Alexeev Hydrodynamic Equations (AHE)

• Alexeev Hydrodynamic Equations (AHE) are a set of moment equations derived from a generalized Boltzmann equation that incorporate finite particle size, kinetic nonlocality, and a material timescale.
• The AHE approach yields an analytical superposition of laminar and turbulent velocity components with a Reynolds-number-dependent relaxation scale, showing improved agreement with experiments compared to classical Navier–Stokes solutions.
• AHE provides mechanistic insights into turbulence by coupling transverse pressure gradients and wall-normal motions, suggesting novel flow control strategies such as targeted wall suction or injection.

Alexeev Hydrodynamic Equations (AHE) are hydrodynamic moment equations derived from Alexeev’s Generalized Boltzmann Equation (GBE) to incorporate finite particle size, kinetic nonlocality, and finite-rate relaxation effects through a material timescale parameter $\tau$. In the recent turbulent-flow literature, AHE are used as an alternative to classical incompressible Navier–Stokes (NS) modeling for stationary mean channel and pipe flows, yielding analytical superpositions of laminar and turbulent velocity components, a Reynolds-number-dependent relaxation scale $\tau=\delta^2 Re$, and, in a further reduction, kink-type analytical structures interpreted as streamwise streaks in wall turbulence (Fedoseyev, 17 Dec 2025, Fedoseyev, 5 Apr 2026).

1. Kinetic derivation and material scales

The AHE originate in Alexeev’s GBE, which extends the classical Boltzmann or Enskog framework by incorporating finite particle size through an interaction volume $v_b$ and a temporal nonlocality represented by a second material derivative of the distribution function. In the form used in the recent channel-flow analyses, the GBE is written as

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$

with

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$

where $f(\mathbf{r},\mathbf{v},t)$ is the one-particle velocity distribution function and $J^B$ is the standard Boltzmann collision integral (Fedoseyev, 17 Dec 2025).

In this formulation, $\tau$ is a material timescale associated with finite interaction-scale kinetics. The turbulent-flow papers consistently introduce the length scale

$\delta_0=\sqrt{\tau_0 \nu},$

and its nondimensional counterpart

$$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$

with $\tau=\delta^2 Re$0 or, depending on the chosen scaling, the corresponding bulk-based Reynolds number. The papers treat $\tau=\delta^2 Re$1 as a fluid property or similarity parameter and use it as the principal additional scale beyond the conventional hydrodynamic variables (Fedoseyev, 17 Dec 2025, Fedoseyev, 18 Aug 2025).

A central claim of the 2025 channel-flow comparison study is that the dimensional scale $\tau=\delta^2 Re$2 is approximately invariant within an experimental family and coincides with the Kolmogorov microscale $\tau=\delta^2 Re$3 in the relevant high-$\tau=\delta^2 Re$4 wall-bounded regimes. That paper reports $\tau=\delta^2 Re$5 for distilled water and nearly the same value for air over a wide pressure range, interpreting this scale as the thickness of the near-wall turbulent region in the analytical solution (Fedoseyev, 17 Dec 2025). This identification is presented as a physical interpretation of the AHE similarity parameter rather than as a universally established consequence of kinetic theory.

2. Hydrodynamic structure and relation to Navier–Stokes

In nondimensional variables, the full incompressible AHE used in the recent literature take the form

$\tau=\delta^2 Re$6

$\tau=\delta^2 Re$7

with the wall pressure condition

$\tau=\delta^2 Re$8

The additional $\tau=\delta^2 Re$9-dependent terms encode second-time-derivative, pressure–time, and mixed pressure–velocity effects that are absent from incompressible NS (Fedoseyev, 5 Apr 2026).

For the turbulent channel-flow analyses, a simplified AHE model is often used:

$v_b$0

with $v_b$1 at solid walls (Fedoseyev, 17 Dec 2025, Fedoseyev, 18 Aug 2025). In this reduced form, the momentum equation has the stationary NS form, but continuity is modified by the pressure Laplacian term. The papers interpret this as a nonclassical correction that retains fluctuation or nonlocal effects even in incompressible mean-flow settings.

A basic structural property emphasized throughout this literature is the reduction to classical incompressible NS in the limit $v_b$2, equivalently $v_b$3 (Fedoseyev, 17 Dec 2025, Fedoseyev, 18 Aug 2025). In that limit the modified continuity relation becomes $v_b$4, the extra kinetic scale disappears, and the analytical AHE constructions collapse to classical Poiseuille behavior when the turbulent weighting parameter $v_b$5 is also taken to zero.

