Analytical Kink-Type Solutions and Streak Formation in Turbulent Channel Flow

Abstract: An analytical framework for turbulent channel flow is developed based on the Alexeev hydrodynamic equations, focusing on the coupled behavior of streamwise and transverse velocity components. The mean streamwise velocity is represented as a superposition of a laminar (parabolic) component and a nonlinear turbulent contribution, yielding velocity profiles that agree with experimental data from channel and pipe flows over a wide range of Reynolds numbers, $3\times10^3 \le Re \le 3.5\times10^7$, with deviations of approximately $1\%$ at moderate Reynolds numbers and up to $3\%$ at the highest Reynolds numbers. The transverse velocity component is analyzed using a simplified form of the governing equations, leading to analytical expressions that capture its dominant spatial structure. The coupling between transverse velocity and streamwise momentum is then examined, revealing that the streamwise turbulent component admits a family of kink-type solutions. These solutions exhibit localized monotonic transitions separating regions of nearly uniform velocity and are interpreted as analytical representations of streamwise streaks. The model predicts characteristic streak properties, including spacing, thickness, intensity, and streamwise extent, which are shown to be consistent in order of magnitude with experimental observations of near-wall streaks. The results provide a unified analytical description of mean velocity profiles, secondary flows, and streak formation in wall-bounded turbulence, and suggest a mechanism linking transverse velocity fluctuations to the emergence of coherent streamwise structures.

• The paper introduces an analytical model based on Alexeev hydrodynamic equations that derives explicit streamwise and transverse velocity profiles with deviations below 3% from experiments.
• A super-exponential turbulent correction and matched asymptotic expansions yield kink-type solutions that accurately predict near-wall streak spacing and structure.
• The unified framework links transverse velocity dynamics with streak formation, offering insights for improved turbulence modeling and control strategies.

Analytical Kink-Type Solutions and Streak Formation in Turbulent Channel Flow

Analytical Framework Based on Alexeev Hydrodynamic Equations

This work formulates an analytical treatment for turbulent channel and pipe flows using the Alexeev hydrodynamic equations (AHE), an extension of the incompressible Navier-Stokes framework with an additional parameter $\tau$ introducing higher-order fluctuation and dissipation scales. The mean streamwise velocity $U(y)$ is decomposed into a weighted laminar (parabolic) component $U^L(y)$ and a nonlinear turbulent correction $U^T(y)$:

$$U(y) = U_0\left[\gamma U^T(y) + (1-\gamma)U^L(y)\right],$$

with $\gamma$ determined by viscous dissipation minimization.

For stationary two-dimensional flow, an explicit analytical solution for $U^T$ is derived from a reduced form of AHE, resulting in a super-exponential profile, while the improved fit to empirical data requires a superposition of two $U^T$ branches corresponding to positive and negative sign choices for the fluctuation scale $\delta$. The model yields velocity profiles with deviations from experimental observations below $1\%$ for $U(y)$0 and less than $U(y)$1 up to $U(y)$2.

Figure 1: Streamwise mean velocity $U(y)$3 in wall units compared to Wei & Willmarth's channel flow data for two $U(y)$4; laminar, turbulent, and composite analytical profiles shown against experiment.

Coupled Transverse Velocity Dynamics

The paper advances beyond traditional turbulence closures by deriving a consistent analytical form for the transverse (wall-normal) velocity component $U(y)$5. Applying a linearized treatment for moderate $U(y)$6, the transverse field is expressed as a sum over sinusoidal modes with temporal oscillatory evolution:

$U(y)$7

where $U(y)$8 encodes the spatial mode and fluctuation timescale. The leading spatial mode reflects experimental findings, with measured $U(y)$9 dynamics and sinusoidal form matching amplitude and structure observed in open-channel and pipe flows.

Figure 2: Comparison of vertical velocity $U^L(y)$0 from experiment, least-squares sinusoidal fit, and analytical mode shape, demonstrating spatial structure agreement.

