viscous incompressible flow

Fluid Dynamics I-1ANSYS, Inc. Proprietary

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Introduction to CFD using Ansys CFX

Fluid Dynamics I:Viscous Incompressible Flow



Viscous Incompressible Flow

• Governing Equations

• Phenomenology

• Analytic Solutions

• Boundary Conditions

• Problem Setup Using CFX



= Convective Flux + Diffusive Flux

Governing Equations: Scalar Transport

• Conservation Principals

– All fluid dynamics transport equations are based on conservation

principles

– Simplest case is transport of a scalar quantity F = some quantity

per unit mass e.g. Mass fraction of some dissolved chemical

species.

– Consider transport of F through a control volume W in a fluid of

density r with given flow velocity field U.

Flow Field UControl Volume W

Area Element dA

FF UJ r




• Integral Conservation Law– Rate of change of total amount of F in W =

Rate of production of F in W

- Amount of F leaving boundary of W per unit time

– S = source of F per unit volume e.g. due to chemical reactions

– J = Convective Flux + Diffusive Flux =

AJ ddVSdVt

F

WWW

r

FFUr

dVSddVt

WWW

FFF

AUrr




• Differential Form– Apply Gauss’ theorem

– Shrink the control volume down to zero:

• This is the Scalar Advection-Diffusion Equation– Fundamental starting point for Computational Fluid Dynamics

0div

FFF

W

dVSt

Urr

dVd WW

JAJ div

St

FFF

Urr div



Governing Equations: Mass Conservation

• Mass Conservation– Assume no mass sources or sinks

• Integral Conservation Law– Rate of change of fluid mass in W =

Amount fluid mass entering boundary

of W per unit time:

AU ddVt

WW

rr



Governing Equations: Mass Conservation

• Differential Conservation Law – Apply Gauss’ theorem and shrink control volume to zero:

• This is the Continuity Equation

• If density = constant (incompressible flow) then

velocity field is divergence free (solenoidal):

• For now, consider incompressible flow only– Compressibility effects are discussed later.

0div

Ur

r

t

0div U



Governing Equations: Momentum Conservation

• Momentum Conservation

• Integral Conservation Law (Newton’s 2nd Law)– Rate of change of momentum in W =

Amount fluid momentum entering boundary of W per unit time

+ Total force on fluid in W

• FB = body force per unit Volume– FB = rg for buoyant flows

• FS = surface force per unit area – See next slide for details

dAFdVFdAUUdVUt

SiBijjii WWWW

rr




• Surface Forces– Surface forces acting on the fluid particle are due to pressure

and viscous stresses.

– The pressure and viscous force acting on the fluid can be

expressed as

– Viscous Stress Tensor for Incompressible Newtonian Fluid:

– Symmetric: and trace free:

W

isi PdAF )pressure( W

jijsi dAF )viscous(

viscositydynamic ,

i

j

j

iij

x

U

x

U

jiij 0ii




• Differential Form

– Apply Gauss’ theorem and shrink control volume to zero

– After some manipulation, the momentum equation may be

expressed as a Vector Advection-Diffusion Equation

– Primary action of viscosity is molecular diffusion of momentum,

e.g. Shear layer:

– Momentum Source = - Pressure Gradient + Body Forces

Bi

ij

iji

j

i Fx

P

x

UUU

xU

t

rr



Pressure Field Definitions (1):

Absolute, Reference and Static Pressure

• STP: Absolute Pressure Pabs = 1 [bar] = 105 [Pa]

– Hence, obtain large truncation errors if absolute pressure is used

to compute pressure gradients

• Define a constant Reference Pressure Pref typical of the

ambient surroundings

• Define the Relative Static Pressure Pstat = Pabs – Pref

• Hence, absolute pressure is the sum of reference and

static pressures:

• CFX solves for the static pressure as it is typically much

smaller than absolute pressure, but has the same

gradient.

