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Improvements of High-Order Unstructured Grid Schemes

through RBF Interpolation

Qiuying Zhao1 and Chunhua Sheng

2

Mechanical, Industrial and Manufacturing Engineering The University of Toledo

Toledo, OH 43606

An effort to improve the numerical accuracy of high-order unstructured grid schemes is

presented in the present study. The basic concept is to construct a high-order polynomial

approximation using both solution variables and their gradients to calculate the Riemann

fluxes on the discontinuous interface. The high-order fluxes are then integrated over each

control volume using the RBF interpolation concept to achieve a higher order accuracy. Two

vortex-dominated viscous flows, a NACA 0015 stationary wing at a subsonic Mach number

and a NACA 0012 hovering rotor at a transonic tip speed, are presented to demonstrate the

efficacy and accuracy of the improved high-order method.

I. Introduction

n the previous work, a new flux reconstruction scheme [1] was introduced in an unstructured grid CFD solver, based on the finite-volume concepts [2-6], using high-order polynomial reconstructions within each control volume. As an extension to the Riemann solver [7], flow variables on the interface are reconstructed using

quadratic and quartic polynomials to provide a third- and fifth-order spatially accurate scheme, respectively. This finite-volume discretization [1] provided both improved accuracy and compact stencils usually seen in discontinuous Galerkin methods [8-11]. The high-order polynomials were constructed based on the upwind-biased formulas using both solution variables and their gradients at the vertex of the control volume and its adjacent cells. Although a high-order projection of fluxes was calculated, a uniform distribution of reconstructed fluxes was assumed across each interface for the sake of simplicity in the finite-volume integration.

In order to preserve the high-order accuracy of the finite-volume scheme, the surface integral of fluxes has to be implemented in a high-order manner corresponding to the overall numerical accuracy of the scheme. Traditionally, this was accomplished through the Gaussian quadrature rule [12] for specific types of surface shapes (such as triangles). However, although the Gaussian quadrature can guarantee an accurate integration for the lower-order equations under the specified order of polynomials in one-dimensional cases, the effect of the Gaussian quadrature for multidimensional cases it is still an open issue [13]. Besides, the quadrature points depend on the shape of the

integration surface, which also restricts the calculation on unstructured meshes. Xiao, et al. [13] and Mousavi, et al. [14] have developed an iterative method to calculate the Guassian quadrature on arbitrary shapes in two-dimensional problems, but introduced additional costs. In the present study, a new approach is introduced based on the Radial Basis Function (RBF) interpolation [15-16] as an alternative way to the Gaussian quadrature integration. The RBF method is a weighted sum of translations of a radically symmetric basis function augmented by a polynomial term. The general theory of RBFs was presented by Buhmann [15] and Wendland [16]. RBFs are popular for interpolating

scattered data as the associated system of linear equations is guaranteed to be invertible under very mild conditions of the data points. The RBF interpolation method has been extensively investigated by Rendall and Allen [17] as an effective way on grid deformations in Fluid-Structure coupled computations. Beckert and Wendland [18] developed a multivariate interpolation scheme for coupling fluid (CFD) and structural models (FE) in three-dimensional spaces using the RBF interpolations. In particular, RBF method requires no data connectivity information, making the selection of the integration points much easier than the Gaussian or multiquadric method. For this reason, RBF

interpolation method is employed in the present study to approximate the surface integral of the high-order fluxes

1 Graduate Research Assistant, MIME, University of Toledo, OH 43606, AIAA Member

2 Associate Research Professor, MIME, University of Toledo, OH 43606, AIAA Associate Fellow

I

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3856

Copyright © 2011 by Chunhua Sheng. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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within the control volume. The geometric sub-cells of each unstructured mesh element are used as the quadrature area and the gravity centers of them as the quadrature points.

The high-order numerical schemes have many practical applications in a wide range of fluid-related computational problems. One particular application is the vortex-dominant flows associated with the aircraft wings

or rotors. The accurate prediction of the trailing tip vortex generated by the wing or rotor is one of the most challenging problems faced by CFD today. The capabilities of newly developed high-order unstructured grid schemes using the RBF interpolation method are investigated and validated in two vortex-dominant viscous flows, a NACA 0015 [19] wing at subsonic Mach number and a NACA 0012 rotor [20] at transonic tip Mach number.

