[American Institute of Aeronautics and Astronautics 40th Fluid Dynamics Conference and Exhibit - Chicago, Illinois ()] 40th Fluid Dynamics Conference and Exhibit - Reduced Order Method

American Institute of Aeronautics and Astronautics

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Reduced Order Method for Deforming Unstructured Grid in CFD

H. Q. Yang 1 CFD Research Corporation, Huntsville, AL 35805, USA

Abstract A highly accurate and efficient solution algorithm for Navier-Stokes equations by unstructured grid for aeroelastic analysis of complex aircraft configurations or for the deformation surface and bodies is important in further design of modern aircraft and space vehicles. To handle the Lagrangian portion of the fluid equation moving mesh, we proposed a reduced order method (ROM) to model the grid motion. By this method, the fluid mesh is modeled as having structural properties: Young's modulus density and Poisson's ratio, but it acts passively to the structural deformation (wing or space vehicle deformation). By utilizing the properties of a maximum 6 degrees of freedom (DOF) of a moving body, the presently proposed method computes the grid deformation using nonlinear large deformation theory to preserve the original grid quality under each DOF, so that no extra computations are required during the unsteady motion, just matrix-vector multiplications. The deformation of unstructured grid around control and moving surfaces is shown to be very efficient with minimum grid distortion.

I. Introduction sing reduced order method to deform unstructured grids was first proposed by the present author [1, 2]. According to this method, the fluid mesh is modeled as having structural properties: Young's modulus,

density and Poisson's ratio, but it acts passively to the structural deformation (wing or space vehicle deformation). The mode shape of both the structure and CFD grid can be determined from a FEM solver. By utilizing the modal superposition property of structural dynamics, the reduced model for the structure and CFD grid can be derived and the efficiency of the grid deformation can be significantly improved by first preparing a specified grid mode shape corresponding to the structural modal function.

The method has been successfully applied to fluid-structure interaction problems using complicated unstructured grids for both solid and fluids. It utilizes the following features of fluid-structure interaction:

1. CFD codes are formulated in a Lagrangian-Eulerian frame, and CSD (Computational Structural Dynamics) are in a Lagrangian frame. It is possible to group the Lagrangian part together and solve simultaneously, and hence bring in a unified approach; and

2. Solid material can sustain shear stress, and hence can reduce distortion of the grid that are encountered in the spring analogy.

As such, one can treat the fluid mesh as part of the CSD region, and solve by Lagrangian formulations. As a result, one has the equilibrium equation for fluid mesh as:

j,gijgit

vσ=

∂

∂ρ (1)

where ( ) k,gkiji,gjj,gigij uuu λδ++η=σ (2)

The ug is the fluid grid deformation, and vg is it velocity. The boundary conditions to be satisfied are the continuity of displacement and velocity of fluid structure interface: vgf-s = vsf-s (3) ugf-s = usf-s (4)

A. Modal Analysis Approach In many aeroelasticity computations, the structural solver is performed using modal analysis. The advantage

of using modal analysis is that usually only a relatively small number of the natural frequencies will be excited to a significant degree. Thus accurate solutions with many fewer degrees of freedom can be obtained. By modal analysis, one assumes that the system response can be written as weighted sum of the mode shape: ( ) ( ) ( ) ( ) nn3322l1 dtqdtqdtqdtqu ++++= K (5)

where di is the modal shape, and qj(t) is a time-dependent weighting factor to be applied to the modal shape. It is clear that the deformation in the domain will simply be the summation of all the modes:

1 Chief Scientist, Research, 2115 Wynn Drive, Huntsville, AL 35805, and Senior AIAA Member.

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40th Fluid Dynamics Conference and Exhibit28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4617

Copyright © 2010 by H. Q. Yang. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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( )( )∑=η

=1igiig dtqu (6)

where dgi is the "mode shape" of the fluid mesh deformation due to the ith modal deformation at the fluid-structure interface.

