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We extend the impulse theory for unsteady aerodynamics, from its classic global form to finite-domain formulation then to minimum-domain form, and from incompressible to compressible flows. For incompressible flow, the minimum-domain impulse theory raises the finding of Li and Lu (J. Fluid Mech., 712: 598-613, 2012) to a theorem: The entire force with discrete wake is completely determined by only the time rate of impulse of those vortical structures still connecting to the body, along with the Lamb-vector integral thereof that captures the contribution of all the rest disconnected vortical structures. For compressible flow, we find that the global form in terms of the curl of momentum, obtained by Huang (Unsteady Vortical Aerodynamics. Shanghai Jiaotong Univ. Press, 1994), can be generalized to having arbitrary finite domain, but the formula is cumbersome and in general the curl of momentum no longer has discrete structure and hence no minimum-domain theory exists. Nevertheless, as the measure of transverse process only, the unsteady field of vorticity may still have discrete wake. This leads to a minimum-domain compressible vorticity-moment theory in terms of density-weighted vorticity (but it is beyond the classic concept of impulse). These new findings and applications have been confirmed by our numerical experiments. The results not only open an avenue to combine the theory with computation-experiment in wide applications, but also reveals a physical truth that it is no longer necessary to account for all wake vortical structures in computing the force and moment.
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High-fidelity wall-resolved large-eddy simulations (LES) are utilized to investigate the flow-physics of small-amplitude pitch oscillations of an airfoil at Re = 100,000. The investigation of the unsteady phenomenon is done in the context of natural laminar flow airfoils, which can display sensitive dependence of the aerodynamic forces on the angle of attack in certain off-design conditions. The dynamic range of the pitch oscillations is chosen to be in this sensitive region. Large variations of the transition point on the suction-side of the airfoil are observed throughout the pitch cycle resulting in a dynamically rich flow response. Changes in the stability characteristics of a leading-edge laminar separation bubble has a dominating influence on the boundary layer dynamics and causes an abrupt change in the transition location over the airfoil. The LES procedure is based on a relaxation-term which models the dissipation of the smallest unresolved scales. The validation of the procedure is provided for channel flows and for a stationary wing at Re = 400,000.
Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for adjoint method to obtain the gradients of all design variables. However, the calculation cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the adjoint-based optimization efficiency, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction about NACA0012 airfoil. The results indicate that with negligible calculation cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.
Recent experiments and simulations have shown that unsteady turbulent flows, before reaching a dynamic equilibrium state, display a universal behaviour. We show that the observed universal non-equilibrium scaling can be explained using a non-equilibrium correction of Kolmogorovs energy spectrum. Given the universality of the experimental and numerical observations, the ideas presented here lay the foundation for the modeling of a wide class of unsteady turbulent flows.
We extend unsteady thin aerofoil theory to model aerofoils with generalised chordwise porosity distributions. The analysis considers a linearised porosity boundary condition where the seepage velocity through the aerofoil is related to the local pressure jump across the aerofoil surface and to the unsteady characteristics of the porous medium. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm--Volterra integral equation which does not admit an analytic solution. Accordingly, we develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading- and trailing-edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to {be unsuitable} for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems, and the separation of circulatory and non-circulatory contributions are presented. Analogues to the classical Theodorsen and Sears functions are computed numerically, which show that an effect of trailing-edge porosity is to reduce the amount of vorticity shed into the wake, thereby reducing the magnitude of the circulatory lift. {Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and K{u}ssner functions for transient aerofoil motions or gust encounters, respectively.}
This article presents an original methodology for the prediction of steady turbulent aerodynamic fields. Due to the important computational cost of high-fidelity aerodynamic simulations, a surrogate model is employed to cope with the significant variations of several inflow conditions. Specifically, the Local Decomposition Method presented in this paper has been derived to capture nonlinear behaviors resulting from the presence of continuous and discontinuous signals. A combination of unsupervised and supervised learning algorithms is coupled with a physical criterion. It decomposes automatically the input parameter space, from a limited number of high-fidelity simulations, into subspaces. These latter correspond to different flow regimes. A measure of entropy identifies the subspace with the expected strongest non-linear behavior allowing to perform an active resampling on this low-dimensional structure. Local reduced-order models are built on each subspace using Proper Orthogonal Decomposition coupled with a multivariate interpolation tool. The methodology is assessed on the turbulent two-dimensional flow around the RAE2822 transonic airfoil. It exhibits a significant improvement in term of prediction accuracy for the Local Decomposition Method compared with the classical method of surrogate modeling for cases with different flow regimes.
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