SENSE, Newest Tool in CFD Design
AeroSoft would like to introduce you to an exciting new CFD post-processing tool called SENSE . With SENSE, we hope to change the way you think about analyzing and interpreting a CFD solution. Instead of allowing you to look at scalar contour plots of Mach number and pressure, velocity vector plots, and boundary layer profiles, SENSE will provide you with the ability to see how your solution will change as you vary one or more design variables, whether those variables are flow parameters (e.g., freestream Mach number or angle of attack) or geometry related variables such as wing-thickness distribution or sweep angle. So, for example, in addition to looking at pressure contour plots, you can look at a plot of the variation in pressure throughout the entire flow field as the freestream Mach number varies. You might also compute the rate at which the skin friction will vary as you change altitude. Not only can SENSE provide you with this valuable insight, it can do so at a fraction of the cost of a single CFD solution. In many cases, sensitivity solutions have been attained at less than 10% of the cost of the underlying CFD solution. The data from SENSE can be used in a variety of ways such as: approximation of near-by flows, innovative flow analysis and re-analysis, guidance for parametric studies, aerodynamic force and moment coefficients, vehicle design and optimization.
The purpose of SENSE is to determine discrete solutions to the linear sensitivity equations. WITH SENSE, users of high-order CFD codes can obtain the derivative of the entire flow field with respect to a generic design parameter. Since the sensitivity equations are linear, sensitivity solutions can be obtained in much less time than required by a typical flow solution. Implemented as a stand-alone application, SENSE works viably with any flow solver needing only a grid of (x,y,z) data points and a solution of conservative variables, both in Plot3D format. Because of the universal support for the Plot3D format in CFD, SENSE is compatible with codes which originate from commercial vendors and government labs, and those written personally by the user.
The numerics implemented in SENSE have been refined so that the user can focus on the end solution and not on how to solve the equations. The input to SENSE has been minimized with much of the "bean counting" procedures taken care of internally. Users may prescribe any number of geometric and/or flow-related design parameters. Attempts have been made to increase the flexibility of the output from SENSE for the user, and suggestions for future releases are always welcome.
The original idea for a stand-alone sensitivity solver resulted from one of many consultation meetings between Gene Cliff, Justin Appel, Max Gunzburger and Andrew Godfrey. The theoretical development of the continuous sensitivity-equation approach is due in large part to Drs. John Burns, Jeff Borggaard, Gunsburger, Cliff and their students through the Interdisciplinary Center for Applied Mathemetics (ICAM) at Virginia Tech.
Perhaps the most powerful application of SENSE is in providing near-by solutions in the design space. Rather than recompute a non-linear Euler or Navier-Stokes calculation, SENSE can be used to output a linear approximation to a near-by case directly. This solution could be used as an initial guess or to analyze important trends in the flow. Together with the sensitivity field, the near-by flow provides a unique insight into the driving mechanism of an applied-CFD engineer's flow problem.
By solving the sensitivity equations for multiple design variables, the user can determine the most important parameters driving the flow. This helps narrow the focus of a parametric study and decreases the turn-around time for a vehicle analysis and design. The post-processing utilities in SENSE can be used to determine which design variable has the largest effect on an objective function. In some cases, design variables can be ignored because of the relatively small influence.
SENSE has been used to compute performance parameters such as lift-curve slope and stability derivatives such as roll and yaw damping. For design-oriented applications, the sensitivity derivatives from SENSE are independent of the objective function. This means that with a single sensitivity solution for each design variable, the gradient of multiple objective functions can be found at once. This contrasts the adjoint-variable approach which determines the gradient of one objective function for multiple design variables.
Solutions to the fluid-dynamic equations provide coefficients for the sensitivity equations. Three-dimensional flow of a turbulent, viscous fluid is governed by a system of non-linear, hyperbolic partial differential equations which can be written in integral form as
(1)
Here, the conservative variables represent the specie mass, momentum, and total energy per unit volume of the fluid. The surface integrals represent the inviscid and viscous fluxes; the source term (W) represents the production and destruction of species through chemical reactions. SENSE currently supports three chemistry models: calorically perfect air, five-species air, and seven-species hydrogen-air mixture. More models will be added as needed.
The objective in the SEM approach is to derive a linear boundary-value problem for the flow sensitivity derivatives. Here, we use the symbol h to denote a generic parameter. To emphasize that the flow solution depends both on position in space and on the parameter, we write Q=Q(x,y,z;h). The sensitivity we seek is then formally given by
(the flow sensitivity) (2)
To derive the sensitivity equations, we formally differentiate the elements of the non-linear, boundary-value problem represented by the flow equations. In the case where the parameters of interest do not explicitly appear in any of the flux functions, we proceed by differentiating Eqn. (1) with respect to h and then interchange orders of differentiation to find
(3)
Note that the governing equations for S(x,y,z;h) are linear and that the conservative flow solution, which the user of SENSE supplies, enters through spatially varying coefficients.
A concept incorporating the SEM approach with
tremendous potential is the fast calculation of aerodynamic
stability derivatives (e.g., lift-curve slope, static-stability
derivative and roll damping). To demonstrate work in the area, we model the
flow around a natural laminar flow airfoil. The flow conditions
(M=0.5, Re=2 million) are extracted from an AGARD set of test
cases for validating CFD codes. The pressure contours and
streamlines at zero-degrees angle-of-attack is shown in Fig. 1(a)
on a 185 x 97 x 2 "C" mesh.
Figure 1(a). Zero-degree flow solution.
The sensitivity to angle of attack at zero-degrees angle-of-attack is shown in Fig. 1(b). Sensitivity pressure contours and streamlines of velocity-vector sensitivity are depicted. From the pressure sensitivity near the nose, we see that an increase in the angle of attack will cause an increase in the lower-surface pressure and a decrease in the upper-surface pressure. The streamlines show that the magnitude of the velocity vector will decrease beneath the airfoil surface and increase along the upper surface. Careful investigation at the trailing edge shows that the momentum of the boundary layer is predicted to decrease. All these indications are consistent with an increase in the angle of attack for attached flow.
Figure 1(b). Sensitivity to angle of attack.
To determine the accuracy of the sensitivity method, we extrapolate the baseline flow to four degrees angle-of-attack using a first-order, Taylor-series expansion. The pressure and streamlines for the near-by approximation is given in Fig. 1(c). The predominant flow features are a rearward movement of the stagnation point at the nose and decreased pressure along the upper surface. The accuracy of the pressure coefficient on the airfoil surface can be evaluated using Fig 1(d).
Figure 1(c). Near-by four-degree flow solution.
Figure 1(d). Comparison of pressure distribution.
The sensitivity project would not have been possible without the funding from three Phase I SBIR projects. Two of them were awarded by NASA Langley Research Center under the direction of Sharon Stack, Perry Newman and Tom Zang. The other SBIR was awarded by Wright-Patterson Air Force Base under the direction of Howard Emsley. ICAM support in this area has been sponsored by the Air Force Office of Scientific Research under Major S. Schreck.
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