti deformace s-1


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jiS , tensor rychlosti deformace s-1 Sc Schmidtovo slo 1 t as s T absolutn teplota K

iu i-t sloka rychlosti ms-1

iu i-t sloka stedn rychlosti ms-1

iu i-t sloka fluktuan rychlosti ms-1 * , uu rychlost definovan stnovou funkc ms-1

u tec rychlost ms-1 V objem m3

ix souadnice v kartzskm systmu x1, x2, x3 nebo x, y, z y kolm vzdlenost od stny m

* , yy bezrozmrn veliina pi odvozovn stnovch funkc 1 * vy bezrozmrn tlouka podvrstvy 1

vy tlouka vazk podvrstvy m

Py vzdlenost bodu P od stny ve smru normly m relaxan faktor 1 teplotn vodivost m2s-1

,, k inverzn turbulentn Prandtlova sla m2s-1 souinitel teplotn roztanosti K-1

ij Kroneckerovo delta 1 rychlost disipace m2s-3

P rychlost disipace v logaritmick vrstv m2s-3 disipan funkce

t turbulentn difusivita m2s-1 von Krmnova konstanta, pomr mrnch tepelnch kapacit 1 souinitel tepeln vodivosti Wm-1K-1

t souinitel efektivn tepeln vodivosti Wm-1K-1


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dynamick viskozita Pas eff efektivn viskozita Pas t turbulentn viskozita Pas

kinematick viskozita m2s-1 t turbulentn viskozita m2s-1 ij celkov tenzor napt Pa

hustota kgm-3 ref referenn hustota kgm-3 k empirick konstanta 1 empirick konstanta 1 h turbulentn Prandtlovo slo 1

asov perioda s vazk napt Pa

ij tenzor vazkch napt Pa w vazk napt na stn Pa t turbulentn napt Pa

druh viskozita Pas obecn promnn Indexy:

i index sloky rychlosti, index iterace 1 i stac index 1

lkj , , suman Einsteinv index 1 pew , , index stny konenho objemu 1

NBBFSNPW , , , , , E, ,

index konenho objemu 1

Preface

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Preface This textbook is addressed to for students of master and doctoral programs of all

faculties who want to learn about the fundamentals of numerical modeling of fluid phenomena

in transfer, ie transfer of mass, momentum (torque), heating, etc. in laminar and turbulent flow.

In applications in teaching CFX and Fluent software will be used.

Numerical modeling of many physical phenomena are closely linked to modeling a form

of motion mathematically. Movement of fluids is associated with the solution of various

problems of the physical model:

laminar and turbulent flow in both simple and complex geometries compressible and incompressible flow steady, unsteady and transient flow heat transfer, natural and mixed convection, radiation transfer of chemical admixtures, including chemical reactions multiphase flow, free surface flow, flow with solid particles, bubbles, respectively. drops combustion and chemical reactions porous flow, etc.

The mathematical model consists of defining the equations describing the above happens.

Given that this is a two-dimensional, axially symmetric or general three-dimensional and time

dependent cases, which are described by system of partial differential equations, which can

be solved by numerical methods. Their use is given by the expansion of knowledge in the field

of flow, turbulence, numerical methods, computer technology

The flow solution is possible to use commercial software systems, such as Fluent, CFX

and others. The task of the user is to build the correct calculation model, which contains some

mathematical, physical and technical principles. For such a model it is necessary to find all

input data in the existing standards, build the input data for a program that can solve

computational model, solving by the terminal, correctly interpret the results for further use in

all phases and carry out effective monitoring of all inputs and outputs. The user must safely

divide all the information on the geometric data (two-dimensional or three-dimensional

features, topology), data on the effect of external forces and physical data (information about

flowing media, its physical properties). Thus, an essential task is to know hydromechanics,

thermodynamics and other sciences to the complexity of the problem.

As regards the computational methods underlying the use of the program, the designer

should know their nature for reliable use in standard cases. For program or Fluent. CFX it is a

need to know in what shape the final volume will work, it follows the choice of network density,

Preface

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approximation schemes, the nature of the time dependence of the quantities and the resulting

time step size, etc.

No less important part there is the evaluation results, particularly difficult in three-

dimensional problems. It is optimal to have at least the approximate value of calculated vriables

, better is to compare the results with the experiment.

Literature

Bojko, M.: Nvody do cvien Modelovn proudn Fluent Blejcha, T.: Nvody do cvien Modelovn proudn CFX.

Introduction to numerical modeling of fluid flow

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1. Introduction to numerical modeling of fluid flow 1.1. Introduction to fluid flow

Fluid flow can be divided according to several aspects 0:

Dividing by the physical properties of fluids

Dividing the flow according to time

steady (stationary) flow does not depend on time tvv ; 0t

unsteady flow flow is, in which variables dependent on time, tzyxvv ,,, ; tsvv ,; tvv .

z (x3)w (u3) v (u2)

y (x2)

u (u1)

x (x1) Fig. 1.1 Coordinate system

In the most general case, the

position of the point is defined by

coordinates zyxX ,, respectively 321 ,, xxxX . Velocity vector is defined by

components wvu ,,u respectively 321 ,, uuuu . The designation is shown in Fig. 1.1

1.2. Transfer of mass, momentum, heat in non isothermal flow of incompressible fluid

Basic laws of physics describing the flow are the conservation laws of mass,

momentum, heat or other scalar variables. They are expressed by Navier Stokes equations

FLUID FLOW

flow of ideal (inviscid) fluid

flow of real (viscous) fluid

laminar turbulent potencial swirling


2

along with continuity equation and describe the laminar and turbulent flow regime. In the case

of non isothermal unsteady incompressible flow they have the following form:

Continuity equation:

Ozw

yv

xu

( 1.2.1)

Navier-Stokes equation:

xfz

uyu

xu

xp

zuw

yuv

xuu

tu

2

2

2

2

2

21

yfz

vyv

xv

yp

zvw

yvv

xvu

tv

2

2

2

2

2

21

zfz

wyw

xw

zp

zww

ywv

xwu

tw

2

2

2

2

2

21

( 1.2.2)

where using schema in Fig. 1.1 u , v a w are the velocity components, p is pressure, is density, kinematic viscosity ane zyxf ,, refers to components of volume or external forces (gravity, centrifugal force).

The equation of heat transfer, ie. the law of conservation of energy is in the form

2

2

2

2

2

2

zT

yT

xT

zwT

yvT

xuT

tT

222

222

2

yw

zv

xw

zu

xv

yu

zw

yv

xu

( 1.2.3)

where pc

is temperature conductivity, is molecular heat conductivity and pc specific

heat.

In terms of variables with three or nine components (components of speed, stress, etc.)

it is appropriate use special abbreviated designation with precisely defined rules, known as

Einstein's summation, see chap. 13, where only one member can express all three

components of velocity respectively nine stresses. The same can be expressed

mathematically for clarity using character sums. Thus, the continuity equation is written simply:

Oxu

xu

xu

3

3

2

2

1

1

resp. Oxun

j j

j 1

resp. Oxu

j

j

( 1.2.4)


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Navier-Stokes equations:

i

n

j j

i

i

n

j j

jii fxu

xp

xuu

tu

12

2

1

1

resp.

i

j

i

ij

jii fxu

xp

xuu

tu 2

21

, ni ,...,1 ( 1.2.5)

where the index i consistently expresses the component of vector and the index j (or other

alphabetically) represents the summation index ( 3 resp. 2,1j ). The equation for heat transfer can be written similarly:

2

2

n

nj j

n

nj j

j

xT

xTu

tT ,

n

j

n

l j

l

l

j

xu

xu

1 1

2

21

resp.

