Microjets-Based Active Control of Store Trajectory in aSupersonic Cavity Using a Low-Order Model
Debashis Sahoo and Anuradha AnnaswamyMassachusetts Institute of Technology, Cambridge, MA, 02139, USA
Farrukh AlviFlorida A&M University and Florida State University, Tallahassee, FL, 32310, USA
Abstract
Recently, there has been a concerted effort in controlling the trajectory of a store departingfrom a cavity in a supersonic flow so as to ensure safe separation. The candidate actuator thathas achieved a safe departure in model-scale wind tunnel tests using minimum flow rate is atandem array of microjets placed in the spanwise direction near the leading edge of the cavity[1]. In this paper, a low order model is derived that captures the dominant mechanisms thatgovern the store trajectory with and without microjets and conditions under which unsafe andsafe departure occur are delineated. This model includes separate components to predict thepitch and plunge motion of the store when it is inside the cavity, when it is passing throughthe shear layer at the mouth of the cavity and when it is completely outside the cavity. Themodel derivations are based on the assumptions of store ejection from the cavity middle, slenderaxi-symmetric body aerodynamics, thin shear layer at the cavity mouth, high Reynolds numberexternal cross-flow, plane shock waves associated with the microjet actuators, no-flow conditioninside the cavity and ignorance of the cavity acoustic field. The model is validated by comparingwith the results of store drop experiments performed under the HIFEX Program at Mach 2.0and 2.46 using a generic sub-scale weapons bay for different control inputs [1]. It was foundthat the model predictions corroborate the main experimental observations that (I) the storedrop is unsuccessful for the microjet-off case, (II) the store departs safely under small microjetpressures, and (III) the store returns back to the cavity under large microjet pressures, undercertain operating conditions. The model is extended to include the effect of the cavity acousticfield through the shear layer thickness and is experimentally validated. In particular, the effectof microjets on the cavity acoustic field and on the shear layer structures is addressed. Finally,the optimal control input to ensure clean store departure for a host of drop conditions is foundusing the low-order model.
1 Introduction
It has been observed that a store, when released from an internal aircraft bay, can return back under
certain flight conditions. Instances of unsuccessful store drops under subsonic [2] and transonic
1
11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference)23 - 25 May 2005, Monterey, California
AIAA 2005-3097
Copyright © 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
[3] flows have been found in the literature. In addition, recent experiments under the DARPA-
funded HIFEX Program [1, 4, 5] have illustrated that the same phenomenon occurs under supersonic
flow conditions as well. It should be noted that unsafe store separation is a dangerous problem for
an aircraft while it drops weapons from its internal bay. To ameliorate this problem, a group of
researchers under the DARPA-funded HIFEX Program [1, 4, 5] have used an active control method
to ensure safe release of a slender axi-symmetric store from a rectangular cavity (bay) with an
external supersonic flow. The effectiveness of various flow control actuators has been studied by the
group, and the most promising one that guarantees safe store separation over a range of operating
conditions while using a minimum amount of mass flow rate was observed to be a tandem array
of microjet flow injectors distributed in the spanwise direction near the leading edge of the cavity.
By varying the pressure at which the microjets are activated, it was observed that successful store
separation was achievable even as the operating conditions changed. An interesting point to note is
that the same actuator has been shown experimentally in [4] and through a low-order model in [6]
to be effective in quietening the shear layer at the cavity opening and reducing of the cavity tones.
Although instances of safe store release have been observed after suitable application of micro-
jets, the underlying mechanism that governs the store separation from a cavity in supersonic flow is
currently unknown. Also unknown are the limiting conditions under which the store can be dropped
from the cavity to achieve safe separation. Finally, the dominant mechanism underlying the im-
pact of microjets on the store release trajectory is also not understood. The need therefore arises
for a model that addresses these issues and that accurately predicts the trend in the store trajectory,
i.e. whether the store is ejected from the cavity safely or returns back into the cavity, under given
initial drop conditions and a given control input. The model should also be reduced-order and para-
metric in order to be applied for developing the optimal control strategy that guarantees safe store
separation over a range of operating conditions. From a preliminary investigation of the dominant
physics of the problem, it can be said that the trend in the store trajectory is potentially influenced
by a host of factors (Figure 1), including the inhomogeneous flow field generated by the wavy shear
layer structures at the cavity opening, the potential induced by the store motion, the shock waves
2
introduced by the microjets and at the store nose, and noise inside the cavity, all of which should be
incorporated in the model. What is examined in this paper, in particular, is an investigation of the
forces and moments on the store due to these factors, a low-order model that captures the dominant
mechanisms, and an optimal control design for safe store separation.
ab
U∞
gravity
shock due to flow injectors
potential induced by store motion
inhomogeneous flow field due to wavy
shear layer structure
presence of
cavity walls
cavity acoustics
Figure 1: Different factors affecting the store release trajectory under supersonic cross-flow and inthe presence of microjet-based actuators. Labels ‘a’ and ‘b’ indicate different rows of microjetslocated near the cavity leading edge.
1.1 Background
The dynamics of a slender body separating from a rectangular cavity under a cross-flow is difficult
to analyze, expensive to experiment upon and is mostly used for military applications. As such,
limited models are available in literature and the pertinent ones are outlined here. The problem of
store release from a cavity under subsonic flow has been extensively analyzed by Shalaev, et. al. in
Ref. [2]. In the said reference, a low-order model of the store dynamics is derived, based on slender
body potential flow theory, and results in a set of ordinary differential equations for the store motion
which is analytically solvable under suitable conditions. This model is shown to predict the store
trajectories observed in subsonic wind tunnel experiments. In Ref. [3], the store separation from
a cavity under transonic flow has been studied by the same authors. A similar model as in [2] is
developed here too, but results in a set of nonlinear differential equations for the store trajectory
that cannot be solved analytically. This model is then used to show conditions under which unsafe
store departure occurs. Finally in [7], the same problem is considered in supersonic flow, with the
treatment being similar to [3]. Ref. [7] predicts the conditions for unsafe store ejection and also
3
discusses the store interaction with shock waves due to the parent body from which the store is
separated.
An alternative approach for external store separation found in the literature [8] is based on grid
testing. In this method, the store aerodynamic load at any position in the grid is assumed to be
a function of only its free stream aerodynamics and the parent aircraft induced aerodynamics at
that flow field position. The free stream aerodynamics is a property of the store alone while the
aircraft induced aerodynamics is determined by the aircraft flow field, with the mutual interference
between the aircraft and the store being considered a secondary effect. The aerodynamic loads are
then fed into a six degrees-of-freedom (6 DOF) program based on Newton’s laws to compute the
external store release trajectory from an aircraft. An improvement over this approach is tried in a
store separation problem for the SH-2G helicopter [9] in which a linear interference effect between
the store and the helicopter is incorporated in the 6 DOF program. To reduce the cost of testing, a
judicious combination of wind tunnel testing and CFD analysis is used. This method is validated by
comparing with experimentally available store release trajectory data. However, the limitations of
this approach are lack of available free stream data for all store shapes for all possible pitch angles,
lack of physical insight into the problem and expensive experiments required.
The goal of this paper is mainly to study the effect of microjets on the store trajectory. For this
purpose, the low-order model developed in Refs. [2, 3, 7] needs to be extended to incorporate the
effects of microjets. One method of realizing the aforementioned goal is to determine the dominant
mechanisms that (a) govern the store trajectory in the absence of microjets, and (b) are introduced
due to the addition of microjets. The extended low-order model should therefore be suitably pa-
rameterized and extensively analyzed so as to provide insight into the underlying mechanisms, and
enable the derivation of sufficient conditions under which (a) the store departs unsafely or safely
without microjets, and (b) the store departs unsafely or safely with microjets. Therefore, the start-
ing point for this work is in the low-order model developed in Refs. [2, 3, 7]. By using a careful
investigation of the force and moment expressions on the store obtained from the said references,
we obtain a set of parametric nonlinear ordinary differential equations that enables us to identify
4
the mechanism and, therefore, the criteria for unsafe store release from a cavity in general. A more
important contribution of the current paper is the extension of the model to include the effect of
microjets on the store trajectory. The model identifies the most relevant microjet-dependent control
parameter and the underlying process by which the store trajectory is influenced, thereby predicting
the microjet pressures for realizing a successful store departure. The model is also used to deter-
mine the optimal microjet pressure distribution for a clean weapon release and this control strategy
is shown to agree with experimental observations made under the DARPA-funded HIFEX Program
[1, 4, 5] for different supersonic Mach numbers. Finally, the model suggests better choices for the
actuator parameters at some of the operating conditions.
2 Store Release Model Development
The model, used to identify the microjet-based optimal control strategy for safe store ejection, is
restricted to the pitch and plunge motion of the store. Additionally, it is assumed that the store
release is an ejected drop from the middle of the cavity. The initial development of the store release
model is further based on the following five major assumptions:
(M1) The store is a slender axi-symmetric body with a small angle of attack.
(M2) The shear layer is thin compared to the cavity length.1
(M3) The external flow is of high Reynolds number (inviscid flow).
(M4) Microjets introduce weak plane shock waves near the leading edge.
(M5) The inside volume of the cavity is quiescent.1
1Assumptions (M2) and (M5) are somewhat relaxed in a model extension discussed in Section 3.3, that takes intoconsideration the shear layer structures. It is to be noted that the shear layer structure is influenced by the feedbackloop between the acoustic waves inside the cavity and the downstream propagating shear layer disturbances at the cavityopening [10]. The model extension, thus, indirectly considers the effect of the cavity acoustics on the store releasetrajectory.
5
The model also incorporates the factors that affect the store plunge and pitch motion given in Figure
1, out of which, the wavy structures, the store induced potential, the leading edge microjets and
gravity can be treated as the primary factors and the cavity walls as boundary conditions. We limit
the scope of the model to first order effects and introduce these factors separately.
The procedure employed for developing the first order model is as follows: Assumptions (M2)-
(M5) allow the use of potential function in solving the velocity field associated with store motion.
Using this method, it is first shown that because of the thin shear layer and the thin store diameter,
the wavy structures and indirectly the cavity acoustics provide negligible influence on aerodynamic
forces and moments on the store. It is then shown that the velocity potential induced by the store
motion as it drops from inside the cavity satisfies the Laplace Equation in the cross-section plane.
Owing to the linearity of the problem, the Laplace Equation is solved separately for the cases when
the store is inside the cavity, when it crosses the shear layer and when it is completely outside the
cavity. The technique used for solving the Laplace Equation is a combination of multipole expansion
and conformal transformation.
The resulting model predicts that when the store just exits the cavity in the no-control case, it
experiences an aerodynamic force (in a direction normal to the external flow) and pitching moment
such that the linear velocity of the store becomes positive and the store starts to return back to the
cavity. When microjets are switched on, the external flow stream is changed so that the force and
moment on the store are altered mainly when the store passes through the shear layer and when it is
completely outside the cavity. These changes cause the store to drop successfully from the cavity.
The model is also used to predict that when the microjet pressures are increased to higher values,
the nonlinearities in the normal force and moment cause the store to return back towards the cavity.
These predictions are then corroborated with experimental results [1, 5]. Finally, the model is used
to develop an optimal control strategy that ensures safe store drop under a variety of drop conditions.
