We are interested in the study, via direct numerical simulations, of active vortex generators. Vortex generators may be used to modify the inner part of the boundary layer or to control separation thus enhancing the performance and maneuverability of aerodynamic configurations. We consider generators that consist of a surface cavity elongated in the stream direction and partially covered with a moving lid that at rest lies flush with the boundary. Streamwise vorticity is generated and ejected due to the oscillatory motion of the lid. The present simulations complement relevant experimental investigations of active vortex generators at NASA Ames and Stanford University (Saddoughi, 1994, and Jacobson and Reynolds, 1993). Jacobson and Reynolds (1993) used a piezoelectric device in water, allowing for small amplitude high frequency oscillations. They placed the lid asymmetrically on the cavity and observed a strong outward velocity at the small gap of the cavity. Saddoughi used a larger mechanically driven device in air to investigate this flow and he observed a jet emerging from the wide gap of the configuration, contrary to the findings of Jacobson and Reynolds. Our task is to simulate the flows generated by these devices and to conduct a parametric study that would help us elucidate the physical mechanisms present in the flow. Conventional computational schemes encounter difficulties when simulating flows around complex configurations undergoing arbitrary motions. Here we present a formulation that achieves this task on a purely Lagrangian frame by extending the formulation presented by Koumoutsakos, Leonard and Pepin (1994). The viscous effects are taken into account by modifying the strength of the particles, whereas fast multipole schemes employing hundreds of thousands of particles allow for high resolution simulations. The results of the present simulations would help us assess some of the effects of three-dimensionality in experiments and investigate the role of two-dimensional vortex generation due to an oscillating lid

Koumoutsakos, P.

English

NASA Technical Reports Server

=' _-" _// 233Cenler for Turbulence Research
Annual Research Briefs 1995
Simulations of vortex generators
By P. Koumoutsakos
1. Motivation and background
We are interested in the study, via direct numerical simulations, of active vortex
generators. Vortex generators may be used to modify the inner part of the boundary
layer or to control separation thus enhancing the performance and maneuverability
of aerodynamic configurations. We consider generators that consist of a surface cav-
ity elongated in the stream direction and partially covered with a moving lid that at
rest lies flush with the boundary. Streamwise vorticity is generated and ejected due
to the oscillatory motion of the lid. The present simulations complement relevant
experimental investigations of active vortex generators at NASA Ames and Stan-
ford University (Saddoughi ,1994, and Jacobson and Reynolds, 1993). Jacobson and
Reynolds (1993) used a piezoelectric device in water, allowing for small amplitude
high frequency oscillations. They placed the lid asymmetrically on the cavity and
observed a strong outward velocity at the small gap of the cavity. Saddoughi used
a larger mechanically driven device in air to investigate this flow and he observed
a jet emerging from the wide gap of the configuration, contrary to the findings of
Jacobson and Reynolds.
Our task is to simulate the flows generated by these devices and to conduct a
parametric study that would help us elucidate the physical mechanisms present
in the flow. Conventional computational schemes encounter difficulties when sim-
ulating flows around complex configurations undergoing arbitrary motions. Here
we present a formulation that achieves this task on a purely Lagrangian frame by
extending the formulation presented by Koumoutsakos, Leonard and Pepin (1994).
The viscous effects are taken into account by modifying the strength of the particles,
whereas fast multipole schemes employing hundreds of thousands of particles allow
for high resolution simulations. The results of the present simulations would help
us assess some of the effects of three-dimensionality in experiments and investigate
the role of two-dimensional vortex generation due to an oscillating lid.
2. Accomplishments
An adaptive computational scheme, based on vortex methods, has been developed
to investigate flows past the two dimensional configuration shown in Fig. 1.
The configuration consists of a cavity partially covered with an oscillating lid
that is placed asymmetrically. In this report we describe our numerical method and
present preliminary results of our direct numerical simulations.
g.1 Mathematical formulation
Two-dimensional incompressible unsteady flow of a viscous fluid may be deter-
mined by the vorticity transport equation as
0_
+ u-Vw = vV2w (1)
Ot
https://ntrs.nasa.gov/search.jsp?R=19960022314 2020-06-16T04:52:14+00:00Z
234 P. Koumoutsakos
dq
;/
_i / z_m, '
x
TOP VIEW
(_F_w O_mmn
Z_y
SIDE VIEW
FIGURE 1. Definition sketch.
where u(x,t) is the velocity, ¢o = w_ = V x u the vorticity and u denotes
the kinematic viscosity . Using the definition of the vorticity and the continuity
(V • u = 0) it can be shown that u is related to w by a Poisson equation
V2u = -V × _o (2)
The vorticity equation (Eq. 1) may be expressed in a Lagrangian formulation by
solving for the vorticity carrying fluid elements (xa) based on the following set of
equations:
dxa
- u(x,,, t)dt
(3)
-- _ //' _72_2
dt
For flow around a non-rotating body, moving with velocity Vb(t), the velocity of
the fluid (u) on the surface of the body (x,) is equal to the velocity of the body:
u(x,,t) = Ub(t) At infinity we have: u(x) --* U_ as Ix] _ cx_ where U_ is the
free stream velocity.
