AIAA2001-2994majdalani.eng.auburn.edu/publications/pdf/2001... · AIAA2001-2994 Three Laminar...

AIAA 2001-2994Three Laminar Solutions for Chamberswith Arbitrary Sidewall Mass Addition Joseph MajdalaniMarquette UniversityMilwaukee, WI 53233

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.

31st AIAA Fluid Dynamics Conference11–14 June 2001

Anaheim, CA

AIAA-2001-2994

—1— American Institute of Aeronautics and Astronautics

Three Laminar Solutions for Chambers with Arbitrary Sidewall Mass Addition

J. Majdalani*

Marquette University, Milwaukee, WI 53233

In this article, we consider the laminar oscillatory flow in a low aspect ratio channel with porous walls. For small-amplitude pressure oscillations, we derive asymptotic formulations for the flow parameters using three different perturbation approaches. The undisturbed state is represented by an arbitrary mean-flow solution satisfying the Berman equation. For uniform wall injection, symmetric solutions are obtained for the temporal field from both the linearized vorticity and momentum transport equations. Asymptotic solutions that have dissimilar expressions are compared and shown to agree favorably with one another and with numerical experiments. In fact, numerical simulations of both linearly perturbed and nonlinear Navier-Stokes equations are used for validation purposes. As we insist on verifications, the absolute error associated with the total time-dependent velocities is analyzed. The order of the cumulative error is established and the formulation based on the two-variable multiple-scale approach is found to be the most general and accurate. The explicit formulations help unveil interesting technical features and vortical structures associated with the oscillatory wave character. A similarity parameter is shown to exist in all formulations regardless of the mean-flow selection.

I. Introduction PROBLEM that has become a classic in fluid mechanics is the study of laminar flows in

channels with porous walls. Over the past four decades, numerous analytical and numerical investigations have, in fact, been devoted to derive exact or approximate mean-flow solutions in the presence of different ranges of cross-flow Reynolds numbers. A number of experimental investigations have also been conducted for validation purposes. Reported results have indicated the presence of rich structures and intellectually challenging flow features depending on the assumptions used, initial conditions, and injection levels. For suction at the porous surface, inflection points were often detected in some ranges of the cross-flow Reynolds number R . These points heralded solution multiplicity and possible flow reversal and instability. Conversely, wall injection promoted unique solutions that became more stable to temporal disturbances with successive increases in R . The existence of single or

*Assistant Professor, Department of Mechanical and

Industrial Engineering. Member AIAA. Copyright © 2001 by J. Majdalani. Published by the

American Institute of Aeronautics and Astronautics, Inc., with permission.

dual solutions was first assessed numerically, and later proved rigorously by a group of capable mathematicians. In principle, most theoretical investigations in channels with porous walls have relied on steady mainstream formulations based on an exact similarity solution that was presented by Berman.1 In fact, assuming a streamfunction that varied linearly with the axial coordinate, Berman was able to collapse the Navier-Stokes equations into a single, nonlinear, fourth-order differential equation. The latter depended on four boundary conditions and a Reynolds number R . This so-called cross-flow Reynolds number was based on the wall injection speed wv , channel half-spacing h , and viscosity ν . Except for infinitely large R , Berman’s differential equation precluded the attainment of exact analytical solutions in finite form. Historically, asymptotic series expansions and numerical simulations were initiated in concert to obtain solutions over various ranges of R . Examples abound and some will be briskly revisited to gain perspective on the problem at hand. In 1953, Berman managed a regular perturbation expansion in R that became asymptotically exact as

0R → . This was followed by a first-order approximation by Sellars2 who addressed the large suction case. For infinitely large injection, Taylor3 exacted an elegant formulation that has come to be

A

AIAA-2001-2994


known as the Taylor flow. Taylor’s solution coincided with the leading-order term of an expansion for

1R >> that was carried out, totally independently, by Yuan.4 In an attempt to cover the entire injection range, Morduchow5 employed numerical curve-fitting concepts to contrive nonunique, user-defined approximations. In the same vein of achieving greater uniformity, White et al.6 produced a convergent power series that was applicable to an arbitrary R . The generality of the latter was offset by the need to calculate the power series coefficients iteratively. An era of improvements and refinements followed subsequently. Archetypal examples are found in Terrill7 who enhanced Sellars’s earlier expansion by reproducing additional terms for large suction. He also extended Berman’s series for 0R → . In order to remove a singularity occurring in Yuan’s large injection formula, Terrill8 introduced exponentially small terms in association with matched asymptotic expansions to capture the inner layer near the core. The main focus was later shifted to consider spatial instabilities that were exceptionally pronounced in the event of fluid withdrawal at the porous surface. Works included those by Varapaev and Yagodkin,9 Raithby and Knudsen,10 Brady,11 and Durlofsky and Brady.12 Almost invariably, the onset of instability was associated with the presence of inflection points and multiple solutions. The issue of solution multiplicity was addressed, in turn, by a number of authors. These included Robinson,13 Skalak and Wang,14 Shih,15 Hastings et al.,16 Lu et al.,17 MacGillvray and Lu,18 and Lu.19 When injection was concerned, Skalak and Wang14 reported unique and stable solutions. The corresponding proof was carried out by both Shih15 and Hastings et al.16 Sporadic studies continued to appear in the literature. The temporal stability of channel flows with porous walls also received attention vis-à-vis the works of Zaturska et al.,20 Taylor et al.,21 and Watson et al.22 Such studies confirmed that steady symmetric flows of the injection type were stable to time-dependent perturbations. The level of complexity is raised considerably when the steady mainstream solution to Berman’s equation is subjected to small-amplitude pressure perturbations. From a practical standpoint, it is inevitable that time-dependent fluctuations will occur during the injection process. The ensuing acoustic environment will thus exhibit small amplitude pressure oscillations. Since no former theoretical treatment seems to have addressed the resulting temporal solution, it is the purpose of this article to develop a general model for oscillatory flows that are bounded by permeable surfaces.

The onset of harmonic pressure oscillations can occur in a number of engineering applications. Examples include filtration, phase sublimation, surface ablation, propellant combustion, biological transport in living organisms, and chemical dispensing through porous media. The source of harmonic waves can be external, user-defined, or internal, decreed by the channel’s natural frequency. An example of the former type has been achieved in long rectangular channels by Ma et al.,23,24 and Barron et al.25 These were motivated by the need to simulate the burning process of solid propellant grains in a hazard-free environment. To that end, both Ma and Barron constructed cold-flow facilities that comprised sections where sublimating carbon dioxide could originate from flat rectangular blocks. The sublimating sections thus emulated transpiring walls along which gases could be expelled into the channel. Wave generation was accomplished by means of electric motors harnessed to either a slider-crank or a Scotch-yoke mechanism. Naturally, the to-and-fro piston motion caused the injectant to vibrate harmonically at user-controlled frequencies. The establishment of self-excited pressure oscillations has been recently described by Avalon et al.,26 and Casalis et al.27 Their facility used air as the working fluid inside a long channel with counterfacing permeable and impermeable walls. Inevitably, small fluctuations in the injection rate triggered a strong acoustic environment that depended on the system geometry. In all three experiments cited above, the placement of a choked orifice or nozzle at the channel’s downstream end caused the oscillation mode character to be of the closed-closed type. The forthcoming model will therefore be tailored to incorporate such oscillation mode shapes. It is hoped that the resulting formulations will be useful in predicting the temporal field arising in physical settings whose mainstream can be described by one of Berman’s mean flow solutions. The strategy we follow is well conceived. In Sec. II, we define the geometry and Berman’s mainstream solution. Fundamental criteria that justify a perturbation scheme are also visited. In Sec. III, we exploit the small injection Mach number and normalized pressure wave amplitude to derive a linearized set of Navier-Stokes equations. The velocity vector is subsequently decomposed, in Sec. IV, into irrotational and solenoidal parts. This allows splitting the linearized equations into two distinct sets that are coupled through existing boundary conditions. The ease of determining the irrotational, pressure-driven response is offset by the ingenuity required in obtaining the solenoidal, vorticity-driven response. The latter is relegated to Sec. V where the vorticity transport equation is summoned and resolved via regular perturbation tools. Both WKB and multiple-scale procedures are evoked in Sec. VI to derive simpler

AIAA-2001-2994


formulations, directly from the momentum equation. In Sec. VII, we compare our asymptotics and numerics, and reap, as a windfall, a new similarity parameter that appears in all formulations at any cross-flow Reynolds number. In closing, we retire with final comments in Sec. VIII.

