Generalised Goman-Khrabrov Dynamic Stall Model

• The Generalised Goman-Khrabrov model is a physics-based dynamic stall model that replaces empirical time constants with delay and vortex-shedding scales derived from flow physics.
• It models the transition between attached and separated flow using a first-order state-space formulation, capturing key dynamic stall phenomena in airfoil performance.
• Validation across various airfoils and pitching kinematics shows the model’s improved universality, though limitations remain for highly irregular or nonmonotonic motions.

Searching arXiv for the generalized Goman–Khrabrov dynamic stall model and closely related work. First, locating the core generalization paper. The generalised Goman–Khrabrov model is a physics-based extension of the Goman–Khrabrov (GK) dynamic stall model in which the original empirical time constants are replaced, or further reinterpreted, using physically derived time scales tied to dynamic stall delay and post-stall vortex-shedding dynamics. In its original form, the GK model is a nonlinear first-order state-space model for airfoil dynamic stall, especially cases dominated by trailing-edge separation and vortex breakdown at high angles of attack. The generalised form preserves the single-state GK structure but seeks broader transferability across airfoils, Reynolds numbers, and pitching kinematics by grounding the delay parameters in identifiable flow physics rather than case-by-case empirical fitting (Ayancik et al., 2021). Subsequent work has shown that this generalisation is effective for linear and some nonlinear monotonic pitch-up motions, while also identifying limitations in the standard effective-angle construction when the pitch rate varies in time (Rezapour et al., 14 Aug 2025).

1. Origin in the Goman–Khrabrov dynamic stall model

The original Goman–Khrabrov model uses a single internal state variable $X(t)$, interpreted as the degree of flow attachment on the airfoil, with $X=1$ corresponding to fully attached flow and $X=0$ corresponding to fully separated flow (Ayancik et al., 2021). Its core evolution equation is

$$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$

where $\tau_1$ is a relaxation time constant, $\tau_2$ is a stall delay time constant, $X_0(\alpha)$ is the static separation-point location as a function of angle of attack, $\alpha(t)$ is the instantaneous angle of attack, and $\dot{\alpha}(t)$ is the instantaneous pitch rate (Ayancik et al., 2021). Once $X(t)$ is known, the lift coefficient is computed using Kirchhoff’s law:

$X=1$0

In the original formulation, the central limitation is that the model requires two empirical parameters, $X=1$1 and $X=1$2, which must be tuned from experiments or simulations. The static function $X=1$3 is obtained by fitting the static lift curve, and then $X=1$4 and $X=1$5 are selected empirically to best match unsteady data. These fitted values are often treated as constant for a given airfoil over many pitching frequencies (Ayancik et al., 2021). The generalised GK model is motivated by the view that this empirical calibration is the main reason the original GK model is not fully generalisable.

2. Physics-based generalisation of the time constants

The central contribution of "All you need is time to generalise the Goman-Khrabrov dynamic stall model" (Ayancik et al., 2021) is the replacement of the empirical time constants with physically derived time scales. The paper identifies two processes: the dynamic stall delay, which replaces $X=1$6, and the post-stall decay rate or vortex-shedding time scale, which replaces $X=1$7.

The stall delay constant $X=1$8 is interpreted as the temporal delay between the instant the static stall angle is exceeded and the onset of dynamic stall. The paper defines the dynamic stall angle as

$X=1$9

where $X=0$0 is the static stall angle, $X=0$1 is the dynamic stall angle, and $X=0$2 is the pitch rate at the time the static stall angle is crossed (Ayancik et al., 2021). For constant pitch-rate ramp-up motion, $X=0$3, so the GK stall delay constant is just the physical delay time. For sinusoidal pitching, the paper gives an explicit relation for $X=0$4 as a function of kinematics once $X=0$5 is known (Ayancik et al., 2021).

A central result is the universal fit for the dynamic stall delay:

$X=0$6

Here, $X=0$7 is a normalised effective unsteadiness, and $X=0$8 is the delay time expressed in convective time units (Ayancik et al., 2021). As the normalised pitch rate increases, the dynamic stall delay decreases, and the delay approaches a minimum asymptotic value of $X=0$9 convective time units. That minimum is interpreted as the time required for the vortex formation stage, which is largely independent of motion rate (Ayancik et al., 2021).

