ϕ0 = porosityΩ = disturbance frequencyω = angular frequency
Subscripts
s = shockw = wall− = just behind the shock
I. Introduction
T RANSITION to turbulence in hypersonic flows is associatedwith amplification of the first and/or second Mack modes.
The first Mack mode is the high-speed counterpart of Tollmien–Schlichting waves, and so a viscous instability, with modes locatedclose to the boundary. The second Mack mode is an inviscidinstability. The second mode is believed to be responsible fortransition to turbulence on hypersonic slender bodies. These modescorrespond to the unstable perturbations occurring in the low-frequency band and the high-frequency band, respectively, of theslow (S) mode identified by Fedorov and Tumin [1]. Recentexperiments have shown that a porous coating greatly stabilizes thesecond Mack mode of the hypersonic boundary layer on sharpslender cones [2–4]. The effect of the porous coating is to reduce thegrowth rates of the secondmode to a level where they are comparablewith those of the first mode (occurring at lower frequencies). Inaddition, the first mode is observed to be slightly destabilized by thepresence of the porous coating. Thus, the firstmodemaynowbemoresignificant in the transition process.The destabilizing effect of these porous walls on the first Mack
mode has been confirmed by recent numerical simulations [5–10].This destabilization has been shown to be significant for a felt-metalporous coating [6].A previous work on the linear instability of the firstMackmodes in
the hypersonic flow over a cone includes the effect of curvature andalso the attached shock [11]. The effect of curvature is significantwith disturbances only existing over a finite range of wave numbersfor a fixed radius. Beyond a critical radius, dependent on theazimuthal wave number, all disturbances are damped. It wasdemonstrated that the effect of the shock, which allows incoming andoutgoingwaves, gives rise tomultiplemodes [11]. This study showedthat the effect of the shock should not be neglected; otherwise, thecorrect effect of curvaturewill not be realized. Themodes that exist inthe absence of the shock are now totally destroyed. The effect ofnonlinearity and curvature in hypersonic boundary-layer flow hasalso been considered [12]. The current study investigates the effectsof a porous layer on the first Mack mode for axisymmetric and
nonaxisymmetric disturbances. This is to confirm the experimentalobservations with a theory that includes the effect of curvature. Theeffect of curvature is of practical importance and has not beenpreviously investigated theoretically for a porous coating. Also, if thesecond mode (and higher modes) is greatly stabilized, then the firstmode may well lead to transition to turbulence, particularly if thedisturbances are triggered by wall roughness, because this mode islocated close to the wall and governed by viscous effects.We consider the effect of a porous wall on the linear instability of
hypersonic flow over a sharp slender cone. In this theoretical andasymptotic investigation for large Mach number and large Reynoldsnumber, the scales used are appropriate to the first Mack-modeinstability,which is governedby a triple-deck structure. The effects ofcurvature and the attached shock are taken into account. The currentstudy considers axisymmetric and nonaxisymmetric disturbances toinvestigate which modes will be more dominant. The effect of theporouswall changes the boundary condition on the normal velocity atthe interface.The porous model of Fedorov et al. for a regular microstructure
is considered [2]. This can be extended to consider a randommicrostructure [3] and to include gas-rarefaction effects in the porouslayer [4]. The regular microstructure comprising of layers of finemesh of square cross section is also considered, motivated by recentexperiments [13]. These cases are characterized by an admittanceAy,which is a function of the disturbance frequency and depends on thephysical properties of the flow and the porous layer. The physicalparameters are chosen to correspond to the relevant experimentalstudies [2–4].In Sec. II, the theoretical models of the porous walls to be
considered are described. The linear-stability problem is presentedfor axisymmetric and nonaxisymmetric disturbances, resulting in adispersion relation for each case. Section III considers neutral andnonneutral solutions of the dispersion relations for the differentporous-wall models. Maximum spatial growth rates are presentedand comparisons made with available experimental results. Theeffects of varying porosity, pore radius, and pore depth are consid-ered. Finally, maximum spatial growth rates for a porous surfaceoptimized for the secondMack-mode stabilization are comparedwiththose for a regular microstructure comprising circular pores. InSec. IV, we draw some conclusions.
II. Methods, Assumptions, and Procedures
The flow of a compressible viscous fluid over a sharp cone with aporous boundary, of semi-angle θc, is considered at hypersonic speedU0 aligned with its axis. The attached shock makes an angle θs withthe cone surface, a situation that is illustrated in Fig. 1. The dashedlines in this figure indicate the triple-deck structure of the disturbedflow. Spherical polars x; θ;ϕ are the natural coordinate system inwhich to describe the basic flow, and here ϕ denotes the azimuthalangle. Furthermore, the radial distance x has been nondimension-alized with respect to L, the distance from the tip of the cone to thelocation under consideration.