II. Model Description

A. The Finite Channel We consider in Fig. 1 the symmetrical influx of a perfect gas into a parallelepiped with two equally permeable walls that are 2h apart. We take the channel length L to be larger than the height 2h and width w . As shown by Terrill,7 assuming that

/ 8w h ≥ enables us to treat the motion as a case of two-dimensional flow. An orthogonal coordinate system can be anchored at the porous wall with x , y , and z denoting the streamwise, normal, and spanwise coordinates (normalized by h ). Due to symmetry about the meridian plane, our investigation can be confined to the range 0 x l≤ ≤ , and 0 1y≤ ≤ , where

/l L h= . In addition, we assume that the flow at the downstream end is choked. In Fig. 1, the bulk fluid motion is represented by streamline patterns that denote the undisturbed state. When sinusoidal pressure oscillations are introduced, a more sophisticated temporal field will emerge whose features are yet to be examined.

B. Limiting Criteria On the one hand, we limit our scope to low speed injection characterized by a wall Mach number

/w sM v a= of order 310− , where sa is the internal speed of sound. Practically, since sa far exceeds wv in most applications, M will be very small. Having identified an ideal candidate for small perturbation expansions, we constrain, on the other hand, the pressure oscillation amplitude A to remain small by comparison to the mean pressure sp at the channel fore end. This enables us to define another small quantity that scales with / sA p . Furthermore, in order to

maintain consistency in perturbation orders, we limit the channel length to 100l < . As we show in the next section, this geometrical restriction is necessary to ensure spatial uniformity in the mean pressure amplitude.

C. Mean-flow Solution The undisturbed state can be assessed from a classic similarity transformation of the Navier-Stokes equations. As demonstrated by Berman,1 when the steady streamfunction Ψ is chosen to vary linearly in the streamwise direction, one can write, without loss in generality, ( , ) ( )x y xF yΨ = − . Subsequently, the two-dimensional Navier-Stokes equations are reduced into ( ) 0ivF R F F FF′ ′′ ′′′+ − = , (1) where, in our notation, / 0wR v h ν= > for injection. The corresponding velocity and vorticity vectors, normalized by wv , can be expressed as

0 0 0ˆ ˆu v= +u i j , and 0 0 0

ˆ= ∇× = Ωu kΩ ,

where 0u xF ′= − , 0v F= , and 0 xF ′′Ω = . (2) These must fulfil the no-slip condition, 0( , 0) 0u x = ,

and the normal influx at the wall, 0( , 0) 1v x = .

Furthermore, symmetry about the meridian plane demands that 0( ,1)/ 0u x y∂ ∂ = , and 0( ,1) 0v x = .

The assortment of four boundary conditions on F translates into, (0)F ′ (1)F= (1)F ′′= 0= , and (0) 1F = . (3) As shown in a survey by Terrill,7 various asymptotic solutions are available for different values of R . Despite our aim geared towards a formulation that would be applicable to an arbitrary F , we are inclined to invoke, occasionally, simple solutions that can be used as vehicles for verification and added physical clarity. To that end, we furnish below two asymptotic formulae by Berman1 and Yuan4 that correspond to either small or large wall injection:

( )

2 3 23 12 2

12

1 (10 ), small ( )

cos ( ), large

y y R RF y

y R Rπ

−

−

− + += +

O

O. (4)

The mean pressure associated with Eq. (2) can be normalized by spγ , where γ is the ratio of specific heats, and then integrated from the ideal momentum equation, 2

0 0 0M p⋅∇ = −∇u u . Recalling that

( / )s s sa pγ ρ= , and that the stagnation pressure condition 0(0,1) 1/p γ= must be satisfied at the fore

end, we gather, after some algebra, 2 2 2 21

20( , ) 1/ ( )p x y M F F FF xγ ′ ′′= − + −

2 21/ ( )M xγ= +O . (5)

z

x

y

0

1

0

10

Fig. 1 The planar channel geometry showing typical mean-flow streamlines at three spanwise locations.

AIAA-2001-2994


Equation (5) indicates that the error associated with a uniform pressure distribution will be less than a few percent as long as 100x ≤ . This justifies the upper limit posted earlier on the channel length.

III. Linearized Navier-Stokes Equations

A. Representation The differential conservation of mass and momentum principles can be expressed in the familiar dimensionless form

( )/ 0tρ ρ∂ ∂ +∇ =. u (6)

( )[ ]/ tρ ∂ ∂ + ∇ =u u. u

( ) ( )[ ]1 4 / 3p R−−∇ + ∇ ∇ −∇× ∇×.u u (7) As we employ asterisks to denote dimensional counterparts, the total instantaneous velocity, pressure, density, spatial coordinates and time are normalized in the following fashion: * / sa=u u , * /( )sp p pγ≡ , * / sρ ρ ρ≡ , * / ,x x h= * /y y h= , and * /st t a h= . (8) Following these transformations, sa /R h ν= will designate the so-called acoustic Reynolds number.

B. Variable Decomposition When small oscillations are introduced at a circular frequency ω , the instantaneous pressure can be represented by a linear sum of its steady and time-dependent fluctuations. Using subscripts for perturbation orders, we follow Lighthill28 in expanding the total pressure into * * * * * * *

0 1( , ) ( , , )p p x y p x y t∗ = + (9) where P is a spatial function of (1)O that will be determined in Sec. IV(D). Normalizing and using

*0 sp p= , one obtains

2 2( , , ) 1/ ( , )exp( i ) ( )mp x y t P x y k t M xγ ε= + − +O 11/ ( , , )p x y tγ ε≅ + , (10) where /m sh aω ω= is the nondimensional frequency, and /( )sA pε γ= is the wave amplitude ratio. Density can be expanded in a similar fashion. If 1ρ is of (1)O ,

we can set *1 1sρ ερ ρ≡ , and write

( )*1 1/ 1s sρ ρ ρ ρ ερ= + = + . (11) In much the same way, velocity lends itself to decomposition given small velocity oscillations about the mean. At the outset, we write * * *

0 01 1wv= + = +*u u u u u . (12)

Setting *1 1saε≡u u , we normalize by sa and reap, for

the total velocity and vorticity vectors, 0 1M ε= +u u u , and 0 1M ε= +Ω Ω Ω . (13)

C. Navier-Stokes Linearization When the perturbed pressure, velocity, and density are inserted back into Eqs. (6)-(7), they precipitate two sets of equations depending on the order of the wave amplitude. On the one hand, the zero-order set reduces to Berman’s nonlinear equation that is automatically satisfied by the mean flow. On the other hand, the first-order set in the wave amplitude can be segregated to provide the fundamental interaction equations that govern the time-dependent field: ( )01 1 1/ t Mρ ρ∂ ∂ +∇⋅ = − ∇⋅u u , (14)

( )0 0 01 1 1 1/ t M ∂ ∂ = − ∇ ⋅ − × − × u u u u uΩ Ω

( )11 1 14 / 3p R− −∇ + ∇ ∇⋅ −∇×

u Ω . (15)

This linear set embodies the influence of mean fluid motion on the oscillatory field.

IV. Vector superposition

A. Flowfield Decomposition The temporal disturbances can be decomposed into separate curl-free and divergence-free components. To that end, the former is taken to be the gradient of a scalar φ , and the latter is taken to be the curl of a vector V . When the circumflex and tilde are used to designate irrotational and solenoidal parts, the time-dependent velocity can be partitioned into 1

ˆ φ= + ≡ ∇ +∇×u u u V . (16) From Eq. (16), the conditions ˆ 0∇× =u and

0∇⋅ =u are self-evident. Similar decomposition of small disturbances into irrotational, pressure-driven responses and rotational, vorticity-driven responses has been accomplished previously by Chu and Kovásznay,29 Carrier and Carlson,30 and others. From Eq. (16), it follows that 1 1= = ∇×uΩ Ω , 1

ˆp p= , and 1ˆρ ρ= . (17)

Clearly, unsteady vortices owe their presence to the rotational response and oscillatory pressures are associated with the irrotational response. The last term in Eq. (17) stems from the known relation, ˆ ˆp ρ= , for a perfect gas undergoing isentropic oscillations.