6
2.1 General Governing Equation for Two Dimensional Model
Before deriving the two dimensional first order model, we consider the general problem of a slender
axi-symmetric body falling through a cross-flow in the absence of viscosity and body forces. This is
given by the full unsteady potential equation[11] for the pitch and plunge store motion as follows:
∇2Φ =1
c2
[∂2Φ
∂t2+
∂Q2
∂t+ �Q · ∇
(Q2
2
)](1)
In addition, the relation between local sound speed and sound speed under free stream conditions is
given by:
c2 = c2∞ − (γ − 1)
[∂Φ
∂t+
1
2
(Q2 − U2
∞)]
. (2)
HereQ is the velocity field,c the sound speed,U∞ the free stream cross-flow velocity,Φ the velocity
potential,γ the ratio of specific heats andc∞ the free stream sound speed. The reference frame for
this particular problem is given in Figure 2.
X�
Y�
U∞g
α�Store
Figure 2: Reference frame for the general problem of a slender axi-symmetric store falling througha cross-flow in the absence of viscosity and body forces.
2.2 Contribution of Wavy Shear Layer Structures
We now consider the effect of the wavy shear layer structures on the normal force and pitching
moment on the store. As mentioned before, the wavy shape of the shear layer structure is influenced
by the feedback loop between the acoustic waves inside the cavity and the shear layer disturbances
at the cavity opening. These wavy structures result in non-uniformity in the flow domain outside
the cavity, that potentially change the aerodynamic loads on the store. To identify the expression for
7
the loads due to the wavy structures, the technique of Ref. [7] is adopted in which a combination
of non-dimensional analysis and the method of perturbations is used to identify the dominant terms
in Eqns. (1) and (2). Then we perform an order of magnitude analysis on the force and moment
expressions to quantify the effect of the wavy structures.
The non-dimensional scheme is given by:
X ′ =X
Lo
, Y ′ =Y
Lo
, Z ′ =Z
Lo
(3)
whereLo is the cavity length. Another useful parameter to consider is the Strouhal numberS = lotoU∞
whereto =
√δlog
is the characteristic time scale for the store plunge motion.
Since we are interested in the first order effect of the presence of the wavy structures on the store
force and moment, the store is neglected and the method of perturbations results in[7]:
Φ = U∞,r
[X + Loδsφ + · · ·
](4)
whereU∞,r is the external cross-flow velocity relative to the wavy shear layer structure and is typ-
ically U∞2
. Also φ is the perturbation velocity potential induced by the wavy structures. Here it is
assumed that the wavy structures are uniform along the cavity width. For a two-dimensional flow
problem, the flow perturbation [12] is of the orderδs, whereδs = δs
Lo� 1 and is the thickness of the
wavy structures.
The dominant terms in Eqns. (1) and (2), accurate toO (δs), are given as[7]:
β2φXX − φY Y = 0 (5)
whereβ2 = M2∞,r − 1.
Assuming that the wavy structure has small slope, the associated boundary condition is given by
8
[7]:∂φ
∂Y ′
∣∣∣∣∣Y ′=0
=dh (X ′)
dX ′ (6)
where the non-dimensional wavy structure is given by:
Y ′w = δsh (X ′) , h (·) = O (1) . (7)
The solution of this problem is given by [7]:
φ =1
βh (X ′ + βY ′) . (8)
Next, the velocity potential from Eqn. (8) is converted into pressure by using the Bernoulli’s
equation. Integrating pressure along the store cross-section and axis gives the normal forceFw and
pitching momentMw on the store. The detailed expressions forFw andMw are given in Ref. [7].
In the current paper, we further estimate the order of magnitude ofFw and Mw from these
expressions. Without any loss of generality, we assume that the wavy structures consist of piecewise
cosine components, such as:
h (X ′) = Wi cos (λiX′ + ζi) + Ri (9)
wherei is an integer,Wi = O (1) andλi = O (10). Then we have:
|Fw| ≤ πµδsδ maxi
(Wiλ
2i
)G0
|Mw| ≤ πµδsδ maxi
(Wiλ
2i
)G1. (10)
whereG0 =∫ xexo
a2 (x) dx, G1 =∫ xexo
a2 (x) x dx, x = xlo
(see Figure 22 in Appendix 5),a = aδlo
,
(xo, xe) are the nose and tail coordinates of the store, andµ = loLo
≤ O (1). Also, Fw andMw
are non-dimensionalized by(ρ∞U2∞)r δsl
2o and (ρ∞U2
∞)r δsl3o respectively, where(ρ∞U2
∞)r is the
dynamic pressure of the external cross-flow velocity relative to the wavy shear layer structure.
9
SinceG0 ≤ O (1), G1 ≤ O (1), δ � 1 andδs � 1, we have|Fw| � 1 and|Mw| � 1. In other
words, because of the slender body and the small thickness of the wavy structures, the effect of the
wavy structures on the normal force and moment on the store is negligible. Thus, the indirect effect
of the cavity acoustics field on the store release trajectory can be neglected. The effect of a thick
shear layer is investigated later in Section 3.3.
The remaining factors that affect the pitch and plunge motion of the store are the store-induced
velocity potential as it drops from the cavity, the shock waves due to microjets and gravity. We
consider the first factor below.
2.3 Store Motion Induced Velocity Potential
To derive the velocity potential associated with the store motion inside and outside the cavity, we
identify the dominant terms of Eqns. (1) and (2) by a combination of non-dimensional analysis and
the method of perturbations as before, following Ref. [7]. The appropriate non-dimensional scheme
is:
X =X
lo, Y =
Y
δlo, Z =
Z
δlo, t =
t
to(11)
wherelo is the store length. Using the method of perturbations, we have [7]:
Φ = U∞lo[X + δ2φ + · · ·
](12)
keeping in mind that the dominant flow perturbation is of orderδ2 for a body of revolution [12].
Hereδ is the slenderness ratio of the store (i.e., maximum store radius/store length).
The dominant terms, accurate toO (δ2) are given by [7]:
φY Y + φZZ = 0 (13)
which represents the Laplace Equation in the transverse (Y − Z) plane. Since the store angle of
10
attack is small by assumption (M1), we can assume thatφ is governed by the two dimensional
Laplace Equation in the cross-flow plane perpendicular to the store axis. Thus, the problem of
a slender axi-symmetric body that falls through an inviscid cross-flow is reduced to solving the
Laplace Equation of the perturbation velocity potential in the store cross-section plane.
After solving for the velocity potential, we use Bernoulli’s equation to convertφ to pressure, then
integrate the pressure along the store surface to get normal force and pitching moment on the store.
The force and moment expressions are given in Ref. [7]. These expressions are further processed
in the current paper to obtain a set of parametric nonlinear ordinary differential equations for the
store trajectory. We develop the appropriate parametric expressions for the force and moment in
three stages: (I) when the store is completely inside the cavity, (II) when the store passes through
the shear layer, and (III) when the store is completely outside the cavity.
2.4 Force and Moment for Store Inside Cavity
Let the store be released with the conditions(Y0, α0, V0, ω0
)from within the cavity. The problem of
the store drop inside the cavity, which is assumed to be quiet, reduces to the fall of a circle towards
the free slip surface in an immovable fluid. The presence of the cavity walls is ignored because
the cavity is sufficiently wider and taller than the slender store diameter. The simplified problem is
solved using the multipole expansion (Laurent series) technique [3].
The resulting normal force and pitching moment on the store, expressed in a parametric form,
are given by:
F = πρ∞δl3o
(g0
dVc
dtlo + g1
dω
dtl2o + g2ω
2l2o + g4V2c + g9ωloVc
)
M = πρ∞δl4o
(h0
dVc
dtlo + h1
dω
dtl2o + h2ω
2l2o + h4V2c + h9ωloVc
)(14)
where ρ∞ is the external free stream density,g0, g1, g4, g9 andh0, h1, h4, h9 are non-dimensional
factors that depend on the store geometry, position and orientation relative to the shear layer (exact
11
expression given in Appendix 2). Also,Vc is the store linear velocity (positive upwards) andω is
the rotational velocity (positive nose-up). The most dominant term in the above equation is the store
velocity (linear or angular velocity term). Assuming the linear velocity term to be dominant, the
vertical motion of the store inside the cavity is given by Newton’s second law, expressed as follows:
msdVc
dt= πρ∞δl3og4V
2c − msg,
dYc
dt= Vc, Is
dω
dt= πρ∞δl4oh4V
2c ,
dα
dt= ω, (15)
wherems and Is are the mass and moment of inertia of the store about its axis,g is gravity,Yc is
the store c.g. position relative to the shear layer line (positive upwards) andα is the angle between
the store axis and the external free stream direction (positive clockwise). The store state (position,
orientation and velocity) is illustrated in Figure 2. Also the factorg4 does not vary significantly
when the store is inside the cavity.
The following velocity expression can be obtained from this equation:
Vc = −√
g
Atanh
t
√gA +
1
2log
√gA− V0√
gA
+ V0
, g4 > 0
= −√
g
Atan
t
√gA − arctan
V0√gA
, g4 < 0 (16)
whereA = ρ∞δl3og4
ms. Figure 3 shows typical evolution ofV while the store falls inside cavity for
various drop conditions.
It is clear that irrespective of the signs ofg4 andV0, Vc eventually becomes negative inside the
cavity, although ifV0 > 0, thenVc stays positive for a certain time duration before turning negative.
In other words, the inner volume of the cavity has a stabilizing influence on the store, in that it
eventually falls towards the shear layer irrespective ofV0. Also h4 > 0 inside the cavity so thatω
andα increase monotonically.
If ω terms in force are significant, a similar analysis can be done to find the drop conditions (i.e.,
V0 andω0) that would ensure that the store falls towards the shear layer. For example, if the store
12
0 0.2 0.4 0.6 0.8 1-0.6
-0.4
-0.2
0
0.2
Vc
Instability region
Stability region
V0 < 0, g
4 > 0
V0 > 0, g
4 > 0
V0 < 0, g
4 < 0
V0 > 0, g
4 < 0
0
Ul∞�� t
Figure 3: Velocity of store while inside cavity for various drop conditions, withVc = Vc
δU∞ . The
values of parameters used are:∣∣∣V0
∣∣∣ = 0.18δU∞, |g4| = 0.75.
is dropped with very highω0 and lowV0, the store may pitch up and move towards the cavity roof
instead of the shear layer.
2.5 Force and Moment When Store Crosses Shear Layer
When the store crosses the shear layer, a part of the store falls inside the cavity with no flow, another
part falls outside the flow facing a uniform flowU∞ and the remaining part faces both flows. This is
illustrated in Figure 4.
X�
Y�
U ∞
α�
Store
Side view
x�
y�
ex�
0x�
1x�
2x�
Inside cavity
Outside cavity
Partially submerged
in flow
Figure 4: Reference frame used when the store crosses the shear layer slip surface.
