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

TFAWS Active Thermal Paper Session 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 Presented By David Dodoo-Amoo Thermal & Fluids Analysis Workshop TFAWS 2017 TFAWS August 21-25, 2017 NASA Marshall Space Flight Center MSFC ∙ 2017 Huntsville, AL

Governing Equations • Conservation of Mass : 𝜖 𝜍𝐵 + 𝜖 𝜍𝑣𝐵 = 0 𝜖𝑢 𝜖𝑦 • Conservation of Momentum : + 𝜖 𝜍𝑣 2 + 𝑞 𝐵 𝜖 𝜍𝑣𝐵 − 𝑞 𝑒𝐵 𝑒𝑦 = 0 𝜖𝑢 𝜖𝑦 • Conservation of Energy : 𝜖 𝜍𝑓 𝑈 𝐵 + 𝜖 𝜍𝑓 𝑈 + 𝑞 𝑣𝐵 = 0 𝜖𝑢 𝜖𝑦 TFAWS 2017 – August 21-25, 2017 2

ҧ ҧ ҧ ҧ ҧ ҧ ҧ ҧ ҧ Governing Equations- Non Dimensional • Conservation of Mass : 𝜍 ҧ 𝑣 ҧ 𝜖 𝐵 + 𝜖 𝜍ത 𝐵 = 0 𝜖 ҧ 𝑢 𝜖 ҧ 𝑦 • Conservation of Momentum : 𝑣 ҧ 𝑒 ҧ ത 𝜍ത 𝜖 𝜍ത 𝐵 + 𝜖 𝑈 − ҧ 𝑈 𝐵 𝑣 2 + 𝜍 ҧ 𝐵 ത 𝑦 = 0 𝜖 ҧ 𝑢 𝜖 ҧ 𝑦 𝛿 𝛿 𝑒 ҧ • Conservation of Energy : 𝜖 𝐵 + 𝜖 𝑓 𝑈 ҧ 𝑣 ҧ 𝑓 𝑈 + ത 𝜍 ҧ 𝜍ത 𝐵 𝑈 = 0 𝜖 ҧ 𝑢 𝜖 ҧ 𝑦 𝑓 𝛿 𝑣 2 𝑓 𝑈 ≝ 𝛿−1 + 2 ത Where TFAWS 2017 – August 21-25, 2017 3

Vector Form 𝜖𝑉 𝜖𝐺 𝜖𝑢 + 𝜖𝑦 − 𝐻 = 0 Where the vectors, U, F and G are defined as 𝜍𝑣𝐵 0 𝜍𝐵 𝜍𝑣𝐵 𝜍𝐵 𝑣 2 + 𝑈 𝜍𝑈 𝑒𝐵 𝑉 = , 𝐺 = , 𝐻 = 𝛿 𝑒𝑦 𝛿 𝜍𝑓 𝑈 𝐵 𝜍𝑣𝐵 𝑓 𝑈 + 𝑈 0 TFAWS 2017 – August 21-25, 2017 4

