Wall Similarity Model (WSM) in Turbulent Flows

• Wall Similarity Model (WSM) is a reduced-order framework that leverages invariant statistical properties and modal decompositions to predict near-wall turbulence and scalar transport.
• WSM employs precise scaling parameters such as δe and Ue along with POD modes to collapse velocity and scalar profiles across various Reynolds numbers and geometries.
• WSM is applied in LES, boundary layer analysis, and scalar dispersion, achieving error reductions of up to 50% compared to traditional models.

The Wall Similarity Model (WSM) constitutes a class of reduced-order modeling approaches and similarity-based scaling laws for turbulence and scalar transport in wall-bounded flows. These models are unified by the recognition that velocity and scalar statistics in regions near the wall can exhibit forms of similarity—either exact or approximate—across varying Reynolds numbers, geometries, and flow regimes when the appropriate scaling parameters and modal decompositions are chosen. WSM frameworks are distinguished by mathematically precise similarity definitions, rigorous extraction of scaling quantities, and their direct physical connection to canonical turbulent near-wall processes, as detailed in recent studies for velocity fluctuations (Hansen et al., 2023), mean velocity profiles (Tuoc, 2010, Weyburne, 2017), and scalar plume dispersion (Amankwah et al., 25 Jan 2026).

1. Formal Definitions of Wall Similarity

Wall similarity in turbulent wall-bounded flows is quantified by the invariance or quasi-invariance of certain statistical measures under changes in Reynolds number, geometry, or wall-bounded regime, once quantities are appropriately normalized or projected.

• Strong Similarity: For the two-point correlation tensor of velocity fluctuations $C_u(y,y';\mathrm{Re}_\tau)$, strong similarity implies factorization of all Reynolds number dependence:

$$C_u(y,y';\mathrm{Re}_\tau) = G(\mathrm{Re}_\tau)\,\widetilde C(y,y')$$

This produces Reynolds-independent shapes of all proper orthogonal decomposition (POD) modes: $$\phi_n(y, \mathrm{Re}_\tau) = \widetilde{\phi}_n(y)$$ for all $n$.

• Weak Similarity: Only the first $N$ POD modes collapse in shape:

$$\phi_n(y; \mathrm{Re}_\tau) = \widetilde{\phi}_n(y), \quad n=1, \dots, N$$

Higher modes retain Reynolds-number dependence (Hansen et al., 2023).

• Zone-based Similarity: Mean velocity scaling in the “inner region” (wall layer + log law region) is achieved by normalizing with the velocity $U_e$ and thickness $\delta_e$ at the wall-layer edge. In turbulent scalar dispersion from ground-level sources near roughness elements, wall similarity manifests as a vertical concentration profile that assumes a stretched-exponential (fat-tailed) instead of Gaussian form, parameterized by a “shape exponent” $s$ reflecting scaling within the wall-bounded region (Amankwah et al., 25 Jan 2026).

2. Derivations and Key Equations

(a) Velocity Fluctuation Model (Inner Wall Law Extension)

The WSM for streamwise velocity fluctuations builds on the observation that the first POD mode $\phi_1(y)$ of $u'(y,t)$ demonstrates weak similarity over a substantial portion of the near-wall region ($0 < y^+ \lesssim 40$). The reconstructed profile is: $$u^+(x, y, z, t) = c_1^*(x, z, t)\;\mathrm{LoW}(y^+) + c_2^*(x, z, t)\;g(y^+)$$ where $\mathrm{LoW}(y^+)$ is the law of the wall (e.g., Reichardt profile), and $g(y^+)$ fits the first POD mode shape (see Section 11 in (Hansen et al., 2023)).

(b) Zonal Similarity Model for Mean Velocity

The three-zone model for wall-bounded shear flows introduces inner scaling using $U_e, \delta_e$ at the wall-layer edge. The similarity variables are

$\eta = y/\delta_e,\quad U^* = U/U_e$

The inner mean velocity is collapsed by $U^* = f(\eta)$, where $f$ transitions from an error function in the wall layer to a log-law segment for $0.2 \lesssim \eta \lesssim 1$ (Tuoc, 2010).

(c) Integral-moment WSM for 2D Boundary Layers

The viscous and outer scales, $\delta_v(x)$ and $\delta_d(x)$, are constructed from integral moments of $d^2[u/u_e]/dy^2$ and $d[u/u_e]/dy$, respectively. Profile similarity requires that the ratio $\alpha(x)=\delta_d(x)/\delta_v(x)$ is constant across streamwise stations (Weyburne, 2017).

(d) Scalar Dispersion WSM

For ground-level sources in boundary layers, the cross-wind integrated concentration $C'(x, z)$ is modeled as

$$\frac{C'(x, z)}{C'_M(x)} = \exp\left[ -\ln 2 \left( \frac{z}{\delta_{cz}(x)} \right)^s \right]$$

where $s = 2 + m - n$ (with $m, n$ as vertical power-law exponents for mean velocity and diffusivity) and $\delta_{cz}$ is the vertical half-width at half maximum (Amankwah et al., 25 Jan 2026).

3. Empirical Identification and Extraction of Similarity Scales

The extraction of similarity and scaling parameters is central. In the zonal similarity framework (Tuoc, 2010):

• $\delta_e$: Identified at the $y^+$ location where mean velocity departs from a log-law fit, or where the viscous stress fraction falls below a chosen threshold (e.g., $4\%$ of total stress).
• $U_e$: Corresponding mean velocity at $y = \delta_e$.

