Mathematical Problems in Engineering

Volume 2018, Article ID 9745862, 20 pages

https://doi.org/10.1155/2018/9745862

## Helicopter Rotor Flow Analysis Using Mapped Chebyshev Pseudospectral Method and Overset Mesh Topology

^{1}School of Automotive & Mechanical Design Engineering, Youngsan University, Yangsan, Gyeongnam, Republic of Korea^{2}Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA

Correspondence should be addressed to Seongim Choi; ude.tv@1iohcs

Received 26 November 2017; Accepted 15 February 2018; Published 15 May 2018

Academic Editor: S. S. Ravindran

Copyright © 2018 Dong Kyun Im and Seongim Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Unsteady helicopter rotor flows are solved by a Chebyshev pseudospectral method with overset mesh topology which employs Chebyshev polynomials for solution approximation and a Chebyshev collocation operator to represent the time derivative term of the unsteady flow governing equations. Spatial derivative terms of the flux Jacobians are discretized implicitly while the Chebyshev spectral derivative term is treated in explicit form. Unlike the Fourier spectral method, collocation points of standard Chebyshev polynomials are not evenly distributed and heavily clustered near the extremities of the time interval, which makes the spectral derivative matrix ill-conditioned and deteriorates the stability and convergence of the flow solution. A conformal mapping of an arcsin function is applied to redistribute those points more evenly and thus to improve the numerical stability of the linear system. A parameter study on the condition number of the spectral derivative matrix with respect to the control parameters of the mapping function is also carried out. For the validation of the proposed method, both periodic and nonperiodic unsteady flow problems were solved with two-dimensional problems: an oscillating airfoil with a fixed frequency and a plunging airfoil with constant plunging speed without considering gravitational force. Computation results of the Chebyshev pseudospectral method showed excellent agreements with those of the time-marching computation. Subsequently, helicopter rotor flows in hovering and nonlifting forward flight are solved. Moving boundaries of the rotating rotor blades are efficiently managed by the overset mesh topology. As a set of subgrids are constructed only one time at the beginning of the solution procedure corresponding to the mapped Chebyshev collocation points, computation time for mesh interpolation of hole-cutting between background and near-body grids becomes drastically reduced when compared to the time-marching computation method where subgrid movement and the hole-cutting need to be carried out at each physical time step. The number of the collocation points was varied to investigate the sensitivity of the solution accuracy, computation time, and memory. Computation results are compared with those from the time-marching computation, the Fourier spectral method, and wind-tunnel experimental data. Solution accuracy and computational efficiency are concluded to be great with the Chebyshev pseudospectral method. Further applications to unsteady nonperiodic problems will be left for future work.

