Numerical Study of a Cascade Unsteady Separation Flow

A CFD solver is developed to solve a 3D, unsteady, compressible Navier-Stokes equations with the Baldwin-Lomax turbulence model to study the unsteady separation flow in a high incidence cascade. The second order accuracy is obtained with the dual time stepping technique. The code is first validated for its unsteady simulation capability by calculating a 2D transonic inlet diffuser flow. Then a 3D steady state calculation is carried out for the cascade at an incidence of 10 . The surface pressure distributions compare reasonably well with the experiment measurement. Finally, the 3D unsteady simulation is carried out with 3 inlet Mach numbers at the incidence of 10 . The separation bubble oscillation and the static pressure oscillation on the leading edge of the blade suction surface exhibit clear periodicity. The details of the leading edge vortex shedding is captured. The inlet Mach number is shown to be the determinant factor in the characteristics of the separation flow. In the subsonic inlet flow region, increasing the inlet Mach number enlarges the separation region and the pressure oscillation intensity. The separation flow is weakened when the inlet flow becomes supersonic. NOMENCLATURE ρ density μ μt molecular, turbulent viscoscity τ stress γ specific heat ratio ω vorticity PhD Student, Email: [email protected] †Associate Professor, Director of CFD Lab ‡Senior Research Supervisor a local speed of sound e energy per unit mass l mixing length p static pressure u v w velocity components x y z Cartesian coordinates t time Cp static pressure coefficient E F G inviscid flux vector Ma Mach number Pr Prandtl number Q conservative variable vector Re Reynolds number R S T viscid flux vector Introduction Flutter in axial turbomachines is a highly undesirable and dangerous self-excited blade oscillation mode that can result in high cycle fatigue blade failure. Modern turbine engines employ transonic fan stages with high aspect ratio blades that are prone to flutter. It is imperative to understand the origins of flutter for reliable and safe operation of these engines. High subsonic and transonic torsional stall flutter occurs near the fan stall limit line at speeds up to about 80% of the design speed and with high incidence. Two potential factors are assumed to trigger the flutter mode, the shock wave motion in the transonic conditions and the large separation on the suction side of the blade surface under high incidence angle [1]. Actually, the shock wave does not appear until very high subsonic inlet Mach number is reached. However, the flutter exists in much 1 Copyright c 2004 by ASME wider Mach number region (appears in smaller Mach number) than where the shock wave exists. The separation flow on the leading edge of the blade suction side is a likely cause for flutter. A series of experiments have been carried out in NASA Glenn Research Center to study the transonic separation flow characteristics of modern airfoils for transonic fans. A low aspect ratio fan blade operating near the stall flutter boundary under high incidence is simulated in the NASA Transonic Flutter Cascade. The unsteady pressure is measured at selected points on the chord line of the cascade surfaces [2]. The flow pattern is visualized using dye oils and schilieren flow visualization methods [1]. The objective of this paper is to numerically study the unsteady characteristics of the NASA transonic cascade separation flow with a incidence angle of 10 in 3D condition. A CFD solver is developed to solve the 3D, unsteady, compressible NavierStokes equations with the Baldwin-Lomax turbulence model [3]. The dual time stepping method is applied to achieve the second order accuracy in time. The unsteady computing capability of the CFD solver is validated by a transonic inlet diffuser flow, where turbulent boundary layer interacts with the shock wave and causes unsteady flow separation. Finally, the unsteady separation flow simulation is carried out at 3 inlet Mach numbers, 0.5, 0.8 and 1.18. The vortex shedding mechanism is analyzed for the case of Mach number 0.5. The characteristics of the separation flow under varying inlet Mach numbers is demonstrated. Numerical Algorithm The governing equations for flow field simulation used in this paper are the Reynolds averaged time-dependent compressible Navier-Stokes equations in generalized coordinate system. The simulation is carried out with finite volume method. The equations are discretized using the third order MUSCL differencing [4]. The linearized equation systems are solved using the Gauss-Seidel line iteration. Upwind differencing is implemented with the Roe scheme [5] and the van Leer scheme [6]. The second order accuracy of time marching is obtained with the dual time stepping technique. Governing Equations For simplicity, the non-dimensional form of the equations in conservation law form are expressed in Cartesian coordinates as the followings. ∂Q ∂ t ∂E ∂x ∂F ∂y ∂G ∂z ∂R ∂x ∂S ∂y ∂T ∂z (1)