The metrics, and how each is measured

Seven metrics. Each is stated, defined by its formula, and followed by the value measured.

  1. Relative L2 difference

    The difference of a field qq between the two codes over all cells, after 30 time steps with the case's own linear solvers. Median of the 42 tutorials.

    ∥qbrae−qOF∥2∥qOF∥2\frac{\lVert q_{\mathrm{brae}}-q_{\mathrm{OF}}\rVert_2}{\lVert q_{\mathrm{OF}}\rVert_2}
    (1)

    alpha 2.7e-12 · p_rgh 1.1e-11 · U 1.1e-08

  2. Normalised L-infinity difference

    The largest pointwise difference of the field over its scale sqs_q, on the same runs. Median of the 42 tutorials.

    max⁡i ∣qbrae,i−qOF,i∣sq\frac{\max_i\,\lvert q_{\mathrm{brae},i}-q_{\mathrm{OF},i}\rvert}{s_q}
    (2)

    alpha 1.4e-10 · p_rgh 1.2e-10 · U 1.1e-08

  3. Engineering quantity difference

    The difference between the codes of a quantity QQan engineer reads off the run: the first impact pressure peak at sensor P1 at 4.89 M cells with both codes' pressure solvers converged to 1e-10, and the arrival of the front at gauge H1.

    εQ=∣Qbrae−QOF∣∣QOF∣\varepsilon_Q=\frac{\lvert Q_{\mathrm{brae}}-Q_{\mathrm{OF}}\rvert}{\lvert Q_{\mathrm{OF}}\rvert}
    (3)

    0.002% (11,102.4 and 11,102.2 Pa) · arrival identical on every mesh

  4. Mass imbalance

    The change of the water volume since the start, at 6 s in the open-topped tank at 0.4 M cells, and before the jet reaches the top.

    ΔVw(t)=Vw(t)−Vw(0)Vw(0),Vw(t)=∑iαi(t) Vi\Delta V_w(t)=\frac{V_w(t)-V_w(0)}{V_w(0)},\qquad V_w(t)=\sum_i \alpha_i(t)\,V_i
    (4)

    −0.96% brae, −0.91% OpenFOAM · 1e-06 to 5e-06 before 0.6 s

  5. Residual reduction

    The final over the initial residual of a pressure solve, in OpenFOAM's normalisation. Median of 3,450 solves of the 42 tutorials at their own tolerances.

    rfinalrinitial,r=∑i∣b−Ax∣i∑i(∣Ax−Axˉ∣i+∣b−Axˉ∣i)\frac{r_{\mathrm{final}}}{r_{\mathrm{initial}}},\qquad r=\frac{\sum_i \lvert b-Ax\rvert_i}{\sum_i \bigl(\lvert Ax-A\bar{x}\rvert_i+\lvert b-A\bar{x}\rvert_i\bigr)}
    (5)

    5.4e-03 brae with the case's solvers · 5.4e-03 serial OpenFOAM

  6. Grid convergence index

    After Celik et al. (2008), on three meshes, with ϕ1\phi_1 and ϕ2\phi_2 the values on the fine and medium mesh, r21r_{21} their refinement ratio and pp the apparent order.

    GCIfine=1.25r21 p−1 ∣ϕ1−ϕ2ϕ1∣\mathrm{GCI}_{\mathrm{fine}}=\frac{1.25}{r_{21}^{\,p}-1}\,\Bigl\lvert\frac{\phi_1-\phi_2}{\phi_1}\Bigr\rvert
    (6)

    2.0% peak kinetic energy (order 1.37) · 0.48% arrival at H1, both codes

  7. Speed-up

    The ratio of the wall clock TT of the whole run to 6 s: OpenFOAM on 26 cores of a Xeon Platinum 8480+ over brae on 1 × H100.

    S=TOFTbraeS=\frac{T_{\mathrm{OF}}}{T_{\mathrm{brae}}}
    (7)

    5.1× at 0.4 M · 7.3× at 1.37 M · 11.9× at 4.89 M cells