II. Methodology

A. Governing Equations

The unsteady three-dimensional Reynolds-averaged Navier-Stokes equations are cast in Cartesian coordinates. The compressible governing equations are expressed in terms of primitive variables and in a conservative flux formulation. In order to solve flows from essentially incompressible regime to supersonic Mach number, a preconditioning matrix is introduced into the time derivative terms of the compressible governing equations,

which can be expressed as [21-23]

where is a constant diagonal matrix that only depends on the reference Mach number . M is the

transformation matrix from conservative variables to primitive variables F is the vector of the flux on the face of the control volume, which can be written in the face

normal direction, . is the normal velocity on the face of a control volume, and is an Eckert

number. , , are the viscous flux components, and is the heat flux term.

B. High-order Reconstruction

Based on the concept of upwind schemes, high-order spatial accuracy can be achieved by introducing more spatial points to calculate the function derivatives in the finite-volume schemes, which is an extension of Godunov‟s approach [24] followed by van Lear [25, 26]. If the solution at the center of the control volume is directly projected

onto the interface on a piecewise constant base, it is equivalent to the first-order spatial accuracy of the Godunov scheme. However the Riemann problems are solved at the interface of the control volumes, in which the projected approximation is decoupled from the physical stage. Therefore in order to produce a high-order spatial accuracy, the piecewise constant approximation within each control volume should be replaced by a higher order representation of the flow variables without modifying the Riemann solver. The method for the second-order upwind scheme using a linear extrapolation is often referred to as the MUSCL approach [25, 26], which stands for Monotone Upstream-

centered Schemes for Conservation Laws, after the name of the first code applying this method developed by van Leer. Similarly, a quadratic representation leads to a third-order scheme; and a quartic representation leads to a fifth order scheme; and so on.

In the previous work [1], the high-order discretization was achieved by extending the concept of the MUSCL scheme in the Riemann solver. The Roe scheme [27] was employed for the inviscid flux computation. Flow variables are extrapolated onto the interface through a quadratic polynomial for the third-order scheme and a quartic

polynomial for the fifth-order scheme. By using both flow variables and their first- and second-order gradients that

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are stored at the vertex of the control volume and its adjacent volumes, the reconstructed flow variable based on a quartic polynomial, for example, can be expressed at the interface of the control volume as

where is the distance vector from the interface to the centroid of the control volume. Constants , , , , and are the basis functions evaluated using the function value and the first and second gradients to define the high-

order polynomial within each control volume [1].

C. RBF Interpolation

To maintain the high-order accuracy of a finite-volume scheme, the numerical integration of surface fluxes must

be implemented in a high-order manner without degrading the over accuracy of the high-order fluxes constructed using polynomial (4). Due to unresolved issues of multi-dimensional integration and the complexity associated with irregular surface shapes for the traditional Gaussian quadrature, a novel interpolation method based on the Radial Basis Function (RBF) method is introduced and investigated here, which can be viewed as an alternative way to the Gaussian quadrature integration over the surface of control volumes for the high-order schemes (see Section C).

RBF method is a weighted sum of translations of a radially symmetric basic function augmented by a polynomial

term. The general theory of RBFs was presented in details by Buhmann [15] and Wendland [16]. RBFs are popular for interpolating scattered data as the associated system of linear equations is guaranteed to be invertible under very mild conditions on the locations of the data points. Rendall and Allen [17] developed a unified mesh deformation method using the RBF method and applied it to fluid-structure interaction calculations. In particular, RBFs do not require that the data lie on regular grid points (connectivity information), making the selection of integration point much easier than the Gaussian or multiquadric method. Since the RFB interpolation can be used to construct a high-

order smooth curve based on any scattered points, the evaluation of quadrature points within each control volume for general high-order schemes can be replaced by a more flexible and lower cost RFB interpolation. Another advantage of the RBF interpolation is its global nature, meaning that all high-order variables reconstructed within each control volume can be used as the basis points to build an interpolation polynomial, while other methods can only used a selected number of points.

The solution of an interpolation problem using RBFs begins with the form of the required interpolation

Here, s(x) is the function value to be evaluated at location x, is the form of a basis function. The index i denotes the basis points of the RBFs and xi is the coordinates on the basis points. The last term p(x) is a polynomial up to a linear degree [18] to ensure that the translational and rotational motions are recovered by the RBF interpolation. The polynomial in the x direction (equivalent results hold in y and z directions) is defined as

The coefficients in Equation (5) are found by requiring exact recovery of the original function, in this case the

basis point values. When the polynomial term is included, the system is completed by an additional requirement (sometimes referred to as a „side condition‟)

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It was found that the selection of different RBF functions and the number of basis points both play an important role in preserving the interpolation accuracy of the high-order polynomial within the control volume. In the current study, Wendland‟s C2 RBF function [17] is employed, which has the following form