B. Reduced Order Method for Moving Body and Control Surfaces For the control surface deployment, the available degrees of freedom are limited: for 3D it is 6:

displacements in x,y,z and rotational angles corresponding to roll, pitch, and yaw. For 2D problems, it reduces to 3: displacements in x and y directions, and rotating pitch angle. This unique property of a small number of DOF makes the preparation of the mode shape very simple. Figure 1 illustrates the reduced order idea for the deforming unstructured grid around a 2D store. Since there are only three DOF for 2D moving body, at any instance, the motion of the moving body surface consists of the components of each DOF (displacement in x and y, and pitching angle). The corresponding mode shape can be calculated a priori, either from a structural solver, or a flow solver, or a black box. These modal shapes will be constant during the simulation, so that the grid movement in the unstructured grid domain will be:

[ ]∑=

=3

1)(

iigig dtqu

(7)

The simplification can be found in the expression for di, which varies along the control surface. For

example, when we control the 2D surface moving flap problem, Figure 4 in the following section, the displacements along the flap all are a function of angle only, so that:

θ

θ

θ dd

d

dd

iy

ix

iy

ix

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

∂

∂∂∂

=⎭⎬⎫

⎩⎨⎧

(8)

This allows equation (5) for the unstructured grid deformation to be a function of angle only. It can be seen from Equation (8) that the displacement of the grid inside the CFD domain is a simple vector-

matrix multiplication. It is very fast and is applicable to any CFD solver, and can be made into a black box. The present ROM for hybrid adaptive Cartesian and moving unstructured grid mesh has the merits of reduced

Figure 1. ROM for the Unstructured Grid around a Moving Body


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grid distortion, minimum modification to an existing code, efficient moving grid methodology with simple matrix multiplication, and the elimination of the need for hole-cutting and donor cell searching.

In following section we will demonstrate the effectiveness of ROM for deforming 2D unstructured grid problems encountered in control surface related problems.

II. Results A. ROM Method for The Motion of Control Surfaces Over a Wing Surface Here we consider the control surface moving over the aircraft wing or other parts of the aircraft with very

close proximity of the control surface edges to the adjacent parts of the aircraft. These types of simulations are difficult to model and are problematic to the available computational tools. Modeling and simulation of these adjacent surfaces in many cases would require contact between opposing surfaces of the mesh, which is computationally challenging.

ROM can significantly simplify the problem. Figure 2 illustrates the reduced order method for deforming an unstructured grid around a 2D version of a flapping wing over an airfoil. As shown in Figure 2, for 2D rigid flapping of a wing, there is only one DOF, namely the rotation angle. The number of DOF does increase when the control surfaces are deformable; however the essential idea stays the same. For now, we will assume the control surface of flap is rigid. Since there is only one DOF, any motion will be the summation of the mode shape multiplied by angle as demonstrated in Figure 2.

Fig. 3. Deformed Unstructured Grid Using ROM Method with a Shape Function Calculated a priori

Figure 2. ROM for the Unstructured Grid around a 2D Moving Flap over an Airfoil


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Two examples are shown in Figure 3 with rotational angles of 55 deg and 5 deg, respectively, by using the modal shape function calculated a priori. The process is just a simple vector-matrix multiplication and it gives a very good quality grid through large angle rotations.

B. CFD Solution Using ROM Method for the Motion of Control Surfaces Over a Wing Surface The problem definition is illustrated in Figure 4. Here a free stream flow at 10m/s is applied to a stationary

NACA0012 foil. On the top of the airfoil is a flapping wing, which can rotate in a sinusoidal function as: Hzfftoo 10),2sin(1560.23 =−= πα

The angle α is an angle relative to the horizontal axis.

Figure 5 shows the transient vorticity distribution from unsteady Navier-Stokes equation solution for the

above flapping wing. The generation and transport of vorticity from the wing cavity region can be clearly seen.

The corresponding pressure field about the flapping wing is shown in Figure 6. The deforming mesh is

essentially automatic and using the ROM is very effective.

