fluid/structure rotationally-rarefied gas harmonic
particle-ladden symmetric
immersed
surface-piercing
submerged
incompressibleinviscid
subsonic rotational blaminar
transonicturbulent
supersonic irrotationaltransitional
hypersonic
ounded/internal conduction
unbounded/external convection
mixed radiation
fluid time space free-surface
viscosity compressibility vorticity domain heat transfer
St Fr
Re Ma Ro Nu
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Chapter 1 Introduction 5
Elements of CFD(A) Numerical Modeling Process Construct a mathematical description of the object to be
modeled
Derive a set of differential, integral or continuum equations
Usually the analytic solutions to these equations are not available due to nonlinearity, complex geometry, complex initial/boundary conditions, etc.
Moreover, no digital computers can actually solve these equations in differential form (i.e., continuous function)
designer must exchange the differential or integral equations for other equations such as finite-difference, finite-element, discrete or something similar
the discretized equations should resemble those of the differential or continuum equations
Elements of CFD(B) Physical, Mathematical and Numerical Problems Are the continuum equations really describe the
phenomena of interest (turbulence, non-Newtonian, surface roughness)?
The continuum equations may over-determine or under-determine the problem (inconsistent or ill-posed)
“Numerical instability” may occur, usually it is only a problem of the discrete equations
“Numerical Dissipation” may be needed to stabilize the problem (numerical diffusion)
The “drugged” model may suffer a distinct reduction in its predictive capability just like a “drugged” individual cannot perform at any level of normal efficiency
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Chapter 1 Introduction 6
Elements of CFD(C) Numerical Accuracy Does not mean “accuracy at any price”, but rather the
opposite to this - “Accuracy at the minimum price”
Obtain highest level of accuracy with minimum number of numerical operations and smallest data storage requirements
(D) Numerical Efficiency Making the model relevant to its applications as cheaply as
possible
Simplify the model to the greatest extent that its envisaged application permits
Each hypothesis, semi-empirical correlation, approximation, or assumption introduce further sources of error or uncertainty
Elements of CFD(E) Validation Preferably against prototype data
Physical model itself usually involves “scale effects”
Numerical model may be subject to similar effects due to ill-specified boundary conditions (roughness, reflective properties, geometry representation, etc.)
(F) Reliability and Flexibility Crucial to the economy and lead-time of a project
The system should be flexible in provide convenient means to setup and test models before and during their running
Fluid Mechanics Fluid Statics - normal stress (pressure) only
Fluid Dynamics - both normal and tangential stresses
Fluid kinematics - Lagrangian and Eulerian
Continuum - macroscopic, statistical
Fluid particle is defined as an infinitesimal portion of the fluid as a continuum, which possesses individuality
( , , , ) ( , )
( , , , ) ( , )
x y z t r t
V V x y z t V r t
Fluid Properties Intensive property - independent of mass
Extensive property - proportional to mass
Defined based on the concept of continnum
Properties are defined macroscopically without knowing the microscopic behaviors
Point function -- not a path function
Property is a quantity that depends on the state of the system, and is independent of the path
Provide a precise description of fluid characteristics as a function of space and time
e.g., iji ,v,,m,,T,p
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Chapter 1 Introduction 9
Fundamental Laws Conservation of geometry of motion and deformation of
matter - translation, rotation, linear and angular deformations (fluid kinematics)
Conservation of Mass - Continuity equation
Conservation of momentum - Newton’s second law
Conservation of Angular Momentum
Conservation of Energy - First Law of Thermodynamics
Entropy, Irreversible flow - Second Law of Thermodynamics
Equation of state, stress/rate-of-strain relations, Fourier law, and other constitutive equations
Reynolds Transport Theorem Kinematics of Moving Control Volume
Control Mass
* Largrangian - system approach
Control Volume
* Eulerian - field approach
Reynolds transport theorem
* Conversion from Lagrangian to
Eulerian description
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Chapter 1 Introduction 10
Lagrangian Description Particle approach Material volume, material surface, material curve
Fluids in material volume (system) will move, distort, and change size and shape, but always consists of the same fluid particles
MV(to)
MV(t)
pathlines
( , )
( , )
V V( , ) etc.
o
o
o
Pathline
r r r t
p p r t
r t
Eulerian Description Field Approach Abandon the tedious and often unnecessary task
of tracking individual particles
Individual particles are not labeled and not distinguished from one another
Focuses attention on what happens at a fixed point (or volume) as different particles goes by
CV(t)
( , )
( , )
V V( , ) etc.
r t
p p r t
r t
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Chapter 1 Introduction 11
Lagrangian vs. EulerianLagrangian Approach tag each individual particle
difficult to track many particles at a time
may be irrelevant to the problem of interest
Eulerian Approach observe fluid particles in a pre-selected, often fixed,
control volume
always focus on regions of interest
easy to setup experiment or computational domain
But the conservation laws need to be derived from particle approach!!!
