Future Research Plan

Brief Description

   As has been already mentioned in the Description of Research, in recent years my research has been concerned with mathematical analysis of aircraft wing models. It is a very substantial long-term project, which on the one hand is of great practical importance, and on the other hand involves deep and highly nontrivial mathematics. My main research plans consist of continuation of the current project. Briefly, those plans include the following.

1. Theoretical part.

   • Complete rigorous asymptotic and spectral analysis of already existing model of a wing in a subsonic, inviscid, incompressible airflow. This is exactly the model that has been developed at Flight System Research Center at UCLA in collaboration with NASA Dryden. As has been states in the Description of Research, this work is currently supported by two NSF grants and one Texas ASP grant. That parts of the project contains several topics for the Ph.D. dissertations.
      

   • Extend current research to a new generation of wing models. These models describe an aircraft wing in a compressible subsonic airflow. They include a model of a long slender wing and a finite rectangular or trapezoid wing. The work on the first of these models has already begun, and currently I am preparing a paper on this topic. The model is governed by a system consisting of a linear hyperbolic equation (which governs the elastic structure, the wing) coupled with a singular integral equation (which governs the air flow around the wing). The integral equation, the famous Possio equation, is obtained by several step reduction of the Euler equation for compressible, inviscid, isentropic fluid. Mathematical tools necessary for analysis of this model include:
      

     1. methods of asymptotic analysis;

     2. spectral theory of nonselfadjoint differential and integral operators;

     3. Sz. Nagy-Foias functional model for nonselfadjoint operators in a Hilbert space;

     4. theory of singular integral equations;

     5. theory of operator-valued meromorphic anlytical functions of a Fredholm type.
         

   • Begin work on rigorous analysis of wing model in transonic airflow.
      

   • Develop a new multidisciplinary graduate program for both applied mathematics and engineering students. This work is currently in progress. The program will include such courses as (a) methods of Asymptotic Analysis, (b) Hilbert space Spectral Theory, (c) Selected topics from Aeroelasticity and Hydrodynamics.
      

2. Computational and Experimental parts.
   
   This part is based on collaboration with aircraft engineers both experimentalists and numerical analysts. It will involve validation of already obtained and future theoretical results in lab and wind tunnel experiments. It will also include designing of new nonlinear models. Close collaboration with FSRC at UCLA and NASA Dryden will be continued. In addition, I am involved in a joint research with faculty members from Mechanical Engineering Department of Texas Tech and Aerospace Engineering Department University of Texas at Austin.
    

Technical Details

        To give more details on my research plans, I have to briefly describe mathematics related to analysis of wing models.

        Aircraft wing not embedded in an airflow is an elastic structure, whose motion governed by a system of hyperbolic partial differential equations. The complexity of the system with the accompanying boundary conditions depends on specific assumption of a particular wing model:

• long slender wing,

• finite rectangular wing,

• wing with or without internal damping,

• wing with intrinsic structural nonlinearities.

        Even in the simplest case of a linearized long slender wing, the system is very complicated (it is known as bending-torsion vibration model).

        If a wing embedded into an air stream, the aforementioned hyperbolic system has to be supplied with right-hand side forcing terms, which are called aerodynamic loads, i.e., generalized forces exerted on the wing by an air stream. The aerodynamic loads can be expressed in terms of the pressure distribution on the wing, which in turn is to be determined from the hydrodynamic equations governing the airflow. Traditional assumption in wing aerodynamics is that the airflow is potential and isentropic. In addition, in the subsonic regime, it is reasonable to use specific linearized version of hydrodynamic equations. All these assumptions lead to a single scalar hyperbolic equation for the flow perturbation potential (small perturbation of the back-ground flow potential). The latter hyperbolic equation is supplied with very special boundary conditions on the wing:

• the flow tangency condition,

• the Kuta-Joukowsky condition, and

• the far-field condition at infinity.

        The aforementioned flow tangency condition involves the unknown functions from the structural part of the system describing the wing vibrations. Therefore, the structural and hydrodynamic equations are coupled.

        The aforementioned model, developed at FSRC at UCLA and NASA Dryden, deals with a long slender wing in subsonic, inviscid, incompressible air flow. In that case, the hydrodynamic loads can be calculated explicitly from the hydrodynamic part of the system and are expressed in the forms of complicated time-convolution type integrals. The latter leads to a single evolution-convolution integro-differential system describing the wing. This system was a subject of research in a series of my  papers, in which I was able to obtain the first in the world literature on aeroelasticity explicit asymptotic formulae for the aeroelastic modes and mode shapes. I have also established the Riesz basis property of the mode shape (see "Description of Research").

        For the case of compressible flow, the hydrodynamic part of the system cannot be solved explicitly. However, it can be reduced to a single singular integral equation - the famous Possio equation (this equation is one-dimensional in the spatial variable for a long slender wing and two or three dimensional for more complicated geometries of the wing). The asymptotic and spectral analysis of structural hyperbolic system coupled with Possio equation present a new challenge for the future research.

        In transonic airflow, the minimal level of complexity of hydrodynamic  equations requires inclusion of nonlinear terms since linear theory cannot explain shock waves. Study of such problem will necessarily involve analysis of bifurcations.

        It is important to point out that all of the above models have been subject of many years extensive  numerical investigation. The distinctive feature of my research is that I was successful in purely analytical approach. Theoretical results provide new insights not available from pure computations. These results are essential in designing flutter control mechanisms.