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Development and Preliminary Assessment of the Open-Source CFD toolkit SU2 for Rotorcraft Flows

Computational aerodynamic analyses of rotorcraft main rotor blades are performed in both hover and forward flight. The open-source SU2 code is used for rotor performance prediction. The core of the code is the set of RANS equations, which are solved for determining the flow. In hover, both steady-state and time-accurate modelling techniques of varying complexity are used and assessed. Simulation specific parameters which have a significant influence on the solution are also addressed. In forward flight, the code is developed to include the main rotor blade kinematics which is a prerequisite for modelling a trimmed rotor. Two databases are used for the validation of the rotor performance prediction. The renowned Caradonna-Tung experimental tests of a model rotor are used to evaluate the pressure distribution along the blade during hover. The extensive aerodynamic and aeroacoustic data survey of the AH-1G Cobra helicopter is used to assess the pressure distribution at different advancing and retreating azimuth angles during forward flight. The prediction capabilities of the solver in terms of rotor performance are demonstrated and are overall in good agreement with the measured data.


Introduction
Rotorcraft have the unique ability to be able to vertically take-off and land.
This allows for their operation in highly demanding and challenging flight scenarios where conventional fixed-wing aircraft cannot pursue. Their frequent use for demanding operations however can lead to dangerous in-flight situations which can be seen from the relatively high number of accident reports from the National Transportation Safety Board (NTSB) [1]. Tools that allow the analysis of rotorcraft are therefore essential for supporting design and for reducing accident rates. Sustained computational development over the past 30 years [2] means that many codes are now capable of modelling rotorcraft behaviour.
Computational advancements have enabled CFD-based methods to simulate and further understand complex rotorcraft aerodynamics which are rich in flow physics.
NASA have a long history of rotorcraft CFD code development with OVER-FLOW [3] and FUN3D [4]. Of the two codes, OVERFLOW has been used more extensively for rotorcraft simulations. OVERFLOW uses body-fitted structured grids near solid surfaces and automatically generated Cartesian grids in the background and is a finite-difference node-based solver. FUN3D uses unstructured grids throughout the domain and is a finite volume node-based solver. Academia also plays a leading role in rotorcraft CFD code development. In-house codes HMB from the University of Glasgow [5] and ROSITA from Politecnico di Milano [6] are both finite-volume solvers and utilize structured Chimera multi-block grids to account for the blade motion. Each of the codes have been part of large-scale collaborations for the assessment of their predictive capabilities such as the GOAHEAD project [7]. Another code developed within academia is the TURNS research code from the University of Maryland [8,9]. The TURNS code has Chimera overset grid capabilities and uses a finite difference numerical algorithm that evaluates the inviscid fluxes using an upwind-biases flux scheme. European research centers also have their own codes which have been heavily developed. French and German research in-stitutes ONERA and DLR have their own multi-block Chimera based rotorcraft codes. ONERA have the elsA solver [10] and DLR have the FLOWer solver [11]. is shown to be increasingly important so to identify a wide variety of physical modelling, discretization, and solution errors [12]. To that end, an open-source code with an active and growing community of users provides a platform for extensive VnV and innovative new solutions.
This work looks to provide the first open-source and validated rotorcraft CFD code by developing the well established SU2 code [13]. The main contribution of this work is the introduction of the blade motion and an approach for modelling forward flight. The contemporary nature of the code means that it can benefit from new and effective numerical techniques. The recent implementation by Gori et al. [14] of a method for dealing with non-conformal boundary interfaces using the supermesh technique is one example of this. Another example of this is radial basis function mesh deformation which can now utilize computationally efficient algorithms for dealing with large-scale problems. The outline of this paper is as follows; the physical modelling of the RANS equations is discussed in Section 2, the numerical implementation is described in Section 3, the validation work and results of an isolated rotor in hover and forward flight are discussed in Section 4, and finally the main talking points of the work are concluded in Section 5.

Physical Modelling
The following section on the physical modelling of the Reynolds-averaged Navier-Stokes equations inside of the SU2 code is primarily a summary of the work from Economon et al. [13] and is here to give context to the numerical implementation in Section 3.

Reynolds-Averaged Navier-Stokes Equations
Within this framework, we are interested in time-accurate turbulent flow around rotor blades with arbitrary motion. Therefore we are concerned with compressible flow governed by the Reynolds-averaged Navier-Stokes (RANS) equations. These mass, momentum, and energy conservation equations can be expressed in arbitrary Lagrangian-Eulerian differential form as, where the fluid density, flow velocity vector, static pressure, temperature, and specific heat are respectively given by the orthodox notation ρ, υ, p, T, c p . Of the remaining variables; E, is the total energy per unit mass;τ is the viscous stress tensor; u Ω is the grid velocity.