Figure 5. Vorticity Field Due To an Oscillating Flapping Wing On Top Of A NACA 0012 Airfoil

Figure 4. Computational Model for Flow over a Flapping Wing Using Reduced Order Method


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It should be noted that ROM method has limitations when grid deformation is very large, in which case the

grid quality can deteriorate and negative volumes may arise. The proposed automatic adaptive Cartesian grid solution technology will be used to insert Cartesian cells into the low quality grid region. The results on the generation of automatic and adaptive overset Cartesian grid will be discussed in a later section.

III. ROM Method for Opening/Closing of Weapons Bay Door Another familiar example occurs inside the internal weapons bays of aircraft. The trend in modern fighter

aircraft is to increase the packing efficiency of the weapons inside the weapons bay and to enhance gliding time through kinematically designed weapons. These requirements have pushed weapons toward marginal aerodynamic stability. At the same time, the flowfield around the weapons bay cavity is naturally unsteady. The dynamic proximity of the weapons inside the cavity offers another challenging problem. In addition, the unsteady flow phenomenon associated with opening and closing the weapons bay door compounds the complexity. As shown in Figure 7, the dynamic motion of the weapons bay doors changes the computational domain drastically; this is challenging for any grid deformation model. By using the present ROM formulation, the grid deformation due to the opening/closing of the doors becomes a simple matrix-vector multiplication.

Opening/closing door is illustrated in Figure 7, where a specified grid mode shape corresponding to each degree of freedom of rigid body motion is first prepared, significantly increasing the grid deformation efficiency. In general good quality grids were obtained with door openings ranging from 30 degrees to 90 degrees.

To demonstrate the capability of the present ROM method in unsteady aerodynamics, we take a cross-section of the weapons bay door for the analysis. The simulation model with an unstructured grid is shown in Figure 8. For the real problem, the flow is moving into the paper, but for this demonstration we will look at the side flow effect, where the flow is moving from the left to the right. The cylinder body represents the weapons, and there is a layer of solid cells representing the door. The initial opening of the door is assumed to be 10%.The singularity of the grid topology will not be considered here, but rather in a later section of the report when the Cartesian grid is invoked. We will concentrate on the fast unstructured grid motion for the door

Figure 6. Pressure Field Due To an Oscillating Flapping Wing On Top Of A NACA 0012 Airfoil.


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opening from 10 degrees to the full 90 degrees. As noted in Figure 8, there is a thin boundary layer grid around the doors, During the Cartesian grid insertion solution, the boundary layer grid will be kept to preserve boundary layer resolving capability. procedure.

The unsteady cross flow feature around the weapons bay door is shown in Figure 9. There is a

distinguishable vortex shed from the leading corner of the door, and there is another vortex shed from the trailing corner of the door. These two vortices have different senses of rotation and form a pair. One can also see the vorticity due to the boundary layer near the top surface. The shedding and transport of the vortex pair is shown in Figure 10. This simulation clearly shows the inherent unsteady nature of the cross flow around the weapon door. The vortex shedding frequency is about 10Hz.

Next we turn our attention to the interaction of the vortex shedding from the corners of the door with the

motion of the door. The vorticity field from this interaction is shown in Figure 11. The door opens with an angular velocity of 8o/s. The initial opening angle of the gap is 10 degrees. The weapons bay door reaches a fully-open position after 10 seconds. The unstructured grid deformation using our Reduced Order Method and solution of the transient flow field can be clearly seen in the figure. The grid is able to keep a relatively good quality during the whole process, and the grid deformation portion of the simulation is very fast due to its algebraic nature. Figure 12 shows the pressure field around the weapons bay door.

Figure 8. CFD Grid and Model for 2D Flow Over A Weapons Bay. Here the Flow is Assumed to be Cross Flow from Left to the Right.

(a). 2D weapon bay door closing/opening

(b) From 30 deg < � < 90 deg, good quality grid using ROM.

Figure 7. Unstructured Grid Deforming Using ROM for Closing/Opening of Weapons Bay Door


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Figure 10. Unsteady Vortex Shedding due to Cross Flow From The Weapons Bay When the Door is Closed.

