Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 108C (2017) 1883–1892
International Conference on Computational Science, ICCS 2017, 12-14 June 2017, Zurich, Switzerland
Numerical Simulation of Rotation of Intermeshing Rotors using Added and Method Numerical Simulation of Eliminated Rotation ofMesh Intermeshing Rotors using Added and Eliminated Mesh Method Masashi Yamakawa 1*†, Naoya Mitsunari 1 and Shinichi Asao 2 1
Kyoto Institute of Technology, Kyoto, Japan
1 College of1*† Industrial Technology, Hyogo, Masashi Yamakawa , Naoya Mitsunari andJapan Shinichi Asao 2 2
[email protected] Kyoto Institute of Technology, Kyoto, Japan College of Industrial Technology, Hyogo, Japan
[email protected] 1
2
Abstract To compute flows around objects with complicated motion like the intermeshing rotors, the unstructured moving Abstractgrid finite volume method was developed. Computational elements are added and eliminated according toflows motion of rotors, to with keepcomplicated the computation domain around rotors which reverse. To compute around objects motion like the intermeshing rotors,mutually the unstructured Also, thegrid geometric conservation satisfied inComputational the method, using fourare dimensional time moving finite volume methodlaw wasisdeveloped. elements added and space eliminated unified domain for control volume. the method,domain accurate computation is carried out reverse. without according to motion of rotors, to keepUsing the computation around rotors which mutually interpolation of physical quantities.law Applying to a in flow a sphere, computation procedure was Also, the geometric conservation is satisfied thearound method, using four dimensional space time established with introduction of concept of a hierarchical grid distinction. Then, the results of application unified domain for control volume. Using the method, accurate computation is carried out without to the flow around intermeshing showed of the method.computation The resultsprocedure also showed interpolation of physical quantities.rotors Applying to a efficacy flow around a sphere, was applicability of the method toofcompute around anygrid complicated motion. established with introduction conceptflows of a hierarchical distinction. Then, the results of application to the flow around intermeshing rotors showed efficacy of the method. The results also showed © 2017 TheCFD, Authors. Published by Elsevierflow, B.V. Moving grid, Finite volume method Keywords: applicability of Simulation, the methodCompressible toof compute flowscommittee around any complicated Peer-review under responsibility the scientific of the Internationalmotion. Conference on Computational Science Keywords: CFD, Simulation, Compressible flow, Moving grid, Finite volume method
1 Introduction
To design and estimate the aerodynamic performance for a helicopter, computational fluid dynamics 1is often Introduction used. In this case, the characteristic part of the computation is how to express the rotation of
rotors, then theand overset gridthe method and the performance sliding mesh for method are oftencomputational adopted. To design estimate aerodynamic a helicopter, fluid dynamics By the way, are known which is hashow a high autonomous stability. is often used. Inintermeshing this case, therotors characteristic partasofa helicopter the computation to express the rotation of In this then system, themethod main rotor cansliding be canceled by reversing directions of rotation of mutual rotors, the torque oversetof grid and the mesh method are often adopted. rotors. it isintermeshing good not onlyrotors at mobility but also efficiencywhich of engine. for computations By Thus, the way, are known as aathelicopter has a However, high autonomous stability. of the intermeshing rotors, it is difficult to adopt the conventional mesh methods using an overset grid In this system, torque of the main rotor can be canceled by reversing directions of rotation of mutual (Steger, 1987)it or using not a sliding In the sliding mesh method, overlapping of domains rotors. Thus, is good only atmesh. mobility butcase alsoofatthe efficiency of engine. However, for computations rotor avoid the computation. Then, in the the caseconventional of overset grid method, a truncation caused of each the intermeshing rotors, it is difficult to adopt mesh methods using an error overset grid (Steger, 1987) or using a sliding mesh. In the case of the sliding mesh method, overlapping of domains of each rotor avoid the computation. Then, in the case of overset grid method, a truncation error caused * † * †
Masterminded EasyChair and created the first stable version of this document Created the first draft of this document Masterminded EasyChair and created the first stable version of this document Created the first draft of this document
1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science 10.1016/j.procs.2017.05.061
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by interpolation of physical quantities will be increasing because of overlapping of most part of each sub-grid. Thus the method may not be suitable for the computation. As for the issue, the unstructured moving grid finite volume method with addition and elimination of computational elements was proposed. In this method, the elements are added and eliminated according to motion of object. Thus the single grid without overlapping of grid and without sliding surface can be represented. To satisfy a geometric conservation law (Obayashi, 1992) on elements adding, eliminating and moving, the finite volume method for space-time unified domain is adopted. As other approaches, ALE method (Donea, 1982) which links motion of grid and flow velocities has also been used. However it is not easy to assure the law perfectly. Using this method which has a function of addition and elimination of computational elements in the unstructured grid system, expression of every object with complicated motions has become possible theoretically. In this paper, application to a flow around spheres passing each other shows efficacy of the method. After then, the method is applied to a flow around intermeshing rotors.
