1. Introduction; Fluid dynamicsusers.abo.fi/rzevenho/iCFD19-RZ1.pdf · 2019-10-10 · Fluid mechanics & Thermodynamics Algorithm development Turbulence modelling Grid generation Åbo

Introduction to Computational Fluid Dynamics 424512 E #1 - rz

Introduction to Computational Fluid Dynamics(iCFD) 424512.0 E, 5 sp

1. Introduction; Fluid dynamics(lecture 1 of 4)

Ron ZevenhovenÅbo Akademi University

Process and Systems EngineeringThermal and Flow Engineering Laboratory

tel. 3223 ; [email protected]


oktober 2019 Åbo Akademi Univ - Process and Systems Engineering Piispankatu 8, 20500 Turku

2 / 66

1.0 Course content / Time table



3 / 66

Positioning CFD (and DEM)

CFD

Engineeringproblem solving

(consulting-type)

Scientificproblem solving

(R&D-type)

Fluid mechanics& Thermodynamics

Algorithm development

Turbulence modelling

Grid generation

Åbo AkademiProcess and Systems Engineering

DEM(discrete element

methods)

Structuremechanics

Mechanics



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Course time table 2019 (5 days week 42)

* 3 x ~½ day exercises & demo’s* 1 CFD software exercise* 1 written exam 24 h

http://users.abo.fi/rzevenho/introCFD.html

LECTURES BY Ron Zevenhoven RZEero Immonen EI Sirpa Kallio SKFredrik Bergström FBDebanga Mondal DM



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Two comments / details to note Computational fluid dynamics, (CFD) is not the

same as ”using a commercial CFD software”.

In this course and course material, as also in the literature, two symbols are typically used for dynamic viscosity:

µ or η (Pa· s = kg /(m· s))

besides this, kinematic viscosityν = /ρ (m2/s) with density ρ (kg/m3)



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1.1 Balance equations

Note: many slides are taken (withoutany or with only little modification) from the material for this course earlierproduced by J. Brännbacka (2006, 2005)



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General balance equationInflow + Generation = Outflow + Accumulation

outgenin Bdt

dBBB

inB

genB

outBdt

dB= inflow = outflow= accumulation

= generation

genoutin BBBdt

dB

Net flux of B

Accumulation = (Outflow – Inflow) + Generation

consumed producedgen BBB



8 / 66

Mass balance

outmdtdm

inm

outin mmdt

dm

gen,out,in, aaaa nnn

dt

dn

Balance of species a (moles, or kilos)



9 / 66

Energy balance Momentum balance

outIdt

dIinI

FIIdt

dIoutin

F1 F2

Momentum is a

vector quantity!

outin EEdt

dE Energy balanceproduction/consumption not possible: 1st law of thermodynamics !!

Momentum balanceI =

mass×velocity

İ = mass flow×velocity



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Momentum balances /1

Momentum* balance / Conservation of Momentum

Linear momentum I of a mass m moving with velocity v= (vx, vy, vz) is defined as I = mv (unit: kg·m/s)

Linear momentum can be transferred between objects: (Σ mivi + Σ Fi) = constant for all directions i

* sv: rörelsemängd see also Ö96



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Momentum balances /2

Momentum balance / Conservation of Momentum

Newton’s law: force = mass × acceleration F = m·dv/dt integrates to t1∫t2 F dt = m(v2-v1) = I2- I1and = F (t2-t1) if force F is constant during time t1 → t2

dI/dt = ΣF + ( İin- İout )

Also a rotational (or angular) momentum balance holds: if there are no external torques rotational (angular) momentumL = mass × velocity × radius = constant(where radius = distance to axis of rotation) – see applications in rotating fluid-handling machinery suchas pumps, turbines, stirrers, ...... and literature on mechanics.



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1.2 The continuity equation



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Equation of Continuity- a differential mass balance

x

y

z

Δx

Δz

Δy (x,y,z)

Consider a cubic balance region Δx Δy Δz

within a flowing fluid

(Front) surfaceΔxꞏΔzfor in / outflowin direction y



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zyxt

yxwyxw

zxvzxvzyuzyu

zzz

yyyxxx

Mass balance for the volume element (unit: kg/s)

tz

ww

y

vv

x

uuzzzyyyxxx

dividing by the volume ΔxΔyΔz gives

Equation of Continuity

...))(()(: 2

xOxO

x

uu

dx

udseriesTaylor xxx



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Equation of Continuity

tz

w

y

v

x

u

In vector notation:

t

u

For constant density of the fluid:

z

y

x

u

u

u

w

v

u

u

u

0 p variable scalar for

p grad T

T

)z

p,

y

p,

x

p(p

)z

,y

,x

(

z

y

x

The vector gradient symbol



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Some vector calculus /1

z

a

y

a

x

a zyx

a

Tzyx

z

y

x

aaa

a

a

a

a

Divergence of a :

