engine: add a small heated flow sample

Demonstrate temperature slices and signed face-average wall heat with cool air passing an isothermal bracket. Register the original CPU-rendered gallery thumbnail and document the coarse transient setup.

Part of #157.

crates/engine/src/cmd/forms.rs

@@ -1971,6 +1971,7 @@ pub const SAMPLE: &[Field] = &[choice(
         ("drop-test", "Drop test"),
         ("rubber-seal", "Rubber seal"),
         ("flow-past", "Flow past a bracket"),
+        ("heated-flow", "Heated flow past a bracket"),
         ("heated-bracket", "Heated bracket"),
         ("bracket-hex", "Hex-meshed bracket"),
         ("crank-arm", "Bicycle crank arm"),

crates/engine/src/sample.rs

@@ -18,6 +18,8 @@
 use fieldcraft_geometry::Shape;
 use serde_json::{Value, json};
 
+mod heated_flow;
+
 /// A sample the app offers (Help › Samples, the empty viewport, `file.sample`, `--sample NAME`).
 pub struct Sample {
     pub name: &'static str,
@@ -92,6 +94,14 @@ pub const SAMPLES: &[Sample] = &[
         prepare: None,
         script: |_| flow_past(),
     },
+    Sample {
+        name: "heated-flow",
+        label: "Heated Flow Past a Bracket",
+        about: "Cool air passing an 80 °C bracket: temperature slices and signed wall heat, a small transient demonstration",
+        design: Some(UPRIGHT_BRACKET),
+        prepare: None,
+        script: heated_flow::script,
+    },
     Sample {
         name: "heated-bracket",
         label: "Heated Bracket",

crates/engine/src/sample/heated_flow.rs

@@ -0,0 +1,58 @@
+//! A small transient heated-flow demonstration using the original upright bracket.
+//! Air at 20 °C passes an 80 °C bracket; no buoyancy or conjugate solid conduction.
+
+use fieldcraft_geometry::Shape;
+use serde_json::{Value, json};
+
+pub(super) fn script(bodies: &[Shape]) -> Value {
+    let walls: Vec<Value> = bodies
+        .iter()
+        .filter(|b| b.name == "Bracket")
+        .flat_map(|b| {
+            b.faces().iter().map(move |f| {
+                json!({
+                    "face": {"body": b.name, "face": f.name},
+                    "condition": {"type": "temperature", "value": 80.0}
+                })
+            })
+        })
+        .collect();
+    let [density, viscosity] = super::AIR;
+    json!([
+        {"command": "model.units", "params": {"system": "mm-t-s-N-MPa"}},
+        {"command": "model.material.create", "params": {"name": "Air", "density": density,
+            "fluid": {"type": "newtonian", "viscosity": viscosity},
+            "thermal": {"conductivity": 0.026, "specific_heat": 1.005e9}}},
+        {"command": "model.fluid_domain.create", "params": {
+            "name": "WarmAir", "material": "Air", "min": [-30, -20, 0], "max": [100, 60, 70], "spacing": 5,
+            "faces": {"x_min": {"type": "velocity", "velocity": [1, 0, 0]}, "x_max": {"type": "pressure", "pressure": 0},
+                "y_min": {"type": "slip"}, "y_max": {"type": "slip"}, "z_min": {"type": "wall"}, "z_max": {"type": "slip"}},
+            "obstacles": ["Bracket"], "initial_velocity": [1, 0, 0],
+            "thermal": {"initial_temperature": 20, "surfaces": walls,
+                "faces": {"x_min": {"type": "temperature", "value": 20}}}
+        }},
+        {"command": "model.step.create", "params": {"name": "HeatedFlow", "procedure": {
+            "type": "fluid", "domain": "WarmAir", "steps": 180, "frames": 6
+        }}}
+    ])
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn heated_flow_sample_solves_and_exposes_temperature_and_wall_heat() {
+        let mut s = crate::Session::default();
+        s.execute("file.sample", &json!({"name": "heated-flow"})).unwrap();
+        s.execute("solve.fluid", &json!({"backend": "cpu"})).unwrap();
+        let report = s.execute("results.slice", &json!({"field": "temperature"})).unwrap();
+        let max = report["contour"]["range"]["max"]["value"].as_f64().unwrap();
+        let min = report["contour"]["range"]["min"]["value"].as_f64().unwrap();
+        assert!(max > 20.1 && max <= 80.01 && min >= 19.99, "{report}");
+        let report = s.execute("results.surface", &json!({"field": "wall_heat_flux"})).unwrap();
+        assert_eq!(report["contour"]["unit"], "mW/mm²");
+        assert!(report["wall_heat"]["faces"].as_array().unwrap().iter().any(|f| f["power"].as_f64().unwrap_or(0.0) < 0.0));
+        assert_eq!(report["frames"].as_array().unwrap().len(), 7);
+    }
+}