A Boundary-Reorganization Memory Prototype for Unsteady Lift, Dynamic Stall Delay, and Wake Persistence

Unsteady lift on pitching and surging aerofoils is controlled by coupled changes in surface pressure, circulation, wake-induced velocity, and separated-flow topology [citation]. Dynamic stall adds delayed separation onset, strong load hysteresis, large transient vortical structures, and delayed recovery [citation]. Time-resolved force, pressure, and flow measurements further show that lift history depends on how vorticity is created, shed, transported, and reorganized in the surrounding field [citation].

The lifting body is treated as a prescribed moving boundary with persistent identity over the benchmark window. The surrounding continuum is supplemented by a localized boundary-reorganization source, a transported scalar memory field, and a memory stress added to the standard momentum balance. Pressure difference, circulation-sensitive load correction, delayed reattachment, and wake persistence are then treated as distinct observables of one declared reorganized continuum state on the benchmark family/window.

Boundary localization and frame indifference restrict the leading source class to effective-incidence-rate, tangential-reorganization, and curvature-asymmetry channels. An energetic budget defines the admissible scalar-memory window. Section averaging first yields an averaged source law on one fixed family. A one-state reduction is admitted only if the projected effective-incidence forcing is non-null on the identification window and the normalized reduction-entry residual satisfies $\epsslv\le \varepsilon_*$TeX\epsslv\le \varepsilon_* on one fixed reduction convention $\Cid$TeX\Cid. The consequences derived below apply only on such admitted family windows. Families requiring multiple leading memory scales, a nonlocal history kernel at baseline order, or an independent wake-side source fall outside the baseline closure.

The comparison window, weighted norm, filter, attached-flow baseline, and lag convention are fixed before extraction; changing any of them defines a different comparison class. A harmonic window identifies $\tau_M$TeX\tau_M from baseline-subtracted excess loops. Ramp and wake windows then test the same $\tau_M$TeX\tau_M against stall delay, delayed reattachment, and wake decay with no observable-wise retuning.

Reduction chain on admitted same-family windows..

The reduction chain is finite on admitted same-family windows: the declared first-order local source basis fixes the three retained boundary-reorganization channels; section averaging produces $\bar S_\Gamma$TeX\bar S_\Gamma; weighted projection onto $\dot\alphaeff$TeX\dot\alphaeff fixes $\chi_\Gamma$TeX\chi_\Gamma, the orthogonal residual $\Rslv$TeX\Rslv, and $\epsslv$TeX\epsslv; admitted windows then yield one reduced memory equation, the excess-loop area law, the peak-frequency rule, the stall-delay scaling, the wake-length scaling, and the pressure--circulation--wake consistency residual. These consequences are not independent fits. They are downstream outputs of the same declared source basis, memory law, reduction convention, and fixed-family identification window.

definition: Moving boundary insertion. Let $\Gam(t)$TeX\Gam(t) be the moving solid--fluid interface of a lifting body with boundary velocity $\mathbf{v}_{\Gam}$TeX\mathbf{v}_{\Gam}. On a benchmark coarse-graining window, $\Gam(t)$TeX\Gam(t) is treated as a moving boundary insertion if its primary role is to alter the admissible organization of the surrounding flow field rather than to be represented as a distributed bulk force source. The boundary conditions remain the usual no-penetration or no-slip constraints appropriate to the benchmark, but the interpretive burden shifts to the ambient field response.

definition: Response-memory field. A scalar field $M(\mathbf{x},t)$TeXM(\mathbf{x},t) is called a response-memory field if it measures, on the benchmark coarse-grained window, how much of the previous boundary-adapted flow organization persists locally after the boundary state or forcing has changed. Positive and negative values correspond to opposite signed memory of the reorganized state.

remark: Physical meaning of the added field. The variable $M$TeXM is a coarse-grained continuum state variable defined on the fluid domain. On the benchmark window it records the persistence of boundary-adapted organization not closed by instantaneous velocity and pressure alone at the intended reduced level. It is introduced only as a benchmark-window continuum response coordinate and is not promoted here to a new material species or to a class-free memory law.

definition: Restricted regime. The construction is restricted to a fixed family of moving lifting boundaries in incompressible or weakly compressible flow, with one characteristic free-stream speed $U_\infty$TeXU_\infty, one characteristic chord or geometric length $c$TeXc, and one coarse-grained time window on which a single memory time $\tau_M$TeX\tau_M is meaningful. The theory is meant for attached, incipiently separated, and moderately separated unsteady regimes in which section-level aerodynamic loading remains a useful observable. Deep fully turbulent wake breakdown, multi-body mutual interference, and arbitrary bluff-body shedding lie outside the baseline claim. All recovery, reduction, and benchmark claims below are read only inside this declared restricted regime.

Over the benchmark family the solid body is treated as a prescribed moving boundary with persistent identity on the time window of interest. Constitutive changes of the body itself are not modelled here. The problem addressed below is the response of the surrounding flow. Pressure difference, circulation correction, downwash, and wake structure are treated as distinct observables of one declared reorganized flow state on the benchmark family/window induced by the moving boundary. The added variable $M$TeXM records the persistence of that state after the instantaneous boundary forcing has changed.

The baseline constitutive extension for the moving-boundary regime augments the bulk momentum balance by one memory stress and advances one advection--relaxation--source law for $M$TeXM.

definition: Baseline bulk-plus-memory law. Let $\Db(\ub)=\tfrac12(\nabla\ub+\nabla\ub^{\mathsf T})$TeX\Db(\ub)=\tfrac12(\nabla\ub+\nabla\ub^{\mathsf T}) be the symmetric strain tensor. The baseline boundary-reorganization memory closure is

\rho(\partial_t\ub+\ub\cdot\nabla\ub)
=
-\nabla p + \mu\Delta\ub + \nabla\cdot\SigM,

\nabla\cdot\ub = 0,

\partial_t M + \ub\cdot\nabla M
=
-\tau_M^{-1} M + \Seps[\ub,\Gam].