For 2D turbulent boundary layers (Weyburne, 2017), integral moments are used for:

• Viscous (inner) thickness: $\delta_v = \mu_1 + 2\sigma_v$, with

$\mu_1 = \nu u_e / u_\tau^2$

$\sigma_v^2$ derived from the second moment of $d^2[u/u_e]/dy^2$.

• Outer (boundary layer) thickness: $\delta_d = a_1 + 2\sigma_d$, with moments derived from $d[u/u_e]/dy$.

In scalar dispersion WSM, vertical profiles at various downstream locations and for a range of roughness/geometry configurations are rescaled by measured $\delta_{cz}$ and fit to extract the best-fit value of $s$ (Amankwah et al., 25 Jan 2026).

4. Comparative Performance and Data Collapse

Direct quantitative comparisons highlight the improved fidelity of WSM-type models over classical approaches across multiple contexts:

Model/Context	Equilibrium Law/Classic Model	WSM/Extended Model
Instantaneous $u^+$ field (channel, $y^+=15$) (Hansen et al., 2023)	$r_{eq} \approx 0.62$ with DNS	$r_{WSM} \approx 0.91$ with DNS
1D spectra at $y^+=15$ (Hansen et al., 2023)	Misses spectral peaks ($\lambda^+\simeq100$)	Recovers peaks to within $\lesssim 5\%$ of DNS amplitude
Mean velocity (inner region) (Tuoc, 2010)	$u_\tau$, $\nu$ scaling: large scatter	$U_e$, $\delta_e$ scaling: collapse to $<3\%$ deviation
Scalar vertical profile (GLS plumes) (Amankwah et al., 25 Jan 2026)	Gaussian ($s=2$): underpredicts near-wall peak	Best-fit $s\approx1.5-1.7$, accurate in fat tails

These results show that WSM approaches, whether in velocity or scalar concentration profiles, enable single- or two-parameter extensions that reproduce DNS or experimental data with errors approximately half those of classical models in the near-wall or wall-attached regime (Hansen et al., 2023, Tuoc, 2010, Amankwah et al., 25 Jan 2026).

5. Physical Interpretation and Regime Dependence

The physical justification for WSMs lies in the structure of wall turbulence:

• The first POD mode $g(y^+)$ characterizes wall-attached eddy motions, peaking at $y^+ \approx 10$–$15$, yielding corrections that directly reconstruct near-wall streaks and energy spectra lost in mean-profile-only models (Hansen et al., 2023).
• Zonal similarity recognizes three distinct regimes in wall turbulence: an outer region dominated by large-scale pressure-driven motions, a wall layer governed by intermittent sweeps/ejections (Stokes-layer scaling), and a log-law intermediate region (Tuoc, 2010).
• In scalar transport from ground-level sources, vertical stretching of the profile ($s<2$) indicates enhanced plume spread and fat tails, directly arising from wall similarity in diffusion properties and the geometry-induced modification of vertical turbulent transport (Amankwah et al., 25 Jan 2026).

6. Applicability, Limitations, and Extensions

Applicability:

• Flows considered: Turbulent channel, pipe, and boundary-layer flows at $\mathrm{Re}_\tau \gtrsim 10^3$ (Tuoc, 2010).
• Newtonian and mildly non-Newtonian fluids (WSM extended to drag-reducing riblet and polymer flows) (Tuoc, 2010).
• Ground-level scalar releases behind wall-mounted obstacles (for WSM in dispersion), with $AR_1 \lesssim 3$ yielding pronounced non-Gaussianity (Amankwah et al., 25 Jan 2026).
• 2D turbulent boundary layers with small streamwise variation in the ratio $\alpha(x) = \delta_d(x)/\delta_v(x)$ (Weyburne, 2017).

Limitations and Open Issues:

• WSM for velocity fluctuations is strictly a two-mode expansion; residual details beyond $g(y^+)$ are not modeled.
• Classical WSM for mean velocity offers no first-principles prediction of $\delta_e$ and $U_e$; these must be determined from measurements or semi-empirical fits (Tuoc, 2010).
• Scalar dispersion WSM is restricted to fully developed GLS regime; not directly applicable to elevated plume (EP) or ground-level plume (GLP) cases without recourse to the Gaussian Dispersion Model (Amankwah et al., 25 Jan 2026).
• For similarity collapse in 2D boundary layers, true similarity (constant $\alpha(x)$) is rare; only select datasets exhibit this property (Weyburne, 2017).

7. Implementation and Practical Guidance

For wall-modeled large-eddy simulation (LES), the extended law of the wall (WSM) requires only two modal coefficients, determined via projection onto $\{\mathrm{LoW}, g\}$ bases. For experimental or numerical datasets of mean velocity, check for constancy of $\alpha(x)$ before applying WSM-based collapse.

For scalar dispersion:

• Fit the vertical profile using the stretched-exponential functional form; extract $s$ empirically in the GLS regime.
• For elevated releases or near-source regions, revert to GDM or analogous image-source extensions.

WSM thus delivers a family of models that exploit deep modal or statistical similarity in wall-bounded turbulence, robustly extending classic near-wall theory and providing improved accuracy for near-wall flows and scalar transport across a broad range of practical scenarios (Hansen et al., 2023, Tuoc, 2010, Amankwah et al., 25 Jan 2026, Weyburne, 2017).