Generalized Reynolds Analogy in Compressible Flows

• GRA is a model that extends Reynolds analogy by establishing an enthalpy–velocity correspondence in compressible turbulent boundary layers.
• It predicts wall shear stress and heat flux directly using physics-based relations and drag models, eliminating empirical corrections.
• Validated against DNS data up to Mach 4, GRA supports advanced wall modeling for hypersonic applications and rough-wall flows.

The generalized Reynolds analogy (GRA) extends the classical Reynolds analogy for turbulent convective heat and momentum transfer to compressible wall-bounded flows, including scenarios with rough walls, high Mach numbers, and varying wall thermal conditions. GRA provides a local, physics-based relationship between total enthalpy defect and velocity, enabling direct modeling of wall shear stress and heat flux without empirical correction factors. It capitalizes on a fundamental similarity between the mean enthalpy and velocity fields outside the roughness sublayer, rigorously accounting for compressibility and wall boundary effects.

1. Theoretical Foundation

GRA rests on a linear enthalpy–velocity correspondence within compressible turbulent boundary layers. The generalized recovery enthalpy is defined as

$H_g = c_p\,T + \frac{r_g}{2}\,u^2,$

where $c_p$ is the specific heat at constant pressure, $T$ is the temperature, $u$ is the local velocity, and $r_g$ is the recovery factor, which remains constant across the layer. The wall-flux-based velocity scale is

$U_w = -Pr\,\frac{q_w}{\tau_w},$

where $Pr$ is the Prandtl number, $q_w$ the wall heat flux, and $\tau_w$ the wall shear stress.

The fundamental GRA relation is

$$\bar H_g - \bar H_w = U_w\,\bar u \Longrightarrow c_p\,(\bar T_{rg} - \bar T_w) = -Pr\,\frac{q_w}{\tau_w}\,\bar u,$$

with $\bar T_{rg} = \bar H_g/c_p$ and $\bar H_w = c_p T_w$. Non-dimensionalizing via the friction temperature ($T_\tau = q_w/(c_p\,\rho_w\,u_\tau)$, $\tau_w = \rho_w\,u_\tau^2$) yields a Walz-type form:

$$\frac{\bar T_{rg} - \bar T_w}{T_\tau} = -Pr\,\frac{\bar u}{u_\tau}.$$

The general recovery factor $r_g$ relates edge values:

$$r_g = \frac{T_w - T_\delta}{u_\delta^2/(2 c_p)} + 2\,\frac{U_w}{u_\delta},$$

where $u_\delta$, $T_\delta$ denote edge velocity and static temperature.

2. Model Formulation for Rough-Wall Compressible Flows

The GRA-based model targets prism-shaped roughness elements. In these flows, two regions are distinguished:

• Roughness Sublayer ($0 < y < k$): Direct element effects dominate; GRA validity is lost.
• Outer Layer ($y > k$): Townsend similarity applies; GRA recovers validity.

Above $k$, the velocity profile is described by a compressible log-law with virtual origin $d$:

$$\frac{d\bar u}{dy} = \frac{u_\tau}{\kappa}\,\frac{1}{y - d}\sqrt{\frac{\rho_w}{\bar\rho(y)}}.$$

Integration gives crest velocity:

$$\bar u_k = \frac{u_\tau}{\kappa} \ln\frac{k-d}{z_0},$$

where $z_0$ is the hydrodynamic roughness length; $d$ and $z_0$ depend on roughness geometry and attenuation factor $a$ (Yang et al. 2016).

Outside the sublayer, the temperature profile is quadratic in velocity:

$$\bar T(y) = b_0 + b_1\,\bar u(y) + b_2\,\frac{\bar u(y)^2}{2}.$$

Coefficients $\{b_0, b_1, b_2\}$ are determined by conditions at wall, edge, and matching location $y_m$.

