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Computational Fluid Dynamics ME 552

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KNUST Visiting Professor Lecture Series Fall 2006. Introduction to Basic Concepts of CFD ... mechanical movement (e.g. of pistons, fans, rudders) ... – PowerPoint PPT presentation

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Title: Computational Fluid Dynamics ME 552


1
Computational Fluid Dynamics- ME 552
  • Patrick F. Mensah, PhD
  • Professor
  • Mechanical Engineering Dept.
  • Southern University and AM College

2
Outline of Lecture
  • Course overview
  • Introduction of basic concepts of CFD

3
Introduction to Basic Concepts of CFD
  • The Need for CFD
  • Applications of CFD
  • The Strategy of CFD
  • Mathematical Description of Physical Phenomena
  • Discretization Methods
  • Assembly of Discrete System and Application of
    Boundary Conditions

4
Introduction to Basic Concepts of CFD
  • CFD is predicting what will happen,
    quantitatively, when fluids flow, often with the
    complications of
  • simultaneous flow of heat,
  • mass transfer (e.g. perspiration, dissolution),
  • phase change (e.g. melting, freezing, boiling),
  • chemical reaction (e.g. combustion, rusting),
  • mechanical movement (e.g. of pistons, fans,
    rudders),
  • stresses in and displacement of immersed or
    surrounding solids

5
Introduction to Basic Concepts of CFD
  • Solution of Discrete System
  • Grid Convergence
  • Dealing with Nonlinearity
  • Direct and Iterative Solvers
  • Iterative Convergence
  • Numerical Stability

6
Applications
  • Used routinely to solve fluid flow problems in
    industries such as
  • Design of devices such as pumps, compressors, and
    engines
  • Aircraft engineers simulate three dimensional
    flows about entire aircraft
  • Simulation of flow over vehicles
  • Bio-medical engineering is a rapidly growing
    field and uses CFD to study the circulatory and
    respiratory systems

7
Applications
  • Computational modeling of heat transfer in
    engineering systems
  • Thermo-mechanical analyses of gas turbine blades
    and hot section
  • Multi-phase flow heat transfer in phase change
    systems boilers, condensers, evaporators,
    cooling of electronic systems, Nuclear reactors
  • New applications in nano and micro devices
  • Fuel cells

8
Applications
  • http//www.cham.co.uk/phoenics/d_polis/d_applic/ap
    plic.htm

9
Strategy of CFD
  • The strategy of CFD is to replace the continuous
    problem domain with a discrete domain using a
    grid. In the continuous domain, each flow
    variable is defined at every point in the domain.
    For instance, the pressure p in the continuous 1D
    domain shown in the figure below would be given
    as p p(x) 0 lt x lt 1
  • In the discrete domain, each flow variable is
    defined only at the grid points. So, in the
    discrete domain shown below, the pressure would
    be defined only at the N grid points. pi p(xi)
    i 1, 2,,N

10
Strategy of CFD
  • In a CFD solution, one would directly solve for
    the relevant flow variables only at the grid
    points. The values at other locations are
    determined by interpolating the values at the
    grid points.
  • The governing partial differential equations and
    boundary conditions are defined in terms of the
    continuous variables p, V etc. One can
    approximate these in the discrete domain in terms
    of the discrete variables pi, Vi etc. The
    discrete system is a large set of coupled,
  • algebraic equations in the discrete variables.
    Setting up the discrete system and solving it
    (which is a matrix inversion problem) involves a
    very large number of repetitive calculations and
    is done by the digital computer.