The memory stress is taken at baseline level to be

\SigM = \mu_M M\Db(\ub),

with one fixed coefficient $\mu_M\ge 0$TeX\mu_M\ge 0. All reduced, recovery, and benchmark statements below are read only relative to this declared baseline bulk-plus-memory law on the fixed family and coarse-grained time window of the restricted regime.

definition: Admissible local source class. The boundary-reorganization source is assigned the admissible baseline form

\Seps[\ub,\Gam]
=
\phieps(d_{\Gam})
\left(
\chi_\alpha\,\dot\alphaeff
+
\chi_s\,\partial_s u_t
+
\chi_\kappa\,U_t\kappa_{\Gam}
\right),

where $d_{\Gam}(\mathbf{x},t)$TeXd_{\Gam}(\mathbf{x},t) is the signed distance to the boundary, $\phieps$TeX\phieps is a narrow positive localization kernel, $u_t$TeXu_t is the tangential fluid velocity, $\partial_s$TeX\partial_s the tangential derivative along the boundary, $\kappa_{\Gam}$TeX\kappa_{\Gam} the local boundary curvature, $U_t\equiv \mathbf{v}_{\Gam}\cdot\tb$TeXU_t\equiv \mathbf{v}_{\Gam}\cdot\tb the signed tangential boundary speed, and $\dot\alphaeff$TeX\dot\alphaeff the effective-incidence rate including any benchmark-specific induced-flow correction. Here $\dot\alphaeff$TeX\dot\alphaeff is a prescribed kinematic input of the benchmark family, fixed before coefficient extraction on the intended family window; it is not retuned observable by observable inside the same family. The coefficients $(\chi_\alpha,\chi_s,\chi_\kappa,\tau_M,\mu_M)$TeX(\chi_\alpha,\chi_s,\chi_\kappa,\tau_M,\mu_M) are fixed once for a fixed geometry family and are not retuned cycle by cycle. All downstream source-class and benchmark-window claims are read only relative to this declared local source class, this fixed family, and the fixed conventions stated where they are used below.

definition: First-order local scalar source basis. On the declared benchmark isotropic window, let $\mathcal B_\Gamma^{(1)}$TeX\mathcal B_\Gamma^{(1)} denote the quotient class of boundary-local scalar source terms that are localized by the same positive kernel $\phieps(d_\Gam)$TeX\phieps(d_\Gam), scalar under Euclidean frame changes, first order in the retained boundary kinematics and near-boundary deformation measures, and read modulo higher-order curvature, acceleration, nonlocal wake feedback, and three-dimensional coupling remainders. Within this quotient class the retained span is

\mathcal B_\Gamma^{(1)}
=
\operatorname{span}\!\left\{
\phieps(d_\Gam)\dot\alphaeff,\
\phieps(d_\Gam)\partial_su_t,\
\phieps(d_\Gam)U_t\kappa_{\Gam}
\right\}.

All source-class statements below are statements inside this declared quotient span. Higher-order, nonlocal, independent acceleration, and fully three-dimensional source terms are not proved absent; they are outside the baseline class retained here.

proposition: Declared three-channel baseline source basis in the benchmark isotropic window. Assume that the source term is (i) localized to a thin neighborhood of the moving boundary, (ii) scalar under Euclidean frame changes, (iii) first order in the declared boundary-kinematic and near-boundary deformation variables, and (iv) evaluated inside the quotient span $\mathcal B_\Gamma^{(1)}$TeX\mathcal B_\Gamma^{(1)} of Definition reference. Then the baseline source retained by the model is represented by reference, up to the out-of-basis remainders specified in Definition reference. A family requiring an additional independent leading channel is not fitted by changing the three coefficients; it leaves the declared baseline constitutive class.

proof. Work in the quotient span $\mathcal B_\Gamma^{(1)}$TeX\mathcal B_\Gamma^{(1)}. Boundary localization supplies the common multiplier $\phieps(d_\Gam)$TeX\phieps(d_\Gam). Euclidean frame indifference excludes vector-valued or frame-oriented terms from the retained scalar source at this order. First-order locality leaves scalar representatives built from the retained boundary kinematics and near-boundary deformation measures. The three representatives that distinguish the retained temporal, tangential, and fore--aft asymmetry classes are

\dot\alphaeff,
\qquad
\partial_su_t,
\qquad
U_t\kappa_{\Gam} .

Terms such as $|\ub|^2$TeX|\ub|^2, $(\partial_su_t)^2$TeX(\partial_su_t)^2, $\kappa_{\Gam}^2$TeX\kappa_{\Gam}^2, independent acceleration corrections, nonlocal wake feedback, and three-dimensional coupling either have the wrong retained parity, are higher order in the declared derivative/geometric counting, or lie outside the quotient span. Hence the retained representative of an admitted leading local scalar source is the localized linear combination displayed in reference, modulo the stated out-of-basis remainders.

remark: Kinematic and geometric source channels. Equation reference collects the leading channels by which the moving boundary biases the surrounding flow reorganization. Pitch history enters through $\dot\alphaeff$TeX\dot\alphaeff, near-wall redistribution through $\partial_s u_t$TeX\partial_s u_t, and boundary geometry through $U_t\kappa_{\Gam}$TeXU_t\kappa_{\Gam}.

corollary: Three retained response-asymmetry channels. The source law reference decomposes the baseline reorganization into three retained channels: (i) an incidence-rate channel $\chi_\alpha\dot\alphaeff$TeX\chi_\alpha\dot\alphaeff, which sets the temporal skew between imposed motion and field response; (ii) a near-wall reorganization channel $\chi_s\partial_su_t$TeX\chi_s\partial_su_t, which measures tangential redistribution adjacent to the inserted boundary; and (iii) a geometry-asymmetry channel $\chi_\kappa U_t\kappa_{\Gam}$TeX\chi_\kappa U_t\kappa_{\Gam}, which encodes the fore-aft or camber bias of the moving boundary. Within the quotient span $\mathcal B_\Gamma^{(1)}$TeX\mathcal B_\Gamma^{(1)}, the retained baseline model is generated by the three displayed scalar channels at the retained derivative/geometric order. Any family requiring a fourth independent leading channel is treated as outside the declared constitutive class and is not represented by refitting the three retained coefficients within the present model.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

proposition: Baseline admissible memory stress in the benchmark isotropic window. Assume that the additional Cauchy stress is local, symmetric, objective, isotropic in the benchmark family, and leading-order linear in the scalar memory field $M$TeXM and the first velocity gradient. Then, under incompressibility, the only admissible leading extra stress is

\SigM=\mu_M M\Db(\ub),

up to higher-order terms and tensorial-memory extensions.

proof. For an incompressible fluid, the objective symmetric tensors built linearly from $\nabla \ub$TeX\nabla \ub are $\Db(\ub)$TeX\Db(\ub) and $(\nabla\cdot\ub)\mathbf I$TeX(\nabla\cdot\ub)\mathbf I. The latter vanishes by reference. Isotropy and symmetry then force the leading extra stress to be proportional to $\Db(\ub)$TeX\Db(\ub). Because the extra stress must vanish when the memory field is absent, the proportionality factor is linear in $M$TeXM at baseline order, yielding reference.

proposition: Kinetic--memory budget and admissibility window. Let $\Omegaf(t)$TeX\Omegaf(t) be the instantaneous fluid region of a benchmark and define the combined kinetic--memory energy

\Ekm(t)
=
\int_{\Omegaf(t)}
\left(
\frac{\rho}{2}|\ub|^2 + \frac{\kappa_M}{2}M^2
\right)dV,

with $\kappa_M>0$TeX\kappa_M>0 a fixed memory-energy weight. Assume impermeable or periodic far-field closure and collect the power delivered by the moving body into $\Pgam(t)$TeX\Pgam(t). Then solutions of reference and reference satisfy

\frac{d}{dt}\Ekm
\le
\Pgam(t)
+
\kappa_M\int_{\Omegaf(t)} M\,\Seps\,dV
-
\int_{\Omegaf(t)} (2\mu+\mu_M M)\,\Db(\ub):\Db(\ub)\,dV
-
\frac{\kappa_M}{\tau_M}\int_{\Omegaf(t)} M^2\,dV .