To evaluate the impact of the Wendland‟s C2 function (8) on the accuracy of polynomial interpolations, the interpolation errors are analyzed by choosing different number of basis points for interpolating a quadric and a quartic polynomial, which represents the third- and fifth-order schemes being investigated in this study. The emphasis here is to find out the resolution characteristics of the Wendland‟s C2 function (8) rather than the formal order of the accuracy (i.e., truncation error). The resolution characteristics means the error bound by which different approximations represent the exact value of a polynomial over the full range of length scales that is realized in a

given mesh. One dimensional formula is used for the purpose of simplicity. From the interpolation point of view, an ideal interpolation scheme should have the property that the values evaluated at the basis points (known values) must satisfy the exact solutions of the polynomial, and the values evaluated at other interpolated points are close enough to the exact solutions within the specific error tolerance. The first condition is automatically satisfied with the Wendland‟s C2 RBF function [17], and the second condition is the issue that needs to be evaluated and addressed here. Figure 1 illustrates the relative errors by choosing different number of basis points on a uniform grid

of 21 discretized grid points within a control volume. Obviously, the interpolated values by RBF exhibited an oscillatory behavior around the basis points. The magnitude of the oscillation, which is also the error bound, has a direct impact on the interpolation accuracy of the RBF method, and thus affects the overall stability and accuracy of the finite-volume schemes. The error bound is significantly reduced with the increase of number of basis points. Numerical tests indicated that at least 14 basis points for the quadratic polynomial, or 16 basis points for the quartic polynomial, should be adopted to maintain an error bound below 0.2% of the exact solution.

D. Surface Integration

With the reconstructed flow variables evaluated at the interface of each control volume, high-order discretization schemes may be formed with the proper integral of the surface fluxes over the control volume. In the previous work [1], however, a simplification was made by assuming the high-order flux evaluated at the center of the interface to be uniform across that surface, which may compromise or degrade the overall accuracy of the high-order schemes.

Traditionally, the high-order surface integration was implemented based on the Gaussian qudrature rule [12]. However, the theory for numerical integration in multi-dimensional spaces is far from complete, since the minimum number of points required for a given number of functions on a given domain is yet unknown [13]. Besides, the selected quadrature points are depended on the shape of the integration surface, which also restricts the implementation and flexibility of the method applied to general unstructured meshes.

In the present study, an alternative way to the Guassian quadrature, which is based on the RBF method, is

introduced and evaluated in the surface integration of a finite-volume scheme. Instead of calculating the Guassian

3rd Order Scheme

Figure 1 Error estimation for Wendland's type RBF function in quadratic and quartic polynomials

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quadrature points on irregular surfaces using a costly iterative method [13], all sub-cell faces that compose of the interface of the control volume are used for the high-order surface integration. The values on these sub-cells are obtained through the RBF interpolation. Figure 2 shows a diagram of the sub-cell surfaces on a two-dimensional mesh. For node-based unstructured schemes, variables are stored at the center nodes (vertex) of the control volume

(dash line). The lines connecting nodes are edges. Values at the centers of all edges of a control volume, which are called basis points, are evaluated by the high-order polynomial reconstruction using Eq. (4). Values at the center of each sub-cell faces are evaluated by the RBF interpolation using Eq. 5. Instead of only using the basis points for the surface flux integration, as did in the previous work [1], all sub-cell points are used to perform the high-order surface integration to improve the overall accuracy of the high-order finite-volume scheme.

III. Results

Two vortex-dominant viscous flows are presented here as validations for the RBF interpolation method. The first case is a stationary wing NACA 0015, which is mounted on a flat wall at a subsonic Mach number [19]. The second case is a rotational rotor NACA 0012 with a transonic tip speed [20]. All viscous meshes are generated with mixed element cells, with the y+ value about one on all solid surfaces to resolve the viscous sublayers. One-equation Spalart-Allmaras turbulence model [28] is used for all computations.

A. NACA0015 Wing

NACA0015 wing is a three-dimensional untwisted wing tested by McAlister, et al. [19]. The wing was positioned at 12 degrees angle of attack and mounted on a flat wall. The incoming free stream Mach number is 0.1235, and Reynolds number is 1.5 million based on the chord length. The computational mesh for the wing has 3.66 million nodes shown in Figure 3. An interior cylindrical surface was built along the trailing vortex trajectory to provide a refined grid resolution for capturing the flow details. In this steady flow computation, a local time step is used with a maximum CFL number of five. One Newton‟s iteration and six to eight symmetric Gauss-Seidel

relaxations are applied in the computations.