The relaxation time constant $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$0 is then reinterpreted as a time scale for the post-stall decay of lift fluctuations driven by repeated large-scale vortex shedding after dynamic stall. The paper gives

$$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$1

and states that, in the discussion, this corresponds to a Strouhal number $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$2, while also noting that $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$3 convective times corresponds to $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$4 (Ayancik et al., 2021). This makes $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$5 independent of motion kinematics and ties it to the characteristic time scale of the separated shear layer and vortex shedding after stall onset.

3. Universality claims and validation domain

The generalised model is validated on three experimental dynamic-stall datasets covering NACA 0015, NACA 0018, and OA209 airfoils, with Reynolds numbers varying from $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$6 to $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$7, and with both sinusoidal pitching motions and linear ramp-up pitching motions (Ayancik et al., 2021). Specific examples mentioned include sinusoidal pitching of the OA209 airfoil at $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$8 and $$\tau_1 \frac{dX(t)}{dt} + X(t) = X_0\!\left(\alpha(t) - \tau_2 \dot{\alpha}(t)\right),$$9, sinusoidal pitching of NACA 0015 at $\tau_1$0, and ramp-up motion of NACA 0015 at $\tau_1$1 (Ayancik et al., 2021).

For the OA209 sinusoidal cases, the paper uses combinations such as mean angle of attack $\tau_1$2, amplitude $\tau_1$3, and reduced frequency $\tau_1$4 (Ayancik et al., 2021). The reported comparison is between the best-fit original GK model and the physics-based GK model with the new time constants. Both versions achieve good overall agreement, with $\tau_1$5 across the tested cases. The physics-based model gives slightly lower global $\tau_1$6 in some cases but predicts the timing of the first lift peak better, reproduces the overall lift evolution well, and captures post-stall decay accurately (Ayancik et al., 2021).

These results underpin the paper’s claim that the GK model has been generalised by removing empirical tuning of its time constants. The new model uses physics-based time scales for dynamic stall delay and post-stall vortex-shedding decay, and those time scales are presented as valid across multiple airfoils, multiple Reynolds numbers, and sinusoidal and ramp-up motions (Ayancik et al., 2021). A plausible implication is that the principal gain in generality lies not in changing the order or structure of the state-space model, but in replacing ad hoc parameter identification with a flow-physics-informed closure.

4. Time-varying pitching kinematics and the limits of the standard generalised form

"Dynamic Stall Characteristics and Modelling of Time-Varying Pitching Kinematics" (Rezapour et al., 14 Aug 2025) revisits the generalised GK model for monotonic nonlinear pitch-up motions and argues that the key challenge is not the universal stall-delay scaling itself, but how unsteady pitching is represented inside the model when the pitch rate varies in time. The study uses a NACA0018 airfoil in the SHARX recirculating water channel at EPFL, with $\tau_1$7, chord $\tau_1$8, span $\tau_1$9, and $\tau_2$0. The airfoil pitches about the quarter-chord from $\tau_2$1 to $\tau_2$2, and the dataset covers 126 cases, repeated five times each (Rezapour et al., 14 Aug 2025).

The paper studies linear pitch-up motions and quadratic pitch-up motions with constant second derivative $\tau_2$3, including accelerating and decelerating motions. Stall onset is identified by the first peak in lift coefficient $\tau_2$4, and the static stall angle for the tripped foil is $\tau_2$5 (Rezapour et al., 14 Aug 2025).

A key experimental result is that dynamic stall is delayed beyond the static stall angle, and that this delay depends primarily on the pitch rate at the moment the static stall angle is crossed, not on whether the motion is linear or nonlinear. For representative cases with identical pitch rate at $\tau_2$6 but different acceleration, decelerating motion stalls earliest at $\tau_2$7, linear motion stalls at $\tau_2$8, and accelerating motion stalls latest at $\tau_2$9 (Rezapour et al., 14 Aug 2025). However, when time is shifted so that $X_0(\alpha)$0 marks the instant the static stall angle is exceeded, the lift peaks align closely in convective time. The paper therefore reports that the stall delay in convective units is nearly invariant across the motion types, even though the stall angle and peak lift vary.