The basic flow has been described previously [11], and so only theimportant features are summarized as follows. Away from the surfaceof the cone, the flow satisfies the (inviscid) Euler equations. Thevelocities are nondimensionalized with respect to U−, in which U−is the magnitude of the fluid velocity just behind the shock.Additionally, the time, pressure, and density are nondimensionalizedwith respect toL∕U−, ρ−U
2−, and ρ−, respectively, inwhich ρ− is the
density just behind the shock. Finally, the basic temperature isnondimensionalized by T−, the temperature just behind the shock.The jump conditions at the shockmust be considered and the velocitycomponents satisfy the Taylor–Maccoll equations [14]. A constant-density approximation may be made to these equations in thehypersonic limit [15]. For the slender cone considered here, thisapproximation agrees well with the exact solution [16].From this hypersonic small-disturbance approximation to the
basic flow [16], an approximate value of the shock angle may beobtained from the expression
sinθs θc sin θc
γ 1
2 1
M2∞ sin2 θc
1∕2
(1)
in which γ is the specific-heat ratio, and M∞ denotes the Machnumber in the freestream. For the weak-shock solutions, relevantto the experiments, this gives excellent agreement with the exactsolution.Note that the termM∞ sin θc isO1. Thus, the shock anglemay be calculated, for a fixed cone angle and freestream Machnumber. A rough indication of the results from Eq. (1) correspondingto the experimental conditions of interest is that θs ≈ θc. This will betaken into account in the formulation of the linear-stability analysis.This solution is not valid close to the surface of the cone, and so a
boundary-layer solution has to be introduced in this region. TheReynolds number of the flow is defined by Re ρ−U−L
∕μ−.Taking the angle of the cone to be small, the governing equations inthe boundary-layer region are given in terms of dimensionlesscoordinates x; r;ϕ [11]. Then, L r is the normal direction tothe cone surface, in which r a on the generator of the cone.The corresponding nondimensional velocities are u; v; w, and thenondimensionalized pressure and density p and ρ, respectively.These satisfy the compressible conservation of mass, Navier–Stokes,and energy equations. The boundary conditions are no slip at thesurface of the cone (coupled to the porous layer) and appropriateconditions at the shock location. The nondimensional temperatureand viscosity at the surface of the cone are taken to be Tw and μw,respectively. The only restriction imposed on the temperatureboundary condition is Tw ≫ 1, which is violated only for situationsinvolving strong cooling on the cone wall [11]. Usually, the walltemperature is taken to be Tw TbTr, in which Tr is the adiabaticwall temperature given by
Tr 1Prp γ − 1
2M2
in which Pr is the Prandtl number and M is the Mach number justbehind the shock. Thus, unless the constant Tb is very small, Tw willbe of Oγ − 1M2 for both adiabatic walls (Tb 1) or isothermalwalls. The analysis is unaffected by the particular choice oftemperature–viscosity law. The choice only affects the bounds placedon various parameters of the problem. Sutherland’s viscosity law[μw ∼ 1 CT1∕2
w , C being a constant] is used henceforth.
A. Porous Boundary
We will present the results corresponding to porous surfaces usedin the previous experimental investigations [3,13,17]. In all cases, theporous-layer admittance Ay can then be expressed in the form
Ay −ϕ0∕Z0 tan hΛh (2)
in which ϕ0 is the porosity of the surface. The porous-layerparameters are nondimensionalized with respect to the boundary-layer displacement thickness δ, and so the nondimensional pore
Fig. 1 Geometry of the cone and shock. The cone is taken to be of semi-
angle θc with the attached shock making an angle θs with the surface of
the cone.
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depth h h∕δ and the nondimensional pore radius rp rp∕δ.Z0 and Λ are the characteristic impedance and propagation constantof an isolated pore, respectively. Fedorov et al. [4] give the followingexpressions for the porous-layer characteristics:
Z0 ρD∕CD
pM
Twp and Λ iωM
Twp
ρDCD
p(3)
in which ω is the disturbance frequency. These are functions of thecomplex dynamic density ρD and complex dynamic compressibilityCD. The precise definitions of these quantities depend on thestructure of the porous wall, and are given in Secs. II.A.1–3 for thecases investigated here.The wall boundary condition, in all cases, is then given by
v Ayp − p− (4)
in which v is the nondimensional normal-velocity componentand p− γ−1M−2.
1. Regular Microstructure
Following previous investigations [2,4], we consider the porouslayer on the cone surface to be a sheet of thickness h perforatedwith cylindrical blind holes of radius rp and equal spacings rp
π∕ϕ0
p. This model takes into account gas-rarefaction
effects. We have
ρD 11−FBν;ζ ; CD 1 γ − 1FBE; ζ
Prp
FBν; ζ Gζ1−0.5Bνζ
2Gζ ; FBE; ζPrp Gζ
Prp
1−0.5BEζPrp2Gζ
Prp
9=;(5)
in which
Bν 2α−1ν − 1Kn; BE γ2α−1E − 1∕γ 1PrKn;
Gζ 2J1ζζJ0ζ
(6)
and ζ rpiωρwR∕μw
p. Here, R is the Reynolds number based on
the boundary-layer displacement thickness, J0;1 are Bessel functionsof the first kind, αν and αE are molecular accommodation coeffi-cients, and Kn is the Knudsen number.
