TFAWS August 21-25, 2017 NASA Marshall Space Flight Center MSFC - - PowerPoint PPT Presentation

TFAWS

MSFC ∙ 2017

Presented By

David Dodoo-Amoo

AN INNOVATIVE METHODOLOGY FOR ERROR ANALYSIS OF THERMO-FLUID SYSTEMS

David Dodoo-Amoo, Julio Mendez, Mookesh Dhanasar, Frederick Ferguson North Carolina A&T State University

Thermal & Fluids Analysis Workshop TFAWS 2017 August 21-25, 2017 NASA Marshall Space Flight Center Huntsville, AL

TFAWS Active Thermal Paper Session

Governing Equations

• Conservation of Mass :

𝜖 𝜍𝐵 𝜖𝑢 + 𝜖 𝜍𝑣𝐵 𝜖𝑦 = 0

• Conservation of Momentum :

𝜖 𝜍𝑣𝐵 𝜖𝑢 + 𝜖 𝜍𝑣2 + 𝑞 𝐵 𝜖𝑦 − 𝑞 𝑒𝐵 𝑒𝑦 = 0

• Conservation of Energy :

𝜖 𝜍𝑓𝑈𝐵 𝜖𝑢 + 𝜖 𝜍𝑓𝑈 + 𝑞 𝑣𝐵 𝜖𝑦 = 0

TFAWS 2017 – August 21-25, 2017

2

Governing Equations- Non Dimensional

• Conservation of Mass :

𝜖 ҧ 𝜍 ҧ 𝐵 𝜖 ҧ 𝑢 + 𝜖 ҧ 𝜍ത 𝑣 ҧ 𝐵 𝜖 ҧ 𝑦 = 0

• Conservation of Momentum :

𝜖 ҧ 𝜍ത 𝑣 ҧ 𝐵 𝜖 ҧ 𝑢 + 𝜖 𝜖 ҧ 𝑦 ҧ 𝜍 ҧ 𝐵 ത 𝑣2 + ത 𝑈 𝛿 − ҧ 𝜍ത 𝑈 𝛿 𝑒 ҧ 𝐵 𝑒 ҧ 𝑦 = 0

• Conservation of Energy :

𝜖 𝜖 ҧ 𝑢 ҧ 𝜍 ҧ 𝑓𝑈 ҧ 𝐵 + 𝜖 𝜖 ҧ 𝑦 ҧ 𝜍ത 𝑣 ҧ 𝐵 ҧ 𝑓𝑈 + ത 𝑈 = 0

Where

ҧ 𝑓𝑈 ≝

ҧ 𝑓 𝛿−1 + 𝛿 2 ത

𝑣2

TFAWS 2017 – August 21-25, 2017

3

Vector Form

TFAWS 2017 – August 21-25, 2017

4

𝜖𝑉 𝜖𝑢 + 𝜖𝐺 𝜖𝑦 − 𝐻 = 0 Where the vectors, U, F and G are defined as

𝑉 = 𝜍𝐵 𝜍𝑣𝐵 𝜍𝑓𝑈𝐵 , 𝐺 = 𝜍𝑣𝐵 𝜍𝐵 𝑣2 + 𝑈

𝛿

𝜍𝑣𝐵 𝑓𝑈 + 𝑈 , 𝐻 =

𝜍𝑈 𝛿 𝑒𝐵 𝑒𝑦

5

𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5)2

Problem ID1: Anderson Nozzle Isentropic Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

6

𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5)2

Problem ID1: Anderson Nozzle Isentropic Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 2.0𝜍 𝑗𝑛𝑏𝑦−1 − 𝜍 𝑗𝑛𝑏𝑦−2 2.0𝑣 𝑗𝑛𝑏𝑦−1 − 𝑣 𝑗𝑛𝑏𝑦−2 2.0𝑈 𝑗𝑛𝑏𝑦−1 − 𝑈 𝑗𝑛𝑏𝑦−2

Outflow Conditions

7

𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5)2

Problem ID2: Anderson Nozzle Shock Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

8

𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5)2

Problem ID2: Anderson Nozzle Shock Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions Outflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑢

= 𝜍𝑗𝑛𝑏𝑦−1 0.1520 𝑈𝑗𝑛𝑏𝑦−1

9

𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 |

Problem ID3: Absolute Nozzle Isentropic Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

