AIAA 2002-0119 40th AIAA Aerospace Sciences ...

AIAA 2002-0119 INJECTION PARAMETERS FOR AN EFFECTIVE PASSIVE CONTROL OF CAVITY FLOW INSTABILITY Aldo Rona and Edward J. Brooksbank Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK

40th AIAA Aerospace Sciences Meeting & Exhibit 14-17 January 2002 / Reno, NV For permission to copy or republish, contact the copyright owner named on the first page. For AIAA-held copyright, write to AIAA Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344

AIAA-2002-0119 INJECTION PARAMETERS FOR AN EFFECTIVE PASSIVE CONTROL OF CAVITY FLOW INSTABILITY Aldo Rona∗ and Edward J. Brooksbank† Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK ABSTRACT

INTRODUCTION A turbulent transonic air stream overflowing a rectangular enclosure develops a large amplitude flow instability. The unsteady flow in the enclosure, or cavity, is characterised by large amplitude pressure oscillations, an unsteady vorticity field, and flow recirculation in the enclosure (Fig. 1). This leads to a sustained aerodynamic loading, pressure drag, and noise. In an ‘open’ cavity, the flow instability is driven by the fluctuations of a shear layer that spans across the enclosure.1 The interaction of the shear layer with the rear bulkhead is part of a feed-back loop which selfsustains the instability. The aim of this work is to assess mass injection at selected positions around the cavity as a method to attenuate the oscillations by modifying the flow-geometry interaction. Cavity flows are an established interest of the academic community.2, 3 At certain flow conditions this simple geometry develops a complex and unsteady flow ∗ Lecturer, † Research

e-mail: [email protected]. Associate, e-mail: [email protected].

c 2002 by A. Rona. Published by the American Copyright Institute of Aeronautics and Astronautics, Inc., with permission.

4D

Leading shock U

Radiated noise

8

A Mach 1.5 turbulent cavity flow is unsteady. Large scale shear layer fluctuations drive a self-sustained flow instability, coupled with large amplitude aerodynamic pressure oscillations. The unsteadiness generates sustained cyclic aerodynamic loading on the cavity walls, drag and noise. Therefore an effective method to suppress the instability is sought. A time dependent numerical model has been developed to investigate the physics of the instability in the Mach 1.5 baseline cavity flow and the effects of mass injection at three different locations as a passive control method to steady the flow. Mass injection from below the cavity front edge in the downstream direction outperforms normal mass injection in the upstream boundary layer and blowing from the rear edge. This better configuration suppresses the instability at a sufficiently high injection flow rate, which can be reduced by using an oscillatory injection back pressure. Driving the injection pressure at an harmonic of the dominant baseline flow eigenfrequency f0 generates unwanted ‘beating’ in the wall pressure fluctuation. This beating is absent when a non-harmonic injection frequency, such as πf0 , is used.

Vortex shedding

y x

6D

D L=3D

6D

Fig. 1 Sketch of the supersonic flow past an open cavity.

pattern. At low speeds, this fluid dynamic instability results from the interaction of the pressure and vorticity fields inside the enclosure. At transonic speeds, large amplitude pressure waves, shocks, and expansions enrich the flow pattern. The complexity of this flow-geometry interaction has stimulated a sustained academic interest, focussed towards clarifying the essential characteristics of self-sustained cavity flow. The research is also driven by a practical interest, as cavity flows develop in aircraft landing wells, automotive components, and between train sections. In the transonic flow regime, aircraft store bays have been a very prominent application in recent years. In fact, current airframe designs make extensive use of internal stores for stealth and store protection during cruise. When the store bay is opened during flight, the stores may be subjected to pressure fluctuations in excess of 150dB re 20µP a and this level of structural excitation could undermine the integrity of a store.4 Safe store separation is also a concern, as light stores may impact the aircraft during store ejection, due to the unsteady flow. Numerical, analytical, and experimental models have been developed to investigate the essential physics of the cavity flow instability and to assess suppression methods. An overview of past work is given in Grace.5 Research work initiated at Cambridge6 and then continued at the University of Southampton7–9 highlighted the use of time-dependent two-dimensional numerical models to capture large-scale convecting instabilities which characterise the unsteady shear layer over the cavity opening. These instabilities are the energy containing eddies in the shear layer and were found to be the main driver of the self-sustained flow resonance. This numerical model was also useful to understand noise production mechanisms and to locate the main noise sources in the flow.10 The research

1 American Institute of Aeronautics and Astronautics

b8 (extrapolated)

Leading shock U

Radiated noise

8

Mass injection

Vortex shedding

y

(a)

b2

0.5D

Mass injection

x

1 b3 2

Downstream bulkhead

6D

Upstream edge

Fig. 3

Upstream bulkhead

Mass injection

(c) Fig. 2 Cavity with mass injection: (a) above the upstream edge, (b) below the upstream edge, (c) below the downstream edge.