B. Governing Responses When Eqs. (16)-(17) are substituted back into Eqs. (14)-(15), two distinct sets of formulae are reproduced. The two sets, given below, remain coupled through existing boundary conditions.

1. Pressure-driven Response ( )0ˆ ˆ ˆ/ t Mρ ρ∂ ∂ +∇⋅ = − ∇⋅u u , (18)

( )0 0ˆ ˆ ˆ ˆ/ t p M ∂ ∂ = −∇ − ∇ ⋅ − × u u u u Ω

AIAA-2001-2994


( )1 ˆ4 / 3R−+ ∇ ∇⋅ u . (19)

2. Vorticity-driven Response 0∇⋅ =u , (20)

( )0 0 0/ t M ∂ ∂ = − ∇ ⋅ − × − × u u u u uΩ Ω

1R−− ∇×Ω (21)

C. Boundary Conditions On the one hand, the no-slip condition demands that the component of velocity parallel to the wall be zero. This implies that 1( , 0) 0u x = , or

( , 0) ( , 0) 0u x u x+ = , thus providing a strong dynamic

coupling between pressure and vortical disturbances at the wall. On the other hand, symmetry along the midsection plane implies that 1( , 1)/ 0u x y∂ ∂ = .

D. Irrotational Solution Bearing in mind that ˆ ˆp ρ= , a standard rearrangement of Eqs. (18)-(19) reduces the set into 2 2 2ˆ ˆ/p t p∂ ∂ −∇ =

( ) ( )20 0 0ˆ ˆ ˆ/ ( )M p t − ∇ ⋅ ∂ ∂ −∇ ⋅ +∇⋅ × u u u u Ω .

(22) Equation (22) possesses a well-known solution at

( )MO . In fact, separation of variables can be used in unison with the rigid wall boundary conditions to reproduce the solution corresponding to longitudinal oscillations in a channel with 1l >> . When expressed in Euler’s notation, the oscillatory pressure reads ( ) ( ) ( )ˆ , cos exp i ( )m mp x t x t Mω ω= − +O . (23) Here / / ,m sh a m lω ω π= = is the dimensionless wave number and 1,2, 3,m = … denotes the oscillation

mode number. The corresponding velocity response can be integrated from Eq. (19) to obtain ( ) ( ) ( )ˆˆ , i sin exp i ( )m mx t x t Mω ω= − +u i O . (24)

E. Rotational Equations The oscillatory vortical response can be expressed as

( )( , )exp i mx y tω= −u u , ( )( , )exp i mx y tω= −Ω Ω (25) where ˆ ˆu v≡ +u i j , and ˆ≡ ∇× = Ωu kΩ . In lieu of Eqs. (20) and (21), we now have 0∇⋅ =u , (26)

( )0 0 0i /S = ∇ ⋅ − × − × u u u u uΩ Ω

/K+∇×Ω , (27) where

w

hS

vω= , and

2

mhK R ωων

= = . (28)

The dynamic similarity parameters in Eq. (27) are the Strouhal number S , and the kinetic Reynolds number K . The latter is the quotient of the global length scale h to the much smaller Stokes layer thickness ( / )ν ω

raised to the second power. Since ( / ) hν ω << , we define 1/Kε ≡ to be our primary perturbation parameter. This parameter belongs to the practical range 8 410 10ε− −< < . Furthermore, since oscillatory flows are characterized by appreciable Strouhal numbers, we define 1/Sσ = < 0.1. Note that ε is always smaller than σ by virtue of their ratio

/ / 1wv h Rσ ε ν= = > , irrespective of frequency. Based on foreknowledge and numerical verifications, we now make the conditional stipulation that /v u must be of ( )MO . Subject to dual confirmations at the conclusion of the forthcoming analysis, this proviso is necessary at this point to overcome technical obstructions arising in the governing equations. Since our final solution is limited by the ( )MO error associated with the pressure response, v can be omitted at the first perturbation level. On that account, the axial component of Eq. (27) reduces to

( )2

20 0i ( )u uu uu v Mx y y

σ ε ∂ ∂ ∂ = + − + ∂ ∂ ∂

O ,

or

( )0 0i ( )u uu v Mx y

σ ε ∂ ∂Ω= − Ω + + ∂ ∂

O . (29)

V. Vorticity Transport Formulation

A. Governing Equation The vorticity transport equation can be obtained by performing the curl on Eq. (27). Moreover, when Euler’s notation is used, one gets, at ( )MO ,

( ) 20 0i σ ε= − ∇× × + × − ∇u uΩ Ω Ω Ω . (30)

In scalar form, this can be rearranged in a manner to place leading-order quantities on the left-hand side:

0 0

0 0 0

i u uy v v x v xσ

∂Ω∂Ω Ω ∂Ω− + = −∂ ∂ ∂

2 2

2 20v x y

εσ

∂ Ω ∂ Ω + + ∂ ∂ . (31)

In Eq. (31), the right-hand side quantities represent the axial convection of mean-flow vorticity by virtue of the oscillatory vortical action, and the viscous diffusion of unsteady vorticity. Both are secondary effects that can be ignored at the zero perturbation level.

AIAA-2001-2994


Vorticity can be expanded in powers of M , via 2

0 1 ( )M Mϖ ϖΩ = + +O , and later substituted into Eq. (31). The leading-order equation that follows, namely,

0 0 0 0

0 0

i0

uy v v xϖ ϖ ϖ

σ∂ ∂

− + =∂ ∂

, (32)

suggests investigating a separable solution of the type 0 ( ) ( )X x Y yϖ = . When this form is inserted into Eq.

(32), we reap 0 0( , ) ( ) exp( i )n

n

nx y c xF λ

λϖ = − Φ∑ , (33)

where the phase lead of the vortical wave is dependent on

( )0 yΦ =0

d( )

y zS

F z− =∫

( )( )[ ]

2 21 13 2

12 24 2

ln 1 ln(1 ) , small

ln tan 1 gd ( ), large

S y y y R

S y S y Rπ ππ π

−

− + − − − − + = −

(34) For the large injection case, 2gd( ) 2arctan( )z z π= − is the Gudermannian function described in Abramowitz and Stegun.31 Note that 0ϖ contains a denumerable set of arbitrary constants nc that must be specified in a manner to satisfy the no-slip condition at the wall, written for vorticity. This requires a careful treatment and is examined next.

B. Pressure-driven Vorticity at the Wall

Recalling that 1Ω = Ω , 1v v= , 1ˆp p= , and that

1( , 0, )u x t must vanish to prevent slippage, the axial

projection of Eq. (15) reduces, at the wall, into

( )0 0 0

ˆ 10 pM vv v vx x R y

∂ ∂ ∂Ω= − − Ω − Ω − − ∂ ∂ ∂

.

(35) Rearranging, and using the fact that

( ) ( )ˆ cos exp i ,m mp x tω ω= − the no-slip condition translates into

ˆ1 (0)p v xvFM x y x

εσ

∂ ∂Ω ∂ ′′Ω = + + −∂ ∂ ∂

=− − + ∂∂+S x ty

Mm msin exp~

( )ω ω εσ

a f a fi ΩO , (36)

which can be recast into

( ) 1( , 0) sin ( )mx S x MR y

ω ∂ΩΩ = − + +∂

O . (37)

It can be realized from Eq. (37) that vorticity originates at the wall where the oscillatory pressure gradient is perpendicular to incoming fluxes. Furthermore, vorticity generation is highest at pressure nodes located

at / (2 1)/(2 )x l m m= − , where the inviscid, pressure-driven u has maximum amplitude.