A key feature of this flow is the trailing tip vortex that lasts to the far downstream in the computational domain, see Figure 3. Predicted strength and trajectory of the trailing vortex is a good indicator to the numerical accuracy of a scheme. The higher order the spatial accuracy, the stronger the trailing vortex predicted in the field, as the numerical diffusion is greatly reduced compared to the conventional second-order scheme. Numerical computations are performed using the second-, third-, and fifth-order discretization schemes with and without the RBF

interpolations. Predicted vortical velocity profiles are analyzed and compared to assess the improvement of the RBF method.

Certainly, even if without the RBF interpolation, results obtained by the third- and fifth-order schemes are less diffused compared with the one obtained by the second-order scheme, as reported in the previous work [1]. Figures

3rd Order Sc heme

Figure 2 Interpolation on control surfaces of the two-dimensional unstructured grid

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4-6 compare the cross-flow velocity contours at 6 chords downstream obtained by various schemes with and without the RBF interpolation. The impact of the RBF interpolation is evident in the third- and fifth-order schemes, but little effect is observed in the second-order scheme. This is understandable because a liner polynomial representation for the second-order scheme will receive little effect as the surface integral is performed over the refined sub-cell faces.

However, the impact of the RBF interpolation is rather significant in the third- and fifth-order schemes. The captured vortex by the third-order scheme using the RBF interpolation is even better than that captured by the fifth-order scheme without using the RBF interpolation. Obviously, the vortex strength predicted by the fifth-order scheme using the RBF interpolation is the strongest.

Figure 3 The unstructured mesh of NACA 0015 stationary wing

NACA 0015 wing

Refined mesh in vortex core region

Viscous boundary layer

Trailing tip vortex

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

East

West

North

vortex

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3rd Order Scheme

Figure 6 Predicted cross section velocity contours by 5th

order scheme at 6 chords downstream

5th

order with RBF 5th

order without RBF

0.50

0.25

0.00

Figure 5 Predicted cross section velocity contours by 3rd

order scheme at 6 chords downstream

0.50

0.25

0.00

3rd

order with RBF 3rd

order without RBF

Figure 4 Predicted cross section velocity contours by 2nd

order scheme at 6 chords downstream

0.50

0.25

0.00

2nd

order with RBF 2nd

order without RBF

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Further quantitative comparisons about the effects of the RBF interpolation are shown in Figure 7. Predicted vortical velocities across the vortex core at several downstream locations are also compared with the experimental data [19]. The peak velocity within the vortex core is severely smeared out by the conventional second-order scheme (with and without RBF) due to excessive numerical diffusions. The prediction accuracy is improved by the third- and fifth-order schemes even if without the RBF interpolation [1]. However with the RBF interpolation, predicted

vortical velocities are much closer to the experimental data, in particular in the fifth-order scheme. The vortex decay rate is very small in the fifth-order solution as the predicted peak velocities show little changes from the wing trailing edge to 6 chords downstream.

Predicted pressure coefficients (Cp) on the surface of the wing with different schemes are presented in Figure 8. Pressure coefficients along the wing span are compared with the experimental data. No significant differences are found among various schemes, including the use of the RBF interpolations. This is because that the trailing tip

vortex, which receives the most impact by the RBF method, only affects the flow field downstream of the wing.

3rd Order

Sche me

Figure 7 Comparison of vortical velocity profiles by various schemes at different downstream locations

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B. NACA0012 Rotor

The second case is about a rotary wing, a NACA 0012 rotor in hover. The experiment test was conducted by

Caradonna, et al. [20]. The rotor has two untwisted, untapered blades constructed with the NACA 0012 airfoil. The diameter of the rotor is 2.286 meter and the aspect ratio of the rotor is 6. Collective pitch abgle for the rotor blades is 8 degrees. The rotating speed of the rotor is 2500 RPM, and the blade tip Mach number is 0.877. The Reynolds number here is 4.6x10

7 based on the blade tip speed and the diameter of the rotor. The unstructured viscous mesh for

the NACA 0012 rotor was generated that has 6.37 million volume nodes shown in Figure 9. The grid points are packed around the blades as well as the downwash of the rotor in order to capture the tip vortex and wake flows.

Computations were carried out in the rotating relative frame, where the flow field can be viewed as a steady state. A local time step was used at a CFL number of 2-10.