2

2

jj

j

xT

xTu

tT

2

21

j

l

l

j

xu

xu

( 1.2.6)

1.3. Partial differential equations and boundary conditions The equation for conservation of mass and momentum, energy equation and the

equation for the transport of chemical admixtures in the general conservative form create a

system of partial differential equations, while all the equations can be formally written

Sxxuxt jjjj

accumulation = convection + diffusion + source

( 1.3.1)

where is a variable and members on the right side are gradually convection, diffusion and source term, so the equation is also called convection-diffusion equation. If is a temperature, admixture or another scalar value, then it is a linear equation of second order. If is a velocity component, this equation can be regarded as a nonlinear equation of second order.

If the effect of diffusion member dominates, this is the equation of elliptic type, in

parabolic type equations the convective transport impact prevails and hyperbolic equations are

significant by pressure changes. Equation (1.3.1) but alone is not sufficient to determine the

function . In order to define this function, you need to know more additional conditions.


4

Let the problem to find the temperature distribution in the body is defined. Suppose that

finding function T (ie. temperature) is a function of subsequent spatial independent variables and time

T=T(t,x)

22

xT

tT

( 1.3.2)

T=T(t,x, y)

22

2

2

yT

xT

tT

T=T(t, x, y, z)

2

2

2

2

2

2

zT

yT

xT

tT

.

If the function T depends only on the spatial coordinates and independent of time, the task

is called stationary. The problem is complicated if the function T dependends on time and possibly the spatial coordinates, as mentioned above. For example the body is heated or

cooled with increasing time, then the case is called transient and function T is a function of successive independent variables.

INDEPENDENT VARIABLES

stationary flow transient flow t

x t, x

x, y t, x, y

x, y, z t, x, y, z


5

Example

T=T1

t

T=T2

x L

D

T(x)=(x)

Fig. 1.2 Definition af region

For example equation of heat

conduction in a rod is

22

xT

tT

For the heat equation in a rod a solution

is looking for in a rectangle txD , , see Fig. 1.2, and must satisfy the initial

condition (an analogy with the solution

of ordinary differential equations).

For the temperature T the initial condition is defined as follows:

LxxxT 00, ( 1.3.3) In the event that the second derivative T with respect to t occurs in the equation, the second

initial condition is defined as xtxT

0, .

The boundary condition defines the temperature at the beginning and end of the field

(edge rod).

tTtLT

tTtT

2

1

,,0

( 1.3.4)

The role of finding a solution of equation (1.3.1) satisfying the boundary and initial

conditions is called a mixed role. If the boundary conditions equal zero, they are called

homogeneous boundary conditions, similar to the initial conditions are zero, they are called

homogeneous initial conditions. Instead of boundary conditions (1.3.4) it can be put another

type of conditions that are also called boundary conditions. Reflection on the boundary and

initial conditions for temperature is valid for a general variable .

1.4. Boundary conditions of incompressible fluid flow

1.4.1. Types of boundary conditions Boundary conditions need not be only a constant value, but they can have values defined

by functions, table, etc. (0):


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constants .consty polynom. funkction ...)( 2210 xAxAAxy , where the coefficients are awarded only to 5

significant digits

derivative according to normal (OUTLET, heat flux) .1konst

xxy

piecewice linear function NN yxyxyxyx ,...,,,,,, 332211 combination of polynom. and piecewice linear

function

Fig. 1.3 Profiles of velocity

Conditions of input and output. For the

two flow boundaries may occur only

following basic combinations of boundary

conditions, (a combination of input velocity

and output velocity can not occur because

the velocity on the second input is calculated

from the equation of continuity). When

considering the energy equation there is

entered in addition the value of temperature,

see Fig. 1.4.

outlet rychlost tlak stat. rychlost tlak stat. tlak tot l. Fig. 1.4 Combination of input and output

boundary conditions

Conditions at the flow border - the input and output flow borders can be defined by three types of boundary conditions

INPUT OUTPUT

velocity u condition of stationary flow

0,0,0

nT

np

nu

Total pressure totp

2

21 upppp statdynstattot

static pressure statp

temperature T

Conditions on the wall - the wall can be stationary (fluid velocity is zero) or moving (eg

rotating or sliding), with friction or without friction, smooth or rough. The temperature is given

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.5 1 1.5 2

u [m.s-1]

y[m

]

u-konst

u-polynom

u-po stechlin. funkce


7

by value or by derivation according to nomal vector to the wall, for example 0

nT is insulated

wall.

Conditions of symmetry - zero normal velocity and zero normal gradients of all search

parameters, see Fig. 1.6.

Conditions of axial symmetry - define the axis of the axially symmetric two-dimensional problems, see Fig. 1.5.

axis of symetry

solved region

Fig. 1.5 Cylinder with axis of symmetry and

plane, where the flow is solved.

plane of symetry

solved region

Fig. 1.6 Cylinder with plane of symmetry

and region, where the flow is solved.

Periodic (cyclic) conditions - used when the flow formations are repeated, they can be

rotational and translational type, which allows the definition of the pressure gradient in the

direction of fluid flow along the field.

All types of conditions can be time dependent, if required by their character. Other

boundary conditions do not concern of the flow as such, but other values arising from the

complexity of the mathematical model as, temperature, heat flow, radiation, mass fractions (or

molecular fraction). additions, etc.

periodical condition periodical condition of rotational type of translation type

solved region solved region Fig. 1.7 Periodical conditions of rotational and translational type


8

1.5. Solution of the flow in sudden expansion section This chapter illustrates how to enter and solve the incompressible flow in a sudden

expansion cross section using Fluent, which is one of the fundamental tasks of flow testing,

examined both experimentally and numerically. The task is: define the physical model, physical

properties of the flowing medium and walls:

define the physical model, physical properties of the flowing medium and walls define mathematical model, boundary condition create geometry and mesh specify the initial and boundary conditions in FLUENT, the calculation evaluate the computed values

Example

The physical model is given by the shape, type of flow and hydraulic parameters of

flow. The scheme of the region is shown in Fig. 1.8 and dimensions with the physical properties

in the table.

Fig. 1.8 Geometry of the problem

Geometry of the region

length of the area L [m] 3.5

the area height d [m] 0.5

expansion ss xdL [m x m] 0.7 x 0.1

Physical properties of the fluid We suppose the flow of incompressible fluid with following parameters:

u

d

L

zOblast proudn d-ds

ds

Ls


9

density of fluid [kg.m-3] 1.2 dynamic viscosity resp. [kg.(m.s)-1] 0.0000171

Boundary condition

Boundary conditions are defined at the input by the velocity and at the output there is

determined the condition of steady flow (OUTFLOW). On the walls it is assumed zero velocity.

input mean velocity su [m.s-1] 3

output OUTFLOW

Reynolds number dusRe

[1]

84705

Mathematical model

From the boundary and physical conditions the Reynolds number can be determined

as the dimensionless criterion for deciding whether the flow is laminar or turbulent. The value

of Reynolds number characterizes the flow in the transition region between laminar and

turbulent flow. It will therefore be solved at beginning as laminar.