Turbulence Models
Here we are concerned with rotorcraft which during forward flight, climb, descent, and manoeuvre can be characterised by unsteady turbulent flows. With this in mind, the solution of the unsteady Reynolds average Navier-Stokes equations need to be solved which requires the inclusion of a turbulence model. Using the Boussinesq hypothesis [15], the effect of turbulence is represented as an increase in the viscosity. The total viscosity is then separated into laminar and turbulent viscosity and can be respectively denoted as µ dyn and µ tur . The laminar viscosity is determined based upon Sutherland's law [16]. The total viscosity as part of the momentum and energy equations in Eq. 1 is then substituted by where the laminar and turbulent Prandtl numbers are given by P r l and P r t .
The turbulent viscosity is computed using a suitable turbulence model which itself is dependent upon the flow state and a new set of variables,ν, to represent the turbulence, such that, µ tur = µ tur (U ,ν). One of the most widely used models for aeronautical attached flows is the one-equation Spallart-Allmaras (SA) turbulence model [17].

Rotating Frame of Reference
The unique ability of rotorcraft to be able to hover as well as to climb and descend vertically in axial flight helps to distinguish them from other aircraft.
During these specific operational flight conditions, the flow around the main rotor can be considered as a steady rotation. Under this assumption, it is then possible to transform the unsteady problem into a steady problem to improve the efficiency of the simulation. This is possible by transforming the system of governing equations in Eq. 1 into a rotating frame of reference which rotates at the constant rotational velocity of the main rotor blades. This modification to the system was implemented by Economon et al. [18] inside the SU2 code and can be written as, where the rotational velocity vector of the rotating frame of reference is specified by ω = {ω x , ω y , ω z } T and the radial distance from the center of rotation is specified by r.

Numerical Implementation
Alongside the physical modelling in the paper from Economon et al. [13] there is also a full description of the numerical implementation. In detail, it discusses the spatial and temporal integration which are the foundation for more application-specific methods such as the harmonic balance and non-conformal boundary interface treatment. The concurrent implementation for modelling rotor blade kinematics will finally be introduced.

Spatial Integration
The flow equations are solved numerically via a finite volume method [19] which is applied on unstructured grids with an edge-based structure. Integrating the governing equations over a control volume and using the divergence theorem to obtain the semi-discretized form gives, Centered or upwind schemes such as the Jameson-Schmidt-Turkel scheme [20] or the approximate Riemann solver of Roe [21] are then used to discretize the convective fluxes. Second-order of accuracy of upwind schemes is achieved by using Monotone UpStream-Centered schemes for conservation Laws (MUSCL) [22] to reconstruct the variables on the cell interfaces. Green-Gauss or weighted least-squares methods are used approximate the spatial gradients of the flow at the cell faces to determine the viscous fluxes.

Time Integration
In conjunction with the spatial discretization, there is also the requirement for temporal discretization of the governing equations and so Equation 5 is further discretized over a control volume, |Ω i |, in time such that it becomes, and a backward Euler scheme can be used to evaluate the solution state at the updated time for steady problems.
The temporal discretization for unsteady simulations is achieved using a dual-time stepping approach which allows for second order accuracy in time [23]. Using this approach, an additional fictitious time is introduced so that the unsteady problem becomes a series of pseudo-steady problems. The fictitious time is introduced in front of the governing equations such that, with the subscript n denoting the physical time level and to achieve secondorder backward difference in time, where the physical and fictitious time are expressed by ∆t and τ respectively.
The convergence of every physical time in pseudo time results in the modified residual equates to R * (U n ) = 0 meaning it is equivalent to finding the state

Harmonic Balance
The high computational cost of time-accurate methods means that it is often desirable to use reduced-order models for simulating unsteady problems.
Adopting the harmonic balance method is beneficial as it can be used for quasiperiodic flows dominated by a specific set of frequencies which need not be integral multiples of each other [24]. The subsequent harmonic balance method was implemented by Ref. [25] in SU2 and will briefly be summarised.
In the harmonic balance method the time operator, D t , is introduced and is approximated using spectral interpolation. Applying the spectral operator to the vector of conservative variables,Ũ , which are now to be evaluated at N time instances, one obtains, and the harmonic balance operator D t can eventually take the form, where H is the spectral operator matrix. While now consideringŨ as the vector of conservative variables evaluated at N time instances, Equation 7 can be rewritten for a single time instance as, where q +1 is the physical time step index and where n is now the time instance.
The expression can then be linearised and a semi-implicit approach can be used to solve for each time instance. An unsteady problem can then be characterized by K frequencies.

