1. 1. Numerical Simulation in Real Time Operation
Purpose of the Numerical Simulation unit
Simulate the whole system - Provide the Control Input for the Small Sector Rigs
Simulation
Provide Control Input
Small Sector Rigs
Whole System

2. 1-(1) Modeling for the Numerical Simulation using d’Alembert Solution & Derived Heat Source Model
Model of the System
Separate Models for the regions Inside & Outside of the Heat Source
Heat Source Model
Equation Sets
Sample Result 1 (No Mean flow, Traveling  Standing Wave)
Initial Condition: Only (+) going Traveling Wave
Sample Result 2 (Mean flow, Standing  Travelling Wave)
Initial Condition: Standing Wave with Mean Flow
Current Set-Up of Real Time System (Ongoing)

3. Model of the System
Separate Models for the regions Inside & Outside of the Heat Source
Model of Heat Source
Linear Acoustic Equation d’Alembert Solution
Region of Heat Source
- 1-D Toroidal Cavity
Assume 1-D (Neglect Geometrical Effect, Only Physical Effects)
- The Heat Sources are located in the small zone, not in the entire domain
The Zone is assumed as an Acoustically Compact Zone
- The Outside the Heat Source Region
No heat Addition  Eventually, Isentropic Process (from Energy Relation)
And the Wave Solutions

4. Model of the System Model of Heat Source + 3. Reaction Depends on
Injector
1. Heat Source affects
Waves
2. Waves & Reaction
affect Heat source
- Perturbation in the Cavity
- The Change in Energy Balance
-> Heat Source perturbed
+ 3. Reaction Depends on
- Chemical Energy in the Cavity
- Pressure & Velocity
-> Heat Source affected

5. Model of the System
Final Equations Set
Where

6. Numerical Simulation Numerical Simulation on the Simple Case
No Heat Source
Compact Heat Source Region
Closed Circular Tube
- 1-D
- Closed Circular Tube
- 1 Heat Source
- In the Region of No Heat Source
- d’Alembert’s Solutions works
- In the Compact Heat Source Region
- Use Previous Equation Set or the Combined Equation

7. Result 1 (No Mean flow, Traveling  Standing Wave)
Initial Condition: Only (+) going Traveling Wave
imin
imax
iheat i
i+1
i-1
Pressure
Transition from Travelling Wave to Standing Wave
Animation

8. Result 1 (No Mean flow, Traveling  Standing Wave)
Transition from Travelling Wave to Standing Wave
t=10
t=0.25
t=10.25
t=0.5
t=10.5
t=0.75
t=10.75
t=1
t=11
t=1.25
t=11.25

9. Result 2 (Mean flow, Standing  Travelling Wave)
Initial Condition: Standing Wave with Mean Flow
imin
imax
iheat i
i+1
i-1
Mean Flow
Pressure
Transition from Standing Waves to Travelling Waves
Animation

10. Result 2 (Mean flow, Standing  Travelling Wave)
Transition from Standing Waves to Travelling Waves
t = ~0.25
t = ~16.25
t = ~0.5
t = ~16.5
t = ~0.75
t = ~16.75
t = ~1
t = ~17
T = ~1.25
T = ~17.25
Standing Wave propagates at Mean Velocity
Traveling Wave propagates at Mean Velocity + Speed of Sound

11. Current Set-Up of Real Time System (Ongoing)
Achieved Fast Operation
Heat Source Eqn:
Linear Acoustic Eqn:
d’Alembert Sol.
DS1103 Board
Numerical
Simulation
ADC
DAC
Control Signal
Input Signal
4 channel inputs
4 channel outputs
Simulation Time ~ 6~7 micro sec
~ 2 micro sec
~ 2 micro sec
Sampling Rate ~ 11.2 micro sec