embedded3 S2-6: the oblique-wall instrument (tests/embedded3_wall_position_oblique.rs: z-flow / in-plane Poiseuille + Couette linear exactness; gate oblique_wall_position_is_second_order) and the closures it found, host + device, default OFF — the transverse centroid correction (RTX_E3_DIFFUSION_TRANSVERSE=1, wall_order bit 7), the fine distance floor (RTX_E3_DISTANCE_FLOOR=fine, bit 8); kernel geometry factored into cut_cv; DFG 2D-1 x-shift knob (RTX_E3_DFG_SHIFT_X); FLAG_X0 0.6 -> 0.25 in the three flag tests (the flag of every record was detached)

Co-Authored-By: Claude Fable 5.1 <[email protected]>

crates/specialized/rtx-cfd/tests/embedded3_dfg_2d1.rs

@@ -47,6 +47,10 @@ fn dfg_2d_1_on_the_device() {
     let nz = NZ;
     let lz = nz as f64 * h;
     let t_end = env_f("RTX_E3_DFG_T", 10.0);
+    // S2-6: the body (and the Δp probes) shifted in x by a fraction of h —
+    // the lift's grid-registration scatter (a y-shift would change the
+    // physical lift: the cylinder is 5 mm off the centreline).
+    let cx = CX + env_f("RTX_E3_DFG_SHIFT_X", 0.0) * h;
     // Explicit stability: CFL 0.3 on U_m and half the viscous limit.
     let dt = (0.3 * h / U_M).min(0.5 * h * h / (6.0 * NU));
     let mut solver = Solver::new(
@@ -81,7 +85,7 @@ fn dfg_2d_1_on_the_device() {
     // traction route (S2-1).
     solver.set_body(Body::extruded(
         rtx_cfd::solvers::incompressible::EmbeddedBody::circle(
-            CX,
+            cx,
             CY,
             0.5 * D + env_f("RTX_E3_DFG_DR", 0.0) * h,
         ),
@@ -101,8 +105,10 @@ fn dfg_2d_1_on_the_device() {
     solver.initialize(&mut field);
     let mask_cells = solver.mask().map_or(0, |m| m.fluid_cells());
     println!(
-        "  DFG 2D-1 ny {ny}: {nx}×{ny}×{nz} = {} cells ({mask_cells} fluid), h {h:.4e}, dt {dt:.3e}, t_end {t_end}",
-        g.cells()
+        "  DFG 2D-1 ny {ny}: {nx}×{ny}×{nz} = {} cells ({mask_cells} fluid), h {h:.4e}, dt {dt:.3e}, t_end {t_end}, centre x/h {:.3} y/h {:.3}",
+        g.cells(),
+        cx / h,
+        CY / h
     );
     let mut device = DeviceStep::new(solver, g);
     device.upload(&field);
@@ -138,8 +144,8 @@ fn dfg_2d_1_on_the_device() {
             let ci = |x: f64| ((x / h).round() as usize).clamp(2, nx - 2);
             let cj = |y: f64| ((y / h).round() as usize).clamp(2, ny - 2);
             let bx = (
-                ci(CX - margin),
-                ci(CX + margin),
+                ci(cx - margin),
+                ci(cx + margin),
                 cj(CY - 0.15),
                 cj(CY + 0.15),
                 0,
@@ -159,10 +165,10 @@ fn dfg_2d_1_on_the_device() {
             let (cd_s, cl_s) = (coef * fs.f[0], coef * fs.f[1]);
             let zc = 0.5 * lz;
             let p_front = mask
-                .pressure_at(&field.p, CX - 0.5 * D, CY, zc)
+                .pressure_at(&field.p, cx - 0.5 * D, CY, zc)
                 .unwrap_or(f64::NAN);
             let p_back = mask
-                .pressure_at(&field.p, CX + 0.5 * D, CY, zc)
+                .pressure_at(&field.p, cx + 0.5 * D, CY, zc)
                 .unwrap_or(f64::NAN);
             let dp = p_front - p_back;
             let (cd, cl, cd_cv, cl_cv) = (coef * fw[0], coef * fw[1], coef * fcv[0], coef * fcv[1]);

