Surface Roughness Effects on Oblique Shock Angle

Real surfaces aren't smooth — manufacturing tolerances, ablation, erosion, and wear roughen them up. This asks whether that roughness, modeled as an equivalent sand-grain height rather than resolved geometry, measurably shifts the oblique shock angle on a supersonic double-wedge.

• M 2.73freestream
• k-ω SSTturbulence model
• 1e-3 mmproduction first-layer height
• 0.6/0.8/0.9mm sand-grain cases

Why a double wedge

One 2D geometry produces all three flow features of interest in a single run: an attached oblique shock at the leading edge (compression), a Prandtl–Meyer expansion over the mid-chord, and a trailing recompression shock.

Double-wedge model at M = 2.0: region 1 freestream, region 2 leading-edge oblique shock, region 3 the Prandtl-Meyer expansion over the mid-chord, region 4 the trailing recompression shock.

Analytical baseline (smooth wall) comes from the standard oblique-shock relation tan θ = 2 cot β · (M₁² sin²β − 1) / (M₁²(γ + cos 2β) + 2) and the Prandtl–Meyer function ν(M) = √((γ+1)/(γ-1)) · atan(√((γ-1)/(γ+1)·(M²-1))) − atan(√(M²-1)), evaluated at M₀ = 2.73, p₀ = 358.1 kPa, T₀ = 455 K, γ = 1.4 — the "should be" answer every CFD case gets checked against.

Reference geometry: (a) single wedge, attached shock at angle β for deflection θ; (b) symmetric double wedge; (c) asymmetric double wedge with two shock angles β and β′; (d) detached bow shock once θ exceeds θ_max for the given M₁ — no attached-shock solution left.

Method

Compressible RANS with the SST k-ω turbulence model — the standard choice for shock/boundary-layer interaction (accurate near-wall behavior via k-ω, less freestream sensitivity via the k-ε blend outside the boundary layer). Before trusting the double-wedge mesh, a benchmark rough-wall flat-plate case was run first and checked for grid independence, residual convergence, and a sane Mach contour — standard practice: validate the setup on a known-answer geometry before spending mesh budget on the real case. Roughness itself is a wall-function correction (equivalent sand-grain height k_s), not a literally rough resolved surface, at k_s = 0.6, 0.8, and 0.9 mm against a smooth-wall baseline.

Mesh independence

Skin friction coefficient and wall shear stress on the validation flat-plate, swept across five first-layer heights (1e-1 mm down to 1e-5 mm):

Cf vs. first-layer height — stabilizes from 1e-3 mm onward.

Wall shear stress vs. first-layer height — same story, flat from 1e-3 mm.

Both metrics are essentially unchanged from 1e-3 mm through 1e-5 mm, which is why 1e-3 mm was picked as the production first-layer height — on the flat part of the curve, not still changing with further refinement.

Sample result

Velocity contour, smooth-wall case — the oblique shock fan off the leading edge (orange/red high-velocity band), flow settling toward the trailing edge. Rough-wall cases (k_s = 0.6/0.8/0.9 mm) follow the same contour format in the full report.

Comparison

Two checks, in order: smooth-wall CFD against the analytical prediction (validates the solver against theory), then rough-wall CFD against smooth-wall CFD (the actual research question) — shock angle, C_f, wall shear stress, and near-wall y⁺.

This report's own conclusions chapter is written as a template pending final numbers ("final conclusions will be updated after completion of the simulation matrix"), even though the results chapter already has the mesh-independence data above plus smooth-wall contours. Treat this as a mid-revision academic report: the raw CFD case files aren't included in the repo, only the PDF and its figures — nothing here is runnable, and roughness results are only as good as the sand-grain wall function's validity range for the given k_s/boundary-layer-thickness ratio.

Remaining work per the report: final double-wedge geometry/freestream numbers, additional roughness heights and higher Mach numbers, and 3D / realistic random-roughness simulations.

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