Hence, if the admitted regime satisfies

2\mu+\mu_M M \ge \nuad>0

throughout the benchmark family, the source-free, quiescent-boundary limit is dissipative and the scalar-memory closure does not autonomously create kinetic--memory energy.

proof. Multiply reference by $\ub$TeX\ub, integrate over $\Omegaf(t)$TeX\Omegaf(t), use incompressibility, and collect all moving-boundary terms into $\Pgam(t)$TeX\Pgam(t). The viscous and memory-stress contributions combine into $-\int (2\mu+\mu_M M)\Db:\Db$TeX-\int (2\mu+\mu_M M)\Db:\Db. Multiply reference by $\kappa_M M$TeX\kappa_M M, integrate, and use incompressibility again to remove the advection contribution from the volume production term. Adding the two identities yields reference. If reference holds, the quadratic velocity-gradient term is nonpositive, and in the absence of source injection and moving-boundary work the remaining right-hand side is strictly dissipative.

remark: Meaning of the admissibility window. The scalar-memory baseline is confined to moderate-memory regimes satisfying reference. Violation of this bound indicates the need for a higher-order or tensorial extension.

proposition: Lift as reorganized surface response. For any benchmark family with lift direction $\yb$TeX\yb, the instantaneous lift is

L(t)
=
\int_{\Gam(t)}
\left[
-p\,\nb\cdot\yb
+
2\mu\,\Db(\ub)\nb\cdot\yb
+
(\SigM\nb)\cdot\yb
\right]ds.

The first two terms are the conventional pressure and viscous contributions, while the third is the traction generated by persisted reorganization memory.

remark: Lift decomposition. Equation reference preserves the conventional pressure and viscous contributions and adds the constitutive memory traction as an additional surface term.

remark: Pressure, circulation, and wake projections. Surface pressure, circulation-sensitive load, and wake persistence are distinct observables of the same reorganized state at baseline order. The baseline claim concerns consistency of sign, lag ordering, and leading relaxation scale, not pointwise equivalence [citation].

proposition: Quasi-steady recovery within the declared restricted regime. Within the declared restricted regime and the declared section-averaged memory-state convention, if the memory time and memory stress vanish at leading order,

\tau_M \to 0,
\qquad
\mu_M \to 0,

then reference and reference reduce to the standard incompressible momentum balance and the lift law reference reduces to the conventional pressure-plus-viscous load integral. No broader recovery claim is made here for response-memory systems outside the declared restricted regime or outside the declared section-averaged memory-state convention. On attached-flow benchmarks, the usual circulation-based and linear unsteady lift baselines are then recovered.

proof. In the limit $\tau_M\to0$TeX\tau_M\to0, the memory field relaxes instantly and remains bounded only if $M\to 0$TeXM\to 0 at leading order when the source is finite. With $\mu_M\to0$TeX\mu_M\to0, the memory stress vanishes from the momentum equation and from the surface traction. The remaining equations are the standard incompressible bulk equations with the benchmark moving boundary conditions. Attached-flow recoveries then reduce to the usual benchmark-specific circulation or linear unsteady lift formulas [citation].

The closure adds one coarse-grained state field and one constitutive stress to the standard momentum balance for the moving-boundary regime.

The previous section gives a field theory. This section extracts consequences that are experimentally accessible and do not require a fully resolved solution to state. The key step is to average the memory field over the boundary layer and near wake of a benchmark airfoil section.

definition: Section-averaged memory state. Let $\bar M(t)$TeX\bar M(t) denote the coarse-grained memory state obtained by averaging $M$TeXM over a thin neighborhood of the airfoil section and weighting by the localization kernel $\phieps$TeX\phieps. On a fixed family of section benchmarks, $\bar M$TeX\bar M is the reduced memory state exported to the observable load law. All reduced load and recovery statements below are read only relative to this declared section-averaged memory-state convention on that fixed family.

definition: Section-averaged source. Let $\langle\cdot\rangle_\Gamma$TeX\langle\cdot\rangle_\Gamma denote the same coarse-grained section average used to define $\bar M$TeX\bar M. The corresponding averaged source is

\bar S_\Gamma(t)
=
\chi_{\Gamma,\alpha}\dot\alphaeff
+
\chi_{\Gamma,s}\langle\partial_s u_t\rangle_\Gamma
+
\chi_{\Gamma,\kappa}\langle U_t\kappa_{\Gam}\rangle_\Gamma .

definition: Reduction convention on the identification window. For a fixed geometry family, let

\Cid=
(\Omid,\lVert\cdot\rVert_{\mathrm{w},\Omid},\mathcal F_{\mathrm{flt}},C_L^{(0)},\varepsilon_* )

denote the reduction convention on the identification window, consisting of a comparison window, a fixed weighted $L^2$TeXL^2 norm on that window,

\lVert f\rVert_{\mathrm{w},\Omid}^2:=\int_{\Omid} w(t)\,|f(t)|^2\,dt,
\qquad
w(t)>0\ \text{for a.e. } t\in\Omid,

a filtering and detrending rule, the attached-flow baseline, and a reduction tolerance. All elements of $\Cid$TeX\Cid are fixed before parameter extraction. Changing any of them defines a different comparison class.

proposition: Projected effective-incidence reduction on the identification window. On a fixed convention $\Cid$TeX\Cid, define the projected coefficient

\chi_\Gamma
:=
\operatorname*{arg\,min}_{\chi\in\mathbb R}
\lVert \bar S_\Gamma-\chi\dot\alphaeff\rVert_{\mathrm{w},\Omid}^2,

