Jie-Zhi Wu

Post-stall flow control on an airfoil by local unsteady forcing

By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at post-stall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by periodic blowing–suction near the leading edge with low-level power input. This unsteady forcing can modulate the evolution of the separated shear layer to promote the formation of concentrated lifting vortices, which in turn interact with trailing-edge vortices in a favourable manner and thereby alter the global deep-stall flow field. In a certain range of post-stall angles of attack and forcing frequencies, the unforced random separated flow can become periodic or quasi-periodic, associated with a significant lift enhancement. This opens a promising possibility for flight beyond the static stall to a much higher angle of attack. The same local control also leads, in some situations, to a reduction of drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for high-α buffet control. The computations are in qualitative agreement with several recent post-stall flow control experiments. The physical mechanisms responsible for post-stall flow control, as observed from the numerical data, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The leading-edge shear layer and vortex shedding from the trailing edge are two basic constituents of unsteady post-stall flow and its control. Since the former has a rich spectrum of response to various disturbances, in a quite wide range the natural frequency of both constituents can shift and lock-in to the forcing frequency or its harmonics. Thus, most of the separated flow becomes resonant, associated with much more organized flow patterns. During this nonlinear process the coalescence of small vortices from the disturbed leading-edge shear layer is enhanced, causing a stronger entrainment and hence a stronger lifting vortex. Meanwhile, the unfavourable trailing-edge vortex is pushed downstream. The wake pattern also has a corresponding change: the shed vortices of alternate signs tend to be aligned, forming a train of close vortex couples with stronger downwash, rather than a Karman street.

Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer and Majdalani Phys. Fluids 15, 1485 2003 by taking the wall expansion ratio to be time invariant and then proceed to reduce the Navier‐Stokes equations into a fourth order ordinary differential equation with four boundary conditions. Using the homotopy analysis method HAM, an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska et al. Fluid Dyn. Res. 4, 151 1988. Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier‐ Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution of t from an initial 0 to a final state 1. In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored.

A THEORY OF THREE-DIMENSIONAL INTERFACIAL VORTICITY DYNAMICS

A three‐dimensional theory of vorticity dynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticity dynamics on an arbitrarily curved rigid wall [J. Fluid Mech. 254, 183 (1993)]. General formulas with arbitrary Reynolds numbers Re are derived for determining (1) how much vorticity exists on an interface S, (2) how much vorticity is created from S and sent into the fluid per unit area in per unit time, and (3) the force and moment acted on a closed interface by the created vorticity thereon. The common feature and fundamental difference between interfacial vorticity dynamics and its rigid‐wall counterpart are analyzed. In particular, on a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface‐deformation stress alone, which results in a weak creation rate of O (Re−1/2) at large...

Absolute and convective instability character of slender viscous vortices

Motivated by the need for effective vortex control, the character of absolute and convective instabilities (AI/CI) of incompressible and high-Mach number slender vortices with axial-velocity deficit is studied. Attention is focused on the disturbance modes which lead to the maximum absolute growth rate, and their dependence on flow conditions such as axial-flow profile, Reynolds number, and Mach number. A significant difference between the AI/CI and temporal-instability characters of the vortices occurs as the axial velocity deficit reduces. These theoretical results are applied to the flow region where vortex breakdown happens. It is found that the breakdown region is absolutely unstable, where waves are dominated by the spiral disturbance with lowest azimuthal wave number, in reasonable agreement with measurement.

A vorticity dynamics theory of three-dimensional flow separation

A theory of three-dimensional incompressible flow separation is presented in terms of the on-wall signatures of the flow. Some long-standing controversial issues are revisited and answers given, such as the inconsistency of the separation criteria based on the topological theory and “open separation,” and whether a separation line is an asymptote or envelope of neighboring skin-friction lines. General criteria for identifying an “open” or “closed” flow separation zone and separation line (including the initial point of the latter), steady and unsteady, are obtained, which apply to a generic smooth curved wall at any Reynolds numbers. The criteria are found to be most clearly given in terms of on-wall signatures of vorticity dynamics. These are then specified to steady boundary layer separation at large Reynolds numbers. A scale analysis under mild assumptions leads to a three-dimensional triple-deck structure near a generic boundary layer separation line. Criteria are presented for “separation watch,” whi...

