Nail K. Yamaleev

Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions

The Lax-Wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities. The discrete operator, however, need not be derived from the divergence form of the continuous equations. Indeed, conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal-norm skew-symmetric summation-by-parts (SBP) spatial operator, yield discrete operators that are conservative. Furthermore, split-form, discretely conservation operators can be derived for periodic or finite-domain SBP spatial operators of any order. Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and are supplied in an accompanying text file.

Third-order Energy Stable WENO scheme

A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. We also present new weight functions which drastically improve the accuracy of the third-order ESWENO scheme. Based on a truncation error analysis, we show that the ESWENO scheme is design-order accurate for smooth solutions with any number of vanishing derivatives, if its tuning parameters satisfy certain constraints. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.

Reduced-Order Model for Efficient Simulation of Synthetic Jet Actuators

A new reduced-order model of multidimensional synthetic jet actuators that combines the accuracy and conservation properties of full numerical simulation methods with the efficiency of simplified zero-order models is proposed. The multidimensional actuator is simulated by solving the time-dependent compressible quasi-1-D Euler equations, while the diaphragm is modeled as a moving boundary. The governing equations are approximated with a fourth-order finite difference scheme on a moving mesh such that one of the mesh boundaries coincides with the diaphragm. The reduced-order model of the actuator has several advantages. In contrast to the 3-D models, this approach provides conservation of mass, momentum, and energy. Furthermore, the new method is computationally much more efficient than the multidimensional Navier-Stokes simulation of the actuator cavity flow, while providing practically the same accuracy in the exterior flowfield. The most distinctive feature of the present model is its ability to predict the resonance characteristics of synthetic jet actuators; this is not practical when using the 3-D models because of the computational cost involved. Numerical results demonstrating the accuracy of the new reduced-order model and its limitations are presented.

Quasi-One-Dimensional Model for Realistic Three-Dimensional Synthetic Jet Actuators

A systematic methodology for approximating realistic three-dimensional synthetic jet actuators by using a reduced-order model based on the time-dependent compressible quasi-one-dimensional Euler equations is presented. The following major questions are addressed: 1) which three-dimensional actuator geometries are amenable to the quasi-one-dimensional approximation; 2) which three-dimensional actuator parameters should be retained in the quasi-one-dimensional model; 3) which actuator flow regions are essentially multidimensional and are not candidates for reduced-order modeling; and 4) which geometrical features practically do not contribute to the fidelity of the actuator solution. Constraints that should be imposed on the actuator geometry and the flow parameters are discussed. The accuracy of the quasi-one-dimensional model is validated by comparing the numerical results with experimental data and full time-dependent Navier-Stokes simulation of the same realistic actuator.

Local-in-time adjoint-based method for design optimization of unsteady flows

We present a new local-in-time discrete adjoint-based methodology for solving design optimization problems arising in unsteady aerodynamic applications. The new methodology circumvents storage requirements associated with the straightforward implementation of a global adjoint-based optimization method that stores the entire flow solution history for all time levels. This storage cost may quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and to approximate the solution of the unsteady adjoint equations and the sensitivity derivative as a combination of the corresponding local quantities computed on each time subinterval. Since each subinterval contains relatively few time levels, the storage cost of the local-in-time method is much lower than that of the global methods, thus making the time-dependent adjoint optimization feasible for practical applications. Another attractive feature of the new technique is that the converged solution obtained with the local-in-time method is a local extremum of the original optimization problem. The new method carries no computational overhead as compared with the global implementation of adjoint-based methods. The paper presents a detailed comparison of the global- and local-in-time adjoint-based methods for design optimization problems governed by the unsteady compressible 2-D Euler equations.

Adjoint-based Methodology for Time-Dependent Optimization

This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.

The Effect of a Gust on the Flapping Wing Performance

The effect of a wind gust on the aerodynamic characteristics of a rigid wing undergoing insect-based flapping motion is studied numerically. The turbulent flow near the flapping wing is described by the 3-D unsteady compressible Reynolds-Averaged Navier-Stokes equations with the Spalart-Allmaras turbulence model. The governing equations are solved using a second-order node-centered finite volume scheme on a hexahedral body-fitted grid that rigidly moves along with the wing. A low-Mach-number preconditioner is used to accelerate the convergence at each time step. The effects of wind gust direction with respect to a wing orientation are investigated. Our numerical results show that the centimeter-scale wing considered in the present study is susceptible to strong downward wind gusts. In the case of frontal and side gusts, the flapping wing can alleviate the gust effect if the gust velocity is less or comparable with the wing tip velocity. For all cases considered, the thrust coefficient returns to its original baseline profile within one full stroke after the gust is removed, thus indicating that the flapping wing can effectively recover from wind gust fluctuations.