Equivalently, on non-null identification windows,

\chi_\Gamma
=
\frac{\langle \bar S_\Gamma,\dot\alphaeff\rangle_{\mathrm w,\Omid}}
{\langle \dot\alphaeff,\dot\alphaeff\rangle_{\mathrm w,\Omid}},
\qquad
\langle f,g\rangle_{\mathrm w,\Omid}:=
\int_{\Omid} w(t)f(t)g(t)\,dt,

provided $\lVert\dot\alphaeff\rVert_{\mathrm w,\Omid}>0$TeX\lVert\dot\alphaeff\rVert_{\mathrm w,\Omid}>0. Define the normalized reduction-entry residual by

\Rslv(t):=\bar S_\Gamma(t)-\chi_\Gamma\dot\alphaeff(t),
\qquad
\epsslv
:=
\frac{\lVert \Rslv\rVert_{\mathrm{w},\Omid}}{\lVert \chi_\Gamma\dot\alphaeff\rVert_{\mathrm{w},\Omid}},

on benchmark windows satisfying $\lVert \chi_\Gamma\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0$TeX\lVert \chi_\Gamma\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0.

The fixed weighted least-squares projection also satisfies the orthogonality relation

\langle \Rslv,\dot\alphaeff\rangle_{\mathrm w,\Omid}=0 .

Because the weight satisfies $w(t)>0$TeXw(t)>0 almost everywhere on $\Omid$TeX\Omid, the admitted identification window excludes the null-input case $\dot\alphaeff\equiv 0$TeX\dot\alphaeff\equiv 0, and the convention uses the fixed weighted $L^2$TeXL^2 norm of reference, the minimizer in reference is unique on the fixed convention $\Cid$TeX\Cid. If the projected forcing vanishes on the identification window, the one-state reduction is not admitted and only the averaged law reference remains available. If $\epsslv\le \varepsilon_*$TeX\epsslv\le \varepsilon_* and the correction budget $O(\dsep+\dthree+\dfam)$TeXO(\dsep+\dthree+\dfam) remains within the fixed family tolerance on $\Omid$TeX\Omid, then the reduced law reference is admissible on that family window. If $\epsslv>\varepsilon_*$TeX\epsslv>\varepsilon_*, the admissible statement is only the averaged law reference; retaining reference by retuning $\chi_\Gamma$TeX\chi_\Gamma, $\tau_M$TeX\tau_M, the norm, the filter, or the baseline counts as failure of the one-state closure on that family.

proof. Let

\langle f,g\rangle_{\mathrm{w},\Omid}
:=
\int_{\Omid}w(t)f(t)g(t)\,dt .

Since $w(t)>0$TeXw(t)>0 almost everywhere and the admitted window has $\lVert\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0$TeX\lVert\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0, the one-dimensional weighted least-squares projection is unique and has the closed form reference. Thus

\bar S_\Gamma=\chi_\Gamma\dot\alphaeff+\Rslv,
\qquad
\langle\Rslv,\dot\alphaeff\rangle_{\mathrm{w},\Omid}=0,

and the normalized residual is exactly the quantity in reference. Equation reference is the fixed-family projection identity used by the reduced comparison. If $\epsslv\le\varepsilon_*$TeX\epsslv\le\varepsilon_*, the tangential and curvature content not captured by the projected effective-incidence input is subleading on $\Cid$TeX\Cid, so the one-state reduction reference is admitted up to the stated correction budget. If the residual exceeds the tolerance, the reduction-entry condition fails and only the averaged law reference remains admitted.

remark: Closed quantities for the reduced comparison. On admitted one-state windows, the reduced comparison is determined by the fixed scalar quantities

\chi_\Gamma,
\qquad
\epsslv,
\qquad
\Ahysex(\deM),
\qquad
\omega_{\mathrm{pk}}\tau_M,
\qquad
\Delta\alpha_{\mathrm{ds}}/(\dot\alphaeff\tau_M),
\qquad
\Lw/(U_c\tau_M).

Each quantity is evaluated under the fixed conventions $\Cid$TeX\Cid and $\Cpcw$TeX\Cpcw. Changing the window, weight, filter, baseline, or readout defines a different comparison class rather than a refinement of the same reduced chain.

proposition: Admitted one-state reduction on fixed-family identification windows. On a fixed section family with weak spanwise drift and controlled three-dimensional corrections, the averaged memory law is

\dot{\bar M} + \tau_M^{-1}\bar M
=
\bar S_\Gamma(t)
+
O(\dsep+\dthree+\dfam).

For harmonic or monotone-ramp motions on the fixed convention $\Cid$TeX\Cid satisfying the reduction-entry condition of reference, this becomes

\dot{\bar M} + \tau_M^{-1}\bar M
=
\chi_\Gamma\,\dot\alphaeff
+
O(\dsep+\dthree+\dfam),

where $\chi_\Gamma$TeX\chi_\Gamma is a fixed effective source coefficient. The corresponding lift law becomes

C_L
=
C_L^{(0)}[\alphaeff,k]
+
a_M\bar M
+
O(\dsep+\dthree+\dfam),

with $C_L^{(0)}$TeXC_L^{(0)} the fixed attached-flow baseline and $k$TeXk the reduced-frequency argument when relevant. The associated baseline-subtracted excess load is

\Delta C_L
\equiv
C_L-C_L^{(0)}[\alphaeff,k]
=
a_M\bar M
+
O(\dsep+\dthree+\dfam).

remark: Reduction scope. Equation reference is admitted only on families for which the tangential and curvature channels are subleading relative to the effective-incidence forcing at leading order in the sense of reference. If $\epsslv>\varepsilon_*$TeX\epsslv>\varepsilon_* on the benchmark window, the averaged law reference is retained.

remark: Baseline scope in loop observables. The fixed attached-flow baseline $C_L^{(0)}$TeXC_L^{(0)} may itself carry family-specific unsteady phase effects. The scalar-memory reduction organizes the excess load $\Delta C_L$TeX\Delta C_L relative to that baseline, and all loop-area statements below refer to the baseline-subtracted response.

remark: Relation to delay-state aerodynamic models. Equation reference shares the single-state relaxation form of Goman--Khrabrov-type models [citation], but here it is obtained by section averaging the field law reference. State-space reduced models furnish alternative low-order representations of unsteady lift [citation]; the present reduction is additionally constrained by wake transport and cross-observable tests. The comparison with Goman--Khrabrov-type or state-space reductions is benchmark-relative only and is not promoted here to a universal replacement theorem for reduced-order unsteady-aerodynamic models.

