cylindrical system (r, θ, z), and spherical system(r, θ, Φ) should be appropriately chosen for a better resolution of the geometry (e.g. cylindrical for circular pipe).

10
Modeling (coordinates)
x
y
z
x
y
z
x
y
z
(r,
,z)
z
r
(r,
,
)
r
(x,y,z)
Cartesian
Cylindrical
Spherical
General Curvilinear Coordinates
General orthogonal
Coordinates

11
Modeling (governing
equations)
•
Navier-Stokes equations (3D in Cartesian coordinates)
2
2
2
2
2
2
ˆ
z
u
y
u
x
u
x
p
z
u
w
y
u
v
x
u
u
t
u
2
2
2
2
2
2
ˆ
z
v
y
v
x
v
y
p
z
v
w
y
v
v
x
v
u
t
v
0
z
w
y
v
x
u
t
RT
p
L
v
p
p
Dt
DR
Dt
R
D
R
2
2
2
)
(
2
3
Convection
Piezometric pressure gradient
Viscous terms
Local
acceleration
Continuity equation
Equation of state
Rayleigh Equation
2
2
2
2
2
2
ˆ
z
w
y
w
x
w
z
p
z
w
w
y
w
v
x
w
u
t
w

12
Modeling (flow conditions)
•
Based on the physics of the fluids phenomena,
CFD can be distinguished into different
categories using
different criteria
•
Viscous vs. inviscid
(Re)
•
External flow or internal flow (wall bounded or not)
•
Turbulent vs. laminar (Re)
•
Incompressible vs. compressible (Ma)
•
Single- vs. multi-phase (Ca)
•
Thermal/density effects (Pr,
, Gr, Ec)
•
Free-surface flow (Fr) and surface tension (We)
•
Chemical reactions and combustion (Pe, Da)
•
etc…

13
Modeling (initial conditions)
•
Initial conditions (ICS, steady/unsteady flows)
•
ICs should not affect final results and only
affect convergence path, i.e. number of
iterations (steady) or time steps
(unsteady) need to reach converged
solutions.
•
More reasonable guess can speed up the
convergence
•
For complicated unsteady flow problems,
CFD codes are usually run in the steady
mode for a few iterations for getting a
better initial conditions

14
Modeling(boundary
conditions)
•
Boundary conditions:
No-slip or slip-free
on walls, periodic,
inlet
(velocity inlet, mass flow
rate, constant pressure, etc.),
outlet
(constant
pressure, velocity convective, numerical beach, zero-
gradient), and non-reflecting (for compressible flows,
such as acoustics), etc.
No-slip walls: u=0,v=0
v=0, dp/dr=0,du/dr=0
Inlet ,u=c,v=0
Outlet, p=c
Periodic boundary condition in
spanwise direction of an airfoil
o
r
x
Axisymmetric

15
Modeling (selection of
models)
•
CFD codes typically designed for solving
certain fluid
phenomenon by applying different models
•
Viscous vs. inviscid
(Re)
•
Turbulent vs. laminar (Re,
Turbulent models
)
•
Incompressible vs. compressible (Ma,
equation of
state
)
•
Single- vs. multi-phase (Ca,
cavitation model, two-
fluid
model
)
•
Thermal/density effects and energy equation
(Pr,
, Gr, Ec,
conservation of energy
)
•
Free-surface flow (Fr,
level-set & surface tracking
model
) and
surface tension (We,
bubble dynamic model
)
•
Chemical reactions and combustion (
Chemical
reaction

16
Modeling (Turbulence and free surface
models)
•
Turbulent models
:
•
DNS
: most accurately solve NS equations, but too expensive
for turbulent flows
•

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- Summer '18