Turbulent force as a diffusive field with vortical sources

In Reynolds-average Navier–Stokes equation it is the divergence of Reynolds stress tensor, i.e., the turbulent force, rather than the tensor itself, is to be simulated and partially modeled. Thus, directly working on turbulent force could bring significant simplification. In this paper a novel exact equation for incompressible turbulent force f is derived: (∂/∂t −ν∇2)f=∇⋅S, where ν is the molecular viscosity and all source terms in tensor S to be modeled are vortical. The dominant mechanism is the advection and stretching (with an opposite sign) of a “pseudo-Lamb vector” by fluctuating velocity field. No coupling with pressure is involved. The equation follows from a study of the mean fluctuating Lamb vector and kinetic energy, which constitute the turbulent force. Both constituents are governed by the same kind of equations as f. This innovative turbulent-force equation is similar to Lighthill’s acoustic analogy and naturally calls one’s attention to studying the vortical sources of turbulent force. The ...

Streaming vorticity flux from oscillating walls with finite amplitude

How to describe vorticity creation from a moving wall is a long standing problem. This paper discusses relevant issues at the fundamental level. First, it is shown that the concept of ‘‘vorticity flux due to wall acceleration’’ can be best understood by following fluid particles on the wall rather than observing the flow at fixed spatial points. This is of crucial importance when the time‐averaged flux is to be considered. The averaged flux has to be estimated in a wall‐fixed frame of reference (in which there is no flux due to wall acceleration at all); or, if an inertial frame of reference is used, the generalized Lagrangian mean (GLM) also gives the same result. Then, for some simple but typical configurations, the time‐averaged vorticity flux from a harmonically oscillating wall with finite amplitude is analyzed, without appealing to small perturbation. The main conclusion is that the wall oscillation will produce an additional mean vorticity flux (a fully nonlinear streaming effect), which is partial...

POST-STALL LIFT ENHANCEMENT ON AN AIRFOIL BY LOCAL UNSTEADY CONTROL Part II. Mode Competition and Vortex Dynamics

The physical mechanisms responsible to post-stall flow control on an airfoil by unsteady forcing, as has been observed by both experiment and computation, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The analysis is based on the data base of a two-dimensional Reynolds-averaged turbulent flow over a NACA-0012 airfoil at different angles of attack and forcing frequencies, reported in a companion paper. A local forcing with proper frequency can monitor the leading-edge shear layer and trailing vortex, so that the instability frequency of the former and the shedding frequency of the latter both shift to a harmonic or subharmonic of the forcing frequency, to form a well-ordered resonance. Depending on the frequency, unforced chaotic flow, forced periodic or quasi-periodi c flow, and forced chaotic flow, are observed. This resonance is associated with the process that the forcing strengthens the entrainment of the lifting vortex, suppresses random secondary and tertiary separations, and squeezes the unfavorable trailing vortex into a narrow region and pushes it downstream.

POST-STALL FLOW CONTROL ON AN AIRFOIL BY LOCAL UNSTEADY FORCING Part I. Lift, Drag, and Pressure Characteristics

By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at poststall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by a local unsteady excitation with low-level power input. In a certain range of post-stall angles of attack and forcing frequency, the unforced random separated flow can become periodic or quasi-periodi c, associated with a strong lift enhancement, which opens a promising possibility for one to fly beyond the static stall till to a much higher angle of attack. The same local control also leads to, in some situations, a reduction of the drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for higha buffet control. The computations confirm the physical principles for controlling a massively separated unsteady flow proposed by Wu, Vakili, and Wu (Prog. Aerospace Sci. 28 (1991), 73-131) and are in qualitative agreement with several recent post-stall flow control experiments. In Part I of this two-part paper, we examine the characteristics of lift, drag, and surface pressure without and with control, focusing mainly on the effect of forcing frequency and angles of attack. The necessary conditions for unsteady control to be successful are identified.