statrefabs PPP




Hydrostatic and Modified Pressure

• Consider constant density flow under gravity

• Hydrostatic Condition (No Flow)

a Pressure Gradient Balances Gravity Force

• Integrate to obtain Hydrostatic Pressure:

• In general coordinates:

0at ,

yPPg

y

Prefr

y

x

g = g(0,-1,0)

gyPP refhyd r

refrefPPP xxg at ,0r

xg rrefhyd PP




Hydrostatic and Modified Pressure

• Flow under gravity is caused by deviations of pressure

from hydrostatic

• Hence, convenient to work with Modified Pressure field

with hydrostatic component removed:

• Relative to this pressure field, the RHS of the

momentum equation can be written independently of

gravity:

• So, for constant density flows, we may ignore gravity,

provided we redefine pressure to be the modified

pressure.

x

hydabs PPP '

'PPP xgg rr



Summary: Navier Stokes Equations

• Combined Momentum + Mass Conservation equations

are known as the Navier-Stokes Equations

• For constant density and constant viscosity flows, these

simplify to:

• Momentum Equation :

– (P = modified pressure, hence no gravity term)

• Mass Equation:

– Solenoidal velocity field

3,2,1 ,

iPUUdivU

tiii rr U

0div U



Non Dimensionalisation and Dynamic Similarity

• Let V0, L0 be typical constant velocity and length scales

for a given flow situation

– E.g. Pipe Flow, L0 = Pipe Diameter, V0 = Average Inlet Velocity

• Define non-dimensional variables as follows:

• Rewrite Momentum Equation in terms of non-

dimensionalised Variables

• = Reynolds Number

/ ,/ ,/ ,/ 2

0

*

00

*

0

*

0

* UPPLtUtLU r xxUU

3,2,1 ,Re

1div ****

*

iPUU

t

U

ii

i U

Re 00

r LU



Non Dimensionalisation and Dynamic Similarity

• Hence, for viscous incompressible flow, the solution

only depends on one dimensionless group:

• Reynolds Number

• n /r Kinematic Viscosity (units L2 T-1).

• Two flows are dynamically similar if they have the same

Reynolds number.

– Solutions can be related to each other by rescaling

– E.g. Wind tunnels.

– Low Re a viscous forces dominate

– High Re a inertial forces dominate

• Two fluids will behave the same if they have the same

kinematic viscosity.

n

r 0000 ReLULU



Viscous Incompressible Flow:

Phenomenology and Some Useful Analytic Solutions



Reynolds experiment

• Flows can be classified as either laminar or turbulent flow with a

transition zone between the regimes

• Osborne Reynolds’ classical dye trace pipe flow experiment

demonstrates the classification of the flow

• Laminar flow: dye streak follows distinct streamline.

• Transitional flow: Periodic time dependent

• Turbulent flow: Chaotic time dependent..

– Mixes the dye due to fluctuating (random) velocity components

• In a smooth pipe of circular cross section, the transition from

laminar to turbulent flow occurs at around Re = 2300.

Low Re: Laminar Transitional High Re: Turbulent



Entrance Region and Fully Developed Flow

• Flow in Conduits e.g. Pipes and Ducts

– Initially flat inlet velocity profile slowly changes shape (develops) as

flow moves downstream

– Sufficiently far downstream, no further change of shape takes place,

and the flow is fully developed

• Due to growth of the boundary layer on the surface of the conduit until it

completely fills the conduit

• Occurs for both laminar and (time averaged) turbulent flow.

• For round pipe of diameter D, entrance length to reach 95% of fully

developed flow conditions:

Inviscid

core

Boundary Layer

Developing flow

(Entrance Length, Le) Fully developed flow

)(turbulent 40/ (laminar) ,Re/30/ DLDL ee



Fully Developed Laminar Duct Flow (1)

• 2D Duct flow. Height h = 2a.