Creating geometry and mesh In a Ansys the exact geometry will be created by method similar to the CAD programs.

In addition, this program will use the possibility to form a mesh. The resulting mesh should

have the area shape taking into account the development of swirl and velocity profile near the

wall, see Fig. 1.9 Computational mesh in longitudinal section.

Fig. 1.9 Computational mesh in longitudinal section


10

The calculation results

For clarity there are introduced the possibilities of evaluation, such as velocity vectors, velocity

profiles and filled isolines.

Fig. 1.10 Vector velocity u

Fig. 1.11 Detail of region with vortex (velocity vectors u).


11

Fig. 1.12 Filled isolines of static pressure

1.6. Water flow between the plates numerical solution Solve the water flow between two infinitely large plates. The physical model is determined by

the shape of the area, the type of flow and hydraulic parameters of flow. Numerical calculation

define in the programming environment ANSYS Fluent. The creation of the computational

domain (geometry) and of computer networks and uses programs DesignModeler ANSYS

Meshing.

x

y

l

inlet

solved domain

lower wall

s

outlet

upper wall


12

Fig. 1.13 Scheme of domain

Water flows inside by speed 0.05 ms-1 and extends into the open air where the relative pressure is equal 0 Pa. Scheme area including the dimensions is shown in Fig. 1.14 and Tab. 1.1. Task for simplicity is given as a 2D model and represents the flow in a rectangular gap of the length and thickness of the gap. Physical properties of the flowing medium are given in Tab. 2.2

Fig. 1.14 Geometry of 2D rectangular computational domain

Tab. 1.1 Geometry of domain

length of domain l [m] 0.5

thickness of the gap s [m] 0.033

Tab. 1.2 Physical properties of water

Density of water [kg.m-3] 998 viscosity [kg.(m.s)-1] 0.001003

Boundary conditions At the entry velocity boundary condition (VELOCITY INLET) is defined and the output is given by the condition of static pressure (OUTLET PRESSURE). On the wall there is a boundary condition of type WALL, where it is assumed zero velocity (is predefined). The boundary conditions are given in Tab. 2.3

. Tab. 1.3 Boundary conditions inlet mean velocity s

u [m.s

-1] 0.05

Outlet static pressure p [Pa] 0

Mathematical model

x

y

l

inlet

fluid

lower wall

s outlet

upper wall


13

Selection of a mathematical model will be resolved in later chapters, now it leaves predefined one (laminar flow model). Creating of geometry and networks To create a domain and network the DesignModeler program and ANSYS Meshing is used. Programs are available in the ANSYS Workbench program manager. After starting Workbench is available environment shown in Fig. 1.15. On the left side of the tools Toolbox - Analysis Systems" the "Fluid Flow"(FLUENT) is selected and draged by the mouse with the left button and moved the item to the desktop (Project Schematic). This is a new project, which can be renamed (eg. 2D-plate, see Fig. 1.15).

In the first phase, the program will create the geometry in DesignModeler using the "Geometry". When the default startup command DesignModeler "Geometry" user creates 3D geometry. If you need to create 2D geometry, it is necessary to check the right panel, under Properties of Schematic A2: Geometry - Geometry Advanced Options - Analysis Type - 2D and that will create 2D geometry. This setting is induced by right-click the command "Geometry" and select "Properties". Then the program starts DesignModeler by click "Geometry".

Fig. 1.15 Software Environment Workbench

After starting the program DesignModeler a graphical environment is displayd (Fig. 1.16). On the left side there is a tree of individual operations to be carried out in the program. In the middle of the graphics window there is the work area and in the upper part there are the


14

individual commands.

Fig. 1.16 Programming environment DesignModeler

To create geometry of Fig. 1.13 is relatively simple because it is a rectangular area. The basic philosophy of geometry creation consists in drawing in the various levels of the coordinate system (XY, XZ, YZ) defined in the the so-called. "Sketch". This means that at first there is defined a "Sketch" in the corresponding plane of the coordinate system and thus in defined "Sketch" is form a rectangle. Creating a "Sketch" in the selected plane of the coordinate system is shown in Fig. 1.17.

Fig. 1.17 Creating Sketch

Subsequently, the rectangle is formed by drawing individual edges, forming a closed unit. To drawing the edges the tool from the "Draw" in the "Sketching" is used. To create a individual entities of rectangle the "Line" is used with help of mouse to mark the two points in the workspace and create the edge (Fig. 1.18). By the same procedure all the edges will be

Working area

Commands

operations

Sketch

Plane of coordinate system


15

continuously created and closed whole rectangle (Fig. 1.19).

Fig. 1.18 Creating of individual edges

Fig. 1.19 Area of computational geometry

Subsequently, the calculation area is defined using the "Dimensions" (horizontal and vertical dimensions), see Fig. 1.20. Dimensions H1 and V2 are specified according to the assignment


16

and subsequent updating of sizes is made by the command "Generate".

Fig. 1.20 Dimensioned computational domain

The subsequent step is to create areas from individual edges using the "Surfaces From Sketches" (Fig. 1.21). The command is available from the pull-down menu in the "Concept". Surfaces can be created from one or more "Sketch". In our case, we have created single "Sketch".

Fig. 1.21 Command Surface From Sketches

The "Surfaces From Sketches" is displayed in the tree and select a "Sketch" command in detail. After correctly choosing the selected "Sketch" is highlighted in light blue color (Fig. 1.22). The resulting creation of area confirm by the command "Generate" .

Command Generate

Dimensions


17

Fig. 1.22 Selecting "Sketch" to create surfaces

The resulting surface after generation is shown in Fig. 1.23.

Fig. 1.23 The resulting surface

The next step is to create a computational grid in ANSYS Meshing, which starts from the Workbench environment like DesignModeler program. When you return to the Workbench environment, ANSYS Meshing is opened using the "Mesh" in created project, see Fig. 1.24.

Command Generate

Surfaces From Sketch

Selection of Sketch for surface


18

Fig. 1.24 Starting ANSYS Meshing

Before starting the ANSYS Meshing must be in the "Geometry" green check mark (tick), as shown in Fig. 1.24. The green check mark indicates that the created geometry is fine. Otherwise, it is necessary to return to the program DesignModeler and make adjustments geometry. Environment ANSYS Meshing is shown in Fig. 1.25. To create a computational grid in 2D geometry the square (rectangles, squares) or triangular (triangles) elements can be exploited. In this application the calculation region is in a rectangular shape so as elements shall be used regular square elements. The possibility of creating a computer network are numerous. In this case, a constant size of the elements is defined. Determining of the elements is based on the assumption that in the transverse direction (y-axis, see Fig. 2.14) it is to have at least 20 elements. Dimension in the transverse direction is 0.033 m, so that the size of the elements is 0.00165 m. This size will be defined for the elements in the longitudinal direction. Setting the parameters for generating a computational mesh with constant element sizes is shown in Fig. 2.27. After defining the size of elements in computational grids the creation of grid is done with the command "Generate Mesh". The resulting computational grid is shown in Fig. 2.27. The process how to get the definition of the grid parameters is also shown in Fig. 2.27.

