The Kutta Joukowski(KJ) theorem, relating the lift of an airfoil to circulation, was widely accepted for predicting the lift of viscous high Reynolds number flow without separation. However, this theorem was only proved for inviscid flow and it is thus of academic importance to see whether there is a viscous equivalent of this theorem. For lower Reynolds number flow around objects of small size, it is difficult to measure the lift force directly and it is thus convenient to measure the velocity flow field solely and then, if possible, relate the lift to the circulation in a similar way as for the inviscid KJ theorem. The purpose of this paper is to discuss the relevant conditions under which a viscous equivalent of the KJ theorem exists that reduces to the inviscid KJ theorem for high Reynolds number viscous flow and remains correct for low Reynolds number steady flow. It has been shown that if the lift is expressed as a linear function of the circulation as in the classical KJ theorem, then the freestream velocity must be corrected by a component called mean deficit velocity resulting from the wake. This correction is small only when the Reynolds number is relatively large. Moreover, the circulation, defined along a loop containing the boundary layer and a part of the wake, is generally smaller than that based on inviscid flow assumption. For unsteady viscous flow, there is an inevitable additional correction due to unsteadiness.
By using a special momentum approach and with the help of interchange between singularity velocity and induced flow velocity, we derive in a physical way explicit force formulas for twodimensional inviscid flow involving multiple bound and free vortices, multiple airfoils, and vortex production. These force formulas hold individually for each airfoil thus allowing for force decomposition, and the contributions to forces from singularities（such as bound and image vortices,sources, and doublets） and bodies out of an airfoil are related to their induced velocities at the locations of singularities inside this airfoil. The force contribution due to vortex production is related to the vortex production rate and the distance between each pair of vortices in production, thus frameindependent. The formulas are validated against a number of standard problems. These force formulas, which generalize the classic Kutta–Joukowski theorem（for a single bound vortex） and the recent generalized Lagally theorem（for problems without a bound vortex and vortex production） to more general cases, can be used to identify or understand the roles of outside vortices and bodies on the forces of the actual body, optimize arrangement of outside vortices and bodies for force enhancement or reduction, and derive analytical force formulas once the flow field is given or known.
The average-passage equation system （APES） provides a rigorous mathematical framework for account- ing for the unsteady blade row interaction through multi- stage compressors in steady state environment by introduc- ing deterministic correlations （DC） that need to be modeled to close the equation system. The primary purpose of this study is to provide insight into the DC characteristics and the influence of DC on the time-averaged flow field of the APES. In Part 1 of this two-part paper, firstly a 3D viscous unsteady and time-averaging flow CFD solver is developed to investi- gate the APES technique. Then steady and unsteady simu- lations are conducted in a transonic compressor stage. The results from both simulations are compared to highlight the significance of the unsteady interactions. Furthermore, the distribution characteristics of DC are studied and the DC at the rotor/stator interface are compared with their spatial cor- relations （SC）. Lastly, steady and time-averaging （employing APES with DC） simulations for the downstream stator alone are conducted employing DC derived from the unsteady re- suits. The results from steady and time-averaging simula- tions are compared with the time-averaged unsteady results. The comparisons demonstrate that the simulation employing APES with DC can reproduce the time-averaged field and the 3D viscous time-averaging flow solver is validated.
The efficiency and mechanism of an active control device ＂＇Spark Jet＂ and its application in shock-induced separation control are studied using large-eddy simulation in this paper. The base flow is the interaction of an oblique shock-wave generated by 8° wedge and a spatially-developing Ma = 2.3 turbulent boundary layer. The Reynolds number based on the incoming flow property and the boundary layer displacement thickness at the impinging point without shock-wave is 20000. The detailed numerical approaches were presented. The inflow turbulence was generated using the digital filter method to avoid artificial temporal or streamwise periodicity. The , merical results including velocity profile, Reynolds stress profile, skin friction, and wall pressure were sys- tematically validated against the available wind tunnel particle image velocimetry （PIV） measure- ments of the same flow condition. Further study on the control of flow separation due to the strong shock-viscous interaction using an active control actuator ＂＇Spark Jet＇＂ was conducted. The single-pulsed characteristic of the device was obtained and compared with the experiment. Both instantaneous and time-averaged flow fields have shown that the jet flow issuing from the actuator cavity enhances the flow mixing inside the boundary layer, making the boundary layer more resis- tant to flow separation. Skin friction coefficient distribution shows that the separation bubble length is reduced by about 35% with control exerted.
Yang GuangYao YufengFang JianGan TianLi QiushiLu Lipeng
Three-dimensional corner separation is a common phenomenon that significantly affects compressor performance. Turbulence model is still a weakness for RANS method on predicting corner separation flow accurately. In the present study, numerical study of corner separation in a linear highly loaded prescribed velocity distribution （PVD） compressor cascade has been investigated using seven frequently used turbulence models. The seven turbulence models include Spalart Allmaras model, standard k-e model, realizable k-e model, standard k-to model, shear stress transport k co model, v2-fmodel and Reynolds stress model. The results of these turbulence models have been compared and analyzed in detail with available experimental data. It is found the standard k-1： model, realizable k-e model, v2-f model and Reynolds stress model can provide reasonable results for predicting three dimensional corner separation in the compressor cascade. The Spalart-Allmaras model, standard k-to model and shear stress transport k-w model overesti- mate corner separation region at incidence of 0°. The turbulence characteristics are discussed and turbulence anisotropy is observed to be stronger in the corner separating region.
The average-passage equation system （APES） provides a rigorous mathematical framework for account- ing for the unsteady blade row interaction through multistage compressors in steady state environment by introducing de- terministic correlations （DC） that need to be modeled to close the equation system. The primary purpose of this study was to provide insight into the DC characteristics and the in- fluence of DC on the time-averaged flow field of the APES. In Part 2 of this two-part paper, the influence of DC on the time-averaged flow field was systematically studied; Several time-averaging computations boundary conditions and DC were conducted with various for the downstream stator in a transonic compressor stage, by employing the CFD solver developed in Part 1 of this two-part paper. These results were compared with the time-averaged unsteady flow field and the steady one. The study indicat;d that the circumferential- averaged DC can take into account major part of the unsteady effects on spanwise redistribution of flow fields in compres- sors. Furthermore, it demonstrated that both deterministic stresses and deterministic enthalpy fluxes are necessary to reproduce the time-averaged flow field.
It is widely accepted that in a turbulent boundary layer （TBL） with adverse pressure gradient （APG） an outer peak usually appears in the profile of streamwise Reynolds stress. However, the effect of APG on this outer peak is not clearly understood. In this paper, the effect of APG is analysed using the numerical and experimental results in the literature. Because the effect of upstream flow is inherent in the TBL, we first analyse this effect in TBLs with zero pressure gradient on flat plates. Under the individual effect of upstream flow, an outer peak already appears in the profile of streamwise Reynolds stress when the TBL continues developing in the streamwise direction. The APG accelerates the appearance of the outer peak, instead of being a trigger.
