High–Speed Wall Functions with –Independent Blending

High–Speed Wall Functions with –Independent Blending
High–Speed Wall Functions with \(y^+\)–Independent Blending (CMPS)

High–Speed Wall Functions with \(y^+\)–Independent Blending (CMPS)

Introduction

Predicting wall shear stress \(\tau_w\) and wall heat flux \(q_w\) in turbulent boundary layers is notoriously challenging because the near–wall region contains very steep velocity and temperature gradients and multiple interacting sub–layers. Directly resolving the viscous sublayer, buffer layer, and logarithmic layer requires meshes with the first node at \(y^+\!\approx\!1\) and dozens of points across the inner layer, which is computationally prohibitive at industrial Reynolds numbers.

The CMPS solver employs \(y^+\)–independent blended wall functions that (i) are valid for any first–node placement, (ii) include semi–local compressible scaling for strong property variations, (iii) provide a blended thermal law for heat transfer, and (iv) incorporate a viscous–heating correction important for high–speed aerothermal problems. The same framework consistently supports adiabatic, isothermal, imposed–flux, external convection, radiation, mixed convection–radiation, and conjugate (FSI) wall thermal boundary conditions.

Core idea. The mean profiles are written in wall units and represented by asymptotic laws (linear and logarithmic) that are smoothly blended through a \(y^+\)–independent transition. These composite laws furnish closed–form relationships between near–wall state \((U,T)\) at the first off–wall node and the unknown wall fluxes \((\tau_w, q_w)\), which the solver determines iteratively.

Inner–Layer Scaling and Semi–Local Transformations

Friction scales and classical wall units

The friction velocity \[ u_\tau \equiv \sqrt{\frac{\tau_w}{\rho_w}}, \] with \(\rho_w\) the wall density, sets the velocity scale of the inner layer. Classical wall units are \[ y^+ = \frac{y\,\rho_w u_\tau}{\mu_w}, \qquad u^+ = \frac{U_\parallel}{u_\tau}, \] where \(U_\parallel\) is the magnitude of the mean tangential velocity relative to the wall, and \(\mu_w\) is the wall viscosity.

Thermal wall units and friction temperature

For heat transfer, \[ T_\tau = \frac{q_w}{\rho_w c_{p,w} u_\tau}, \qquad T^+ = \frac{T - T_w}{T_\tau}, \] with \(c_{p,w}\) the wall specific heat. Molecular and turbulent Prandtl numbers, \(\Pr\) and \(\Pr_t\), determine the relative roles of diffusion and turbulent transport.

Semi–local compressible scaling

With strong property variation, CMPS employs semi–local (van–Driest–type) transformations: \[ y^\ast = y\,\sqrt{\frac{\rho}{\rho_w}}\,\frac{\rho_w u_\tau}{\mu}, \qquad u^\ast = \frac{U_\parallel}{u_\tau \sqrt{\rho/\rho_w}}, \] and analogously for temperature, \[ T_\tau^\ast = \frac{q_w}{\rho c_p u_\tau \sqrt{\rho/\rho_w}}, \qquad T^\ast = \frac{T - T_w}{T_\tau^\ast}. \] All wall laws below are expressed in terms of the semi–local variables \((y^\ast, u^\ast, T^\ast)\) for robustness in compressible and hypersonic regimes.

Tangential projection at the wall

Let \(\hat{\boldsymbol{n}}\) be the outward wall normal and \(\boldsymbol{I}\) the identity tensor. The tangential projector is \(\boldsymbol{P}_t=\boldsymbol{I}-\hat{\boldsymbol{n}}\hat{\boldsymbol{n}}^{\!\top}\). With fluid and wall velocities \(\boldsymbol{u}\) and \(\boldsymbol{u}_w\), \[ \boldsymbol{u}_t = \boldsymbol{P}_t\,(\boldsymbol{u}-\boldsymbol{u}_w), \qquad U_\parallel = \|\boldsymbol{u}_t\|. \] This \(U_\parallel\) is the quantity entering \(u^+\) or \(u^\ast\).