Starting ANSYS Meshing


19

Fig. 1.25 ANSYS Meshing

Fig. 1.26 Setting the size of elements of computational grid

The last step before defining the mathematical model is designation and naming of boundary conditions on the borders of each model, which in this case corresponds to the individual edges of the rectangle. Boundary conditions must conform to the conditions shown in Fig. 1.13.


20

Fig. 1.27 Identification and description of boundary conditions

Identification and naming of edges as boundary conditions is performed using the "Create Named Selection" (Fig. 1.28). The command is activated by using the mouse, select the appropriate earliest edge (eg. Input - Fig. 1.13) and right-click from the menu commands, choose "Create Named Selection". The given boundary condition can then be arbitrarily named. Names of boundary conditions for subsequent clarity in ANSYS Fluent will be defined by Fig. 1.13. The designation and the naming of all boundary conditions is shown in Fig. 1.28.

Fig. 1.28 Identification of boundary conditions


21

After the creation computational grid, labeling and naming of all boundary conditions go back to the Workbench environment. ANSYS Fluent starts using "Setup" (Fig. 1.29). Before running the "Setup" it is necessary to ascertain whether the items "Geometry" and "Mesh" have a green check mark. If not, then you must update the project with the command "Update Project".

Fig. 1.29 Starting ANSYS Fluent

Work in Fluent The first step is to run Fluent, where you define size of the area (2D) and in particular,

whether the calculation will be carried out with conventional or double precision, see Fig. 1.30. Size of the area is automatically detected, which in this case is a 2D. Calculation of double precision is selected by checking the "Options", see Fig. 1.30.

Fig. 1.30 Input parameters when running Fluent.

Work continues by confirmation of choosing OK. Follows checking network, showing all

Update of project

Start of ANSYS Fluent


22

borders and throughout the area command "Display". Select of all items in the "Surface" appears boundary conditions and network. The network is to illustrate very rough. Practically, it will be much smoother and compressed toward the walls.

Fig. 1.31 Rendering grid and boundary conditions

It is necessary to check the dimensions of the network "Mesh-Scale" where you see the real size area. If the computational domain is created in other dimensions (mm, cm, ...), you can use the "Scaling" and "Specify Scaling Factor" convert dimensions to the units meters (m), see Fig. 1.32.

Fig. 1.32 Check the dimensions of grid ("Mesh Scale")

Another control relates to the number of grid cells "Mesh-Info-Size". It appears in the text box line with information on the number of cells (Cells), faces (Faces) and nodes (Nodes) of grid, see Fig. 1.33. The "Mesh Info Size" is available in the pull-down menu under Mesh.


23

Fig. 1.33 View the number of cells, areas and nodes.

Follows the checking of the existence of negative volumes in the network by command "Check Mesh", see Fig. 1.34, which occurs in complex geometries and in this case it is necessary to create a network again.

Fig. 1.34 Checking the network.

If all settings are correct, you proceed the menu from left to right and top to bottom. Follows command "General", where you set the type of "Solver", and especially "Steady" or "Transient" for the task stationary or time-dependent. It is necessary to check whether the problem solves 2D planar "Planar", axially symmetric "Axisymetric" or axially symmetric rotation "Axisymetric Swirl". In Fig. 1.35 we can see, that there is possible to define the external force (eg. Gravity) with acceleration "Gravity" in any direction and to change the physical units "Units" from SI unit system to another unit or only selected variables.


24

Fig. 1.35 The "General"

In the "Models" (see Fig. 1.36) there is defined physical nature of the task according to the very visual offers, whether it is a multi-phase flow "Multiphase", flow with heat "Energy", flow of admixtures "Species", flow with discrete phases "Discrete Phase" or what type of flow "Viscous. Here you can define laminar flow, turbulent flow using various turbulence models and you can also solve the special case of ideal fluid flow "inviscid". Comments will be made in the relevant chapters later. In this application, define the laminar flow model.

Fig. 1.36 Choosing a model for the solution of flow

In Materials their physical properties can be defined, first flowing medium "Fluid" and, secondly wall material "Solid", see Fig. 1.37. If the material does not appear in the main menu "Materials", then it must be connected in the "Create / Edit Material" button "Fluent Database"


25

where you will select and use the "Copy" to join the list of materials. Physical properties (Density, Specific Heat, Thermal Conductivity, Viscosity, ...) are shown at the bottom of the menu and can be changed according to the requirements of the investigator.

Fig. 1.37 Entering the appropriate materials and physical properties

The flowing material is water, so the material will be loaded into the system from the "Cell Zone Conditions" command "Edit", see Fig. 1.38. Especially when inserting new materials, it is necessary to check their choice and confirm here, otherwise there will remain first in the list of material.


26

Fig. 1.38 Definitions of suitable material.

Follows the definition of boundary conditions. For each condition it is necessary to check and correct these items. Offers are more complicated in the case of more complex models of flow. Setting of input and output boundary conditions is shown in Fig. 1.39 and Fig. 1.40.


27

Fig. 1.39 Input boundary condition.

Fig. 1.40 Output boundary condition.

Subsequently, the initialization of the current flow field is made, ie. Definition of initial conditions


28

inside the entire area. The values are defined based on the input boundary conditions, see Fig. 1.41.

Fig. 1.41 Initialization calculated from input boundary conditions

After initialization, the iterative calculation starts. You must specify the number of iterations (Number of iterations). Default value is 0. The entered value is high enough, eg. 10000, when it is assumed that the convergence is reached (Fig. 1.42). Convergence can be monitored both graphically and numerically. List of residuals is activated commands "Display / Residuals / Residuals Monitors".


29

Fig. 1.42 Set the number of iterations and start calculating

Progress of residuals is shown in Fig. 1.43. Residual values (relative error) for each computed variable (pressure - continuity, velocity in the direction x - x-velocity and the velocity in the y direction - the y-velocity) must be less than 0.001. On reaching this accuracy the calculation is itself terminated.

Fig. 1.43 Progress of residuals

The results of the calculation with a constant mean velocity at the inlet


30

The inlet velocity boundary condition is defined therefore the constant value of the mean velocity

05.0su ms-1 . Subsequently, the initialization of the flow field is done and the numerical calculation starts. To illustrate the evaluation the velocity vectors, velocity profiles and filled isolines are displayd.

The velocity vectors are defined in each cell of the computational domain by command "Display /

Vectors / SetUp / Vectors", where it is possible to define vectors of selected quantity (the most common

is velocity and is also predefined) painted by different variable (eg. Temperature). The "Scale" will

allow to increase the size of the vector and "Skip" to skip a given number of vectors to be less dense.

Thus for Scale = 10 and Skip = 10 result is apparent from Fig. 1.44.

Fig. 1.44 Velocity Vectors with Scale=10 and Skip=10 (u [ms-1])

For better overview it is possible to create the helper sections of given coordinates in which the vectors

appear. Create a section of the value of x = 0.1 m can be made eg. by typing "Surface / Iso-suface /

Surface of Constant / Mesh / X-Coocordinate" and set the Iso-Values 0.1 and call it such X-coordinate-

0.1, see Fig. 1.45.