#### 1. Introduction

Challenges in helicopter rotor flow analysis are several. Blade wake interactions need to be solved accurately requiring higher mesh resolution in the wake-trailing regions. Moving boundaries of the rotating blades need to be handled dynamically in time [1–5]. A typical local, finite difference method for the time integration of the flow governing equations of the URANS (unsteady Reynolds-averaged Navier-Stokes equations), such as BDF2 (2-step backward differentiation formula) or BDF3 (3-step backward differentiation formula), of the dual-time-stepping method [6, 7] requires a small time step for solution stability. Although the pseudo-time integration for the inner iteration and a multigrid technique can make the solution converge faster, the time-marching computation is still expensive even with highly parallel computation as it needs to fully resolve unsteady transient flows before flows reach a periodic steady-state. On the other hand, a spectral method for time integration has been proposed [8–16] as an alternative to a time-marching, dual-time stepping method with much reduced computational cost at equivalent solution accuracy. With the flow solution approximation by a discrete Fourier series, a time derivative term of the governing equations reduces to a spectral derivative term and removes time dependence of the flow solutions. The time-spectral derivative is the multiplication of the spectral derivative matrix and solution vectors at all time instances. The flow solutions advance in a pseudo-time domain with periodic steady-state remaining throughout the iterations until flows converge. Variants of the time-spectral method have been developed by various researchers. Hall et al. [8] originally developed a harmonic balance method (HBM) to solve internal flows of a two-dimensional compressor and use a discrete Fourier series for the solution approximation. The improved HBM by Thomas et al. [9, 10] also approximates flow variables by a discrete Fourier series and represents the time derivative term of the governing equations in Fourier coefficients but solves the FFT-transformed governing equations directly in the time domain by taking the inverse transform of the frequency domain governing equations. This approach is more convenient as the flow solutions are integrated directly in time with residuals checked simultaneously in the time domain. Based on the same principle, McMullen et al. [13, 14] developed a nonlinear frequency domain (NLFD) method which is slightly different from the HBM in the details of the implementation. Although the flow approximation by the Fourier series and the formulation of the governing equations in the frequency domain are the same as the HBM, the flow solutions are directly solved in the frequency domain, while the convergence is checked by the residuals reformulated in the time domain by taking the IFFT of the conservative flow variables. Gopinath and Jameson [12] proposed a time-spectral method which replaces the time derivative term of the governing equations through the spectral derivative matrix operator using Fourier spectral differentiation. Similar to the HBM developed by Thomas et al. [9, 10], it integrates the spectral derivative term of the governing equations directly in the time domain. Ekici et al. [11] solved helicopter rotor flows using the harmonic balance method with periodic boundary conditions both in time and in space, which makes the computation domain more compact than that of the time-spectral method and contributes to further decreasing computation time and memory in comparison to the time-marching computation. Choi et al. [17, 18] used the time-spectral CFD computation method for fluid-structure interaction analysis of the rotor flows in unsteady forward flight of the UH60-A helicopter. Adjoint sensitivity analysis has been carried out in the time-spectral form of the URANS equations to compute the sensitivity of aerodynamic performance metrics (drag, lift, torque, etc.) with respect to hundreds of shape design parameters. Despite the rigid blade assumption, aerodynamic shape optimization using time-spectral and adjoint sensitivity analysis was validated with the FSI analysis afterward with good performance and shown to be effective in solving unsteady (i.e., periodic steady) flows. A parameter study on the accuracy and computation efficiency of flow solutions with respect to varying number of harmonics has been carried out at various flight conditions. A Chebyshev pseudospectral method [19–22] uses Chebyshev polynomials for the solution approximation of the boundary-value problems, either periodic or nonperiodic, and the spectral derivative matrix is derived correspondingly to continuous differentiation. Standard Chebyshev polynomials are defined in the interval of and are a member of an orthogonal polynomial set. They can approximate function values with great precision and are applied practically to mathematics, physics, and engineering areas. Dinu et al. [19] applied the Chebyshev pseudospectral method to an aeroelastic problem and solved unsteady aerodynamic forces accurately. Niu et al. [20] solved unsteady cylinder flows with the Chebyshev pseudospectral method applied to the spatial derivative terms of the momentum and energy conservation equations. Flow solutions demonstrated great agreements with those of the time-marching computation which uses the multistage Runge-Kutta method. Khater et al. [21] applied the method to the spatial derivative term of the flow governing equations of the Burgers equations in various forms and excellent agreements were observed in comparison to the exact solutions. Im et al. [23] applied a Chebyshev collocation operator to the time derivative term of the URANS equations and employed a conformal mapping function to improve the numerical stability of the method. They applied the method to the oscillating airfoil with a prespecified frequency and compared the computation results with those of the dual-time stepping method and the harmonic balance method. In the current study, helicopter rotor flows in a hovering condition and forward level-flight are solved using the mapped Chebyshev pseudospectral method applied to the time integration of the URANS equations. To accurately solve the blade-vortex interaction and vortex wake flows trailing from the rotor blades, two key approaches are made: conformal mapping of an arcsin function and the overset mesh topology. The arcsin function is used for the conformal mapping of the uneven collocation points of the standard Chebyshev polynomials and yields points more evenly distributed over the physical time interval of . Improved numerical stability is shown with respect to varying control parameters of the function. The application of the overset mesh topology is particularly advantageous with the mapped Chebyshev pseudospectral method, as the overset mesh construction and a hole-cutting procedure is carried out once and for all for subtime intervals corresponding to mapped collocation points of the Chebyshev polynomials. Cost for identifying hole-cutting topology between the near-body and background grids is reduced considerably when compared to the time-marching computation where it is carried out at each physical time step with the size often limited to a small value due to the numerical stability. The ratio of the number of subtime intervals between the time-spectral and the finite difference method is about : , and the corresponding cost saving is more than an order of magnitude. Despite some disadvantage of increase in computation memory per solution iteration due to the storage of meshes and flow solutions at all subtime intervals, the mapped Chebyshev pseudospectral method has shown cost effectiveness in the present study. In the current work, the overset mesh topology will be used for the mapped Chebyshev pseudospectral method to solve helicopter flows of Caradonna and Tung’s rotors in both hovering and unsteady nonlifting level flights. Aerodynamic results including section Cp distribution, thrust, and torque show excellent agreements in all computations and with wind-tunnel experimental data.