10

𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 |

Problem ID3: Absolute Nozzle Isentropic Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 2.0𝜍 𝑗𝑛𝑏𝑦−1 − 𝜍 𝑗𝑛𝑏𝑦−2 2.0𝑣 𝑗𝑛𝑏𝑦−1 − 𝑣 𝑗𝑛𝑏𝑦−2 2.0𝑈 𝑗𝑛𝑏𝑦−1 − 𝑈 𝑗𝑛𝑏𝑦−2

Outflow Conditions

11

𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 |

Problem ID4: Absolute Nozzle Isentropic Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

12

𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 |

Problem ID4: Absolute Nozzle Isentropic Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions Outflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑢

= 𝜍𝑗𝑛𝑏𝑦−1 0.1520 𝑈𝑗𝑛𝑏𝑦−1

13

𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5

Problem ID5: Feng Nozzle Isentropic Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

14

𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5

Problem ID5: Feng Nozzle Isentropic Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions Outflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 2.0𝜍 𝑗𝑛𝑏𝑦−1 − 𝜍 𝑗𝑛𝑏𝑦−2 2.0𝑣 𝑗𝑛𝑏𝑦−1 − 𝑣 𝑗𝑛𝑏𝑦−2 2.0𝑈 𝑗𝑛𝑏𝑦−1 − 𝑈 𝑗𝑛𝑏𝑦−2

15

𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5

Problem ID6: Feng Nozzle Shock Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 ൗ 0.59 𝜍𝐵 1.0

𝑦𝑐𝑏𝑠

0 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 ൗ 0.59 𝜍𝐵 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 < 1.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 ൗ 0.59 𝜍𝐵 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

𝑦𝑐𝑏𝑠 ≥ 1.5

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

16

𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5

Problem ID6: Feng Nozzle Shock Flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Inflow Conditions Outflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 𝜍 𝑗𝑛𝑏𝑦−1 0.48 𝑈 𝑗𝑛𝑏𝑦−1

17

𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4)

Problem ID7: Hoffmann Nozzle Isentropic flow

Initial Conditions

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

𝜍 𝑣 𝑈 𝑗

𝑢=0

= 0.395 1.5 × 0.6897 0.6897 0 ≤ 𝑌𝑐𝑏𝑠 ≤ 1

18

𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4)

Problem ID7: Hoffmann Nozzle Isentropic flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Outflow Conditions

𝜍 𝑣 𝑈 1 = 0.395 1.5 × 0.6897 0.6897 𝜍 𝑣 𝑈 𝐽𝑛𝑏𝑦

𝑜

= 2 𝜍 𝑣 𝑈 𝐽𝑛𝑏𝑦−1

𝑜

− 𝜍 𝑣 𝑈 𝐽𝑛𝑏𝑦−2

𝑜

Inflow Conditions

19

𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4)

Problem ID8: Hoffmann Nozzle Shock flow

Initial Conditions

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

𝜍 𝑣 𝑈 𝑗

𝑢=0

= 0.395 1.5 × 0.6897 0.6897 0 ≤ 𝑌𝑐𝑏𝑠 ≤ 1

20

𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4)

Problem ID8: Hoffmann Nozzle Shock flow

TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems

Outflow Conditions

𝜍 𝑣 𝑈 1 = 0.395 1.5 × 0.6897 0.6897

Inflow Conditions

𝜍 𝑣 𝑈 𝐽𝑛𝑏𝑦

𝑜

= 𝜍 𝑗𝑛𝑏𝑦−1 0.5081 × 𝑈𝑗𝑛𝑏𝑦 𝜍 𝑗𝑛𝑏𝑦−1

21

𝐵 𝑦 = 2.0 𝑔𝑝𝑠 − 1.0 ≤ 𝑦 < −0.5 , 0.5 ≤ 𝑦 ≤ 1.0 𝐵 𝑦 = 1.0 + 𝑡𝑗𝑜2 𝜌𝑦 𝑔𝑝𝑠 − 0.5 ≤ 𝑦 < 0.5 ,

Problem ID9 : Feng 2nd Nozzle Isentropic Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 0.3 1.0 𝑦𝑐𝑏𝑠 𝑦𝑐𝑏𝑠 < −0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 0.7 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

− 0.5 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 1.5 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 ≤ 1.0