Lagrangian vs. Eulerian
MV(t)
MV(t+dt)
(u,v,w)
Linear and angular deformations of material volume
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Chapter 1 Introduction 12
dS
dS*
bdt
S t*( )
S t dt*( )t*( )
t dt*( )
bndSˆ
t t t
dS b t ndS b n t dS
* *( ) ( )
ˆ ˆ
Reynolds Transport Theorem
Control volume *(t) – may be fixed or deformableControl surface S*(t)
Reynolds Transport Theorem
tt t t
t
Ft t t b n t dS F r t t F r t t
t
F r t t d F r t t d F r t t d
F F F r t t d F r t t
t t
* *
*
* *
( )( ) ( )
( )
ˆ( ) ( ) ; ( , ) ( , )
( , ) ( , ) ( , )
( , ) ( , )
se
t
tt t t t
t
cond order as t, tt
0t
d
dF r t d F r t t d F r t d
dt t
F r t Ft d F r t d t d
t t t
* * *
*
( )
0( ) ( ) ( )
0( ) ( )( )
1( , ) lim ( , ) ( , )
1 ( , )lim ( , )
t t S t
d F r tF r t d d F r t b ndS
dt t* * *( ) ( ) ( )
( , )ˆ( , ) ( , )
total time derivative partial time derivative boundary movement
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Chapter 1 Introduction 13
dS
dS*
bdt
S t*( )
S t dt*( )t*( )
t dt*( )
bndS
Reynolds Transport Theorem
t t S t
d F r tF r t d d F r t b ndS
dt t* * *( ) ( ) ( )
( , )ˆ( , ) ( , )
Liebniz’s TheoremTotal time derivative of an integral with time-dependent limits equals partial time derivative of the integral plus a term that accounts for the motion of the integration boundary
Liebniz’s Theorem
MV t CV t CS t
D dFd Fd F V b ndS
Dt dt( ) ( ) ( )
ˆ
CV t CV t CS t
CV t CV t
MV t MV t
d Farbitrary control volume Fd d Fb ndS
dt t
d Ffixed control volume b = 0 Fd d
dt t
D Fmaterial volume, b = V Fd d FV
Dt t
( ) ( ) ( )
( ) ( )
( ) ( )
ˆ
,
MS t
ndS( )
ˆ
Reynolds Transport Theorem
Lagrangian (MV) Eulerian (CV & CS)
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Chapter 1 Introduction 14
dS
n
n
n
I
III
II
SII
SIIIV
V
MS(t), CV(t)=CV(t+t)
MS(t+t)
CS(t) = MS(t) = SII + SIII
MV(t) = CV(t) = I + II
MV(t +t) = I + III
CV(t +t) = CV(t)
Fixed Control Volume
( ) ( ) ( )
( , ) ( , ) ( , )MV t CV t CS t
D dF r t d F r t d F r t V ndS
Dt dt
tMV t MV t t MV t
t MV t t MV t t
t MV t t MV t
DF r t d F r t t d F r t d
Dt t
F r t t d F r t d t
F r t d F r t d t
0( ) ( ) ( )
0 ( ) ( )
0 ( ) ( )
1( , ) ( , ) ( , )
1( , ) ( , )
1( , ) ( , )
lim
lim
lim
t MV t t MV t CV t
t I III I II
F r t t F r t d t
F r t d F r t d F r t d F r t d t
0 ( ) ( ) ( )
0
1( , ) ( , )
1( , ) ( , ) ( , ) ( , )
lim
lim
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Chapter 1 Introduction 15
tMV t CV t III II
D dF r t d F r t d F r t d F r t d
Dt dt t0( ) ( )
1( , ) ( , ) ( , ) ( , )lim
MV t CV t CS t
D dF r t d F r t d F r t V ndS
Dt dt( ) ( ) ( )
( , ) ( , ) ( , )
II II
III III
MV t CV t CS t
d V t nds across S
d V t nds across S
D dF r t d F r t d F r t V ndS
Dt dt( ) ( ) ( )
( ) ( )
( ) ( )
( , ) ( , ) ( , )
Rate of change of F in MV
Rate of change of F in CV
Convective Transport (flux) of F across CS
n
n
VCV
MV(t+t)
: Absolute fluid velocity with respect to a fixed coordinate system
: absolute velocity of the control volume
: relative fluid velocity (with respect to a moving control volume)
CV(t+t)
MV(t) = CV(t)
rV
V
V
CV
rV
rV
CVr VVV
Moving Control Volume
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Chapter 1 Introduction 16
r
MV t CV t CS t
D dF r t d F r t d F r t V ndS
Dt dt( ) ( ) ( )
( , ) ( , ) ( , )
Lagrangian description following the system
Eulerian description following control volume
sys MV
Let F then Fd d;
r
MV CV CS
D d d d V ndS
Dt dt
*
* The system always consists of the same mass
* The control volume may change in size and shape
Fddt
dFd
t ;only )t(fFd
Divergence Theorem
CS CV
A ndS Ad
Conversion of surface integral to volume integral
;
( )
( )
CS CV
CS CV
let A FV then
FV nds FV d or
V nds V d
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Chapter 1 Introduction 17