Vector a :

Velocity vector u or u : Twvuu



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Som vector calculus /2

z

w

y

v

x

u

w

v

u

z

y

x

uudiv

auua:note

)cos(.u.a

cwbvau

w

v

u

c

b

a

ua

u and a between angle

Vector cross productRotation of a vector u

Vector dot (or scalar) productDivergence of a vector u

y

u

x

vx

w

z

uz

v

y

w

w

v

u

z

y

x

uurot

auua:note

)sin(.u.a

ubva

wavc

vcwb

w

v

u

c

b

a

ua

u and a between angle

For Cartesian(x,y,z)coordinates

Similar for spherical, cylindrical, ...systems



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Coordinate systems

.

.

Left: Cartesian (x,y,z)Centre: Cylindrical (r,θ,z) with r2 = x2 + y2

Right: Spherical (r,φ,θ) with r2 = x2 + y2 + z2

Cartesian, cylindrical, spherical

here, e = unit vector with length 1



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1.3 The equation of motion



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Equation of Motion- a differential momentum balance

FIII

outin

dt

d

uI m

uI m

Momentum

Momentum flow


Example: Power output wind power plants

w1 undisturbedinput velocity

w2 velocity at a distance of ~ Dp

after the propeller“1”

“2”

Output power-POutput axle force-F

wp velocity throughthe propeller

“1” “2”

m : mass flow if fluid through the propeller

oktober 2019 RoNz 21Åbo Akademi Univ - Process and Systems Engineering Piispankatu 8, 20500 Turku

Dp

4

2p

p

DA

2

2212

121 wmPwm

See course 424514 Fluid and particulate systems


Velocity wp through the propeller

222

1212

1 wmPwm 2axle1 wmFwm

0paxle pAF

Combining

pVP

results in

axlepp F

P

A

Vw

122

1

12

21

222

1

axlep ww

wwm

wwm

F

Pw

Fluid velocity through propeller

22




Betz’ Law (1926)

P is power extracted from the wind

Power of undisturbed flow through area Ap

Efficiency of power extraction from wind

with maximum P/P0 =16/27 at w2/w1=1/3


RoNz 23

pair A

wwwwP

2

)(

2212

221

pair AwP 3

10 2

)1(12

1

1

221

22

0 w

w

w

w

P

P




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Momentum balance x-momentum

x

y

z

Δx

Δz

Δy (x,y,z)

xxx FII

dt

dIout,in, x

yxuwyxuw

zxuvzxuv

zyuuzyuuII

zzz

yyy

xxxxx

out,in,

Unit İ: kg/m3ꞏm/sꞏm/sꞏm2 = (kg/s)ꞏ(m/s)



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The ”sum of forces” term

F

Surface forces

Body force

Pressure forces acting on the surfaces

Stress forces acting on the surfaces

Gravity force acting on the enclosed volume

pF

gp FFFF

gF



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zypzypFxxxxp

,

zyxgF xxg ,

gravity force

pressure forces

Pressure, gravity

and similar for y and z direction

x

y

z

Δx

Δz

Δy (x,y,z)



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Stress forces

Definition : yx is the stress (force per area) in the x-directionexcerted on a surface perpendicular to the y-axis by the fluidin the lesser y-axis location, and similarly for xx and zx.

x

y

z

(x,y,z)

xx|x xx|x+Δx

zx|z+Δz

zx|z

Viscous stresses acting on the surfaces of the balance region. The arrows indicate the directions in which the stress forces act on each surface of the balance region if all stress tensors have positive signs.



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Stress forces and accumulation termBy the definitions of , we get for the stress forces

zyxzyx

yx

zxzyF

zxyxxx

zzzxzzx

yyyxyyxxxxxxxxx

,

The accumulation of momentum is

t

uzyx

t

I x



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The equation of motion x, y &z directions

xzxyxxx gzyxx

puw

zuv

yuu

xt

u

yzyyyxy gzyxy

pvw

zvv

yvu

xt

v

zzzyzxz gzyxz

pww

zwv

ywu

xt

w

gτuuu

pt

Compactly written in vector and tensor notation



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Some vector calculus /3T

z

s

y

s

x

ss

z

a

z

a

z

a

y

a

y

a

y

a

x

a

x

a

x

a

zyx

yyx

zyx

a

Gradient of a scalar

Gradient of a vector field gives a tensor

zzyzxz

zyyyxy

zxyxxx

bababa

bababa

bababa

abThe dyadic product ab of two vectors a and bresults in a tensor with the components

which is the equivalent of the vector product abT if a and b are column vectors



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Vector calculus /4

The divergence of a tensor is a vector, given by

z

τ

y

τ

x

τ

z

τ

y

τ

x

τ

z

τ

y

τ

x

τ

τττ

τττ

τττ

zzyzxz

zyyyxy

zxyxxx

zzzyzx

yzyyyx

xzxyxx

τ

The divergence of a stress tensor is a force vector



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Example: Flow in vertical pipe