Figure 8 Predicted pressure coefficients by different schemes along the wing span

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Like the previous stationary wing case, the key feature of the hovering rotor is the trailing vortices and wakes generated by the rotor blades in the downwash region, which have a great impact on the aerodynamic loading of the rotor. For rotors in forward flights, the rotor wake will cause the blade-vortex interaction phenomenon. Therefore,

the accurate prediction of the rotor trailing vortex and wake is essential to the rotor aerodynamic design and analysis. Shown in Figures 10-12 are the iso-swirl parameters (value of 1.0) predicted by the second-, third-, and fifth-order schemes with and without using the RBF interpolation, respectively. Noticeable difference was found in the solutions obtained by the third- and fifth-order schemes over the conventional second-order scheme. Improved vortex strength and trajectory were revealed in the high-order solutions even if no RBF method was used. In addition, improved prediction to this vortical flow is clearly demonstrated in the helicity contours as shown below.

Figures 13-15 show the vortex strength represented by helicity contours predicted using various schemes with and without the RBF interpolation. In general, the helicity strength is stronger in the higher-order solutions comparing to the lower-order ones. With the RBF interpolation implemented in various schemes, the helicity strength is further enhanced in both third- and fifth-order solutions, similar to the effect observed in the previous fixed wing case. This validation demonstrates the applicability of the current high-order schemes using the RBF interpolation, which is suitable for vortex dominated flows in both fixed and rotary wings.

The pressure distributions on the rotor blade for different schemes are provided in Figures 16-18. Unlike the previous low Mach number flow, a shock is generated at around 80% blade span location in this transonic flow. Overall, similar patterns of the surface pressure are found for all solutions. However, noticeable differences (noises) are observed near the shock region, in particular in the third- and fifth-order solutions (with and without using the RBF interpolation). These noises are typical for high-order schemes as they tend to have less numerical dissipation than the second-order scheme. The surface pressure distributions are plotted in Figure 19 along the blade spans.

However, a similar pattern is obtained in all schemes with and without the RBF interpolation.

Figure 9 The mesh resolution of NACA0012 rotor

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3rd

order with RBF 3rd

order without RBF

Figure 11 Predicted iso-swirl parameters of NACA0012 Rotor by 3rd

order scheme

2nd

order with RBF 2nd

order without RBF

Figure 10 Predicted iso-swirl parameters of NACA0012 Rotor by 2nd

order scheme

5th

order without RBF 5th

order with RBF

Figure 12 Predicted iso-swirl parameters of 1 of NACA0012 Rotor by 5th

order scheme

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Figure 15 Predicted helicity contours of NACA0012 Rotor by 5th

order scheme

5th

order without RBF 5th

order with RBF

0.3

0.0

-0.3

Figure 14 Predicted helicity contours of NACA0012 Rotor by 3rd

order scheme

3rd

order without RBF 3rd

order with RBF

0.3

0.0

-0.3

Figure 13 Predicted helicity contours of NACA0012 Rotor by 2nd

order scheme

2nd

order without RBF 2nd

order with RBF

0.3

0.0

-0.3

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1.3

0.8

0.3

Figure18 Predicted pressure distribution on NACA0012 Rotor by 5th

order scheme

5th

order without RBF 5th

order with RBF

1.3

0.8

0.3

3rd

order without RBF 3rd

order with RBF

Figure 17 Predicted pressure distribution on NACA0012 Rotor by 3rd

order scheme

1.3

0.8

0.3

2nd

order without RBF 2nd

order with RBF

Figure 16 Predicted pressure distribution on NACA0012 Rotor by 2nd

order scheme

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IV. Conclusion

An improvement to high-order finite-volume schemes was introduced and developed based on arbitrary unstructured meshes. By introducing the Radial Basis Function interpolation procedure into the surface integration, a significant improvement was achieved for integrating the reconstructed fluxes in finite-volume schemes. This new concept has been investigated and validated on two vortex-dominated viscous flows including a fixed wing and a rotating rotor. However, certain issues, including the extra computational costs of the RBF interpolation and its effect on the formal order of accuracy, remain unsolved. Nevertheless, the RBF interpolation method presented in

this study offered an alternative way to the traditional Gaussian quadrature for the high-order unstructured grid schemes, which is much more straightforward to implement on any unstructured finite-volume schemes than the latter.

Acknowledgments

Special thanks are expressed to Dr. Christian Allen of the University of Bristol, U.K. for his insights, suggestions and discussions throughout this work.

Figure 19 Predicted pressure coefficients by different schemes along the bade span of NACA0012 Rotor

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