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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Course Outline 1
MAEEM Dept., UMR

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Course Outline 1
MAEEM Dept., UMR
Password to access files will be emailed to
you Course Information
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Course Outline 1
MAEEM Dept., UMR

• Course Outline
• Ordinary differential equations (ODE)
• Numerical techniques for solving ODEs
• Example Flow in constant area pipe with heat
  addition and
• friction
• Partial differential equations, classification
• Discretization of derivatives
• Errors and analysis of stability
• Example Unsteady heat conduction in a rod
• Example Natural convection at a heated
  vertical plate
• Discretization techniques

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Course Outline 2
MAEEM Dept., UMR

• Course Outline (continued)
• Couette flow
• The shock tube problem
• Introduction to packaged codes
• Grid generation
• Problem setup
• Solution
• Turbulence modeling

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Introduction
1 MAEEM Dept., UMR
ODEs and PDEs may be discretized-approximated- as
a set of algebraic equations and
solved Discretization methods for ODEs are well
known e.g., Runge-Kutta methods for initial
value problems and shooting methods for BV
problems PDEs involve more than 1 independent
variable e.g., x, y, z, t in Cartesian
coordinates for time-dependent Problems PDEs can
be discretized using finite difference
Methods
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Introduction 2 MAEEM
Dept., UMR
PDEs can also be discretized in integral form,
known as finite volume methods Sometimes
coordinate transformation is necessary before
discretization
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 0a MAEEM
Dept., UMR
Flow with heat addition and friction Ref Hill
Peterson, Mechanics and Thermodynamics of
Propulsion, Addison-Wesley
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 0b MAEEM
Dept., UMR
Flow with heat addition and friction Ref Hill
Peterson, Mechanics and Thermodynamics of
Propulsion, Addison-Wesley
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 0c MAEEM
Dept., UMR
Flow with heat addition and friction Ref Hill
Peterson, Mechanics and Thermodynamics of
Propulsion, Addison-Wesley
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 1 MAEEM Dept.,
UMR
Flow with heat addition and friction Ref Hill
Peterson, Mechanics and Thermodynamics of
Propulsion, Addison-Wesley
CV
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 2 MAEEM Dept.,
UMR
Flow with heat addition and friction
CV
Perfect gas flows from left to right in a
constant area duct Heat addition and/or friction
may be present Flow properties will change
during the process Equations can be solved
analytically when either heat addition or
friction is present, but not both
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 3 MAEEM Dept.,
UMR
Flow with heat addition
Stagnation enthalpy change (Conservation
of Energy/First Law)
(1)
Conservation of mass
(2)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 4 MAEEM Dept.,
UMR
Flow with heat addition
Momentum
(3)
Stagnation Enthalpy
(4)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 5 MAEEM Dept.,
UMR
Flow with heat addition
Integration yields

(5)
Equation of state
Speed of sound
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 6 MAEEM Dept.,
UMR
Flow with heat addition
Above equations can be combined to yield the
following

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 7 MAEEM Dept.,
UMR
Flow with heat addition

The above equation the following adiabatic flow
relation can be used to get stagnation
temperature ratio
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 8 MAEEM Dept.,
UMR
Flow with heat addition

Note Final Mach number depends on initial Mach
number and final stagnation temperature.
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 9 MAEEM Dept.,
UMR
Flow with heat addition
Reference conditions Note that the stagnation
conditions change due to heat addition For given
initial conditions, Mach 1 conditions, denoted
by () can be used for reference Thus

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 10 MAEEM
Dept., UMR
Flow with heat addition

The above equation shows that, for given initial
conditions, fluid properties are only a function
of the local Mach number
21
Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 11 MAEEM
Dept., UMR
Flow with heat addition

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 12 MAEEM
Dept., UMR
Flow with heat addition
Calculation procedure, given p01, T01, M1, and
q Determine T0 using T01 and M1 Determine
T02/T0 using

Calculate M2, p2, p02
Flow with heat addition Observe in figure, for
subsonic and supersonic cases Heat addition
drives M towards 1. Results in thermal
choking. There is a loss of stagnation
pressure.
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 13 MAEEM
Dept., UMR
Flow with heat addition and friction

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 14 MAEEM
Dept., UMR
Flow with heat addition and friction

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 15 MAEEM
Dept., UMR
Flow with friction
Momentum equation

where c is the circumference and A is the
cross-section area. cdx is the curved surface
area of the tube of length dx. t0 is the wall
shear stress (N/m2)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 16 MAEEM
Dept., UMR
Flow with friction
The energy equation in this case is h0 h
u2/2 constant or dh udu 0

Shear stress correlation for fully developed pipe
flow
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 17 MAEEM
Dept., UMR
Flow with friction
where e is the rms roughness of the pipe wall cf
is the skin coefficient The equations of
continuity, momentum and energy can now be
combined with the perfect gas equation of state
to get the equations for flow with friction

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 18 MAEEM
Dept., UMR
Flow with friction
See Hill Peterson for detailed derivation of
the following equation

Note the behavior of the flow for subsonic and
supersonic cases. In both cases, Mach number
tends towards 1. Condition is called friction
choking
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 19 MAEEM
Dept., UMR
Flow with friction
Integrating and applying the limit between M M
and M 1 yields the following result for length
to choke, L

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 20 MAEEM
Dept., UMR
Flow with friction

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 21 MAEEM
Dept., UMR
Flow with heat addition and friction
The continuity equation is the same as
before. The momentum equation has the shear
stress term. The energy equation has the heat
addition (q) term. These equations can now be
combined with the perfect gas equation to get
the differential equation which does not have
a closed-form solution. Solution can be obtained
by numerical integration.

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 22 MAEEM
Dept., UMR
(1)
(2)

(3)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 23 MAEEM
Dept., UMR
(4)

(5)
Momentum
(6)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 24 MAEEM
Dept., UMR
From (1)

(7)
Substitute in ( 4 )
(8)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 25 MAEEM
Dept., UMR

Combine (3) and (6)


(9)
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 26 MAEEM
Dept., UMR


(10)

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 27 MAEEM
Dept., UMR

(11)


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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 28 MAEEM
Dept., UMR

Substitute (10) and (11) in (9)

2nd RHS term

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 29 MAEEM
Dept., UMR


(12)

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 30 MAEEM
Dept., UMR

(11)

Substitute in (8) using (10) and (11)

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 31 MAEEM
Dept., UMR


LHS factor

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 32 MAEEM
Dept., UMR



or
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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 33 MAEEM
Dept., UMR
Flow with heat addition and friction
The final form of the equation is as follows (see
handout for details).

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac ODEs 34 MAEEM
Dept., UMR
Flow with heat addition and friction
The above ODE can be integrated by using methods
such as Runge-Kutta or using software packages
such as Matlab which has routines for solving
ODEs

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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Runge-Kutta MAEEM
Dept., UMR

Runge-Kutta
Method See hand out for theory. 4th order
method Let the first order ODE be represented
as



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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Runge-Kutta MAEEM
Dept., UMR


The 4th order RK-Gill algorithm is then given by


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Computational Fluid Dynamics (AE/ME 339)
K. M. Isaac Runge-Kutta MAEEM
Dept., UMR


The method can be used for several simultaneous
first-order equations as well as a single
higher-order equation. See Carnahan, Luther and
Wilkes for details.