was then extended to evaluate the effects of modifying the baseline geometry with spoilers, ramps, and mass injection at the leading edge, to attenuate the instability.10–12 The numerical work at the University of Southampton predicted appreciable levels of reduction in the wall pressure fluctuations when the upstream and downstream baseline geometry was appropriately modified. Mass injection at the cavity leading edge was also a promising technique; mass injection through the surface upstream of the cavity was shown to reduce the mean pressure drag coefficient and reduce pressure fluctuations. This was confirmed by independent parallel work in the US.13 Recent efforts with mass injection techniques concentrated on actuators located at the cavity leading edge, to modify the separating boundary layer characteristics and change its receptivity.4, 14, 15 This work concerns the application of mass injection through perforated walls at three different locations around the cavity, as shown in Figs. 2(a-c): at the upper surface upstream of the cavity, through the upstream bulkhead tangential to the free stream, and through the downstream bulkhead, tangential to the free stream. In the current work, geometry modifications are replaced by mass injection, which, in an aircraft application, can be turned off when the store bay doors are closed.

A rectangular enclosure is tested at transonic flow conditions, as shown diagrammatically in Fig. 3. The cavity length to depth ratio is 3 and the inlet flow Mach number is 1.5. The geometry and inflow parameters are designed to match the experimental conditions of a selected test in Zhang.6 At these conditions, the flow develops large amplitude fluctuations and the fluctuating shear layer re-attaches on the downstream edge,

b6 6D

Baseline cavity.

giving an ‘open’ cavity regime. A turbulent boundary layer develops above the upstream cavity edge. All dimensions are normalised by the 15mm cavity depth D. At the computational domain inlet boundary b1 the boundary layer thickness δ99 is 0.333D. Above the boundary layer, the uniform free stream speed U∞ , Mach number M∞ , density ρ∞ , static pressure p∞ , and static temperature T∞ are 425.2m/s, 1.5, 0.9373kg/m3, 53.801kN/m2, and 200K respectively. All cavity flow results are normalised by the above free stream values and D/U∞ normalises time.

NUMERICAL METHOD The numerical method that is being used is an extension of the scheme by Rona & Dieudonn´e,16 where a laminar cavity model at a similar flow regime was presented. The extended method is reported in Rona & Bennett.17 The short-time averaged Navier-Stokes equations18 are the flow governing equations. Turbulence closure is obtained by the use of the k − ω two-equation model of Wilcox19 with the addition of a cross-diffusion term by Kok.20 The laws for the conservation of mass, momentum, total energy, and the k − ω two-equation turbulence closure can be expressed in the following compact vector form: ∂W + ∇ · (F i + F t ) + S = 0 ∂t

(1)

where W is the conservative variables vector, F i and F t are respectively the inviscid and turbulent flux vectors, and S is a source term for the turbulence model. These are: W

INFLOW CONDITIONS

D

b4 3 L=3D

4D

Pressure tap 1: top of upstream bulkhead Pressure tap 2: 0.333D from upstream bulkhead Pressure tap 3: 2.333D from upstream bulkhead Pressure tap 4: top of downstream bulkhead

(b)

0.5D

4 b5

b7 (extrapolated)

Downstream edge b1 (inflow)

0.5D

Fi Ft

= = =

T

(ρ, ρu, ρes , ρk, ρω)

(2) T

(ρu, ρuu + pI, ρuhs , ρuk, ρuω) (0, − (τ + t) ,

(3)

− (q + q t ) − (τ + t) · u − (µl + σ ∗ µt ) ∇k, − (µl + σ ∗ µt ) ∇k, − (µl + σµt ) ∇ω)T S

=

(4)

(0, 0, 0, β ∗ ρkω − t : ∇u, β ∗∗ ρω 2 − α (ω/k) t : ∇u −σd (ρ/ω) max [∇k · ∇ω, 0])


T

(5)

µt

= ρk/ω

(6)

τ

= µl (∇u + u∇ − 2/3I∇ · u)

(7)

t q

= µt (∇u + u∇ − 2/3I∇ · u) − 2/3Iρk (8) = (µl /P r) Cp ∇T (9)

qt hs

= (µt /P rt ) Cp ∇T = es + p/ρ

(10) (11)

es

= Cv T + u · u/2 + k

(12)