Harmonic-response law and excess hysteresis area

proposition: Harmonic memory response. For harmonic effective incidence

\alphaeff(t)=\alpha_0+\alpha_1\sin(\omega t),

with $\deM\equiv \omega\tau_M$TeX\deM\equiv \omega\tau_M, the reduced memory state is

\bar M(t)
=
\chi_\Gamma\alpha_1
\frac{\deM^2\sin(\omega t)+\deM\cos(\omega t)}{1+\deM^2}
+
O(\dsep+\dthree+\dfam).

Hence the baseline-subtracted excess load--incidence hysteresis area is

\Ahysex
=
\oint \Delta C_L\,d\alphaeff
=
\pi a_M\chi_\Gamma\alpha_1^2\frac{\deM}{1+\deM^2}
+
O(\dsep+\dthree+\dfam).

proof. Substituting reference into reference and solving the linear first-order equation gives reference. The part of $\bar M$TeX\bar M in phase with $\alphaeff$TeX\alphaeff modifies the effective stiffness of the excess load curve, while the quadrature part generates excess loop area. Inserting reference into reference and integrating over one forcing period yields reference.

remark: Meaning of reference. The observable $\Ahysex$TeX\Ahysex isolates the memory-induced quadrature response relative to the fixed attached-flow baseline. If the memory time vanishes, the quadrature part disappears and so does the excess loop area.

corollary: Peak-frequency identification of the memory time. At fixed forcing amplitude $\alpha_1$TeX\alpha_1, the excess loop area $\text{ref}$TeX\cref{eq:Ahys} is extremized at

\deM=1
\qquad\Longleftrightarrow\qquad
\omega_{\mathrm{pk}}=\tau_M^{-1},

with maximal magnitude

|\Ahysex|_{\max}=\frac{\pi}{2}|a_M\chi_\Gamma|\alpha_1^2.

proof. Let

h(\deM)=\frac{\deM}{1+\deM^2},\qquad \deM>0.

Then

h'(\deM)=\frac{1-\deM^2}{(1+\deM^2)^2},
\qquad
h''(1)<0.

Thus the unique positive stationary point is $\deM=1$TeX\deM=1, where the loop-area factor has its maximum magnitude. Substituting $\deM=1$TeX\deM=1 into reference gives reference.

remark: Identification from harmonic data. A family of moderate-amplitude harmonic tests identifies $\tau_M$TeX\tau_M from the frequency at which the baseline-subtracted excess loop area is maximal, before stall-onset or wake-decay data are used.

Dynamic-stall and reattachment delay

definition: Attachment-support indicator. Near the static stall threshold, define the attachment-support indicator

\Aatt(t)
=
\alphaqs-\alphaeff(t)+\sigma_M\bar M(t),

where $\alphaqs$TeX\alphaqs is the static critical angle of the same family and $\sigma_M>0$TeX\sigma_M>0 is a fixed support coefficient. Equation reference is the minimal affine support functional consistent with the admitted one-state reduction on a fixed family window; any fixed monotone rescaling defines an equivalent thresholding convention without changing the baseline dependence on $\dot\alphaeff\tau_M$TeX\dot\alphaeff\tau_M. At baseline order, attached or weakly separated flow persists while $\Aatt>0$TeX\Aatt>0, and large-scale separation begins when $\Aatt\le 0$TeX\Aatt\le 0; the sign convention is aligned with leading-edge suction/support diagnostics used to mark unsteady separation onset [citation].

proposition: Baseline dynamic-stall delay scaling on admitted ramp windows. For monotone ramp-up motions on one fixed family on which $\dot\alphaeff$TeX\dot\alphaeff is approximately constant over the pre-stall window and the memory state has entered its moderate-ramp plateau, the excess stall angle obeys the baseline scaling

\Delta\alpha_{\mathrm{ds}}
\equiv
\alpha_{\mathrm{ds}}-\alphaqs
\sim
\sigma_M\chi_\Gamma\dot\alphaeff\,\tau_M.

The scaling is stated for a benchmark family with fixed static threshold and correction budget. During pitch-down, the same indicator predicts delayed reattachment because the memory term changes sign more slowly than the incidence, so that an angle below $\alphaqs$TeX\alphaqs is necessary but not sufficient for immediate recovery.

proof. In the moderate-ramp regime, the section-averaged state solves $\dot{\bar M}+\tau_M^{-1}\bar M\approx \chi_\Gamma\dot\alphaeff$TeX\dot{\bar M}+\tau_M^{-1}\bar M\approx \chi_\Gamma\dot\alphaeff, so after a short transient $\bar M\sim \chi_\Gamma\dot\alphaeff\tau_M$TeX\bar M\sim \chi_\Gamma\dot\alphaeff\tau_M. Substituting this into reference and imposing $\Aatt=0$TeX\Aatt=0 at onset gives reference at scaling level. On pitch-down, the sign of $\dot\alphaeff$TeX\dot\alphaeff reverses but the memory field relaxes over the finite time $\tau_M$TeX\tau_M, so the support indicator does not switch sign instantaneously; the resulting reattachment threshold lies below the static threshold until the memory has decayed.

remark: Cross-family normalization. For cross-family comparisons, recent Goman--Khrabrov analyses suggest replacing the raw rate $\dot\alphaeff$TeX\dot\alphaeff by a normalized instantaneous pitch-rate variable [citation]. That normalization is not built into the baseline closure; the constitutive scaling reference is retained on one fixed family.

remark: Why reattachment is delayed. Time-resolved measurements place reattachment below the static threshold and show staged recovery rather than an instantaneous switch [citation].

Wake persistence as transported memory

proposition: Wake-memory length on admitted convective wake windows. After the moving boundary has passed and the local source term is negligible, the far-wake memory field satisfies the transport--relaxation equation

\partial_t M + U_c\partial_x M = -\tau_M^{-1}M,

where $U_c$TeXU_c is the prescribed convective speed on the wake window. For wake segments convected close to the free stream, $U_c\approx U_\infty$TeXU_c\approx U_\infty at leading order. For a statistically steady convected wake, the downstream memory envelope is

M(x)\propto e^{-x/\Lw},
\qquad
\Lw = U_c\tau_M.

proof. Equation reference follows from reference when the source is negligible and the far-wake convection speed is represented by the prescribed convective speed $U_c$TeXU_c on the wake window under test. A steady convected profile $M(x)$TeXM(x) then satisfies $U_c M'(x)=-\tau_M^{-1}M(x)$TeXU_c M'(x)=-\tau_M^{-1}M(x), whose solution is reference; when the wake is convected close to the free stream one may further set $U_c\approx U_\infty$TeXU_c\approx U_\infty at leading order.

remark: Wake as reorganized memory. The same leading time scale that organizes excess load hysteresis also organizes convective wake decay. When the convective speed can be measured directly, $U_c$TeXU_c should be used in reference rather than defaulting automatically to $U_\infty$TeXU_\infty.