0

L

dr

r

z

?rz

Question:

(here in cylindrical coordinates)

How to proceed, simplify this?



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1.4a Newtonian fluids; viscosity; laminar / turbulent flows



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Fluids will (try to) resist a change in shape, as will occur in fluid flowsituations where different fluid elements have different velocities

Note the definition of a fluid: a fluid is a substance that deformscontinuously under the application of a shear stress

Internal friction in fluid flow /1

xy

Picture T06

ndeformatioonaccelerati gives force

amF



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Consider fluid flow betweenplates:– The no-slip condition implies

that at the wall the velocity of the fluid is the same as the wallvelocity *), for a fixed wallvfluid = 0 at the wall

– Between the plates a velocityprofile exists: vx = vx(y)

– Shear stresses, τfluid, arise due to velocity differences betweendifferent fluid elements


*) this applies always except for very low pressuregases, for example in the upper atmosphere

xy

Picture T06



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For a fluid between plates with width W (m), distance d (m) the shear force F = (Fx,Fy,Fz) = (Fx,0,0) (unit: N) to pull the fluid at velocity v = (vx,vy,vz) = (vx,0,0) gives a shear stress τyx

(unit: N/m2) in the fluid at y = d which is equal to:

with τyx as stress in direction ”x” in a plane for constant ”y”

Pic

ture

: http

://w

ww

.phy

sics

.uc.

edu/

~si

tko/

Col

lege

Phy

sics

III/9

-Sol

ids&

Flu

ids/

Sol

ids&

Flu

ids.

htm

xy z

Wx

x, wall→fluid

yΔ

vΔη

dy

dvηyxLW

F

surface

Fxx

dy

fluidwall,xwallfluid,x τ

Lvx = 0 @ y = 0



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This defines the dynamic viscosity η (unit: Pa· s = kg· m-1· s-1)

! Note sign: τyx at y = y0 is the shear stress of fluid elements with y < y0 on the fluid elements with y > y0. As a result Fx > 0 if dvx/dy < 0 !

Pic

ture

: http

://w

ww

.phy

sics

.uc.

edu/

~si

tko/

Col

lege

Phy

sics

III/9

-Sol

ids&

Flu

ids/

Sol

ids&

Flu

ids.

htm

xy z

Wx

x, wall→fluid

yΔ

vΔη

dy

dvηyx

LW

F

surface

F

xx

dy

fluidwall,xwallfluid,x

τ

Lvx = 0 @ y = 0

with τyx as stress in direction”x” in a plane for constant ”y”



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Non-Newtonian fluids

For non-Newtonian fluids, viscosity is a function of the velocity gradient: τyx = η(dvx/dy)· dvx/dy

For example Bingham fluids (toothpaste, clay) or pseudo-plastic(Ostwald) fluids (blood, yoghurt).

Picture: BMH99

PTG



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ViscosityViscosity is a measure of a

fluid's resistance to flow; it describes the internal friction of a moving fluid.More specifically, it defines

the rate of momentumtransfer in a fluid as a resultof a velocity gradient.Dynamic viscosity η

(unit: Pa.s) is related to a kinematic viscosity, ν (unit: m2/s) via fluid density ρ (kg/m3) as: ν = η/ρ

Picture T06 Picture: KJ05

η


For circular tube flow, the laminar → turbulent flow transition occurs at Reynolds number Re ~ 2300, with dimensionless number defined as Re = ρ·<v>· d/ηfor ρ = fluid density (kg/m3), <v> = fluid average velocity (m/s), d = tube diameter (m) and η = fluid dynamic viscosity (Pa·s)


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Laminar ↔ turbulent fluid flow

Pictures: T06

Osborne Reynolds’s dye-streakexperiment (1883) for measuring laminar → turbulent flow transition

laminar: Re < 2100

laminar → turbulent

turbulent: Re > 4000

dv

η

vρ

ratio force Re2



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Example: a liquid film on a vertical wall /1

A stationary laminar flow of water (at 1200 kg/h) runsdown a vertical surface (with width W = 1 m).Give– the expression for the shear stress distribution,– the expression for the velocity profile, and– the expression for volumetric flow rate V (m3/s)

and calculate– film thickness d – velocity <vy> averaged over the film thickness– maximum velocity vy,max

Data: dynamic viscosity for water η = 10-3 Pa.sdensity for water ρ = 1000 kg/m3

gravity g = 9.8 m/s2

.

