10 Wall Functions and Thermal Boundary Treatment

CMPS uses wall closures to connect the cell-centred RANS state to wall shear and heat flux without requiring the first cell centre to remain at one prescribed \(y^+\). The currently documented branch uses continuous viscous/logarithmic blends for momentum and temperature. The same wall quantities are differentiated in the implicit boundary assembly when they depend on active solution variables.

10.1 Tangential relative velocity

Let \(\mathbf{u}_P\) be the fluid velocity at the wall-adjacent cell and \(\mathbf{u}_w\) the wall velocity. With wall normal \(\mathbf{n}\), define the relative tangential velocity \[\mathbf{u}_t =\left(\mathbf{I}-\mathbf{n}\otimes\mathbf{n}\right) (\mathbf{u}_P-\mathbf{u}_w), \qquad u_t=|\mathbf{u}_t|.\] The wall distance \(y\) is the normal/near-wall distance supplied by the wall-distance infrastructure.

10.2 Velocity scale and wall distance

For the GE-RANS wall treatment, a blended velocity scale is \[u_*= \left[ \frac{\mu u_t/y+\sqrt{C_\mu}\,\rho k}{\rho} \right]^{1/2}, \qquad C_\mu=0.09, \](10.1) and \[y^+=\frac{\rho u_*y}{\mu}. \](10.2) This construction retains a molecular contribution near the wall and a turbulence-related contribution when \(k\) is active.

10.3 Continuous velocity blend

The viscous and logarithmic limits are \[u^+_{\ell}=y^+, \qquad u^+_{\log}=\frac{1}{\kappa}\ln\left[E_w\max(y^+,0.2)\right],\] with \(\kappa=0.4187\) and \(E_w=9.793\) in the documented current branch. The continuous blend is \[u^+ =\frac{u^+_{\ell}u^+_{\log}} {\left[(u^+_{\ell})^4+(u^+_{\log})^4\right]^{1/4}}. \](10.3) It approaches the viscous law for small \(y^+\) and the logarithmic law for sufficiently large \(y^+\) without a discrete switch.

10.4 Thermal wall law

Define the molecular Prandtl number \[\Pr=\frac{\mu c_p}{k_g},\] with turbulent Prandtl number \(\Pr_t\). A Kader-type continuous thermal blend (Kader 1981) uses \[\Gamma_T =\frac{0.01(\Pr y^+)^4}{1+5\Pr^3y^+}, \](10.4) \[P =9.24\left[\left(\frac{\Pr}{\Pr_t}\right)^{0.75}-1\right] \left[1+0.28\exp\left(-0.007\frac{\Pr}{\Pr_t}\right)\right], \](10.5) and \[T^+ =e^{-\Gamma_T}\Pr y^+ +e^{-1/\Gamma_T}\Pr_t \left[ \frac{1}{\kappa}\ln(E_wy^+)+P \right]. \](10.6) The corresponding unblown gas-side heat-transfer coefficient is \[h_0=\frac{\rho c_pu_*}{T^+}. \](10.7) The same expression spans the conductive sublayer, buffer layer, and logarithmic region.

10.5 Wall shear and heat flux

A wall shear vector has the generic form \[\boldsymbol{\tau}_w=-\tau_w\frac{\mathbf{u}_t}{u_t},\] where \(\tau_w\) is supplied by the active wall-function relation. Its face-integrated momentum contribution is \(\boldsymbol{\tau}_wA_f\).

For a stationary thermal wall with no blowing, the convective part of the gas-side heat flux is represented as \[q''_{g,0}=h_0(T_g-T_w), \](10.8) with optional viscous-heating correction where enabled. Adiabatic, prescribed-temperature, prescribed-flux, external-convection/radiation, and conjugate configurations require different closure equations for \(T_w\) and \(q''_w\); they must all reduce to a dimensionally consistent energy flux in \(\mathrm{W\,m^{-2}}\).

10.6 Turbulence wall quantities

For \(k\)–\(\omega\)-class closures, the nondimensional specific dissipation is \[\omega^+=\omega\frac{\nu}{u_*^2}, \qquad \nu=\frac{\mu}{\rho}.\] The near-wall treatment blends viscous and logarithmic asymptotes rather than switching abruptly at one grid threshold. Turbulence production, turbulent viscosity, and the wall-normal distance must be recomputed consistently whenever the solution, turbulence model, or multigrid level changes.

10.7 Implicit boundary assembly

A boundary heat flux \(\Phi_T=A_fq''_w\) contributes to the total-energy residual. If it depends on active cell variables, \[\delta\Phi_T =\sum_j\frac{\partial\Phi_T}{\partial q_j}\,\delta q_j.\] CMPS uses AD in the active boundary path so the local thermal/momentum response can enter the coupled Jacobian. This is particularly important for wall laws in which \(u_*\), \(y^+\), \(h_0\), turbulence, and wall temperature are mutually dependent.

10.8 Current implementation qualification

The velocity and Kader-type thermal relations above are established in the current CMPS wall/material-response documentation and are used by the boundary-attached charring model described in Chapter 11. The broader non-adiabatic wall/CHT implementation nevertheless has an open issue identified in the August 2026 source audit: one viscous-heating correction is dimensionally inconsistent in a laminar branch and is not assembled with equal-and-opposite balance on the conjugate interface in the affected GE-RANS path. Moving-wall traction work is also missing from the carrier total-energy boundary flux.

Accordingly, this chapter documents the intended/implemented wall-function structure but does not claim that every non-adiabatic or moving-wall energy balance is presently validated. The exact open items are summarized in Chapter 20.