corollary: Cross-observable prediction from one memory time. If $\tau_M$TeX\tau_M is identified from $\omega_{\mathrm{pk}}^{-1}$TeX\omega_{\mathrm{pk}}^{-1} through reference on the declared harmonic identification window, and if the ramp and wake windows remain in the same admitted family with the same fixed reduction convention and no observable-wise retuning, then the fixed-family reduction exports the predictions

\Delta\alpha_{\mathrm{ds}}
\sim
\sigma_M\chi_\Gamma\frac{\dot\alphaeff}{\omega_{\mathrm{pk}}},
\qquad
\Lw
\sim
\frac{U_c}{\omega_{\mathrm{pk}}},

up to the stated correction budget. Equivalently, on an admitted wake window with measured or separately prescribed $U_c$TeXU_c, a measurement of $\Lw$TeX\Lw gives the leading-order cross-check $\omega_{\mathrm{pk}}\sim U_c/\Lw$TeX\omega_{\mathrm{pk}}\sim U_c/\Lw for the same $\tau_M$TeX\tau_M. If the ramp or wake window leaves the admitted family, or if the wake speed, baseline, filter, norm, readout convention, or $\tau_M$TeX\tau_M is refitted on the wake window, the relation is a re-identification and not a cross-observable prediction of the one-memory law.

The same memory layer is then tested against baseline-subtracted excess load hysteresis, stall delay, delayed recovery, and the far wake.

Family-level identification proceeds in two stages: reduction-entry admissibility is checked first on the fixed convention $\Cid$TeX\Cid, and only admitted windows proceed to harmonic identification and cross-window testing. Because the same field law exports the same memory time into baseline-subtracted excess load loops, stall delay, delayed reattachment, and wake decay, the closure can be tested at family level. The quantitative comparison uses two coupled objects: the reduction-entry witness

(\chi_\Gamma,\Rslv,\epsslv)

on the fixed convention $\Cid$TeX\Cid, and the fixed-family residual package

\left(
\omega_{\mathrm{pk}}\tau_M-1,\;
\Ahysex-\pi a_M\chi_\Gamma\alpha_1^2\frac{\deM}{1+\deM^2},\;
\Delta\alpha_{\mathrm{ds}}-\sigma_M\chi_\Gamma\dot\alphaeff\tau_M,\;
\Lw-U_c\tau_M,\;
\Rpcw
\right).

Both objects are read only after $\Cid$TeX\Cid, $\Cpcw$TeX\Cpcw, the attached-flow baseline, the filter, the weighted norm, and the admitted family windows have been fixed. A successful comparison is the simultaneous satisfaction of the reduction-entry witness and this residual package on one admitted family under one identified $\tau_M$TeX\tau_M, not a collection of observable-wise fits.

Independent identification and cross-window test..

The harmonic data used to identify $\tau_M$TeX\tau_M are not reused as an independent test of the ramp or wake predictions. Once $\tau_M$TeX\tau_M, $\chi_\Gamma$TeX\chi_\Gamma, $a_M$TeXa_M, the attached-flow baseline $C_L^{(0)}$TeXC_L^{(0)}, the filter, the weighted norm, and the readout convention have been fixed on the declared identification window, the ramp, reattachment, wake, and pressure--circulation--wake checks are evaluated without observable-wise retuning. If the same data segment is reused for both extraction and cross-window testing without a declared data split or holdout, the result is an internal consistency diagnostic rather than a family-level closure test.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

definition: Pressure--circulation--wake readout. For a fixed family and normalization time $T$TeXT, let

\Cpcw=
(\mathcal M_p,\mathcal M_c,\mathcal M_w,
\mathcal F_{\mathrm{pcw}},\mathcal R_t,
\Sigma_{\mathrm{sgn}}^{\mathrm{rule}},
\Delta_{pc}^{(0)},\Delta_{cw}^{(0)},T,\epspcw)

be the fixed readout consisting of pressure, circulation-sensitive, and wake markers $(\mathcal M_p,\mathcal M_c,\mathcal M_w)$TeX(\mathcal M_p,\mathcal M_c,\mathcal M_w), a fixed filter and lag estimator $\mathcal F_{\mathrm{pcw}}$TeX\mathcal F_{\mathrm{pcw}}, a reference-time rule $\mathcal R_t$TeX\mathcal R_t, a sign-comparison rule $\Sigma_{\mathrm{sgn}}^{\mathrm{rule}}$TeX\Sigma_{\mathrm{sgn}}^{\mathrm{rule}}, two admissible benchmark lag offsets $\Delta_{pc}^{(0)}$TeX\Delta_{pc}^{(0)} and $\Delta_{cw}^{(0)}$TeX\Delta_{cw}^{(0)}, the normalization time $T$TeXT, and the tolerance $\epspcw$TeX\epspcw. Given data reduced under $\Cpcw$TeX\Cpcw, let

\delta t_p^{\mathrm{obs}},\qquad
\delta t_c^{\mathrm{obs}},\qquad
\delta t_w^{\mathrm{obs}}

be the observed lag outputs and let $\Sigma_{\mathrm{sgn}}^{\mathrm{obs}}\in[0,1]$TeX\Sigma_{\mathrm{sgn}}^{\mathrm{obs}}\in[0,1] be the observed sign-mismatch fraction. Define the consistency residual

\Rpcw
:=
\max\!\left\{
\frac{\left|\bigl(\delta t_p^{\mathrm{obs}}-\delta t_c^{\mathrm{obs}}\bigr)-\Delta_{pc}^{(0)}\right|}{T},\;
\frac{\left|\bigl(\delta t_c^{\mathrm{obs}}-\delta t_w^{\mathrm{obs}}\bigr)-\Delta_{cw}^{(0)}\right|}{T},\;
\Sigma_{\mathrm{sgn}}^{\mathrm{obs}}
\right\}.

All objects in $\Cpcw$TeX\Cpcw are fixed before cross-observable testing and are held fixed thereafter. The maximum in reference is a fixed conservative aggregation rule for the readout $\Cpcw$TeX\Cpcw; another fixed equivalent norm would define a different readout convention without changing the constitutive content of the closure. When direct circulation reconstruction is unavailable, $\mathcal M_c$TeX\mathcal M_c may be an admitted circulation-sensitive proxy whose sign and lag ordering have been benchmarked not to reverse the corresponding circulation ordering on the benchmark window.

Pressure--circulation--wake consistency gate..

