Lees-Doroditsyn Compressible Boundary Layer Equations
Transcript of Lees-Doroditsyn Compressible Boundary Layer Equations
Finite Difference Solutions to the Lees-DorodnitsynCompressible Boundary Layer Equations
Chelsea Onyeador
May 12, 2020
2.29 Final Project
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Motivation
• Compressible, and specifically hypersonic, boundary layers can have complex velocity and temperature profiles
• Lees-Doroditsyn transformation results in decreased need for boundary layer scaling
• Boundary layer analysis is necessary to assess surface heating, skin friction, and external flow displacement effects
• Developing robust methods for modeling hypersonic boundary layers
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TSL L-D Coordinate Transformation
𝜉 = න0
𝑥
𝜌𝑒𝑢𝑒𝜇𝑒 𝑑𝑥 η =𝑢𝑒
2𝜉න0
𝑦
𝜌 𝑑𝑦
𝐶𝑓′′ ′ + 𝑓𝑓′′ =2𝜉
𝑢𝑒𝑓′ 2 −
𝜌𝑒𝜌
𝑑𝑢𝑒𝑑𝜉
+ 2𝜉 𝑓′𝜕𝑓′
𝜕𝜉−𝜕𝑓
𝜕𝜉𝑓′′
𝜕𝑝
𝜕𝜂= 0
𝐶
𝑃𝑟𝑔′
′
+ 𝑓𝑔′ = 2𝜉 𝑓′𝜕𝑔
𝜕𝜉+𝑓′𝑔
ℎ𝑒
𝜕ℎ𝑒𝜕𝜉
− 𝑔′𝜕𝑓
𝜕𝜉+𝜌𝑒𝑢𝑒𝜌ℎ𝑒
𝑓′𝑑𝑢𝑒𝑑𝜉
− 𝐶𝑢𝑒2
ℎ𝑒𝑓′′ 2
𝐶 =𝜌𝜇
𝜌𝑒𝜇𝑒, 𝑔 =
ℎ
ℎ𝑒, and, 𝑓′ =
𝑢
𝑢𝑒
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Lees-Dorodnitsyn Equations• For 2D, Laminar, Compressible Boundary Layers
• Effectively Parabolic
• 5th Order system of coupled equations
• Introduce F, U, S, H, and Q to simplify into 5 - 1st order equations
◦ 𝐹 = 𝑓 =𝜑
2𝜉(stream function relation)
◦ 𝑈 =𝜕𝑓
𝜕𝜂=
𝑢
𝑢𝑒(velocity relation)
◦ 𝑆 =𝜕𝑈
𝜕𝜂=
𝜕2𝑓
𝜕𝜂2(shear stress relation)
◦ 𝐻 = 𝑔 =ℎ
ℎ𝑒=
𝑇
𝑇𝑒=
𝜌𝑒
𝜌(enthalpy/temperature relation)
◦ 𝑄 =𝜕𝑔
𝜕𝜂=
𝜕𝐻
𝜕𝜂(heat transfer relation)
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Flow Parameters
FLUID MODELS
• Viscosity Sutherland’s Law
◦ 𝜇 =𝜇𝑟𝑒𝑓
𝑇
𝑇𝑟𝑒𝑓
32𝑇𝑟𝑒𝑓+𝑆𝑟𝑒𝑓
𝑇+𝑆𝑟𝑒𝑓
• Calorically Perfect Gas
◦ ℎ = 𝑐𝑝𝑇
◦ 𝛾 = 1.4
• Constant Pr
◦ 0.71 (Van Driest), 0.75, or 1
• Isothermal wall
SPECIFIED PARAMETERS
◦ 𝜉𝑚𝑎𝑥
◦ 𝜂𝑒◦ 𝑇𝑒 → ℎ𝑒◦ 𝑀𝑎𝑒 𝜉 → 𝑢𝑒
◦ℎ𝑤
ℎ𝑒→ ℎ𝑤
◦ 𝑃𝑒 = 𝜌𝑒
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Flow Parameters
FLUID MODELS
• Viscosity Sutherland’s Law
◦ 𝜇 =𝜇𝑟𝑒𝑓
𝑇
𝑇𝑟𝑒𝑓
32𝑇𝑟𝑒𝑓+𝑆𝑟𝑒𝑓
𝑇+𝑆𝑟𝑒𝑓
• Calorically Perfect Gas
◦ ℎ = 𝑐𝑝𝑇
◦ 𝛾 = 1.4
• Constant Pr
◦ 0.71 (Van Driest), 0.75, or 1
• Isothermal wall
SPECIFIED PARAMETERS
