computational fluid dynamics me g515 - bits...

BITS Pilani Dubai Campus

COMPUTATIONAL

FLUID DYNAMICS ME G515

BITS Pilani, Dubai Campus

BASICS OF COMPUTATIONAL FLUID DYNAMICS ANALYSIS


Overview

Introduction

History of CFD

Basic concepts

CFD Process

Derivation of Navier-Stokes Duhem Equation

Example Problem

Applications


Components of Fluid Mechanics

Fluid Mechanics

Fluid Statics Fluid Dynamics

Laminar Turbulent

Newtonian Fluid Non-Newtonian Fluid

Ideal Fluids Viscous Fluids

Compressible

Flow

Incompressible Flow

BASIC CONCEPTS

CFD Solutions for specific Regimes

Rheology


Fluid (gas and liquid) flows are governed by partial differential equations

which represent conservation laws for the mass, momentum, and energy.

Computational Fluid Dynamics (CFD) is the art of replacing such PDE

systems by a set of algebraic equations which can be solved using digital

computers.


What is fluid flow?

Fluid flows encountered in everyday life include

• Meteorological phenomena (rain, wind, hurricanes, floods, fires)

• Environmental hazards (air pollution, transport of contaminants)

• Heating, ventilation and air conditioning of buildings, cars etc.

• Combustion in automobile engines and other propulsion systems

• Interaction of various objects with the surrounding air/water

• Complex flows in furnaces, heat exchangers, chemical reactors etc.

• Processes in human body (blood flow, breathing, drinking . . . )

• and so on and so forth


Introduction

What is CFD?

Prediction fluid flow with the complications

of simultaneous flow of heat, mass transfer,

phase change, chemical reaction, etc using

computers.


CFD is a branch of Fluid dynamics

So what really is Engineering Fluid Dynamics in the first place? Lets look at

some examples:

We are interested in the forces (pressure , viscous stress etc.) acting on

surfaces (Example: In an airplane, we are interested in the lift, drag, power,

pressure distribution etc)

We would like to determine the velocity field (Example: In a race car, we

are interested in the local flow streamlines, so that we can design for less

drag)

We are interested in knowing the temperature distribution (Example: Heat

transfer in the vicinity of a computer chip)

What is CFD/FD ?

What is CFD/FD ?

• Roughly put, in Engineering fluid dynamics,

• we would like to determine certain flow

properties in a certain region of interest, so that

the information can be used to predict the

behaviour of systems, to design more efficient

systems etc..


Since 1940s analytical solution to most fluid dynamics problems was

available for idealized solutions. Methods for solution of PDEs

were conceived only on paper due to absence of personal computer.

Daimler Chrysler was the first company to use CFD in Automotive

sector.

Speedo was the first swimwear company to use CFD.

There are number of companies and software's in CFD field in the

world. Some software's by American companies are FLUENT,

TIDAL, C-MOLD, GASP, FLOTRAN, SPLASH, Tetrex, ViGPLOT,

VGRID, etc.

History of CFD


Compressible and Incompressible flow

A fluid flow is said to be compressible when the pressure variation in the flow field is large enough to cause substantial changes in the density of fluid.

Viscous and Inviscid Flow

jjiiii qpf

dt

dq,,

~1

In a viscous flow the fluid friction has significant effects on the solution where the viscous forces are more significant than inertial forces

0)()(

v

yu

x


Steady and Unsteady Flow

Whether a problem is steady or unsteady depends on the frame of reference

Laminar and Turbulent Flow

Newtonian Fluids and Non-Newtonian Fluids

In Newtonian Fluids such as water, ethanol, benzene and air, the plot of shear stress versus shear rate at a given temperature is a straight line


Initial or Boundary Conditions

Initial condition involves knowing the state of pressure

(p) and initial velocity (u) at all points in the flow.

Boundary conditions such as walls, inlets and outlets

largely specify what the solution will be.


Discretization Methods

Finite volume

method

Finite Element

method

0

FdAQdv

t

e

ii QdvWR

• Where Q - vector of conserved variables

• F - vector of fluxes

• V - cell volume

• A –Cell surface area

Ri=Equation residual at an element vertex

Q- Conservation equation expressed on element basis

Wi= Weight Factor


Finite difference method

Boundary element method

0

z

H

y

G

x

F

t

Q

The boundary occupied by the fluid is divided into surface mesh

Q – Vector of conserved variables

F,G,H – Fluxes in the x ,y, z directions


CFD PROCESS

Geometry of problem

is defined .