PTG



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Example: a liquid film on a vertical wall /2Answer: For this steady-state process:

The vertical force balance for a volume element withlength dy as shown gives Fgravity = Fshear

with vy = vy,max @ x=d: vy,max = ½ρgd2/η

For the average velocity <v> with V = <v>·d·W:

)½()(

)( :gintegratin

,)(

)(with

0)(0)(

2

00

xxdg

dxgxd

dxdx

dvxv

gxd

dx

dvgxd

dx

dv

gxddyWgdyWxd

xxy

y

yyxy

xyxy

dy

3

0

22

0

3 gives and

3)½(

1)(

1

g

Vd

dW

Vv

gddxxxd

g

ddxxv

dv

y

dd

yy

The data gives: d = 0.47 mm, <vy> = 0.71 m/s; vy,max = 1.07 m/s



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Example: Flow in vertical pipe /2

dr

0

L

r

zdr

dwrz

Newtonian fluid with dynamic viscosity µ:



44 / 66

Example: shear stress concentric cylinders /1

Oil with viscosity η = 0.05 Pa· s fills a 0.4 mm gap between twocylinders of which the inner onerotates whilst the outer one is fixed.

The diameter of the inner cylinder is 8 cm, the length is 20 cm.

How much power is required to rotate the inner cylinder at 300 rpm? Picture: KJ05

Question ÖS96-4.1



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Example: shear stress concentric cylinders /2

The velocity of the inner cylinder is vr = r· ω with angular velocityω = (300/60)· 2π s-1 = 10· π s-1 , at r = r1 = 0.04 m gives vr = 10π· 0.04 = 0.4· π m/s

The shear stress on the inner cylinder is τ = Fvisc/A = η· dvr/dr ≈ η· vr /(r2-r1) *) and r2-r1 = 0.4 mm

The viscous work rate Wvisc = Fvisc· vr = τ· vr·A at r = r1 which gives with A = 2π· r1· L

Wvisc = 2π· L· η· r1· (r1· ω)2/(r2 - r1) = 2π· L· η· r1· vr

2/(r2 - r1) = 992 W Picture: KJ05Question ÖS96-4.1

*) The space between the two cylinders is very small and may be approximatedby a flat plate

.

.



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1.4b Fluid element deformation and rotation



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Deformation of fluid region

dydtdy

dududtdx

yx

-yx

y

x



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Rotation of fluid region

y

x

dxdudtdydty

u

dydvdtdxdtx

v

y

u

x

v

!! ~rotation a

for corrected bemust

dy

du

dy

duyx

y

u

x

v

ifrotation no



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Tangential stress components

x

v

y

uxyyx

y

w

z

vyzzy

z

u

x

wzxxz

y

u

x

v,

x

w

z

u,

z

v

y

wurot

y

u

x

vx

w

z

uz

v

y

w

w

v

u

z

y

x

uurot

Note:

rot u is also known as”vorticity”

Turbulence is characterised byfluctuating vorticity.



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Normal stress components

u

3

22

x

uxx

u

3

22

y

vyy

u

3

22

z

wzz

otherwise this corrects for changing density: compression or expansion of the volume element content

constantisdensityf iterm 03

2 u



51 / 66

The stress tensor (Newtonian fluid)

u

u

u

τ

3

22

3

22

3

22

z

w

y

w

z

v

z

u

x

wy

w

z

v

y

v

x

v

y

uz

u

x

w

x

v

y

u

x

u



52 / 66

The stress tensor /2

δuτ3

2

z

w

y

w

x

w

z

v

y

v

x

v

z

u

y

u

x

u

z

w

z

v

z

u

y

w

y

v

y

u

x

w

x

v

x

u

δuuuτ

3

2T

Here, δ = Kronecker delta: δij = 1 if i=j; δij = 0 if i≠j, i,j = x,y,z



53 / 66

The stress tensor /3 3 compressive stresses (sv: tryckspänningar) xx, yy and zz and 6 shear stresses (sv: skjuvspänningar) xy, xz, yz, zx, yx and zy

τyx is in x-direction in plane of constant y



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1.4c The Navier-Stokes equations



