MeccanicaDOI 10.1007/s11012-013-9828-2

Linear stability of particle laden flows: the influenceof added mass, fluid acceleration and Basset history force

Joy Klinkenberg · H.C. de Lange · Luca Brandt

Received: 19 February 2013 / Accepted: 17 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract Both modal and non-modal linear stabilityanalysis of a channel flow laden with particles is pre-sented. The particles are assumed spherical and solidand their presence modelled using two-way coupling,with Stokes drag, added mass and fluid accelerationas coupling terms. When the particles considered havea density ratio of order one, all three terms becomeimportant. To account for the volume and mass ofthe particles, a modified Reynolds number is defined.Particles lighter than the fluid decrease the criticalReynolds number for modal stability, whereas heavierparticles may increase the critical Reynolds number.Most effect is found when the Stokes number definedwith the instability time scale is of order one. Non-modal analysis shows that the generation of stream-wise streaks is the most dominant disturbance-growthmechanism also in flows laden with particles: the tran-sient growth of the total system is enhanced propor-tionally to the particle mass fraction, as observed pre-viously in flows laden with heavy particles. Whenstudying the fluid disturbance energy alone, the op-timal growth hardly changes. We also show that the

J. Klinkenberg (B) · H.C. de LangeTechnische Universiteit Eindhoven, Postbus 513,5600 MB, Eindhoven, The Netherlandse-mail: j.klinkenberg@tue.nl

J. Klinkenberg · L. BrandtLinné Flow Centre, KTH Mechanics, 100 44, Stockholm,Sweden

Basset history force has a negligible effect on stabil-ity. The inclusion of the extra interaction terms doesnot show any large modifications of the subcritical in-stabilities in wall-bounded shear flows.

Keywords Linear stability analysis · Modalanalysis · Non-modal analysis · Particle-laden flows

1 Introduction

Particle laden flows are found in our environment andengineering applications, e.g. sand in the atmosphere,soot in gas flows and turbines. Indeed, significant ef-forts have been recently devoted to the study of par-ticles in turbulence as reviewed in Refs. [1] and [2].Already in the early ‘60, it has been shown that addingdust to a pipe flow reduces the drag [3]. As an ex-planation for this phenomenon, it has been proposedthat the interaction between fluid and particles dampsthe initiation and growth of disturbances which thenleads to turbulent structures. Turbulent structures en-hance drag, thus, if turbulence is delayed, drag can bereduced. Drag reduction in flows with light and heavyparticles has been demonstrated experimentally lateron in several papers [4–6]. Numerically, it is found thatmicro-bubbles, which can be modeled as rigid sphereswhen the bubbles are small enough, reduce drag in aturbulent flow [7, 8]. On the other hand, simulationsof drag reduction by means of heavy particles are pre-sented in [9].

Meccanica

These findings motivate us to investigate whetherthe laminar-turbulent transition can be delayed usingparticles, while we studied the case of heavy particlesin previous papers [10, 11]. Here we denote by heavyparticles those whose density is significantly largerthan the fluid density and the interaction between fluidand particles is given by the Stokes Drag only. Whenlighter particles are considered, instead, Stokes dragalone is not sufficient to describe the fluid-particle in-teraction. The present study focuses therefore on theeffect of the other interaction terms that are importantwhen the particles have density of the same order ofthe fluid, which we denote ‘light particles’ throughoutthe paper.

Most of the research on stability of particle ladenflows discusses heavy particles. Saffman in 1962 [12]showed theoretically that adding dust to a gas mightstabilize the flow. Michael in 1964 [13] confirmed theresults by Saffman by showing neutral stability curvesfor several particle sizes. His findings show that anoptimal particle relaxation time exists for maximumstabilization. Both Saffman and Michael considered aplane parallel Poiseuille flow in which the base flowparticle velocity equals the base flow of the fluid. Boththe fluid and the particles are modeled in a Eulerianframework. In addition, particles are considered spher-ical and homogeneously distributed.

