{
  "version": 1,
  "topic": "FA09",
  "title": "Mesh convergence and residuals in an annular CFD calculation",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Manufactured refinement sequence with a known limit",
    "premise": "Teaching quantity Qh = 1 + h², exact limit Q = 1, h = 1/n. Every manufactured discrete residual is 10⁻¹⁰.",
    "formula": "Qh = 1 + 1/n²; relative error = (Qh − 1)/1",
    "variables": "n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Resolution count n = 16",
        "explanation": "Teaching quantity Qh = 1 + h², exact limit Q = 1, h = 1/n. Every manufactured discrete residual is 10⁻¹⁰."
      },
      {
        "label": "Select the relation",
        "calculation": "Qh = 1 + 1/n²; relative error = (Qh − 1)/1",
        "explanation": "n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study."
      },
      {
        "label": "Substitute the values",
        "calculation": "At n = 16: Qh = 1 + 1/256 = 1.00390625; error = 0.390625%; residual remains 10⁻¹⁰",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Quantity of interest Qh = 1.00390625; Discrete residual = 1e-10; Relative error (%) = 0.390625",
        "explanation": "Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks."
      }
    ],
    "result": "Quantity of interest Qh = 1.00390625; Discrete residual = 1e-10; Relative error (%) = 0.390625",
    "interpretation": "Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks.",
    "control": "Resolution count n",
    "columns": [
      "Resolution count n",
      "Quantity of interest Qh",
      "Discrete residual",
      "Relative error (%)"
    ],
    "rows": [
      [
        4,
        1.0625,
        1e-10,
        6.25
      ],
      [
        8,
        1.015625,
        1e-10,
        1.5625
      ],
      [
        16,
        1.00390625,
        1e-10,
        0.390625
      ],
      [
        32,
        1.0009765625,
        1e-10,
        0.09765625
      ],
      [
        64,
        1.000244140625,
        1e-10,
        0.0244140625
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Resolution count n = 4. Quantity of interest Qh: 1.0625; Discrete residual: 1e-10; Relative error (%): 6.25.",
      "Resolution count n = 8. Quantity of interest Qh: 1.015625; Discrete residual: 1e-10; Relative error (%): 1.5625.",
      "Resolution count n = 16. Quantity of interest Qh: 1.00390625; Discrete residual: 1e-10; Relative error (%): 0.390625.",
      "Resolution count n = 32. Quantity of interest Qh: 1.00097656; Discrete residual: 1e-10; Relative error (%): 0.09765625.",
      "Resolution count n = 64. Quantity of interest Qh: 1.00024414; Discrete residual: 1e-10; Relative error (%): 0.0244140625."
    ],
    "provenance": "Original manufactured convergence exercise with an explicitly known limit. No CFD runtime result or measured convergence order is claimed.",
    "questions": [
      {
        "question": "Reproduce the baseline Quantity of interest Qh without rounding intermediate values.",
        "answer": "At n = 16: Qh = 1 + 1/256 = 1.00390625; error = 0.390625%; residual remains 10⁻¹⁰. Quantity of interest Qh = 1.00390625. Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Resolution count n changes from 4 to 64. Other controls remain as stated in the premise. Quantity of interest Qh changes from 1.0625 to 1.00024414."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. n is a nominal resolution count. This table is manufactured, not a recorded annular CFD mesh study. Identical small residuals coexist with different discretization errors, exposing the distinction between the two checks."
      }
    ]
  },
  "recorded": null
}