31

Fig. 1.45 Creating helper section for x = 0.1 m

In individual sections and boundaries the velocity profiles are defined by the command "Display /

Vectors / SetUp / Vectors". In the "Surfaces" used planes are selected, see Fig. 1.46. Then the vectors

are plotted by command "Display", see Fig. 1.47.

Fig. 1.46 Menu for the creation of the velocity vectors in the individual sections

Fig. 1.47 Velocity vector in the individual sections ( u [ms-1])


32

The evaluation shows that the length of the computational domain leads to gradual formation of the

parabolic velocity profile. To achieve the desired flow profile (from the previous solution) the

computational domain is short. The contours of velocity magnitude are shown in Fig. 1.49 and rendered

with command "Display / Contours / Contours of / Velocity". Next, specify whether the magnitude of

velocity, velocity component or other variables related to velocity will be drown. "Levels" defines the

number of isosurfaces. By checking the "Filled" in the "Options" the filled isolines appear, otherwise it

is peers, see Fig. 1.48.

Fig. 1.48 Menu for the creation of filled-velocity magnitude isosurfaces

Fig. 1.49 The contours of velocity magnitude in the computational domain ( u [ms-1])

Similarly the plotting of isolines of static pressure is setting in Fig. 1.50.


33

Fig. 1.50 Filled isolines of static pressure in the computational domain statp [Pa]

Further evaluation presents the velocity profiles in the various sections from inlet to outlet with a step 0.05 m along the length of the computational domain, see Fig. 1.52. This figure is very instructive, if it is necessary to compare the profiles of quantities in input, output, or in other sections of the field. It is set with command Display/Plot/Y Axis Function/Velocity/Velocity Magnitude. In the "Plot Direction" we put the value 1 for X, if we plot the dependence on X or value 1 for Y if we plot the dependence on Y. Other variables are equal zero, see Fig. 1.51.

Fig. 1.51 Menu for the creation of velocity profile

The results indicate the formation of the velocity profile from the constant velocity value at the inlet to the parabolic velocity profile at the outlet from the area (Fig. 1.52). The first image is a copy of the screen and is suitable for fast orientation in the result. Far better result is obtained by checking in the "Options / Write to File", which will offer a filename (file is in text format). This file is then read and modify in Excel. The result is in the second part of the picture.


34

Fig. 1.52 Velocity profiles in different sections of the computational domain ( u [ms-1])

Another evaluation of the progress of static pressure along the length of the computational domain is at the Fig. 1.53. Static pressure is evaluated in the axis of the computational domain.


35

Fig. 1.53 Progress of static pressure evaluated along the axis of the tube ( statp [Pa])

Geometry and computational grid generation

36

2. Geometry and computational grid generation 2.1. The term "grid" and its importance for the mathematical

modeling The grid is a system of calculated field distribution in the partial consecutive cells in 2D

or 3D cells in two-dimensional or three-dimensional space ([23]). We can say that

computational grid coverage is the basis of mathematical modeling. Since a separate

mathematical model (a system of mathematical relations) is only a "passive" tool, which takes

effect only when it is applied to a specific problem (computational network coverage).

If you talk about mathematical models that are based on the numerical solution of

partial differential equations and thus require input and boundary conditions, we can say that

the possibility of realizing the tasks are very limited power of computer technology. This applies

several principles here:

calculation is even more difficult (slower), the more equations in the mathematical model

included in the calculation (according to the demands and complexity of the model);

calculation is even more difficult, the more the cell is calculated;

calculation is even more difficult, the less good network is prepared for calculation.

2.2. Workbanch, grid elements Workbanch is much broader than just the physical drawing and networking. Fig. 2.1

schematically illustrates the basic structure of the engagement of other programs into the

network.


37

Fig. 2.1 Function of program Workbanch.

Numerical finite volume method is based on a system of non-overlapping elements,

finite volumes. Originally, the finite volume method is based on finite volumes of curvilinear

rectangles, rectangles in two-dimensional case and blocks or general hexahedron in three-

dimensional problems (see Fig. 2.2). Such a grid is called a structured grid. The fundamental

rule is that the elements boundary must adjacent with the single element boundary, so we can

not arbitrarily refine grid (it is analogous to the method of finite differences, including the

possibility of using indexing). Also, the resulting computational region is then a block or a

rectangle. Today there is starting to take a new approach of building a so-called unstructured

grid. The final volume is a 3D cuboid, tetrahedron, and pyramid prismatic element, the benefits

have been validated in problems of elasticity, solved by finite element method.

FLUENTBasic structure of programWorkbanch- FLUENT

FLUENT 1. Geometry, generation of grid2. Physical model 3. Boundary conditions 5. Physical properties 6. Parameters of calculation 7. Calculation 8. Evaluation, result interpretation

Meshing -Definition of geometry -2D/3D grid

other CAD/CAE systems

TGRID -2D/3D grid -hybrid grid

grid

geometry grid

2D/3D grid grid on the border of region

grid on the border of region /in volume


38

Fig. 2.2 Shape of finite volume

The above listed elements can now be combined to obtain the optimal grid in the walls vicinity

where the rectangles and blocks are used (to get more accuracy) and in other areas where

there are no large gradients of solved variables due to existence of the boundary layer, we

apply the remaining elements. They provide an easy change of grid density, see Fig. 2.3

Fig. 2.3 Using of different types of elements

2.3. Criteria for grid quality assessment Grid quality is assessed through:

cell size (with respect to the modeled process and the requirement for accuracy of calculation)

suitability of structure in the cell compartment (eg grid refinement of interesting places in terms of fluid flow) with respect to the particular type of task

quality of the cells (asymmetry - Skewness, the ratio of edges (faces) elements - Acpect Ratio, etc.)

block tetrahedron pyramid prizmatic element


39

The most important criterion for cells quality assessment is the asymmetry when

assessing how much the cell shape closes to the ideal regular geometric shape in accordance

with the appropriate grid schema. If the cell is deformed in any way, its quality is worse.

Generally, the quality of each cell is expressed by dimensionless number in the range 0 - 1,

where 0 is the best result and 1 is worst result on the contrary, that is problematic cell for

calculations. This value is called the "degree of cell skew " (or "skewness measure"), or the

degree of deformation.

Fig. 2.4 The principle of assessing the cell

quality for 2D triangular elements ( "TRI")

Fig. 2.5 The principle of assessing the cell

quality for 3D grid formed by tetrahedrons (

TET)

To determine the quality of 2D cells, respectively. its rate of deformation is the following

relationship:

optimal

realoptimal

SSS

TRImeasureSkewness)( ( 2.3.1 )

where optimalS presents optimal cell area, realS real cell area (which may or may not be

optimal) and )(TRImeasureSkewness is term for "peace cell deformation" relating to the

2D triangular diagram of the grid. For other schemes we use similar logic. The resulting value

should not exceed 0.85. Should this happen, you need a cell or a grid structure to edit, order

not to endanger feasibility and accuracy of the calculation.