11
Mathematical Description of Physical Phenomena
  • Based on
  • Conservation Laws

12
Governing Equations of Fluid Dynamics
  • CFD is based on the fundamental equations of
    fluid dynamics. The equations are mathematical
    statements of fundamental physical principles
  • Mass is conserved
  • Newtons law Fma
  • Energy is conserved

13
Road map for Relationship Between Fundamental
Physical Laws and Fluid Flow Models
14
Models of the Flow
  • Approach to obtaining basic equations of fluid
    flow motion
  • Choose the appropriate fundamental physical
    principles from the law of physics such as
  • Mass is conserved
  • Fma (Newtons second law)
  • Energy is conserved
  • Apply these physical principles to suitable model
    of the flow
  • From this application, extract the mathematical
    equations which embody such physical principles

15
How to apply physical principles to suitable
model of moving fluid and its visualization
  • Finite control volume fixed in space
  • Finite control volume moving with the fluid
  • Infinitesimal fluid element fixed in space
  • Infinitesimal fluid element moving along a
    streamline with velocity V

16
The Substantial Derivative (Time Rate of Change
Following a Moving Fluid Element)
17
Substantial Derivative
  • Physically is that time rate of change of
    material property following a moving fluid
    element
  • Mathematical expressed in terms of the
  • local derivative- which is physically the time
    rate of change of material property at a fix
    location
  • and convective derivative, which is physically
    the time rate of change due to movement of the
    fluid element from one location to another in the
    flow field where properties are spatially
    different

18
Divergence of the Velocity
  • Physically it is the time rate of change of the
    volume of a moving fluid element per unit volume

19
Differential Control Volume , dx, dy, dz, for
convection and diffusion of chemical species in
rectangular coordinates
20
Mathematical Description of Physical Phenomena
  • Conservation of Chemical Species
  • Let mi denote the mass fraction of a chemical
    species. In the presence of a velocity field u,
    the conservation of mi is expressed as
  • Where the first term on the LHS denotes the rate
    of change of the mass of the chemical species per
    unit volume

21
Mathematical Description of Physical Models
  • Conservation of Chemical Species
  • ?umi is the convection flux of the species, i.e.,
    flux carried by general flow field ?u.
  • Ji stands for the diffusion flux, normally caused
    by gradients of mi
  • On the RHS Ri is the rate of generation of
    chemical species per unit volume due to chemical
    reactions

22
Mathematical Description of Physical Models
  • Conservation of Chemical Species
  • Diffusion flux can be expressed by Ficks Law of
    diffusion
  • Where ?i is the diffusion constant
  • Hence the conservation equation of chemical
    species is written as

23
Differential Control Volume , dx, dy, dz, for
convection and diffusion of chemical species in
rectangular coordinates
24
Conservation of Momentum
25
Conservation of Momentum on a Fluid Element
  • Note for Cartesian coordinate system x1 x, x2y
    and x3z
  • Normal Stress (sii ) and shear stress (sij)
    relations
  • s11sxx s12sxys13sxz
  • s21syxs22syys23syz
  • s31szxs32szys33szz

26
Conservation of Momentum on a Fluid Element
  • From the conservation of momentum principle
  • Net force on the fluid element equals its mass
    times acceleration

27
Conservation of Momentum on a Fluid Element
  • x-momentum gain by convection
  • x-momentum due to surface forces
  • Viscous forces
  • Pressure forces
  • Body forces (gravitational)

28
Conservation of Momentum on a Fluid Element
  • Note that

29
Conservation of Momentum on a Fluid Element
  • Summation of forces

30
Mathematical Description of Physical Models
  • Conservation of Momentum x-direction
  • ? is the viscosity p is the pressure, Bx is the
    x-direction body force per unit volume, and Vx
    stands for the viscous terms that are in addition
    to those expressed by div (? grad u)

31
The Energy Equation
  • Physical Principle Energy is conserved

32
Conservation of Energy for a Control Volume

33
Mathematical Description of Physical Phenomena
  • Conservation of Energy

34
Mathematical Description of Physical Phenomena
  • h is the specific enthalpy k is the thermal
    conductivity, Sh is the volumetric rate of heat
    generation. The term div (k grad T) represents
    the influence of conduction

35
Mathematical Description of Physical Phenomena
  • The General Differential Equation

36
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37
Vector Differential OperationsReview
  • Gradient Field
  • Divergence
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