Figure 9. Instantaneous Vorticity Field for Cross Flow over a Weapons Bay Door.


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Figure 12. Pressure Field During Weapons Door Opening Due To Cross Flow

Figure 11. Vorticity Field Due To Interaction of Cross Flow and Weapon Door Opening. The Door is Opening at an Angular Rate of 8 Deg/Sec.


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IV. Demonstration of ROM Method for NACA 0012 Pitching Airfoil We consider the ROM method applied to a NACA 0012 pitching airfoil, which has been used to assess the

unsteady flow prediction capability of USM3D [3], and the overset unstructured grid capability of UMS3D [4]. The test condition is from an AGRAD report by Landon [5] with AGARD CT case I that corresponds to the test at a free stream Mach number of 0.6, and a Chord Reynolds number of 4.8x106. The angle-of-attack variation is prescribed as follows:

α(t)=αmean+αampl sin(2πft) (9) Where:

αmean =2.89 deg., αampl =2.41 deg., and f=50.34 Hz (10) Since the airfoil pitches with respect to the quarter chord length, the pitching angle is the only available

DOF. This makes ROM a perfect tool to deform the grid. The grid around the NACA 0012 wing is shown in Figure 13(a). The far field is set at 15 Chord lengths away from the airfoil surface. Only the region near the airfoil is subject to the grid deformation as shown in Figure 13(b). The red-colored lines can be taken as the overlap region in the overset Cartesian grid method. The angular mode shape functions for x and y are shown in Figure 13(c) and (d), respectively. Using these two functions, the grid deformation at any instance is just a vector multiplication of shape function with the angle.

This grid deformation was been implemented into a CFD code, and Figure 14 shows the pressure counters

and grid at the highest and lowest pitching angles of attack. For the present case, there is no noticeable overhead due to moving mesh.

Figure 13. NACA 0012 Pitching Airfoil.


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(a) Pressure and grid at the peak of the angle of attack.

(b) Pressure and grid at the lowest point of the angle of attack

Figure 14. Use of ROM for Pitching NACA0012 Airfoil Problem.

V. Conclusions By utilizing the properties of a maximum 6 degrees of freedom (DOF) of a moving body, the present

Reduced Order Method computes the grid deformation using nonlinear large deformation theory to preserve the original grid quality under each DOF, so that no extra computations are required during the unsteady motion, just matrix-vector multiplications. The deformation of unstructured grid around various control and moving surfaces has been shown to be very efficient with minimum grid distortion.

VI. Acknowledgements The present effort was supported through SBIR Program under contract number No: N68335-08-C-0452 funded by NAVYAIR and SBIR Program under contract number No. FA9302-09-M-0015 funded by the Air Force Flight Test Center at Edwards Air Force Base.

VII. References 1Yang, H. Q. “A Reduced Order method for Grid Deformation in Aeroelasticity Analysis”, AIAA-09-0890,

AIAA Science Meeting, 2009. 2Essam F. Sheta, H. Q. Yang, and S. D. Habchi, “Solid Brick Analogy for Automatic Grid Deformation of

Fluid-Structure Interaction”, AIAA-2006-3219, 36th AIAA Fluid Dynamics Conference, & Exhibits, June 5-8, San Francisco, CA, 2006.

3Pandya, M. J. Frink, N.T., Abdol-Hamid, K.S., and A Chung, J.J.: "Recent Enhancements to USM3D Unstructured Flow Solver for Unsteady Flows", AIAA 2004-5201, August, 2004.

4Pandya, M.J., Frink, N.T., and Noack, R.W.: "Progress Toward Overset-Grid Moving Body Capability for USM3D Unstructured Flow Solver,” AIAA 2005-5118, June 2005.

5Landon, R., “Data Set 3 NACA 0012 Oscillatory and transient pitching”, AGRAD Report 702, AGARD, January, 1982.

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[American Institute of Aeronautics and Astronautics 40th Fluid Dynamics Conference and Exhibit - Chicago, Illinois ()] 40th Fluid Dynamics Conference and Exhibit - Reduced Order Method