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AERE 546 lecture 14;15;16;17;18;19;20 QUESTIONS AND ANSWERS
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5

September 2025
Algebraic grid generation;(Review finite-diff. and metrics); Equations on curvilinear grids: “metric” tensor; Finite volume method; Hyperbolic Equations; Summary of explicit methods:
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AerE 546 Lecture 14 Algebraic grid generation
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7

September 2025

2025/2026

A+
Algebraic grid generation A. A simple, algebraic grid. Grid is a set of points connected by lines (Mesh is the set of points joined into cells.) GRID MESH General idea: mesh surface, propagate into interior -- sometimes by solving elliptic boundary value problem for x(i,j). Choose points on wall, connect them, place tic marks along connecting line = grid; i.e., define surfaces by ( xin(i),yin(i) ) Specify x,y inner and outer then: x(s) = xin + ( xout - xin ) s (0≤ s ≤1) DO i=1,I ...
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AerE 546 Lecture 15 Equations on curvilinear grids: “metric” tensor
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4

September 2025

2025/2026

A+
Recall 2-surface method. Grid lines are curved; can use finite diff or finite vol. Former here; will be lectures on latter, but FYI. Actually, grid is a set of points organized into cells. Equations on curvilinear grids: “metric” tensor A. Computational and physical space Grid is not in x-y direction: Grid generation produces [x(i,j),y(i,j)]. Now use that to solve equations. Map from computational to physical. Think of i,j as a grid in ξ - η space. For example δf/δξ = (f(i+1,j...
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AerE 546 Lecture 16 (Review finite-diff. and metrics)
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7

September 2025

2025/2026

A+
(Review finite-diff. and metrics) Second derivatives: Evaluate ∂2ψ/∂2x as ∂xF with F = ∂xψ Outside, then in, but iterated central would give 5 i-points. Wrong: δξ(F) =( Fi+1 - Fi-1 )/2 with F= δξψ → [(ψi+2 -ψi) - (ψi - ψi-2 )]/4. Instead use: ( Fi+1/2 - Fi-1/2 )/2 → [(ψi+1 -ψi) - (ψi - ψi-1 )]/4. Fi+1/2 = value on cell face Fi+1 is data at cell center. Or recall formulating equations in conservation form: Flux in - Flux out + source = rate of change inside c....
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AerE 546 Lecture 17 Finite volume method
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5

September 2025

2025/2026

A+
Finite volume method Natural for unstructured meshed. Also, working with fluxes leads into hyperbolic equations. A. Recall origin of p.d.e.ʼs Donʼt take second step: apply discretization to the integral balance. Control volume is polyline (polyhedron in 3-D). Divergence theorem (Gaussʼ) is used in f.v. method. Rationale: f = ∫df/dx dx but now f ! ∫F·dS and df/dx dx ! ∫ ∇·F dV =∫ ∇·F dS dxn =∫ dF/dxn dxn dS = ∫n·FdS Divergence theorem <-> fundamental theorem...
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AerE 546 Lecture 18 Hyperbolic Equations
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5

September 2025

2025/2026

A+
Hyperbolic Equations Physics: Convection and wave propagation; sound in compressible flow. ∂tq = -∇·Fq . Now consider convective (dominated) flux uq. A. Examples: Linearized compressible potential flow (1-M2) ∂x2ϕ + ∂y2ϕ = 0 M>1 is hyperbolic: recall asymptotes=radiation # # # # # c.f. Mach waves Shallow water waves (long waves; non-dispersive): k = 2π/λ; ω = 2π/T ∂t 2h - gH∂x2h = 0 (or ∇2h) ; a = √gH = |ω/k| ; ω = ±k√gH Pictures: expansions and shocks;...
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Exam (elaborations)

AerE 546 Lecture 18 Hyperbolic Equations
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7

September 2025

2025/2026

A+
Physics: Convection and wave propagation; sound in compressible flow. ∂tq = -∇·Fq . Now consider convective (dominated) flux uq. A. Examples: Linearized compressible potential flow (1-M2) ∂x2ϕ + ∂y2ϕ = 0 M>1 is hyperbolic: recall asymptotes=radiation # # # # # c.f. Mach waves Shallow water waves (long waves; non-dispersive): k = 2π/λ; ω = 2π/T ∂t 2h - gH∂x2h = 0 (or ∇2h) ; a = √gH = |ω/k| ; ω = ±k√gH Pictures: expansions and shocks; .gif animations. Will...
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Exam (elaborations)

Summary of explicit methods:AerE 546 Lecture 20
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10

September 2025

2025/2026

A+
Iowa State University AERE 546 Summary of explicit methods: Euler !! upwind = central + ε·diffusion, first order accurate Runge-Kutta !! recall RK is `stable for convectionʼ, time-stepping adds dissipation: ! ! ! variations (low storage, higher damping) are used in CFD. Lax-Wendroff minimum dissipation (ε=C), 2nd order accurate in space and time MacCormick! ! two-step method, similar to L-W Generally CFL < 1, or CFLstab → time-step restriction ∆t < min[ CFLstab ∆x / a NO...
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