Rudyak et. al. [14] and Asmolov and Manuilovic[15] extended the linear stability analysis in channeland boundary layer flow by improving the numericalaccuracy using a different technique based on integra-tion in the complex plane. Modal analysis is also con-sidered in Klinkenberg et al. [10]. These authors showthat stabilization due to heavy particles arises when theratio of particle relaxation time and the period of theinstability, the stability Stokes number, is of order one.

Non-modal analysis is a relatively new, but an im-portant tool to predict instabilities. Nowadays it isunderstood that a perturbation in a shear flow canexperience significant transient energy growth [16–19]. This growth is responsible for the initial linearamplification of disturbances which leads to subcrit-ical transition to turbulence. Non-modal effects cantherefore explain the discrepancy observed betweenthe critical Reynolds number for linear instability andthe experimental observations of transition in wall-bounded shear flows. One needs therefore to consideralso non-modal analysis to gain insight into the stabil-ity of wall-bounded flows seeded with light particles.

Klinkenberg et al. [10] investigated non-modal growthin channel flow seeded with heavy particles and foundthat particles have little effect on the flow stability.

The general equations for particle-laden flowsgiven by Maxey and Riley in 1983 [20] provide anoverview of the particle-fluid interaction forces. Theseforces consist of the Stokes drag, the added mass, thefluid acceleration force (also known as pressure cor-rection force), buoyancy and the Basset history term.The starting point of their analysis is the equationof motion proposed by Tchen [21] and modified byCorrsin and Lumley [22]. Besides these interactionforces, the Saffman lift force [23] may also be rele-vant, as discussed by several authors [24–26] and morerecently by Boronin and Osiptsov [27]. In addition tothese different interaction terms, the effect of finiteparticle volume, the volume the particles have, is in-vestigated by Vreman [28] and Boronin [29]. In thepresent paper Stokes drag, added mass, fluid accelera-tion and Basset history force are considered, limitingthe validity of our analysis to small particles.

Although the different forces between fluid and par-ticles have been known for a long time and have beendiscussed for turbulent flows in several papers, e.g.Calzavarini et.al. [30], transitional flows with lightparticles have not been often investigated in literature.Relevant work on transition in particle-laden flows ispresented by Matas et. al. [31, 32]. These authors per-formed experiments in a pipe flow seeded with neu-trally buoyant particles. To control transition to startat Re ≈ 2100 in a clean flow, they inserted a ringat the pipe entrance. Particles of four different sizeswere injected into the flow and the Reynolds number atwhich transition starts, the transitional Reynolds num-ber, investigated. All particle sizes stabilized the flowfor large concentrations, i.e. the transitional Reynoldsnumber increased. For smaller concentrations (volumefraction ≤ 0.2), large particles destabilized the flow,while the smaller particles (d/D ≤ 70) stabilized theflow. Interestingly, in the range of small particles, theresults became independent of the particle diameter.

The aim of this paper is to investigate modal andnon-modal stability of particle suspensions for a widerange of density ratios. To perform our analysis, weadopt the model introduced in Ref. [20], rewritten intoa Eulerian framework. While this continuous approachis likely to fail in turbulent flows due to particle clus-tering and singularities in the particle field, it can stillbe used for laminar flows and perturbation expansions,

Meccanica

such as in linear stability calculations [33]. Instead ofmodeling both fluid and particles in a Eulerian frame-work, the particles can also be solved by Lagrangiantracking. Meiburg et al. [34] investigated the behaviorof heavy particles in a 2D-mixing layer using two-waycoupling in a Eulerian-Lagrangian framework. Theyfound a vorticity-based understanding of the propaga-tion of a traveling wave.

2 Flow model and stability analysis

Few approximations exist for the coupling betweenfluid and solid phase depending on the particle size andvolume fraction, see Elghobashi [35]. As a first stepin the analysis of the stability of particle laden flows,we neglect collisions and hydrodynamic coupling andtherefore use the two-way coupling model. The two-way coupling includes Stokes drag, added mass, fluidacceleration and Basset history force. We are first in-terested in the influence of added mass and fluid accel-eration on the stability and therefore neglect buoyancy.The Basset history force accounts for the developmentof the boundary layer on the particles and its effect isstudied in Sect. 4.