Quasi 1-D Problems Problem ID1: Anderson Nozzle Isentropic Flow Initial Conditions 𝑢=0 𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5) 2 1.0 𝜍 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 5 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID1: Anderson Nozzle Isentropic Flow Inflow Conditions 𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5) 2 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝑜 2.0𝜍 𝑗 𝑛𝑏𝑦 −1 − 𝜍 𝑗 𝑛𝑏𝑦 −2 𝜍 2.0𝑣 𝑗 𝑛𝑏𝑦 −1 − 𝑣 𝑗 𝑛𝑏𝑦 −2 𝑣 = 2.0𝑈 𝑗 𝑛𝑏𝑦 −1 − 𝑈 𝑗 𝑛𝑏𝑦 −2 𝑈 𝑗 𝑛𝑏𝑦 6 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID2: Anderson Nozzle Shock Flow Initial Conditions 𝑢=0 𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5) 2 1.0 𝜍 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 7 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID2: Anderson Nozzle Shock Flow Inflow Conditions 𝐵 𝑦 = 1.0 + 2.2(𝑦 − 1.5) 2 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝜍 𝑗 𝑛𝑏𝑦 −1 𝑢 𝜍 𝑣 = 0.1520 𝑈 𝑗 𝑛𝑏𝑦 𝑈 𝑗 𝑛𝑏𝑦 −1 8 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID3: Absolute Nozzle Isentropic Flow Initial Conditions 𝑢=0 1.0 𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 | 𝜍 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 9 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID3: Absolute Nozzle Isentropic Flow Inflow Conditions 𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 | 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝑜 2.0𝜍 𝑗 𝑛𝑏𝑦 −1 − 𝜍 𝑗 𝑛𝑏𝑦 −2 𝜍 2.0𝑣 𝑗 𝑛𝑏𝑦 −1 − 𝑣 𝑗 𝑛𝑏𝑦 −2 𝑣 = 2.0𝑈 𝑗 𝑛𝑏𝑦 −1 − 𝑈 𝑗 𝑛𝑏𝑦 −2 𝑈 𝑗 𝑛𝑏𝑦 10 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID4: Absolute Nozzle Isentropic Flow Initial Conditions 𝑢=0 1.0 𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 | 𝜍 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 11 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID4: Absolute Nozzle Isentropic Flow Inflow Conditions 𝐵 𝑦 = 1.0 + 5.3 𝑏𝑐𝑡| 𝑦 − 1.5 | 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝜍 𝑗 𝑛𝑏𝑦 −1 𝑢 𝜍 𝑣 = 0.1520 𝑈 𝑗 𝑛𝑏𝑦 𝑈 𝑗 𝑛𝑏𝑦 −1 12 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID5: Feng Nozzle Isentropic Flow Initial Conditions 𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝑢=0 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 1.0 𝜍 𝑔𝑝𝑠 𝑦 > 1.5 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 13 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID5: Feng Nozzle Isentropic Flow 𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 Inflow Conditions 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝑜 2.0𝜍 𝑗 𝑛𝑏𝑦 −1 − 𝜍 𝑗 𝑛𝑏𝑦 −2 𝜍 2.0𝑣 𝑗 𝑛𝑏𝑦 −1 − 𝑣 𝑗 𝑛𝑏𝑦 −2 𝑣 = 2.0𝑈 𝑗 𝑛𝑏𝑦 −1 − 𝑈 𝑗 𝑛𝑏𝑦 −2 𝑈 𝑗 𝑛𝑏𝑦 14 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID6: Feng Nozzle Shock Flow Initial Conditions 𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝑢=0 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 1.0 𝜍 𝑔𝑝𝑠 𝑦 > 1.5 𝑣 0.59 𝜍𝐵 = 0 ≤ 𝑦 𝑐𝑏𝑠 < 0.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 𝑦 𝑐𝑏𝑠 𝑢=0 1.0 − 0.366 𝑦 𝑐𝑏𝑠 − 0.5 𝜍 𝑣 0.59 𝜍𝐵 = 0.5 ≤ 𝑦 𝑐𝑏𝑠 < 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 1.0 − 0.167 𝑦 𝑐𝑏𝑠 − 0.5 𝑦 𝑐𝑏𝑠 𝑢=0 0.634 − 0.3879 𝑦 𝑐𝑏𝑠 − 1.5 𝜍 0.59 𝜍𝐵 𝑣 = 𝑦 𝑐𝑏𝑠 ≥ 1.5 ൗ 𝑈 𝑦 𝑐𝑏𝑠 0.833 − 0.3507 𝑦 𝑐𝑏𝑠 − 1.5 𝑦 𝑐𝑏𝑠 15 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID6: Feng Nozzle Shock Flow 𝐵 𝑦 = 1.0 + 2.2 𝑦 − 1.50 2 Inflow Conditions 𝑔𝑝𝑠 𝑦 ≤ 1.5 𝐵 𝑦 = 1.0 + 0.2223 𝑦 − 1.50 2 𝑔𝑝𝑠 𝑦 > 1.5 𝑢 𝜍 1.0 𝑣 = 2.0𝑣 2 − 𝑣 3 𝑈 𝑗=1 1.0 Outflow Conditions 𝑜 𝜍 𝑗 𝑛𝑏𝑦 −1 𝜍 0.48 𝑣 = 𝑈 𝑗 𝑛𝑏𝑦 −1 𝑈 𝑗 𝑛𝑏𝑦 16 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID7: Hoffmann Nozzle Isentropic flow Initial Conditions 𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4) 𝑢=0 0.395 𝜍 𝑣 = 0 ≤ 𝑌 𝑐𝑏𝑠 ≤ 1 1.5 × 0.6897 𝑈 𝑗 0.6897 17 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID7: Hoffmann Nozzle Isentropic flow 𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4) Inflow Conditions 0.395 𝜍 𝑣 = 1.5 × 0.6897 𝑈 1 0.6897 Outflow Conditions 𝑜 𝑜 𝑜 𝜍 𝜍 𝜍 𝑣 𝑣 𝑣 = 2 − 𝑈 𝐽𝑛𝑏𝑦 𝑈 𝐽𝑛𝑏𝑦−1 𝑈 𝐽𝑛𝑏𝑦−2 18 TFAWS 2017 – August 21-25, 2017

Quasi 1-D Problems Problem ID8: Hoffmann Nozzle Shock flow Initial Conditions 𝐵 𝑦 = 1.5643 + 0.3883tanh(8𝑦 − 4) 𝑢=0 0.395 𝜍 𝑣 = 0 ≤ 𝑌 𝑐𝑏𝑠 ≤ 1 1.5 × 0.6897 𝑈 𝑗 0.6897 19 TFAWS 2017 – August 21-25, 2017