To determine the quality of the 3D cells corresponding to grid diagram consisting of

tetrahedrons similar relationship is applied:

optimal

realoptimal

VVV

TETmeasureSkewness)( , ( 2.3.2 )

where optimalV presents optimal cell volume, realV real cell volume and

)(TETmeasureSkewness is a term for "peace cell deformation" relating to the 3D

tetrahedron diagram of the grid. The resulting value should not exceed 0.9. Should this happen,

Optimal area (equilatteral)

Aktual area

Circumscribed circle

Optimal cell (equilateral)

Aktual cell

Teoretical circumscribed sphere area


40

you need a cell or a grid structure to edite.

Fig. 2.6 Approximate color range to assess the quality of grid cells

(0 - highest quality cells, 1 - the lowest quality cell)

The quality of the network can be tested and count in preprocessor Gambit 2.2.30, using

"Examine Mesh". After entering this command window it is available to provide several options

to control local or global mesh quality across the computing field. The test result is a grid cells

in color hue, which corresponds to the approximate range from color quality level of the cell.

2.4. Pklad vytvoen st Pro ilustraci je prezentovno

vytvoen geometrickho modelu a st pro een proudn ve ventilu pi maximlnm vyuit dostupnch

prostedk pro konstrukci prvk (0). Model je vytvoen v CAD software Solid Edge (), kter

jako takov nem na program

Fluent, lpe eeno na program Gambit, dnou nvaznost.

Proto se tento model musel

exportovat

Fig. 2.7 ez 3D modelem otevenho pojistnho ventilu

(ry se ipkou zna smr proudu kapaliny)

(uloit) ve formtu, kter je pro Gambit srozumiteln a doke nast potebn data o geometrii modelu. Proto byl proveden export modelu ze Solid Edge do souboru o formtu IGES a STEP,

co jsou dva nejpouvanj soubory pro obecn exportovn dat, tedy pedevm geometrie modelu. Po importu do Gambitu peklada tohoto programu bohuel nebyl schopen nast celou geometrii. Proto se geometrie penesla jako mezioperace do programu Ideas, ze kterho se dle exportovala do IGES formtovho souboru a nsledn opt do programu Gambit.


41

Model je een jako konstruktrsk soust, co znamen z hlediska proudn jako skopka kolem fluidnho jdra, viz , se kterm pracuje

matematick program Fluent.

Nejdve je nutno provst separaci tohoto jdra od skopky pomoc booleanovskch pkaz. Odstrann vech objem sousti, kter nejsou v bezprostednm kontaktu

Fig. 2.8 ez modelem s vytvoenou oblast proudn

s kapalinou (pruiny, tsnn, plastov obal atd.) se provede opt pomoc booleanovskch funkc (prnik) a zstane jen objem vnitn, tedy objem fluidn (), ve kterm bude eeno proudn.

Fig. 2.9 Vnitn objem pojistnho ventilu Fig. 2.10 Vytvoen zjednoduen objem s naznaenou oblast pro zhutn

Geometrie je periodick, je tedy mono vytvoit zjednoduen objem ( pvodnho objemu), dle se zkrt vstupn pvod a provede se vyitn geometrie o velmi sklonn hrany, viz . Toto zjednoduen pomh usnadnit a pedevm urychlit vpoet.

2.4.1. Vytvoen st Vzhledem ke sloitosti 3D geometrie byla zvolena s s tetra/hybridnmi prvky (3D

tystny), kter je nestrukturovan s velikostmi hran jednotlivch element 0.17 a bylo vytvoeno zahutnm kolem ostrch hran uitm size function v mst, jak je naznaeno na . S takto vytvoenou st se vygenerovalo 1 500 000 bunk.


42

Fig. 2.11 Vytvoen s s naznaenm zhutnm kolem zvolench hran

Kvalita vytvoen st v tto oblasti byla posouzena uitm vyhodnocen kriteria

nesoumrnosti (skewness) a zobrazena v detailu graficky, viz .

Fig. 2.12 Detail st s vyhodnocenm kvality st.

V tomto ppad je vtina bunk sovan oblasti mlo nebo stedn deformovan, avak nkolik bunk se pohybuje v zn velk deformace. To vak nevad, protoe pi prci ve Fluentu bude tato sov domna pevedena na mnohostnn buky s jinou kvalitou st z dvodu vraznho urychlen vpotu.


43

2.4.2. Definition of the boundary conditions in Gambit

Gambit is also a tool to define the properties of the walls in connection with the boundary conditions and defining the internal areas for multiphase flow, or for walls with heat transfer. So you can define two types of boundary conditions:

conditions on the boundary the conditions for the continuity areas (continuum types), the specification of the area

of flow or solids.

Creation of boundary conditions is realized by specifying the properties of the selected model areas for further work, thus creating pressure input, output, creating two symmetrical walls (rotating part) and other walls that have no other function than to just enclose the "fluid" volume, see .

Tlakov vstup Pressure inlet

Tlakov vpus Pressure outlet symetrick podmnka Symmetry 1,2.

zbyl stny Wall Fig. 2.13 Boundary conditions

This kind of gid is exported to a file with suffix .msh and thus all work in the pack are ended.

Programov systm Fluent

44

3. Fluent software system 3.1. Overview of methods for solving of partial differential

equations The differential method is the oldest well-known method for solving differential equations,

which is used in the illustrative example of heat conduction in a rod. It consists in replacing the

derivatives by using differential Taylor rates, deriving differential equations and their solutions

The finite volume method is briefly told in three basic points

dividing the area into discrete volumes using general curvilinear network definition of individual unknowns in finite volume and discretization numerical solution of discretized equations

Fluent defines discrete final volumes using non-staggered scheme, where all variables are stored in the centers of finite volumes.

Currently begins to form the finite element method in fow solving, which consists of those

points

multiplying the differential equation by basis functions the dividing the area into triangular or quadrangular elements in the two-dimensional (2D)

area or tetrahedrons resp. hexahedrons in the three-dimensional (3D) area

integration over a finite element based on the variational principle minimization of residual

Special method is a spectral method which is suitable for periodic flow in a single regions

(Taylor vortices arising in the gap between concentric cylinders, one of which rotates).

Other chapters are devoted only finite volumes.

3.2. Integration by finite volume method Integration of differential equations is easily explained when using Cartesian coordinates

and for the simplicity the equations of one spatial independent variables which can be thought

as the flow in the three-dimensional space, where all variables in the derivative direction y

and z are zero. Flow is also stationary (time independent).