2._ Particle (vortex) methods
The present numerical method is based on the discretization of the above equa-
tions in a Lagrangian frame using particle (vortex) methods. The vorticity field
Simulation_ of vortex generator_ 235
is considered as a discrete sum of the individual vorticity fields of the computa-
tional particles, having core radius e, strength F(t), and an individual distribution
of vorticity determined by the function _7,, so that:
N
 (x,0 = (4)
n=l
In the context of vortex methods the right-hand side of Eq. 3 is replaced by
integral operators..The velocity field may be determined from the vorticity field
using the Green's function formulation for the solution of Poisson's equation (Eq.
2).
1 fK(x - y) x w dy + U0(x,t) (5)u 27r
where U0(x, t) accounts for the presence of the body and U_, and K(z) = z/Iz[ 2.
The use of the Biot-Savart law to compute the velocity field guarantees the enforce-
ment of the boundary condition at infinity.
The Laplacian operator may be approximated by an integral operator ( Mas-
Gallic, 1987) as well:
dy (6)
These integrals are discretized using a quadrature having as quadrature points the
locations of the particles. Fast multipole algorithms with a computational cost
scaling as O(N) (Greengard and Rohklin, 1987) have been efficiently implemented,
allowing for high resolution simulations using a few millions of particles (grid points).
The no-slip boundary condition is enforced by formulating the physical mecha-
nism it describes. A vorticity flux (Ow/On) may be determined on the boundary
in a way that ensures the no-slip condition is satisfied. We implement a fractional
step algorithm that allows for the calculation of this vorticity flux by extending
the formulation presented (Koumoutsakos, Leonard and Pepin 1994) to account for
multiple bodies. It is shown then that this mechanism of vorticity generation can
be expressed by an integral operator as well, so that the vorticity field is modified
by:
dw f Owt _ u H(xa,y) _nn (y) dy (7)
where the kernel H(x) is described in [5].
The accuracy of the method relies on the accuracy of the quadrature rule. The
convergence properties of vortex methods with a finite core dictate that the particles
must overlap at all times (Beale, 1986). When particles cease to overlap in the
present scheme, a procedure is implemented (Koumoutsakos and Leonard, 1995)
that restores the particle overlap while conserving the moments of the vorticity
field.
236 P. Koumoutsakos
_.3 b3"actional _tep algorithm - boundary condition_
A fractional step algorithm is implemented that accommodates the enforcement
of the no-slip boundary condition. Let us assume that at the n-th time step (cor-
responding to time t - 6t) the vorticity field has been computed (respecting the
no-slip boundary condition) and we seek to advance the solution to the next time
step (time t). The following two-step procedure is implemented:
*Step 1 (kinematics-no through flow):
Particles are advanced via the Biot-Savart law and their strength is modified based
on the scheme of particle strength exchange. In order to account for the presence
of the body the no-through flow needs to be enforced. This is accomplished by
distributing vortex sheets on the surface of the bodies. For a doubly connected
domain as that shown in Fig. lb the potential flow problem is solved using the
following set of equations:
ILl2 + LogiC- x,la¢ = u.(x,)
J-LI2 xp -- _
8a
L/2 t fL/ 0 (xp,T) .
L/2 g(_,t)d_ _- -v_ J-L 20_(Xp) dxpdT (8b)
f_Zl2 x( ) Logl¢- + 7 (C) o- Logl¢- CIdC = u,(¢) (Sc)
L/2
where g(_),xp and 7w(¢),¢ denote vortex sheets and location of points on the
surface of the plate and the cavity respectively.
The above set of equations, when discretized using a panel method, results in a
well posed system of equations (Koumoutsakos and Leonard, 1995), which can be
solved iteratively. Note that Eq. 8b guarantees the solvability of the equations and
the uniqueness of the pressure distribution on the surface of the bodies.
*Step 2 (dynamics - no slip):
The no-slip boundary conditions are enforced in this stage by a vorticity (not
particle) creation algorithm. The vortex sheet that is distributed on the surface of
the body enters the fluid, thus generating a vorticity flux at the surface of the body.
This vorticity flux accounts for the modification of the strength of the particles so as
to enforce the no-slip boundary condition. Algorithmically, in this step, the diffusion
equation is solved on a Lagrangian frame with Neuman boundary conditions. Please
see Koumoutsakos et al. (1994) for further details.
_.4 Result_
A series of simulations have been carried out to determine the important param-
eters of the flow. It has been speculated that the parameters of the flow such as
the frequency of oscillation (f), the gap diameter (d,, dt for the small and the large
Simulations of vortex generators
D_r_tl_a
II,mzo _ OIDGllllttl.lall __
FIGURE 2. Sketch of flow type I.
237
gap respectively), and the viscosity of the flow may be combined so as to produce
a coefficient, called the Stokes number:
to characterize the different physical phenomena of the flow. We have performed a
series of computations by varying the above parameters of the flow along with the
amplitude of the oscillations in an attempt to match the respective experimental
cases. The simulations have shown a dramatic difference in the flow behavior for
various parameters of the configuration. In the following we present the types of
flows that are being observed (and described below) computationally and experi-
mentally, along with the respective Stokes numbers.