C. Inviscid Vorticity At the wall, Eq. (37) must be equated to Eq. (33) in order to specify the separation eigenvalues. Thus, since

(0) 1F = , and 0(0) 0Φ = , then

( ) ( )0 , 0 sinn

n

n mx c x S xλ

λϖ ω= = −∑

( ) ( )( )

2 1

0

12 1 !

nnm

n

xS

nω +∞

=

−≡ −

+∑ , (38)

whence

0ϖ = ( )( )

2 10

0

1( ) exp( i )

2 1 !

nn

mn

S xFn

ω∞

+

=

− − − Φ + ∑ .

(39) From Sec. II(C), we recall that xFΨ = − , and thus identify the infinite series between braces to be a sine function of Ψ . At the outset, we let

( , )mZ x y ≡ ( , )m x yω Ψ , and simplify Eq. (39) into 0 0( , ) sin( )exp( i )mx y S Zϖ = − Φ . (40)

D. Inviscid Streamfunction At this juncture, one can resort to the oscillatory streamfunction to replace the velocity components via

/u yψ= ∂ ∂ and /v xψ= −∂ ∂ . At the outset, the vorticity equation becomes,

2 2

2 2x yψ ψ∂ ∂Ω = − −

∂ ∂. (41)

We then proceed heuristically by suggesting the ansatz 0 0( , ) ( ) ( , )cx y y x yψ ψ ϖ= . This is based on the

assumption that ψ must possess the same axial dependence as Ω . When inserted back into Eq. (41), a balance between leading-order quantities of comparable size yields 2 2

c Fψ σ= 20 0sin( )exp( i )mF Zψ σ⇒ = − Φ . (42)

Subsequently, the inviscid, vortical component of the oscillatory velocity can be deduced by simple differentiation. The result is ( )3

0ˆ ˆ( , ) i sin( ) cos( ) exp im mx y F Z MF Z = + − Φ u i j .

(43)

E. Viscous Corrections Following the same perturbative scheme, we set an ansatz for each of the velocity and vorticity terms. Assuming that both u and Ω must possess the same axial dependence as their inviscid counterparts, we set ( )0( , ) ( )sin( )exp ic mu x y u y Z= − Φ ,

( )0( , ) ( )sin( )exp ic mx y y ZϖΩ = − Φ . (44)

AIAA-2001-2994


The viscous correction multipliers, cu and cϖ , must be found in a manner to satisfy the complete vorticity transport equation. In fact, when we substitute Eq. (44) into Eq. (31), we note the cancellation of several terms. Balancing the remaining leading-order quantities demands that 3 3d /dc cy S Fϖ ε ϖ−+ 1 2( , ) 0cF F u M σ−′′+ + =O , (45) where 3 2 3

wS hvξ ε ω ν −= = appears as a dynamic similarity parameter. At this point, a relationship between cu and cϖ is needed to make any headway. On that account, we resort to Eq. (29) and find ( )1 2i ( , )c cu F F Mσ ξσ ϖ σ−= + +O . (46) Inserting this formula into (45) leads to: ( )3 2d /d i ( , ) 0c cy F F Mϖ ξ σ ϖ σ− ′′+ + + =O , (47) which, after some algebra, gives ( ) expc y Cϖ ζ= , (48) It may be instructive to note that, for consistency in perturbation orders, one must suppress the imaginary argument in ζ which is of effective 2( )σO . The reason is that this spurious argument that appears at

( )σO must be added eventually to the leading-order 1

0 ~ σ−Φ . Thus, ζ can be evaluated from

3

0( )

yF z dzζ

ξ−= − =∫

( ) ( ) ( )

2 4 2

2 2 2 2

14 2 2

4 3 4 18 12 1ln , small

27 2 9 (3 ) 18

ln tan 1 sec tan , large

y y yR

y y y

y y y Rπ π ππ

− − +− − − − − + +

(49) where 1y y= − . (50)

F. Vorticity Correction The integration constant C can be specified from Eq. (37). In fact, when Eq. (48) is implemented to update the vorticity boundary condition at the wall, we observe, at 2( , )M σO , that

201 (0) i (0)C ξσ ζ ′− − Φ ′

[ ] ( )0sin ( , 0) exp (0) i (0) sinm mZ x S xζ ω × − Φ = − (51) where

3(0) (0)Fζ ξ ξ−′ = − = − ; 10 (0) (0)SF S−Φ = − = −′ ;

0(0) (0) 0ζ = Φ = , F∀ . (52) Direct substitution into Eq. (51) gives

2(1 i ) ( )C Sξσ σ− = +O . When the superscripts ‘r ’ and ‘ i ’ are used to denote real and imaginary parts, the constant C can be expressed by

3

2 2r SCS ξ

=+

, 2

2 2i SCSξ

ξ=

+. (53)

Straightforward substitution into Eq. (48), Eq. (44), and Eq. (25) yields, at last, ( )0( , , ) sin( )exp i im mx y t C Z tζ ωΩ = − Φ − . (54)

G. Axial Velocity Correction In much the same way, the corrective multiplier cu can be deduced from Eq. (46), viz. ( )1i exp i expcu F F C Uσ ξσ ζ ζ−= + ≡ , (55) where ( )/r r iU C F C Fσ ξσ= + ,

( )= /i i rU C F C Fσ ξσ− , (56) so that u is soluble by backward substitution into Eq. (44) and Eq. (25). At length, we find that ( )( , , ) exp i mu x y t u tω= −

( )0i sin( )exp i im mU Z tζ ω= − Φ − . (57)

H. Normal Velocity Correction The normal component v can be elicited in a manner to satisfy mass conservation. In practice, this process is expedited by assuming that ( )0( )cos( )exp i im mv g y Z tζ ω= − Φ − . (58)

Direct substitution into Eq. (20) produces ( )g y . In fact, satisfaction of / / 0v y u x∂ ∂ + ∂ ∂ = demands that

2g MUF= . Therefore, ( )2

0( , , ) cos( )exp i im mv x y t MUF Z tζ ω= − Φ − , (59) which lends support to the earlier stipulation, in Sec. IV(E), contending that / ( )v u M=O .

I. The Total Time-dependent Solution For the reader’s convenience, we retrace our steps and summarize the results of the current procedure. The real components of time-dependent velocity and vorticity are hence revisited below. Using the superscript ‘V ’ to refer to the vorticity transport formulation, we list ( ) ( )1 sin sinV

m mu x tω ω=

( ) ( )sin cos exp sinr imU U xFϕ ϕ ζ ω− − ,

0mtϕ ω= +Φ (60) ( ) ( )2

1 cos sin exp cosV r imv MF U U xFϕ ϕ ζ ω= − + ,

(61) ( ) ( )1 cos sin exp sinV r i

mC C xFϕ ϕ ζ ωΩ += − . (62)

AIAA-2001-2994


Note that the first term in Eq. (60) may be envisaged as the inviscid, irrotational response to the oscillatory pressure while the second may be held accountable for the viscous, vortical response due to the presence of solid boundaries. Naturally, the second response decays asymptotically with increasing distance from the wall.

VI. Momentum Transport Formulation

A. Governing Equation In the previous section, the rotational velocity was obtained from the vorticity and vorticity transport equations following a number of successive approximations. At present, u will be derived directly from the momentum equation, written at ( )MO . To that end, we first rearrange Eq. (29) into

2

2(i ) ( )u S u ux F u F Mx F y y

σ σ ε ∂ − ∂ ∂ ′= + − + + ′∂ ∂ ∂

O ,

(63) and then call for separation of variables in order to investigate a solution of the form ( , ) ( ) ( )u x y X x Y y= . Substitution into Eq. (63) renders,

dd

x XX x

=

2

2

d d (i )d d n

S Y YF F YF Y y y

ε σ σ λ ′− − + + = ′

,

(64) where nλ must be positive for a nontrivial solution. Integration of the x − equation is straightforward. The result is ( ) n

n nX x c xλ= , where nc is an integration constant tied to nλ . Owing to the linearity of Eq. (63), the general solution becomes ( , ) ( )n

n n

n

u x y c x Y yλ

λ=∑ (65)

where nλ must be determined from the no-slip boundary condition at the wall written for the velocity. As specified in Sec IV(C), this condition is fulfilled when, ( )( , 0) i sin mu x xω= − . Writing out the

MacLaurin series expansion for the sine function, and equating the result to Eq. (65) gives,

( ) ( )( )