crates/specialized/rtx-cfd/tests/embedded3_flag_geometry.rs

@@ -4,7 +4,11 @@
 use rtx_cfd::solvers::incompressible::embedded3::{Body, Boundaries, Grid, Mask, Side};
 
 const H: f64 = 0.41;
-const FLAG_X0: f64 = 0.6;
+/// The flag's ROOT: the cylinder's rear (the Turek–Hron flag runs from the
+/// cylinder to its tip A at x = 0.6). Until 2026-09-18 this was 0.6 — the
+/// flag sat DETACHED, its root where the benchmark's tip is; every flag
+/// record before that date is of that geometry.
+const FLAG_X0: f64 = 0.25;
 const FLAG_LEN: f64 = 0.35;
 const FLAG_HALF: f64 = 0.01;
 const FLAG_SPAN: f64 = 0.2;

crates/specialized/rtx-cfd/tests/embedded3_flag_reference_2d.rs

@@ -19,7 +19,11 @@ const L: f64 = 2.5;
 const CX: f64 = 0.2;
 const CY: f64 = 0.2;
 const R_CYL: f64 = 0.05;
-const FLAG_X0: f64 = 0.6;
+/// The flag's ROOT: the cylinder's rear (the Turek–Hron flag runs from the
+/// cylinder to its tip A at x = 0.6). Until 2026-09-18 this was 0.6 — the
+/// flag sat DETACHED, its root where the benchmark's tip is; every flag
+/// record before that date is of that geometry.
+const FLAG_X0: f64 = 0.25;
 const FLAG_LEN: f64 = 0.35;
 const FLAG_HALF: f64 = 0.01;
 const AMP: f64 = 0.084;

crates/specialized/rtx-cfd/tests/embedded3_flag_wake.rs

@@ -31,7 +31,11 @@ const L: f64 = 2.5;
 const CX: f64 = 0.2;
 const CY: f64 = 0.2;
 const R_CYL: f64 = 0.05;
-const FLAG_X0: f64 = 0.6;
+/// The flag's ROOT: the cylinder's rear (the Turek–Hron flag runs from the
+/// cylinder to its tip A at x = 0.6). Until 2026-09-18 this was 0.6 — the
+/// flag sat DETACHED, its root where the benchmark's tip is; every flag
+/// record before that date is of that geometry.
+const FLAG_X0: f64 = 0.25;
 const FLAG_LEN: f64 = 0.35;
 const FLAG_HALF: f64 = 0.01;
 const FLAG_SPAN: f64 = 0.2;

