Post on 13-Jul-2020
transcript
HC Chen 1/13/2020
Chapter 1 Introduction 1
Chapter 1
Computational Fluid Dynamics: An Introduction
Fluid Mechanics1. Experimental Fluid Dynamics (EFD) Wind tunnels (subsonic, transonic, supersonic,
low-turbulence, pressurized, …)
Towing tank, wave tank, large circulation channel, cavitation tunnel, flume, hydraulic model, …
Field experiments
2. Theoretical Fluid Dynamics
3. Computational Fluid Dynamics (CFD) Numerical algorithms, Computers, Graphics,
Animations
HC Chen 1/13/2020
Chapter 1 Introduction 2
Limitation of EFD Fundamental limitations of ground-base
experimental facilities Limited by model size, pressure, velocity,
Reynolds number, temperature, and the type of atmosphere that can be simulated
Wall and support interference limit the accuracy of simulation (especially in transonic facilities)
Stream nonuniformities preclude accurate simulation of boundary layer transition
Lengthy, expensive, risky, incomplete (simultaneous velocity, pressure, and turbulence measurement?)
Examples of EFD LimitationWeather forecast Events cannot be repeated (chaotic) Steady-state meaningless
Hypersonic vehicles Impossible to do experiments, EFD used to validate CFD for
hypersonic vehicles
Safety / Stability High-angle-of-attack maneuvering
Combustion High temperature, high reaction rates, chemical reactions, …
Turbulence Simultaneous measurement of velocity and pressure
fluctuations, two-point correlation, ...
HC Chen 1/13/2020
Chapter 1 Introduction 3
Advantages of CFD CFD complements experimental and theoretical fluid
dynamics
Alternative cost-effective means of simulating real fluid flows (lower energy consumption also)
significant reduction of lead time in design and development
simulate flow conditions not reproducible in model tests (weather forecast, extremely high pressure/temperature, …)
provide more detailed and comprehensive information
CFD is increasing more cost-effective than wind-tunnel and wave-tank testing
Limitations of CFDPhysical Modeling turbulence, non-Newtonian fluid, multi-phase
flow, initial and boundary data representations
Geometry Modeling Roughness, geometry details, multi-phase interface
Numerical Modeling Truncation and round-off errors, convergence and
stability, resolution of a wide range of length and time scales
HC Chen 1/13/2020
Chapter 1 Introduction 4
Typical Practical Problems Complex geometry, simple physics
Simple geometry, complex physics
Ultimate goal - complex geometry with complex physics
More powerful computers
More memory capacity
More accurate discretization schemes
Massive Parallel Computing
User-friendly pre- and post-processing software
Typical Practical Problems Classification of fluid flows
single phase 1D, 2D, 3DNewtonian steady
multiple-phase axisymmetricnon Newtonian transient
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
User-friendly pre- and post-processing tools
HC Chen 1/13/2020
Chapter 1 Introduction 7
Overview of CFD
efficiency
transferheat
separation flow
flowrate
moments
forces lift/drag
,
,,
etc. law, sFourier'
strain-of-eStress/rat
state of Equation
Energy of onConservati
Momentum of onConservati
Mass of onConservati
puuu
T
p
WVU
iji
for each numerical element
discrete nodal values
for each fluid particle (continuous function)
Discretization - Numerical Methods Auxiliary (initial/boundary) conditions
Overview of CFD
finite number of discrete
nodal values
continuous function at every point
Discretization - Numerical Methods Auxiliary (initial/boundary) conditions
Partial Differential Equations
Discretization
Finite-difference Finite-volume Finite-element
Spectral Boundary-element
System of Algebraic Equations
Numerical Solutions
Matrix solvers
HC Chen 1/13/2020
Chapter 1 Introduction 8
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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)
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
( ) ( )
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
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
HC Chen 1/13/2020
Chapter 1 Introduction 25
Energy Equation Heat and Work Interactions
Differential Form
Mechanical Energy Equation (multiply the momentum equation by )
MS MV MV MS
CV CV CV CV
q ndS Qd V fd t n VdS
qd Qd f Vd T V d
RHS
1
2
De V V T V f V q Q
Dt
V
1
2
D DVV V V f V V T
Dt Dt
Energy Equation Thermal energy equation
Fourier’s law for conduction
Energy equation
DeT V q Q
Dt
T q
De
p V T QDt
Compression work
Dissipation Conduction Heat generation
:2
V 2 D D
HC Chen 1/13/2020
Chapter 1 Introduction 26
Energy Equation Incompressible flow, constant
Cartesian coordinates
: 2v
DTC 2 D D T Q
Dt
2 2 2
v 2 2 2
T T T T T T TC u v w Q
t x y z x x x
222 2
22
u v w u v2 2 2
x y z y x
u w v w
z x z y
Generic Conservation Equation Integral form
Vector form
Cartesian coordinates
CV CS CS CV
d V ndS ndS q dt
V qt
j
j j j
u qt x x x
Γ:diffusivity for ; q :source/sink of