ME 423 - COMPUTATIONAL
FLUID MECHANICS AND
HEAT TRANSFER
Spring 2018/2019
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Instructor: Dr. Osama Ibrahim
Phone: 2498-5789
E-mail: dr.oibrahim@gmail.com
Location: 14th KH building – 1st floor - Room 47
Office Hours: Sunday, Tuesday, Thursday from 11:00 - 1:00PM
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Description
ME 423: Computational fluid dynamics and heat transfer (3,0,3).
Classification of partial differential equations. Numerical techniques for solving fluid dynamics and
heat transfer problems. Finite difference, finite element, boundary element, and finite control
volume methods. Numerical solutions of parabolic, elliptic, and hyperbolic equations in fluid
dynamics and heat transfer.
Textbook
Computational Fluid Mechanics and Heat Transfer, by Dale A. Anderson and John C. Tannehill,
Taylor & Francis, Inc., 3rd Edition, 2011.
References
Computational Fluid Dynamics For Engineers, by Klaus A. Hoffmann and Steve T. Chiang, A
Publication of Engineering Education System, Wichita, Kansas, USA, 1998.
Computational Fluid Dynamics: The Basics with Applications, by John D. Anderson, Jr., McGraw-
Hill, Inc., 1995
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Prerequisites by Topics:
1. Heat Transfer
2. Fluid Dynamics
3. Numerical Analysis
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Heat Convection Latif M. Jiji
Topics:
Preliminaries.
Conservation equations.
Diffusion problems.
Advection-diffusion problems.
Pressure-velocity coupling flows.
Unsteady diffusion.
Solution of discretized equations.
Boundary conditions.
Other discretization methods.
Special topics: Phase change
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Evaluation Methods:
1) Computer Assignments +HW+Q 25%
2) Project 15%
3) 2 Midterm Exams 20%
4) Final Exam 40%
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Mathematical Background
Review of the following mathematical definitions which are needed in
the differential formulation of the basic laws.
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Mathematical Background
In Cartesian coordinates
In cylindrical coordinates
In spherical coordinates
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Mathematical Background
The divergence of a vector ܸ is a scalar defined as
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Heat Convection Latif M. Jiji
Mathematical Background
The gradient of a scalar, such as temperature T, is a vector given by
In general this quantity is a function of the four independent variables x,
y, z and t. Thus in Cartesian coordinates we write
Consider a variable of the flow field designated by the symbol f, which is a
scalar quantity such as temperature T, pressure p, density ρ , or velocity
component u.
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Mathematical Background
The total differential of f is the total change in f resulting from changes in x,
y, z and t.
Dividing through by dt
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Mathematical Background
Convective derivative Local derivative
Setting f = u, for example
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Mathematical Background
Setting f = T, for example
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DIFFERENTIAL FORMULATION OF THE BASIC LAWS
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Three fundamental laws must be satisfied.
Conservation of mass, momentum, and energy,
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Conservation of Mass: The Continuity Equation
Consider an element dxdydz as a control volume in the flow field
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Conservation of Mass: The Continuity Equation
Consider an element dxdydz as a control volume in the flow field
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Conservation of Mass: The Continuity Equation
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Heat Convection Latif M. Jiji
Conservation of Mass: The Continuity Equation
Substituting and dividing through by dxdydz, gives
An alternate form is obtained by differentiating the product terms to obtain
Represent the total derivative of ρ and the last three terms represent
the divergence of the velocity vector ܸ
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Conservation of Mass: The Continuity Equation
In Cartesian coordinates
In cylindrical coordinates
In spherical coordinates
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Conservation of Momentum: The Navier-Stokes Equations of Motion
• Momentum is a vector quantity
• Conservation of momentum (Newton’s law of motion) provides three equations
• Application of Newton’s law of motion to the element shown in the figure below gives
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Conservation of Momentum: The Navier-Stokes Equations of Motion
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Conservation of Momentum: The Navier-Stokes Equations of Motion
Next we determine the sum of all external forces acting on the element in the
x-direction. We classify external forces as:
(i) Body force. This is a force that acts on every particle of the material or
element. Examples include gravity and magnetic forces.
(ii) Surface force. This is a force that acts on the surface of the
element. Examples include tangential forces (shear) and normal
forces (pressure and stress).
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Conservation of Momentum: The Navier-Stokes Equations of Motion
(i) Body force in x-direction
where gx is gravitational acceleration
component in the plus x-direction
(ii) Surface force in the x-direction.
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Conservation of Momentum in the x-Direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
x-direction
y-direction
z-direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
Application of the moment of momentum principle to a differential element gives
where µ is a property called viscosity and p is the hydrostatic pressure.
A fluid that obeys the above equations is referred to as Newtonian fluid
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Conservation of Momentum: The Navier-Stokes Equations of Motion
x-direction
y-direction
z-direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
In a vector form,
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Assuming constant viscosity and constant density
The continuity equation
The momentum equation: vector form
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Assuming constant viscosity and constant density
The continuity equation
=0
Conservation of Momentum: The Navier-Stokes Equations of Motion
x-direction
y-direction
z-direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
Cylindrical Coordinates
r-direction
θ-direction
z-direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
Spherical Coordinates
r-direction
θ-direction
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Conservation of Momentum: The Navier-Stokes Equations of Motion
Spherical Coordinates
φ-direction
The operator ▽ in spherical coordinates is defined as
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Conservation of Energy: The Energy Equation
Cartesian Coordinates
Consider an element dxdydz
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Conservation of Energy: The Energy Equation
Cartesian Coordinates
The coefficient of thermal expansion β is a property of material defined as
The dissipation function ϕ is associated with energy dissipation due to
friction. It is important in high speed flow and for very viscous fluids.
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Conservation of Energy: The Energy Equation
Cartesian Coordinates
Assumptions
Continuum,
Newtonian fluid,
Negligible nuclear, electromagnetic and radiation energy transfer.
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Simplified Form of the Energy Equation
Incompressible fluid and constant heat capacity
Constant thermal conductivity k
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Simplified Form of the Energy Equation
With the following assumptions:
Incompressible fluid, constant heat capacity and constant thermal conductivity.
Using the definition of total derivative and operator
Using the definition of total derivative and operator ,
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Simplified Form of the Energy Equation
Cylindrical Coordinates
With the following assumptions:
Incompressible fluid, constant heat capacity and constant thermal conductivity.
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Spherical Coordinates
With following assumptions Incompressible fluid, and constant thermal conductivity.
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