The measured stall-delay data collapse onto the same power law previously found for linear pitch-up motions:

$X_0(\alpha)$1

The relevant pitch rate is defined as

$X_0(\alpha)$2

that is, the instantaneous pitch rate at the moment the geometric angle first exceeds the static stall angle (Rezapour et al., 14 Aug 2025). The authors argue that this is the proper characteristic pitch rate for predicting stall delay because the stall-delay data for nonlinear motions collapse onto the same universal decay law as linear motions when plotted versus $X_0(\alpha)$3.

The same paper reformulates the generalised time scales through a decomposition of the total stall delay,

$X_0(\alpha)$4

with $X_0(\alpha)$5 depending on the pitch rate and $X_0(\alpha)$6 being a kinematics-independent vortex-formation time (Rezapour et al., 14 Aug 2025). The asymptotic relaxation time is identified as

$X_0(\alpha)$7

This differs numerically from the $X_0(\alpha)$8 asymptote reported in the earlier generalisation paper (Ayancik et al., 2021). The data block attributes both values to their respective papers, so the numerical discrepancy should be understood as a difference between the reported scaling laws rather than harmonised model constants.

5. Effective-angle formulation and modified generalised GK model

In the generalised GK model as used in the 2025 study, unsteady pitching is represented through the effective angle of attack

$X_0(\alpha)$9

For linear pitch-up, $\alpha(t)$0 is constant, so this introduces a constant lag and works well. For nonlinear pitch-up, however, $\alpha(t)$1 changes continuously, and the lag term therefore varies during the stall-development window (Rezapour et al., 14 Aug 2025). The paper argues that this standard effective-angle formulation incorrectly lumps together two processes that do not vary in the same way: the reaction delay is kinematic-dependent, whereas the vortex-formation or relaxation delay is an intrinsic physical process that proceeds independently once initiated.

According to the reported validation, the original generalised GK model captures the attached-flow lift evolution well for all motions, predicts stall onset timing accurately for linear pitch motions, and, across the full dataset, both original and modified formulations achieve high overall correlation with experiments, with $\alpha(t)$2 (Rezapour et al., 14 Aug 2025). However, for nonlinear motions, the original generalised GK model mispredicts stall onset timing, post-stall lift evolution, and peak lift timing and magnitude for accelerating and decelerating cases. The paper states more specifically that decelerating motion is predicted with stall too early and post-stall lift deviates significantly, whereas accelerating motion is predicted with stall too late and the post-stall response also deviates. The model’s stall-delay errors can be as large as about one convective time, and these errors grow with motion acceleration (Rezapour et al., 14 Aug 2025).

To address this limitation, the paper proposes splitting the lag in the effective angle of attack into separate reaction and vortex-formation contributions:

$\alpha(t)$3

The interpretation given is that the reaction delay part uses the full time-varying pitch-rate history through $\alpha(t)$4, while the vortex-formation lag is tied to the pitch rate at the stall-triggering instant, $\alpha(t)$5, and is treated as invariant during the later development (Rezapour et al., 14 Aug 2025). This modification preserves the original model’s behavior for linear motions, improves post-stall lift prediction for linear motions slightly, and substantially improves both stall timing and post-stall prediction for nonlinear accelerating and decelerating motions.

This suggests that the principal structural weakness of the standard generalised GK model is not its use of physics-based time scales per se, but the way those scales are injected into the state equation through a single instantaneous-rate lag. The later modification retains the first-order GK architecture while changing the interpretation of the forcing argument supplied to the static separation function.

6. Broader formulations and application-specific extensions

The GK framework has also been extended in application-specific ways outside the time-scale generalisation literature. "Pitch-axis supermanoeuvrability in a biomimetic morphing-wing UAV" (Pons et al., 2022) uses a modified Goman-Khrabrov dynamic stall model for all lifting surfaces in a multibody flight simulator. The model is described as building on prior work by Reich et al. and Pons and Cirak, and it is extended to cover the full flight envelope of the UAV aerofoils, including reverse flow and leading-edge versus trailing-edge behavior, to incorporate control-surface deflection on the stabilizer aerofoil, and to be integrated into a multi-section, multibody flight simulator (Pons et al., 2022).