3. Reduction to channel flow and analytical mean-profile construction

The canonical application is fully developed pressure-driven flow in a plane channel with streamwise coordinate $v_b$6, wall-normal coordinate $v_b$7, and spanwise coordinate $v_b$8, bounded by two parallel walls. The recent AHE papers assume statistically steady mean flow, dependence only on $v_b$9, constant streamwise pressure gradient $$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$0, no slip at the wall, and symmetry at the centerline. In the normalized half-channel used in the derivations, the laminar reference profile is the parabolic Poiseuille solution

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$1

or, in the more general notation,

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$2

For stationary NS, the continuity constraint forces $$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$3, and the resulting stationary solution is laminar; the 2025 comparison paper states explicitly that no stationary turbulent solution exists within the NS framework under these assumptions (Fedoseyev, 17 Dec 2025).

The AHE-based construction introduces a superposition of laminar and turbulent components,

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$4

with $$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$5. In the simplest analytical version, the turbulent transverse component is obtained from a linearized third-order equation and yields representative branches

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$6

while the corresponding streamwise turbulent contribution takes the super-exponential form

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$7

The total mean profile is then

$$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$8

on the normalized interval used in the papers (Fedoseyev, 5 Apr 2026, Fedoseyev, 17 Dec 2025).

The weighting coefficient $$\frac{Df}{Dt}-\frac{D}{Dt}\!\left(\tau\,\frac{Df}{Dt}\right)=J^B,$$9 is determined by a minimum-dissipation criterion. The functional minimized in the 2025 studies is

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$0

and the reported optimal values lie typically in the range $$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$1–$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$2 for both water and air datasets (Fedoseyev, 17 Dec 2025). A later refinement replaces the single super-exponential term by a paired construction,

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$3

chosen so that the wall derivative is preserved while improving high-$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$4 agreement (Fedoseyev, 18 Aug 2025).

4. Transverse dynamics and the emergence of kink-type streak solutions

A distinctive development in the 2026 channel-flow paper is the use of AHE to derive analytical transverse-velocity dynamics and, through coupling, kink-type solutions for the streamwise turbulent component. In the reduced two-dimensional channel setting, the wall-normal velocity $$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$5 satisfies, after elimination of the pressure gradient and linearization for small $$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$6,

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$7

Separation of variables gives sinusoidal eigenfunctions in $$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$8 under no-penetration wall conditions and oscillatory temporal behavior. The resulting general solution is written as

$$\frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\frac{\partial}{\partial \mathbf{r}}+\mathbf{F}\cdot\frac{\partial}{\partial \mathbf{v}},$$9

with coefficients determined by initial data (Fedoseyev, 5 Apr 2026).

The paper then assumes that the streamwise turbulent component evolves quasi-steadily relative to the faster transverse oscillations. Averaging over fast oscillations and retaining a dominant spatial mode yields the effective transverse velocity

$f(\mathbf{r},\mathbf{v},t)$0

The streamwise turbulent balance reduces to

$f(\mathbf{r},\mathbf{v},t)$1

Writing $f(\mathbf{r},\mathbf{v},t)$2 gives

$f(\mathbf{r},\mathbf{v},t)$3

and therefore

$f(\mathbf{r},\mathbf{v},t)$4

$f(\mathbf{r},\mathbf{v},t)$5

In the small-$f(\mathbf{r},\mathbf{v},t)$6 limit, Laplace asymptotics convert this integral into a sequence of error-function transitions,

$f(\mathbf{r},\mathbf{v},t)$7

which the paper interprets as a family of kink-type solutions. These are localized monotonic transitions of thickness $f(\mathbf{r},\mathbf{v},t)$8 that separate nearly uniform plateaus and are proposed as analytical representations of streamwise streaks (Fedoseyev, 5 Apr 2026).

The same paper notes that, although it does not derive a closed-form $f(\mathbf{r},\mathbf{v},t)$9 expression directly, the error-function profile is equivalent up to scaling to a tanh-type kink. The mode number $J^B$0 controls the spacing, with

$J^B$1

and kink centers at $J^B$2. In this construction, the periodicity of the effective wall-normal transport fixes the spacing, while diffusion fixes the thickness.

5. Similarity scaling, wall units, and comparison with experiments

The AHE channel and pipe studies consistently express results in wall variables,

$J^B$3

with $J^B$4 in fully developed channel flow and $J^B$5 or equivalent friction-factor definitions (Fedoseyev, 5 Apr 2026, Fedoseyev, 17 Dec 2025). Within this normalization, the streak spacing predicted by the kink construction becomes

$J^B$6

Matching the canonical near-wall streak spacing $J^B$7 gives

$J^B$8

and for $J^B$9 the 2026 paper notes $\tau$0, which it describes as consistent with observations. The same analysis associates the streak thickness with $\tau$1, reporting $\tau$2 in fitted cases and streamwise extents of order $\tau$3–$\tau$4 wall units, with observed near-wall streak lengths of order $\tau$5 wall units (Fedoseyev, 5 Apr 2026).