2.1 Governing equations

The governing equations are the Navier Stokes Equa-tions with the addition of the different fluid-particle in-teraction terms, written in a Eulerian framework. Thedimensional equation for the total particle field reads[36]:

dupi

dt= ρf

ρp

Dui

Dt− 1

2

ρf

ρp

[dupi

dt− Dui

Dt

]

+ K

mp

(ui − upi), (1)

with ui the fluid velocity, upithe particle velocity

and i = 1 − 3. mp = 4/3πr3ρp is the particle mass,ρp and ρf the density of the particle and the fluid,mf = 4/3πr3ρf the fluid mass ideally contained bythe volume of one particle, and K = 6πrμ the Stokesdrag term with μ the fluid kinematic viscosity and r

the particle radius. The notation D/Dt is used for thetotal derivative following a fluid element, while d/dt

is used for the total derivative following a moving par-ticle. The terms on the right-hand side of Eq. (1) are

the fluid acceleration, added mass and Stokes dragrespectively. The particle relaxation time is definedas τ = mp

K. In the particle momentum equation the

particle-particle interaction has been neglected. Al-though such interactions are important when the vol-ume fraction is large, we chose to neglect these andrestrict the maximum particle volume fraction, as dis-cussed in Sect. 2.2. To account for particle-particle in-teractions, a viscosity and pressure could be definedin the particle momentum equation. More informationon the modeling of particle-particle interactions in atwo-fluid model can be found in [37].

The counterpart is the momentum equation for thefluid

ρf

Dui

Dt= −mf N

Dui

Dt+ 1

2mf N

[dupi

dt− Dui

Dt

]

− KN(ui − upi) − ∂p

∂xi

+ μ∂2ui

∂x2j

, (2)

with i = 1 − 3, N the number of particles per unit vol-ume.

When every term is made non-dimensional with thechannel half-width L, centerline velocity U , we obtainthe following non-dimensional particle and fluid mo-mentum equations:

dupi

dt= ξ

Dui

Dt− 1

2ξ

[dupi

dt− Dui

Dt

]+ 1

SR(ui − upi

),

(3)

(1 − Φ)Dui

Dt= − ∂p

∂xi

+ 1

R

∂2ui

∂x2j

− f ξDui

Dt

− 1

2f ξ

[Dui

Dt− dupi

dt

]

+ f

SR(upi

− ui), (4)

with i = 1 − 3. The different dimensionless numbersare defined in Table 1. These are the mass concentra-tion f , the mass of particles for unit volume of thesuspension over the density of the fluid, the densityratio ξ and the volume concentration Φ , the volumeoccupied by the particles over the total volume of thesuspension. The particle relaxation time τ is eithernon-dimensionalized with the viscous time scale of theflow (S) or the advective time scale (SR). Note that thefluid equation (2) is defined using the total volume. To

Meccanica

Table 1 Definition of the non-dimensional numbers

fmpN

ρfMass concentration

ξρf

ρpDensity ratio

Φ f ξ Volume concentration

Rρf UL

μReynolds number

S ντ

L2 (= 29

r2

L2ρp

ρf) Viscous Relaxation time

SR UτL

Stokes number

account for the particle volume we have to multiplythe density with (1 − Φ).

The dimensionless numbers in Table 1 are notindependent of each other and only 4 of them arestrictly necessary. The volume fraction, density ra-tio and Mass fraction are coupled through Φ = f ξ .Also the Reynolds number and dimensionless relax-ation time S can be coupled into the Stokes numberSR. Although not all the non-dimensional numbers re-ported in the table are independent, we define all ofthem to clarify how they change when varying someof the basic features of the flow or of the particles.For instance, we may wish to investigate the impor-tance of either the density ratio, the size of the particlesor the volume fraction independently and thus changethe different non-dimensional number in a less obvi-ous way. The size of the particles, as example, appearsin several of these dimensionless numbers.

