Compressible Couette Flow

Compressible Couette Flow (aerodynamics/high-speed/compressible-couette-flow)

Use when you must compute the exact constant-property solution of the shear-driven plate gap in a high-Mach Couette flow: a stationary cold plate at y = 0 held at the static temperature T_e and an insulated (adiabatic) moving plate at y = h dragged at the edge velocity Ue = Me * a(T_e). The velocity profile is exactly linear, the temperature profile follows the Crocco energy integral over that linear profile, and the moving plate is not cooled, so its surface floats to the recovery temperature and every joule of viscous dissipation leaves through the stationary plate. This is the canonical closed-form anchor of the aeroheating vein, useful as a laminar-regime heating estimate for a plate gap and as a CFD verification case. It pairs with aerodynamics/high-speed/flat-plate-skin-friction-heating (the EXTERNAL boundary layer on a plate with regime-dependent recovery factors) and aerodynamics/high-speed/aerodynamic-heating (stagnation-point heating): those are different geometries and different recovery physics, while this leaf owns the INTERNAL shear-driven gap with the identity r = Pr. Fully closed form, pure stdlib.

Domain quick reference

• Edge state: sound speed of the stationary-plate gas a_e = sqrt(gamma R T_e), moving-plate velocity Ue = Me * a_e (edge_velocity). Constants: GAMMA = 1.4, R = 287.0, and the derived CP = GAMMA * R / (GAMMA - 1) = 1004.5 J/(kg K), PR = 0.72, MU = 1.8e-5 Pa s; conductivity is derived k = mu * cp / Pr, never an input. Sutherland-free, uniform mu and k.
• Recovery factor: r = Pr exactly, the constant-property Couette identity (recovery_factor). The external-plate laminar sqrt(Pr) and turbulent Pr^(1/3) values do NOT apply inside the gap.
• Moving-plate temperature: T_aw = T_e * (1 + r (gamma - 1) Me^2 / 2) (adiabatic_wall_temperature), identical to T_e + Pr Ue^2 / (2 cp) because the plate is insulated and its temperature is an output, not an input.
• Velocity profile: u(y) = Ue * y / h, exactly linear (velocity_profile).
• Temperature profile: the Crocco energy integral T(y) = T_e * (1 + r (gamma - 1) Me^2 (2 eta - eta^2) / 2) with eta = y / h (temperature_profile), so T(0) = T_e, T(h) = T_aw and dT/dy(h) = 0.
• Gradient: dT/dy = T_e * r (gamma - 1) Me^2 (1 - eta) / h (temperature_gradient), positive at the stationary plate, zero at the insulated moving plate.
• Wall shear: tau_w = mu Ue / h, constant across the gap because the profile is linear (wall_shear).
• Wall heat flux into the stationary plate: q_w = mu Ue^2 / h (wall_heat_flux), positive = heat into the cold plate; equals k * dT/dy(0), tau_w * Ue and the dissipation integral mu * (Ue / h)^2 * h, the energy-balance close.
• Gap Reynolds number: Re_gap = rho Ue h / mu (couette_solution), the laminar-regime indicator; transition in plane Couette flow near Re ~ 1500. tau_w and q_w are independent of rho.
• Units are SI: K, m/s, Pa, W/m2, kg/m3, m.
• NACA-TR-824 frames the high-speed laminar boundary-layer and heat-transfer context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the plate-gap state: stationary-plate static temperature T_e, plate Mach number Me, Prandtl number Pr, constant viscosity mu, edge density rho and gap height h, with the moving plate insulated. Run the plate-gap edge-state traverse: the edge sound speed a_e = sqrt(gamma R T_e) and the edge velocity Ue = Me * a_e with edge_velocity.
2. Apply the constant-property recovery identity with recovery_factor: the recovery factor of the insulated moving plate is r = Pr, not the external-plate sqrt(Pr) or Pr^(1/3) values.
3. Float the insulated moving plate with adiabatic_wall_temperature to its recovery temperature T_aw = T_e * (1 + r (gamma - 1) Me^2 / 2), and cross-check the equivalent energy form T_e + r Ue^2 / (2 cp).
4. Walk the profiles across the gap with velocity_profile (linear u = Ue y / h), temperature_profile (Crocco energy integral) and temperature_gradient (analytic dT/dy, zero at the insulated moving plate).
5. Read the constant wall shear with wall_shear: tau_w = mu Ue / h, uniform across the gap.
6. Compute the heat flux into the stationary plate with wall_heat_flux: q_w = mu Ue^2 / h, and close the dissipation energy balance q_w = k * dT/dy(0) = tau_w * Ue, where k = mu * cp / Pr. The moving plate is adiabatic, so its flux is zero.
7. Assemble the full solution with couette_solution, which returns the dict with keys Ue, T_aw, tau_w, q_w, r, Re_gap; the gap Reynolds number Re_gap = rho Ue h / mu frames the laminar regime (density enters only there).
8. Confirm the deterministic checks with the contract test scripts/test_compressible_couette_flow.py.

