22.589 Homework 4. Due 10/7/13
1. Calculate the term
L
M il Ni Nl dx
0
Ni
1
0
l
i
r
For uniform grids and linear shape functions.
2. Set up the cavity with a moving lid problem as described in the attachment, and
the Echo360 tutorial. Use a Reynolds numbe

Transient Formulations
Suppose that the application of the finite-element technique to a steady-state problem
results in the matrix formulation
K B
The corresponding transient problem contains the additional term
and application
t
of the finite element t

The Momentum Equation
Newtons Second Law of Motion:
Newton's second law of motion for an inertial observer (one that is not rotating or
accelerating) can be expressed in the form
1
F
ma
gc
where gc is a constant which depends on the system of units used.

The Transport of Momentum
Pressure in a Stationary Fluid.
Atoms and molecules in random thermal motion transport momentum and give rise to two
macroscopic forces; PRESSURE and VISCOSITY. As a consequence of the vector nature of
momentum the laws governing

Weak Formulation of the Heat Conduction Equation
Consider the steady-state heat conduction equation in the form
CV T k T R 0
This equation is defined in a control volume , enclosed by a surface S.
S
n
dS
d
Multiply this equation by a finite function of

Integration by Parts (Partial Integration).
Consider the chain rule applied to the product of two functions of x
d
dq
d
q q
dx
dx
dx
Rearrange
dq d
d
q q
dx dx
dx
Integrate
dq
d
d
dx q dx q dx
dx
dx
dx
a
a
b
b
a
b
Evaluating the total integral gives the

Boundary and Initial Conditions
The heat conduction equation is a partial differential equation (PDE) whose independent
variables are time and position. As with any differential equation, its solution requires the
specification of additional information,

22.589 Finite Element in Thermofluids, Fall 2013. Assignment # 3
Due: Mon 9/30/13
1. Show by algebra that
u
u
P
W E S
2
2 2
2
Is the same as
2*
u *
u *
W
E S
P
2
2
where
* = 1+ Pe
2.
Transient four pipes tutorial.
Recall t

Upwinding in Finite Elements
Streamline Upwind Petrov-Galerkin Stabilization
Compared with finite-volumes, the introduction of upwinding techniques in the finite-element
method is much more mathematical, and much less physical in nature. The obvious appro

Upwind Differencing
A popular remedy for the suppression of convection instabilities is upwind differencing or
upwinding
To introduce the idea of upwind differencing it is instructive to review the finite-volume
technique (where upwind differencing got st

Convection Instabilities.
In a previous section we saw that a typical internal node was governed by the following equation
u
2
l
i
2
(l)
l
u
2
r S
(r)
r
i
This equation can be expressed in the form
Ai i = Al l + Ar r Si
where
Ai
2
u
A

1D Assembly Procedure
The procedure for calculating the finite-element equations describe in the previous section can be
expressed as a computer algorithm, known as the assembly procedure.
Application of the Galerkin technique to the one—dimensional conve

Galerkin Technique in 1D
Consider the weak form
L
ﬁpu wig + I‘d—Wﬂ - wS] dx = —w(L)qL + w(0)q(0)
0 dx dx dx
This equation is exact, and if enforced for all choices of weighting function w(x) , would yield
the exact solution to the problem.
We will look fo

Evaluation of the Element Equations
Recap the form of the 1D linear basis function associated with node i.
The basis function is non-zero only in the elements that contain node i. The interval 1 to i is one
such element, as is the interval i to r.
For a

1D Basis Functions
Discrete Formulations of Continuous Problems
Physical quantities are generally continuous in the sense that they are defined at an
inﬁnite number of points in a region of space. They are also deﬁned by
continuous mathematical constructs

Weak Formulation of the 1D Problem
Recall the heat conduction equation
In order to present some of the essentials of the ﬁnite element technique, it is instructive to consider
a steady-state, one-dimensional analogue that represents the transport of a gen

The Heat Conduction Equation
The heat conduction equation is a mathematical formulation of the principle of
conservation of energy. Consider an arbitrary, fixed, constant shape control volume, ,
enclosed by a surface S. The control volume is contained ins

Continuity Equation
The conservation of mass can be expressed in words as follows;
'The rate of increase of mass in control volume must equal the net rate of
mass flow into the volume across the closed boundary surface S.'
This statement assumes that mass

Heat Conduction
Internal Energy
Consider a region of solid material macroscopically at rest. The solid consists of a
collection of atoms or molecules that exert forces on each other. It is instructive to
visualize the solid as a lattice of particles conne

Convective Fluxes.
Thermofluids is concerned with the flow, or transfer, of three basic quantities; heat,
mass, and momentum. The most fundamental way of quantifying the flow of a
quantity is by its flux at a point.
The flux of a physical quantity is defi

Gradient, Divergence, Laplacian
To refresh your knowledge of fields and vector calculus, refer to any undergraduate text
on advanced calculus or advanced engineering math.
Fields:
A field is a physical quantity that is defined in some region of space and

The Divergence Theorem of Gauss.
If a volume, V, is bounded by the simple closed surface, S, and if a differentiable vector
r
field, q , is defined within V and on S, then
r r
r
.q dV = q.n dS
V
S
where the unit vector normal to the surface, n , is defin

22.589 Finite Element Methods in Thermofluids.
Four pipes tutorial. Problem Definition.
In this lab we will introduce the use of the COMSOL finite element package to solve a steadystate heat conduction problem
A bundle of four long pipes carries ice/water

22.589 Finite Element Method in Thermofluids
Computational Fluid Dynamics.
The use of computers and numerical techniques to solve problems in engineering and
science is now widespread, if not always routine. The most notable successes are in the
fields of