Large Gap Small Gap Flow Type
12.600(Exp.) 2.960 (Exp.) I
34.010 14.680 I
53.238 14.589 I
4.652 (Exp.) 0.930 (Exp.) II
15.597 1.794 II
13.870 3.381 II
46.515 9.744 II
In what we call flow type L a positive (inducing an outward velocity) vortex dipole
establishes itself in the small gap region just outside the cavity. The pair is produced
by the downward motion of the plate. It is continuously fed by the downward motion
of the oscillation cycle in such a way as to overcome its erosion by diffusion. The self
induced velocity of the dipole, on the other hand, is balanced by the upward motion
of the plate, thus establishing a quasi-steady vorticity distribution at the small gap
of the configuration. On the large gap side, the distance of the plate from the cavity
walls is such that no strong vorticity is being ejected from the cavity walls. The
vorticity that is produced at the tip of the plate has a zero mean strength, and
diffusion acts to reduce its strength rapidly. Hence as the plate is oscillating no
238 P. Kouraout_ako_
FIGURE 3. Sketch of flow type II.
dominant vortical features are established at this end of the lid. As we reduce the
frequency of oscillations, we may observe that dipoles are being ejected from the
small gap of the cavity at the downward motion of the plate, while a new pair is
being formed at the small gap. However,the dipoles that emerge from the gaps are
quickly eroded by diffusion and do not significantly alter the character of the flow.
A strikingly different pattern is observed in flow type II. In this configuration, the
lid is relatively closer to both edges of the cavity and the amplitude of oscillations
is about an order of magnitude in absolute value larger than the previous cases.
Hence vortex dipoles are being formed on both gaps of the cavity. The stronger
pairs are initially formed in the area of the small gap. Due to the low frequency
of the plate oscillations, the upwash motion of the plate does not overcome the
self-induced velocity of the vortical pairs and vorticity is ejected from both gaps.
However, as the dipoles that are being formed at the large side of the cavity are
more asymmetric, with the clockwise vorticity being more dominant, their path
arches towards the cavity lid. This establishes, at later times, a large clockwise
vortical region over the cavity. This vortex in turn, further modifies the behavior
of the flow as it induces an additional downwash velocity on the small side of the
gap. The effect of this clockwise (negative) vortex on the flow at the small gap
side is twofold: (a) It diverts the positive vortex that is formed at the tip of the lid
during the downward motion of the plate in a direction parallel to the lid and (b) it
induces an additional downwash velocity. Thus the dipole strength is progressively
reduced at the small gap, resulting in a blockage of the vorticity production and
fluid ejection from the small gap. The final configuration with the large negative
vortex over the cavity lid appears to be stable, thus establishing a flow field with a
main jet of fluid emanating form the large gap side of the cavity.
The above described mechanisms may offer an initial tentative 'two-dimensional
explanation' to the different behavior exhibited by the flow in the experiments of
Jacobson and Reynolds (1995) and Saddoughi (1994). A more systematic study (see
below) of the configuration is in order while future three-dimensional simulations
would reveal the full mechanism of vorticity generation and flow ejection from the
gaps observed in the experiments.
Simulations of vortex generator_ 239
Stage Z
8rage ZZ
FIGURE 4. Proposed simulated configurations.
3. Conclusions and future work
We have presented simulation of flows past complex configurations undergoing
arbitrary motions using a Lagrangian computational scheme based on high resolu-
tion viscous vortex methods. The results of these computations attempt to elucidate
some of the intricate behavior that has been observed in related experimental works
on flows past active vortex generators.
In order to investigate further the mechanisms of vorticity generation and the
observed flow patterns, we are in the process of conducting further detailed simu-
lations. More specifically (Fig. 4) each stage of our study would attempt to isolate
and examine a different aspect of vorticity generation and destruction which appear
in these vortex generators.
(i) In the first stage we are in the process of conducting simulations of a free
oscillating plate. This study would help us establish the generation of vorticity at
the cavity lids as well as demonstrate the effect of the frequency of oscillation in the
generation of vorticity and the interplay of vorticity generation and destruction due
to the plate oscillation and diffusion. A theoretical analysis would be conducted to
examine the limit of very high frequency oscillations.
(ii)In the second stage of our study, we would examine the interaction of the
plate with a corner. This study would help us understand the vorticity formation
240 P. Koumoutsakos
at both gaps of the full confguration. By varying the distance of the plate from the
cavity we would be able to determine distances as well as frequencies and amplitudes
of oscillation for which a vortex dipole is being formed, and we would be able to
determine when the effects of the cavity is negligible.
(iii) In the third stage, the full configuration would be examined in a more careful
and systematic manner based on our gained insight from the studies of the more
simplified configurations.
We are also in the process of developing three-dimensional codes (Koumout-
sakos, 1995) for the study of the full configuration that would elucidate the three-
dimensional aspects of the flow.
Acknowledgments
I wish to acknowledge many insightful discussions with Dr. Nagi N. Mansour.
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