Because of the linearity property of solutions of the Laplace Equation, the problem of the store
crossing the shear layer can be reduced to three unit problems: one is a circle that falls towards
the slip surface in an immovable fluid (problem (I)), another one is a circle partially immersed in a
13
flow (problem (II)), and the last one is a circle that falls from a free surface into a uniform stream
(problem (III)). Problem (I) has been solved in the previous section and problem (III) is solved in
the following section. Problem (II) can be solved using conformal transformation [3] of the store
region partially immersed in external cross-flow to an ‘equivalent’ flat plate in the complex plane
σ = ξ + iη as illustrated in Figure 5. The mapping of the store region submerged in the external
Back view
,Y y
,Z zH a
cc−
β
( )a Physical plane
Body 0Y >
Body 0Y <
η
ξbb−
( )b Transformed plane
Body 0Y >
Body 0Y <
b+− b+
Y >
Y <
Y >
Y <
(ζ) (σ)
Figure 5: Conformal transformation required to solve for the velocity potential associated with thestore region partially immersed in the external flow.
cross-flow is given by [3]:
ζ = f (σ,X, t) = cRn + 1
Rn − 1, n =
π − β
π,R =
σ + b
σ − b, b =
c
n(17)
while the mapping for the store region inside the cavity is:
ζ+ = f+ (σ,X, t) = cRm
+ + 1
Rm+ − 1
,m =β
π,R+ =
σ + b+
σ − b+, b+ =
c
m(18)
To verify the transformation of the store cross-section into a flat plate, we solve Eqn. (17) to obtain
the transformed(Y, Z) from (ξ, η). The result is:
Y (ξ)|η=0 = −2cQn
Dsin πn, Z (ξ)|η=0 = c
Q2n − 1
D,D = Q2n − 2Qn cos πin + 1, Q =
ξ + b
ξ − b(19)
It can be shown that:
(Y (ξ)|η=0 −
c
tan π (1 − n)
)2
+ Z (ξ)|2η=0 =
(c
sin π (1 − n)
)2
(20)
thus showing that the transformed(Y, Z) form an arc from the flat plate(ξ, η = 0). This type of
14
transformation has been used for the flow over a log given in Milne-Thomson [13] for slip line
boundaries. Also forn > 12, the flow velocities have a singularity of type(σ2 − b2)
12−n at the points
where the free surface intersects the body. Since this singularity is integrable,F andM are not
singular.
After the transformation to a flat plate,F and M due to the store partially immersed in the
external flow are computed. To this, the contributions of the store portion completely inside and
completely outside the cavity are added. Before we can predict the force and moment as the store
crosses the shear layer, let us study problem (III) when the store is outside the cavity.
2.6 Force and Moment for Store Outside Cavity
The problem of the store drop outside the cavity reduces to the fall of a circle in a free stream. If
the store is far from cavity, then the stream can be considered unbounded (the no-cavity model). If
the cavity depth is small, then the stream can be considered to be bounded by a rigid wall (the zero-
depth-cavity model). Also if the store is close to the shear layer, then the stream can be considered to
be bounded by a free surface (the near-cavity model). These problems are solved using the multipole
(Laurent series) expansion technique [3]. Finally the presence of shear layer structures is to be taken
into account if the thickness of the shear layer is not small (the thick shear-layer model). In the latter
case, the shear layer structure was obtained from flow visualization experiments, and the technique
developed in Section 2.2 was used to estimate the effect of the shear layer structures on the store
release trajectory.
The steps for solving the problems for velocity potential and converting the potential to force
and moment on the store is given in Ref. [3]. Here we use the same steps to obtainF andM , the
expressions of which are further manipulated so that they are represented in a parametric form:
F = πρ∞δl3o
(g0
dVc
dtlo + g1
dω
dtl2o + g2ω
2l2o + U2∞g3α
2 + g4V2c + U∞δg5ωlo + U2
∞δg6α+
U∞δg7Vc + U∞g8αωlo + g9ωloVc + U∞g10αVc + δ2U2∞g11
)
15
M = πρ∞δl4o
(h0
dVc
dtlo + h1
dω
dtl2o + h2ω
2l2o + U2∞h3α
2 + h4V2c + U∞δh5ωlo + U2
∞δh6α+
U∞δh7Vc + U∞h8αωlo + h9ωloVc + U∞h10αVc + δ2U2∞h11
)(21)
whereg0, · · · , g11, h0, · · · , h11 depend on the model used (i.e., the no-cavity, zero-depth-cavity or
near-cavity model or the model with wavy structures) and are given in Appendix 4. It may be noted
that this equation also gives the general expressions forF andM when the store crosses the shear
layer, with the correspondingg0, · · · , g11, h0, · · · , h11 being outlined in Appendix 3.
Because of the high speed of the external flow and because the store is inclined to the flow,
the component of the external flow normal to the store is dominant compared to the store velocity.
Hence the dominant terms inF andM , when the store is external to the cavity, containα, and is of
the form:
F = πρ∞U2∞δl3o
(g3α
2 + δg6α),
M = πρ∞U2∞δl4o
(h3α
2 + δh6α). (22)
The second order dependence ofF andM on α makes it difficult to analyze the dominant trend in
the store motion outside the cavity. Hence, we linearize Eqn. (22) aboutα = 0 to get:
F = πδ2l3oβf (α) ,
M = πδ2l4oβfm (α) ,
β = ρ∞U2∞,
f (α) = g6α,
fm (α) = h6α, (23)
It should be mentioned here that the error incurred in using assumption (M1) to develop the above
expression forF is less than 10% forα < 40◦. Using Eqn. (23) and following Eqn. (15), the
16
governing equation for the store trajectory outside the cavity becomes:
msdVc
dt= πρ∞U2
∞δ2l3og6α − msg,dYc
dt= Vc,
Isdω
dt= πρ∞U2
∞δ2l4oh6α,dα
dt= ω. (24)
After the store exits the shear layer, let its state be denoted by(Y1, α1, V1, ω1
). Whenα1 is large,
sinceg6 > 0 in Eqn. (24), the linear dependence off (α) on α results in largeF . This increases the
likelihood of F to overcome gravity and the store to return back towards the cavity. This is further
demonstrated from the expression forVc which is obtained by solving Eqn. (24) withV1 as the initial
condition. As a result we get:
Vc = B(−ω1 cos Λt + Λα1 sin Λt
)− gt + Bω1 + V1 (25)
whereΛ2 = πρ∞U2∞δ2l4o |h6|, B = log6
|h6| . The linear velocity expression consists of a sinusoidal term,
a linear term and an offset. From an order of magnitude analysis, it can be shown that the offset term
is the dominant one. For a given(Y1, α1
), if
ω1
V1
> − 1
B, (26)
thenVc becomes positive when1Λ
arccos(1 + V1
Bω1
)< t < 2π
Λ− 1
Λarccos
(1 + V1
Bω1
). The implication
is that over this time interval, the store returns towards the cavity. The above discussions indicate that
the store dynamics outside the cavity can be characterized by a stable region with linear boundary,
where a stable region is defined as the set of(Y1, α1, V1, ω1
)such that the store exhibits a safe drop.
The stability analysis performed so far, particularly Eqn. (26), is based on the simplified dynam-
ics of the store in Eqn. (24). To verify if this stability boundary is valid in general, we evaluate
the four models (i.e., the no-cavity, zero-depth-cavity, near-cavity models and the model with wavy
structures) and determine the conditions(V1, ω1
)that result in an unsafe store departure for a partic-
ular(Y1, α1
). This is shown in Figure 6 and validates the linear stability boundary predicted above.
17
It should also be noted that the near-cavity model predicts the most conservative stability region.
Therefore, out of the four models, the near-cavity one is used for determining the optimal control
strategy for successful store release in Section 5.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-0.25 -0.2 -0.15 -0.1 -0.05 0
Far from cavity model
Zero-depth cavity model
Near-cavity model
Thick shear layer case
0l
U
ωδ
∞
�
�
1V
Uδ ∞
�
Stable region
Unstable region
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-0.25 -0.2 -0.15 -0.1 -0.05 0
Far from cavity model
Zero-depth cavity model
Near-cavity model
Thick shear layer case
0l
U
ωδ
∞
�
�
V
Uδ ∞
�
Stable region
Unstable region
1
Figure 6: Stability region predicted by the four outside-cavity models in the no-control case. Theinitial conditions for the models are:Y1 = Y1
δlo= −1.63 andα1 = α1
δ= 0.01 with V1 = V1
δU∞andω1 = ω1 lo
δU∞ shown in the above plot. The uncertainty in the stability boundaries are as follows:∆V1 = ±0.005, ∆ω1 = ±0.125. The cross-flow Mach number is 2.46, Reynolds number based onstore length is 4.9 million, cavity aspect ratio is 5 and store length is half of cavity length.
We note that the thick shear layer model is discussed in Section 3.3 and evaluated in Section 4.3
so as to determine the impact of the shear layer thickness and the store drop location on the quality
of the store trajectory.
3 Analysis of the Complete Store Drop
The low order model developed in Sections 2.4-2.6 is now used to analyze the store trajectory as it
is dropped from inside the cavity to the point where it has either exhibited successful drop or it starts
to return back into the cavity. This analysis is performed for two cases: (I) when microjets are off,
and (II) when the microjets are switched on. Finally, a store drop model extension that considers the
wavy shear layer structures at the cavity opening is discussed at the end of this section.
18
3.1 No microjets present
Let the store be dropped from inside the cavity with the conditions(Y0, α0, V0, ω0
), with α0 positive
and small,V0 < 0 andω0 > 0 but not large enough so that the store hits the cavity roof immediately.
It follows from Section 2.4 thatω, α and∣∣∣Vc
∣∣∣ inside the cavity increases monotonically by the time
it reaches the shear layer. Also, from Eqns. (15) and (23), it follows that when the store crosses the
shear layer, the portion of the store inside the cavity experiences a normal force proportional toVc2
while the portion outside experiences a force proportional tof (α). The latter is dominant because
of the high external flow speed. Now from Eqn. (23) (withh6 < 0) and from the fact that the tail
side is first exposed to the external stream when the store exits the cavity, it follows that there is an
increase inM in the nose-down (negative) direction and a consequent decrease inω. Despite this, if
the store is released with a largeω0 when compared toV0, thenω1 is large when the store exits the
cavity and Eqn. (26) is satisfied, thus implying an unsuccessful drop. It also follows that if the store
is released with a smallω0, thenω1 is small and Eqn. (26) is not satisfied. In that case, the store
exhibits a clean departure. In summary, if the store is dropped with a smallω0 when compared to
V0, the store exhibits a safe departure from inside the cavity.
It is instructive to compare the analysis performed here with that done for subsonic and transonic
flows [2, 3]. Because of the high speed of supersonic flow, the dominant term inF andM are
dependent onα when the store is external to the cavity. Hence it is relatively easy to analyze its
stability. On the other hand, in the subsonic and transonic cases, the terms containingVc, ω and their
time derivatives are equally important and hence it is more difficult to analyze the trend in store
trajectory outside the cavity. However, it should be noted that unsafe store departure depends on the
nature of the drop conditions in all flow cases; in the subsonic and transonic cases, unsafe departure
occurs if eitherV0 is small orα0 is large whereas in the supersonic case, we note that unsuccessful
departure occurs if∣∣∣ ω1
V1
∣∣∣ is large.
19
3.2 Microjets Present
Next we analyze the store drop phenomenon when microjets are switched on. The following dom-
inant effects are observed due to the introduction of microjets: (i) flow deceleration(
U∞,2
U∞,1< 1
),
(ii) flow compression(
ρ∞,2
ρ∞,1> 1
), (iii) flow turning (away from the cavity by an angleθ), and (iv)
introduction of plane shock waves at the cavity leading edge. Figure 7 gives a pictorial description
of these effects.
,2U∞
θα Store
Shear layer
Flow
injectors
Plane shock
,1U∞,1ρ ,∞
,2ρ ,∞
ψ
Figure 7: Shock geometry outside the cavity due to application of microjet-based flow injectors atthe leading edge and upstream end.
Since the store is released in the middle of the cavity, it is not expected to intersect the plane
shock line; therefore effect (iv) is not directly considered any further in the discussions below. How-
ever, effects (i)-(iii) influenceF andM on the store and their magnitudes are determined by the ratio
of microjet-cross-flow momentum ratioCµ defined below2:
Cµ =14(ρAU2N)µ
12(ρU2)∞
(DoδBL
) . (27)
The dependence of effects (i)-(iii) onCµ was experimentally determined in the Florida State Univer-
sity (FSU) setup and is explained in Appendix 1. The main idea is twofold: (a) the flow visualization
studies done using the FSU cavity setup established theCµ-dependence of the shock orientation an-
2The subscriptµ refers to the mean microjet properties at the microjet nozzle exit, subscript∞ refers to the meanundisturbed external flow properties,ρ andU are the mean dimensional density and flow speed,A is the microjet cross-section area,N is the number of microjets,Do is the cavity width as shown in Figure 22 (in Appendix 5) andδBL is thedimensional boundary layer thickness at the cavity leading edge. Also,
(ρU2
)µ
depends on the pressure at which themicrojets are activated, thereby establishing a direct relationship between the microjet pressure andCµ.