In addition to the momentum equations, conserva-tion of mass is necessary to close the system; for par-ticles and fluid these read

∂f

∂t= − ∂

∂xi

(f upi) (5)

∂ui

∂xi

= 0, (6)

with i = 1 − 3. To study linear stability, a smallperturbation (u′) to a base flow (U ) is introduced,where the base flow is considered to be the parallelPoiseuille flow, U = U(y) = 1 − y2, with y ∈ [−1,1].The particle base flow is equal to the fluid base flow,independent of the number of particles. Substitut-ing ui = U + u′

i , upi= U + u′

pi, p = P + p′ and

f = f ′ + f0 in (4)–(6), linearized stability equationsare derived in a standard way [19]. f0 is considereda homogeneous mass fraction of particles and f ′ is asmall deviation from that homogeneous condition. Thelinearized equations read (primes are omitted except

for f ′ and f = f0):

(1 − Φ)∂ui

∂t= − ∂p

∂xi

− (1 − Φ)Uj

∂ui

∂xj

− (1 − Φ)uj

∂Ui

∂xj

+ 1

R

∂2ui

∂x2j

+ f

SR(upi

− ui) + f · (AM + FA)

(7)

∂upi

∂t= −Uj

∂upi

∂xj

− upj

∂Ui

∂xj

+ 1

SR(ui − upi

)

+ AM + FA (8)

∂f ′

∂t= −∇ · (f ′U + f upi

)(9)

∂ui

∂xi

= 0. (10)

With i = 1 − 3, AM and FA the Added Mass andFluid Acceleration, with subscripts f and p , denotingfluid and particle respectively:

AM = −1

2ξ

(∂

∂t(ui − upi

) + Uj

∂

∂xj

(ui − upi)

+ (ui − upi)∂Ui

∂xj

), (11)

FA = −ξ

(∂ui

∂t+ Uj

∂ui

∂xj

+ uj

∂Ui

∂xj

), (12)

with i = 1 − 3. For the configuration considered, theequation for the particle mass fraction f ′ (9) is decou-pled from the rest of the system. As a consequence,Squires theorem can be extended to this case and acomplex Orr-Sommerfeld equation can be derived.Reference [29] This has been done in Ref [12, 13]for heavy particles, but the approach can be extendedto incorporate the added mass and fluid acceleration.However, we are here interested in the non-modal sta-bility of the full three-dimensional problem and there-fore consider the corresponding initial value problem.The equations for the fluid velocities are rewritteninto wall-normal velocity v and wall-normal vorticityη = ∂u

∂z− ∂w

∂x, analogous to the standard Orr-Sommer-

feld-Squire system used for parallel single phaseflows. This is done by eliminating the pressure fromEq. (7) and by solving for ∂u

∂z− ∂w

∂x. The correspond-

ing total system of particle and fluid equations then

Meccanica

reads(1 + 1

2f ξ

)∂

∂t�2 v − 1

2f ξ

∂

∂tQ

={−

(1 + 1

2f ξ

)[U

∂

∂x�2 −U ′′ ∂

∂x

]− f

SR�2

+ 1

R�4

}v

+ 1

2f ξ

[−U ′′ ∂vp

∂x− U ′ ∂

∂x� ·�up + U

∂

∂xQ

]

+ f

SRQ (13)

(1 + 1

2f ξ

)∂η

∂t− 1

2f ξ

∂

∂t

(∂up

∂z− ∂wp

∂x

)

= −(

1 + 1

2f ξ

)(U

∂η

∂x+ U ′ ∂v

∂z

)

+ 1

2f ξ

(U ′ ∂vp

∂z+ U

∂

∂x

(∂up

∂z− ∂wp

∂x

))

+ f

SR

(∂up

∂z− ∂wp

∂x− η

)+ 1

R�2 η (14)

with

Q = ∂2vp

∂x2+ ∂2vp

∂z2− ∂2up

∂x∂y− ∂2wp

∂z∂y. (15)

The boundary conditions of this system are v = η =up = vp = wp = 0 at top and bottom walls. In theAppendix, the matrix representation of this system canbe found which is used to solve the eigenvalue prob-lem.