where Cp and Cv are the specific heats at constant pressure and constant volume. P r and P rt are the Prandtl number and turbulent Prandtl number. The turbulence model closure coefficients 9 3 , β ∗∗ = 40 , σ = 21 , σ ∗ = 23 , σd = 12 α = 95 , β ∗ = 100 are as specified in Kok.20 The equation of state, p = ρRT , completes the governing equations. The flow governing equations in discrete form are integrated over the computational domain to obtain a finite volume approximation of the flow field. The integration method follows from previous work.16 The upwind cell centred method implemented by Mensink21 is adopted to estimate the inviscid fluxes at the cell interfaces. Second order accuracy in space is obtained through the use of the Monotone Upwind Scheme for Conservation Laws (MUSCL) interpolation technique, together with the minmod limiter. For the presented work, the approximate Riemann solver of Roe22, 23 is used. Flux integration is performed over the finite volume boundary by an explicit two-step Runge-Kutta method by Hu et al.24 Standard Runge-Kutta coefficients (1.0, 0.5) give a second order formal accuracy. The two-step low storage implementation detailed in Hu et al.24 is adopted. A constant time step ∆t = 0.00435D/U∞ is used to facilitate the prediction post-processing. This corresponds to a maximum Courant number of 0.4 on the computational grid that is used in the present work. The computational domain is shown in Fig. 3. The domain extends to x = −6D in the upstream direction, x = 9D downstream, and y = −1D and y = 4D in the normal direction. The domain is discretised by a rectangular regular mesh of 200 by 200 unit cells above the cavity and by 40 by 40 cells inside the en-

30 25

u = u / uτ

20

+

where ρ, u, p, k, and ω are the fluid density, flow velocity vector, pressure, short-time averaged kinetic energy, and specific kinetic energy dissipation rate. T I is the identity matrix and () the transpose operator. The laminar viscosity µl is estimated using Sutherland’s law. The eddy viscosity µt , viscous stress tensor τ , turbulent stress tensor t, heat flux vectors (q, q t ), the stagnation enthalpy hs and specific stagnation energy es are defined in the following auxiliary relationships:

15

Spalding’s Law + 3  (κu+)2- (κu y+ = u+ + e(-κB) e(κB) - 1 - κu+ -  )  2 3  

10 5 10

0

10

1

10

2

10

3

10

y = ρuτ y / µl +

4

10

5

Fig. 4 Boundary layer profile approaching the cavity at x = −1.0D.

closure. A fixed turbulent boundary layer is defined at the inlet boundary b1 which is located sufficiently upstream so as not to interfere with the separating shear layer motion above the cavity. No slip conditions are imposed at the solid walls and Spalding’s law of the wall is used to estimate the tangential flow speed at the first interior point, with the von Karman constant κ = 0.41 and B = 5.0. The exterior ghost cell tangential velocity ue is computed using Eqs. 13-15 to give the correct friction velocity uτ at the wall: h + i y˜ =

u ˜ = ue

=

e

− max 0.0,

y

1.0 − y˜ y˜ + κu+ ui (1 − 2.0˜ u)

−5 20

(13)

(14) (15)

where ui is the tangential velocity of the first interior point. The turbulent kinetic energy and specific dissipation rate at the wall are computed according to a compressible model by Wilcox.19 A turbulent boundary layer develops over the bulkhead upstream of the cavity enclosure. The numerical method models the essentially steady boundary layer flow through the log-layer. Figure 4 shows the predicted boundary layer profile 1.0D upstream of the cavity leading edge. In the absence of mass injection, the velocity profile follows the Van Driest law of the wall reasonably well. Injection is considered at three locations around the cavity: through the upstream bulkhead and normal to the free stream, Fig. 2(a), tangential to the flow in the downstream direction, Fig. 2(b), and through the downstream bulkhead, Fig. 2(c). In the baseline case, Fig. 3, the cavity walls are solid. In the modified geometry of Figs. 2(a-c), perforated cavity walls are used. Injection is performed through surfaces of constant void-ratio and a variable back pressure p0 is imposed below the surface. The wall normal velocity vw is then estimated according to the linear form of the Darcy pressure-velocity law, for which vw =

σ (p0 − pw ) ρ∞ U ∞


(16)

where σ is the geometric porosity.25 pw is the static pressure extrapolated from the conditions adjacent to the wall and p0 is the static pressure below the perforated wall. The back pressure p0 is determined at each computational time step from h i e cos (2πf t) p0 = p∞ K + K (17)

Equation 17 imposes a time-dependent back pressure that has a dominant time-mean component, determined by K, and an harmonic perturbation of ame about the mean. f is the frequency of the plitude K harmonic perturbation and t is the cavity flow computational time (sec.). In order to evaluate the performance of different cavity flow control configurations, the cost of controlling the flow is to be included in the assessment. While comparing the cost of installing and operating flow suppression devices on an aircraft is beyond the scope of this fundamental study, some useful discussion can be presented, based on the injected mass flow rate. Specifically, the level of flow instability suppression attained by each flow control device is discussed with respect to the mass flow rate supplied to the perforated wall, which represents the cost of flow control. The space averaged mass flow rate through the perforated wall is estimated from Z 0.5D 1 ρvw dl (18) hρvw i = ρ∞ U ∞ D 0

where l ≡ x for Fig. 2(a) and l ≡ y for Figs. 2(bc). The time average and the Root Mean Square of hρvw i are obtained over several cavity characteristic times D/U∞ . In this preliminary discussion, the time average (hρvw i) and Root Mean Square (hρvw iRMS ) components are regarded as making equal contributions to the cost of flow control. Mass injection is known from previous work26 to affect the time averaged wall pressure distribution around the enclosure and hence the cavity drag. The form drag coefficient of the cavity can be estimated as Z 0 p (L, y) − p (0, y) Cd = dy (19) 1 2 −D 2 ρ∞ U ∞ where L is the cavity streamwise length, defined in Fig. 3. In this work, the effects of mass injection on the time mean drag coefficient Cd are monitored in order to ascertain any reduction in form drag with flow control.

