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

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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.

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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

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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

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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

in Figure 22.

X�

Y�

U∞α�

Store

Side view

0L�

0H�

0D�

Y�

Z�

x�

y�

z�

y�

r�

θH�

( )Radius a x� �

Back view

ex�

0x�

0x�

Figure 22: Reference frames (both fixed to the store and inertial) for computingF andM on store.

49