2 1

0

1(0) i

2 1 !n

nnm

n nn

n

xc x Y

nλ ω

λ

+∞

=

−≡ −

+∑ ∑ , (66)

which precipitates, 2 1, 0,1,2,n n nλ = + = … ,

( ) ( )( )

2 11i

2 1 !

nnm

nc nω +−

= −+

, (0) 1nY = , (67)

wherefore,

( ) ( )( )

2 1

0

1( , ) i ( )

2 1 !

nnm

nn

xu x y Y y

nω +∞

=

−= −

+∑ . (68)

The velocity eigenfunction Yn is left to be determined from (64) via

( )2

2

d di 2 1 0

d dn n

nY Y

F n F Yy y

ε σ σ ′− + + + =

0 1y≤ ≤ , (69) with boundary conditions ( )0 1nY = and (1) 0nY =′ . (70) By virtue of Eq. (3), Eq. (69) admits a regular singularity at the core where (1) 0F = . Evidently, the presence of a small multiplier in the highest derivative evokes the possibility of a perturbation treatment. In fact, both WKB and two-variable multiple-scale expansions appear promising. For example, the two-variable derivative expansion technique was shown in a previous work32 to result in locally valid solutions corresponding to inner, outer, and transition scales. As described therein, a uniform two-scale expansion could also be attained using a hybrid technique. The latter was based on the choice of a composite scale that could reproduce the inner, outer, and transition scales in their respective domains. In the current investigation, we shall use a different strategy. Instead of constructing the so-called composite scale from our foreknowledge of inner, outer, and transition scales, we shall leave the modified variable unspecified while carrying out the two-scale expansion. At the conclusion of the asymptotic analysis, physical arguments will be employed to solve for the undetermined coordinate transformation. Before initiating the asymptotic work, we define 1

Nu to be the numerical solution of the linearized momentum equation, given by Eq. (29). In fact, 1

Nu can be determined by coupling the numerical solution to Eq. (69), specified in Ref. 32 with Eqs. (68), (25), (16), and (24).

B. The WKB Approach While seeking an asymptotic solution for Eq. (69), two cases must be considered separately depending on the order of the secondary perturbation parameter σ . Our motivation is this:

1. The Traditional Outer Expansion For (1)σ =O , the leading-order term of the outer solution o

nY can be resolved via

( )[ ]2 2 0o on nFY i n F Yσ σ′ ′− + + + = ,

(0) 1onY = , 2 2 1

0( ) exp i ( )d

yo nnY y F S F z z+ − = ∫ (71)

with 2 2 1

0[ ( )] cos ( )d

yo nnY y F S F z z+ − ℜ = ∫ . (72)

AIAA-2001-2994


Recalling from Eq. (3) that all solutions to Berman’s equation must satisfy (1) 0F = , we realize that the

2 2nF + factor in the outer solution decays rapidly as 1y → . Clearly, the decay is more rapid at higher

eigenvalues. As a result, the remaining boundary condition at the origin is automatically satisfied by the first derivative. This obviates the need for an inner solution at this order. As made obvious in Eq. (72), the exponential term in o

nY represents an oscillatory behavior that is rapid for large S . Since S could be large in real applications, the rapid oscillations that occur on a shorter length scale preclude the possibility of a uniformly valid solution. This becomes apparent in the first-order correction when Yn

o is written at 2( )εO : 2 2

0exp( i )o nnY F += − Φ

( )( 2

01 2 1

yS S nε× + − + +∫

( )[ ]22 1FF n F′′ ′× + + ( ) i 4 3n SF ′+ + )3dF z− (73) Realizing that the correction term contains a part of

3( )SεO , nonuniformity is expected when S becomes large. At this point, a WKB strategy becomes more suitable.

2. The WKB Ansatz For small σ , rapid oscillations occur on a short length scale, and a slow drift takes place on the scale

(1)x =O . The leading-order equation that delivers the WKB ansatz is 0n nFY iSY− + =′ , (0) 1nY =

1

0( ) exp i ( )d

y

nY y S F z z− ⇒ = ∫ (74)

Equation (74) evokes setting 0( ) ( )exp( i )nY y g y= − Φ and revisiting Eq. (69). At the outset, we segregate terms of 2( )SεO and harvest

( )3 3 12 1g S F n F F gε − − ′ ′+ − +

2 2 1 1 2i (2 ) ( )S F g F F g SF g Sε ε ε− − −′ ′ ′′= − + =O . (75) The WKB solution can be extracted therefrom, namely,

2 2 20 0( ) exp( i ) ( )W n

nY y F Sζ ε+= − Φ +O ,

30 0

( )dyF z zζ ξ −= − ∫ ; (76)

where the superscript ‘W ’ refers to the WKB formulation. The dimensionless group

3 2 3S hξ ε ω ν ϑ−= = appears to be in control of the exponential rate of decay as 1y → . Note that ( )W

nY y is uniformly valid and its derivative satisfies the boundary condition at the core.

3. The WKB Solution

Equation (76) can be inserted into Eq. (68) to give ( ) ( )( )

2 1

0 00

1( , ) i exp( i )

2 1 !

nnm

n

xFu x y F

nω

ζ+∞

=

−= − − Φ

+∑

0 0i exp( i )sin( )mF xFζ ω= − − Φ . (77) This formula can be used in conjunction with Eqs. (25), (24), and (16) to construct the total oscillatory velocity component, 1( , , ) i sin( )exp( i )m mu x y t x tω ω= − 0 0i exp( i i )sin( )m mF t xFζ ω ω− − Φ − . (78) Thus, in the real domain, the WKB solution offers 1 ( , , ) sin( )sin( )W

m mu x y t x tω ω= 0 0sin( )exp sin( + )m mF xF tω ζ ω− Φ . (79)

C. The Multiple-scale Approach A two-variable multiple-scale procedure requires specifying two virtual coordinates, an outer scale 0y , and an inner scale, 1y , that enables us to resolve rapidly changing phenomena.

1. Nonlinear Coordinate Transformation Conventional inner scale choices include linear transformations that bear 1 ( )y yδ ε= , where ( )δ ε is usually conjectured from foreknowledge, trial-and-error, or a scaling analysis. In the problem at hand, such transformations fail to provide uniformly valid, two-scale expansions. On the contrary, we find it rather useful to introduce a nonlinear variable transformation of the form, 1 ( )y s yε= , where ( )s y is a general function that can accommodate nonlinear scaling assortments. We find this choice to provide the necessary freedom needed to achieve a balance in the governing equation between diffusive, convective, and inertial terms.

2. The Generalized Two-scale Expansion Following the procedure described by Majdalani,32 we introduce two virtual coordinates, 0y y= , and

1 ( )y s yε= . The difference here is that s will remain unspecified until the conclusion of the asymptotic analysis. Owing to this virtual transformation, functions and derivatives may be expanded in a manner to discard 2( )εO and smaller quantities: (0) (1) 2

0 0 01 1 1( , ) ( , ) ( , ) ( )nY y y Y y y Y y yε ε= + +O ,

0 0 1

d dd d

sy y y y

ε∂ ∂= +∂ ∂

, 2 2

2 20

d( )

dy yε∂= +

∂O . (80)

After inserting these expansions into Eq. (69), terms of the same order in ε can be segregated to decouple the original equation into

( )(0)

(0)

0

2 1 i 0Y F Sn Yy F F

′∂ − + + = ∂ , (81)

AIAA-2001-2994


( )(1)

(1)

0

2 1 iY F Sn Yy F F

∂∂

′ − + +

(0) 2 (0)

20 01

dds Y S Yy y F y

∂ ∂∂ ∂

= − + . (82)

Boundary conditions given by (70) translate into

( )(0) 0 1Y = , ( )(0)

0

1 0Yy

∂ =∂

. (83)

Integration of the zero-order equation yields ( ) ( )[(0)

0 1 1 1( , ) exp 2 1Y y y a y n= +

( ) ( )0 10 01 10

ln ( ) i ( )dy

F y S F z z a y yχ−+ ≡∫ , (84)

In preventing the asymptotic series from containing terms whose quotient between two successive orders can be unbounded, the right-hand side of Eq. (82) must vanish:

( )3 31

1

d2 1

da

s F S S ny

− ′− + − + +

[ ] ( ) 2 2(2 1) i 4 3FF n F S n F′′ ′ ′× + + + +

)1 0a χ× = (85)

The resulting first-order differential equation in 1a can be integrated in closed form. Expressed in the original laboratory coordinate, we write ( ) ( )( 3 2

01 exp 1 2 1a y a F nξη σ−= − + +

( )[ ] ( ) )22 1 i 4 3F F n F n Fσ′′ ′ ′+ + + + .