If near-field pressure asymmetry, circulation-sensitive excess load, and wake persistence are projections of the same reorganized state on a fixed family, then under one fixed readout $\Cpcw$TeX\Cpcw the observed sign pattern and pairwise lag offsets satisfy

\Rpcw = O(\dsep+\dthree+\dfam).

At baseline order the three observables are distinct projections of the same reorganized state, so a fixed readout can constrain only their sign pattern and pairwise lag offsets up to the stated separation, three-dimensional, and family corrections. The constants $\Delta_{pc}^{(0)}$TeX\Delta_{pc}^{(0)} and $\Delta_{cw}^{(0)}$TeX\Delta_{cw}^{(0)} encode admissible benchmark offsets rather than exact lag equality. If the circulation-sensitive channel is represented by an admitted proxy $\mathcal M_c$TeX\mathcal M_c, the statement refers to the prebenchmarked proxy offsets and sign ordering rather than to an independently reconstructed circulation field. Equation reference packages the deviations from those fixed offsets into a single residual. If $\Rpcw$TeX\Rpcw exceeds $\epspcw$TeX\epspcw without leaving the family correction budget, the one-state memory closure fails on that family.

Family test sequence..

For a fixed geometry family in the admissible regime, after verifying on the fixed convention $\Cid$TeX\Cid that the reduction-entry residual satisfies $\epsslv\le \varepsilon_*$TeX\epsslv\le \varepsilon_*, the scalar-memory closure is tested as follows:

- Harmonic window: subtract the fixed attached-flow baseline $C_L^{(0)}$TeXC_L^{(0)}, measure the excess load--incidence loops under moderate harmonic forcing, and infer $\tau_M$TeX\tau_M from $\omega_{\mathrm{pk}}^{-1}$TeX\omega_{\mathrm{pk}}^{-1} using reference; - Ramp window: use the same $\tau_M$TeX\tau_M in reference and in the attachment-support law to predict dynamic-stall delay and delayed reattachment; - Wake window: use the same $\tau_M$TeX\tau_M in reference to predict the far-wake e-folding length, measuring $U_c$TeXU_c directly when the wake window permits it and using $U_c\approx U_\infty$TeXU_c\approx U_\infty only when that approximation is justified; - Cross-channel check: when surface-pressure and circulation- or impulse-sensitive estimates are available, reduce all three channels under the fixed readout $\Cpcw$TeX\Cpcw, compute the observed lag offsets and sign-mismatch fraction, and require $\Rpcw\le \epspcw$TeX\Rpcw\le \epspcw up to the stated correction budget.

If any of these steps requires observable-wise retuning of $\tau_M$TeX\tau_M, $\chi_\Gamma$TeX\chi_\Gamma, the norm, the filter, the baseline, or the readout inside the same fixed family, or if the cross-channel check fails without leaving the stated correction budget, the scalar-memory closure fails for that family.

The sequence separates harmonic identification from ramp, wake, and cross-channel checks. Harmonic baseline-subtracted excess loops determine $\tau_M$TeX\tau_M. Ramp and wake data then test whether the same $\tau_M$TeX\tau_M survives across a different forcing class and a different observable class. When dense pressure or near-wake measurements are available, they also provide a direct check of lag offsets and sign ordering [citation].

Thin-airfoil harmonic--ramp--wake benchmark

definition: Reference thin-airfoil family. Let $\Famthin$TeX\Famthin denote a fixed family consisting of one fixed thin or moderately thick two-dimensional aerofoil section, one fixed pitch axis, one fixed Reynolds--Mach regime, one attached-flow baseline $C_L^{(0)}$TeXC_L^{(0)}, one reduction convention $\Cid$TeX\Cid, and one wake readout. The family contains three benchmark windows: a moderate harmonic window for excess-loop identification, a monotone ramp-up and pitch-down window crossing the same static threshold, and a downstream wake window on which the convective speed $U_c$TeXU_c is measured or independently specified. Experimental families of this type already exist in harmonic and ramp dynamic-stall data, in time-resolved pressure and velocity measurements of reattachment, and in near-wake diagnostics [citation].

Within $\Famthin$TeX\Famthin, the identification step is fixed by the baseline-subtracted excess loop family $\Ahysex(\omega)$TeX\Ahysex(\omega) on $\Omid$TeX\Omid. The peak frequency $\omega_{\mathrm{pk}}$TeX\omega_{\mathrm{pk}} determines $\tau_M=\omega_{\mathrm{pk}}^{-1}$TeX\tau_M=\omega_{\mathrm{pk}}^{-1}. The same $\tau_M$TeX\tau_M is then exported to the ramp and wake windows:

\Delta\alpha_{\mathrm{ds}}^{\mathrm{pred}}
=\sigma_M\chi_\Gamma\dot\alphaeff\,\tau_M,
\qquad
\Lw^{\mathrm{pred}}=U_c\tau_M,

while delayed reattachment is read from the same attachment-support law reference with no change of $\tau_M$TeX\tau_M. Agreement on the fixed family requires that the harmonic reduction-entry residual, the ramp delay, the reattachment delay, the wake length, and the pressure--circulation--wake residual all satisfy the same fixed readouts and windows.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The quasi-steady recovery of reference returns the standard pressure-plus-viscous load representation and the fixed attached-flow unsteady baseline. The construction remains classical and continuum-mechanical throughout; it introduces no event-counting, latching, or record-bearing variable. The added state enters as a constitutive extension of classical unsteady aerodynamics [citation]. Where attached-flow circulation theory is applicable, the reduced correction $a_M\bar M$TeXa_M\bar M is interpreted as a delayed correction to that baseline. Because the fixed attached-flow baseline may itself carry family-specific unsteady phase effects, the scalar-memory reduction is tested through the excess load $\Delta C_L$TeX\Delta C_L and the excess loop area $\Ahysex$TeX\Ahysex, not through the total loop area.

The reduced law reference shares the single-state relaxation structure of Goman--Khrabrov-type delay models [citation], and recent analyses support the use of identifiable delay scales only on restricted regimes and fixed kinematic families [citation]. State-space reduced models furnish alternative low-order representations of unsteady lift [citation]. Here the reduced state is inherited from the field law and is tested through a common export of the leading time scale to excess load, stall delay, delayed reattachment, and wake decay.

Recent measurements and simulations of laminar-bubble bursting, disturbance-sensitive stall onset, vorticity-distribution effects, synchronous surging and pitching, and delayed reattachment provide benchmark families with coupled near-boundary and wake dynamics [citation]. Families requiring multiple leading time scales, tensorial memory, explicit spanwise dynamics at baseline order, or an independent wake-side state fall outside the one-state closure.

Reduction-entry test..