• Assume velocity profile independent of x

• Continuity: (no slip)

• V-Momentum: (independent of y)

• U-Momentum:

y

x

a

-a

0

x

V

x

U

0 0 0

V

y

V

y

V

x

U

x

-y

P

y

U

)( y

-0 xPPP




• U-Momentum:

– RHS = function of x only,

– LHS = function of y only.

• Hence:

• Integrates to:

• Fully Developed Parabolic Profile

x

-y2

2

PU

constant 'x

P

P

at 0 ,'2

2

ayUPy

U

a

12

a'2

22

yPU




• Write in terms of Mean Velocity:

• Hence, constant inlet velocity profile develops into a

parabolic profile with maximum velocity

• Mass Flow Rate per Unit Length in z-direction:

• Mass flow rate is determined by pressure drop, and

vice-versa

2

2

a1

2

3 yUU

UU

2

1

a

a

Udya

U

3

a' 2

PU

UU 5.1 max

a

a

UdyM r

3

a'2 3

rPM



Other Analytic Solutions for 2D Duct Flow

• Couette Flow = Wall Driven Flow

– Plane duct of height h.

– Bottom wall stationary, top wall moving

– Zero Pressure Gradient a Linear Velocity Profile

– Non-Zero Pressure Gradient (Wall and Pressure Driven Flow):

h

yUU wall

Uwall

max2

'U

h

yhyy

PU

Uwall



Fully Developed Pipe Flow (Poiseuille Flow) (1)

• Consider fully developed laminar flow in a circular pipe of

radius a (Poiseille Flow)

• Follow same logic as for plane duct flow, working in cylindrical coordinates.

• U velocity momentum equation:

y

x

a

r

x

P-

r

1P

r

Ur

r



Fully Developed Pipe Flow (Poiseuille Flow) (2)

• Integrate and express in terms of mean velocity

• In this case, the velocity maximum at the pipe axis is twice the

mean velocity.

• Mass Flow Rate = Mass per unit time passing through pipe)

2

2

a1 2

rUU

8

a' 2

PU

20

a

rdrUM r 8

a' 4

rPM



Flow Separation and Reattachment (1)

• Boundary layer separation occurs when

– the shear stress at the wall is zero or negative

– A sharp sudden expansion is encountered.

– Separated flows can reattach

Flow past an object Backward facing step

separation

reattachment

recirculation

zone



CFX Setup:Fluid Domains



Domain Creation: Selecting Materials

• Create a name for the fluid to be used

• Select the material to be used in the domain – limited default choice.

• Additional Materials are available by clicking

• Further additional materials may be imported from material libraries

Useful libraries for incompressible flow.



Domain Creation: Creating/Editing Materials

• Create by clicking toolbar icon and specify an appropriate name

• Or, right click “Materials” in the Outline Tree

• Edit materials by double clicking Material object in outline tree

• Use Pure Substance for materials

with constant properties

• Other options are for mixtures,

multiphase mass transfer, and

combustion problems (see later).



Domain Creation: Material Properties

• Constant Property Isothermal

flow only requires Density and

Dynamic Viscosity

– Several choices of units are

available.

• Molar Mass and Reference State

are optional

– Used for mass transfer and

chemical reactions

• Dynamic Viscosity appears under

Transport Properties button



Domain Creation – General Options

• Morphology

– Used for multiphase flow

• Leave as default option = Continuous

Fluid for single phase problems.

• Reference Pressure

– Represents the absolute pressure

datum from which all relative

pressures are measured

Pabsolute = Preference + Prelative

• See earlier discussion.

• Called the Gauge Pressure in FLUENT

• Buoyancy

– May be ignored for constant

property flows

• See Heat Transfer Lecture for details.