(86) where 0a is a pure constant of integration that must be specified from the boundary condition at the wall. The effective scale functional ( )yη that appears in Eq. (86) is the ratio of the scale function and its first derivative, namely, ( ) ( )/ ( )y s y s yη ′≡ . (87) The rest is straightforward substitution into Eq. (84). Recalling that the overall solution is sought at ( )MO , there is no justification for retaining other than (0)Y , and Eq. (80) reduces, at ( )εO , into 2 2

0 expnnY a F +=

( )( ) 3 2 21 1n nF F F Fξη σ λ λ− ′′ ′− + + +

( ) 1 2 3

0i d 4 3

yS F z n F Fξσ η− − ′+ + + ∫ . (88)

3. Specifying the Undetermined Scale Note that both real and imaginary arguments in the exponential above display leading-order quantities followed by smaller terms of 2( )σO . Suppressing these secondary corrections reduces Eq. (88) to a form that bears a striking resemblance to the WKB solution

given by Eq. (76). In fact, the leading-order terms will be identical to those in W

nY when

3 3

0( ) ( ) ( )d

yy F y F z zη −= ∫ . (89)

From (0) 0η = and ( )0 1nY = we conclude that (0) 0s = and 0 1a = . As a result, Eq. (88) will satisfy

both boundary conditions stated in Eq. (83). Having uncovered the effective scale functional in general form, the corresponding undetermined scale can be specified from Eq. (87). The result is

3

1

30 0

0

( )d( ) | exp ( )d | | exp |

( )d

y y

zF z z

s y z zF

ητ τ

−−

−= =∫ ∫

∫.

(90) It may be safe to say that the dependence of the scaling transformation on the Berman function F is by no means intuitive. This may explain the difficulty in guessing beforehand the modified coordinate transformation that leads to uniform two-scale expansions. Moreover, we are no longer surprised that the traditional linear transformations of the form

1 ( )y yδ ε= could not prevail.

4. Comparison for the Large Injection Case For the sake of argument, we consider the large injection case corresponding to 2cos( )F yπ= . Despite the complexity of Eqs. (89) and (90), we are able to manage exact expressions for η and s . In fact, after some algebra, we find that

( )[ ] 212 4 2 2( ) cos( ) ln tan 1 tan( ) cos ( )y y y y yπ π π π

πη = + + (91) ( ) ( ) ( )2 2 4( ) sec tan ln tan 1s y y y yπ π π= + + . (92) Obviously, Eq. (92) precludes the possibility of conjecturing this appropriate coordinate transformation in advance, as required by standard procedures. The most striking result is, perhaps, the agreement between our current didactic formulation and the hybrid formulation obtained by Majdalani.32 In the previous work, a composite scale

1.51.5( ) ys y yy −= was initially constructed in a manner to reproduce asymptotically the apparent inner, outer, and transition scales. Usage of the composite scale thus reduced the number of spatial scales to two. This space-reductive step was necessary for the success of the multiple-scale procedure. After devising the composite scale, the effective scale functional ( )1.5 1( ) / 1 1.5 1.5 lny y y yy yη − = + − was derived and then inserted into the solution. In the current analysis, the process is somewhat reversed. Here η has to be determined first. Only then could the space-reductive scaling transformation be determined. It may be instructive to note that the nonlinear scaling transformation obtained here is also capable of

AIAA-2001-2994


reproducing near 0, 1y = the local scales proposed by

Majdalani.32 To gain further reassurance, Eqs. (91) and (92) are compared in Fig. 2 with the values of η and s obtained previously using the composite scale technique.32 It is interesting to note the qualitative agreement which has led, undoubtedly, to two uniformly valid formulations. Furthermore, Eq. (88) can be readily applied to the large injection case. The result will be

( ) 2 2 3cos exp secnnY θ ξη θ+=

( )2 2 221 1 (2 sin cos2 )n nπ σ θ θ × − + + −

( ) 1 2 22 3

2i gd 2 sec tan ( )S nπ θ πξσ η θ θ ε− + − + + O . (93) A standard error analysis can be performed on Eq. (93) to verify the agreement with its counterpart, given by Eq. (5.22) of the former article.32 Since the current approach follows precisely the steps outlined in,32 they are omitted here. As found previously, results show at

0n = an error of ( )εO that depreciates incrementally with 1n ≥ . A similar analysis can be applied to small injection with 2 33 1

2 21F y y= − + . In fact, for the small R case, results are omitted because they involve long expressions that can be easily reproduced using a symbolic program. For illustration, the corresponding η is included in Fig. 2a.

5. The Multiple-scale Solution Having determined nY , Eq. (88) can be substituted back into Eqs. (68) and (25), and then added to Eq. (24) via Eq. (16). At length, we obtain,

( ) ( )( )

2 1

10

1( , , ) i exp

2 1 !

nnm

n

xFu x y t F

nω +∞

=

−= −

+∑

( )[ ] ( 3 2 21 2 1 (2 1)F n F F n Fξη σ− ′′ ′− − + + +

( ) )1 2 3

0i ( )d 4 3 i

y

mS F F n F tς ς σ ξη ω− − ′+ + + − ∫

i sin( )exp( i ) ( )m mx tω ω ε+ − +O . (94) Equation (94) clearly displays leading-order quantities and quantities of 2( )σO in the u component. Upon further examination, it appears that the infinite power series can be reduced to a more convenient form without loss in accuracy. In fact, since the error associated with 1n ≥ terms is smaller than the error at

0n = , corrections of 2( )σO can be retained at 0n = and dismissed in 1n ≥ terms. When this is accomplished, an equivalent expression for 1u is begotten at ( )MO . Using the superscript ‘U ’ in reference to the multiple-scale outcome based on the ‘undetermined’ scaling transformation, we write, in the real domain

( ) ( )1

irrotational part

( , , ) sin sinUm mu x y t x tω ω=

( ) ( )solenoidal part

wave propagationwave amplitude

sin exp sinm mF xF tω ζ ω− +Φ (95)

where 3 2 2 2 1

0 1 2 ( )F F F F Fζ ζ ζ ξη ξσ η− − −′′ ′= + = − + + , (96) 1 2 3

0 1 0( )d 3

yS F z z F Fξσ η− − ′Φ = Φ +Φ = − + ∫ .

(97) By comparison to 1

Wu , 1Uu is slightly more accurate

since it reduces to Eq. (79) when the first-order corrections 1ζ and 1Φ are abandoned. A sample comparison between velocity formulations is illustrated in Fig. 3 where numerical and asymptotic predictions are shown to coincide for three different sets of

0

0.05

0.10

0.15

0.20

0.25

η U (large R) η C (large R,32) η U (small R)

(a)

η (y)

0 0.2 0.4 0.6 0.8 1.0

10-510-310-1101103105 (b)

s(y)

sec θ tan θ + gd–1 θ y(1–y)–3y3/2/2

y

Fig. 2 For the large injection case, we compare in (a) and (b) the currently derived effective scale functional η and its corresponding scaling transformation s to existing composite scale results32. Indices ‘U ’ and ‘C ’ refer to the two-variable scheme based on either ‘undetermined’ or ‘composite’ scales. For small injection, η is shown in (a) as well.

AIAA-2001-2994


parameters corresponding to 25S = , and 410 ,K = 510 , and 610 . This parametric spread can be achieved,

for instance, by reducing the kinematic viscosity by two orders of magnitude while holding constant ω , wv , and h . Clearly, increasing viscosity promotes shorter penetration depths of rotational waves. In all three cases, the velocity profiles satisfy the no-slip condition at the wall and reduce to the irrotational amplitude along the core. Figure 3 also shows results acquired from numerical simulations of the nonlinear Navier-Stokes equations. Such results were obtained, totally independently, through personal communication33. In fact, they were based on the fully implicit, finite volume code described in.34 The small discrepancies between asymptotic and computational data may be attributed to the finite space and time discretization errors, finite grid resolution, and small Navier-Stokes nonlinearities that escape our analytical formulae.

