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Environmental Computational
Fluid Dynamics (E-CFD)ENG4084M
Lecturer: Dr. Songdong Shao
Office: Chesham C0.12
E-mail: [email protected]
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Lecture-1
Introduction of Module Hydrodynamic Equations
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Topic 1: Introduction of Module
Module Descriptor
Assessment CFD and E-CFD
CFD Website and Resources
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Learning Outcome
Theoretical Competence: HydrodynamicEquations, Numerical Methods and Solutions, etc
Practical Skills through Computer Simulation:Numerical Coding, Results Analysis by Using
Plotting Software, etc
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Module Contents
Fundamental hydrodynamic equations, such as Navier-Stokes equations and shallowwater equations
Different turbulent models
Basic numerical schemes, such as finite difference, finite
volume and particle methods Widely used engineering models such as RANS model,
SWE model and SPH model
How to write code, carry out numerical simulations and
analyze numerical results by using plotting software How to use a commercial software to solve practical
problem in environmental flows
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Assessment!!!
Coursework 1: Make a numerical program, run it on acomputer, plot figures and analyze results
Coursework 2: Similar to Coursework 1, but originality in
developing numerical scheme and analysis is moreemphasized at this stage
Notes: The above two assessment is 50% each; no wordslimit; the first assessment will be submitted in the middle of
term and the second will be submitted at the end of term
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CFD and E-CFD
CFD is to use the knowledge of Fluid Mechanics andComputational Methods to study a flow phenomenonthrough the use of a computer simulation
E-CFD is one kind of CFD in which the fluids are generally
regarded as INCOMPRESSIBLE. Especially E-CFD focuson flows with a free surface
Numerical Movie 1 (by other person)
Numerical Movie 2 (by myself)
Numerical Movie 3 (by myself)
http://../Numerical%20Movies/water_dry_states.mpghttp://../Numerical%20Movies/deck.ppthttp://../Numerical%20Movies/water-entry.ppthttp://../Numerical%20Movies/water-entry.ppthttp://../Numerical%20Movies/deck.ppthttp://../Numerical%20Movies/water_dry_states.mpg -
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CFD website
www.cfd-online.com
Large job database
Announcement of CFD events, workshopsand conferences
CFD discussion forum
Comprehensive CFD resources includingCFD books, visualizations, numericalsoftware to download, etc
http://www.cfd-online.com/http://www.cfd-online.com/http://www.cfd-online.com/http://www.cfd-online.com/ -
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Reading List
(1) Computational Fluid Dynamics: The Basics with ApplicationsBy John David AndersonPublisher: McGraw HillISBN: 0070016852 (Highly recommended)
(2) Computational Methods for Fluid DynamicsBy Joel H. Ferziger and Milovan Peric
Publisher: Springer Verlag
ISBN: 3540653732 (Second highly recommended)
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Topic 2: Hydrodynamic Equations
Continuity Equation
Euler Equation Navier-Stokes Equation
Partial Differential Equations
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Continuity Equation
Consider a small box dxdydz Define three directions: x, y, z Define velocity U, V, W at center
point A in three directions
Define fluid density at center A
In x direction, the rate of mass flow into the box through left face is
dydzdx
x
UU
dx
x)
2)(
2(
The sum is
dydzdx
x
UU
dx
x)
2)(
2(
dxdydzx
Udydz
dx
x
UU
dx
xdydz
dx
x
UU
dx
x
)()
2)(
2()
2)(
2(
In x direction, the rate of mass flow out of the box through right face is
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Continuity Equation
Similarly, the sum of rate of mass flow in y and z directions is
Time rate of change of mass in small box dxdydz is
dxdydzy
V
)(dxdydz
z
W
)(
dxdydzz
Wdxdydz
y
Vdxdydz
x
U
)()()(
dxdydzt
dxdydzdxdydzt
)(
(1)
and
Total sum of rate of mass flow in box dxdydz is
(2)
According to Universal Mass Conservation law (1) = (2)
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Continuity Equation
Universal form:
dxdydzt
dxdydzzWdxdydz
yVdxdydz
xU
)()()(
0)()()(
z
W
y
V
x
U
t
For a steady flow: 0)()()(
z
W
y
V
x
U
For an incompressible flow: 0
z
W
y
V
x
U
0)( V
0 V
),,( WVUVVector form:steady flow
incompressible flow
0)(
V
t
universal flow
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Euler Equation -Inviscid Fluids
Consider a small box dxdydz Define three directions: x, y, z Define velocity U, V, W at center
point M in three directionsDefine pressure and density at M
In x direction, the acting forces includepressure forceandbody forceNo shear stress for ideal fluids
Write the Newtons Second Law maFdt
dUdxdydzdxdydzfdydz
dx
x
ppdydz
dx
x
pp x
)
2()
2(
x
pf
z
UW
y
UV
x
UU
t
U
dt
dUx
