Computational Fluid Dynamics ME 552

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Computational Fluid Dynamics- ME 552

• Patrick F. Mensah, PhD
• Professor
• Mechanical Engineering Dept.
• Southern University and AM College

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Outline of Lecture

• Course overview
• Introduction of basic concepts of CFD

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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

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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

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Introduction to Basic Concepts of CFD

• Solution of Discrete System
• Grid Convergence
• Dealing with Nonlinearity
• Direct and Iterative Solvers
• Iterative Convergence
• Numerical Stability

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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

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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

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Applications

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

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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

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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.

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Mathematical Description of Physical Phenomena

• Based on
• Conservation Laws

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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

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Road map for Relationship Between Fundamental
Physical Laws and Fluid Flow Models
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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

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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

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The Substantial Derivative (Time Rate of Change
Following a Moving Fluid Element)
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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

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Divergence of the Velocity

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

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Differential Control Volume , dx, dy, dz, for
convection and diffusion of chemical species in
rectangular coordinates
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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

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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

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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

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Differential Control Volume , dx, dy, dz, for
convection and diffusion of chemical species in
rectangular coordinates
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Conservation of Momentum
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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

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Conservation of Momentum on a Fluid Element

• From the conservation of momentum principle
• Net force on the fluid element equals its mass
  times acceleration

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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)

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Conservation of Momentum on a Fluid Element

• Note that

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Conservation of Momentum on a Fluid Element

• Summation of forces

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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)

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The Energy Equation

• Physical Principle Energy is conserved

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Conservation of Energy for a Control Volume

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Mathematical Description of Physical Phenomena

• Conservation of Energy

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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

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Mathematical Description of Physical Phenomena

• The General Differential Equation

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Vector Differential OperationsReview

• Gradient Field
• Divergence