3. Treatment of Roughness and Compressibility

Roughness effects are incorporated via parameters from the drag model (Yang et al. 2016): exponential attenuation $a$, virtual origin $d$, roughness length $z_0$, and crest velocity $\bar u_k/u_\tau$. This approach dispenses with empirical Stanton or friction coefficient corrections; wall stress and flux are obtained self-consistently.

Compressibility is treated with Van Driest transformation:

$$\bar u_{VD} = \int_0^{\bar u} \sqrt{\frac{\bar\rho}{\rho_w}}\,du,$$

to transform velocity for application of the log-law. Outer-layer matching height $y_m$ (typically $y_m^+ \sim 300$) is used for profile closure.

4. Solution Procedure

The coupled solution involves:

1. Initial guess of friction velocity $u_\tau$.
2. Compute $a$, $d$, $z_0$, and $\bar u_k/u_\tau$ from the drag model.
3. Define crest velocity and integrate the log-law to $y_m$.
4. Set quadratic temperature–velocity fit at $y_m$.
5. Evaluate friction temperature at crest:

$$T_\tau = [\bar T_k - T_w + r_g\,\frac{\bar u_k^2}{2c_p}]\,\frac{\bar u_k}{u_\tau Pr},$$

where $\bar T_k$ is the profile temperature at crest.

1. Update heat flux: $q_w = c_p\,\rho_w\,u_\tau\,T_\tau$.
2. Iterate on $u_\tau$ until convergence in $\tau_w$ and $q_w$.

Inputs include roughness geometry, freestream/wall conditions (Mach number, pressure, recovery temperature, wall temperature ratio), fluid properties, and outer-layer values at $y_m$.

5. Validation and Performance

A priori validation against DNS data for Mach 2 and Mach 4, both adiabatic and isothermal, demonstrates high model fidelity:

Case	Wall-Shear Error $\epsilon_{\tau_w}$	Heat-Flux Error $\epsilon_{q_w}$
M2A	+1.8%	—
M2I	−0.7%	+5.5%
M4A	−10.8%	—
M4I	−20.9%	−0.4%

In adiabatic cases, $q_w \approx 0$ as expected. By comparison, classic semi-empirical correlations (Hill et al., 1980) yield heat-flux errors $\epsilon_{q_w} \sim 200\%$. The GRA-based model reproduces mean velocity and temperature profiles from roughness crest up to $y_m$ and predicts $q_w$, $\tau_w$ within a few percent at Mach 2, with increased error at Mach 4 primarily due to transformation inaccuracies (Cogo et al., 9 Jan 2026).

6. Physical Mechanisms and Applicability Limits

The direct near-wall disruption of enthalpy–momentum coupling by roughness is strictly confined to the roughness sublayer. Outside this region, enthalpy and velocity fields regain a similarity analogous to smooth-wall behavior, and the GRA becomes asymptotically valid. No evidence supports a requirement for ad hoc adjustment of friction or Stanton numbers for rough surfaces. This suggests that the GRA framework, when paired with a robust drag model, is capable of physically-consistent prediction of heat and momentum transfer for compressible turbulent flows over engineered roughness.

A plausible implication is that GRA offers a universal interface between wall models and outer-layer CFD solvers for a wide range of compressible, wall-bounded turbulent flows. However, its accuracy will depend critically on the validity of underlying velocity transformation (e.g., Van Driest) and the fidelity of roughness drag characterization.

7. Significance for Wall Modeling and Turbulence Closure

GRA furnishes a direct link between wall shear stress and heat flux in the high-speed regime, enabling predictive wall modeling for rough surfaces in the absence of empirical tuning. Its adoption in coupled CFD–wall model frameworks may reduce uncertainty in engineering calculations of hypersonic vehicle skin friction and heating. The generalization underscores the universality of nonlinear coupling between thermal and velocity fields in turbulent wall-bounded flows, facilitating transfer of incompressible flow information to compressible counterparts (Cogo et al., 9 Jan 2026).