1 Local Acceleration +Convective Acceleration
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Euler Equation -Inviscid Fluids
Similarly in y and z directions
In 1775, put forward by Euler: Relating force and motion for ideal fluids,both incompressible and compressible
For a static fluid
For a static fluid under gravity only, integrate
01
z
pg
Cgzp
y
pf
dt
dVy
1
z
pf
dt
dWz
1
01
x
pfx
0
1
y
pfy
0
1
z
pfz
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N-S Equation -Real Fluids
For a plane normal to z axis, the actingsurface forces are characterized by
Similarly, for a plane normal to x or y axis, the acting surfaceforces are characterized by
Normal stress zzp
Tangential stress zx zy
Normal stress xxp
Tangential stress xzxy
Normal stress yyp
Tangential stress yzyx
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N-S Equation -Real Fluids
Following the same procedureas deriving the Euler equation
and considering additionalshear stresses, by using theNewtons second law:
dt
dUdxdydzdxdz
dy
ydxdz
dy
y
dxdydz
zdxdy
dz
zdxdydzfdydz
dx
x
ppdydz
dx
x
pp
yx
yx
yx
yx
zx
zx
zx
zxx
xx
xx
xx
xx
])
2
()
2
[(
])2
()2
[()2
()2
(
)(1
zyx
pf
dt
dU zxyxxxx
Momentum Equation
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N-S Equation -Real Fluids
)(1
zxy
pf
dt
dV zyxyyyy
Similarly, the momentum equations in y andz directions can be derived as:
)(1
yxz
pf
dt
dW yzxzzzz
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Incompressible Newtonian Fluids
Following Newtons law of viscosity(Newtonian fluids)
Also, normal stressin different directions can be represented by(Incompressible fluids)
We can have and prove
dy
du
)(x
V
y
Uyxxy
)(
z
U
x
Wzxxz
)(
z
V
y
Wzyyz
x
Uppxx
2
y
Vppyy
2
z
Wppzz
2
In an ideal or static fluid, normal stress ordynamic pressure is the same in all directions
pppp zzyyxx
In a real fluid in motion, normal stress or dynamicpressure may not be the same in all directions
pppp zzyyxx
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N-S Equation
Incompressible Newtonian Fluids
By taking the pressure and shear stress relationships for an IncompressibleNewtonian fluid into the previous general N-S equations, we have
)(1
2
2
2
2
2
2
z
U
y
U
x
U
x
pf
z
UW
y
UV
x
UU
t
U
dt
dUx
)(1
2
2
2
2
2
2
z
V
y
V
x
V
y
pf
z
VW
y
VV
x
VU
t
V
dt
dVy
)(
12
2
2
2
2
2
z
W
y
W
x
W
z
p
fz
W
Wy
W
Vx
W
Ut
W
dt
dWz
French Engineer: Navier (1821) + British Engineer: Stokes (1845)
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N-S Equations for Incompressible Flow
Continuity Equation 0 V
VfV 21
P
dt
d
Momentum Equation
),,( WVUV ),,( zyx ffff
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Solutions of N-S Equations
Momentum equations + Continuity equation = 4 Equations
But, it is very difficult to get theoretical solutions due tononlinear partial differential equations, so we have touse theNUMERICAL METHODS
Pressure P + Velocities U, V, W = 4 Unknowns
New Module: Computational Fluid Dynamics
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Initial and Boundary Conditions
No-penetration boundary for a inviscid flow
Initial condition: At the beginning of computation, thedistributions of flow velocity and pressure, etc
0
tt ),,(00 zyxPP tt ),,(00
zyxtt VV
Boundary condition: Throughout the computation, the controlconditions at the physical and numerical boundaries, e.g.
0VNo-slip boundary for a viscous fluid
0nV
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Mathematical Behavior of Partial DifferentialEquations based on Characteristic Lines
Elliptic Type, e.g.Laplaces equation
Parabolic Type, e.g. Heatconduction equation
Hyperbolic Type, e.g.
Wave equation
2
2
x
T
t
T
0
x
uc
t
u2
22
2
2
x
uc
t
u
02
2
2
2
yx
Hyperbolic and Parabolic types have a marching direction,while Elliptic type has no marching direction.
Different types of equation have different solution methods. N-S equations have a mixed behavior and can be simplified
into these model equations under certain conditions
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Supporting Readings
Computational Fluid Dynamics: The Basics with ApplicationsBy John David AndersonPublisher: McGraw HillISBN: 0070016852
Chapter 1-3
Computational Methods for Fluid DynamicsBy Joel H. Ferziger and Milovan PericPublisher: Springer VerlagISBN: 3540653732
Chapter 1
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Shallow Water Equations
For more details, search website using: Shallow Water Equation
Example:http://en.wikipedia.org/wiki/Shallow_water_equations
Particularly suitable for Environmental Free Surface Flowswith an open space, such as river, lake, estuary, coast
SWE is based on 3-D N-S equations, by assuming:
Horizontal scale of flow is much larger than vertical scale Pressure distribution is hydrostatic in the vertical direction
Vertical velocity variation is zero, i.e. no change in vertical velocity Integration of 3-D N-S equation along vertical line and introducing
drag force on lower boundary
http://en.wikipedia.org/wiki/Shallow_water_equationshttp://en.wikipedia.org/wiki/Shallow_water_equations