Volume occupied by

fluid is divided into

discrete cells.


CFD PROCESS cont.. Physical modeling is defined. Boundary conditions are defined which involves specifying of fluid behavior and properties at the boundaries. Equations are solved iteratively as steady state or transient state. Analysis and visualization of resulting solution.

post processing

Density

ρ

Physics of Fluid

Fluid = Liquid + Gas

lecompressib variable

ibleincompress const

Substance Air(18ºC) Water(20ºC) Honey(20ºC)

Density(kg/m3) 1.275 1000 1446

Viscosity(P) 1.82e-4 1.002e-2 190

Viscosity μ: resistance to flow of a fluid

)(3

Poisem

Ns

Conservation Law

in out M inm outm

outin mmdt

dM

outin mm

0dt

dMMass

Momentum

Energy

Navier-Stokes Equation I

Mass ConservationContinuity Equation

0

i

i

x

U

Dt

D

Compressible

0, Dt

Dconst

0

i

i

x

UIncompressible

Navier-Stokes Equation II

Momentum ConservationMomentum Equation

V

j

IV

i

ij

III

j

II

i

j

i

I

jg

xx

P

x

UU

t

U

k

kij

j

i

i

j

ijx

U

x

U

x

U

3

2

I : Local change with time

II : Momentum convection

III: Surface force

IV: Molecular-dependent momentum exchange(diffusion)

V: Mass force

Navier-Stokes Equation III

Momentum Equation for Incompressible Fluid

j

i

j

ji

j

i

jg

x

U

x

P

x

UU

t

U

2

2

k

k

i

ij

j

i

i

j

ii

ij

x

U

xx

U

x

U

xx

3

2

0

i

i

x

U

2

2

2

2

i

j

i

i

ji

j

i

ij

x

U

x

U

xx

U

x

Navier-Stokes Equation IV

Energy ConservationEnergy Equation

V

i

j

ij

IV

i

III

i

i

II

i

i

I

x

U

x

T

x

UP

x

TUc

t

Tc

2

2

I : Local energy change with time

II: Convective term

III: Pressure work

IV: Heat flux(diffusion)

V: Irreversible transfer of mechanical energy into heat

Discretization

Discretization Methods Finite Difference

Straightforward to apply, simple, sturctured grids

Finite Element

Any geometries

Finite Volume

Conservation, any geometries

Analytical Equations Discretized Equations

Discretization

Finite Volume I

General Form of Navier-Stokes Equation

q

xU

xt i

i

i

TU j ,,1

S

i

V i

dSndVx

Integrate over the

Control Volume(CV)

Local change with time Flux Source

VS

i

i

i

V

dVqdSnx

UdVt

Integral Form of Navier-Stokes Equation

Local change

with time in CV

Flux Over

the CV Surface

Source in CV

Finite Volume II

Conservation of Finite Volume Method

VS

i

i

i

V

dVqdSnx

UdVt

A B

A B

Finite Volume III

;VdVm p

Vi

Approximation of Volume Integrals PU

eU

EU

Interpolation

0)( if

0)( if

eE

eP

e

nUU

nUUU

Upwind

Central

PE

PeeePeEe

xx

xxUUU

)1(

wesnkSPdSPdVP k

k

kSV ii

,,,

Approximation of Surface Integrals ( Midpoint Rule)

VudVumu PP

V

ii

i


The Navier-Stokes equations are the fundamental partial

differentials equations that describe the flow of incompressible

fluids.

Two of the alternative forms of equations of motion, using

the Eulerian description, were given as Equation (1) and

Equation (2) respectively:

jjiijji

i fqqt

q,,

)(

.1

,, jjiijij

ii fqqt

q

dt

dq

(1)

(2)

DERIVATION OF NAVIER-STOKES-DUHEM EQUATION


If we assume that the fluid is isotropic ,

homogeneous , and Newtonian, then :

.~2)

~( ijijkkij p (3)

DERIVATION (Cont’d)

Substituting Equ(3) into Equ(2), and utilizing the Eulerian

relationship for linear stress tensor we get :

