We present applications of modal analysis techniques to study, model, and control canonical aerodynamic flows. To illustrate how modal analysis techniques can provide physical insights in a complementary manner, we selected four fundamental examples of cylinder wakes, wall-bounded flows, airfoil wakes, and cavity flows. We also offer brief discussions on the outlook for modal analysis techniques, in light of rapid developments in data science.

https://arc.aiaa.org/doi/pdf/10.2514/1.J058462

Modal Analysis of Fluid Flows: Applications and Outlook

Low Speed Aerodynamics

Unsteady aerofoil flows are often characterized by leading-edge vortex (LEV) shedding. While experiments and high-order computations have contributed to our understanding of these flows, fast low-order methods are needed for engineering tasks. Classical unsteady aerofoil theories are limited to small amplitudes and attached leading-edge flows. Discrete-vortex methods that model vortex shedding from leading edges assume continuous shedding, valid only for sharp leading edges, or shedding governed by ad-hoc criteria such as a critical angle of attack, valid only for a restricted set of kinematics. We present a criterion for intermittent vortex shedding from rounded leading edges that is governed by a maximum allowable leading-edge suction. We show that, when using unsteady thin aerofoil theory, this leading-edge suction parameter (LESP) is related to the term in the Fourier series representing the chordwise variation of bound vorticity. Furthermore, for any aerofoil and Reynolds number, there is a critical value of the LESP, which is independent of the motion kinematics. When the instantaneous LESP value exceeds the critical value, vortex shedding occurs at the leading edge. We have augmented a discrete-time, arbitrary-motion, unsteady thin aerofoil theory with discrete-vortex shedding from the leading edge governed by the instantaneous LESP. Thus, the use of a single empirical parameter, the critical-LESP value, allows us to determine the onset, growth, and termination of LEVs. We show, by comparison with experimental and computational results for several aerofoils, motions and Reynolds numbers, that this computationally inexpensive method is successful in predicting the complex flows and forces resulting from intermittent LEV shedding, thus validating the LESP concept.

Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding

Experimental studies of the flow topology, leading-edge vortex dynamics and unsteady force produced by pitching and plunging flat-plate aerofoils in forward flight at Reynolds numbers in the range 5000–20 000 are described. We consider the effects of varying frequency and plunge amplitude for the same effective angle-of-attack time history. The effective angle-of-attack history is a sinusoidal oscillation in the range to with mean of and amplitude of . The reduced frequency is varied in the range 0.314–1.0 and the Strouhal number range is 0.10–0.48. Results show that for constant effective angle of attack, the flow evolution is independent of Strouhal number, and as the reduced frequency is increased the leading-edge vortex (LEV) separates later in phase during the downstroke. The LEV trajectory, circulation and area are reported. It is shown that the effective angle of attack and reduced frequency determine the flow evolution, and the Strouhal number is the main parameter determining the aerodynamic force acting on the aerofoil. At low Strouhal numbers, the lift coefficient is proportional to the effective angle of attack, indicating the validity of the quasi-steady approximation. Large values of force coefficients () are measured at high Strouhal number. The measurement results are compared with linear potential flow theory and found to be in reasonable agreement. During the downstroke, when the LEV is present, better agreement is found when the wake effect is ignored for both the lift and drag coefficients.

Unsteady force generation and vortex dynamics of pitching and plunging aerofoils

A low-order point vortex model for the two-dimensional unsteady aerodynamics of a flat plate wing section is developed. A vortex is released from both the trailing and leading edges of the flat plate, and the strength of each is determined by enforcing the Kutta condition at the edges. The strength of a vortex is frozen when it reaches an extremum, and a new vortex is released from the corresponding edge. The motion of variable-strength vortices is computed in one of two ways. In the first approach, the Brown–Michael equation is used in order to ensure that no spurious force is generated by the branch cut associated with each vortex. In the second approach, we propose a new evolution equation for a vortex by equating the rate of change of its impulse with that of an equivalent surrogate vortex with identical properties but constant strength. This impulse matching approach leads to a model that admits more general criteria for shedding, since the variable-strength vortex can be exchanged for its constant-strength surrogate at any instant. We show that the results of the new model, when applied to a pitching or perching plate, agree better with experiments and high-fidelity simulations than the Brown–Michael model, using fewer than ten degrees of freedom. We also assess the model performance on the impulsive start of a flat plate at various angles of attack. Current limitations of the model and extensions to more general unsteady aerodynamic problems are discussed.

