Data-assimilated low-order vortex modeling of separated flows

Vortex models have been used for decades as computationally efficient tools to investigate unsteady aerodynamics. However, their utility for separated flows—particularly when such flows are subjected to incident disturbances—has been hindered by the tradeoff between the model's physical fidelity and its expectation for fast prediction (e.g., relative to computational fluid dynamics). In this work, it is shown that physical fidelity and speed can be simultaneously achieved by assimilating measurement data into the model to compensate for unrepresented physics. The underlying inviscid vortex model captures the transport of vortex structures with a standard collection of regularized vortex elements that interact mutually and with an infinitely thin flat plate. In order to maintain a low-dimensional representation, with fewer than $O\left(100\right)$ degrees of freedom, an aggregation procedure is developed and utilized in which vortex elements are coalesced at each time step. A flow state vector, composed of vortex element properties as well as the critical leading-edge suction parameter, is advanced within an ensemble Kalman filter (EnKF) framework. In this framework, surface pressure is used to correct the states of an ensemble of randomly initiated vortex models. The overall algorithm is applied to several scenarios of an impulsively started flat plate, in which data from a high-fidelity Navier-Stokes simulation at Reynolds number 500 are used as a surrogate for the measurements. The assimilated vortex model efficiently and accurately predicts the evolving flow as well as the normal force in both the undisturbed case (a separated flow) as well as in the presence of one or more incident gusts, despite lack of a priori knowledge of the gust's characteristics.

Figure 1

Comparison of vortex element distributions (left) and surface pressure coefficient distributions (right) predicted by a vortex blob model with different ${\text{LESP}}_c$ values after one convective time. [(a), (b)] ${\mathrm{LESP}}_c=3.0$, [(c), (d)] ${\mathrm{LESP}}_c=0.3$, and [(e), (f)] ${\mathrm{LESP}}_c=0.8$. In each surface pressure coefficient distribution panel, the top of the panel represents the leading edge of the plate, the bottom represents the trailing edge, and the horizontal axis represents the time (in convective units).

Figure 2

Comparison of the surface pressure distribution over time along an impulsively translating plate at ${60}^{\circ }$ between (a) high-fidelity simulation, (b) a basic discrete vortex element model, (c) a vortex element model with aggregation, and (d) the proposed EnKF-assisted model. Convective time is defined as $t^{*}=tU/c$.

Figure 3

Comparison of vorticity distribution for the ${60}^{\circ }$ impulsive translation case at $t^{*}=3$ (left column) and $t^{*}=5$ (right column), predicted by [(a), (b)] high-fidelity simulation, [(c), (d)] a basic discrete vortex element model, [(e), (f)] a vortex element model with aggregation, and [(g), (h)] the proposed EnKF-assisted model.

Figure 4

Comparison of the history of the normal force coefficient (normalized by $\rho U^{2}c/2$) on the plate at ${60}^{\circ }$ predicted by the high-fidelity truth simulation ($—$), a basic discrete vortex element model ($—$), a vortex element model with aggregation ($—$), and an EnKF-assisted vortex element model ($—$).

Figure 5

Number of vortex elements used in vortex models of the impulsively translating plate at ${60}^{\circ }$. Basic vortex element model ($—$); aggregated vortex element model ($—$); EnKF-assisted vortex element model ($—$).

Figure 6

Time history of the ensemble mean value of ${\text{LESP}}_c$ for the impulsively translating plate at ${60}^{\circ }$ case.

Figure 7

Ensemble variances for the ${60}^{\circ }$ impulsive translation case. Variances for (a) element $x$ position, (b) element $y$ position, (c) element strength, and (d) critical LESP.

Figure 8

Vorticity distribution for the ${20}^{\circ }$ impulsive translation case at $t^{*}=3$ (left column), $t^{*}=4$ (center column), and $t^{*}=5$ (right column), from [(a)–(c)] high-fidelity truth simulation at Reynolds number 500, [(d)–(f)] the EnKF-assisted model without inflation, and [(g)–(j)] the EnKF-assisted model with inflation.

Figure 9

Histories of the surface pressure distribution along an impulsively translating plate at ${20}^{\circ }$, from (a) high-fidelity truth simulation at Reynolds number 500, (b) EnKF-assisted model without covariance inflation, and (c) EnKF-assisted model with covariance inflation.

Figure 10

Ensemble variances for the ${20}^{\circ }$ impulsively translating plate, with no covariance inflation ($—$) and with covariance inflation ($—$). Variances for (a) element $x$ position, (b) element $y$ position, (c) element strength, and (d) critical LESP.

Figure 11

Comparison of the histories of the normal force coefficient for an impulsively translating plate at ${20}^{\circ }$ from high-fidelity truth simulation ($—$) and an EnKF-assisted vortex model without covariance inflation ($—$) and with covariance inflation ($—$).

Figure 12

Number of vortex elements used in the EnKF-assisted model over time for the impulsively translating plate at ${20}^{\circ }$, without covariance inflation ($—$) and with covariance inflation ($—$).

Figure 13

Time history of the ensemble mean value of ${\text{LESP}}_c$ for the ${20}^{\circ }$ impulsively translating plate, without covariance inflation ($—$) and with covariance inflation ($—$).

Figure 14

Histories of the surface pressure distribution along an impulsively translating plate at ${20}^{\circ }$, subjected to incident gusts at $t^{*}=3$ and $t^{*}=4$, from high-fidelity truth simulation at Reynolds number 500 (top) and the EnKF-assisted model with covariance inflation (bottom).

Figure 15

Vorticity distribution for an impulsively translating plate at ${20}^{\circ }$ subjected to incident gusts, at $t^{*}=3$ (left column), $t^{*}=4$ (center column), and $t^{*}=5$ (right column), predicted by [(a)–(c)] high-fidelity truth simulation at Reynolds number 500, and [(d)–(f)] the EnKF-assisted model with covariance inflation.

Figure 16

Comparison of the histories of the normal force coefficient for an impulsively translating plate at ${20}^{\circ }$ subjected to two incident gusts at $t^{*}=3$ and $t^{*}=4$, from high-fidelity truth simulation ($—$) and an EnKF-assisted vortex model ($—$).

Figure 17

Time history of the ensemble mean value of ${\text{LESP}}_c$ for ${20}^{\circ }$ impulsively translating plate, subjected to incident gusts at $t^{*}=3$ and $t^{*}=4$.

Figure 18

Ensemble variances for the ${20}^{\circ }$ impulsively translating plate, subjected to incident gusts at $t^{*}=3$ and $t^{*}=4$, with no covariance inflation ($—$) and with covariance inflation ($—$). Variances for (a) element $x$ position, (b) element $y$ position, (c) element strength, and (d) critical LESP.