20
gleψ andθ (Figure 7), and (b) the effects (i)-(iii) were quantified using results of (a) and plane shock
theory [11]. The resulting dependence of (i)-(iii) onCµ is given in Figure 8. It should be noted that
all these effects are introduced when the store exits the cavity, whereas they do not affect the mean
flow field inside the cavity.
0.04 0.06 0.08 0.1 0.12
1.09
1.095
Cµρ ∞
, 2ρ ∞
,10.04 0.06 0.08 0.1 0.12
0.545
0.55
0.555
0.56
U
0.04 0.06 0.08 0.1 0.127.4
7.6
7.8
8
8.2
8.4
θ, d
eg
∞, 2
∞,1
U
Cµ
Cµ
Figure 8: Variation of external flow properties withCµ due to application of microjets.
0 1 2 3 4 5 6-5
-4
-3
-2
-1
0
1x 10
-3
norm
al f
orce
Microjets offθ = 5.73 degθ = 7.45 degθ = 7.94 deg
Store crossing shear layer
0 1 2 3 4 5 6-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
mom
ent
Microjets offθ = 5.73 degθ = 7.45 degθ = 7.94 deg
Store crossing shear layer
U∞�� t0l
U∞�� t0l
Figure 9: Model-predictedF and M evolution with time for differentθ when the store passes
through the shear layer. The other external flow parameters used are:ρ∞,2
ρ∞,1= 1.818, andU∞,2
U∞,1=
0.915. Also, the cross-flow uncontrolled Mach number is 2.46, Reynolds number based on storelength is 4.9 million, cavity aspect ratio is 5 and store length is half the cavity length. A smoothingfunction is used to remove noise in the force and moment predictions of the model.
Now F andM have been derived and analyzed in Section 2 for general values ofρ∞ andU∞. In
order to study the effect of microjets, we varyCµ, which in turn introduces the changes (i)-(iii). This
in turn changesF andM as outlined in Section 2. An important point to note here is the change in
F andM as a function ofθ. In Figure 9, the responses ofF andM as a function of time and asθ
is varied, is illustrated. It is clear from the figure that a dramatic change in the slope ofM occurs
21
whenθ is increased. In particular,M becomes negative whenθ is below a certain value and positive
for higher value ofθ, which essentially indicates onset of the store returning towards the cavity, as
explained below. A similar trend is observed forF . It should also be noted that the variations ofF
andM with effects (i) and (ii) are more gradual.
To further analyze the dependence ofM on θ, we consider the general expression forM given
by Eqn. (21) when the store crosses the shear layer. Keeping the dominant terms in the equation and
with microjets on, we have:
M = πρ∞δl4o
h4V
2c︸ ︷︷ ︸
I
+ h9ωloVc︸ ︷︷ ︸II
+ U2∞h3 (α − θ)2︸ ︷︷ ︸
III
+ U2∞δh6 (α − θ)︸ ︷︷ ︸
IV
. (28)
These terms are plotted in Figure 10. It is clear that when the store touches the shear layer from
0 2 4 6-10
-8
-6
-4
-2
0
2x 10 -3 θ = 7.45 deg
IIIIIIIV
0 2 4 6-10
-8
-6
-4
-2
0
2x 10 -3 θ = 7.94 deg
IIIIIIIV
U∞�� t0l
U∞�� t0l
Figure 10: Different components of the model-predictedM (from Eqn. (28)) and their evolution withtime for differentθ when the store passes through the shear layer. The other external flow parameters
used are:ρ∞,2
ρ∞,1= 1.818, andU∞,2
U∞,1= 0.915. Also, the cross-flow uncontrolled Mach number is 2.46,
Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 and store length ishalf of cavity length. A smoothing function is used to remove noise in the moment prediction of themodel.
inside the cavity, the dominant effect is the introduction of terms III and IV inM , both of which
depend onα − θ. Because of this feature, a drastic change inM is observed at the instant when the
store hits the shear layer. At this point, ifα is greater thanθ, there is a negative pitching moment
because the store tail is exposed to the external flow. However, ifθ is increased by introducing higher
microjet pressures such thatα is less thanθ at this point, then there is a positive pitching moment.
22
It follows that when the microjets are switched on such thatθ is less than a certain value,M
increases in the negative direction and causesω to decrease when the store crosses the shear layer.
At this point, Eqn. (26) is not satisfied which implies that the store exhibits a safe drop. When
microjet pressures are increased, there is a further increase inθ. If θ exceeds the limiting value,M
increases positively and correspondingω increases when the store crosses the shear layer. Increase
in ω1 causes Eqn. (26) to be satisfied leading to an unsuccessful drop. Also this trend is independent
of ω0. The variation of the store state as it crosses the shear layer in the absence and presence of
microjets is given in Figure 11.
0 5 10 15 20-0.23
-0.22
-0.21
-0.2
-0.19
-0.18
-0.17
Vc
NOCMCHC
U∞�� t0l
Outside cavity
0 5 10 15 200.5
0.52
0.54
0.56
0.58
0.6
0.62
0.64
ω
NOCMCHC
Outside cavity
U∞�� t0l
Figure 11:Vc andω evolution with time for different control inputs when the store passes throughthe shear layer. The control inputs are specified in Table 1. Also, the cross-flow uncontrolled Machnumber is 2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 andstore length is half of cavity length.
In summary, the introduction of microjets can ensure safe departure for anyω0 as long asθ is
less than a certain value. However, forθ greater than this limit, which occurs for large microjet
pressures, the store tends to return back to the cavity.
3.3 Model Extension
In previous sections, we discussed the store drop trajectory when the shear layer at the cavity opening
is thin. In Section 3.3.1, we extend the store drop model to include the effect of the wavy shear layer
structures on the store drop trajectory. In Section 3.3.2, this extension is further used to study the
effect of the drop location of the store inside the bay on its trajectory.
23
3.3.1 Effect of Thick Shear Layer
When the shear layer is thick, we cannot discount the effect of the wavy shear layer structures on
the storeF andM . In this case, we use the technique developed in Section 2.2 for estimating the
effect of the shear layer structures on the storeF andM . The latter are parameterized as given
in Eqn. (14) and the detailed expressions ofF andM are given in Appendix 4. The unknown in
these expressions is the shear layer structure. As noted before, the feedback loop between the tones
inside the cavity and the convective shear layer structures at the mouth sustains the acoustic field
and the shape of the structures [10]. The addition of momentum flux at the cavity leading edge, due
to microjets, introduces an active damping factor as explained in [6] and disrupts the feedback loop.
The result is the reduction of cavity noise and change in the wavy shear layer shape.
The reduction of cavity noise is illustrated through experiments conducted on a rectangular cav-
ity of aspect ratio 5.1 under Mach 2.0 flow in the Florida State University (FSU) wind tunnel facility
[10] using different microjet pressures. A typical tonal reduction result is shown in Figure 12.
Another interesting result is illustrated in Figure 13. The figure shows saturation behavior in the
2 4 6 8 10110
120
130
140
150
160
Frequency (kHz)
dB (
re: 2
0 µP
a)
No control30 psig
Figure 12: Experimentally observed SPL spectra recorded by the pressure transducer located atthe middle of the FSU cavity leading edge. The two spectra are for the microjets-off and 30 psigmicrojet pressure respectively. External flow conditions:M = 2.0 and Re= 3 million (basedon cavity length). Cavity dimension:L/D = 5.1. Uncertainty in experimental SPL:±0.5dB.Uncertainty in frequency:±40Hz.
noise reduction ability of microjets with increase in microjet pressure. Both suggest that for the
FSU cavity, OASPL reduction at the leading edge is saturated for microjet pressure 30 psig and
24
above.
0
1
2
3
4
5
6
7
8
9
10
10 20 30 40 50 60 70 80Microjet pressure, psig
Decre
ase
in
OA
SP
L l
ev
el,
dB
No microjet
Figure 13: Experimentally observed OASPL reduction recorded by the pressure transducer locatedat the middle of the FSU cavity leading edge. External flow conditions:M = 2.0 and Re= 3million (based on cavity length). Cavity dimension:L/D = 5.1. Uncertainty in experimentalOASPL:±1.0dB.
As for the shear layer structure at the cavity mouth, it was obtained in this paper from flow
visualization experiments conducted using the same FSU setup. The mean flow field inside and
around the cavity was recorded for different microjet pressures (or equivalently, for different values
of Cµ) using the technique of Particle Image Velocimetry (PIV). From the mean flow field, the
shear layer structure was identified. The specific shear layer structure for the FSU cavity is shown
in Figure 14. Using the above experimental data, we can obtain the shear layer structure for any
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
X’
δ s
Cµ = 0
Cµ = 0.119 Cµ = 0.439
Shear layer structure
Figure 14: Shear layer structure for different microjet momentum ratios, obtained from FSU flowvisualization studies. The notationsX ′ andδs are explained in Eqns. (3) and (4). In addition, theflow conditions for the FSU setup are:M = 2.0 and Re= 3 million (based on cavity length). Cavitydimension:L/D = 5.1.
arbitraryCµ by interpolation. The effect of shear layer thickness on the store release trajectory is
examined by comparing the model predictions to experimental results in Section 4.3.
It should be noted that the aforementioned model extension pertains to only the shear layer
25
thickness and thus considers the indirect effect of the cavity acoustics. Extension due to direct
unsteady effects such as the acoustic waves inside the cavity and the convective shear layer waves is
ignored.
3.3.2 Effect of Initial Drop Location of Store Inside Bay
The interesting point to note from Figure 14 is that the shear layer thickness not only varies with
change inCµ but also with the streamwise locationX ′. The change in the shear layer thickness with
X ′ causes non-uniformity in the flow outside the cavity that also varies withX ′, as illustrated in
Eqn. (8). Thus, the drop location of the store inside the cavity is expected to change its trajectory.
This change can be estimated using the model extension developed in Section 3.3.1 and is discussed
in more detail in Section 4.3 where the model predictions are compared with experimental results.
In particular, the change in the store trajectory was studied when the store was dropped from: (a)
near the leading edge of the cavity(X ′c = 0.4), (b) near the middle of the cavity(X ′
c = 0.58), and
(c) near the trailing edge(X ′c = 0.76). Here, ‘X ′
c’ refers to the store c.g. streamwise coordinate.
The above study is limited by the assumptions that the store does not intersect the plane shock
waves due to the microjets during its drop, and that the streamwise coordinate of the store is not
changed during its motion. In the former case, the diffractive effects associated with the shock-line-
store intersection are ignored while the latter assumption is made because the streamwise motion of
the store during its drop is insignificant compared to the transverse motion and the streamwise drag
force on the store is not considered in this paper.
4 Experimental Validation
In this section, we compare the low order model predictions with results from experiments per-
formed under the HIFEX Program [1, 4, 5].
26
4.1 Experimental Details
A series of free drops of a slender axi-symmetric store from a sub-scale generic weapons bay cavity
was conducted under the HIFEX Program at various supersonic cross-flow Mach numbers [1, 5].