The one-state reduction may be used only on identification windows satisfying $\lVert \chi_\Gamma\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0$TeX\lVert \chi_\Gamma\dot\alphaeff\rVert_{\mathrm{w},\Omid}>0 and $\epsslv\le\varepsilon_*$TeX\epsslv\le\varepsilon_* on one fixed convention $\Cid$TeX\Cid. If the projected forcing vanishes on $\Omid$TeX\Omid, or if $\epsslv>\varepsilon_*$TeX\epsslv>\varepsilon_*, the admissible statement is only the averaged law reference. Any attempt to restore reference by retuning $\chi_\Gamma$TeX\chi_\Gamma, $\tau_M$TeX\tau_M, the norm, the filter, the baseline, or the window inside the same family counts as failure of the one-state closure.

Recovery test..

When $\tau_M\to 0$TeX\tau_M\to 0 and $\mu_M\to 0$TeX\mu_M\to 0, the closure must reduce to the standard classical benchmark for the same geometry family. If it does not recover the standard quasi-steady or attached-flow unsteady baseline, it fails immediately.

Constitutive-admissibility test..

The admitted regime must satisfy the effective dissipation-floor condition $2\mu+\mu_M M\ge \nuad>0$TeX2\mu+\mu_M M\ge \nuad>0. If fitting the data requires systematic excursions outside that window while keeping the scalar baseline form, the constitutive closure is rejected.

Source-class test..

The localized source law must remain within the admissible class of reference. If a family can be matched only by introducing leading nonlocal source terms, direct lift forcing, or object-wise source coefficients outside the stated correction budget, the baseline source class fails.

Family-parameter invariance..

The coefficients $(\chi_\alpha,\chi_s,\chi_\kappa,\tau_M,\mu_M,a_M,\sigma_M)$TeX(\chi_\alpha,\chi_s,\chi_\kappa,\tau_M,\mu_M,a_M,\sigma_M), together with the conventions $\Cid$TeX\Cid and $\Cpcw$TeX\Cpcw, must be fixed for one geometry family and must not be retuned cycle by cycle or observable by observable. If object-wise or trajectory-wise retuning is required to maintain agreement, the closure is rejected on that family.

Hysteresis-collapse test..

For a fixed family under moderate harmonic forcing, on identification windows satisfying the reduction-entry condition $\epsslv\le\varepsilon_*$TeX\epsslv\le\varepsilon_*, the baseline-subtracted excess loop area must collapse according to reference when plotted against the memory Deborah number $\deM$TeX\deM, up to the stated correction budget. Persistent deviations outside that budget reject either the one-state reduction or the scalar-memory hypothesis on the fixed family.

Cross-observable identification test..

A memory time identified from baseline-subtracted excess harmonic loops must also organize dynamic-stall delay and wake-memory length through reference, and the same fixed conventions must keep the pressure--circulation--wake residual within the tolerance $\epspcw$TeX\epspcw. If the load, stall, and wake data require unrelated leading time scales, or if the residual $\Rpcw$TeX\Rpcw fails inside the same fixed family, the one-field closure fails.

Stall-delay test..

For monotone ramp-up motions on one fixed family inside the admitted regime, the dynamic-stall delay must scale with $\dot\alphaeff\tau_M$TeX\dot\alphaeff\tau_M at baseline order as in reference. If stall delay is independent of the identified memory time or depends on motion history in a way that cannot be absorbed into the stated correction budget, the proposed attachment-support law fails.

Wake-persistence test..

Downstream memory observables must decay on the length scale $\Lw\sim U_c\tau_M$TeX\Lw\sim U_c\tau_M at baseline order, with the convective speed prescribed for the wake window under test. If wake persistence requires a completely independent leading time scale unrelated to the load-hysteresis time scale, the single-memory-field closure fails.

Reattachment test..

During pitch-down, an angle below the static critical angle must be necessary but not sufficient for immediate recovery whenever the memory field remains finite. If recovery is entirely instantaneous in a fixed family that otherwise shows strong lift hysteresis, the proposed memory interpretation is contradicted.

Domain limitation..

The scalar-memory baseline applies only to the moving-boundary regime. Deep fully three-dimensional stall, multi-body wake interference, or object families requiring multiple leading memory scales lie outside the baseline claim.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

A localized boundary-reorganization source, a transported memory field, and a constitutive memory stress furnish a family-bound classical continuum prototype for unsteady lift excess, dynamic-stall delay, delayed reattachment, and wake persistence around prescribed moving boundaries on the admitted fixed-family windows studied here. Inside the declared first-order local source basis, boundary localization and frame indifference retain the effective-incidence, tangential-reorganization, and curvature-asymmetry channels, while the kinetic--memory budget yields the dissipation floor $2\mu+\mu_M M\ge \nuad>0$TeX2\mu+\mu_M M\ge \nuad>0 on the admitted regime. Section averaging yields an averaged source law on every fixed family window, but a one-state reduction is admitted only when the projected effective-incidence forcing is non-null and the reduction-entry witness satisfies $\epsslv\le \varepsilon_*$TeX\epsslv\le \varepsilon_* on one fixed convention.

On admitted harmonic, monotone-ramp, and wake windows of one fixed family, a single $\tau_M$TeX\tau_M organizes, at baseline order, baseline-subtracted excess hysteresis area, stall delay, delayed recovery, and wake decay, while the pressure--circulation--wake residual $\Rpcw$TeX\Rpcw checks cross-observable lag and sign consistency between pressure, circulation-sensitive excess load, and wake persistence. No claim is made that unsteady moving-boundary families generally admit a one-state memory closure. Accordingly, on admitted fixed-family windows one leading memory time is exported consistently across load, stall, recovery, and wake observables without observable-wise retuning, and the one-state closure is admitted only when the reduction-entry residual, the harmonic identification, the stall-delay scaling, the wake-memory length, and the pressure--circulation--wake residual are simultaneously satisfied as one fixed-family residual package.

The present result remains a family-bound continuum prototype. It introduces no event-counting, latching, or record-bearing variable, does not define a new material species, does not provide a universal turbulence or fully three-dimensional separation closure, and does not claim a class-free memory law for arbitrary moving-boundary flows.

Baseline order excludes four additions. Direct load forcing is excluded because the claim is a field-mediated export to both surface-load and wake observables. Leading nonlocal source terms are excluded because they introduce an independent history kernel or spatial support larger than the benchmark thin boundary layer. Antisymmetric or purely tensorial memory stresses are excluded because the Cauchy stress is symmetric and the baseline state is scalar. An independent wake-side source is excluded because it would introduce a second leading state. Higher-order, nonlocal, or multi-state extensions are not treated here.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.