• Domain Motion

– Rotating Frames of Reference

• See lecture on High Speed Flow,



Domain Creation – Fluid Models

• Heat Transfer

– Specify whether a heat transfer model

is used to predict the temperature

throughout the flow

– Discussed in Heat Transfer Lecture

• Turbulence

– Specify whether a turbulence model is

used to predict the effects of turbulence

in fluid flow

– Discussed in Turbulence Lecture



CFX Setup:Boundary Conditions



Defining Boundary Conditions

• Defining Boundary Conditions involves:

– Identifying the location of the boundaries

• e.g., inlets, outlets, walls, symmetry

– Supplying information on the variables at the boundaries

• Variable Values and/or Flux Values

• Data required at a boundary depends upon the boundary condition

type and the physical models employed

• You must be aware of types of the boundary condition available and

locate the boundaries where the flow variables have known values or

can be reasonably approximated

Poorly defined boundary conditions can have a

significant impact on your solution



Available Boundary Condition Types

• Inlet

– Flow must come into the domain

• Outlet

– Flow must come out of the domain

• Opening

– Flow may come in or out of the domain

• Wall

– Flow may neither exit nor enter the domain

• Symmetry

• In this lecture, we discuss details relevant to incompressible isothermal fluid flow.

Inlet

Opening

Outlet

Wall

Symmetry



Inlets

• Inlets are used for regions where inflow is expected.

• Artificial walls are created if flow tries to go out of an inlet.

• Only Subsonic Options available for incompressible fluids

• Two main types:

– Velocity Specified Inlets

– Pressure Specified Inlets

Inlet Inflow

allowed

Outflow

disallowedArtificial

Wall



Velocity Specified Inlets

• Normal Speed

– Flow Direction is normal to inlet boundary

• Individual Velocity Components

– Cartesian or Cylindrical

• Mass Flow Rate

– Constant (uniform) normal speed is deduced from specified mass flow rate

– Flow direction may be normal to boundary, or in user specified direction

Drop down menu for

choice of unitsCEL Button:

Permits specification

of inlet BC as an

algebraic expression



Pressure Specified Inlets

• Static Pressure

– Actual pressure at inlet relative to the Reference Pressure

• Total Pressure

– Static pressure if fluid at inlet is brought isentropically to rest

– Inlets that draw flow in from the atmosphere often use Total Pressure = 0 e.g. Open window.

• Flow Direction Options

– Normal to boundary

– Specified normal vector components

– Zero Gradient = Unspecified

• Determine from solution

2

1 2Ur statictotal PP

staticreferenceabsolute PPP



Outlets

• Outlets are used for regions where outflow is expected.

• Artificial walls are created if flow tries to go out of an inlet.

• Only Subsonic Options available for incompressible fluids

• Two main types:

– Velocity Specified Outlets

– Pressure Specified Outlets

Outlet Outflow

allowed

Inflow

disallowedArtificial

Wall



Velocity Specified Outlets

• Specified Normal Speed or Individual Velocity Components

– Use with caution!

– Discontinuous jumps may occur in solution for inaccurately specified outlet velocity profiles.

• Specified Mass Flow Rate

– Zero velocity gradient adjusted to achieve specified mass flow rate

– Reduces to fully developed flow at the end of a sufficiently long uniform conduit



Pressure Specified Outlets

• Static Pressure

– Pressure at outlet relative to the Reference Pressure

– Zero Gradient Conditions applied to velocity

– Fully developed flow at end of a sufficiently long uniform conduit

• Average Static Pressure

– Use when you know the mean outlet pressure, but the outlet pressure distribution is not uniform

– Allows the outlet pressure profile to vary, with the average value constrained to a specific value

• Total Pressure

– Disallowed

– Unconditionally unstable as an outlet BC

• Flow Direction

– Unspecified

– Determined from solution



Openings

• Artificial walls are not erected at

Openings

– Both inflow and outflow are allowed

• Necessary to specify information

that is used if the flow becomes

locally inflow, e.g. Temperature

• Two main types:

– Pressure Specified Openings

– Velocity Specified Openings

Opening

Inflow

allowed

Outflow

allowed

inTT

0

n

T



Openings: Details

• Specified Velocity

– Individual components only

– Can’t specify normal speed, as direction is not known a priori

• Static Pressure and Direction

– Pressure at opening relative to the Reference Pressure

• Opening Pressure and Direction

– Total Pressure at inflow

– Static Pressure at outflow

– Required for numerical stability

• Entrainment

– Flow direction not specified

– Toggle between Static and Opening Pressure by ticking Pressure Option button



Symmetry

• Used to reduce computational effort in problem when the geometry

is symmetric.