For mean velocity profiles, the principal validation range reported for the AHE superposition is $\tau$6. The 2026 paper states deviations of approximately $\tau$7 at moderate Reynolds numbers and up to $\tau$8 at the highest Reynolds numbers in comparisons with channel and pipe experiments (Fedoseyev, 5 Apr 2026). The 2025 comparison paper reports that AHE solutions agree significantly better with experiments than NS solutions over Reynolds numbers from $\tau$9 to $\delta_0=\sqrt{\tau_0 \nu},$0, including Wei & Willmarth channel data, van Doorne & Westerweel pipe data, Pasch channel data, and Princeton Superpipe measurements by Zagarola and Smits (Fedoseyev, 17 Dec 2025).

A further refinement in the 2025 “Improved Analytical Solution” paper reports that the maximum discrepancy was reduced from $\delta_0=\sqrt{\tau_0 \nu},$1 to $\delta_0=\sqrt{\tau_0 \nu},$2 for Reynolds numbers of order $\delta_0=\sqrt{\tau_0 \nu},$3, and from $\delta_0=\sqrt{\tau_0 \nu},$4 to $\delta_0=\sqrt{\tau_0 \nu},$5 for Reynolds numbers up to $\delta_0=\sqrt{\tau_0 \nu},$6, with comparisons spanning Nikuradse’s smooth-pipe data, Wei’s channel data, Zagarola’s Superpipe data, van Doorne’s stereoscopic PIV study, and Pasch’s channel measurements (Fedoseyev, 18 Aug 2025). In the Superpipe series, that paper gives parameter ranges $\delta_0=\sqrt{\tau_0 \nu},$7 and $\delta_0=\sqrt{\tau_0 \nu},$8.

The same body of work also compares the AHE profiles with the classical logarithmic law,

$\delta_0=\sqrt{\tau_0 \nu},$9

The 2025 comparison study states that the AHE profile matches the experimental wall-normal structure across the viscous sublayer, buffer, inner, and outer regions, whereas the log law incurs errors of approximately $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$0 in the buffer layer and approximately $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$1 in the outer region for the cited Wei & Willmarth data (Fedoseyev, 17 Dec 2025).

6. Mechanistic interpretation, control implications, and limitations

The AHE literature presents a specific mechanism for turbulence generation in the reduced mean equations: nonzero transverse velocity is required to sustain a nontrivial streamwise turbulent component. In the simplified channel model,

$$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$2

and the associated transverse pressure gradient satisfies

$$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$3

The 2025 comparison paper interprets this coupling to mean that transverse pressure gradients and the resulting wall-normal motions are the source of turbulence in the stationary mean equations; when $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$4, the model returns to the laminar state (Fedoseyev, 17 Dec 2025).

This interpretation leads directly to a control proposal in the same paper: wall suction or injection through an array of small holes can be used to drive $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$5 toward zero and suppress turbulence. The reported required suction or injection speeds are typically $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$6–$$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$7 times smaller than the streamwise mean velocity, and this proposal is linked to classic boundary-layer control observations cited there (Fedoseyev, 17 Dec 2025). A plausible implication is that AHE recast mean-flow control as direct manipulation of the cross-stream transport term rather than as modification of an eddy-viscosity field.

Relative to mainstream wall-turbulence theory, the AHE account differs from linear transient-growth or lift-up explanations and from self-sustaining-process formulations. The 2026 streak paper states explicitly that classical theories attribute streaks to streamwise vortices and nonlinear vortex–streak regeneration, whereas the AHE route derives streak-like structures from $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$8-dependent pressure–velocity coupling and a quasi-steady advection–diffusion problem with analytically solvable transverse modes (Fedoseyev, 5 Apr 2026). The trade-off, also emphasized in that paper, is the reliance on strong simplifications: incompressible two-dimensional mean flow, dependence only on $$\delta=\frac{\sqrt{\tau_0\nu}}{L_0}, \qquad \tau=\tau_0\frac{U_0}{L_0}=\delta^2 Re,$$9, linearization of the $\tau=\delta^2 Re$00 equation, neglect of some $\tau=\delta^2 Re$01-terms in the $\tau=\delta^2 Re$02 equation, and omission of fully three-dimensional time-dependent turbulent interactions.

The stated scope of the current AHE results is smooth-wall channel and pipe flow. The papers do not claim complete capture of spanwise variability, full nonlinear saturation, or the temporal regeneration cycle associated with canonical near-wall turbulence. Their own suggested extensions include nonlinear saturation in the transverse dynamics, multimode coupling, use of more complete AHE closures beyond the simplified continuity correction, and generalization to rough walls or non-Newtonian fluids (Fedoseyev, 18 Aug 2025, Fedoseyev, 5 Apr 2026).