In that formulation, the force coefficient for each aerodynamic section $\alpha(t)$6 and force type $\alpha(t)$7 is written as a convex blend of attached and separated-flow models:

$\alpha(t)$8

and the mixing parameter evolves according to

$\alpha(t)$9

Here $\dot{\alpha}(t)$0 is the dynamic mixing parameter, $\dot{\alpha}(t)$1 is the relaxation delay, $\dot{\alpha}(t)$2 is the stall-delay or effective-angle-delay parameter, and $\dot{\alpha}(t)$3 is the static mixing function describing the transition between attached and separated flow (Pons et al., 2022). The paper reports that 5 aerodynamic stations per lifting surface are enough for convergence to within 1%, resulting in 25 aerodynamic DOFs plus the 12-DOF multibody dynamics.

The paper adopts

$\dot{\alpha}(t)$4

and states validity limits for the dynamic stall model of maximum reduced frequency $\dot{\alpha}(t)$5 and maximum reduced pitch rate $\dot{\alpha}(t)$6 (Pons et al., 2022). These are not the same as the physics-based generalisation of Ayancik et al.; rather, they constitute a phenomenological GK-style extension tailored to morphing-wing UAV simulation.

The distinction is important. In the dynamic-stall literature, the term generalised Goman–Khrabrov model most directly refers to the replacement of empirical time constants with physics-based delay laws (Ayancik et al., 2021), and to later modifications of that construction for nonlinear kinematics (Rezapour et al., 14 Aug 2025). In UAV simulation, however, “modified” or “generalized” GK often refers more broadly to extensions of the same first-order attachment-state idea to full-envelope aerodynamics, reverse flow, leading-edge and trailing-edge partitioning, control-surface dependence, and multibody coupling (Pons et al., 2022).

7. Interpretation, scope, and limitations

Across the cited works, the generalised GK model remains a first-order, low-order, state-space description of dynamic stall. Its central variable is a scalar representation of attachment or separation, and its principal strength is that it can express delayed transition between attached and separated flow with very low computational cost (Ayancik et al., 2021, Rezapour et al., 14 Aug 2025). The physics-based generalisation does not replace this structure; it seeks to make the delay parameters more transferable by linking them to measurable or broadly universal flow time scales.

The strongest claim supported by the evidence is that the dynamic stall delay can be expressed through a universal scaling based on the pitch rate at the moment the airfoil crosses the static stall angle, and that the post-stall decay can be tied to a kinematics-independent vortex-shedding time scale (Ayancik et al., 2021, Rezapour et al., 14 Aug 2025). A second supported claim is that, for monotonic nonlinear pitch-up motions, the stall-delay timing remains universal when referenced to $\dot{\alpha}(t)$7, but the original generalised effective-angle construction may fail because it uses the instantaneous pitch rate throughout the motion rather than separating the reaction stage from the invariant vortex-formation stage (Rezapour et al., 14 Aug 2025).

The model’s limitations are also explicit. The 2025 study states that the modified effective-angle formulation is intended for monotonic pitch-up motions with smooth nonlinear acceleration or deceleration and is not expected to handle highly irregular pitch histories with plateaus, sharp disturbances, or nonmonotonic patterns during the critical stall-development interval, because first-order models cannot fully capture the vortex dynamics in such cases (Rezapour et al., 14 Aug 2025). The UAV study likewise notes neglected out-of-plane effects, unmodeled attached-flow unsteady effects, incomplete capture of potential static hysteresis for the deflected stabilizer aerofoil, and validity limits set by reduced frequency and pitch-rate thresholds (Pons et al., 2022).

The generalised Goman–Khrabrov model therefore occupies a specific position within dynamic stall modelling. It is more physically grounded and more transferable than the empirically tuned original GK formulation, yet it remains deliberately low-order and phenomenological. Its development shows that substantial gains in generality can be obtained by reinterpreting the time constants in terms of dynamic stall delay and vortex-formation physics, while later work indicates that further generalisation depends on how those time scales are embedded in the effective-angle dynamics rather than on the time scales alone (Ayancik et al., 2021, Rezapour et al., 14 Aug 2025).