RESULTS AND DISCUSSION The test cases in the current study are identified by a four-character extension of the form WXYZ, where W refers to the position of the perforated wall, X refers to the time mean back pressure ratio K, Y refers to the

0 W none X Y Z

0.0 0.0 0.0

1 upper wall 1.25 0.05 πf0

Table 1

2 upstream bulkhead 1.5 0.1 4.0f0

3 downstream bulkhead 2.0 2πf0

4 2.5 -

Summary of test cases.

e of the harmonic back presperturbation amplitude K sure imposed beneath the perforations, and Z refers to the harmonic back pressure excitation frequency f . Cav0000 is the baseline case with no perforated wall. The other values of WXYZ reported in this paper are detailed in Table 1. Baseline case

Cav0000: The flow in the baseline cavity features a shear layer spanning over the entire enclosure and reattaching at the downstream bulkhead. The shear layer is characterised by a coupled motion of shear layer flapping in the transverse direction, due to the shear layer instability, and of vortex convection in the streamwise direction, with vortices impinging on the rear bulkhead. The interaction of the shear layer with the cavity downstream edge establishes a feed-back loop resulting in a self-sustained stationary flow oscillation in the cavity (Fig. 5), with the largest amplitude pressure oscillations occurring at the downstream edge (Fig. 6). Noise is also radiated away from the cavity, and can be seen as a series of pressure waves running along the leading edge shock to the far-field. These pressure waves originate from the downstream edge and the snapshot of Fig. 5(a) captures the departure of one of such upstream travelling pressure waves. The vortices within the shear layer convect downstream and impinge against the downstream edge. A portion of the mass flow convected by the vortices is injected into the cavity as a vortex approaches the downstream edge, as shown in Fig. 5(a). This mass injection phase is followed by mass ejection, shown in Fig. 5(c), when fluid leaves the enclosure, circumventing the downstream edge. The mass flow ejection is stretched along the trailing edge wall as it convects towards the outflow boundary b7 (Fig. 3). Mass injection with a steady back wall pressure

Initial flow unsteadiness: This section reports the predicted changes in the cavity flow instability when a steady back pressure is applied across the perforated surfaces shown in Figs. 2(a-c). Mass injection is applied after the cavity flow has reached a self-sustained instability without injection, so that the initial flow conditions for all test cases with mass injection are identical to the ones for the baseline flow displaying stationary flow resonance. In selecting these conditions, it was considered that any effective suppression


(a)

(a)

(b)

(b)

(c)

Fig. 7 Cav1300: Density contours of a Mach 1.5 cavity flow with mass injection above the upstream edge: (a) t = 0, (b) t = 0.5T . ∆ρ = 0.03.

Cav0000 Cav1300 Cav2300 Cav3300

(d)

Cd

CdRMS

hρvw i ρ∞ U∞ D∞

hρvw iRM S ρ∞ U∞ D∞

0.4221 0.4354 0.1035 0.18

1.807 1.6988 0.4932 1.5084

0.03605 0.02508 0.04455

0.00174 0.00044 0.00692

Table 2 Comparison of drag and mass flow rate for a Mach 1.5 cavity flow with and without mass injection with a steady back pressure. Fig. 5 Cav0000: Baseline cavity density contours over one time period T : (a) t = 0, (b) t = 0.25T , (c) t = 0.5T , (d) t = 0.75T . ∆ρ = 0.03.

0.5 2

pw / (ρ∞U∞)

0.4 0.3 0.2

tap 1, upstream edge tap 2, cavity floor, X = 0.333D tap 3, cavity floor, X = 2.333D tap 4, downstream edge

0.1 0

5206

5208

5210

tU∞ / D

5212

5214

Fig. 6 Cav0000: Baseline cavity surface pressure histories over two periods (2T ).