For brevity, this paper considers the experiments that were performed at two Mach numbers, 2.0
and 2.46. In addition, the bay model has the aspect ratioLo
Ho= 5 and Lo
Do= 5, with Ho being the
depth andDo the width of cavity. The Reynolds number of the flow based on the store length is
approximately 4.9 million and the speed of flow at Mach 2.46 is 579 m/s. The store is half the cavity
length, its associatedδ is 0.0615, and the store geometry is illustrated in Figure 15. Moreover, the
−0.4 −0.2 0 0.2 0.40
0.5
1
x
a(x)
Figure 15: Radius profile of the store. Herex = xlo
anda = aδlo
. Also x = 0 corresponds to the storec. g.
flow-injection actuator consists of two rows of sonic microjets along the leading edge and two rows
near the upstream end of the cavity, situated aboutL/8 distance from the leading edge. HereL is
the cavity length, the microjets are of 400µm in diameter and each row consists of 50 microjets.
The store was dropped fromYc = Yc
δlo= 1.3 from the middle of the cavity with nearly zero angle
of attack and the store trajectory outside the cavity was recorded in high speed video [1, 5]. The
drop tests were conducted using three different control inputs: no control (NC), microjet-control on
(MC), and high-microjet-pressure-control (HC). Table 1 shows the changes in the external flow for
the three control inputs and the store drop results are discussed below.
4.2 Comparison with the Model
As mentioned in Section 2.6, we have four models (i.e., the no-cavity, zero-depth-cavity, near-cavity
models and the model with wavy structures) to describe the trajectory outside the cavity. To choose
27
Item NC MC HC MC-NC, % HC-MC, %ρ∞,2
ρ∞,11.000 1.818 1.826 81.8 0.4
U∞,2
U∞,11.000 0.915 0.914 -8.5 -0.1
ρ∞,2U2∞,2
ρ∞,1U2∞,1
1.000 1.523 1.524 52.3 0.1
θ, deg 0 7.827 7.947 − 1.5
Table 1: Changes in external flow properties with control. Three control inputs are considered:no control (NC), microjet-control on (MC) and high-microjet-pressure-control (HC). MC refers to100-250 psig microjet pressure and HC refers to 200-250 psig microjet pressure. The first numberhere refers to the rows of microjets closest to the leading edge and the second number to the rowsat the upstream end of the cavity. Also, the cross-flow uncontrolled Mach number is 2.46, Reynoldsnumber based on store length is 4.9 million, cavity aspect ratio is 5 and store length is half of cavitylength.
the appropriate one, we varied the initial conditions(V1, ω1
)keeping
(Y1, α1
)fixed as a starting
point and performed stability analysis using all the models, as shown in Figure 6. MATLAB’s
ode45.m was used to solve for the store governing equations in the simulations. From these studies,
we identified that the near-cavity model provided the most conservative stability region for the store,
and was the model of choice for subsequent model simulations.
Next we use a combination of the inside-cavity model, the model of the store passing through
shear layer, and the near-cavity model to obtain the complete trajectory after the store is released
from inside the cavity. Using this tool, we predict the store release trajectory in the NC case and
then consider two specific experiments in the MC and HC cases, where the corresponding flow
deceleration, flow compression, and flow turning angle are calculated as outlined in Section 3.2, and
the resulting store release trajectory under these flow conditions and the same drop conditions as the
NC case are determined. The model-predicted trajectories are compared with experiments in Figure
16. The figure shows unsafe store separation for the no-control and high microjet pressure cases
and safe separation for the low microjet pressure case. The figure also indicates that the prediction
of the experimental store trajectories by the low order models is somewhat poor. However it should
be noted that the models correctly predict the variation of˙Yc as the store drops outside the cavity,
which in turn implies that the models accurately foretell whether the drop is successful or not. This
validates the models and illustrates the usefulness of the models in designing an optimal control
28
0
Ylδ
−�
�
0
Ylδ
−�
�
0
Ylδ
−�
�
0
Ul∞�� t
0
Ul∞�� t
0
Ul∞�� t
10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
Experiment
Model
0 20 40 60 80 1001
2
3
4
5
6
7
Experiment
Model
0 20 40 60 80 1000
2
4
6
8
10
12
Experiment
Model
(a) NC (b) MC (c) HC
Figure 16: Store trajectory prediction by the near-cavity model versus experimental observation for:(a) no control (NC), (b) microjet-control on (MC) and (c) high-microjet-pressure-control (HC). Also(t, Yc
)= 0 correspond to the instance when the store just exits the cavity from the middle of the
bay, and increasing−Yc corresponds to the store moving away from the bay. The cross-flow Machnumber is 2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 andstore length is half of cavity length. The initial conditions for the model are given in Table 2.
Item Inside cavity (Point 0) While exiting cavity (Point 1)No control (NC)
Yc 1.30 -1.63α 0.01 8.96Vc -0.18 -0.22ω 0.62 0.59
Microjet-control on (MC)Yc 1.30 -1.63α 0.01 9.10Vc -0.18 -0.22ω 0.62 0.57
High-microjet-pressure-control (HC)Yc 1.30 -1.63α 0.01 9.49Vc -0.18 -0.22ω 0.62 0.60
Table 2: Initial conditions for the store trajectory analysis used in Figure 16. The cross-flow Machnumber is 2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 andstore length is half of cavity length.
strategy that ensures clean store departure.
To explain the behavior depicted in the figure, we refer to Section 3 where we have analyzed the
aerodynamic loads on the store for given drop conditions and microjet pressures. The point to note
is that the store drop is successful when∣∣∣ ω1
V1
∣∣∣ is small and unsuccessful otherwise. When microjets
are off and the store just exits the cavity,ω1 is large because of largeω0 at the drop point. The
largeω1 in comparison toV1 results in the store to exhibit a unsuccessful drop. When microjets are
29
U∞�� t
0
Ul∞�� t
0
Ul∞�� t
0
Ul∞�� t
0
Ylδ
−�
�
0
Ul∞�� t
0l
(a) (b)
(c) (d)
(e) (f)
0 20 40 60 80-0.3
-0.2
-0.1
0
0.1V c
NCMCHC
Instability region
Stability region
0 20 40 60 80-1
-0.5
0
0.5
1
ω
NCMCHC
0 20 40 60 80-2
0
2
4
6
8
NCMCHC
Shear layer line
0 20 40 60 80-20
0
20
40
60
80
α, d
eg
NCMCHC
Outside cavityOutside cavity
Outside cavity
Outside cavity
0 20 40 60 80-0.01
-0.005
0
0.005
0.01
0.015
norm
al f
orce
NCMCHC
Outside cavity
0 20 40 60 80-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
mom
ent
NCMCHC
Outside cavityU∞�� t0l
Figure 17: Model-predicted normal force, pitching moment and state of the store after being droppedfrom inside the cavity to when the store either returns back into the cavity or drops successfully in thethree control cases: no control (NC), microjet-control on (MC) and high-microjet-pressure-control(HC). The cross-flow Mach number is 2.46, Reynolds number based on store length is 4.9 million,cavity aspect ratio is 5 and store length is half of cavity length. The model used for outside-cavitytrajectory prediction is the near-cavity one, with the initial conditions given in Table 2. A smoothingfunction is used to remove noise from force and moment predictions of the model.
switched on, the external flow is compressed, decelerated and turns away from the cavity with an
angleθ. There exists a linear relationship betweenM andα − θ as soon as the store hits the shear
layer such thatM becomes negative whenθ is small and positive whenθ is large. In the case of MC,
θ is small and the result is an increase inM in the negative direction and decrease inω, when the
store exits the cavity. Lowω1 in comparison toV1 results in the store to exhibit a safe drop. When
the microjet pressures are increased in the case of HC,θ exceeds the limiting value. The result is
that M and thereforeω increase positively when the store crosses the shear layer. This results in
largeω1 in comparison toV1, thus leading to an unsuccessful drop. Figure 17 illustrates the force
30
and moment as well as the state of the store after being dropped from inside the cavity to when the
store either returns back into the cavity or drops successfully in the three control cases.
4.3 Comparison with Extended Model
In this section, we discuss the results of incorporating the thick shear layer wavy structures and the
effect of the initial drop location of the store inside the bay on the store drop trajectory. As explained
in Section 2, the effect of the cavity acoustics is indirectly taken into consideration in this model.
4.3.1 Effect of Thick Shear Layer
0
Ylδ
−�
�
0
Ylδ
−�
�0
Ylδ
−�
�
0
Ul∞�� t
0
Ul∞�� t
0
Ul∞�� t
(a) NC: Store dropped in cavity middle
(b) MC: Store dropped in cavitymiddle
(c) HC: Store droppedin cavity middle
20 40 60 80 1001
2
3
4
5
6
7
Experiment
Model (thick SL)
20 40 60 80 1000
2
4
6
8
10
12
14
Experiment
Model (thick SL)
20 40 60 80 1000
2
4
6
8
10
12
Experiment
Model (thick SL)
Figure 18: Store trajectory prediction by the thick shear layer model versus experimental observationfor: (a) no control (NC), (b) microjet-control on (MC) and (c) high-microjet-pressure-control (HC).Also
(t, Yc
)= 0 correspond to the instance when the store just exits the cavity from the middle
of the bay(X ′c = 0.58), and increasing−Yc corresponds to the store moving away from the bay.
The cross-flow Mach number is 2.46, Reynolds number based on store length is 4.9 million, cavityaspect ratio is 5 and store length is half of cavity length. The initial conditions for the model aregiven in Table 3.
The thick shear layer model-predicted trajectories are compared with experiments in Figure 18.
The figure shows unsafe store separation for the no-control and high microjet pressure cases and safe
separation for the low microjet pressure case, the initial drop conditions being similar in the three
cases as illustrated in Table 3. Comparison of Figures 18(a) and 16(a) shows that the store trajectory
prediction of the no-control case is improved by including the thick shear layer structures. However,
the quality of prediction of the MC and HC cases is not affected by introduction of the shear layer
thickness. One possible reason is that the shear layer structure near the leading edge becomes flatter
when microjet pressure is increased, as illustrated in Figure 14. Also, from Eqn. (8), it is clear that
31
the presence of the shear layer structure at point(X ′, Y ′ = 0) creates non-uniformity in the flow
outside the cavity at point(X ′ + βY ′, Y ′). In other words, the shear layer structures present near
the leading edge affect the store trajectory when it is dropped near the middle of the cavity. The
flatter structures resulting from higher microjet pressures diminish the effect of the wavy shear layer
structures on the storeF andM . In summary, the impact of the wavy structures is observable in the
no-control case and not in the microjets-on cases.
Item Inside cavity (Point 0) While exiting cavity (Point 1)No control (NC)
Yc 1.30 -1.63α -44.27 33.58Vc -0.04 -0.16ω 3.47 3.89
Microjet-control on (MC)Yc 1.30 -1.63α -43.27 35.43Vc -0.04 -0.19ω 3.47 3.95
High-microjet-pressure-control (HC)Yc 1.30 -1.63α -43.27 32.58Vc -0.04 -0.17ω 3.48 3.91
Table 3: Initial conditions for the store trajectory analysis used in Figure 18. The cross-flow Machnumber is 2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 andstore length is half of cavity length.
4.3.2 Effect of Initial Drop Location of Store Inside Bay
The model extension developed in Section 3.3.1 allows us to study the effect of the initial drop
location of the store inside the bay on its drop trajectory. The result is given in Figures 18 and 19,
with the initial drop conditions being similar in all the cases as illustrated in Tables 18 and 4. It is
clear that when the store is dropped from near the leading edge of the bay, Figure 19(a) shows unsafe
store separation for the no-control case, and safe separation for the low and high microjet pressure
cases. This is opposed to the store drop behavior when the store is released from the middle of the
cavity, in which case store separation becomes unsafe with increase in microjet pressure. In the case
32
when the store is dropped from near the trailing edge, the store makes a safe exit even in the absence
of control, as shown in Figure 19(b). These predictions were corroborated by HIFEX experiments
performed for the store drops near the leading and trailing edges of the cavity, in addition to the
drops from the bay middle that were discussed in Sections 4.1 and 4.2.