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The Navier-Stokes equationsThe divergence of the stress tensor for constant viscosity(grad µ =0) and density (div u =0):

uτuτ :generalmore2

guuuu

2pt

Inserting all into the equation of motion:

Note:

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

22

,,

)()(

z

w

y

w

x

w

z

v

y

v

x

v

z

u

y

u

x

u

uzyx

uuu



56 / 66

The Navier-Stokes equations /2

xgz

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u

2

2

2

2

2

21

ygz

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

v

2

2

2

2

2

21

zgz

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w

2

2

2

2

2

21

Note: if there is no free surface then gravity (which is a constant !) can be combined with pressure, p, to give so-called ”modified pressure”



57 / 66

The Navier-Stokes equations /3 Making the N-S eqns. dimensionless by using characteristic

velocity v0, characteristic length L, gives dimensionless variables for velocity, length, time and pressure:

changing the N-S eqns. to

This allows for simplifications for Re <<1 and Re >> 1:

vρ

p*p,

L

vt*t,

L

x*x,

v

u*u

flow potential :1Re for

flow creeping :1Re for

gρpuuut

ρ

gρuμput

ρ

*g*uRe

*p*g*uLvρ

μ*p*u*u*u

t

Eulerequations



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The Navier-Stokes equations /4 Cartesian, cylindrical, spherical



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1.5 The general differential balanceequations



60 / 66

The general balance equation for additive specific property

x

y

z

Δx

Δz

Δy (x,y,z)

Jx|x Jx|x+Δx

Rt

J

Accumulation Outflow -Inflow

Generation

J = flux= flow in or outper m2 surface



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Flux of chemical species χ

diff,conv, JJJ

Convection :

uJ conv,

Diffusion (Fick’s law) :

Γdiff,J

For a species (substance matter) χ



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Balance of chemical species χ

RJJt

diff ,conv ,

Rt

u

Rt

u

or:



63 / 66

Steady heat transport (simplified energy equation)

hSTkh u

hp

Shc

kh u

For ideal gases and solids:

p

h

p c

ST

c

kT u

and, if cp is constant:

T = temperatureh = enthalpyk = thermal

conductivitycp= specific

heat



64 / 66

Balance of x-direction momentum(Navier-Stokes for x direction)

xx Vgx

puuu

t

u

Vx contains the additional viscous terms (not common…)

where u = (u,v,w)



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The general form of the balance/transport equation

St

u

Accumulation (unsteady) term

Convection term (net outflow due to convection)

Diffusion term (net inflow due to diffusion)

Source term (generation)

where Φ = u (momentum); or cpꞏT (heat); or 1 (mass), and for mass transfer concentration c (mol/m3) can be used instead of ρ

Г = diffusivity (ν = µ/ρ; or k/(ρ.cp); or D)



66 / 66

Sources / further reading #1 BMH99: Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena” Wiley, 2nd edition (1999)

BSL60: R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley (1960)

B06: J. Brännbacka ”Introduction to CFD” course material Åbo Akademi University (2006)

HKTJ07: K. Hanjalić, S. Kenjereš, M.J. Tummers, H.J.J. Jonker “Analysis and modelling of physical transport phenomena” VSSD, Delft, the Netherlands (2007) ÅA library 13 hardcopies (ASA): https://abo.finna.fi/Record/alma.896997

KJ05: D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids Engineering”, Wiley (2005)

S10: O. Zikanov ” Essentional Computational fluid dynamics” Wiley & Sons (2010) ÅA library: https://ebookcentral.proquest.com/lib/abo-ebooks/detail.action?docID=819001

T06: S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006)

Z13: R. Zevenhoven ”Principles of process engineering” (Processteknikens grunder), course compendium 424101 Åbo Akademi University (August 2013) Chapter 6 Can be downloaded here: http://users.abo.fi/rzevenho/PTG%20Aug2013.pdf

Z18: R. Zevenhoven ”Fluid and particulate systems” course material 424521 Åbo Akademi University (version 2018) http://users.abo.fi/rzevenho/kursRZ.html#FPS

Ö96: G. Öhman ”Massöverföring”, Åbo Akademi University (1996) §8.1 – 8.2

ÖS96: G. Öhman, H. Saxén ”Värmeteknikens grunder”, Åbo Akademi University (1996)

Ö01: G. Öhman ”Strömningsmekanik/Fluid mechanics”, Åbo Akademi University (2001)

Documents

1. Introduction; Fluid dynamicsusers.abo.fi/rzevenho/iCFD19-RZ1.pdf · 2019-10-10 · Fluid mechanics & Thermodynamics Algorithm development Turbulence modelling Grid generation Åbo