device ought to be able to recover from a saturated flow instability which establishes within about 20D/U∞ seconds from the cavity being exposed to transonic flow. This corresponds, in an aircraft, to less than one second from the store bay doors being opened. An initial set of tests was performed to determine which of the three positions for mass injection, Figs. 2(a-c), would have the potential to deliver the largest suppression with the lowest cost of flow control. Three tests, Cav1300, Cav2300 and Cav3300, were performed with a constant perforated wall back pressure K = 2.0 without any harmonic perturbation,

e = 0. K

Cav1300: With mass injection above the upstream edge, the cavity flow does retain a dominant selfsustained shear layer driven instability, similar to the baseline flow. Figs. 7(a-b) show two snapshots of the density contours at different times during the dominant mode period T . Injecting mass above the upstream edge results in a thickening of the shear layer. This reduces the gradient of mass flux ∂ρuy /∂y and of momentum flux ∂ρux uy /∂y across the separated flow over the cavity. A lower momentum flux feeds a Kelvin-Helmholtz instability in the shear layer and reduces the vorticity ∇ × ρu produced by the shear flow. The addition of the mass above the upstream edge has introduced a leading edge shock upstream of the perforated wall, due to the flow turning anticlockwise, away from the cavity edge and towards the far-field. This shock also contributes to thickening the boundary layer approaching the cavity by raising the sonic line away from the wall. This allows pressure waves to propagate upstream through the boundary layer, below the sonic line, up to the base of the shock. These disturbances perturb the separating boundary layer and affect the mass flow rate through the perforations. Specifically, the mass flow rate is reduced as a pressure wave travels over the injection point, giving an unsteady hρvw i and hρvw iRMS is 0.00174, as detailed in Table 2.


The presence of a strong shock over the upstream leading edge is undesirable. In fact, the shock increases the cavity wave drag and causes a clockwise pitching moment as the wall pressure increases along the cavity perimeter, downstream of the shock. In the far-field, the shock wave would be experienced as noise. The intensity of the unsteady pressure waves generated at the downstream edge of the cavity has been reduced slightly. This is indicated by the reduced density contour packing across the upstream travelling pressure wave which is shown approaching the leading edge in Fig. 7(b), compared to the same flow feature in Fig. 5(c). The shear layer is still in vortex shedding mode, impinging on the downstream bulkhead, and still appears to be aligned with the streamwise direction. A more detailed analysis of this test case is given by Zhang et al.26 Cav3300: Locating a perforated wall at the back of the cavity aims to provide a cushioning mechanism for the impinging shear layer and also to lift the shear layer middle line above the downstream edge. Model flow predictions with this flow control configuration are shown in Figs. 8(a-b). The cavity flow is still characterised by large scale oscillations with alternating phases of mass ejection (Fig. 8(a)) and injection (Fig. 8(b)) at the downstream edge. The radiation of pressure waves from the cavity has been reduced in intensity with respect to the baseline flow, as can be seen by the reduced contour packing of the waves propagating from the rear edge. During the mass ejection phase of the cycle, there is an excessive amount of mass ejection from the cavity, compared to the baseline case of Fig. 5(c). During the mass injection phase, the shear layer impinges against the downstream wall, creating a compression wave that opposes mass injection through the perforations. Once this pressure wave leaves the downstream bulkhead, a low pressure region remains, thus more mass from the perforated wall can flow through the perforations. This mass increases the volume of fluid ejected from the cavity in Fig. 8(a). During mass ejection, the addition of fluid through the perforated wall appears not to perform any clearly identifiable active suppression of the cavity instability. This results in larger flow structures trailing behind the cavity in Figs. 8(a-b) compared to the baseline case of Figs. 5(a-d). Cav2300: Mass injection below the upstream edge has the effect of lifting the shear layer mean line, as shown in the density contour predictions of Figs. 9(ab). The positive streamwise velocity induced below the shear layer by the perforated wall has significantly reduced the vorticity in the separated boundary layer and the shear layer across the cavity is now in a flapping mode as opposed to a vortex shedding mode. The radiation of pressure waves from the cavity down-

(a)

(b)

Fig. 8 Cav3300: Density contours of a Mach 1.5 cavity flow with mass injection from the downstream edge: (a) t = 0, (b) t = 0.5T . ∆ρ = 0.03. (a)

(b)

Fig. 9 Cav2300: Density contours of a Mach 1.5 cavity flow with mass injection below the upstream edge: (a) t = 0, (b) t = 0.5T . ∆ρ = 0.03.

stream edge has been significantly reduced as indicated by sparse density contours above the enclosure. Quasisteady Mach waves stem from the upstream and downstream edges, which are set at approximately the Mach −1 to the free stream direction. The angle arcsin M∞ reduction in pressure wave amplitude would be experienced as a reduction in noise intensity in the far field. The predicted normalised wall pressure histories at selected locations around the cavity perimeter are shown in Fig. 10. Surface wall pressure fluctuations at the cavity walls have been significantly reduced in amplitude and are more sinusoidal than in the baseline flow (Fig. 6). The time averaged form drag generated by the cavity with and without a steady back pressure mass injection have been estimated from Eq. 19. The predicted form drag coefficient Cd for each test case is shown in Table 2 alongside with the mass flow rate representing the cost of controlling the flow. From Table 2, it can be seen that Cav2300 displays the


0.4

0.4

Cav0000 Cav1300 Cav2300 Cav3300

2

2

0.3 0.2

tap 1, upstream edge tap 2, cavity floor, X = 0.333D tap 3, cavity floor, X = 2.333D tap 4, downstream edge