2 Numerical Approach 2.1 Governing Equation For large deformation of computational domain around rotors by inte rmeshing motion, computational elements are added and eliminated. In this case, addition and elimination of elements are carried out dynamically. However nervous grid arrangement in boundary layer might affect to such the dynamical change of element shape. Thus, inviscid flow should be suitable. Then, three-dimensional Euler equation is adopted as a governing equation. The following equation written in conservation law form is
∂q ∂E ∂F ∂ G + + + = 0 .(1) ∂ t ∂x ∂y ∂z Where, ⎛ ρu ⎞ ⎛ ρ ⎞ ⎜ 2 ⎟ ⎜ ⎟ ρ u ⎜ ρu + p ⎟ ⎜ ⎟ q = ⎜ ρv ⎟, E = ⎜ ρuv ⎟, ⎜ ⎟ ⎜ ⎟ ⎜ ρuw ⎟ ⎜ ρw ⎟ ⎜ u (e + p ) ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ e ⎠
⎛ ρv ⎞ ⎛ ρw ⎞ ⎜ ⎜ ⎟ ⎟ ⎜ ρuv ⎟ ⎜ ρuw ⎟ F = ⎜ ρv 2 + p ⎟, G = ⎜ ρvw ⎟ . ⎜ ⎜ 2 ⎟ ⎟ ⎜ ρvw ⎟ ⎜ ρw + p ⎟ ⎜ v (e + p ) ⎟ ⎜ w(e + p) ⎟ ⎝ ⎝ ⎠ ⎠
(2)
The unknown variables ρ, u, v, and e represent the gas density, velocity components in the x and y directions, and total energy per unit volume, respectively. The working fluid is assumed to be perfect, and the pressure p is defined by
e=
p 1 + ρ (u 2 + v 2 ) , γ −1 2
where, the ratio of specific heats γ is typically taken as being 1.4.
(3)
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2.2 Added and Eliminated Mesh Method If computational domain or body boundary stands still, general mesh refinement method can be used for addition and elimination of computational elements. However, when objects move intricately such as the intermeshing rotors, vertices are also moved as well as adding and eliminating elements. In this case, it should be important how to satisfy a geometric conservation law. To assure the law, control volume on the space-time unified domain (x, y, z, t) which is four-dimensional volume for threedimensional flows was adopted. Then using divergence form, equation (1) can be rewritten to
~~ . ∇F = 0ࠉ
(4)
The unstructured moving grid approach is based on a cell centered finite-volume method. Thus, flow variables are defined at a cell center. Then, the control volume of unified domain is shown in figure 1. The tetrahedron at n-time step tetrahedron is expressed using four-grid points ( R1n , R n2 , R 3n , R n4 ). Then another one at n+1-time step is also represented using grid points ( R1n +1 , R n2 +1 , R3n +1 , R n4 +1 ). Here, the space-time unified control-volume is defined by motion of tetrahedron from n to n+1 time step. Therefore, the control-volume is structured by eight vertices ( R1n , R n2 , R3n , R n4 , R1n +1 , R n2 +1 , R3n +1 , R n4 +1 ). After rewriting equation (4) in surface integral form using the Gauss theorem, it is discretized as a follow: 4
{(
) }
~ =0 . q n (n~t )5 + q n +1 (n~t )6 + ∑ E n +1 2㸪F n +1 2㸪G n +1 2㸪q n +1 2 ⋅ n l =1
(5)
l
Figure 1: Control-volume in space-time unified domain
When a new element is added, it is expressed as division of elements with adding a vertex. Placing a new vertex on the center of the gridline, one element is divided to two elements as shown in figure 2. In the case of adding new element, to satisfy the geometric conservation law, the control volume is also deformed to a complicated polyhedron, as shown in figure 3. Here, left hand side volume is estimated same as the last one. On the other hand, right hand side volume in figure 3 is dealt with the element which appeared suddenly at n+1 time-step. In this case, the control volume is defined as a space which is formed by sweeping of the element from n to n+1 time step. Here, the element shape changes from triangle-plate to tetrahedron-volume. Thus, the control volume is structured by vertices R2n, R3n, R4n, R2n+1, R3n+1, R4n+1 and R5n+1 in figure 3. In this case, the volume at n-step is zero, thus the discretization should be equation (6). On the other hand, when an element is eliminated, it is represented by merging vertex. Here, the eliminating element is defined as element j in figure 4. Then, it is eliminated by moving vertex. When the time-step advances from n to n+1, the volume of the element j become to zero. Thus, the elements i and j must be dealt with as union. When the pentahedron which is structured by vertices (R1n, R2n, R3n,