Note finally that the Reynolds number of the flowis defined as R = ρf UL

μ. While this can be seen as

a Reynolds number for the fluid alone, a modifiedReynolds number can be defined for the total systemas:

Rm = (1 + f − Φ)ρf UL

μ,

thus based on the total density. This modified Reynoldsnumber Rm is used in the modal analysis, whereas theReynolds number R is used when presenting resultsfrom non-modal stability.

2.2 Model limitations

The model used here implies some limitations on theplausible values of the particle parameters. The param-

eters to be considered are the volume fraction, the den-sity ratio and the size of the particles. When two-waycoupling is used, the volume fraction should not be toolarge, because collisions and hydrodynamic couplingare not modelled. This limitation is best illustrated us-ing the average particle spacing: l

D∼ ( π

6Φ)1/3, with l

the average particle spacing and D the diameter of theparticle. For a volume fraction of Φ = 0.01, the aver-age particle spacing l

D∼ 3.74. Although the physical

particle spacing below which particle-particle interac-tions play a role is not clearly defined, in this paper welimit the volume fraction to Φ = 0.01.

The limits on the density ratio are given by the factthat particles cannot be too light since we neglect sur-face tension and deformability. Here we will there-fore limit the results to the case ξ ≤ 2. In additionbuoyancy effects become relevant when ξ = 1. Theseforces will be however neglected in this study to sim-plify the problem and to be able to focus on the ef-fect of the other interaction forces (added mass andacceleration). Buoyancy effects, and related phenom-ena like sedimentation, are left as future work. Notealso that neutrally buoyant small particles have no ef-fect on the flow dynamics in the limit considered here;neutrally buoyant particles however have significanteffects on the flow for volume fractions larger than0.01 [31, 38].

The third limiting parameter is the size of the par-ticles, r/L, where r is the radius of the particle and L

the channel half-height. For large particles finite-sizeeffects needs to be included, as in the so-called Faxencorrection. In addition, we need sufficiently many par-ticles to assume a continuous homogeneous particledistribution. Given a specific volume fraction, the sizeand the number of particles are coupled: the smallerthe particle, the more particles are needed to obtainthis volume fraction. This dependence is shown inFig. 1 where the particle volume is given as a functionof the number of particles for different particle sizes.Two limits are also shown, Φ = 0.01 and N = 1000:with a volume fraction Φ = 0.01 and a minimum num-ber of N = 1000 particles, the maximum size of theparticles becomes r/L = 0.0134. This is therefore theorder of magnitude for the maximum size allowed bythe model adopted for this work.

Although these are the physical limitations of ourmodel, in some of the following figures we may alsoshow data outside this range to emphasize the effectof the solid dispersed phase on the stability of channelflow.

Meccanica

Fig. 1 Particle volume versus number of particles, for differentparticle sizes. N0 is the number of particles per unit volume. Thetwo limits (Φ = 0.01 and N = 1000) are shown as thin blacklines. The intersection of these lines coincides with a particlesize r/L = 0.0134. Larger particles fall outside the limits of theapproximations used in this paper

2.3 Modal stability analysis

To study linear stability, we assume wave-like pertur-bations of the following form:

q = q̂(y)ei(αx+βz−ωt),

with q = (v, η,up, vp,wp)T . In the expression above,α and β define the streamwise and spanwise wavenum-ber of the perturbation respectively and ω is a com-plex frequency. The temporal problem is consideredhere: when �(ω) > 0, the perturbation will grow ex-ponentially in time. Conversely, when all �(ω) < 0,all disturbances decay asymptotically, i.e. the flow isstable. The point where ωi = 0, is called neutrally sta-ble. When computing ωi in a range of wavenumbersα and Reynolds numbers, a neutral stability curve canbe obtained. This curve defines the range where expo-nentially unstable waves can be found. As mentionedearlier, the neutral stability curve can be computed as-suming two-dimensional perturbations, since a mod-ified version of Squire’s theorem holds for particleladen flows [12, 29].

