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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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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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)(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

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