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R4n and R5n) sweeps space and changes a shape to a tetrahedron, the control volume is formed. Then the control volume is built by vertices R1n, R2n, R3n, R4n, R5n, R1n+1, R2n+1, R3n+1 and R4n+1) in figure 5. Then the discretization become to equation (7). 4
~} = 0 q nj +1 {(n~t )6 }j + ∑ {(E n +1 2㸪F n +1 2㸪G n +1 2㸪q n +1 2 ) ⋅ n l
(6)
l =1
6
~} = 0 qin {(n~t )7 }i + q nj {(n~t )7 }j + qin +1{(n~t )8 }i + ∑ {(En +1 2㸪F n +1 2㸪G n +1 2㸪q n +1 2 )⋅ n l l =1
Figure 2: Adding new element
Figure 3: Control volume for adding element
Figure 4: Eliminating element
Figure 5: Control volume for eliminating element
(7)
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The flux vectors are evaluated using the Roe flux difference splitting scheme with MUSCL approach, as well as the Venkatakrishnan limiter. Then, to solve the implicit algorithm, the LU-SGS implicit scheme is adopted. Then these deformations of the control volumes are carried out after confirmation of the geometric conservation law by capturing uniform flows on moving, adding and eliminating grid.
2.3 Modification of the Element According to adding and eliminating, sometimes the element shape largely deforms. However, extremely skewed element may have a bad influence for calculation accuracy. Thus it should be modified. But, additional appearance and disappearance of element for modification is not suitable. Thus, only gridline is replaced without moving grid point. In detail, placing a new grid point on center of the replacing gridline, temporary element is added, as shown in figure 6. Then, by moving the new grid point, the temporary element is eliminated. On the procedure, replacement of gridline is carried out. When two of three interior angles of triangle on modifying element satisfy the equation (8) simultaneously, the element is estimated for replacing gridline. Furthermore, when the modified element also satisfies the condition, the replacement is canceled. θ tri . (l ) < 35.0 [deg]
Figure 6: Replacement of the gridline
3 Application to Passing-Each-Other of Spheres 3.1 Computational Conditions In this chapter, a flow around spheres passing each other is computed as a test problem. Computational domain is shown in figure 7. As an initial condition, both spheres are placed keeping 10L distance for x direction and 0.2L distance for y direction. The sphere speed is defined as following equation. Computational mesh is generated using MEGG3D supported by JAXA (Ito, 2002). Number of initial cells is 871,170. However, the MEGG3D is used for making initial mesh only. Then, all deformations of the meth are carried out by an in-house code.
Figure 7: Computational domain for spheres passing each other
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⎧10t u1 = ⎨ ⎩1.0
t < 0.1 t ≥ 0 .1
(9)
If addition and elimination of element is carried out for extremely fine element around the sphere surface, computation cost would increase. Then, colorized blue domain which three times the diameter of the sphere is moved according to the sphere traveling, as shown in figure 8 (right). However, when another sphere invades into the domain, the inside elements are also added and eliminated. By addition and elimination, the balance of coarseness and fineness of placing elements around the sphere may be destroyed. Furthermore, extreme skewness element may be generated. To avoid the problem, we classified sizes of elements according distance from the sphere surface. Then layering unstructured meshes are defined as averages of initial element, as shown in figure 8 (left). The elements on the red parts are in 24th grade. Using the color classified element, sizes of additional and eliminated elements are modified. Then, the size of the deformed mesh is limited between half to double of the initial mesh.