crates/specialized/rtx-cfd/tests/embedded3_wall_position_oblique.rs

@@ -0,0 +1,378 @@
+//! S2-6 instrument: the cut wall's effective position on an OBLIQUE wall
+//! with the flow IN the plane of the cut. Poiseuille flow (body force `F`
+//! along the tangent) in a channel between two embedded parallel planes
+//! `y = s x + c0` and `y = s x + c0 + w`; the sides carry the exact
+//! solution, z is periodic. The exact field is `U(r) t`, `U = F/(2μ)
+//! (G²/4 − r²)`, constant pressure, and the 5-point Laplacian is exact on
+//! it, so on full faces of a central window `u/t_x + F r²/(2μ)` is the
+//! constant `F G_eff²/(8μ)`: the effective gap from the u faces and from
+//! the v faces separately, each an effective wall offset per wall in
+//! units of h. Unlike the z-directed flat-wall test this one exercises the
+//! own-direction coupling of cut faces, the convective terms' cut-face
+//! values and the projection next to the wall (the spurious pressure is
+//! printed).
+use rtx_cfd::solvers::incompressible::ConvectionScheme;
+use rtx_cfd::solvers::incompressible::embedded3::{
+    Body, Boundaries, Field, Fluid, Grid, Parameters, Side, Solver, WallScheme,
+};
+
+const MU: f64 = 0.1;
+const LX: f64 = 2.0;
+const LY: f64 = 2.0;
+const W: f64 = 0.5;
+
+/// The cut wall's parameters: the environment's (`None`), or the S2-6
+/// closures forced on / off for the gate.
+fn parameters(s26: Option<bool>) -> Parameters {
+    let mut p = Parameters {
+        corrector_steps: 2,
+        tolerance: 1e-10,
+        convection_scheme: ConvectionScheme::Upwind,
+        wall_scheme: WallScheme::CutCell,
+        boundaries: Boundaries {
+            z0: Side::Periodic,
+            z1: Side::Periodic,
+            ..Boundaries::default()
+        },
+        ..Parameters::default()
+    };
+    if let Some(on) = s26 {
+        p.diffusion_centroid = true;
+        p.wall_distance_oblique = on;
+        p.diffusion_transverse = on;
+        p.distance_floor_fine = on;
+    }
+    p
+}
+
+struct Reading {
+    /// Effective wall offset per wall from the fitted profile of the u / v faces.
+    off_u: f64,
+    off_v: f64,
+    /// The driving force the profile's curvature implies, over F.
+    force_u: f64,
+    /// 1 − (the fitted streamwise pressure slope)/F: must equal `force_u`.
+    force_p: f64,
+    /// RMS of the pressure about its linear fit, over F·G: full cells, cut cells.
+    p_full: f64,
+    p_cut: f64,
+}
+
+/// Least squares of `y = a − b x`: returns (a, b).
+fn fit(points: &[(f64, f64)]) -> (f64, f64) {
+    let n = points.len() as f64;
+    let (sx, sy) = points
+        .iter()
+        .fold((0.0, 0.0), |s, p| (s.0 + p.0, s.1 + p.1));
+    let (mx, my) = (sx / n, sy / n);
+    let (sxx, sxy) = points.iter().fold((0.0, 0.0), |s, p| {
+        (s.0 + (p.0 - mx) * (p.0 - mx), s.1 + (p.0 - mx) * (p.1 - my))
+    });
+    let slope = sxy / sxx;
+    (my - slope * mx, -slope)
+}
+
+fn reading(n: usize, slope: f64, c0: f64, along_z: bool, s26: Option<bool>) -> Reading {
+    // The driving force (`RTX_E3_OBLIQUE_F`): the problem is linear in it
+    // but for the convective terms, so a small value switches them off.
+    #[allow(non_snake_case)]
+    let F: f64 = std::env::var("RTX_E3_OBLIQUE_F")
+        .ok()
+        .and_then(|v| v.parse().ok())
+        .unwrap_or(1.0);
+    let h = 1.0 / n as f64;
+    let (nx, ny, nz) = ((LX * n as f64) as usize, (LY * n as f64) as usize, 2);
+    let norm = (1.0 + slope * slope).sqrt();
+    let (tx, ty) = (1.0 / norm, slope / norm);
+    let gap = W / norm;
+    let r_of = move |x: f64, y: f64| ((y - slope * x - c0) - 0.5 * W) / norm;
