1. ME/AE 339 Computational Fluid Dynamics K. M. Isaac Topic1_ODE
11/9/2018
ODE

2. Please email me ( isaac@umr.edu ) the following Information:
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Course Outline MAEEM Dept., UMR
Please me ( ) the following Information:
Mailing address
Phone number
Fax number
Any other information you want me to know
11/9/2018
ODE

3. Password to access files: will be emailed to you Course Information
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Course Outline MAEEM Dept., UMR
Password to access files: will be ed to you
Course Information
11/9/2018
ODE

4. Ordinary differential equations (ODE)
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Course Outline 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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ODE

5. Course Outline (continued) Couette flow
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Course Outline 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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ODE

6. ODEs and PDEs may be discretized-approximated-
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Introduction 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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ODE

7. PDEs can also be discretized in integral form,
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Introduction 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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ODE

8. 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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ODE

9. 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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ODE

10. 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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ODE

11. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition and friction
Ref: Hill & Peterson, Mechanics and Thermodynamics
of Propulsion, Addison-Wesley CV
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ODE

12. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

13. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
Stagnation enthalpy change (Conservation of
Energy/First Law)
(1)
Conservation of mass
(2)
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ODE

14. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
Momentum
(3)
Stagnation Enthalpy
(4)
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ODE

15. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
Integration yields
(5)
Equation of state
Speed of sound
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ODE

16. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
Above equations can be combined to yield the following
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ODE

17. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

18. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
Note: Final Mach number depends on initial Mach number and
final stagnation temperature.
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ODE

19. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

20. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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
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ODE

21. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
11/9/2018
ODE

22. Flow with heat addition
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

23. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
and friction
Subscript h is for heat addition and subscript f is for friction
Three processes to attain choking are shown: 1) constant entropy process (vertical direction), 2) heating process, and 3) friction process
Note stagnation pressure falls for both heating and friction cases
Travel in either direction is possible for isentropic and and heat addition cases
Friction process is possible only in one direction
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ODE

24. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Flow with heat addition
and friction
11/9/2018
ODE

25. where c is the circumference and A is the cross-section
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

26. The energy equation in this case is h0 = h + u2/2 = constant
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

27. where e is the rms roughness of the pipe wall
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

28. Note the behavior of the flow for subsonic and supersonic
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

29. Integrating and applying the limit between M = M and M = 1 yields the
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

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

31. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

32. (1) (2) (3) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(1)
(2)
(3)
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ODE

33. Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(4)
(5)
Momentum
(6)
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ODE

34. From (1) (7) Substitute in ( 4 ) (8)
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
From (1)
(7)
Substitute in ( 4 )
(8)
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ODE

35. Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Combine (3) and (6)
(9)
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ODE

36. (10) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(10)
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ODE

37. (11) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(11)
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ODE

38. Substitute (9) for dp/p in (10)
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
Substitute (9) for dp/p in (10)
2nd RHS term:
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ODE

39. (12) Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(12)
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ODE

40. Substitute in (8) using (9), (10), (11) and (12)
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
(11)
Substitute in (8) using (9), (10), (11) and (12)
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ODE

41. LHS factor: Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs MAEEM Dept., UMR
LHS factor:
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ODE

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

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

44. Flow with heat addition and friction
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
ODEs 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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ODE

45. Runge-Kutta Method 4th order method:
Computational Fluid Dynamics (AE/ME 339) K. M. Isaac
Runge-Kutta MAEEM Dept., UMR
Runge-Kutta Method
4th order method:
Let the first order ODE be represented as
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ODE

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

47. The method can be used for several simultaneous first-order equations
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 Applied Numerical Methods, Carnahan, Luther and Wilkes,
or some other numerical methods book for details.
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ODE

48. Program Completed University of Missouri-Rolla
Copyright 2002 Curators of University of Missouri
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ODE