Non-Conformal Boundary Interface Treatment
The numerical discretization of complex rotorcraft geometries is a non-trivial task. It requires the main and tail rotor blades to move in relative motion to the fuselage. To permit this, the computational domain should be split into separate sub-regions, thus introducing non-conformal boundary interface.
Suitable treatment of each artificial boundary interface is achieved through the supermesh technique [26] implemented by developers Gori et al. [14] of SU2.
The supermesh acts an additional auxiliary grid between two non-conformal interfaces and is of size n − 1 dimensional elements where n is the dimension of the computational domain. The algorithm used to construct the supermesh of two generic parent mesh is shown in Fig. 1 and subsequently outlined: Identify the nodes of the control volume element E B contained within E A .

Determine the points of intersection between the edges of E
3. Decompose the overlap region into triangles. 4. Calculate the overlapping area of E B inside of E A and the weight W i .
this procedure is repeated for each of the overlapping neighbouring elements and its contribution to the numerical flux is assembled by Where the number of supermesh faces that are mapped to each control volume is denoted by N sf . The flux balance across the supermesh interface is obtained through incorporating the neighbouring cells either side of the boundary into Eq. 5, which can then be re-written in the semi-discrete form in the cell center of each control volume, k as, where the number of internal faces and boundary faces of each control volume is represented by N f and N bf . The area of the face is represented by ∆S.
The vector of fluxes are represented as F . The conservative variables are once again represented by U and the subscripts p and q denote the neighbouring and boundary cells.

Main Rotor Blade Kinematics
The method in which conventional main-rotor/tail-rotor helicopters generate their vertical and propulsive forces, as well as moments to control the attitude and position of the helicopter in three-dimensional space, will now be introduced and its implementation within SU2 discussed. A schematic depicting the blade motion is shown in Fig. 2. Unlike fixed-wing aircraft, the rotor blades alone must provide both the lifting forces and the control meaning the aerodynamics and dynamics of the blades are coupled so that it is hard to understand one without the other. During hover and axial flight, the velocity variation along the blade is azimuthally axisymmetric and depends solely upon the relationship of ωr. In forward flight, however, this is not the case and a component of the free-stream velocity alters the blade velocity such that it now depends upon its azimuthal position, ψ as given by, where M n represents the blade normal Mach number and M ∞ is the freestream Mach number. The advance ratio, µ, can be considered as the relationship of M ∞ /M tip . A repercussion of local variations in velocity is that each blade is now required to be individually controlled to eliminate rolling and pitching moments. With this, each blade is typically connected to a rotor head by a set of hinges which allows the blades to move independently. In reality, the rotor blades are also highly elastic in nature, however, for simplification this work considers the blades to be rigid with the ramifications of this being understood.
Rigid blades are then able to move with respect to the hinge positions. Each of the hinges will now be introduced: a flap hinge allows the blade to move in the plane containing the blade and the shaft; a pitch hinge allows the blade to move around the quarter chord and spanwise axis; finally a lead-lag hinge allows the blade to move in-plane forward or backwards. The flapping β, lead-lag δ, and pitching θ motion can then be described as a function of the azimuthal position ψ of the blades as they rotate, where β 0 denotes the coning angle and θ 0 denotes the collective pitch applied equally to all the blades. The lateral and longitudinal components of the motion are represented by subscripts c and s respectively.
A rotation matrix then transforms the blade Cartesian coordinates, x, from the fixed frame of reference into the hub frame of reference. The hub reference system is then transformed into each of the blade reference systems and the blade motion laws described in Equation 14 are applied. If we now consider the definition of the reference system where; the rotation occurs around the z-axes, the flapping occurs around the y-axes, the lead-lag motion around the z-axes, and the pitching occurs around the x-axes as depicted in Fig. 2 the following transformation matrices can be introduced to prescribe the blade motion, The transformation from the fixed → hub reference system using C rot is implemented by a rigid rotation of the entire grid as the azimuthal rotation of the blades far exceeds the blade deflections. The rate of rotation is currently constrained and is maintained constant such that, The transformation from the hub → blade reference system can then be combined and applied to the blade surface. This is achieved through deforming the mesh at each physical time step and is required as each blade has its own independent motion. The subsequent transformation matrix, C flap-leadlag-pitch can then be introduced, cos β cos δ sin β sin θ − cos β cos θ sin δ cos θ sin β + cos β sin δ sin θ sin δ cos δ cos θ − cos δ sin θ − cos δ sin β cos β sin θ + cos θ sin β sin δ cos β cos θ − sin β sin δ sin θ The motion can then be applied to the coordinates, x = {x, y, z} T at an arbitrary point, i on the blade as, where x hinge is the position of the hinge about which the flapping, lead-lag and pitching occurs. Furthermore the grid velocity is updated using a second order finite difference scheme, Rotation Around Z-Axes