20 40 60 80 1001
2
3
4
5
6
7NCMCHC
(a) Store dropped nearcavity leading edge
30 40 50 60 70 80 90 1001
2
3
4
5
6
7
0
Ylδ
−�
�
0
Ylδ
−�
�
0
Ul∞�� t
0
Ul∞�� t
(b) NC: Store dropped near cavity trailing edge
Model (thick SL) Model (thick SL)
Figure 19: Store trajectory prediction by the thick shear layer model when: (a) the store is droppedfrom near the leading edge(X ′
c = 0.4), and (b) the store is dropped from near the trailing edge(X ′
c = 0.76), in the no control (NC), microjet-control on (MC) and high-microjet-pressure-control(HC) cases. Also
(t, Yc
)= 0 correspond to the instance when the store just exits the cavity, and
increasing−Yc corresponds to the store moving away from the bay. The cross-flow Mach number is2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 and store lengthis half of cavity length. The initial conditions for the model are given in Table 4.
5 Optimization of Control Input
Having developed, validated and analyzed models for predicting the store trajectory inside and out-
side the cavity, the next step is to find the optimal control input that would ensure a successful store
drop. A suitable control parameter that can be optimized is the microjet momentum ratioCµ as
defined in Eqn. (27). The optimal input is defined as the minimum value ofCµ which satisfies the
user-specified microjet mass flow constraint and which guarantees safe release under a variety of
drop conditions. For this purpose, the drop conditions from Table 2 were taken as the baseline and
a combination of inside-cavity and outside-cavity models were used to determine whether the store
exits the cavity successfully for different values of the control input, with the drops being performed
from the cavity middle. The procedure was repeated for a different Mach number of external flow
33
Item Inside cavity (Point 0) While exiting cavity (Point 1)No control (NC): Store dropped near cavity leading edge
Yc 1.30 -1.63α -44.27 23.64Vc -0.05 -0.17ω 3.15 3.56
Microjet-control on (MC): Store dropped near cavity leading edgeYc 1.30 -1.63α -43.27 22.33Vc -0.05 -0.18ω 3.15 3.57
High-microjet-pressure-control (HC): Store dropped near cavity leading edgeYc 1.30 -1.63α -43.27 21.01Vc -0.05 -0.19ω 3.13 3.61
No control (NC): Store dropped near cavity trailing edgeYc 1.30 -1.63α -43.47 23.64Vc -0.05 -0.17ω 3.13 3.56
Table 4: Initial conditions for the store trajectory analysis used in Figure 19. The cross-flow Machnumber is 2.46, Reynolds number based on store length is 4.9 million, cavity aspect ratio is 5 andstore length is half of cavity length.
that was also tested under the HIFEX Program. The result is shown in Figure 20.
It is clear that a general trend of successful drops atCµ < 0.09 and failed drops atCµ > 0.09
are predicted for both the Mach number cases. The successful drop at Mach 2.0 forCµ = 0.08 is
corroborated by a series of store drop experiments performed under the HIFEX Program [1, 5]. It
can also be observed that for Mach 2.0, a much smaller value ofCµ than that used experimentally
suffices for ensuring a successful drop whereas for Mach 2.46, failure can occur for smallCµ and the
optimalCµ lies close to where the successful drop experiments were conducted. Once the optimalCµ
is determined, a suitable microjet-based actuator corresponding to this value ofCµ can be designed,
which may be based on sonic or converging-diverging nozzles, the latter having the advantage of
lower mass flux for the same momentum flux.
34
0.04 0.06 0.08 0.1 0.12
Su
cc
essfu
l d
rop
Un
success
ful
dro
p
2.46, 11.6 psiM q∞ ∞= =
2, 13.9 psiM q∞ ∞= =
0.04 0.06 0.08 0.1 0.12
Cµ
= =
= =
Observed in HIFEX Program
Observed in HIFEX Program
Observed in HIFEX Program
Figure 20: Successful/Unsuccessful store drop from cavity middle versus microjet momentum ratioCµ. The model used for prediction of the trajectory that lies outside the cavity is the near-cavity one.The cross-flow Mach number is 2.46, Reynolds number based on store length is 4.9 million, cavityaspect ratio is 5 and store length is half of cavity length. The drop conditions for this simulation are:Y0 = 1.30, α0 = 0.01, V0 = −0.18, ω0 = 0.62 andY1 = −1.63.
6 Summary
Recent results obtained in [1, 4, 5] have demonstrated that successful store separation from a cavity
under supersonic flow can be achieved by the introduction of an active control input. This input
is applied using a tandem array of microjet-based-transverse-fluid-injection system placed near the
leading edge of the cavity. The microjet-based actuator reduces the amplitude of tones inside the
cavity, modifies the shear layer structures at the mouth, and introduces shock waves outside the
cavity. The flow field outside the cavity is modified through the associated shock waves and the
change in shear layer structures, thereby affecting the store forces and moments.
This paper focuses on the development of a low-order model to predict the store trajectory under
given drop conditions and control input. The model was used to obtain the optimal control strategy
to ensure a successful store departure using a given setup. The model has three components – one
for when the store is inside the cavity, one for when the store passes through the shear layer, and the
last one for when the store is completely outside the cavity. The low-order model was developed
based on the slender store geometry, thin shear layer at the cavity mouth, high Reynolds number
external cross-flow, plane shock waves associated with the microjets, no-flow condition inside the
cavity and ignorance of cavity acoustic field.
35
The model was validated using a series of store drop experiments performed under the HIFEX
Program [1, 4, 5] at Mach 2.0 and 2.46 using a generic sub-scale weapons bay, with the store being
dropped from the middle of the cavity. The trajectory information was obtained from high speed
video. Three observations were recorded: (I) the store drop was unsuccessful for microjet-off case,
(II) the store dropped safely when microjets were switched on, and (III) the store returned back to
the cavity when the microjet pressures were increased. These features were seen in the model as
well.
The low-order model developed in this paper provides the following explanation for the experi-
mental observations. The relevant points to be kept in mind are that: (a)F outside the cavity depends
linearly onα such that the store exits successfully if∣∣∣ ωVc
∣∣∣ is small at the bay exit, and (b)M , when
the store passes through the shear layer, depends linearly onα which changes sign depending on the
external flow turning angleθ. The reason for effect (a) is that smallω results in lowα and therefore
F does not grow high enough to overcome gravity and the store falls out of the cavity safely. Be-
cause of high speed and no turning of the external flow in the no-control case, the main effect felt
by the store is (a). Thus, if the store is dropped with a large initialω, it leaves the cavity with a large
ω in comparison toVc. This violates (a) and results in an unsuccessful drop, confirming observation
(I). On the other hand, if the drop condition is such thatω is low in comparison toVc when the store
exits the cavity, the store will exhibit a safe drop.
With microjets on, the external flow turns away from the cavity. If the turning angleθ is less
than a certain value,M increases in the negative direction. This results in decrease inω as the
store crosses the shear layer and satisfies effect (a). Thus, the store exhibits a safe drop,confirming
observation (II). Higher microjet pressures cause a small increase inθ. If the flow turning angle
exceeds the limiting value,M and consequentlyω increase in the positive direction, when the store
crosses the shear layer. This violates (a), leading to an unsuccessful drop and confirming observation
(III).
The model was extended in this paper to include the effect of the wavy shear layer structures
on the store trajectory, therefore indirectly introducing the effect of the cavity acoustic field. The
36
extended model was then used to predict the store drop trajectory when it is released from near the
leading edge, the middle or the trailing edge of the bay. It was seen from both the extended model
and the HIFEX experiments that when the store is released near the bay leading edge, the store drops
unsafely in the no-control situation while it exhibits safe departure when microjets are switched on
and even when microjet pressure is increased. When the store is dropped close to the trailing edge,
it exhibits a safe departure and does not need any control.
Finally, the optimal control input to ensure clean store departure for a host of control inputs was
found, with the drops being performed from the cavity middle. Expressed in terms of the microjet
momentum ratioCµ, the optimal control input was observed to match that used experimentally for
certain flight conditions, and was predicted to be even smaller at other conditions.
7 Acknowledgement
This research was supported by a DARPA grant (Technical monitor: Dr. Steven Walker) through a
subcontract from The Boeing Company (Technical monitor: Dr. William Bower), and their coop-
eration is highly appreciated. The authors would like to acknowledge A. Ghoniem, J. Paduano, K.
Willcox and J. Peraire of MIT for their invaluable comments and suggestions throughout the course
of the store model development. The authors also gratefully acknowledge Val Kibens and Norm
Malmuth for their comments and suggestions which significantly improved the paper.
References
[1] Bower, W. W., Kibens, V., Cary, A. W., Alvi, F., Raman, G., Annaswamy, A., and Malmuth, N.,
“High-Frequency Excitation Active Flow Control for High-Speed Weapon Release (HIFEX),”
AIAA-2004-2513, presented at 2nd AIAA Flow Control Conference, June 2004.
37
[2] Shalaev, V. I., Fedorov, A. V., and Malmuth, N. D., “Dynamics of Slender Bodies Separating
from Rectangular Cavities,”AIAA Journal, Vol. 40, No. 3, 2002.
[3] Shalaev, V., Malmuth, N., and Fedorov, A., “Analytical Modeling of Transonic Store Separa-
tion from a Cavity,”AIAA Paper No. 0004, 2003.
[4] Kibens, V., Alvi, F., Annaswamy, A., Raman, G., and Bower, W. W., “High-Frequency Exci-
tation (HIFEX) Active Flow Control for Supersonic Weapon Release,”presented at 1st AIAA
Flow Control Conference, June, 2002.
[5] Alvi, F., Annaswamy, A., Bower, W., Cain, A., Kibens, V., Krothapalli, A., Lourenco, L.,
and Raman, G., “Active Control of Store Trajectory in a Supersonic Cavity,”Presented by A.
Annaswamy, Reactive Gas Dynamics Laboratory Seminar Series, Massachusetts Institute of
Technology, March 2004.
[6] Sahoo, D., Annaswamy, A. M., Zhuang, N., and Alvi, F. S., “Control of Cavity Tones in Super-
sonic Flow,”AIAA 2005-0793, Presented at the 43rd AIAA Aerospace Meeting and Exihibit,
Reno, Nevada, 10-13 January, 2005.
[7] Malmuth, N. and Shalaev, V., “Theoretical Modeling Of Interaction of Mulitple Slender Bodies
In Supersonic Flows,”AIAA Paper No. 1127, 2004.
[8] Davids, S. and Cenko, A., “Grid Based Approach to Store Separation,”AIAA Paper No. 2418,
2001.
[9] Wei, F.-S. J. and Gjestvang, J. A., “Store Separation Analysis of Penguin Missile From the
SH-2G Helicopter,”AIAA Paper No. 0992, 2001.
[10] Zhuang, N., Alvi, F. S., Alkislar, M. B., Shih, C., Sahoo, D., and Annaswamy, A. M., “Aeroa-
coustic Properties of Supersonic Cavity Flows and Their Control,”AIAA 2003-3101, 2003.
[11] Liepmann, H. W. and Roshko, A.,Elements of Gasdynamics, Dover Publications, Inc., New
York, 2001.
38
[12] Ashley, H. and Landahl, M.,Aerodynamics of Wings and Bodies, Dover Publications, Inc.,
New York, 1985.