2. Mesh Microstructure
Following Lukashevich et al. [13], we consider the porous coatingon the cone surface to comprise of several layers of stainless-steelwire mesh. A similar model to the one described previously for aregular microstructure is employed. Following Kozlov et al. [18], wehave different expressions for the complex dynamic density andcompressibility. The expressions for the porous-layer characteristicsfor a square-mesh microstructure are [18]
ρD 1∕1 − Fζ; CD 1 γ − 1F~ζFζ 1 ζ2
P∞m0
2
γ2mβ2m
1 − tan hβm
βm
F~ζ 1 ~ζ2P∞
m0
2
γ2m ~βm2
1 − tan h ~βm
~βm
9>>>>>=>>>>>;
(7)
in which
γm π
m 1
2
; βm
γ2m − ζ2
q; ~βm
γ2m − ~ζ2
q(8)
The characteristic size of an isolated pore is given by
ζ
iωρw ~a
2
μwR
sand ~ζ
Prp
ζ (9)
Here, ~a is the half-pore width, and the flow parameters are chosen tofit the experimental conditions [13]. Following Lukashevich et al.[13], rarefaction effects are neglected for this porous model.
3. Random Microstructure
Following Fedorov et al. [3], we consider the porous layer on thecone surface to have a random microstructure. This was consideredbecause it closely resembles the surface of thermal protection tilesused to protect reentry vehicles from aerodynamic heating. A similarmodel to the one used for the regular microstructure is employed.Wehave different expressions for the complex dynamic density andcompressibility. Fedorov et al. [3] give the following expressionsfor the porous-layer characteristics for flow over a felt-metalmicrostructure:
ρD a∞h1 gλ1
λ1
i; CD γ − γ−1
1gλ2 λ2
gλi 1 4a∞μwλi
σϕ0r2p
q; λ1 ia∞ρwω
ϕ0σ ; λ2 4Prλ1
9=;(10)
where the characteristic pore size is
rp πd
21 − ϕ02 − ϕ0(11)
Here, d is the fiber diameter; σ is the flow resistivity, and its value ischosen to fit the experimental data for flow over the felt metal. Thetortuosity a∞ is taken to be unity. Following Fedorov et al. [3],rarefaction effects are neglected for this porous model.
B. Linear-Stability Problem
The linear stability of the basic flow described previously for aslender cone with M ≫ 1 and Re≫ 1 is investigated in the weak-interaction region following a triple-deck formulation [19,20]. This iswith respect to the interaction parameter M∞R
−1∕6, in whichM∞R
−1∕6 ≪ 1 [21,22]. This ensures that we are considering alocation far from the tip of the cone to ensure that theviscous–inviscidinteraction between the boundary layer and the inviscid flow is small.The conditions to be satisfied at the shock by a disturbance to thisbasic flow must be specified, and these have been derived in detail[23]. The requisite constraints were obtained by considering thelinearized jump conditions at the shock for infinitesimal wavesbeneath the shock; a similar procedure was adopted for flow over awedge [19]. Although the basic flow is not uniform in the regionbelow the shock, the jump conditions may still be evaluated at theundisturbed position of the shock [23]. The condition satisfied by thepressure amplitudes of the two acoustic waves (which are incidentand reflected from the shock) is found to be similar to that for awedge [19].Attention is focused at a location on the surface of the cone
with nondimensional radius a a∕L. It is assumed thataRe3∕8M1∕4μ−3∕8w T
−9∕8w a ∼O1 denotes the scale of the
radius at this point; thus, we have chosen sin θc ∼ θc∼Re−3∕8M−1∕4μ3∕8w T
9∕8w . We note that, in terms of the interaction
parameter χ M∞a∕L, our analysis corresponds to a moderate
inviscid interactionwith χ ∼O1. Because fromEq. (1), θs ≈ θc, theshock is located in the upper deck of the triple-deck structure. Theangle of the cone and the flow parameters enable the shock angle andthe corresponding value of a to be obtained. Then, if rs denotes thescaled location of the shock, the ratio a∕rs may be obtained simplyfrom geometric arguments. We find for a slender cone that
a
rs≈
sin θctan θs sin θc
(12)
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where we have taken cos θc ≈ 1. Because θs ≈ θc, a roughapproximation gives a∕rs ≈ 0.5.Our study is confined to the question of the stability of the flow at a
location on the body where the boundary-layer thickness isORe−1∕2L , which is thin compared to the local radius of the cone,allowing the subsequent analysis to capture the effects of curvature onthe stability problem. For smaller values of radius a, the problemreduces to that of the planar case (flow over a wedge). The analysis issomewhat simplified if nonparallel effects can be neglected, and thisis justifiable if the Newtonian assumption γ − 1 ≪ 1 is made [19].This condition arises from ensuring that the streamwise wavelengthof the disturbances is much less than the distance from the apex of thecone. Thus, for simplicity, this condition is taken to hold in thefollowing analysis, although it can be easily relaxed for moreinvolved studies. This condition leads to restrictions on the lowerbounds of M [11]. The experimental conditions for the poroussurfaces of interest have been shown to be in the range of validity ofour asymptotic analysis [24].It is convenient to scale out some of the parameters in the problem,