Analytical Kink-Type Solutions for Streak Representation

One of the central contributions is the analytical derivation of kink-type solutions for the turbulent streamwise component $U^L(y)$1. Using the quasi-steady limit for $U^L(y)$2 under the rapid transverse mode variation, the equation

$U^L(y)$3

is analyzed. For the dominant transverse mode, the solution is constructed via matched asymptotic expansions, revealing a sequence of spatially localized, monotonic "kinks" (error function-like transitions) of thickness $U^L(y)$4, separating nearly uniform streamwise velocity plateaus.

The kink solutions accurately predict streak spacing via

$U^L(y)$5

with $U^L(y)$6 determined by Reynolds number and matching experimental observations of near-wall streak separation, typically $U^L(y)$7 wall units at moderate $U^L(y)$8.

Figure 3: Kink-type solutions for varying mode parameter $U^L(y)$9; each panel shows the spatial structure of $U^T(y)$0 highlighting localized transitions.

Quantitative Agreement with Experimental Data

The analytical model is validated against comprehensive data sets from Wei & Willmarth (channel), Pasch et al. (channel, LDV), and Princeton Superpipe (pipe, very high $U^T(y)$1). Across all data, the composite solution composed of the analytical $U^T(y)$2 and $U^T(y)$3 tracks measured profiles to within experimental uncertainty, with notable improvement over log-law-based closures at both moderate and high $U^T(y)$4. For Superpipe data, deviation remains under $U^T(y)$5 even at $U^T(y)$6, except where surface roughness may dominate.

Figure 4: Streamwise velocity profiles showing composite model prediction versus experimental measurements in both channel and pipe geometry, spanning a broad $U^T(y)$7 regime.

The model's transverse velocity field and associated streak structure also closely reproduce observed features: the amplitude and wavelength of secondary velocity are in accord with open channel and secondary flow literature, and streak intensity and spacing are quantitatively consistent with wall-resolved PIV and LDV measurements.

Theoretical and Practical Implications

This analytical formulation achieves several advances:

• Unified treatment: It provides closed-form, parameter-minimized expressions capturing both mean velocity and coherent structure formation, linking secondary flow and streamwise streaks in a single hydrodynamic closure.
• Predictive streak properties: Spacing, thickness, and amplitude of streaks emerge analytically, making explicit their dependence on $U^T(y)$8, fluctuation scale $U^T(y)$9, and mode parameter $$U(y) = U_0\left[\gamma U^T(y) + (1-\gamma)U^L(y)\right],$$0, unlike empirical wall-scaling rules.
• Quantitative fit: Deviations below $$U(y) = U_0\left[\gamma U^T(y) + (1-\gamma)U^L(y)\right],$$1 with multiple independent data sets, encompassing both smooth and rough wall regimes.

These results suggest the AHE-based approach could inform analytical wall models for RANS/LES, control-oriented reduced order models, and provide insight into streak and vortex formation mechanisms central to near-wall mixing and drag generation.

Potential for Future Developments

Further extensions could account for nonlinear saturation of transverse velocity, more realistic wall constraints, and generalizations to arbitrary wall-bounded geometries. The connection between $$U(y) = U_0\left[\gamma U^T(y) + (1-\gamma)U^L(y)\right],$$2 scaling and underlying molecular or multi-scale structure remains an avenue for theoretical refinement. The analytical characterization of coherent structures within a unified hydrodynamic system may find applications in turbulence modeling, flow control, and theoretical studies of transitional and fully developed shear flows.

Conclusion

This paper presents an analytical model for turbulent channel flow with explicit solutions for streamwise and transverse velocity fields. The approach yields accurate mean velocity profiles, analytically predicts streak spacing and structure, and connects secondary motion to coherent streak formation via kink-type solutions. Strong quantitative agreement with benchmark experiments is demonstrated. The results clarify the mechanistic role of transverse velocity in structuring streamwise streaks and highlight potential for analytical extensions to complex wall-bounded flows.