2.4 Non-modal stability analysis

Transient disturbance energy growth may appear whenthe eigenvectors of the system are non-normal. This isalso the case in systems that are asymptotically stable.To investigate transient growth, non-modal analysis isnecessary. Non-modal analysis determines the largest

possible growth of a perturbation in a finite time in-terval, also called optimal growth. The initial distur-bance yielding optimal growth is called an optimal ini-tial condition.

The discretized governing linear equations can bewritten in compact form as:

∂q

∂t= Lq. (16)

The largest possible growth at time t is the norm of theevolution operator, or propagator, T = exp(tL). Thispropagator takes any initial condition from t = 0 to aspecified final time t . The maximum amplification isdefined as:

maxq0

‖q‖‖q0‖ = max

q0

‖ exp(tL)q0‖‖q0‖

= ‖ exp(tL)‖ ≡ G(t). (17)

The norm used should be relevant to our problem.Therefore we use the kinetic energy of the full sys-tem defined as the kinetic energy of the fluid and ofthe particles together:

Ekin = 1

2

(mf u2

i + mpu2pi

), (18)

with mf and mp the mass of the fluid and the particlesrespectively.

A matrix M can be constructed to compute the ki-netic energy. This matrix M is applied directly to thevector q = [v,η,up, vp,wp]T to give the kinetic en-ergy integrated over the volume V

E(t) = 1

2

∫Ω

qH MqdV. (19)

With this definition the optimal growth is defined asthe 2-norm of the modified propagator

maxq0

‖q‖E

‖q0‖E

= maxq0

‖Fq‖2

‖Fq0‖2

= maxq0

‖F exp(tL)F−1Fq0‖2

‖Fq0‖2

= ‖F exp(tL)F−1‖2 ≡ G(t) (20)

where F is the Cholesky factorization of M = FFH .As in our previous study, we are not only interested

in optimizing the total energy of the system, but also

Meccanica

wish to investigate the optimal growth when perturb-ing only the fluid or particle velocity. In this case, wedo not consider the total kinetic energy of the system,but only a part of it, depending on the initial conditionand final state chosen. This separation can be achievedby including either fluid or particle energy when com-puting the optimal growth. [10] The optimization canbe written as

G(t) = ‖qout (t)‖Eout

‖qin(0)‖Ein

= ‖T qin(0)‖Eout

‖qin(0)‖Ein

= ‖FoutT qin(0)‖2

‖Finqin(0)‖2

= ‖FoutT F−1in Finqin(0)‖2

‖Finqin(0)‖2

= ‖FoutT F−1in ‖2

= ‖FoutC exp(tL)BF−1in ‖2 (21)

Here, the propagator T = C exp(tL)B is rewritten toinclude the input and output matrices. The input isqin = Bq , while qout = Cq is the output we are in-terested in. The energy norm must be separated like-wise, Min = FinF

Hin is applied to qin to measure the

input energy while Mout = FoutFHout gives the output

energy. In the classic non-modal analysis discussed,Fin = Fout and C = B = I .

2.5 Numerical method

The discretization in y-direction of the equations isdone using the Chebyshev collocation method. [39].Most computations are performed using ny = 37, withny the number of collocation points. Several caseshave also been computed with ny = 67 to validate theaccuracy of the results.

For the transient growth computation, we made useof the following energy matrix M :

M =

⎛⎜⎜⎜⎜⎜⎝

(−D2

k2 + 1)Iw 0 0 0 00 1

k2 Iw 0 0 00 0 f Iw 0 00 0 0 f Iw 00 0 0 0 f Iw

⎞⎟⎟⎟⎟⎟⎠

.