6. Other Components From the axial component u , the normal velocity v can be summoned from mass conservation. To expedite the process, we proceed by setting an ansatz of the form

( ) ( )( , , ) ( )cos exp exp im mv x y t G y xF tω ζ ω = − +Φ , (98) where ( )G y is a subsidiary function that must be determined in a manner to satisfy

/ / 0u x v y∂ ∂ + ∂ ∂ = . After some algebra, we find that 3

0G Mv= . Henceforth,

( ) ( )3( , , ) cos exp exp im mv x y t MF xF tω ζ ω = − +Φ . (99)

Equation (99) reconfirms our initial contention that / ( )v u M=O . At this point, we may want to present

an accurate expression for time-dependent vorticity from the velocity formulation. Indeed, straightforward differentiation begets

( ) ( )( , , ) sin exp im mx y t S xF tω ζ ω Ω = − − +Φ . (100) The total time-dependent fluctuations can now be expressed, in the real domain, by

( ) ( )31 cos exp cosU

m mv MF xF tω ζ ω= +Φ , (101)

( ) ( )1 sin exp cosUm mS xF tω ζ ωΩ − +Φ= . (102)

Despite their disparate formulations, the current expression for vorticity can be shown to coincide with its counterpart in Sec. V(I). At the outset, it is reassuring to find that the exponential decay of temporal vorticity is decreed by the same agents irrespective of the perturbation technique.

VII. Discussion

A. The oscillatory Wave Character Equation (95) represents a travelling wave that begins at the wall and decays in the vicinity of the core. Near the wall, both irrotational and solenoidal amplitudes can couple to produce an overshoot of about twice the irrotational core amplitude. This phenomenon is sometimes referred to as the Richardson velocity overshoot35 and is common to oscillatory flows. The decaying wave amplitude is a by-product of two mechanisms, viscous dissipation and mean-flow convection. Viscous damping is apparent in the exponential term and is strongly influenced by

2 3wh vξ ν ω −= , F∀ . While depreciating with F as

0

½

1

-1 0 1

(a)

-1 0 1

(b)

-1 0 1

(c)

Fig. 3 For the first oscillation modes, we compare profiles of the oscillatory velocity 1u at two successive times separated by a phase difference of 90 degrees. To the accuracy of the graph asymptotic solutions (fullcurves) are indistinguishable from numerical simulations of the linearized momentum equation (brokencurves). Additional numerical computations of the nonlinear Navier-Stokes equation (chain curves) are shown as well. Here 25S = , / ½x l = , and 1m = . Nonlinear finite-volume computations rely on a 40×300 mesh resolution. They are shown after nine iteration cycles for (a) 410K = , (b) 510K = , and (c) 610K = .

AIAA-2001-2994


1y → , the convective term varies sinusoidally in the downstream direction. Most of all, Eq. (95) exhibits a traditional form comprising two distinct components: an inviscid, pressure-driven response, and a viscous, vorticity-driven response. Such constitution is typically associated with oscillatory flows. The reader may consult, for instance, the classic survey by Rott where similar equations are presented for flows over impermeable walls.35 The current dependence on the axial position can be attributed to the body’s finite length. This dependence is not encountered in conventional analyses of oscillatory flows in infinitely long channels. For a typical test case, the oscillatory velocity 1u is shown in Fig. 4 at equispaced locations within the channel for the first three oscillation modes. Apparently, the rotational amplitude is influenced by the sinusoidal pressure-driven amplitude. Rotational effects are felt deeper as we move downstream due to the mean-flow convection of unsteady vorticity. In fact, lines of zero vorticity that originate at pressure antinodes mark their presence in the velocity profiles while drifting downstream. This explains the m premature decay in the velocity profiles located downstream of the thm internal pressure antinode. Following the premature decay, the rotational amplitude recuperates some strength before resuming its normal depreciation. Figure 5 illustrates the effect of reducing wall injection by one order of magnitude. This is accomplished by increasing the Strouhal number while maintaining, at 1m = , a constant kinematic Reynolds number. In essence, both the rotational wavelength and depth of penetration diminish when wall injection is suppressed.

B. Standard Error Analysis Before concluding our discussion, it may be instructive to verify the order of the error associated with the asymptotic formulations, 1

Uu , 1Vu , and 1

Cu . The latter refers to the multiple-scale solution obtained by substituting η for the composite scale presented in32. Following the same procedure described by Bosley36 and used in,32 we define mE to be the maximum absolute error between 1u given asymptotically and

1Nu computed numerically. For example, to determine

mE associated with 1Uu , we calculate over an

oscillation cycle, m( , , )E m S ε =

1 100 1

max ( , , , , , ) ( , , , , , )N U

x ly

u x y t m S u x y t m Sε ε≤ ≤≤ ≤

− . (103)

Assuming a logarithmic error variation of the form m( , , ) ( , )E m S m S αε β ε= , (104) we then evaluate the slope α (representing the order of the error) from the log-log plot of mE versus ε . As borne out in Fig. 6, for all three asymptotic formulations, 1α → asymptotically regardless of S . The maximum absolute error associated with 1

Uu appears to be the smallest. This clear asymptotic behavior indicates that the error is in fact of ( )εO . Since Mε < , the error associated with the asymptotic solutions does not contribute to the overall error of

( )MO entailed in our mathematical model.

0 0.2L 0.4L 0.6L 0.8L L

y

0

½

1

Fig. 4 For the first three oscillation modes, we showprofiles of the oscillatory velocity 1u at two successive times separated by a phase difference of90 degrees. The profiles are shown at evenly spacedlocations within the chamber and correspond to alarge injection case with 1 1 25S m hL M mπ − −= =and 2 1 1 610sK m h a L mπ ν− −= = .

0 0.2L 0.4L 0.6L 0.8L L

y

0

½

1

Fig. 5 The effect of varying wall injection is captured by maintaining constant viscosity and frequency, and depreciating the injection at the wall by one order of magnitude. As in the previous figure, we show selectively profiles of 1u corresponding to

1m = , 2 5/ 10K hω ν= = , and, from top to bottom, / 10, 20,wS h vω= = and 100 . Clearly, both the

rotational wavelength and depth of penetration diminish when wall injection is reduced.

AIAA-2001-2994


VIII. Concluding Remarks Using three different perturbation schemes, we have developed asymptotic formulations to the oscillatory flow in a channel with wall injection. One formulation was obtained from the vorticity transport equation while the two others were determined from the momentum equation using WKB and multiple-scale expansions. The generalized formulations were derived for an arbitrary mean-flow solution of the Berman type. The asymptotic solutions were shown to agree with one another and with numerical solutions to both linearly perturbed and nonlinear momentum equations. A similarity parameter that controls the rate of viscous damping has appeared in all three formulations. The explicit formulae obtained were useful in revealing (fairly rich) vortical structures caused by mean and time-dependent flow interactions. An absolute error

analysis indicated that the two-scale solution based on a nonlinear coordinate transformation was the most accurate. Overall, the ( )εO error associated with the asymptotic solutions did not aggravate the ( )MO error attendant on the mathematical model. The most striking result was, perhaps, the good agreement between asymptotic predictions and numerical simulations of the nonlinear Navier-Stokes equations. Another interesting result is the manner in which the modified coordinate transformation was specified. From a practical standpoint, the generalized solutions we obtained increase our repertoire of useful formulations for periodic flows that occur in a number of engineering applications.