namely μw, Tw, and λ, in which the last quantity denotes theboundary-layer skin friction. For axisymmetric flow, the Machnumbermay be scaled out of the linear-stability problem.A summaryof the triple-deck analysis for axisymmetric and nonaxisymmetricdisturbances is given in Michael and Stephen [24].Previous scaling is applied to the resulting linear-disturbance
equations for axisymmetric disturbances [19,25]. Analytic solutionsof these equations yield an eigenrelation relating the streamwisewave number α and frequency Ω of the disturbance, namely
Here, ξ0 −i1∕3Ωα−2∕3, Aiξ is the Airy function, Knz andInz are the usual modified Bessel functions, and Ay Re−1∕8μ1∕8w λ1∕4T3∕8
w M2 − 13∕8AY . The angular frequency ofdisturbance propagation through the pore is ω R∕ReRe1∕4μ−1∕4w λ3∕2T−3∕4
w M2 − 11∕4Ω. The eigenrelation cor-responding to a solid wall is recovered if AY 0.A similar analysis is applied for nonaxisymmetric disturbances
with azimuthal wave number n. The scaling is different in terms ofMach number [11], and in particular for the wall admittance anddisturbance frequency Ay Re−1∕8μ1∕8w λ1∕4T3∕8
w M5∕4AY andω R∕ReRe1∕4μ−1∕4w λ3∕2T−3∕4
w M−1∕2Ω. The resulting eigenrela-tion for nonaxisymmetric disturbances is given by
Ai 0ξ0R∞ξ0
Aiξ dξ
iα1∕3AY
in2
αa2
IniαrsKniαa − IniαaKniαrsIniαrsK 0n iαa − I 0n iαaKniαrs
(14)
III. Results and Discussion
To present our results in regimes of practical interest, we will usethe flow parameters from the relevant experimental studies [2,3,13].The cone angle and Mach number from the experiments willdetermine the shock angle θs and the scaled radius a. Then, the scaledshock location rs is determined as described in Eq. (12). Theexperimental conditions correspond to a∕rs 0.57; thus, we chooseto present our results for this case.The values for flow of a perfect gas were chosen as M 5.3,
Tw Tad, γ 1.4, Pr 0.71, Re1 15.2 × 106, T− 56.4 K,and αν αE 0.9. These values correspond to the ones used inprevious numerical studies [17]. The experiments were conducted ona cone of 0.5 m in length.
A. Neutral Modes
We now consider neutrally stable solutions of the dispersionrelations (13) and (14) corresponding to the axisymmetric case andthe nonaxisymmetric case, respectively. The presence of the shockallows for multiple modes of solutions.
1. Regular Microstructure
The current results for the regular microstructure comprising aregular array of cylindrical pores of circular cross section arecompared to the results for a solid wall for axisymmetric andnonaxisymmetric modes. For the regular microstructure [4], theresults presented as follows are for the porous-layer parametersrp 28.5 μm,ϕ0 0.2, and h ≫ rp. The last relation implies thatΛh → ∞, and so Eq. (2) simplifies to Ay −ϕ0∕Z0. The effectof pore depth is investigated in Sec. III.C. The porous-layercharacteristics chosen correspond to experimental values [4]. Theflow conditions match the experimental conditions [13]. Here, a coneof angle θc 7 degwas usedwith a flow ofMach numberM∞ 6.The effects of rarefaction are included and the Knudsen number iscalculated from the expression
Kn μwM
rpR
2πγTw
p
giving Kn 0.494.Figure 2 shows the neutral values of frequency Ω as a function of
radius a for the first four axisymmetric modes for a∕rs 0.57. Weidentify increasing mode numbers with higher neutral values ofΩ, asindicated by the arrow in Fig. 2 and subsequent figures. The scaledradius varies linearly with L. The range 0 < a < 5 corresponds to0 < L < 0.02 m for axisymmetric modes for all the wall modelsconsidered. The dashed lines are for the porous wall and the solidlines correspond to the results for a solid wall [11]. The flow isunstable in regions above these curves. For the higher modes for thesolutions for α, there is no discernible difference between the results(not shown). However, the neutral solutions for Ω are lower for aporouswall than those for a solidwall, particularly for larger values ofa. Thus, the flow over the porous surface will become unstable forlower frequencies than those for the solid wall.Figure 3 shows the neutral solutions for Ω as a function of a for
nonaxisymmetric disturbances for a∕rs 0.57 and n 1 and n 2 for a solid wall and a regular microstructure. For n 1, the range0 < a < 5 corresponds to 0 < L < 14 m. Again, the porous wall hasonly a small effect on the neutral values of α (not shown). Thebehavior of the first mode is different from the higher modes withΩ → 0 as a → 0. We see from Fig. 3 that the porous wall has a verysmall effect on the neutral values of Ω for this first mode. However,the porous wall has a significant effect on the higher modes, leadingto lower neutral values of Ω. Thus, the effect of the porous wall is
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5a
3 3.5 4 4.5 5
Ω
Fig. 2 Neutral values of frequency Ω for the first four axisymmetric
modes as a function of radiusawithn 0 anda∕rs 0.57: – – –, regularporous wall; —— , solid wall.
