# and Dimitrios G. Stamos

Title | Background | Progress | Objectives | Responsibilities

### Title :

Development of a flow visualization tool for use in the aerodynamic design when using the Sensitivity Equation Method.

### Background :

The general optimization problem can be stated as the problem of minimizing a scalar function
f(x,u)

subject to some constrains g(x,u). Note that g(x,u) is a vector function. In the above x represents the state variable vector and u the design variable vector. It is

f : X x U -> R
and g : X x U -> X

In the airfoil optimization or in general in the aerodynamic design of flying bodies usually the f(x,u) function represents scalar functions such as the Drag, the Drag to Lift ratio, the negative of the Lift etc. The constraint function g(x,u) represents the governing equations of the flow. Usually they are the Euler equations for inviscid flow or the Navier-Stokes equation for viscous flow. Research at ICAM (Interdisciplinary Center for Applied Mathematics) focuses at solving the optimization problem using the Sensitivity Equation Method. We refer to the sensitivity as the partial derivatives of the conserved or primitive variables of the flow (pressure, density, velocity, energy etc.) with respect to some parameter of the flow such as angle of attack, airfoil thickness, Mach-number etc. It was shown numerically that when we have a basic flow around an airfoil with constant conditions then the flow resulting from a small perturbation in the parameters of the flow can be approximated by a Taylor series. That means that the perturbed flow solution equals the basic flow plus the product of the sensitivity of the flow times the perturbation of the parameter. Mathematically the above can be written as

Q(da)=Q(0) + (dQ/da)*da

It is clear that the above is an efficient and very fast method on getting perturbed flow solution because all it requires is a matrix multiplication and an addition (when the sensitivities are known). A designer can use the above method to visualize a flow for conditions other than the design conditions. This allows him to early identify regions where the design has limited performance and make the appropriate changes.

### Progress :

As of 10.01.97 a program exists that calculates the sensitivities for a given flow. Also the Euler solution for a flow can be consider as given. So at begin of the project all the raw data are given. Both member of the team do not possess any experience with the visualization programs that are going to be used and also do not possess any experience with web publishing and programming.

### Objectives :

• Organize the data (flow solution and sensitivities) for 2D flows and convert them in a readable format for the applications.
• Design a graphic environment that will allow the designer to perturb some parameters and visualize the basic flow, the sensitivities and the resulting flow.
• Sub task is to provide the interface between the design environment and MATLAB, so that the calculation of the new flow can be performed.
• Effort will also be concentrate to overlaying different properties of the flow in the same graph.
• Depending on the time left, we intend to extent the above idea to 3D bodies design. That means that 3D flows have to be visualized. On those 3D flow representations we will provide the user with the ability to do slices in the flow domain.
• Learn how to use several web publishing and web programming tool such as html, JAVA etc.
• Do a web presentation of the project and provide background information that will clarify the basic ideas which led to the project.

### Tools :

A preliminary consideration is to use AVS or PV-Wave for the imaging processes. MATLAB will be used for the calculation requirements and several wed tool (html, JAVA, etc.) for the presentation.

### Responsibilities :

This will be a team project. The team consist of Iossif Mugtussidis and Dimitrios Stamos.

Iossif Mugtussidis will be responsible for the background information, the data and the development of the design tool.

Dimitrios G. Stamos will be responsible for the web implementation and presentation of the project.

Interaction between the two members will be inevitable and in great extent desired.