Asymptotic Inner–Layer Laws (Velocity and Temperature)

Viscous sublayer (\(y^\ast \lesssim 5\))

Momentum is diffusion–dominated: \[ \mu\,\frac{\partial U_\parallel}{\partial y} \approx \tau_w \;\Rightarrow\; U_\parallel(y) \approx \frac{\tau_w}{\mu}\,y, \qquad u^\ast_\nu(y^\ast) = y^\ast. \] Heat transfer is conduction–dominated. Using a Prandtl–dependent slope, \[ T^\ast_\nu(y^\ast,\Pr) = \frac{\Pr}{\kappa}\,y^\ast, \] with \(\kappa\approx0.41\) the von Kármán constant.

Logarithmic region (\(y^\ast \gtrsim 30\))

Momentum balance with a mixing–length–based eddy viscosity gives \[ u^\ast_{\log}(y^\ast) = \frac{1}{\kappa}\,\ln y^\ast + B, \] with \(B\approx5.2\). Similarly, the turbulent heat flux yields a thermal log law, \[ T^\ast_{\log}(y^\ast,\Pr_t) = \frac{1}{\kappa_T}\,\ln y^\ast + B_T, \] with \(\kappa_T\) and \(B_T\) dependent on \(\Pr_t\) (typically \(\Pr_t\approx0.85\text{–}0.95\)).

Thermal transition parameters used in CMPS

The code employs three Prandtl–dependent auxiliaries in the thermal wall law: a slope/offset modifier \(P(\Pr)\); a thermal damping function \(\Gamma_T(y^\ast,\Pr)\); and an intersection level \(U_c^+(\Pr)\) where viscous and turbulent thermal sublayers meet. These modulate the composite \(T^\ast\) so the slope/curvature match data across \(\Pr\).

\(y^+\)–Independent Blending (Velocity and Temperature)

Smooth composite laws

CMPS blends the linear and log–layer laws with smooth weights \(w_v(y^\ast)\) and \(w_t(y^\ast)=1-w_v(y^\ast)\): \[ u^\ast(y^\ast) = w_v\,u^\ast_\nu + w_t\,u^\ast_{\log}, \qquad T^\ast(y^\ast) = w_v\,T^\ast_\nu + w_t\,T^\ast_{\log}. \] A practical family is \(w_v(y^\ast)=\exp[-(y^\ast/y_{cr})^{n}]\), \(w_t=1-w_v\), with crossover \(y_{cr}\) and sharpness \(n\). By construction, the limits recover the exact viscous and log behaviors.

Thermal modifiers \(P(\Pr)\), \(\Gamma_T\), and \(U_c^+\)

These parameters ensure realistic onset and blending of the thermal turbulent contribution across \(\Pr\), preventing premature or delayed transitions and smoothing the composite slope.

From Composite Laws to Fluxes

Wall shear stress via an inner–layer equation

Given the first off–wall value \(U_P=U_\parallel(y_P)\) at distance \(y_P\), CMPS solves \[ U_P = u_\tau \, u^\ast\!\big(y^\ast(u_\tau)\big), \qquad y^\ast(u_\tau) = y_P \sqrt{\frac{\rho}{\rho_w}}\,\frac{\rho_w u_\tau}{\mu}, \] for \(u_\tau\) by Newton’s method. With \(u_\tau\) converged, \(\tau_w=\rho_w u_\tau^2\).

Thermal transfer coefficient from the composite thermal law

From \[ T_P - T_w = T_\tau^\ast \, T^\ast(y^\ast,\Pr,\Pr_t), \qquad T_\tau^\ast = \frac{q_w}{\rho c_p u_\tau \sqrt{\rho/\rho_w}}, \] one obtains \[ q_w = h (T_P - T_w), \qquad h = \frac{\rho c_p u_\tau}{\sqrt{\rho/\rho_w}}\;\frac{1}{T^\ast(y^\ast,\Pr,\Pr_t)}. \]

Viscous–heating correction \(T_c\) (high–speed aerothermal)