,,,,

~~~1jjijijii

i qqpfdt

dq

(4)

( for compressible fluids )


For incompressible fluid flow the Navier-Stokes-

Duhem equation is:

jjiiii qpf

dt

dq,,

~1


If the fluid medium is a monatomic ideal gas, then :

~

3

2~


Navier stokes equation for compressible flow of

monatomic ideal gas is :

,,,,

~~

3

11jjijijii

i qqpfdt

dq



EXAMPLE PROBLEM

Neglecting the gravity field, describe the steady two- dimensional

flow of an isotropic , homogeneous,

Newtonian fluid due to a constant pressure gradient between two

infinite, flat, parallel, plates. State the necessary assumptions.

Assume that the fluid has a uniform density.


The Navier – stokes equations for incompressible flow is:

jjiiijiji qpfqq

dt

dq,,,

~1

Since the flow is steady and the body forces are

neglected, the Navier-stokes equation becomes:

jjiijij qpqq ,,,

~1

SOLUTION (Cont’d)


The no slip boundary conditions for viscous flow are:

0iq at ay 2

Using the boundary conditions ( q2= 0 at y2=+/- a )

Thus, the first Navier-stokes equations becomes

1

2

2

1

2

dy

dp

dy

qd

SOLUTION (Cont’d)


Integrating twice, we obtain

22

2

1

12

1ay

dy

dpq

The results, assumptions and boundary conditions of this

problem in terms of, mathematical symbols are as follows:

Constant 0if

0

t

0

3

y

22

2

1

12

1ay

dy

dpq

SOLUTION (Cont’d)


HOMEWORK PROBLEM

• Using the Navier-Stokes equations investigate the flow (qi) between

two stationary, infinite, parallel plates a distance h apart. Assuming

that you have laminar flow of a constant-density, Newtonian fluid

and the pressure gradient is constant (partial derivative of P with

respect to 1).


Types of Errors and Problems

Types of Errors:

Modeling Error.

Discretization Error.

Convergence Error.

Reasons due to which Errors occur:

Stability.

Consistency.

Conservedness and Boundedness.


Applications of CFD

1. Industrial Applications:

CFD is used in wide variety of disciplines and industries,

including aerospace, automotive, power generation, chemical

manufacturing, polymer processing, petroleum exploration,

pulp and paper operation, medical research, meteorology, and

astrophysics.

Example: Analysis of Airplane

CFD allows one to simulate the reactor

without making any assumptions about the

macroscopic flow pattern and thus to

design the vessel properly the first time.


Application (Contd..)

2. Two Dimensional Transfer Chute Analyses Using a

Continuum Method:

Fluent is used in chute designing tasks like predicting flow shape,

stream velocity, wear index and location of flow recirculation

zones.

3. Bio-Medical Engineering:

The following figure shows pressure

contours and a cutaway view that

reveals velocity vectors in a blood

pump that assumes the role of heart

in open-heart surgery.

Pressure Contours in Blood Pump



4. Blast Interaction with a Generic Ship Hull

Results in a cut plane for the interaction of an

explosion with a generic ship hull: (a) Surface

at 20msec (b) Pressure at 20msec (c)

Surface at 50msec and (d) Pressure at

50msec

The figure shows the

interaction of an explosion

with a generic ship hull.

The structure was modeled

with quadrilateral shell

elements and the fluid as a

mixture of high explosives

and air. The structural

elements were assumed to

fail once the average strain

in an element exceeded 60

percent



5. Automotive Applications:

Streamlines in a vehicle without (left) and with rear center and B-pillar ventilation (right)

In above figure, influence of the rear center and B-pillar ventilation on the

rear passenger comfort is assessed. The streamlines marking the rear

center and B-pillar ventilation jets are colored in red. With the rear center

and B-pillar ventilation, the rear passengers are passed by more cool air. In

the system without rear center and B-pillar ventilation, the upper part of the

body, in particular chest and belly is too warm.


The following are the details of conducting practicals

Five experiments are to be conducted in the CAD lab using ANSYS fluent software

1. Laminar flow through a circular pipe with constant radius

2. Turbulent flow though a circular pipe with constant radius

3. Compressible flow through a CD Nozzle

4. Steady flow over a rotating cylinder

5. Unsteady flow over rotating cylinder.

Four experiential are to be conducted using

Vapor Refrigeration test rig,

Wind tunnel and

Smoke analyzer.

computational fluid dynamics me g515 - bits...

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