[13] Milne-Thomson, L. M.,Theoretical Hydrodynamics, The McMillan Company, New York, fifth
edition, 1968.
Appendix 1: Control Input as a Function of Microjet Pressure
The control action consists of the introduction of microjets near the leading edge of the cavity. After
being introduced in the transverse direction, microjets turn and start mixing with the external cross-
flow, forming a virtual solid body. This obstruction to the cross-flow leads to formation of shock
waves near the cavity leading edge.
Empirically, the shock strength is governed by the microjet-to-external-flow momentum ratio [6].
In order to find the empirical relationship for a given microjet pressure profile, flow visualization
experiments were conducted on a rectangular cavity of aspect ratio 5.1 under Mach 2.0 flow in the
Florida State University (FSU) wind tunnel facility [10]. This setup used 12 microjets of diameter
400µm at the leading edge. To match the 12 microjet FSU setup with the HIFEX test [1, 5], the
momentum ratio parameterCµ defined in Eqn. (27) is used. The numerator in Eqn. (27) is the mean
momentum flux of all the active microjets while the denominator is the mean momentum flux of the
external flow through the cross-section area of the virtual solid body formed by the microjets. The
thickness of the body is same as the penetration height of the microjets into the external flow and the
latter is of the same order as the boundary layer thickness at the point of injection of microjets into
the cross-flow. Also the factor14
is to take into account the blockage of several microjet holes in the
HIFEX setup owing to faulty construction. Additionally, in Eqn. (27),δBL ∼ 0.015Lo for the FSU
cavity setup while for the HIFEX setup,δBL ∼ 0.017Lo, whereLo is the cavity length as shown in
Figure 22 (in Appendix 5).
The plane shock geometry resulting from introduction of microjets was studied for different
39
momentum ratios using the FSU setup. The results are adapted from Zhuang, et. al. [10] and shown
in Table 5. From this table, the shock geometry corresponding to an arbitraryCµ can be determined
Cµ 0.119 0.306 0.573Shock-to-free-stream 37 38 42
inclination angleAngle θ (Figure 7) 8 9 12
Table 5: Plane shock geometry for different microjet momentum ratios (Cµ) corresponding to theFSU setup. External flow conditions:M = 2.0 and Re= 3 million (based on cavity length). Cavitydimension:L/D = 5.1.
by interpolation. Using the shock geometry and plane shock wave theory [11], the external flow
properties (density, speed and flow orientation) under the cavity can be determined.
Appendix 2: Expression for Forces and Moments on Store When
Inside Cavity
When the store is inside the cavity,F andM expressions are given by Eqn. (14). Following the non-
dimensional scheme of Appendix 5, we give below the expressions forg0, g1, g4, g9 andh0, h1, h4, h9
that are present in Eqn. (14). The expressions are given to accuracyO(
1256
).
g0 = −π∫ xe
xo
a (x)2(2q4 + 2q6 + 1 − 2q2
)dx,
g1 = π∫ xe
xo
a (x)2 x(2q4 + 2q6 + 1 − 2q2
)dx,
g2 = 4 π∫ xe
xo
a (x) x2(1 + q2
) (4q8 + 8q6 + 3q2 − 1
)q3 dx,
g4 =∫ xe
xo
(3
16a (x)4 +
1
2a (x)2 H2 − H4
)πa (x)4 −
4q3(3q2 − 1
) (1 + q2
)πa (x) dx,
g9 =∫ xe
xo
(− 3
16a (x)4 − 1
2a (x)2 H2 + H4
)πxa (x)4 +
32q9(2q2 + 1
) (3q2 − 2
) (1 + q2
)πxa (x) dx,
h0 = −π∫ xe
xo
a (x)2 x(2q4 + 2q6 + 1 − 2q2
)dx,
h1 = π∫ xe
xo
x2a (x)2(2q4 + 2q6 + 1 − 2q2
)dx,
40
h2 = 4π∫ xe
xo
x3a (x)(1 + q2
) (4q8 + 8q6 + 3q2 − 1
)q3 dx,
h4 =∫ xe
xo
(3
16a (x)4 +
1
2a (x)2 H2 − H4
)πxa (x)4 −
4xq3(2q2 − 1
) (1 + q2
) (12q8 + 4q6 − 2q4 − q2 + 1
)πa (x) dx,
h9 =∫ xe
xo
(− 3
16a (x)4 − 1
2a (x)2 H2 + H4
)πx2a (x)4 +
32q9(2q2 + 1
) (3q2 − 2
) (1 + q2
)πx2a (x) dx,
q =a
2H. (29)
Appendix 3: Expression for Forces and Moments For Store Por-
tion Partially Immersed in External Flow
When the store is passing through the shear layer, the contribution of the store portion inside the
cavity to F andM is given in Appendix 2, while that of the portion outside the cavity is given in
Appendix 4 (Eqn. (33)). The contribution of the store portion partially immersed in external flow
is given by Eqn. (21) withg0, · · · , g11, h0, · · · , h11 given to accuracyO(
1256
)below. The notation
follows the non-dimensional scheme of Appendix 5.
g0 = −∫ x2
x1
a (x)2 [Φ1 (n) + Φp (m)] dx,
g1 =∫ x2
x1
a (x)2 x [Φ1 (n) + Φp (m)] dx,
g2 =∫ x2
x1
x2 [a (x) P1 (x, t) − Pp (x, t)] +x2a (x)2
[d
dmΦp (m) − d
dnΦ1 (n)
]π√
a (x)2 − H2dx,
g3 =∫ x2
x1
a (x) P1 (x, t) dx,
g4 =∫ x2
x1
a (x) P1 (x, t) − Pp (x, t) +a (x)2
[d
dmΦp (m) − d
dnΦ1 (n)
]π√
a (x)2 − H2dx,
g5 =∫ x2
x1
a (x)da
dx(x) x
a (x) d
dnΦ2 (n)
π√
a (x)2 − H2− P12 (x, t)
+ Φ1 (n) a (x)2 dx +
a (x2)2 Φ1 (n2) x2 − a (x1)
2 Φ1 (n1) x1,
41
g6 = −∫ x2
x1
a (x)da
dx(x) P12 (x, t) dx + a (x2)
2 Φ1 (n2) − a (x1)2 Φ1 (n1) ,
g7 =∫ x2
x1
a (x)da
dx
P12 (x, t) − a (x) d
dnΦ2 (n)
π√
a (x)2 − H2
dx + a (x1)
2 Φ1 (n1) − a (x2)2 Φ1 (n2) ,
g8 =∫ x2
x1
a (x) x
2P1 (x, t) − a (x) d
dnΦ1 (n)
π√
a (x)2 − H2
dx,
g9 = 2∫ x2
x1
x [Pp (x, t) − a (x) P1 (x, t)] +a (x)2 x
[ddn
Φ1 (n) − ddm
Φp (m)]
π√
a (x)2 − H2dx,
g10 =∫ x2
x1
a (x)
a (x) d
dnΦ1 (n)
π√
a (x)2 − H2− 2P1 (x, t)
dx,
g11 = a (x1)2 da
dx(x1) Φ2 (n1) −
∫ x2
x1
a (x)da
dx
2
P2 (x, t) dx − a (x2)2 da
dx(x2) Φ2 (n2) ,
h0 = −∫ x2
x1
a (x)2 x [Φ1 (n) + Φp (m)] dx,
h1 =∫ x2
x1
a (x)2 x2 [Φ1 (n) + Φp (m)] dx,
h2 =∫ x2
x1
x3 [a (x) P1 (x, t) − Pp (x, t)] +x3a (x)2
[d
dmΦp (m) − d
dnΦ1 (n)
]π√
a (x)2 − H2dx,
h3 =∫ x2
x1
xa (x) P1 (x, t) dx,
h4 =∫ x2
x1
x [a (x) P1 (x, t) − Pp (x, t)] +a (x)2 x
[d
dmΦp (m) − d
dnΦ1 (n)
]π√
a (x)2 − H2dx,
h5 =∫ x2
x1
x2a (x)da
dx(x)
a (x) d
dnΦ2 (n)
π√
a (x)2 − H2− P12 (x, t)
dx
+x22a (x2)
2 Φ1 (n2) − x12a (x1)
2 Φ1 (n1) ,
h6 = −a (x1)2 Φ1 (n1) x1 + a (x2)
2 Φ1 (n2) x2 −∫ x2
x1
Φ1 (n) a (x)2 − xa (x)da
dx(x) P12 (x, t) dx,
h7 =∫ x2
x1
a (x)
[a (x) Φ1 (n) + x
da
dx(x) P12 (x, t)
]− a (x)2 x da
dx(x) d
dnΦ2 (n)
π√
a (x)2 − H2dx +
a (x1)2 Φ1 (n1) x1 − a (x2)
2 Φ1 (n2) x2,
h8 =∫ x2
x1
2x2a (x) P1 (x, t) − a (x)2 x2 ddn
Φ1 (n)
π√
a (x)2 − H2dx,
h9 = 2∫ x2
x1
x2 [Pp (x, t) − a (x) P1 (x, t)] +a (x)2 x2
[ddn
Φ1 (n) − ddm
Φp (m)]
π√
a (x)2 − H2dx,
42
h10 =∫ x2
x1
−2xa (x) P1 (x, t) +a (x)2 x d
dnΦ1 (n)
π√
a (x)2 − H2dx,
h11 =∫ x2
x1
−a (x)
[da
dx(x)
]2
P2 (x, t) x + a (x)2 da
dx(x) Φ2 (n) dx +
x1a (x1)2 da
dx(x1) Φ2 (n1) − x2a (x2)
2 da
dx(x2) Φ2 (n2) ,
P1 (x, t) = −2.2n5 + 8.9n4 − 7.8n3 − 2.8n2 + 3.1n − 0.67,
P2 (x, t) = 46543.4n3 − 11980.9n2 + 758.1n +π
2(n < 0.13) ,
= −7.8n3 + 8.8n2 + 0.91n − 0.35 (n ≥ 0.13) ,
P12 (x, t) = −10n5 + 38n4 − 36n3 + 5n2 − 0.66n + 0.19,
Pp (x, t) = sin (πm)
[1 − 1 + πm (1 + 2m2) cot (πm)
6m3
](n < 0.9) ,
= 501427.6n2 − 969470.9n + 468042.7 (n ≥ 0.9) ,
Φ1 (n) = −23n7 + 59n6 − 27n5 − 30n4 + 28n3 − 7.8n2 + 0.071n + 2,
dΦ1 (n)
dn= −161n6 + 354n5 − 135n4 − 120n3 + 84n2 − 15.6n + 0.071,
Φ2 (n) = 5.8n5 − 28n4 + 34n3 − 9.6n2 + 0.3n − 2.4,
dΦ2 (n)
dn= 29n4 − 112n3 + 102n2 − 19.2n + 0.3,
Φp (n) = 23n7 − 100n6 + 160n5 − 100n4 + 6.9n3 + 13n2 + 0.12n − 0.0013,
dΦp (n)
dn= 161n6 − 600n5 + 800n4 − 400n3 + 20.7n2 + 26n + 0.12. (30)
Herem,n are defined in Eqn. (18).
Appendix 4: Expression for Forces and Moments on Store When
Outside Cavity
When the store is completely outside the cavity, there are four models developed in this paper to
explain the forces and moments on the store (the no-cavity model, the zero-depth-cavity model,
the near-cavity model, thick-shear-layer model). The resultingF andM expressions are given by
43
Eqn. (21) withg0, · · · , g11, h0, · · · , h11 given to accuracyO(
1256
)below. The notation follows the
non-dimensional scheme of Appendix 5.