◦ 𝜉𝑚𝑎𝑥
◦ 𝜂𝑒◦ 𝑇𝑒 → ℎ𝑒◦ 𝑀𝑎𝑒 𝜉 → 𝑢𝑒
◦ℎ𝑤
ℎ𝑒→ ℎ𝑤
◦ 𝑃𝑒 = 𝜌𝑒
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Boundary Conditions• 5 Required for a 5th Order system:
• Wall:
◦ 𝐹𝑤𝑎𝑙𝑙 = 0
◦ 𝑈𝑤𝑎𝑙𝑙 = 0
◦ 𝐺𝑤𝑎𝑙𝑙 = 𝐺𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑒𝑑
• Edge:
◦ 𝑈𝑒𝑑𝑔𝑒 = 1
◦ 𝐺𝑒𝑑𝑔𝑒 = 1
• Initial condition @ 𝜉 = 0
◦ 1D Flat plate similarity solution (calculated with Newton-Raphson method)
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Residual Form
F′ − 𝑈 = 0
𝑈′ − 𝑆 = 0
𝐶𝑆 ′ + 𝐹𝑆 +2𝜉
𝑢𝑒𝐻 − 𝑈 2
𝑑𝑢𝑒𝑑𝜉
+ 2𝜉𝜕𝐹
𝜕𝜉𝑆 − 𝑈
𝜕𝑈
𝜕𝜉= 0
𝐶
𝑃𝑟𝑄
′
+ 𝐹𝑄 − 2𝜉 𝑈𝜕𝐻
𝜕𝜉+𝑈𝐻
ℎ𝑒
𝜕ℎ𝑒𝜕𝜉
− 𝑄𝜕𝐹
𝜕𝜉+𝐻𝑢𝑒ℎ𝑒
𝑈𝑑𝑢𝑒𝑑𝜉
+ 𝐶𝑢𝑒2
ℎ𝑒𝑆 2 = 0
𝐻′ − 𝑄 = 0
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𝐹, 𝑈, 𝑆, 𝐻, 𝑄 are unknowns
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Finite Difference StencilBOX SCHEME
• Space-march in 𝜉 direction
• 2nd Order
• Centered Difference in both directions
• Similar to Crank Nicholson
3-POINT BACKWARD DIFFERENCE
• Space-march in 𝜉 direction
• 2nd Order
• Centered Difference in 𝜂
• 3-pt Backward Difference in 𝜉
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𝑖
𝑗 +1
2𝑗
𝑗 + 1
𝑖 +1
2
𝑖 + 1𝜉
𝜂
𝑖 − 2
𝑗 +1
2𝑗
𝑗 + 1
𝑖
𝜉
𝜂
𝑖 − 1
space marchspace march
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Box Stencil
𝑋 = 𝑋𝑖+12
𝑗+12 =
1
4𝑋𝑖𝑗+ 𝑋𝑖
𝑗+1+ 𝑋𝑖+1
𝑗+ 𝑋𝑖+1
𝑗+1
𝑋′ = 𝑋′𝑖+12
𝑗+12 =
𝑋𝑖+12
𝑗+1− 𝑋
𝑖+12
𝑗
Δ𝜂=
12 𝑋𝑖+1
𝑗+1+ 𝑋𝑖
𝑗+1−12 𝑋𝑖+1
𝑗+ 𝑋𝑖
𝑗
Δ𝜂
𝜕𝑋
𝜕𝜉= ቤ𝜕𝑋
𝜕𝜉𝑖+12
𝑗+12
=𝑋𝑖+1
𝑗+12 − 𝑋
𝑖
𝑗+12
Δ𝜉=
12 𝑋𝑖+1
𝑗+1+ 𝑋𝑖+1
𝑗−12 𝑋𝑖
𝑗+1+ 𝑋𝑖
𝑗
Δ𝜉
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𝑖
𝑗 +1
2𝑗
𝑗 + 1
𝑖 +1
2
𝑖 + 1𝜉
𝜂
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3pt - Backward Stencil
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𝑋 = 𝑋𝑖
𝑗+12 =
1
2𝑋𝑖𝑗+ 𝑋𝑖
𝑗+1
𝑋′ = 𝑋′𝑖
𝑗+12 =
𝑋𝑖𝑗+1
− 𝑋𝑖𝑗
Δ𝜂
𝜕𝑋
𝜕𝜉= ቤ𝜕𝑋
𝜕𝜉𝑖
𝑗+12
=
32𝑋𝑖
𝑗+12 − 2𝑋
𝑖−1
𝑗+12 +
12𝑋𝑖−2
𝑗+12
2Δ𝜉=
32 𝑋𝑖
𝑗+1+ 𝑋𝑖
𝑗− 2 𝑋𝑖−1
𝑗+1+ 𝑋𝑖−1
𝑗+12 𝑋𝑖−1
𝑗+1+ 𝑋𝑖−1
𝑗
4Δ𝜉
𝑖 − 2
𝑗 +1
2𝑗
𝑗 + 1
𝑖
𝜉
𝜂
𝑖 − 1
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3pt - Backward Stencil
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𝑋 = 𝑋𝑖
𝑗+12 =
1
2𝑋𝑖𝑗+ 𝑋𝑖
𝑗+1
𝑋′ = 𝑋′𝑖
𝑗+12 =
𝑋𝑖𝑗+1
− 𝑋𝑖𝑗
Δ𝜂
𝜕𝑋
𝜕𝜉= ቤ𝜕𝑋
𝜕𝜉𝑖
𝑗+12