+    let speed = move |x: f64, y: f64| {
+        let r = r_of(x, y);
+        if r.abs() < 0.5 * gap {
+            F / (2.0 * MU) * (0.25 * gap * gap - r * r)
+        } else {
+            0.0
+        }
+    };
+    let mut solver = Solver::new(
+        Fluid {
+            density: 1.0,
+            viscosity: MU,
+            reference_velocity: 1.0,
+            reference_length: 1.0,
+        },
+        parameters(s26),
+    );
+    // `along_z`: the same channel with the flow along the periodic z (the
+    // cross-direction diffusion of w alone: no pressure, no convection).
+    solver.set_boundary_velocity(move |x, y, _z, _t| {
+        let s = speed(x, y);
+        if along_z {
+            (0.0, 0.0, s)
+        } else {
+            (s * tx, s * ty, 0.0)
+        }
+    });
+    solver.set_momentum_source(move |_, _, _, _| {
+        if along_z {
+            (0.0, 0.0, F)
+        } else {
+            (F * tx, F * ty, 0.0)
+        }
+    });
+    solver.set_body(Body::from_sdf(move |x, y, _z, _t| {
+        0.5 * gap - r_of(x, y).abs()
+    }));
+    let g = Grid::cubic(nx, ny, nz, h);
+    let mut field = Field::new(g);
+    for k in 0..=nz {
+        for j in 0..ny {
+            for i in 0..nx {
+                if along_z {
+                    field.w[g.wface(k, j, i)] = speed((i as f64 + 0.5) * h, (j as f64 + 0.5) * h);
+                }
+            }
+        }
+    }
+    for k in 0..nz {
+        if along_z {
+            break;
+        }
+        for j in 0..ny {
+            for i in 0..=nx {
+                field.u[g.uface(k, j, i)] = tx * speed(i as f64 * h, (j as f64 + 0.5) * h);
+            }
+        }
+        for j in 0..=ny {
+            for i in 0..nx {
+                field.v[g.vface(k, j, i)] = ty * speed((i as f64 + 0.5) * h, j as f64 * h);
+            }
+        }
+    }
+    solver.initialize(&mut field);
+    let dt = 0.5 * h * h / (6.0 * MU);
+    let steps = (2.0 / dt).ceil() as usize;
+    for _ in 0..steps {
+        solver.advance(&mut field, dt);
+    }
+    let mask = solver.mask().expect("mask");
+    // The sides pin the flow RATE (the exact profile), so a displaced wall
+    // appears as a streamwise pressure slope: F_eff = F − dp/ds, and on the
+    // full faces of a central window `u/t_x = F_eff/(2μ) (G_eff²/4 − r²)`
+    // exactly (the 5-point Laplacian is exact on it). Fit both constants.
+    let window = |x: f64| (x - 0.5 * LX).abs() < 0.3;
+    let (mut pu, mut pv) = (Vec::new(), Vec::new());
+    let (mut pf, mut pc) = (Vec::new(), Vec::new());
+    for j in 0..ny {
+        for i in 0..nx {
+            let (xu, yu) = (i as f64 * h, (j as f64 + 0.5) * h);
+            let fu = g.uface(0, j, i);
+            let (xw, yw) = ((i as f64 + 0.5) * h, (j as f64 + 0.5) * h);
+            let fw = g.wface(0, j, i);
+            if along_z {
+                if window(xw) && r_of(xw, yw).abs() < 0.4 * gap && mask.a_w(fw) >= 1.0 {
+                    pu.push((r_of(xw, yw).powi(2), field.w[fw]));
+                }
+            } else if window(xu) && r_of(xu, yu).abs() < 0.4 * gap && mask.a_u(fu) >= 1.0 {
+                pu.push((r_of(xu, yu).powi(2), field.u[fu] / tx));
+            }
+            let (xv, yv) = ((i as f64 + 0.5) * h, j as f64 * h);
+            let fv = g.vface(0, j, i);
+            if !along_z
+                && slope > 0.0
+                && window(xv)
+                && r_of(xv, yv).abs() < 0.4 * gap
+                && mask.a_v(fv) >= 1.0
+            {
+                pv.push((r_of(xv, yv).powi(2), field.v[fv] / ty));
+            }
