Finite Difference Solutions to the Lees-DorodnitsynCompressible Boundary Layer Equations

Chelsea Onyeador

May 12, 2020

2.29 Final Project

1

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

8

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

9

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

5/12/2020 C. ONYEADOR

𝑖

𝑗 +1

2𝑗

𝑗 + 1

𝑖 +1

2

𝑖 + 1𝜉

𝜂

10

3pt - Backward Stencil

5/12/2020 C. ONYEADOR

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

11

3pt - Backward Stencil

5/12/2020 C. ONYEADOR

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:

12

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

13

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)

14

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

5/12/2020 C. ONYEADOR

𝛿∗ = න0

𝜂𝑒

1 −ρ

ρ𝑒𝑈

𝜕𝑦

𝜕𝜂𝑑𝜂 = න

0

𝜂𝑒

1 −ρ

ρ𝑒𝑈

𝐻 2𝜉

𝑢𝑒𝜌𝑒𝑑𝜂

16

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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5/12/2020 C. ONYEADOR 20

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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