Figure 8: Classified domain (left) and moving domain (right) around the sphere
3.2 Results Both spheres are approaching to each other (t=3.2), close together (t=4.8), and moving away from each other (t=6.4). The meshes at each time and their enlarged view around the sphere moving downward direction are shown in figure 9. When the both spheres are close together, by merging fine elements around sphere the circumstance can be expressed. Furthermore, when they are moving away, meshes around spheres are reconstructed. Then we can see similar meshes when they are approaching each other. Thus, meshes around spheres passing each other can be represented by single mesh. Similarly, figure 10 shows the pressure contours at each time and their enlarged view, when spheres are approaching to each other (t=1.6), close together (t=5.0), and moving away from each other (t=6.4). We can confirm that the moving spheres with a bow shock make a complicated interaction of shocks when they are close together. In future work, the parameter which can indicate the shape regularity property of the deformed element will be taken.
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t = 3.2
t = 4.8
t = 6.4 Figure 9: Mesh around spheres (left) and their enlarged view (right)
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t = 1.6
t = 5.0
t = 6.4 Figure 10: Pressure contours around spheres (left) and their enlarged view (right)
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4 Application to Intermeshing Rotors Finally, the method is applied to a flow around the intermeshing rotors. As the computational model, Kaman K1200 KMAX (Mansur, 2011) which has two rotors was adopted. The angle formed by two axes of rotors is 28 degrees. Then, the attack angle of blade changes linearly from root (5 degrees) to tip (10 degrees). The phased difference of the rotor is 90 degrees. The rotors are rotated at 273 rpm which is maximum rotation speed. At that time, blade tip speed arrives at Mach number = 0.61.
Initial state
90 degrees rotation Figure 11: Model showing two different rotor blades separation angles and meshes
The relative position of red and green blades and its meshes are shown in figure 11. The upper figure is initial condition and the lower one is state after rotation of 90 degrees. Furthermore, each mesh around rotor is colorized by same color of each blade. Then, the colorized meshes themselves are moved
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according to rotation of blades. The colorized blue indicates mesh around the fuselage, and it is fixed. By comparison with meshes at different time, we can see that moving mesh can be carried out keeping mesh quality. Then, the isosurface of velocity around the helicopter on hovering state is shown in figure 12. Extending downwash as time passes can be seen, thus, the possibility of the complicated computation was shown.
t = 0.0
t = 95.0 Figure 12: Isosurface of velocity (v = 0.035)
t = 190.0
5 Conclusions The unstructured mesh method adding and eliminating elements is developed and applied to flows in this paper. The results of computation for a flow around spheres passing each other showed that an efficacy and an availability of the method for complicated change of computational domain geometry. Furthermore, application to a flow around the intermeshing rotors showed that a practicability of the method and a promising feature for complicated flows.
Acknowledgment This work was supported by JSPS KAKENHI Grant Number 16K06079 and 16K21563.
References Steger, J. et al. (1987). On the use of composite grid schemes in computational aerodynamics, Computer Methods in Applied Mechanics and Engineering, 64, 301-320. Obayashi, S. et al. (1992). Freestream Capturing for Moving Coordinates in Three Dimensions, AIAA Journal, Vol.30, pp.1125-1128. Donea, J. et al. (1982),, Arbitrary Lagrangian-Eulerian finite element method for transient fluidstructure interactions, Computer Methods in Applied Mechanics and Engineering, 33, 689-723. Ito, Y. et al. (2002). Surface Triangulation for Polygonal Models Based on CAD Data, Internal Journal for Numerical Methods in Fluids, Vol. 39, Issue 1, 75-96. Mansur, M.H. (2011). Full Flight Envelop Inner-Loop Control Law Development for the Unmanned K-MAX®, the American Helicopter Society 67th Annual Forum, Virginia Beach, VA.