0.1 0

pw / (ρ∞U∞)

0.42

pw / (ρ∞U∞)

0.5

4862

4864

4866

tU∞ / D

0.38 0.36 0.34 0.32 0

4868

Fig. 10 Cav2300: Surface pressure histories over two periods (2T ) with mass injection below the upstream edge.

lowest drag among these test cases and requires the smallest mass flow rate. This reduction in drag is brought about by a reduced difference between the time mean pressure on the upstream and downstream walls. An indication of this is given by the sampled time mean wall pressure distributions in Fig. 11 which indicate a lower difference in pw between the upstream (0, 0) and downstream (3D, 0) cavity edges with respect to the baseline case. In Fig. 11, pw is plotted versus the cavity perimetical co-ordinate s that runs anti-clockwise along the enclosure perimeter from (x = 0, y = 0), as shown in Fig. 12. In this co-ordinate system, (0, 0) → (s = 0) and (3D, 0) → (s = 5). The mean wall pressure distribution for Cav2300 on the cavity floor is flatter than the baseline, suggesting a reduction and/or a relocation of the standing vortex which sits close to the back wall in the baseline flow.12, 27 Mass injection below the upstream edge has lead to a reduced cavity flow unsteadiness by (i) lifting of the shear layer mean line above the enclosure and (ii) reducing the vorticity in the shear flow. At the selected inflow conditions, this appears to be the most promising position at which to locate the perforated wall in order to attenuate flow oscillations at a lower cost of mass injection and with a better cavity form drag. In the remainder of this paper, the effects varye and f have ing the mass injection parameters K, K been investigated for this perforated wall position. Cav2400: Imposing a steady back pressure ratio of K = 2.5 leads to a steady cavity flow, as shown by the density contours of Fig. 13. The mean line of the shear layer has rotated counter-clockwise, rising above the downstream edge and now runs from the upstream edge to a re-attachment point downstream of the rear edge. Unsteady pressure waves generated at the downstream edge have been suppressed, thereby reducing the noise radiated to the far field. While this mass injection configuration is successful in steadying the cavity flow, the cost of this in terms of the mass flow rate added through the perforations is

1

2

s/D

3

4

5

Fig. 11 Normalised mean wall pressure distribution for a Mach 1.5 cavity flow with and without mass injection with a steady back pressure. (0, 0)

(3D, 0)

s (0.333D, -D)

Fig. 12 walls.

(2.333D, -D)

Location of pressure tappings along cavity

Fig. 13 Cav2400: Density contours of a Mach 1.5 cavity flow with injection from the upstream wall, showing a steady flow. K = 2.5, ∆ρ = 0.03.

high. It is therefore of interest to consider the addition of an harmonic perturbation to the perforated wall back pressure p0 in order to lower the amount of steady mass flow required to suppress the cavity instability. Mass injection with a 5% amplitude harmonic back wall pressure

Perturbation frequency: The model baseline cavity flow has a dominant mode instability at a frequency of f0 = 5658Hz that compares favourably with the measured value of 5900Hz given in Zhang.7 A 5% amplitude harmonic back wall pressure is imposed behind a perforated wall located below the upstream edge, e = 0.05. The inas shown in Fig. 2, by setting K jection frequency is chosen as f = πf0 , in order to prevent the cavity ‘locking-in’ to the dominant mode e = 0.05 frequency f0 . Four test are performed with K and with a steady back pressure ratio in the range 1.25 ≤ K ≤ 2.5: These are Cav2111, Cav2211, Cav2311, and Cav2411. Figs. 14 and 15 show the predicted mean and Root Mean Square (RMS) pressure distributions around the cavity for these tests.


0.6 Cav2111 Cav2211 Cav2311 Cav2411

0.55 0.5

2

pw / (ρ∞U∞)

0.4

2

pw / (ρ∞U∞)

0.42

0.45

0.38 0.36

0.4

0.35

0.34

0.3

0.32

0.25 0

1

2

s/D

3

4

5

Fig. 14 Mean pressure distributions around the cavity: Cav2111, Cav2211, Cav2311, and Cav2411.

5300

tU∞ / D

5400

Fig. 17 Cav2211: Pressure history at the downe = 0.05, f = πf0 . stream edge. K = 1.5, K 0.6

Cav2111 Cav2211 Cav2311, Cav2411

0.55 0.5

2

pw / (ρ∞U∞)

0.08

2

prms / (ρ∞U∞)

0.1

0.45

0.06 0.04

0.4

0.35 0.3

0.02

0.25 0

0

1

2

s/D

3

4

5

Fig. 15 Rms pressure distributions around the cavity: Cav2111, Cav2211, Cav2311, and Cav2411. 0.6

0.5

2

pw / (ρ∞U∞)

0.55

0.45 0.4

0.35 0.3 0.25 5300

tU∞ / D

5400

Fig. 16 Cav2311: Pressure history at the downe = 0.05, f = πf0 . stream edge. K = 2.0, K