No-cavity model:
g0 = −G0, g1 = G1, g5 = G0 + xea (xe)2 , g6 = a (xe)
2 , g7 = −a (xe)2 ,
h0 = −G1, h2 = G2, h5 = x2ea (xe)
2 , h6 = −G0 + xea (xe)2 , h7 = G0 + xea (xe)
2 ,
G0 =∫ xe
xo
a2 (x) dx,G1 =∫ xe
xo
a2 (x) x dx,G2 =∫ xe
xo
a2 (x) x2 dx. (31)
Zero-depth-cavity model:
g0 = −∫ xe
xo
a2(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
g1 =∫ xe
xo
a2x(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
g2 =∫ xe
xo
4ax2(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
g3 =∫ xe
xo
4a(15q8 + 8q6 + 1 + 2q2 + 6q4
)q3 dx,
g4 =∫ xe
xo
4a(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
g5 =∫ xe
xo
(−2q2
) (35q4 + 116q6 + 4 + 12q2
)ada
dxx + a2
(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx +(
2q (xe)4 + 2q (xe)
2 + 1 + 6q (xe)6 + 16q (xe)
8)a (xe)
2 xe,
g6 =∫ xe
xo
(−2q2
) (15q4 + 46q6 + 6q2 + 2
)ada
dxdx
+(2q (xe)
4 + 2q (xe)2 + 1 + 6q (xe)
6 + 16q (xe)8)a (xe)
2 ,
g7 =∫ xe
xo
2q2(35q4 + 116q6 + 4 + 12q2
)ada
dxdx
+(−2q (xe)
2 − 6q (xe)6 − 1 − 16q (xe)
8 − 2q (xe)4)a (xe)
2 ,
g8 =∫ xe
xo
8ax(2 + 4q2 + 40q6 + 15q4 + 15q8
)q3 dx,
g9 = −∫ xe
xo
8ax(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
g10 = −∫ xe
xo
8a(2 + 4q2 + 40q6 + 15q4 + 15q8
)q3 dx,
g11 =∫ xe
xo
2q(1 + 14q6 + 2q2 + 5q4
)a
(da
dx
)2
dx +
44
(−10q (xe)
7 − 4q (xe)5 − 2q (xe) − 2q (xe)
3)a (xe)
2 da
dx
∣∣∣∣∣xe
,
h0 = −∫ xe
xo
a2x(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
h1 =∫ xe
xo
x2a2(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
h2 =∫ xe
xo
4x3a(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
h3 =∫ xe
xo
4ax(4q4 + q2 + 1
) (8q8 + 3q6 + q4 + q2 + 1
)q3 dx,
h4 =∫ xe
xo
4ax(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
h5 = −∫ xe
xo
2q2(35q4 + 116q6 + 4 + 12q2
)ada
dxx2 dx +(
2q (xe)4 + 2q (xe)
2 + 1 + 6q (xe)6 + 16q (xe)
8)a (xe)
2 x2e,
h6 =∫ xe
xo
(−2q2
) (15q4 + 46q6 + 6q2 + 2
)ada
dxx +
(−2q2 − 2q4 − 6q6 − 16q8 − 1
)a2 dx
+(2q (xe)
4 + 2q (xe)2 + 1 + 6q (xe)
6 + 16q (xe)8)a (xe)
2 xe,
h7 =∫ xe
xo
2q2(35q4 + 116q6 + 4 + 12q2
)ada
dxx + a2
(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx −(
2q (xe)4 + 2q (xe)
2 + 1 + 6q (xe)6 + 16q (xe)
8)a (xe)
2 xe,
h8 =∫ xe
xo
8ax2(q2 + 1
) (−12q8 + 27q6 + 13q4 + 2q2 + 2
)q3 dx,
h9 = −∫ xe
xo
8ax2(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
h10 = −∫ xe
xo
8ax(q2 + 1
) (−12q8 + 27q6 + 13q4 + 2q2 + 2
)q3 dx,
h11 =∫ xe
xo
2q(1 + 14q6 + 2q2 + 5q4
)a
(da
dx
)2
x + 2q(5q6 + 1 + q2 + 2q4
)a2 da
dxdx
−2xeq (xe)(1 + q (xe)
2 + 2q (xe)4 + 5q (xe)
6)a (xe)
2 da
dx
∣∣∣∣∣xe
. (32)
Near-cavity model:
g0 = −∫ xe
xo
a2(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
g1 =∫ xe
xo
a2x(2q4 + 6q6 + 1 + 2q2 + 16q8
)dx,
g2 =∫ xe
xo
4ax2(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
g3 =∫ xe
xo
4a(15q8 + 8q6 + 1 + 2q2 + 6q4
)q3 dx,
45
g4 =∫ xe
xo
4a(6q2 + 72q6 + 3 + 15q8 + 24q4
)q3 dx,
g5 =∫ xe
xo
(−4q2
) (−2 + 21q6 + 6q2
)ada
dxx + a2
(4q8 + 2q4 − 2q2 + 2q6 + 1
)dx +(
−2q (xe)2 + 4q (xe)
8 + 2q (xe)6 + 1 + 2q (xe)
4)a (xe)
2 xe,
g6 =∫ xe
xo
(−4q2
) (−1 + 3q2
)ada
dxdx
+(−2q (xe)
2 + 4q (xe)8 + 2q (xe)
6 + 1 + 2q (xe)4)a (xe)
2 ,
g7 =∫ xe
xo
4q2(−2 + 21q6 + 6q2
)ada
dxdx
+(2q (xe)
2 − 2q (xe)6 − 2q (xe)
4 − 1 − 4q (xe)8)a (xe)
2 ,
g8 =∫ xe
xo
8ax(3q4 − 2 + 4q2 + 9q6 − 17q8
)q3 dx,
g9 = −∫ xe
xo
8ax(−3 + 6q4 − 17q8 + 17q6 + 6q2
)q3 dx,
g10 = −∫ xe
xo
8a(3q4 − 2 + 4q2 + 9q6 − 17q8
)q3 dx,
g11 = −∫ xe
xo
2q(1 + 2q4 − 2q2
)a
(da
dx
)2
dx −
2q (xe)(−1 + 3q (xe)
6 + q (xe)2)a (xe)
2 da
dx
∣∣∣∣∣xe
,
h0 = −∫ xe
xo
a2x(4q8 + 2q4 − 2q2 + 2q6 + 1
)dx,
h1 =∫ xe
xo
x2a2(4q8 + 2q4 − 2q2 + 2q6 + 1
)dx,
h2 =∫ xe
xo
4x3a(−3 + 6q4 − 17q8 + 17q6 + 6q2
)q3 dx,
h3 =∫ xe
xo
4ax(2q2 − 1 − 17q8 + q6
)q3 dx,
h4 =∫ xe
xo
4ax(−3 + 6q4 − 17q8 + 17q6 + 6q2
)q3 dx,
h5 = −∫ xe
xo
4q2(−2 + 21q6 + 6q2
)ada
dxx2 dx +(
−2q (xe)2 + 4q (xe)
8 + 2q (xe)6 + 1 + 2q (xe)
4)a (xe)
2 x2e,
h6 =∫ xe
xo
(−4q2
) (−1 + 3q2
)ada
dxx +
(−1 + 2q2 − 2q4 − 2q6 − 4q8
)a2 dx
+(−2q (xe)
2 + 2q (xe)4 + 2q (xe)
6 + 4q (xe)8 + 1
)a (xe)
2 xe,
h7 =∫ xe
xo
4q2(−2 + 21q6 + 6q2
)ada
dxx + a2
(4q8 + 2q4 − 2q2 + 2q6 + 1
)dx −(
2q (xe)2 − 2q (xe)
6 − 2q (xe)4 − 1 − 4q (xe)
8)a (xe)
2 xe,
46
h8 =∫ xe
xo
8ax2(3q4 − 2 + 4q2 + 9q6 − 17q8
)q3 dx,
h9 = −∫ xe
xo
8ax2(−3 + 6q4 − 17q8 + 17q6 + 6q2
)q3 dx,
h10 = −∫ xe
xo
8ax(3q4 − 2 + 4q2 + 9q6 − 17q8
)q3 dx,
h11 =∫ xe
xo
(−2q)(1 + 2q4 − 2q2
)a
(da
dx
)2
x + 2q(q2 − 1 + 3q6
)a2 da
dxdx,
+2xeq (xe)(−q (xe)
2 − 3q (xe)6 + 1
)a (xe)
2 da
dx
∣∣∣∣∣xe
. (33)
Thick-shear-layer model: When the shear layer is thick, we cannot discount the effect of the
wavy shear layer structures on the force and moment on the store. In this case, again assuming
piecewise cosine components for the wavy structures as in Eqn. (9), we get the different parameters
of F andM expressions (given in Eqn. (14)) as follows:
g7 = −βδs
2δ
∑i
Wiλ2i [cos (λi {X∗
c + βY ∗c } + ζi)kc,i + sin (λi {X∗
c + βY ∗c } + ζi) ks,i],
h7 = −βδs
2δ
∑i
Wiλ2i [cos (λi {X∗
c + βY ∗c } + ζi)kcx,i + sin (λi {X∗
c + βY ∗c } + ζi) ksx,i],
g11 = −a (xe)2 δs
2δWjλj {sin (λj {X∗
c + βY ∗c } + ζj)cos (λjµxe) +
cos (λj {X∗c + βY ∗
c } + ζj) sin (λjµxe)} ,
h11 = −xea (xe)2 δs
2δWjλj {sin (λj {X∗
c + βY ∗c } + ζj)cos (λjµxe) +
cos (λj {X∗c + βY ∗
c } + ζj) sin (λjµxe)} . (34)
The rest of the parameters are zero. It is to be noted that this contribution of the wavy structures is
added to Eqn. (31) to get the complete expressions for normal force and moment in the thick shear
case. Also, the indexj in Eqn. (34) corresponds toxe and
kc,i =∫x
cos (λiµx) a (x)2 dx,
ks,i =∫x
sin (λiµx) a (x)2 dx,
47
kcx,i =∫x
cos (λiµx) a (x)2 x dx,
ksx,i =∫x
sin (λiµx) a (x)2 x dx,
kcxx,i =∫x
cos (λiµx) a (x)2 x2 dx,
ksxx,i =∫x
sin (λiµx) a (x)2 x2 dx. (35)
Here the integrations are carried over that portion of the store which corresponds to the piecewise
cosine component represented by the indexi of the wavy structure. This is illustrated in Figure 21.
U∞
( ) ( ) { }* * 'cos , Integeri i i ih X W X W iλ ζ= + + ∈
( )Wavy structure shape piecewise cosine component
corresponding to the store position :
λ ζ
Figure 21: Piecewise cosine wavy structure for the shear layer. The cosine component chosen herecorresponds to theX ′ coordinate of a point on the store surface, withX ′ = X ′
c + µx and thenon-dimensionalization scheme is same as Eqn. (3).
Appendix 5: Non-dimensional Scheme Used in Different Store
Drop Models
The complete set of non-dimensional parameters involved in computingF andM is outlined below:
X =X
lo, Y =
Y
δlo, Z =
Z
δlo, t =
U∞t
lo, x =
x
lo, y =
y
δlo, z =
z
δlo, a =
a
δlo,
Ho =Ho
δlo, Do =
Do
δlo, α =
α
δ,H =
H
δlo= Yc − αx, Vc =
Vc
δU∞, ω =
ωloδU∞
, (36)
whereYc, Vc andω are the vertical coordinate, vertical velocity and angular velocity of the store
c. g. respectively. The reference frame for integration is attached to the store axis and is illustrated
48