Low-order phenomenological modeling of leading-edge vortex formation

An inviscid theoretical method that is applicable to non-periodic motions and that accounts for large amplitudes and non-planar wakes (large-angle unsteady thin airfoil theory) is developed. A pitch-up, hold, pitch-down motion for a flat plate at Reynolds number 10,000 is studied using this theoretical method and also using computational (immersed boundary method) and experimental (water tunnel) methods. Results from theory are compared against those from computation and experiment which are also compared with each other. The variation of circulatory and apparent-mass loads as a function of pivot location for this motion is examined. The flow phenomena leading up to leading-edge vortex shedding and the limit of validity of the inviscid theory in the face of vortex-dominated flows are investigated. Also, the effect of pitch amplitude on leading-edge vortex shedding is examined, and two distinctly different vortex-dominated flows are studied using dye flow visualizations from experiment and vorticity plots from computation.

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

Direct force measurements and qualitative flow visualization were used to compare flow field evolution versus lift and drag for a nominally two-dimensional rigid flat plate executing smoothed linear pitch ramp manoeuvres in a water tunnel. Non-dimensional pitch rate was varied from 0.01 to 0.5, incidence angle from 0 to 90°, and pitch pivot point from the leading to the trailing edge. For low pitch rates, the main unsteady effect is delay of stall beyond the steady incidence angle. Shifting the time base to account for different pivot points leads to collapse of both lift/drag history and flow field history. For higher rates, a leading edge vortex forms; its history also depends on pitch pivot point, but linear shift in time base is not successful in collapsing lift/drag history. Instead, a phenomenological algebraic relation, valid at the higher pitch rates, accounts for lift and drag for different rates and pivot points, through at least 45° incidence angle.

/pdf/unsteady-pitching-flat-plates-3wg4huf8aq.pdf

Unsteady pitching flat plates

Airfoil aerodynamic loads are expected to have quasi-steady, linear dependence on the history of input disturbances, provided that small-amplitude bounds are observed. We explore this assertion for the problem of periodic sinusoidal streamwise gusts, by comparing experiments on nominally 2D airfoils in temporally sinusoidal modulation of freestream speed in a wind tunnel vs. sinusoidal displacement of the airfoil in constant freestream in a water tunnel. In the wind tunnel, there is a streamwise unsteady pressure gradient causing a buoyancy force, while in the water tunnel one must subtract the inertial load of the test article. Both experiments have an added-mass contribution to aerodynamic force. Within measurement resolution, lift and drag, fluctuating and mean, were in good agreement between the two facilities. For incidence angle below static stall, small-disturbance theory was found to be in good agreement with measured lift history, regardless of oscillation frequency. The circulatory component of fluctuating drag was found to be independent of oscillation frequency. For larger incidence angles, there is marked departure between the measured lift history and that predicted from Greenberg's formula. Flow visualization shows coupling between bluff-body shedding and motion-induced shedding, identifiable with lift cancellation or augmentation, depending on the reduced frequency. Isolating the buoyancy effect in the wind tunnel and dynamic tares in the water tunnel, and theoretical calculation of apparent-mass in both cases, we arrive at good agreement in measured circulatory contribution between the two experiments whether the flow is attached or separated substantiating the linear superposition of the various constituents to total lift and drag, and supporting the idea that aerodynamic gust response can legitimately be studied in a steady freestream by oscillating the test article.

Kenneth Granlund

Papers

Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding

Unsteady force generation and vortex dynamics of pitching and plunging aerofoils

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

Unsteady pitching flat plates

Airfoil longitudinal gust response in separated vs. attached flows