Quasi 1-D Problems

22

𝐵 𝑦 = 2.0 𝑔𝑝𝑠 − 1.0 ≤ 𝑦 < −0.5 , 0.5 ≤ 𝑦 ≤ 1.0 𝐵 𝑦 = 1.0 + 𝑡𝑗𝑜2 𝜌𝑦 𝑔𝑝𝑠 − 0.5 ≤ 𝑦 < 0.5 ,

Problem ID9: Feng 2nd Nozzle Isentropic Flow

Inflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 2.0𝜍 𝑗𝑛𝑏𝑦−1 − 𝜍 𝑗𝑛𝑏𝑦−2 2.0𝑣 𝑗𝑛𝑏𝑦−1 − 𝑣 𝑗𝑛𝑏𝑦−2 2.0𝑈 𝑗𝑛𝑏𝑦−1 − 𝑈 𝑗𝑛𝑏𝑦−2

Outflow Conditions

Quasi 1-D Problems

23

𝐵 𝑦 = 2.0 𝑔𝑝𝑠 − 1.0 ≤ 𝑦 < −0.5 , 0.5 ≤ 𝑦 ≤ 1.0 𝐵 𝑦 = 1.0 + 𝑡𝑗𝑜2 𝜌𝑦 𝑔𝑝𝑠 − 0.5 ≤ 𝑦 < 0.5

Problem ID10: Feng 2nd Nozzle Shock Flow

Initial Conditions

𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 0.3 1.0 𝑦𝑐𝑏𝑠 𝑦𝑐𝑏𝑠 < −0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 1.0 − 0.366 𝑦𝑐𝑏𝑠 − 0.5 0.7 1.0 − 0.167 𝑦𝑐𝑏𝑠 − 0.5

𝑦𝑐𝑏𝑠

− 0.5 ≤ 𝑦𝑐𝑏𝑠 < 0.5 𝜍 𝑣 𝑈 𝑦𝑐𝑏𝑠

𝑢=0

= 0.634 − 0.3879 𝑦𝑐𝑏𝑠 − 1.5 1.5 0.833 − 0.3507 𝑦𝑐𝑏𝑠 − 1.5

𝑦𝑐𝑏𝑠

0.5 ≤ 𝑦𝑐𝑏𝑠 ≤ 1.0

Quasi 1-D Problems

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𝐵 𝑦 = 2.0 𝑔𝑝𝑠 − 1.0 ≤ 𝑦 < −0.5 , 0.5 ≤ 𝑦 ≤ 1.0 𝐵 𝑦 = 1.0 + 𝑡𝑗𝑜2 𝜌𝑦 𝑔𝑝𝑠 − 0.5 ≤ 𝑦 < 0.5 ,

Problem ID10: Feng 2nd Nozzle Shock Flow

Inflow Conditions

𝜍 𝑣 𝑈 𝑗=1

𝑢

= 1.0 2.0𝑣2 − 𝑣3 1.0 𝜍 𝑣 𝑈 𝑗𝑛𝑏𝑦

𝑜

= 𝜍 𝑗𝑛𝑏𝑦−1 0.43 𝑈 𝑗𝑛𝑏𝑦−1

Outflow Conditions

Quasi 1-D Problems

Locations where Exact Solutions are known ൗ 𝑈0 𝑈∗ , ൗ 𝑄0 𝑄∗ , 𝑁∗

𝑼𝟏 𝑗𝑡 𝑑𝑝𝑜𝑡𝑢𝑏𝑜𝑢, ሶ 𝒏 𝑗𝑡 𝑑𝑝𝑜𝑡𝑢𝑏𝑜𝑢

• Pointwise (Throat)
• Distributed ( Entire Nozzle )

Problem ID1 and ID2

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Problem ID3 and ID4

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Problems ID5 and ID6

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Problem ID7 and ID8

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Problem ID9 and ID10

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Problem ID1 and ID2

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Problem ID3 and ID4

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Problem ID5 and ID6

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Problem ID7 and ID8

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Problem ID9 and ID10

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Grid Independence Grid Comparison

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Grid Independence Grid Comparison

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Table 1. Table of Error for Isentropic Problems

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Table 2. Table of Error for Shock Problems

CONCLUSIONS

• The results show that the maximum error was 6.78%

and occurred at the throat of the nozzle with an acute throat gradient.

• The standard solution can therefore be used in place of

an analytical solution for the purpose of error analysis

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