The equation of continuity is defined as follows:


45

0xu ( 3.2.1)

momentum equation

Sxu

xxpuu

x

1

( 3.2.2)

equation for general scalar variable

S

xxu

x

( 3.2.3)

Fig. 3.1 Coordinate scheme using special notation of cells for 1D and 3D model instead of

subscripts, where N North, S South, E East, W West, F Front, B Back

By integrating these equations over the final volumes the initial differential equations

are converted into volume integral (dV=dx.dy.dz, dA=dy.dz), using divergence theorem (

Szyx

V

zyx dxdzadxdzadydzadxdydzz

ay

ax

a

on surface integral (in

geometry the capital letters indicate the center of finite volume and small letters indicate

boundaries, i.e. walls between the finite volumes ) and using discretization on the resulting

algebraic form as follows:

weV AV

uAuAdAudxdydzxudV

xu ( 3.2.4)

Integrating the continuity equation leads to the shape 0 we uAuA Physical expressions on the left side indicate the flow rate difference

0 we QQ ( 3.2.5) By integrating the conservation of momentum equation it is obtained


46

Sxu

xxpuu

x

1

VSAx

uuAx

uuAppuQuQw

WPw

e

PEewewwee

1

( 3.2.6)

and the equation for the scalar quantity is adjusted using the same procedure to form

VSAxx

QQw

WPw

e

PEewwee

( 3.2.7) In the previous equations there are used both coefficients and variables defined in the centers

of finite volumes and on the walls of these volumes (e.g. speed in equation (3.2.6)). This is a

disadvantage and it is absolutely necessary in the next to unify storage quantities only in the

centers of finite volumes. If this value will be determined on the wall, the interpolation scheme

for the interpolation of this magnitude in the center of the cell must be used. For illustration the

simplest scheme is used, i.e. the arithmetic mean and differential scheme is simplified. For

example equation (3.2.6) can be modified as follows:

VSAx

uux

uuApp

uuQuuQ

w

WPw

e

EPewe

WPw

PEe

1

22

( 3.2.8)

and equation for general variable:

VSAxx

QQw

WPw

e

EPe

WPw

EPe

22

Then it is possible for this general equation in one-dimensional case to express P using the values in neighboring finite volumes following modifications

VSQxAQ

xA

xA

xAQQ

Ww

wwE

e

eeP

ww

ee

we

2222 CWWEEPP SAAA ( 3.2.9)

where PWEP SAAA . Previous equation can be further adjusted in general

Ci

iii

PiP SASA ( 3.2.10)

where the sum is performed over the neighboring cells (in 1D case is i=E, W; in 3D case i=N, S, E, W, F, B,). Ai are coefficients, which contain contributions from convection, diffusion and

source members and SC and SP are the linearized source and S = SC + SP.

P. The sign is

visible from z . Equations solved in Fluent are an extension of the previous three-dimensional


47

curvilinear coordinate system. Each iteration consists of the steps that are shown in diagram on .

solution of the equation for conservation of momentum

solution of the continuity equation (pressure correction) updating velocity, pressure

check of convergence

updating of property fluid

solutions of the equations for scalar variables updating turbulent quantities updating scalars

END START

Fig. 3.2 Diagram of solution algorithm in Fluent 0

and are described as follows

equations of motion for the unknown velocity components are solved using the pressure values in order to update the velocity field

velocity specified in the previous point can not satisfy the continuity equation, thus the pressure correction and subsequent correction of velocity field shall be determined

using new values of velocity the equation for the turbulent energy k and dissipation is solved

another equation to determine the temperature and other scalar variables are solved physical properties of liquids (eg. viscosity) must be updated checking the convergence

3.3. Choice of interpolation scheme FLUENT stores the components of velocity and scalar quantities in geometric centers of

finite volumes defined by grid. Because the calculation process, the required values of these


48

variables on the border of finite volume are used. These values are obtained by interpolation,

while you can choose between the following three variants differing order of accuracy

(ascending)

power interpolation quadratic upwind interpolation (QUICK) second-order interpolation / central difference QUICK During large changes in pressure and flow rozpotat should be available to compute with the lowest order of accuracy (which is predefined) and after a few iterations to use higher order of accuracy (for vortex flow with heat transfer, dissipation, etc.)

3.4. Convergence. 3.4.1. Residuals

During flow simulation using Fluent it is very important to obtain convergent solutions. The measure of convergence are residuals, which represent the maximum difference between two corresponding values at the same grid point in two consecutive iterations. Residuals are evaluated for all values computed in each iteration step and displayed for the selected variables.

Fig. 3.3 Iteration in numerical steady calculation

Strictly speaking the measure is the sum of the changes of computed variables in the equation for all cells in the area. After discretization the equation a conservative one-dimensional in shape for the general variable is derived as

i-th iteration

i+1-th iteration

Pi

Pi+1


49

CWWEEPP SAAA Residual R is than defined by sum for all P

P

PPCWWEE ASAAR ( 3.4.1) R is the non-standard residual, which has a physical dimension corresponding to the dimension of each member of the equation numerically. The pressure and velocity residuals, for example, may differ by orders of magnitude. Therefore, the commonly used standardized residual is defined as follows:

P

PP

PPPCWWEE

A

ASAAR

( 3.4.2)

Residuals can be evaluated graphically in each iteration step, see Figure 3.4, and the decreasing value of the residuals indicates good convergent task. Numerically exact values can be observed in tab.3.1..

Fig. 3.4 Dependence of pressure and enthalpy residual on the number of iterations

Normalized reziduals


50

number of

iterations

pressure velocity u velocity v entalpy

5.00000E+00 2.647l0E-01 3.62767E-01 4.01625E-01 3.49503E-01

1.00000E+01 4.21854E-02 7.31087E-02 6.09502E-02 1.15607E-01

1.50000E+01 1.61787E-02 5.57186E-02 6.77023E-02 6.28091E-02

2.00000E+01 9.91924E-03 4.11899E-02 5.52667E-02 4.24032E-02

2.50000E+01 7.78245E-03 3.71804E-02 5.02612E-02 3.19044E-02

3.00000E+01 6.71127E-03 3.33559E-02 4.61688E-02 2.55360E-02

3.50000E+01 4.96045E-03 3.13033E-02 4.34564E-02 2.11992E-02

4.00000E+0l 6.07668E-03 3.01096E-02 4.09786E-02 1.80783E-02

4.50000E+01 5.21358E-03 2.85215E-02 3.89507E-02 1.56768E-02

5.00000E+01 6.70681E-03 2.67667E-02 3.67708E-02 1.38577E-02

atd.

5.50000E+02 5.67326E-04 1.93322E-03 1.06027E-03 1.25243E-04

5.55000E+02 6.90138E-04 1.85336E-03 9.84238E-04 1.18761E-04

5.60000E+02 5.32427E-04 1.78220E-03 9.68897E-04 1.12885E-04

5.65000E+02 4.20846E-04 1.68717E-03 1.02076E-03 1.07433E-04

5.70000E+02 3.76113E-04 1.62817E-03 1.07420E-03 1.02499E-04

5.75000E+02 3.26542E-04 1.52597E-03 1.07393E-03 9.78981E-05

5.80000E+02 3.00249E-04 1.47025E-03 1.08095E-03 9.36884E-05

5.85000E+02 2.94075E-04 1.31286E-03 1.07175E-03 8.96829E-05

5.90000E+02 2.54086E-04 1.16047E-03 l.05932E-03 8.59168E-05

5.95000E+02 2.33645E-04 1.02260E-03 1.05061E-03 8.23961E-05

6.00000E+02 2.12518E-04 9.10742E-04 1.04899E-03 7.90365E-05

tab. 3.1

It is also possible to assess at what point is the highest residual value. Residuals are used to

evaluate convergence. Generally, the solution converges very well when the normalized

residuals are reduced to the value of the order of 1.10-3 and enthalpy residuals to the value of

the order 1.10-6.

3.4.2. Convergence acceleration Convergence is influenced by many factors such as initial estimate, a large number of

cell, relaxing factor, etc.