Fixed Control VolumeFor arbitrary (shape and size) but fixed control volume (independent of time)
CV CV
d FFd d
dt t
MV CV
D FFd FV d FV V F F V
Dt t
F F DFFV V F F V F V
t t Dt
( ) ; ( )
( )
( ) ( ) ( )MV t CV t CV t
D DFFd d F Vd
Dt Dt
Conservation of Mass Conservation of Mass
Lagrangian description
MV
F m d;
MV
Dm
DtD
dDt
0
0
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Chapter 1 Introduction 18
For arbitrary, but fixed control volume
Differential Form
MV CV CS
CV CV
CV
D dd d V ndS
Dt dt
d V dt
V dt
( )
( ) 0
Continuity Equation
Vt
( ) 0
Lagrangian
Eulerian
Continuity Equation
Incompressible FlowD
Dt0
Vt
0
V 0
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Chapter 1 Introduction 19
Constant density
Differential Form
MV
MV CV CV
F Vol dV constant
Dd V ndS Vd
Dt
1;
Conservation of Volume
V 0
Continuity Equation Incompressible flow
Cartesian Coordinates
i
i
u v w
x y z
u
x
0
0
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Chapter 1 Introduction 20
MV MV
s b
MV
Momentum M ( V d V d
Let F V or V
DV d F F
Dt
)
( )
Lagrangian
Momentum Equations Newton’s Second Law of Motion
Lagrangian description for control mass
Change of Momentum = Total Forces
Surface forces and body forces
Momentum Equations Lagrangian description
MV MV MS
DV d f d t n dS
Dt( )
Rate of change of momentum in MV
Body force Surface force
Surface force:
Stress tensor:ji
ij aaTT
TnnTt
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Chapter 1 Introduction 21
Momentum Equations Eulerian description
Use Reynolds transport theorem
CV CS CV CS
dV d V V ndS fd T ndS
dt
Rate of change of momentum in CV
Body force Surface force
Momentum flux across CS
Momentum Equations For arbitrary but fixed control volume
Differential Form (Newtonian or non-Newtonian
fluids)
CV CV CV CV
CV
V d VV d fd Tdt
V VV f T dt
( ) ( )
( ) ( ) 0
V VV f Tt
DV f T
Dt
( ) ( )
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Chapter 1 Introduction 22
Newtonian Fluid Stress is linearly proportional to rate-of-strain
Stress tensor has nine components
Rate-of-strain (deformation) tensor also has nine components
T jiij
j i
uu1 1D V V ; D
2 2 x x
jk iij ij
k j i
2T p V I D ideal gas
3
uu uT p
x x x
( ) 2 ; ( )
Navier-Stokes Equations
Constant viscosity and
Compressible
Incompressible
3
2
DV f p V V
Dt21
3
DVf p V
Dt2
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Chapter 1 Introduction 23
Navier-Stokes Equations Incompressible flow
Cartesian Coordinates
x
y
u u u u p u u u u v w f
t x y z x x y z
v v v v p v v v u v w f
t x y z y x y z
w w w w u v w
t x y z
2 2 2
2 2 2
2 2 2
2 2 2
z
p w w w f
z x y z
2 2 2
2 2 2
i i ij i
j i j j
u u u p u f
t x x x x
2
Conservation of Energy First Law of Thermodynamics
Both Q and W are path functions (process dependent)
but the net into the system is a point function
dE is a total differential (a thermodynamic property)
dV
dS
n
Heat
Work
Energy E
DEQ W
Dt
W
Q
Q W
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Chapter 1 Introduction 24
Energy Equation
MV
MS MV MV MS
DE De V V d
Dt Dt
q ndS Qd V f d t n VdS
1
2
Rate of change of energy in MV
Heat added to the system
Power added to the system
Lagrangian
description
Conduction heat flux
Heat generation
(radiation, ...)
Power produced by body force
Power produced by surface force
(Internal & Kinetic)
QDE
DtW
Energy Equation
CV CS
CV
CV
DE de V V d e V V V ndS
Dt dt
e V V e V V V dt
V e V V dt
e V V V e V Vt
1 1
2 2
1 1
2 2
1
2
1 1
2 2
CV
CV
d
De V V d
Dt
1
2
Reynolds transport theorem
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Chapter 1 Introduction 25
Energy Equation Heat and Work Interactions
Differential Form
Mechanical Energy Equation (multiply the momentum equation by )