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destabilizing. Moreover, this effect increases for the higher modes.Similar results have been obtained for n 3 (not shown).
2. Mesh Microstructure and Random Microstructure
The neutral results for the felt-metal microstructure are comparedto those for ameshmicrostructure. FollowingLukashevich et al. [13],we consider the mesh microstructure to correspond to ϕ0 0.8and ~a 0.05 mm. In their experiments, the porosity of the meshmicrostructure is larger than that of the regular array of circular pores.
This will lead to a larger destabilizing effect on the first Mack mode(see Sec. III.B). We show our results for h ≫ ~a.Figure 4 shows the neutral values of Ω as a function of a for the
mesh microstructure and the felt-metal microstructure for the firstfour axisymmetric modes. We find that the felt metal has a largerdestabilizing effect than the mesh microstructure, resulting in lowervalues of neutral frequency.Figure 5 shows the neutral solutions for Ω as a function of a for
nonaxisymmetric disturbances forn 1 anda∕rs 0.57 for ameshmicrostructure and a random microstructure for the first five modes.The neutral values of Ω for the mesh microstructure are lower thanthose for the regular microstructure comprising circular pores as aresult of the porosity being larger.We see from Fig. 5 that the randommicrostructure has a significant destabilizing effect, with the neutralvalues much lower than those for the mesh microstructure.
B. Spatial Growth Rates
We now concern ourselves with an examination of the spatialevolution of disturbances so that we concentrate on solutions ofEqs. (13) and (14) with Ω real and α complex. If α αr iαi, thenαi > 0 is indicative of stability, whereas αi < 0 denotes spatialinstability.Figures 6–8 show the dependence of the spatial growth-rate
parameter αi on the mode frequency Ω for a regular microstructurecompared to a solid wall for n 0, n 1, and n 2, respectively.The results are shown for a∕rs 0.57 and a 0.6, 1.0, 1.5, and 2.0.There is a complete family of modes, as we saw in our account ofneutral disturbances, and it is clear that for eachmember of the family,there is a cutoff frequency Ωc such that for Ω < Ωc that particularmode is stable, but it becomes unstable if Ω > Ωc. The effect of theneutral values being altered by the presence of the porous wall for allwall models leads to a variation in the growth rates. The maximumvalues of (−αi) for the porouswall are increased greatly from those ofa solid wall for each mode.In addition, we also observed that the growth rate for the porous
modes does not rapidly approach zero at high frequencies, as is thecase for the solidwall. Figure 6 shows, that for the porouswall and thesolid wall, the first mode has the largest growth rate for axisymmetricdisturbances for smaller values of a. However, for larger values of a,the second mode has the largest maximum growth rate. As aincreases, the growth rates decrease.From Fig. 7, we see that for nonaxisymmetric modes with n 1,
the growth rate of the highermodes is greatly enhanced by the regularporous wall. It is the last mode shown, which has the highest growthrate. The results for αi presented in Fig. 8 for n 2 are similar tothose for n 1.Figure 9 shows the results for αi for the felt-metal microstructure
with ϕ0 0.75. These are compared to the values for a meshmicrostructure with ϕ0 0.8. The results are presented for n 0;
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ω
a
a) n = 1
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ω
a
b) n = 2
Fig. 3 Neutral values of frequencyΩ for the first five nonaxisymmetric
modes as a function of radius a with a∕rs 0.57: – – –, regular porous
wall; —— , solid wall.
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ω
a
Fig. 4 Neutral values of frequency Ω for the first four axisymmetric
modes as a function of radius a with n 0 and a∕rs 0.57: – – –, mesh
microstructure; —— , random microstructure.
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ω
aFig. 5 Neutral values of frequencyΩ for the first five nonaxisymmetric
modes as a function of radius a with n 1 and a∕rs 0.57: – – –, mesh
microstructure; —— , random microstructure.