=
32𝑋𝑖
𝑗+12 − 2𝑋
𝑖−1
𝑗+12 +
12𝑋𝑖−2
𝑗+12
2Δ𝜉=
32 𝑋𝑖
𝑗+1+ 𝑋𝑖
𝑗− 2 𝑋𝑖−1
𝑗+1+ 𝑋𝑖−1
𝑗+12 𝑋𝑖−1
𝑗+1+ 𝑋𝑖−1
𝑗
4Δ𝜉
𝜉 = 𝜉𝑖𝑢𝑒 = 𝑢𝑒𝑖 = 𝑢𝑒 𝜉𝑖ℎ𝑒 = ℎ𝑒𝑖 = ℎ𝑒 𝜉𝑖
ቤ𝜕ℎ𝑒𝜕𝜉
=𝜕ℎ𝑒𝜕𝜉
𝑖
=𝜕ℎ𝑒𝜕𝜉
𝜉𝑖
ቤ𝜕𝑢𝑒𝜕𝜉
=𝜕𝑢𝑒𝜕𝜉
𝑖
=𝜕𝑢𝑒𝜕𝜉
𝜉𝑖
𝑖 − 2
𝑗 +1
2𝑗
𝑗 + 1
𝑖
𝜉
𝜂
𝑖 − 1
Analytical Evaluations:
Finite difference approximations:
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Newton-Raphson Solver• Necessary for Non-linear system
• 10 – Diagonal Sparse Matrix
• Analytically calculated jacobian
• Utilized MATLAB built in matrix solver
• Quadratic convergence
• Convergence criteria based on magnitude of max residual
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Matrix pattern
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Newton vs. MATLAB FSolve• Compared the Newton-Raphson Method against MATLAB’s nonlinear solver
• Fsolve is a quasi-Newton method
• Assumed both solution methods had similar accuracies
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Newton-Raphson MATLAB FSolve
Runtime 6-7 sec 116 sec
# of Iterations 3-4 2
Implementation Complexity ~650 lines ~125 lines
# of Function Evaluations 525 243
(Solved for 30 x 30 system)
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Selected Results
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Error Analysis• L2 Error Analysis of 𝛿∗ 𝜉 , displacement
thickness
• Compared to solution with fine resolution
• Roughly 2nd order convergence, as expected
• Comparable performance of Box Scheme and 3-pt scheme
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𝛿∗ = න0
𝜂𝑒
1 −ρ
ρ𝑒𝑈
𝜕𝑦
𝜕𝜂𝑑𝜂 = න
0
𝜂𝑒
1 −ρ
ρ𝑒𝑈
𝐻 2𝜉
𝑢𝑒𝜌𝑒𝑑𝜂
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Conclusions• Newton-Raphson (in this case) is significantly preferable to FSolve
• Stability is not a significant concern for the backwards difference scheme
• Oscillations are not present in the box scheme
• There is no clear advantage to using the box scheme instead of the 3-pt backwards difference
• Further sampling of the solution space may be necessary to determine overall behavior of solution methods
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Thank youQuestions?
