1. introduction; fluid dynamicsusers.abo.fi/rzevenho/icfd19-rz1.pdf · 2019-10-10 · fluid...
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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]
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1.0 Course content / Time table
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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
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Mass balance
outmdtdm
inm
outin mmdt
dm
gen,out,in, aaaa nnn
dt
dn
Balance of species a (moles, or kilos)
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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
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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
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Dp
4
2p
p
DA
2
2212
121 wmPwm
See course 424514 Fluid and particulate systems
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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
See course 424514 Fluid and particulate systems
oktober 2019 Åbo Akademi Univ - Process and Systems Engineering Piispankatu 8, 20500 Turku
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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
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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
See course 424514 Fluid and particulate systems
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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
Internal friction in fluid flow /2
*) this applies always except for very low pressuregases, for example in the upper atmosphere
xy
Picture T06
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Internal friction in fluid flow /3
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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Internal friction in fluid flow /4
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
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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:
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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
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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
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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1.4c The Navier-Stokes equations
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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”
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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The Navier-Stokes equations /4 Cartesian, cylindrical, spherical
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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1.5 The general differential balanceequations
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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Balance of chemical species χ
RJJt
diff ,conv ,
Rt
u
Rt
u
or:
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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)
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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)
Introduction to Computational Fluid Dynamics 424512 E #1 - rz
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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)