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-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 100
α i
Ωa) a = 0.6 b) a = 1.0
c) a = 1.5 d) a = 2.0
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
Fig. 6 Spatial growth-rate parameters αiΩ for the first four nonneutral axisymmetric modes for a∕rs 0.57: – – –, regular porous wall; —— , solid
wall.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50 60
α i
Ω
a) a = 0.6
-0.5
0
0.5
1
1.5
2
0 10 20 30 40 50
α i
Ω
b) a = 1.0
c) a = 1.5 d) a = 2.0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 10 20 30 40 50
α i
Ω-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 5 10 15 20 25 30 35 40
α i
Ω
Fig. 7 Spatial growth-rate parameters αiΩ for the first five nonneutral nonaxisymmetricmodes fora∕rs 0.57 andn 1: – – –, regular porouswall;—— , solid wall.
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-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 10 20 30 40 50 60
α i
Ω
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60
α i
Ω
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30 40 50
α i
Ω
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30
α i
Ω
a) a = 0.6 b) a = 1.0
c) a = 1.5 d) a = 2.0
Fig. 8 Spatial growth-rate parameters αiΩ for the first five nonneutral nonaxisymmetricmodes fora∕rs 0.57 andn 2: – – –, regular porouswall;—— , solid wall.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
a) a = 0.6 b) a = 1.0
c) a = 1.5 d) a = 2.0Fig. 9 Spatial growth-rate parameters αiΩ for the first four nonneutral axisymmetric modes for a∕rs 0.57: – – –, mesh microstructure; —— ,
random microstructure.
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a∕rs 0.57; and a 0.6, 1.0, 1.5, and 2.0. We find that the growthrates of all modes are increased greatly for the felt-metalmicrostructure compared to the mesh microstructure. For the felt-metal microstructure, the first mode has the largest growth rate,switching to the secondmode as the radius a increases. This occurs ata larger value of a than for the mesh microstructure. These resultsagree with numerical results for a random microstructure [26].Figure 10 shows the spatial growth rates for n 1 for the mesh
microstructure and the felt-metal microstructure. The effect of thefelt-metal microstructure is to give a very large increase in themaximum growth rates, approximately four times in magnitude,compared to the mesh microstructure.We end this discussion on the effects of a porouswall on the spatial
growth rates by considering the effects of rarefaction for a regularmicrostructure comprising an array of circular pores. In Fig. 11, weshow the neutral values of Ω as a function of a for the first fivenonaxisymmetric modes with n 1 and a∕rs 0.57 for Kn 0andKn 0.494. We see that the effect of nonzero Knudson numberis destabilizing, with lower values of Ω compared to Kn 0,particularly for the higher modes. The effect of rarefaction on thespatial growth rates is illustrated in Fig. 12, showing the resultscorresponding to Fig. 11. The growth rates are increased for nonzeroKnudson number, with larger increases for the higher modes. Wenote, that for the Mack second mode, the effect of rarefaction isstabilizing [4]. However, rarefaction has been shown to give largerfirst-mode destabilization and less second-mode stabilization forhypersonic flow over a flat plate [26].
C. Maximum Spatial Growth Rates
We investigate the effects of changing the properties of the porouswall on the linear stability of hypersonic flow over a sharp cone. Wechoose to concern ourselves with an examination of the effect of thephysical properties of the porous layer on the maximum spatialevolution of disturbances. Thus, we consider the modewhich has the
largest value of (−αi). The effects of porosity, pore radius, and poredepth on the maximum spatial growth rates σmax are considered forazimuthal wave number n 1, a∕rs 0.57, and a 0.8.The effect of increasing the porosity ϕ0 (by decreasing the pore
spacing) for fixed pore radius rp 30 μm is illustrated in Fig. 13.We see, that as the porosity increases, the maximum spatial growthrate σmax increases. Here, σmax is the maximum value of −αi for thedifferent modes for particular values of a∕rs.In Fig. 14,we show the effect of the pore radius rp on themaximum
spatial growth rate σmax for ϕ0 0.25. We see that a larger poreradius leads to larger maximum spatial growth rates.The previous results presented have been obtained by assuming an
infinite pore depth. The effect of finite pore depth is demonstrated inFig. 15 for porosity ϕ0 0.25 and rp 30 μm. We see that as h∕rp
-4
-3
-2
-1
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-1.5
-1
-0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 10 20 30 40 50 60 70 80 90 100
α i
Ω
a) a = 0.6 b) a = 1.0
c) a = 1.5 d) a = 2.0Fig. 10 Spatial growth-rate parameters αiΩ for the first five nonneutral nonaxisymmetric modes for a∕rs 0.57 and n 1: – – –, mesh
microstructure; —— , random microstructure.