CMPS augments the wall energy balance with a temperature–like correction \(T_c\). In a laminar/viscous form (or when turbulence is off): \[ T_c = \tfrac{1}{2}\,\frac{\mu}{k_\ell}\,\|\boldsymbol{u}_t\|^2. \] In RANS, a composite \(T_c(\Pr,\rho,u_\tau,y^\ast,U_\parallel,U_c^+,\Gamma_T,k)\) is used with correct limits. The unified wall heat flux is \[ q_w = h\,(T_P - T_w) + \frac{T_c}{T^\ast}. \]

Wall Thermal Boundary Conditions (Unified Treatment)

Let \(h\) be as above and \(S_{vh}=T_c/T^\ast\). CMPS computes \(T_w\) per BC:

Adiabatic: \(0 = h\,(T_P - T_w) + S_{vh} \Rightarrow T_w = T_P + S_{vh}/h = T_P + T_c/(h\,T^\ast)\).

Isothermal: \(T_w=T_{\text{set}}, \; q_w = h\,(T_P - T_{\text{set}})+S_{vh}\).

Prescribed heat flux: \(q_w=q_{\text{set}}, \; T_w = T_P + (q_{\text{set}}-S_{vh})/h\).

External convection: \(h\,(T_P - T_w)+S_{vh} = \alpha_a\,(T_\infty - T_w)\Rightarrow T_w = (h\,T_P+\alpha_a T_\infty+S_{vh})/(h+\alpha_a)\).

Radiation: solve by Newton: \[ f(T_w) = \varepsilon_w \sigma\,(T_w^4 - T_{ar}^4) - h\,(T_w - T_P) + \frac{T_c}{T^\ast}, \quad f'(T_w) = 4\,\varepsilon_w \sigma\,T_w^3 - h. \] Start with \(T_w^{(0)}=\tfrac{1}{2}(T_{ar}+T_P)\).

Mixed convection–radiation: Newton on \[ f(T_w) = \varepsilon_w \sigma\,(T_w^4 - T_{ar}^4) + \alpha_a\,(T_w - T_\infty) - h\,(T_w - T_P) + \frac{T_c}{T^\ast}, \] with derivative \(f'(T_w) = 4\,\varepsilon_w \sigma\,T_w^3 + \alpha_a - h\).

Conjugate (FSI): with \(\alpha_s=k_s/d_s\): \[ T_w = \frac{\alpha_s\,T_s + h\,T_P + T_c/T^\ast}{\alpha_s + h}. \]

Resulting heat flux: \(q_w = h\,(T_P - T_w) + T_c/T^\ast\) (unless imposed).

Wall Function for the Specific Dissipation Rate \(w^+\)

In CMPS, \(\omega\) is expressed as \[ w^+ = \omega \,\frac{\nu}{u_\tau^2}, \qquad \nu = \frac{\mu}{\rho}. \] Asymptotes: \[ w_{\mathrm{vis}}(y^+) = \frac{6}{\beta_1 (y^+)^2}, \qquad w_{\mathrm{log}}(y^+,\beta_s) = \frac{1}{\sqrt{\beta_s}\,K\,y^+}. \] Calibrated blend: \[ w_{\mathrm{vis},c} = C_w\,w_{\mathrm{vis}}, \quad w_p = w_{\mathrm{vis},c}\!\left[1+\left(\frac{w_{\mathrm{log}}}{w_{\mathrm{vis},c}}\right)^{C_{\exp}}\right]^{1/C_{\exp}}. \] ALL-\(y^+\) option with damping \(\Gamma_k(y^+)\): \[ w_{\mathrm{vis},m} = w_{\mathrm{vis},c}\!\left[1+\left(\frac{w_{\mathrm{log}}}{w_{\mathrm{vis},c}}\right)^{C_{\exp}}\right]^{1/C_{\exp}}, \] \[ w^+ = e^{-\Gamma_k} w_{\mathrm{vis},m} + e^{-1/\Gamma_k}\left(\frac{1}{w_{\mathrm{log}}^2}+\frac{1}{w_{\mathrm{vis},m}^2}\right)^{-1/2}. \] In inner variables: \[ P_k^+ = \frac{P_k \nu}{u_\tau^4}, \quad k^+=\frac{k}{u_\tau^2}, \quad \varepsilon^+ = w^+ k^+, \] and near-wall equilibrium enforces \(P_k^+\approx \varepsilon^+\).