Cav2311 and Cav2411: Cav2411 models an essentially steady flow, similar to Cav2400 which has the same high injection wall back pressure ratio of K = 2.5. Cav2311 also predicts essentially a steady flow, although a significant time is required for the instability to be suppressed, as shown by the predicted wall pressure history of Fig. 16. It is noted that adding a 5% harmonic perturbation has reduced the value of the steady wall back pressure required to suppress the flow instability from a setting of K = 2.5 to one of K = 2.0. Cav2111 and Cav2211: The model flow with a reduced wall back pressure ratio of K = 1.5 is unsteady, as shown by the predicted wall pressure history Fig. 17. The amplitude of the pressure fluctuations

5300

tU∞ / D

5400

Fig. 18 Cav2111: Pressure readings at the downe = 0.05, f = πf0. stream edge. K = 1.25, K

at the cavity trailing edge initially decreases, as compared to the baseline case, reaching a minimum at about tU∞ /D = 5260. The wall pressure amplitude then increases and asymptotes to a constant value at tU∞ /D > 5300, when the flow becomes stationary. This effect is not present in the test case with the lowest mean back pressure ratio, K = 1.25, for which the wall pressure history is documented in Fig. 18. In fact, in Cav2111, the predicted downstream edge pressure amplitude reduces monotonically from the baseline case, preserving a stationary oscillatory flow instability similar to the baseline flow. Predictions of the time averaged drag coefficient and of the mean injected mass flow rate when flow control is applied with a 5% harmonic perturbation are presented in Table 3. The Cav2311 model predicts a drag coefficient lower than the baseline case and lower than the best case with a steady injection wall back pressure, Cav2300. The mass flow rate used to control the flow is also lower than in Cav2300. This confirms that adding an harmonic perturbation to the back pressure leads to a more effective suppression of the cavity instability at a lower cost of flow control.

Mass injection with a 10% amplitude harmonic back wall pressure

Three tests were performed with a back wall prese = 0.1: Cav2121, Cav2221, and sure ratio of K Cav2321. The purpose of the tests was to anal-




0.4207 0.2943 0.0908 0.0464

1.3166 1.3334 0.0 0.0

0.007375 0.0198 0.022018 0.02707

0.004697 0.0043 0.0 0.0

0.6 0.55

0.45

2

5300

tU∞ / D

5400

0.6

0.38

0.55

0.36

0.5

2

pw / (ρ∞U∞)

pw / (ρ∞U∞)

0.3 0.25

Fig. 21 Cav2321: Pressure readings at the downe = 0.1, f = πf0 . stream corner; K = 2.0, K

Cav2121 Cav2221 Cav2321

0.4

0.4

0.35

Table 3 Comparison of drag and mass flow rate for test cases Cav2111, Cav2211, Cav2311, and e = 0.05. Cav2411; K 0.42

0.5

2

Cd RMS

pw / (ρ∞U∞)

Cav2111 Cav2211 Cav2311 Cav2411

Cd

0.34

0.45

0.32 0

1

2

s/D

3

4

5

0.4

0.35 0.3

Fig. 19 Mean pressure distributions around the cavity: Cav2121, Cav2221, and Cav2321.

0.08

5300

tU∞ / D

5400


Cav2121 Cav2221 Cav2321

2

0.6

0.06

0.55 0.5

2

0.04

pw / (ρ∞U∞)

prms / (ρ∞U∞)

0.1

0.25

0.45

0.02 0

0

1

2

s/D

3

4

5

Fig. 20 RMS pressure distributions around the cavity: Cav2121, Cav2221, and Cav2321.

yse the effects of increasing the perturbation amplitude on the attenuation of the cavity flow instability. Figs. 19 and 20 show the predicted time mean and Root Mean Square pressure distributions around the cavity perimeter from these tests. The cavity flow instability is suppressed with a back pressure ratio of K = 2.0, as shown by the predicted pressure history at (3D, 0) of Fig. 21. The time traces are similar to Cav2311 in Fig. 16 but the instability is suppressed in a shorter time. Cav2221 and Cav2121 follow the same trends as the corresponding cases Cav2211 and Cav2111. The pressure history for Cav2221, Fig. 22, shows a pressure amplitude minimum shortly after the onset of mass injection, similar to the prediction for Cav2211. This minimum is not present in the model wall pressure results for Cav2121, in Fig. 23. From Table 4, the time averaged drag coefficients obtained with a 10% amplitude wall back pressure are higher than those predicted with a 5% amplitude back pres-

0.4

0.35 0.3 0.25 5300

tU∞ / D

5400


sure. However, the corresponding injected mass flow rates are lower than the previous test at 5% amplitude p0 . While the increased cavity form drag may be a concern for some applications, mass injection with an harmonic pressure amplitude of 10% seems to be able to steady the flow more rapidly and with a lower injection mass flow rate. This configuration is therefore more advantageous for flow control. In the remainder of this paper, the effects of varying the excitation e = 0.1. frequency f are considered for K Effects of the injection frequency f