#### 2. Mapped Chebyshev Pseudospectral Method

A Chebyshev pseudospectral method uses orthogonal Lagrange polynomials to evaluate state and control parameters of a given function to approximate or reconstruct. Either Legendre-Gauss-Lobatto (LGL) or Chebyshev-Gauss-Lobatto (CGL) points are used to evaluate function derivatives. However, the LGL or CGL points are heavily clustered around the end points of the interval, and the corresponding spectral derivative matrix becomes ill-conditioned with a large condition number. If the original function shows strong nonlinearity around the center of the interval where only a few CGL points are available for the spectral interpolation, the prediction of function value becomes very inaccurate around the region. The uneven distribution of the collocation points also causes numerical instability of the Chebyshev pseudospectral method [24–28]. Conformal mapping based on the arcsin function is applied to the CGL points to reduce the condition number of the spectral derivative matrix and improve the stability.

##### 2.1. Chebyshev Collocation Operator [29]

An arbitrary function defined in time over the interval of can be represented as Chebyshev polynomials with GCL points:A vector of polynomial coefficients is written asA vector of function derivatives can be written similarly asthen the coefficient vector of the function derivative is represented asA variable of in (7) is defined only when the value of becomes an even number and is zero for the odd number. For the interval of , not , the elements of are divided by . Therefore, the time derivative of the arbitrary function is summarized aswhere represents the Chebyshev collocation operator and is a matrix of size .

##### 2.2. Conformal Mapping

The purpose of the conformal mapping is to mitigate the problem of ill-conditioning of the spectral derivative matrix. Collocation points or nodes in a given time interval are rewritten by the mapping function of , where is the standard Chebyshev collocation points represented as a cosine mesh.To have an interval bounded by , not by , a linear transformation is applied asThen, the function derivative of is written by the chain rule,For the conformal mapping function of , the arcsin function is used which was originally suggested by Kosloff and Tal-Ezer [25, 26]. A distribution characteristic is controlled by the constants of with the value of close to one resulting in more even distribution whereas smaller values push the collocation points towards the two ends of the interval. Analytic form of isVariables of and , shown in (11), are the functions of , , and time and can be written as below over the time interval of , where is typically set to be zero for the interval of time.The interval size of is shown in Figure 1. The graph shows the distribution of of inverse sine function. The number of mapping points is 37, where . The distribution of the inverse sine mapping function is consistent and uniform and is less sensitive to the values. A derivative function of the mapping function, , is as follows:It is a diagonal matrix of size with zero off-diagonal terms. Therefore, the mapped Chebyshev collocation operator with the mapping function applied is represented asThe size of the matrix shown in (16) is , and it represents the mapped Chebyshev pseudospectral collocation operator mapped to by a tangent function. The value of corresponds to the interval of as takes into account the . Table 1 shows the values of the condition number for the standard and the mapped Chebyshev spectral derivative matrices with the total number of collocation points increasing from 24 to 96 by a factor of 2, where the values of and are optimized for the lowest condition number for each case. Reduction of the condition number by an order of magnitude is shown in all cases in Table 1 with the application of the conformal mapping of the arcsin function shown in (16).