Turbulence Wall Quantities in Inner Variables

\(w^+ = \omega \nu/u_\tau^2\) makes \(\omega\) finite and consistent as \(y^+\to 0\). With \(k^+=k/u_\tau^2\) and \[ P_k \approx \tau_t \, \frac{\partial U_\parallel}{\partial y}, \qquad P_k^+ = \frac{P_k \nu}{u_\tau^4}, \] CMPS uses the blended gradients \(\partial u^\ast/\partial y^\ast\) to preserve \(P_k^+\approx \varepsilon^+\) through the buffer layer.

Linearization and Assembly for the Coupled System

Let \(A\) be face area. The wall heat flux \(\Phi_T=A\,q_w\) enters the discrete energy equation as a Robin term. CMPS linearizes \(\Phi_T\) with respect to local unknowns (e.g., \(T_P\), quantities depending on \(u_\tau\), \(y^\ast\), \(T^\ast\)), contributing consistent Jacobian entries; the frozen part updates the RHS, improving convergence of the fully coupled solve.

Engineering Measures and High–Speed Indicators

From \(\tau_w\) and \(q_w\), CMPS reports \[ C_f = \frac{2\,\tau_w}{\rho_\infty U_\infty^2}, \qquad St = \frac{q_w}{\rho_\infty c_p U_\infty (T_\infty - T_w)}. \] Also \[ Ec = \frac{U_\infty^2}{c_p (T_\infty - T_w)}, \] with high \(Ec\) and \(C_f\) indicating strong viscous heating (where \(T_c\) is essential).

Algorithmic Summary (CMPS Wall Treatment)

  1. Compute \(\boldsymbol{u}_t, U_\parallel\), local properties \((\rho,\mu,k_\ell,c_p)\), wall distance \(y\) and normal.
  2. Form \(y^\ast\), build \(u^\ast(y^\ast)\); solve for \(u_\tau\) by Newton.
  3. Evaluate \(\Pr\), thermal modifiers \(P(\Pr), \Gamma_T, U_c^+\); build \(T^\ast(y^\ast)\).
  4. Compute \(h\) from \(T^\ast\); build \(T_c\) (laminar form or RANS composite), set \(S_{vh}=T_c/T^\ast\).
  5. Solve \(T_w\) for the chosen thermal BC (closed form or Newton); for FSI use the conjugate formula.
  6. Evaluate \(q_w=h(T_P-T_w)+T_c/T^\ast\) (unless imposed).
  7. Assemble \(\Phi_T=A\,q_w\) with consistent Jacobian contributions.

Remarks on Robustness and Applicability

  • \(y^+\)–independence: same laws whether \(y^+=1,10,200\).
  • Compressibility: semi–local \((y^\ast,u^\ast,T^\ast)\) collapse inner–layer behavior at high \(M\).
  • Thermal BCs: unified \(T_w\) framework covers adiabatic, isothermal, \(q\)–imposed, convection, radiation, mixed, and conjugate.
  • Turbulence consistency: \(w^+,k^+,P_k^+\) tied to the same inner gradients as \(u^\ast,T^\ast\).
  • High–speed heating: \(S_{vh}=T_c/T^\ast\) captures viscous work; reduces to the laminar form in the appropriate limit.

Summary

CMPS implements a wall–function framework that blends viscous and log–layer asymptotes with \(y^+\)–independent weights, uses semi–local compressible scaling, derives \(h\) from the blended \(T^\ast\), augments the energy balance with a viscous–heating correction, and uniformly treats all common thermal BCs (analytic or Newton). Inner–scaled turbulence variables \((w^+,k^+,P_k^+)\) keep momentum, heat transfer, and turbulence closures consistent from incompressible to hypersonic regimes without excessive near–wall refinement.

© CMPS — Wall functions documentation page