Worked example

Canonical case from the wave-42 spec: T_e = 300.0 K, Me = 3.0, Pr = 0.72, mu = 1.8e-5 Pa s, rho = 1.0e-3 kg/m3, h = 0.01 m, gamma = 1.4, R = 287.0 (cp = 1004.5 derived). Real module outputs:

• Edge state: a_e = 347.188709 m/s, Ue = Me * a_e = 1041.566128 m/s.
• Recovery identity: r = Pr = 0.720000 exactly; the numerical recovery check (T_aw - T_e) * 2 * cp / Ue^2 = 0.720000, equal to Pr within 1e-12. The external-plate values sqrt(Pr) = 0.8485 and Pr^(1/3) = 0.8961 would be wrong here.
• Moving-plate temperature: T_aw = 688.800000 K, a 388.8 K recovery rise; the energy form T_e + Pr Ue^2 / (2 cp) gives 688.800000 K, matching within 1e-12.
• Wall shear: tau_w = mu Ue / h = 1.874819 Pa (1.8748190313 full precision), constant across the gap; the viscous work rate tau_w * Ue = 1952.748 W/m2.
• Temperature field and heat flux: k = mu * cp / Pr = 0.025113 W/(m K) (0.0251125 full precision), dT/dy at the stationary plate = 77760.000 K/m, q_w = k * dT/dy = 1952.748 W/m2 into the stationary plate, identical to the closed form mu Ue^2 / h; the moving plate is adiabatic with dT/dy(h) = 0 and q(h) = 0 W/m2.
• Energy balance: the dissipation integral mu (du/dy)^2 over the gap equals mu Ue^2 / h = 1952.748 W/m2, exactly the flux into the stationary plate.
• Gap Reynolds number: Re_gap = rho Ue h / mu = 578.648, laminar plane Couette (transition near Re ~ 1500).
• Profiles at quarter stations: T = 300.000, 470.100, 591.600, 664.500, 688.800 K; u = 0.000, 260.392, 520.783, 781.175, 1041.566 m/s; temperature rises monotonically from the cold plate to the insulated moving plate and the velocity is exactly linear (u(h/2) = Ue/2).

Verification

• Confirm edge_velocity(3.0, 300.0) = 1041.566128 m/s and the sound speed a_e = Ue / Me = 347.188709 m/s.
• Confirm recovery_factor returns Pr exactly for any positive pr and that (T_aw - T_e) * 2 * cp / Ue^2 recovers Pr within 1e-12.
• Confirm adiabatic_wall_temperature(300.0, 3.0, 0.72) = 688.800000 K and its equality with T_e + Pr Ue^2 / (2 cp).
• Confirm the profile endpoints T(0) = T_e, T(h) = T_aw with dT/dy(h) = 0 (insulated moving plate), the quarter-station values and the linear velocity profile.
• Confirm the dissipation energy balance closes: q_w = k * dT/dy(0) = mu Ue^2 / h = tau_w * Ue within 1e-9 relative.
• Confirm every non-positive Mach number, temperature, Prandtl number, viscosity, density, gap height, velocity, and every y outside [0, h] raises ValueError in the corresponding function.
• Run the contract test offline: python3 scripts/test_compressible_couette_flow.py (33 tests, deterministic).

Related leaves

• aerodynamics/high-speed/flat-plate-skin-friction-heating: the EXTERNAL flat-plate boundary layer with regime-dependent recovery factors sqrt(Pr) / Pr^(1/3), reference-temperature property evaluation and running-length Reynolds numbers; the adjacent geometry, different mechanism.
• aerodynamics/high-speed/aerodynamic-heating: stagnation-point convective heating with the Sutton-Graves correlation and the radiation-equilibrium wall temperature; the other corner of the aeroheating vein.
• aerodynamics/high-speed/bow-shock-standoff: shock-layer geometry ahead of a blunt body, the compressible context that sets edge states for gap and plate heating estimates.

Pitfalls

• Carrying the external-plate recovery factor into the gap: inside constant-property Couette flow the recovery factor is r = Pr exactly, not the laminar sqrt(Pr) = 0.8485 or turbulent Pr^(1/3) = 0.8961 values that flat-plate-skin-friction-heating uses, so using those inflates T_aw (for the worked example, 0.8485 would give a 722.3 K moving-plate temperature against the true 688.8 K).
• Fixing the moving-plate temperature instead of floating it: the moving plate is insulated, so its surface temperature T_aw is an OUTPUT of the energy balance. A prescribed moving-plate temperature is an externally cooled case with a different (non-Crocco) profile, outside this leaf's scope.
• Treating the heat flux as leaving the moving plate: with the moving plate adiabatic, all viscous dissipation leaves through the cold stationary plate, so q_w at the stationary plate equals the full dissipation integral mu (Ue / h)^2 * h; there is no second heat sink.
• Applying Sutherland or temperature-dependent properties: the leaf is constant-property by construction (uniform mu, k = mu cp / Pr derived), the laminar-regime statement; variable-viscosity Couette solutions and turbulent gaps need boundary-layer or CFD methods.
• Reading Re_gap as a driver of tau_w or q_w: in constant-property Couette flow rho enters only through Re_gap, so changing the density changes the regime indicator, never the shear or the flux.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_compressible_couette_flow.py

The test covers the wave-42 worked-example contract (Ue, T_aw, tau_w, q_w, r, Re_gap and the quarter-station profiles), the recovery identity r = Pr and its numerical recovery check, the energy-form equivalence of T_aw, the Crocco profile endpoints and analytic gradient with the adiabatic moving plate, the forward-difference gradient cross-check, linear velocity profile stations, the constant wall shear with its linear-profile slope, the dissipation energy-balance close q_w = k * dT/dy(0) = tau_w * Ue, the sign conventions, monotonic temperature rise, the exact dict key order, determinism under repeated calls, and ValueError rejection of every non-physical input class.

Compliance

• Standards referenced, not reproduced: NACA-TR-824 (reference-only per standards-map.yaml) frames the high-speed laminar boundary-layer and heat-transfer context; the compressible Couette relations above are standard engineering methodology, summary-only.
• compliance: STANDARDS-REF, gated: false.