To convergence accelerate, it is proposed to use an initial estimate of the variables

important for flow which is the best way to start solving task successfully. Otherwise, all


51

variables defined by initialization, are often considered to be zero at the beginning of the

calculation. The most important examples of setting the initial conditions are:

temperature for solving of heat transfer problems when using the equation of state velocity in a large number of cells temperature and velocity in soution of natural convection flow with chemical reaction, when it is available to set the temperature and the mass

fraction of species

An important technique to accelerate convergence is the step by step technique (step by step

from simple to more complex tasks). To solve the problem of heat transfer it is good to start

the calculation from izothermal flow, in case of reacting flow to start the solution without the

inclusion of additives. The problem is defined at first complex and then would be necessary to

select the variables for which initial state will be resolved .

3.4.3. Relaxation Due to nonlinearity of differential equations is not generally possible to obtain values of

all variables by solution of originally derived approximations of difference schemes.

Convergence can be achieved using relaxation, which reduces the changes of each variable

in each iteration. Simply said, the new value 1iP, in the finite volume containing point P depends on the old value from the previous iteration

iP, , the new value from the current iteration

vypiP ,1, (or calculated changes iPvypiPP ,,1, and relaxation parameter 1,0 follows

iPvypiPiP .,1.1, 1 resp.

PiPiP ,1, ( 3.4.3)


52

Fig. 3.5 Definition of relaxation parameter

These relaxation parameters can be set for all computed variables. Especially for velocity they

are defined very small, the order of tenths to hundredths. It is desirable during the calculation

to change these values and accelerate the convergence, ie. if the residual changes are large

in the transition from one iteration to another, to set a small relaxation factor and thus to damp

non-linearity, if the residual changes become constant, you should increase the relaxation

factors.

0 1

Turbulent flow of real fluid

53

4. Turbulent flow of real fluid 4.1. Classification of real fluid flow

Flow of real fluids can be classified as laminar or turbulent flow. In turbulent flow it was found (based on experimental measurements) that on walls of

the pipe or wrapped body the laminar layer of liquid motion creates. The laminar sublayer has

a thickness a several tenths of millimeters. Just above the laminar sublayer it is a transition

layer between the laminar sublayer and turbulent core, which is another area of turbulent flow.

Laminar sublayer and transition layer forms a turbulent boundary layer. Consider the simplest case - a thin plate parallel to the fluid flow, see Figure 1.4 The pressure is constant

throughout the volume of fluid. The liquid is attached on the plate, because due to the influence

of viscosity the fluid layer is braked near the surface of the plate. Fluid velocity increases with

increasing the distance from the wall to the undisturbed flow velocity value v . Thickness of

the "brake" liquids, ie the boundary layer thickness is equal to zero at the leading edge and at

the trailing edge is maximum. In the boundary layer and the area around the nozzle plate the

stream lines are not parallel lines, but form a slightly diverging beam. Velocity component

perpendicular to the plate is much smaller and can be neglected.

Fig. 4.1 Boundery layer in case of plate wrapping 0

In front there is the laminar boundary layer, in the back there is the turbulent boundary layer,

between them the transition zone exists. Instantaneous border of turbulent boundary layer -

full irregular curve - is changing with time. Mean thickness of the turbulent boundary layer is

drawn by dashed line. The criterion for determining the transition from the laminar to turbulent

boundary layer is again a critical Reynolds number, whose value varies with the level of

turbulence flow. Usually is defined by 510.5Re k

kxv ,

kx is the distance from the leading

edge where the laminar boundary layer is transferred to turbulent. It can be seen that the

determination of flow is not simple and straightforward and depends on the experience of the

investigator.


54

In the case of one-dimensional flow in the pipe the transition to turbulence is given

by experimentally determined critical Reynolds number Re , defined by dvsRe where s

v

is the mean velocity in the pipe, d its diameter and kinematic viscosity. The critical value kritRe for a pipe of circular cross-section is the 2320. By kritReRe the ordered laminar flow

in the pipeline develops, the movement happens in layers and the fluid particles do not move

across section. If kritReRe , the flow is turbulent. At higher Reynolds numbers, the fluid

particles held disordered movement in all possible directions. This motion is irregular, random

related to motion of molecules of gas, but unlike molecules of the fluid particles can disintegrate

and thus lose their identity. The movement of particles perpendicular to the wall increases the

momentum flux to the wall and therefore the pressure drop in the flow direction is much greater

than in laminar flow. As a result of fluid mixing the velocity differences in different places are

much smaller in cross section than in laminar flow at larger distances from the wall.

The flow can be visualized by different methods to observe the differences in laminar

and turbulent flow see fig 4.2. In turbulent boundary layer the turbulent (coherent) vortex structures characteristic just for of turbulent flow can be defined.

Fig. 4.2 Comparison of laminar and turbulent boundary layer 0

Characteristics of turbulent flow are:

random motion of particles of fluid, ie volumes containing a large number of molecules. Motion of the particles consists of organized medium motion and random

fluctuations, which implies an analogy between the behavior of molecules (Brownian

movement of molecules) and the behavior of fluid particles. Due to fluctuations the

molecule can displace from the area of higher macroscopic velocity to the area of

smaller macroscopic velocity. When impacting to another molecule it is slowing down

and the second molecule ran faster and her hand over part of its momentum. Opposite

is the case, the molecule passes from the area of lower velocity to the area of higher

velocity and its momentum on impact increases. This leads to sharing of momentum


55

between the fluid with many different areas of velocity, which is reflected in increasing

resistance to flow as an internal fluid friction.

shear stress, resulting in turbulent flow is not only determined by internal friction in the fluid

and velocity gradient, as is the case of laminar flow (Newton's law dydv ), but by the change

in momentum of macroscopic particles, as a result of their penetration between adjacent layers.

This disordered movement causes the additional turbulent strain.

turbulent viscosity, which is not a physical constant of fluid, as the molecular viscosity of laminar flow is, but it is defined as a complex functional dependence on fluid flowing

through the state and position of the point being considered, ie the sharing of

momentum fluctuations and distance from the wall. Therefore, the velocity profile for

turbulent flow in comparison with laminar is more flat (no parabolic character).

diffuse character of turbulence, when velocity gradients induced by turbulent fluctuations of velocity are a source of tension and viscous dissipation of energy. This

increases the internal energy of fluid at the expense of the kinetic energy of turbulence.

Turbulence therefore needs a constant supply of energy to cover these losses quickly

otherwise expire.

4.2. Turbulent flow Flow is generally called turbulent if its variables exhibit chaotic fluctuations in both space

and time, see Fig. 4.3 .

Fig. 4.3 Fully developed turbulent flow - velocity as a function of time

The first work in the field of turbulent motion of fluids, one of the fascinating phenomena of

nature, has been dating from r.1883 and written by Osborne Reynolds, see [1]. Equations

describing such a flow have been known decades. Unfortunately, the problem of turbulence


56

with regard to physics is still not resolved. Although currently the significant progress was done,

particularly in the area of nonlinear dynamical systems and chaos theory, a complete solution

of turbulence is not expected in the near future. However, interest in turbulence is not only

inspired by

fluent manual

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