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Ω
aFig. 11 Neutral values of frequencyΩ for nonaxisymmetric modes as a
function of radius a for a regular microstructure with n 1 and
a∕rs 0.57: – – –, Kn 0.494; —— , Kn 0.
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increases, themaximum spatial growth rate increases, but it levels offforh∕rp > 20. In the experiments, the perforated sheet was relativelythick with h 0.45–0.5 mm and rp 25–30 μm, givingh∕rp 15–20. This verifies our earlier assumption of takingh∕rp ≫ 1 in our calculations. Previous calculations determined thatmaximum stabilization of the second Mack mode was achieved forh∕rp 3 [13].
In Fig. 16, we show the effect of pore depth for a meshmicrostructure. An increase in the depth corresponds to more layersof the fine mesh. We find that for h 0.15 mm (corresponding tothree layers ofmesh), themaximumgrowth rate is amaximum for thefirst Mack mode. Previous results [13] showed that h 0.15 mmgave themaximum stabilization of the secondMackmode. So, this isthe worst case for the first Mack mode.In Fig. 17, we present the maximum spatial growth rates to
correspond to the experimental conditions of Maslov [17] fora regular microstructure with rp 25 μm, ϕ0 0.2, andRe 10 × 106. Here, the dimensional spatial growth rate is shownas a function of the distance along the cone for the mode giving thelargest growth rate for n 0, 1, 2, 3 and a∕rs 0.57. The dashedlines correspond to the porouswall and the solid lines correspond to asolid wall. We see, that for the nonaxisymmetric modes, the growthrates are significantly larger for the porous wall compared to the solidwall. However, the size of the growth rates is much smaller than thoseobtained for the second Mack mode [17].Previous parametric studies of regular porous coatings and mesh
coatings have been carried out with the focus on the stabilization ofMack’s second-mode instability [2,13,27,28]. These studies revealthat the porous-layer performance can be optimized by controllingthe porosity and porous-layer thickness. These parametric studiesindicate that optimal porous coatings have thickness h∕rp ≈ 3–3.5.Our results indicate that the porous-layer thickness in this range forregular porous coatings also provides optimal first Mack-modestabilization. Parametric studies also show that high porosityprovides maximum second Mack-mode stabilization. However,numerical studies [29] reveal that porous coatings with too closely
-1
-0.5
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30 35 40 45 50
α i
ΩFig. 12 Spatial growth-rate parameters αiΩ for the first five
nonneutral nonaxisymmetric modes for a regular microstructure for
a∕rs 0.57, n 1, and a 0.6: – – –, Kn 0.494; —— , Kn 0.
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
σ max
, mm
-1
Φ0
Fig. 13 Maximum spatial growth rates σmax for varying porosityϕ0 for
a regular microstructure for n 1, a∕rs 0.57, rp 30 μm, and
Fig. 14 Maximum spatial growth rates σmax for varying pore radius rp
for a regular microstructure for n 1, a∕rs 0.57, ϕ0 0.25, anda 0.8.
0.0015
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
2 4 6 8 10 12 14 16 18 20
σ max
, mm
-1
h/rp
Fig. 15 Maximum spatial growth rates σmax for varying pore-depth
ratio h∕rp for a regular microstructure for n 1, a∕rs 0.57,ϕ0 0.25, rp 30 μm, and a 0.8.
0.002
0.0025
0.003
0.0035
0.004
0.0045
0.005
0.05 0.1 0.15 0.2 0.25 0.3 0.35
σ max
, mm
-1
h*, mm
Fig. 16 Maximum spatial growth rates σmax for varying pore depth h
for n 1, a∕rs 0.57, ϕ0 0.25, and a 0.8 for a mesh micro-
structure for ~a 0.05 mm.
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spaced pores trigger a new shorter wavelength instability whosegrowth rate can be larger than that of the second Mack mode. Theauthors have attempted to optimize the design of porous coatingsbased on the acoustic-scattering properties of the porous layer. Theypropose a porous coating with fixed low porosity comprising ofspanwise grooves. Each porous cavity has a depth h, half-width b,and spacing that varies along the longitudinal length of the cone. Theregular porous model of Eq. (5) can be used to study this model bymaking the following changes [29]:
ζ biωρwμw
R
s; FB; ζ tanζ
ζ1 − Bζ tanζ
The effect of this new design on the first Mack-mode instability isexamined. In Fig. 18, maximum unstable growth rates of the firstazimuthal mode n 1 are compared using this porousmodel and theregular porous model both with porosity ϕ0 0.2. The regularporous model is assumed to be infinitely thick, and the pore radius isfixed at rp 25 μm. FromFig. 18,we see that, at smaller streamwisedistance, the new design leads to lower amplification of unstabledisturbances and with increasing streamwise distance the differencebetween the growth rates of the twomodels becomes very small. Thisnovel design corresponds to porous coatings with low porosity andlarge cavity aspect ratio (2b∕h) (i.e., thinner coatingswith less pores).These types of coatings are easier tomanufacture and incorporate intothermal protection systems in hypersonic vehicles [29]. Recent
studies have investigated the optimum shape, size, and spacing for theporous holes to obtain maximum reduction in growth rates for thesecond mode [30].