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Appendix
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𝜉 = 𝜉𝑖+12
𝑢𝑒 = 𝑢𝑒𝑖+12
= 𝑢𝑒 𝜉𝑖+12
ℎ𝑒 = ℎ𝑒𝑖+12
= ℎ𝑒 𝜉𝑖+12
ቤ𝜕ℎ𝑒𝜕𝜉
=𝜕ℎ𝑒𝜕𝜉
𝑖+12
=𝜕ℎ𝑒𝜕𝜉
𝜉𝑖+12
ቤ𝜕𝑢𝑒𝜕𝜉
=𝜕𝑢𝑒𝜕𝜉
𝑖+12
=𝜕𝑢𝑒𝜕𝜉
𝜉𝑖+12
Analytical Evaluations:
𝜉 = 𝜉𝑖𝑢𝑒 = 𝑢𝑒𝑖 = 𝑢𝑒 𝜉𝑖ℎ𝑒 = ℎ𝑒𝑖 = ℎ𝑒 𝜉𝑖
ቤ𝜕ℎ𝑒𝜕𝜉
=𝜕ℎ𝑒𝜕𝜉
𝑖
=𝜕ℎ𝑒𝜕𝜉
𝜉𝑖
ቤ𝜕𝑢𝑒𝜕𝜉
=𝜕𝑢𝑒𝜕𝜉
𝑖
=𝜕𝑢𝑒𝜕𝜉
𝜉𝑖
Analytical Evaluations:
Chapman Rubesin factor• 𝐶 = 𝐶 𝑔
• 𝐶𝑖+
1
2
𝑗+1
2 = 𝐶 𝑔𝑖+
1
2
𝑗+1
2
• 𝐶𝑖+
1
2
𝑗+1
2 =1
4𝐶 𝑔𝑖
𝑗+ 𝐶 𝑔𝑖+1
𝑗+ 𝐶 𝑔𝑖
𝑗+1+ 𝐶 𝑔𝑖+1
𝑗+1
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Similarity Equations
𝐶𝑓′′ ′ + 𝑓𝑓′′ = 0
𝐶
𝑃𝑟𝑔′
′
+ 𝑓𝑔′ + 𝐶𝑢𝑒2
ℎ𝑒𝑓′′ 2 = 0
F′ = 𝑈
𝑈′ = 𝑆
𝐶𝑆 ′ + 𝐹𝑆 = 0
𝐻′ = 𝑄
𝐶
𝑃𝑟𝑄
′
+ 𝐹𝑄 + 𝐶𝑢𝑒2
ℎ𝑒𝑆 2 = 0
𝑔 = 𝐻
𝑓 = 𝐹
2 - Coupled 2nd and 3rd Order Diff. Eqns 5 - Coupled 1st Order ODEs (Drela Notation)
Where 𝜙′ =𝜕𝜙
𝜕𝜂
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