+            let c = g.cell(0, j, i);
+            let (xc, yc) = ((i as f64 + 0.5) * h, (j as f64 + 0.5) * h);
+            if window(xc) && mask.cell_active(c) && r_of(xc, yc).abs() < 0.5 * gap + h {
+                let s_along = xc * tx + yc * ty;
+                let full = r_of(xc, yc).abs() < 0.5 * gap - 1.5 * h;
+                if full {
+                    pf.push((s_along, field.p[c]));
+                } else {
+                    pc.push((s_along, field.p[c]));
+                }
+            }
+        }
+    }
+    let profile = |points: &[(f64, f64)]| {
+        if points.is_empty() {
+            return (f64::NAN, f64::NAN);
+        }
+        let (a, b) = fit(points);
+        let gap_eff = 2.0 * (a / b).sqrt();
+        (0.5 * (gap - gap_eff) / h, 2.0 * MU * b / F)
+    };
+    let (off_u, force_u) = profile(&pu);
+    let (off_v, _) = profile(&pv);
+    let (p0, minus_slope) = fit(&pf);
+    let rms = |points: &[(f64, f64)]| {
+        (points
+            .iter()
+            .map(|(s, p)| (p - (p0 - minus_slope * s)).powi(2))
+            .sum::<f64>()
+            / points.len().max(1) as f64)
+            .sqrt()
+            / (F * gap)
+    };
+    Reading {
+        off_u,
+        off_v,
+        force_u,
+        force_p: 1.0 + minus_slope / F,
+        p_full: rms(&pf),
+        p_cut: rms(&pc),
+    }
+}
+
+/// The linear-exactness mode: in-plane Couette flow `u = K dist t` over ONE
+/// embedded oblique wall (no force, constant pressure, the sides carry the
+/// exact field). A scheme exact on linear fields returns the wall position
+/// to round-off; the fitted zero of the profile on full faces is the offset.
+fn couette(n: usize, slope: f64, c0: f64, s26: Option<bool>) -> (f64, f64) {
+    const K: f64 = 1.0;
+    let h = 1.0 / n as f64;
+    let (nx, ny, nz) = ((LX * n as f64) as usize, (LY * n as f64) as usize, 2);
+    let norm = (1.0 + slope * slope).sqrt();
+    let (tx, ty) = (1.0 / norm, slope / norm);
+    let dist = move |x: f64, y: f64| (y - slope * x - c0) / norm;
+    let speed = move |x: f64, y: f64| K * dist(x, y).max(0.0);
+    let mut solver = Solver::new(
+        Fluid {
+            density: 1.0,
+            viscosity: MU,
+            reference_velocity: 1.0,
+            reference_length: 1.0,
+        },
+        parameters(s26),
+    );
+    solver.set_boundary_velocity(move |x, y, _z, _t| {
+        let s = speed(x, y);
+        (s * tx, s * ty, 0.0)
+    });
+    solver.set_body(Body::from_sdf(move |x, y, _z, _t| dist(x, y)));
+    let g = Grid::cubic(nx, ny, nz, h);
+    let mut field = Field::new(g);
+    for k in 0..nz {
+        for j in 0..ny {
+            for i in 0..=nx {
+                field.u[g.uface(k, j, i)] = tx * speed(i as f64 * h, (j as f64 + 0.5) * h);
+            }
+        }
+        for j in 0..=ny {
+            for i in 0..nx {
+                field.v[g.vface(k, j, i)] = ty * speed((i as f64 + 0.5) * h, j as f64 * h);
+            }
+        }
+    }
+    solver.initialize(&mut field);
+    let dt = 0.5 * h * h / (6.0 * MU);
+    let steps = (2.0 / dt).ceil() as usize;
+    for _ in 0..steps {
+        solver.advance(&mut field, dt);
+    }
+    let mask = solver.mask().expect("mask");
+    // u/t_x = K (dist − δ): fit on full faces of the central window, two to
+    // six cells off the wall.
+    let mut points = Vec::new();
+    let mut worst: f64 = 0.0;
+    for j in 0..ny {
+        for i in 0..nx {
+            let (x, y) = (i as f64 * h, (j as f64 + 0.5) * h);
+            let f = g.uface(0, j, i);
+            let dd = dist(x, y);
