Derivation of the viscous Moore-Greitzer equation for aeroengine flow

Abstract

The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scales argument, based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast-time and local scale, describes this turbulent flow. The next order equations, produce an axi-symmetric swirl and a flow-pattern analogous to Rayleigh-Bénard convection rolls in Rayleigh-Bénard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter, describes these global flow modulations, when averaged over the small (local) spatial scales, the fast time scale and the time scale of the vortex rotations. Thus a more general system of spatially global, slow-time equations is obtained. This system can be solved numerically without any approximations. The viscous Moore-Greitzer equation is obtained when small inertial terms are dropped from these slow-time, spatially global equations, averaged once more in the axial direction. The new equations are simulated with two different simplifying assumptions and the results compared with simulations of the viscous Moore-Greitzer equations.

Exact internal controllability for a hyperbolic problem in a domain with highly oscillating boundary

“…Let ϕ ε be the corresponding solution of problem (4). Then, there exists a positive constant C, independent of ε such that…”

“…Another interesting application is the air flow through compression systems in turbo machines such as jet engine. For example, such a system is modelled by the Viscous-Moore-Greitzer equation derived from Scaled Navier-Stokes equations (see [4,32,33]). Here the pitch and size of the rotorstator pair of blades in the engine provides a small parameter compared to the size of the engine which is oscillatory as well as rotating (moving).…”

Self-excited Oscillations and Fuel Control of a Combustion Process in a Rijke Tube

“…Asymptotic expansion is used to derive the homogenized equation, however the method is formal and the second step is needed to prove the convergence as the ratio of the size of the period to the size of a sample of the medium goes to zero [1]. Using homogenization, a modified viscous Moore-Greitzer (vMG) equation modelling the airflow through the compression system in turbomachines with respect to the combustion is developed by Birnir et al [5]. The presented homogenization utilizes a separation of different spatial scales in the engine and the different time scales in the flow.…”

Multiscale Homogenization of the Navier-Stokes Equation

“…(2) We equip (2) with the initial data u ε (x, 0) = u 0 ε (x) which is assumed to be bounded in L 2 (Ω). We will throughout the paper assume that the forcing is of the form…”

“…For rotating jet engine flow the eddy viscosity tensor is explicitly derived in [2] and for the Navier-Stokes equation the details can be found in [7] and [8] under this assumption. These calculations will not be repeated here but the consequence of this approximation is that we can write the global equation as:…”