A final series of tests were performed to analyse the effects of using different frequencies for the perforated wall back pressure. The purpose was to analyse


Cav2121 0.4279 1.4147 Cav2221 0.3209 1.258 Cav2321 0.0937 0.0



0.007176 0.017752 0.0211

0.005735 0.004295 0.0

Table 4 Comparison of drag and mass flow rate for test cases Cav2121, Cav2221, and Cav2321; e = 0.1. K

0.6 0.55 0.5

2

Cd RMS

pw / (ρ∞U∞)

Cd

0.45 0.4

0.35 0.3 0.25

0.6

5300

tU∞ / D

5400

Fig. 25 Cav2222: Pressure readings at the downe = 0.1, f = 4.0f0 . stream corner; K = 1.5, K

0.5

2

pw / (ρ∞U∞)

0.55

0.45 0.4

0.6

0.35

0.55 0.5

2

pw / (ρ∞U∞)

0.3 0.25

0.45

5300

tU∞ / D

5400

0.4

0.35

Fig. 24 Cav2122: Pressure readings at the downe = 0.1, f = 4.0f0 . stream corner; K = 1.25, K hρvw i ρ∞ U∞ D∞


0.3058 0.3037 0.4061 0.4022

1.3595 1.3627 1.2856 1.2784

0.0200 0.0199 0.0076 0.0076

0.0053 0.0053 0.0055 0.0054

Table 5 Comparison of drag and mass flow rate for test cases Cav2222 and Cav2122; f = 4.0f0 , and test cases Cav2223 and Cav2123; f = 2πf0 .

whether harmonic perturbations of the back pressure at frequencies other than f = πf0 would further attenuate the cavity flow instability and possibly drive the remaining unsteady flow cases, Cav2221 and Cav2121, to a steady state. The frequencies chosen were f = 4.0f0 , and f = 2πf0 . The former was chosen as an harmonic of the baseline flow main instability mode frequency, in order to force the cavity instability to a higher mode. The latter was chosen as a higher non-harmonic of the baseline flow frequency. Wall pressure histories for f = 4.0f0 are given in Figs. 24 and 25. Forcing the flow at this frequency has introduced ‘beating’ into the model flow wall pressure. The beating frequency is significantly lower than the baseline flow dominant instability mode and may be a concern on the structural integrity of aircraft and stores that are not designed for it. Predictions of wall pressure history for Cav2123 and Cav2223 are reported in Fig. 26 and Fig. 27. The results are similar to the corresponding test cases Cav2121 & Cav2122 and Cav2221 & Cav2222. It is noted that Cav2223 shows a longer transient after the flow is control is introduced, before settling into a stationary self-sustained oscillation.

5300

tU∞ / D

5400

Fig. 26 Cav2123: Pressure readings at the downe = 0.1, f = 2πf0 . stream corner; K = 1.25, K 0.6

0.55 0.5

2

Cd RMS

pw / (ρ∞U∞)

Cav2223 Cav2222 Cav2123 Cav2122

Cd

0.3 0.25

0.45 0.4

0.35 0.3 0.25 5300

tU∞ / D

5400

Fig. 27 Cav2223: Pressure readings at the downe = 0.1, f = 2πf0 . stream corner; K = 1.5, K

CONCLUDING REMARKS

A numerical simulation of the time dependent unsteady flow past a supersonic cavity has been performed. The numerical model includes mass injection at three locations around the cavity perimeter and aims to attenuate or suppress the shear layer driven instability in the flow. A parametric study has been conducted to determine the best mass injection configuration for passive flow control among the configurations tested. The parameters considered are the injection location, the steady and unsteady back pressure amplitude, and the injection frequency. Injecting mass below the upstream edge has been identified as the best location among the three po-


sitions considered, as the control flow modifies the boundary layer receptivity without increasing the cavity wave drag. This supports the present development path at the Wright Patterson Air Force Base and at the North Carolina State University where streamwise mass injection from the upstream edge was investigated in recent studies.4, 28 The suppression of the cavity shear layer instability with a steady injection wall back pressure is predicted only at a high injection mass flow rate. Introducing a small harmonic perturbation in the pressure applied upstream of the perforated wall decreases the steady pressure required to steady the flow. This leads to a significant reduction in the injected mass flow rate. Doubling the perturbation amplitude produces a faster decay of shear layer instability towards an essentially non-oscillatory flow. Forcing mass injection at a non-harmonic frequency of the baseline flow dominant mode frequency avoids ‘beating’ in the flow.

ACKNOWLEDGEMENTS The support of EPSRC grant GR/N23745 is acknowledged. This work was performed using the University of Leicester Mathematical Modelling Centre’s supercomputer which was purchased through the EPSRC strategic equipment initiative.

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