+            if (x - 0.5 * LX).abs() < 0.3 && dd > 2.0 * h && dd < 6.0 * h && mask.a_u(f) >= 1.0 {
+                points.push((dd, field.u[f] / tx));
+                worst = worst.max((field.u[f] / tx - K * dd).abs() / (K * h));
+            }
+        }
+    }
+    let (a, minus_b) = fit(&points);
+    // y = a − (−b) x with b the slope: the zero sits at dist = −a / b.
+    let b = -minus_b;
+    (-a / b / h, worst)
+}
+
+#[test]
+#[ignore = "S2-6 instrument: linear exactness of the cut wall on an oblique wall (in-plane Couette; a minute on the host)"]
+fn oblique_wall_linear_exactness() {
+    for slope in [0.0, 0.25, 0.5, 1.0] {
+        for n in [16usize, 32] {
+            let (off, worst) = couette(n, slope, 0.53, None);
+            println!(
+                "  couette slope {slope:.2} n {n}: wall offset {off:+.4} h (negative = inside the body); worst full-face error {worst:.4} of K·h"
+            );
+        }
+    }
+}
+
+#[test]
+#[ignore = "S2-6 instrument: the oblique cut wall's effective position with in-plane flow (minutes on the host)"]
+fn oblique_wall_effective_position() {
+    for (slope, c0) in [
+        (0.0, 0.53),
+        (0.0, 0.77),
+        (0.25, 0.53),
+        (0.5, 0.53),
+        (1.0, 0.53),
+    ] {
+        // `RTX_E3_OBLIQUE_N=64` adds a finer rung to the in-plane mode.
+        let extra: Option<usize> = std::env::var("RTX_E3_OBLIQUE_N")
+            .ok()
+            .and_then(|v| v.parse().ok());
+        let mut runs = vec![(16usize, false), (32, false), (16, true), (32, true)];
+        if let Some(n) = extra {
+            runs = vec![(n, false)];
+        }
+        for (n, along_z) in runs {
+            let r = reading(n, slope, c0, along_z, None);
+            println!(
+                "  slope {slope:.2} c0 {c0} n {n} {}: wall offset {:+.4} h (u faces) {:+.4} h (v faces), positive = inside the fluid; F_eff/F {:.5} (profile) {:.5} (pressure slope); pressure about its fit: {:.2e} full cells, {:.2e} near-wall cells (of F·G)",
+                if along_z {
+                    "z-flow (w faces)"
+                } else {
+                    "in-plane"
+                },
+                r.off_u,
+                r.off_v,
+                r.force_u,
+                r.force_p,
+                r.p_full,
+                r.p_cut
+            );
+        }
+    }
+}
+
+/// The gate (S2-6): with the oblique distance, the transverse centroid
+/// correction and the fine floor the cut wall is linear-exact to 0.02 h on
+/// an oblique wall (without them it sits 0.05–0.07 h inside the body at
+/// every h), and the z-directed Poiseuille offset halves per rung at
+/// slope ½ (without them: −0.087 → −0.083 h).
+#[test]
+fn oblique_wall_position_is_second_order() {
+    for slope in [0.5, 1.0] {
+        let (fixed, _) = couette(32, slope, 0.53, Some(true));
+        let (before, _) = couette(32, slope, 0.53, Some(false));
+        println!(
+            "  couette slope {slope}: offset {fixed:+.4} h with the S2-6 closures, {before:+.4} h without"
+        );
+        assert!(fixed.abs() < 0.02, "slope {slope}: {fixed}");
+        assert!(
+            before.abs() > 2.0 * fixed.abs(),
+            "the instrument lost its contrast"
+        );
+    }
+    let coarse = reading(16, 0.5, 0.53, true, Some(true)).off_u;
+    let fine = reading(32, 0.5, 0.53, true, Some(true)).off_u;
+    println!("  z-flow slope 0.5: offset {coarse:+.4} h at n 16, {fine:+.4} h at n 32");
+    assert!(coarse.abs() < 0.04, "n 16 offset {coarse}");
+    assert!(
+        fine.abs() < 0.65 * coarse.abs(),
+        "the offset does not halve: {coarse} → {fine}"
+    );
+}