D. Nonlinear Effects
The linear-stability analysis presented will not be valid forlarger disturbances. Thus, it is important to determine the effect ofnonlinearity on the stability of hypersonic boundary-layer flowover asharp slender cone with a porous wall. This has been investigatedexperimentally for second-mode disturbances using the bicoherencemethod [31,32]. We have carried out a lengthy theoretical weaklynonlinear analysis for the first Mackmode for hypersonic flow over aporous surface [24]. The results are presented in [24], and show thatthe effect of nonlinearity is greatly affected by the presence of aporous wall.
E. Wall Admittance
Because it has been observed experimentally and confirmedtheoretically and numerically that a porous microstructuredestabilizes the first Mack mode, it would be desirable to design amicrostructure that stabilizes the second Mack mode while notappreciably destabilizing the first Mack mode. In their efforts todetermine the types of microstructure that will lead to reducedamplification of the first Mack mode, Wang and Zhong investigatedthe effect of the phase angle of the wall admittance on the growth ofthe first Mack mode [8]. For their numerical investigations on a flatplate, they discovered that a smaller phase angle of the walladmittance leads to a weaker destabilization of the first Mack mode.The effect of wall admittance was also investigated theoretically byCarpenter and Porter [33]. They considered a thin porous sheetstretched over a plenum chamber. When the phase of the walladmittance was very close to π∕2, they discovered that Tollmien–Schlichting waves were completely stabilized. Thus, future studiesshould be focused on determining the wall parameters that willminimize the destabilizing effect of a porous wall on the first Mackmode. This will enhance the current knowledge of how the firstMackmode in hypersonic boundary layers may be controlled.
IV. Conclusions
The authors have presented neutral results for the wave numberand frequency of linear disturbances for hypersonic flow over apassive porous wall, with scales appropriate to the first instabilitymode for axisymmetric and nonaxisymmetric disturbances. Theporous-wall parameters were chosen to correspond to previoustheoretical and numerical studies. For the values chosen here, theauthors find that the neutral values of streamwise wave number αare slightly reduced from those corresponding to a solid wall.However, the neutral values of the disturbance frequency Ω fornonaxisymmetric disturbances can be substantially lower than thosefor a solid wall, particularly for larger values of radius a. In addition,the spatial growth rates presented demonstrate that the porous wallhas a destabilizing effect on the nonaxisymmetricmodes. The growthrates are significantly larger than those for the solid wall.The effect of rarefaction for a regular microstructure was shown to
be destabilizing.The authors have also investigated the effect of the porous-wall
parameters on the maximum spatial growth rate. The effect ofporosity, pore radius, and pore depth on the maximum spatial growthrates was also considered. It was shown that increasing each of thesequantities leads to larger growth rates, but they level off.The authors have made some comparisons of the dimensional
growth rates for the first-mode disturbances enhanced by the poroussurface with those obtained from previous numerical results for thesecond mode. It was found that the first-mode growth rates weresmaller than the second-mode ones.The authors have determined the effect of regular microstructures,
randommicrostructures, and mesh microstructures on the first Mackmode in a hypersonic boundary layer. The formulation will allowfor alternative porous walls to be investigated in a straightforward
0
0.002
0.004
0.006
0.008
0.01
0.012
0.1 0.2 0.3 0.4 0.5
σ max
, mm
-1
L*,m
n=1n=2n=3n=0
Fig. 17 Maximum spatial growth rates σmax as a function of L for aregular microstructure for n 0, 1, 2, 3; a∕rs 0.57; ϕ0 0.2; andrp 25 μm: – – –, regular porous wall; —— , solid wall.
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.1 0.2 0.3 0.4 0.5
σ max
, mm
-1
L*,m
Fig. 18 Maximum spatial growth rates σmax as a function of L for
a∕rs 0.57, ϕ0 0.2, and n 1: – – –, regular microstructure for
rp 25 μm; —— , spanwise grooves.
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manner. Investigations to determine the thickness of porous coatingstominimise the growth of the firstMackmodes have been carried out.A porous coating comprising spanwise grooves was also considered.
Acknowledgments
Thisworkwas sponsored by theU.S. Air ForceOffice of ScientificResearch, Air ForceMateriel Command, U.S. Air Force, under grantnumber FA8655-08-1-3044. The U.S. government is authorizedto reproduce and distribute reprints for governmental purposesnotwithstanding and copyright notation thereon. Vipin Michaelacknowledges additional financial support from the School ofMathematics, University of Birmingham for his Ph.D. studies.Discussions with A. Ruban are gratefully acknowledged.
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