References 1Berman, A. S., “Laminar Flow in Channels with Porous Walls,” Journal of Applied Physics, Vol. 24, No. 9, 1953, pp. 1232-1235. 2Sellars, J. R., “Laminar Flow in Channels with Porous Walls at High Suction Reynolds Numbers,” Journal of Applied Physics, Vol. 26, No. 4, 1955, pp. 489-490. 3Taylor, G. I., “Fluid Flow in Regions Bounded by Porous Surfaces,” Proceedings of the Royal Society, London, Series A, Vol. 234, No. 1199, 1956, pp. 456-475. 4Yuan, S. W., “Further Investigation of Laminar Flow in Channels with Porous Walls,” Journal of Applied Physics, Vol. 27, No. 3, 1956, pp. 267-269. 5Morduchow, M., “On Laminar Flow through a Channel or Tube with Injection: Application of Method of Averages,” Quarterly Journal of Applied Mathematics, Vol. 14, No. 4, 1957, pp. 361-368. 6White, F. M., Jr., Barfield, B. F., and Goglia, M. J., “Laminar Flow in a Uniformly Porous Channel,” Transactions of the American Society of Mechanical Engineers: Journal of Applied Mechanics, Series E, Vol. 25, 1958, pp. 613-617. 7Terrill, R. M., “Laminar Flow in a Uniformly Porous Channel,” The Aeronautical Quarterly, Vol. 15, 1964, pp. 299-310. 8Terrill, R. M., “Laminar Flow in a Uniformly Porous Channel with Large Injection,” The Aeronautical Quarterly, Vol. 16, 1965, pp. 323-332. 9Varapaev, V. N., and Yagodkin, V. I., “Flow Stability in a Channel with Porous Walls,” Fluid Dynamics (Izvestiya Akademii Nauk SSSR, Meckanika Zhidkosti i Gaza), Vol. 4, No. 5, 1969, pp. 91-95. 10Raithby, G. D., and Knudsen, D. C., “Hydrodynamic Development in a Duct with Suction and Blowing,” Transactions of the American Society of Mechanical Engineers: Journal of Applied Mechanics, Series E, Vol. 41, 1974, pp. 896-902.

10-6

10-5

10-4

10-2

10-1

S = 200 100 50

10

20

V U

(a)

Em

10-8 10-7 10-6 10-5 10-410-6

10-5

10-4

10-2

10-1

Em

C U

(b)

slope →

1

ε

Fig. 6 We compare at 1m = the maximum absoluteerror entailed in the momentum transportformulation based on the undetermined scale (U ) to (a) the vortical transport solution (V ), and (b) the solution based on the composite scale (C ).

AIAA-2001-2994


11Brady, J. F., “Flow Development in a Porous Channel or Tube,” The Physics of Fluids, Vol. 27, No. 5, 1984, pp. 1061-1067. 12Durlofsky, L., and Brady, J. F., “The Spatial Stability of a Class of Similarity Solutions,” The Physics of Fluids, Vol. 27, No. 5, 1984, pp. 1068-1076. 13Robinson, W. A., “The Existence of Multiple Solutions for the Laminar Flow in a Uniformly Porous Channel with Suction at Both Walls,” Journal of Engineering Mathematics, Vol. 10, No. 1, 1976, pp. 23-40. 14Skalak, F. M., and Wang, C.-Y., “On the Nonunique Solutions of Laminar Flow through a Porous Tube or Channel,” SIAM Journal on Applied Mathematics, Vol. 34, No. 3, 1978, pp. 535-544. 15Shih, K.-G., “On the Existence of Solutions of an Equation Arising in the Theory of Laminar Flow in a Uniformly Porous Channel with Injection,” SIAM Journal on Applied Mathematics, Vol. 47, No. 3, 1987, pp. 526-533. 16Hastings, S. P., Lu, C., and MacGillivray, A. D., “A Boundary Value Problem with Multiple Solutions from the Theory of Laminar Flow,” SIAM Journal on Mathematical Analysis, Vol. 23, No. 1, 1992, pp. 201-208. 17Lu, C., MacGillivray, A. D., and Hastings, S. P., “Asymptotic Behaviour of Solutions of a Similarity Equation for Laminar Flows in Channels with Porous Walls,” IMA Journal of Applied Mathematics, Vol. 49, 1992, pp. 139-162. 18MacGillivray, A. D., and Lu, C., “Asymptotic Solution of a Laminar Flow in a Porous Channel with Large Suction: A Nonlinear Turning Point Problem,” Methods and Applications of Analysis, Vol. 1, No. 2, 1994, pp. 229-248. 19Lu, C., “On the Asymptotic Solution of Laminar Channel Flow with Large Suction,” SIAM Journal on Mathematical Analysis, Vol. 28, No. 5, 1997, pp. 1113-1134. 20Zaturska, M. B., Drazin, P. G., and Banks, W. H. H., “On the Flow of a Viscous Fluid Driven Along a Channel by Suction at Porous Walls,” Fluid Dynamics Research, Vol. 4, No. 3, 1988, pp. 151-178. 21Taylor, C. L., Banks, W. H. H., Zaturska, M. B., and Drazin, P. G., “Three-Dimensional Flow in a Porous Channel,” Quarterly Journal of Mechanics and Applied Mathematics, Vol. 44, No. 1, 1991, pp. 105-133. 22Watson, P., Banks, W. H. H., Zaturska, M. B., and Drazin, P. G., “Laminar Channel Flow Driven by Accelerating Walls,” European Journal of Applied Mathematics, Vol. 2, 1991, pp. 359-385. 23Ma, Y., Van Moorhem, W. K., and Shorthill, R. W., “Innovative Method of Investigating the Role of Turbulence in the Velocity Coupling Phenomenon,”

Journal of Vibration and Acoustics-Transactions of the ASME, Vol. 112, No. 4, 1990, pp. 550-555. 24Ma, Y., Van Moorhem, W. K., and Shorthill, R. W., “Experimental Investigation of Velocity Coupling in Combustion Instability,” Journal of Propulsion and Power, Vol. 7, No. 5, 1991, pp. 692-699. 25Barron, J., Majdalani, J., and Van Moorhem, W. K., “A Novel Investigation of the Oscillatory Field over a Transpiring Surface,” AIAA, Paper 98-2694, June, 1998. 26Avalon, G., Casalis, G., and Griffond, J., “Flow Instabilities and Acoustic Resonance of Channels with Wall Injection,” AIAA, Paper 98-3218, July, 1998. 27Casalis, G., Avalon, G., and Pineau, J.-P., “Spatial Instability of Planar Channel Flow with Fluid Injection through Porous Walls,” The Physics of Fluids, Vol. 10, No. 10, 1998, pp. 2558-2568. 28Lighthill, M. J., “The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in the Stream Velocity,” Proceedings of the Royal Society, London, Series A, Vol. 224, 1954, pp. 1-23. 29Chu, B.-T., and Kovásznay, L. S. G., “Non-Linear Interactions in a Viscous Heat-Conducting Compressible Gas,” Journal of Fluid Mechanics, Vol. 3, No. 5, 1957, pp. 494-514. 30Carrier, B. T., and Carlson, F. D., “On the Propagation of Small Disturbances in a Moving Compressible Fluid,” Quarterly of Applied Mathematics, Vol. 4, No. 1, 1946, pp. 1-12. 31Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions, National Bureau of Standards, New York, 1964, p. 77. 32Majdalani, J., “A Hybrid Multiple Scale Procedure for Boundary Layers Involving Several Dissimilar Scales,” Zeitschrift für angewandte Mathematik und Physik, Vol. 49, No. 6, 1998, pp. 849-868. 33Roh, T. S., Private communication, The California Institute of Technology, 1998. 34Roh, T. S., Tseng, I. S., and Yang, V., “Effects of Acoustic Oscillations on Flame Dynamics of Homogeneous Propellants in Rocket Motors,” Journal of Propulsion and Power, Vol. 11, No. 4, 1995, pp. 640-650. 35Rott, N., “Theory of Time-Dependent Laminar Flows,” in High Speed Aerodynamics and Jet Propulsion –Theory of Laminar Flows, edited by F. K. Moore Princeton University Press, Princeton, New Jersey, 1964, Vol. IV, pp. 395-438. 36Bosley, D. L., “A Technique for the Numerical Verification of Asymptotic Expansions,” SIAM Review, Vol. 38, No. 1, 1996, pp. 128-135.

AIAA2001-2994majdalani.eng.auburn.edu/publications/pdf/2001... · AIAA2001-2994 Three Laminar...

Documents

Transcript of AIAA2001-2994majdalani.eng.auburn.edu/publications/pdf/2001... · AIAA2001-2994 Three Laminar...