Adjoint-Based Optimization of Three-Dimensional Flapping-Wing Flows

Optimization of the three-dimensional unsteady viscous flow near a flapping wing in hover is performed using a time-dependent adjoint-based methodology. The sensitivities of a penalized lift coefficient to wing shape and kinematic parameters are computed using a time-dependent discrete-adjoint formulation. The unsteady discrete-adjoint equations required for the calculation of the sensitivity derivatives are integrated backward in time over the entire interval of interest. The gradient of the objective functional obtained using the adjoint formulation is then used to update the values of shape and kinematic design variables. The efficiency of this adjoint-based methodology is demonstrated by optimizing the shape and kinematics of a hovering wing. The numerical results show that the highest improvement in the wing performance is obtained by using combined optimization of the wing shape and kinematics.

Adjoint-based shape and kinematics optimization of flapping wing propulsive efficiency

Optimization of the 3-D unsteady viscous flow near a flapping wing is performed using a time-dependent adjoint-based methodology developed in [AIAA 2008-5857 and AIAA J. Vol.48, No.6, pp.1195-1206, 2010]. Sensitivities of the thrust and propulsive efficiency to wing shape and kinematic parameters are computed using the time-dependent discrete adjoint formulation. The unsteady discrete adjoint equations required for calculation of the sensitivity derivatives are integrated backward in time over the entire interval of interest. The gradient of the objective functional obtained using the adjoint formulation is then used to update the values of shape and kinematic design variables. The efficiency of this adjoint-based methodology is demonstrated by optimizing shape and kinematics of a wing undergoing insect-based flapping motion. Our numerical results show that the highest improvement in the thrust and propulsive efficiency is obtained by using the combined optimization of wing shape and kinematics. I. Introduction Insects and small birds represent fully functional examples of efficient small-scale flying devices. However, copying of wing kinematics and shape of flying animals is far from being sufficient to design and build effective, highly maneuverable, agile micro air vehicles (MAVs). Indeed, the current state-of-the-art materials, micro-scale actuators, propulsion systems, and power sources are different and in most cases less efficient than those created by Mother Nature over millions years of evolution. This lack in efficiency of currently available MAV components indicates that a different region of the design space than that associated with flying insects and animals should be explored to be able to maximize the performance of flapping-wing microsystems. Therefore, designs inspired by flying animals can be used only as a preliminary conceptual design that requires further optimization for constructing efficient and agile flying micro-scale platforms optimized for size, weight, speed, and maneuverability. This is a very challenging optimization problem that involves hundreds or even thousands kinematics and shape design variables and is governed by highly unsteady vortex-dominated turbulent flows. Therefore, efficient, mathematically rigorous optimization techniques based on optimal control theory should be used for solving this class of problems. In spite of significant progress in modeling and computational fluid dynamics (CFD) analysis of flappingand rotary-wing platforms [1-5], questions related to optimal design of efficient micro air vehicles (MAV) have not yet been properly addressed especially in three dimensions because of the complicated physical phenomena and computational cost involved. Various parametric and sensitivity studies (e.g., see [1]) have revealed that there is an essentially nonlinear relationship between the major wing kinematic parameters (amplitude, frequency, phase shift angle), shape parameters (wing planform, twist, and thickness), and global flow parameters (the Reynolds, Strouhal, and Mach numbers). Conventional parametric studies, which estimate the sensitivity to each individual design variable independently, do not take into account this nonlinear relationship between the main parameters determining the MAV performance. Furthermore, parametric studies are extremely computationally expensive because of the very large dimensionality of the design space and therefore impractical for optimization and design of efficient flapping-wing microsystems. Several attempts have recently been made to use genetic algorithms based on low-fidelity models [6], highfidelity models [7], and experimental apparatus [8] for optimization of flapping-wing flows. Since these stochastic optimization techniques require thousands of evaluations of the objective functional and consequently thousands of solves of the unsteady flow equations for each design variable, all these approaches have been limited to optimization of 2-D flows with a very small number (less than 4) of design variables. Gradient-based methods provide a powerful alternative for optimization of flapping airfoils and wings. Culbreth et al. [9] uses a finite difference method coupled with a 3-D Navier-Stokes solver to evaluate the

Nonlinear model reduction for unsteady discontinuous flows

Abstract We develop a new nonlinear reduced-order model (ROM) based on proper orthogonal decomposition (POD), which can be used for quantitative simulation of not only smooth flows, but also flows with strong discontinuities. The new model is derived using a Galerkin projection of the fully conservative, nonlinear discretized 2-D Euler equations onto the POD basis constructed for each conservative variable. This approach can be interpreted as a variant of the spectral method with a truncated set of basis functions. A system of ordinary differential equations (ODEs) derived using this model reduction technique resembles the major nonlinear and conservation properties of the original discretized Euler equations. The new reduced-order model also preserves the stability properties of the discrete full-order model equations, so that no additional stabilization is required unlike conventional POD-based models that are susceptible to numerical instabilities. The performance of the new POD ROM is evaluated for 2-D compressible unsteady inviscid flows over a wide range of Mach numbers including trans- and supersonic flows with strong shock waves.