Radial Basis Function Mesh Deformation
To allow the rotor blades to move independently requires the mesh to deform at each physical time step. Rendall and Allen, to the authors' knowledge, were the first and only group to demonstrated how Radial Basis Functions (RBF) could be used to account for the blade motion [27]. The outlook of using RBF mesh deformation for rotor blade motion was promising however one major drawback was the high computational cost for large meshes. Despite this, RBF mesh deformation methods are robust and preserve high-quality mesh even during large deformations. They also have the unique property that they do not require the grid connectivity meaning that even for three-dimensional problems they are relatively simple to implement.
The term radial basis function refers to a series of functions whose values depends on their distance to a supporting position. In the most general of forms, radial basis functions can be written as, φ(r, r i ) = φ ( r − r i ), where the distance corresponds to the radial basis centre, r i . This distance is frequently taken as the Euclidean distance, meaning it becomes the spatial distance between two nodes.
An interpolation function, f (r) can be introduced as a method for describing the displacement of a set of nodes in space and can be approximated by a weighted sum of basis functions. However, the interpolation relies on the weight coefficients of the basis points, α. The interpolation takes the form The primary expense of RBF mesh deformation is associated to the solving the linear system to obtain the weight coefficients. To address the concerns of the computation cost of RBF for rotor blades, multi-level greedy surface point selection algorithms [28] and volume point reduction methods [29] for large scale problems are introduced which greatly improve the efficiency. These numerically enhanced methods were implemented by Ref. [30] in SU2 and are now used within this work.

Hover
The  Table 1.  An assessment of the most influential parameters during hover simulations are displayed in Fig. 5. Each of the parameters are compared against the measured data at r/R = 0.80%. The spatial mesh resolution is assessed in Fig. 5a.
Four levels of mesh refinement are used ranging from coarse → very fine. The Finally the flow regime is evaluated in Fig. 5g. It depicts the influence of inviscid, laminar and turbulent flow on the pressure distribution. As excepted there appears to be significant difference between inviscid flow and laminar and turbulent flow regimes.
A contour map of the pressure coefficient on both the suction and pressure sides of the blades for time-accurate problems is shown in Fig. 6. The approach using unsteady rigid motion is displayed at 10 • azimuth increments in Fig. 6a.
The approach using the harmonic balance is displayed at 3 time instances with the input frequencies of ω = (0, ±ω 1 ) in Fig. 6b. Both sets of results show that the pressure coefficient during hover is axisymmetric about the out-of-plane axis.
The iso-surface of the Q criterion visualizing the near-field wake and blade tip vortices during hover are displayed in Fig. 7. The wake exhibits no interaction effects due to the preceding blade tip vortices passing above previous blade tip vortices and due to the isolated blades being modelled without a hub or test stand.

Forward Flight
The extensive report and data survey from Cross and Watts on tip aerodynamics and acoustics [33] is used for forward flight performance validation. The flight tests chosen for the present validation work are detailed in Table 2.
Two subsets of the flight tests are used. The first being a low-speed test which correlates to counter number 2157 from the database and the second being a high-speed test which correlates to counter number 2152 from the database.
During the 150 km/h low-speed condition the rotor is operated at a tip Mach number of 0.65 and an advance ratio of 0. 19. The collective pitch of 11.7 • is set to be congenial with the measured rotor thrust of C T = 0.00464 and the cyclic pitch is trimmed to eliminate pitching and rolling moments. The sine and cosine cyclic pitch angles are respectively set to 1.7 The spatial discretization is achieved using a single-zone mesh of the two main rotor blades. The outer far-field is placed 5 radii away from the blades.

Conclusion
This work provides a unique approach for the modelling of rotorcraft aerody- (c) r/R = 0.89%.                                   Tables   1  Hover test conditions taken from the Caradonna-Tung experiment 15   2 Forward flight conditions taken from the TAAT data survey . . . 18