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Preconditioning Methods to Accelerate and Improve Solution of Compressible Flow around Rotor

Kamali Moghadam, Ramin | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 42915 (45)
  4. University: Sharif University of Technology
  5. Department: Aerospace Engineering
  6. Advisor(s): Hejranfar, Kazem
  7. Abstract:
  8. In the present study, the numerical simulation of the compressible inviscid flow around helicopter rotor is performed using the solution of the preconditioned Euler equations. Three preconditioners proposed by Eriksson, Choi and Merkel, and Turkel are implemented in two- and three-dimensional upwind Euler flow solvers on unstructured meshes. The mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn and their eigenvalues and eigenvectors are compared with each others. For this aim, these preconditioning schemes are expressed in a unified formulation. A cell-centered finite volume Roe's upwind method is used for the discretization of the preconditioned Euler equations. The accuracy and performance of these preconditioning schemes are examined by computing steady low Mach number flows over several two- and three-dimensional geometries for different conditions. The study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low Mach number flows. Although the preconditioning methods implemented provide nearly the same results in accuracy, they give different performances in convergence rate. It is demonstrated while the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality, the condition number of the system of equations plays an important role and it determines the convergence characteristics of solutions. Different cutoffs are also implemented in two- and three-dimensional preconditioned Euler flow solvers for removing the singularity of preconditioning schemes near the stagnation regions where Mach number approaches. The accuracy, performance and robustness of these cutoffs are examined in computing low speed flows over several two- and three-dimensional geometries for the three preconditioning schemes studied. A cutoff is proposed based on the averaged velocity of the neighboring cells in the stagnation region and the performance of this cutoff is examined in comparison with other cutoffs. Preconditioned characteristic boundary conditions (BCs) are also implemented at artificial boundaries for the solution of the two- and three-dimensional preconditioned Euler equations at low Mach number flows. The preconditioned compatibility equations and the corresponding characteristic variables (or the Riemann invariants) based on the characteristic forms of preconditioned Euler equations are mathematically derived for three preconditioners proposed by Eriksson, Choi and Merkel, and Turkel. The accuracy and performance of the preconditioned characteristic BCs applied at artificial boundaries are evaluated in comparison with the non-preconditioned characteristic BCs and the simplified BCs in computing steady low Mach number flows over several two- and three-dimensional geometries for different conditions. The sensitivity of the solution to the size of computational domain and the variation of the angle of attack for each type of BCs is also examined. Indications are that the preconditioned characteristic BCs implemented in the preconditioned system of Euler equations greatly improve the convergence rate of the solution of low Mach number flows compared to the other two types of BCs. The preconditioned Euler equations are also formulated in a rotating coordinate system to solve the inviscid flows around the helicopter rotor in hover using three mentioned preconditioneing schmes. The preconditioning parameter and the cutoff are modified for the formulation in the rotating coordinate system. The preconditioned characteristic boundary conditions (BCs) are also mathematically derived for the formulation in the rotating coordinate system and they are implemented in the preconditioned Euler flow solver. Calculations are carried out for two isolated rotors, the Caradonna-Tung rotor and the 7A rotor, in hover for different tip Mach numbers and the aerodynamic characteristics and computed surface pressure distribution of the blades are compared with the experimental data. The results indicate that the use of the preconditioning schemes for the solution of the preconditioned Euler equations in conjunction with the preconditioned characteristic BCs implemented at the farfield boundary greatly enhance the robustness and performance of the Euler flow solver for simulating the helicopter rotor flow
  9. Keywords:
  10. Rotors ; Low Velocity ; Compressibility ; Euler Equation ; Preconditioning ; Low Mach Number ; Helicopter Rotor Flow

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