• Individual symmetry planes must be planar, and have a unique unit

normal vector

• Used in CFX to model 2D and axisymmetric flows

– Mesh must be one cell thick in z-direction, with symmetry planes specified either side.

– For axisymmetric flows, use a wedge geometry with angle around 3 degrees.

• Beware of the Coanda effect!

– Symmetric geometries sometimes have asymmetric flows!

No Symmetry Plane Symmetry Plane

Coanda effect

not allowed



Specifying Well Posed Boundary Conditions:

Mass Flow vs Pressure Drop

• Recall fully developed duct flow.

– Pressure drop determines mass flow rate and vice versa:

• This principle is true in general for incompressible flow.

– Velocity/Mass flows driven flows determine pressure drops, e.g.

– Pressure driven flows determine mass flow rate, e.g.

– If you try to specify both pressure drop and mass flow rate, the problem will in general be ill posed.

– Hence, for incompressible flow, inlets/outlets allow velocity orpressure to be specified, but not both

3a'2 3 rPM

U P P M P

Pin Pout M

U, Pin Pout posed ill



• When there is 1 Inlet and 1 Outlet

– Most Robust: Velocity/Mass Driven Inlet

• Velocity/Mass Flow at Inlet;

• Static Pressure at outlet.

• The inlet pressure is an implicit result of the prediction

– Robust: Velocity/Mass Driven Outlet

• Total or Static Pressure at Inlet;

• Velocity/Mass Flow at outlet.

• The outlet pressure and inlet velocity are determined by the solution.

– Less Robust: Pressure Driven Flow

• Total or Static Pressure at Inlet;

• Static pressure at outlet.

• The system mass flow is part of the solution.

• Ideally, at least one boundary should specify Pressure (either Total

or Static)


Robustness




Accuracy

• Inlet Boundary Conditions– As with physical models, the accuracy of the CFD solution is only

as good as the initial/boundary conditions provided to the

numerical model.

– Example: Flow in a duct with sudden expansion

• If flow is supplied to domain by a pipe, you should use a fully-

developed profile for velocity rather than assume uniform

conditions.

poor better

Fully Developed

Inlet Profile

Computational

Domain

Computational

Domain

Uniform Inlet

Profile




Accuracy

• Mass flow inlets result in a uniform velocity profile over

the inlet

– Fully developed flow is not achieved

– You cannot specify a mass flow profile

• Mass flow outlets allow a natural velocity profile to

develop based on the upstream conditions

• Pressure specified boundary conditions allow a natural

velocity profile to develop



• Outlet boundaries near recirculation zones

• Poor Location: Over recirculation zone

– An outlet will build artificial walls to prevent inflow

– Use an opening to allow inflow

– Inaccurate due to assumption of zero velocity gradient


Positioning of Outlets and Openings (1)

Opening




• Better Location: Short distance downstream of

recirculation zone

– Some inaccuracies due to use of zero velocity gradient boundary

condition and/or constant pressure

– Specify average pressure or outlet mass flow rate to predict

pressure profile

– Ideally, use accurate pressure profile data if known (difficult)



Outlet




• Preferred Location: Long distance downstream

of recirculation zone

– Ideally, sufficiently far away for flow to become fully developed

– Computationally more expensive

– Extra expense can be mitigated by expanding the mesh

sufficiently far downstream



Outlet



• Outlets on sides of vessels

• Same considerations apply

• Poor:

• Better:

• Ideal:



Outlet

Outlet

Outlet

Documents

viscous incompressible flow