(22)

In the expression above, Iw is the diagonal matrix per-forming spectral integration in y direction. This matrixcan be easily factorized using a singular value decom-position (SVD): M = UΣUH = FFH .

3 Results

3.1 Modal analysis

The results for modal stability analysis are given inFig. 2, where we display the critical total Reynoldsnumber versus the dimensionless, viscous particle re-laxation time S in (a). Particles heavier than the fluidincrease the critical Reynolds number. The largest sta-bilization is found when considering heavy particlesand the only relevant interaction term is the Stokesdrag. Lighter particles, but still heavier than the fluid,also increase the critical Reynolds number. Particleslighter than the fluid (ξ > 1 and still rigid in ourmodel) behave oppositely to heavy particles and de-crease the critical Reynolds number. The results in

Fig. 2 Critical Reynolds number as a function of dimension-less relaxation time S and of stability Stokes number SRωr forlight particles using both added mass and fluid acceleration:ξ=[0.5, 2], f = 0.1. The results for heavy particles with largedensity ratio (Stokes drag only) is also given as reference

Meccanica

Fig. 2(b) show that the largest critical Reynolds num-ber is found when the stability Stokes number SRωr ≈1, where the stability Stokes number is defined as theStokes number times the period of the disturbancewave. The stability Stokes number therefore, is the ra-tio between the dimensionless, advective particle re-laxation time and the timescale of the disturbance.

To gain further understanding on the modal sta-bility of particle-laden flows, we examine the criti-cal Reynolds number as a function of density ratio inFig. 3(a). The mass fraction is prescribed together withthe dimensionless particle size and we can clearly dis-tinguish the effects of both parameters: (i) The massfraction influences the values of the maximum criticalReynolds number. (ii) The particle size influences thedensity ratio at which the maximum Reynolds numberis reached. When the particle size increases, the maxi-mum critical Reynolds number is shifted to lighter par-ticles. The latter finding can be explained by the defi-

nition of S = 29

r2

L21ξ

, which is a function of the particlesize and density ratio. The largest stabilization is in-deed found for stability Stokes number of order one,as shown previously.

In Fig. 3(b) we display the critical Reynolds num-ber versus the dimensionless particle size, r/L. If weconsider small particles, the critical Reynolds num-ber is not affected by the inclusion of particles. Whenparticle size increases, we see the increased criti-cal Reynolds number for values in the range r/L =0.001 < r/L < 0.01. The size at which the criticalReynolds number is maximum is related to the densityratio: the particle relaxation time S depends linearlyon ξ = Φ/f . Note that the data for heavy particles areobtained using Stokes drag only. Fixing the same vol-ume and mass fraction as for light particles ξ = Φ/f ,the size is defined by r/L = √

9/2Sξ as a function ofdimensionless relaxation time S. This definition alsoexplains why small particles hardly have an effect. Forsmall r/L, also S is small and we have seen before(Fig. 2) that at small dimensionless relaxation timesparticles do not affect modal stability.

The effect of added mass and fluid acceleration onthe critical Reynolds is examined in Fig. 4. The re-sults in the figure are obtained using Stokes drag andeither added mass or fluid acceleration. The addedmass term shows the same trend as obtained withStokes drag only: the critical Reynolds number in-creases, see Fig. 4(a). When the particles becomeheavier, the stability curve computed with added mass

Fig. 3 Critical Reynolds number as a function of the density ra-tio ξ for two mass fractions and particle sizes (a) and the criticalReynolds number as a function of particle size r/L for two massfractions and Φ = 0.01 (b). In (b) we report also the results forheavy particles with large density ratio when only Stokes dragis used

overlaps with that for heavy particles. The figure indi-cates that the lighter the particle, the lower the criticalReynolds number when considering only added mass.Fig. 4(b) shows the results obtained considering fluidacceleration and Stokes drag. For ξ = 0.1 the criticalReynolds number is very close to that for heavy parti-cles. For larger values of ξ , the critical Reynolds num-ber decreases. Particles lighter than the fluid destabi-lize the flow and reduce the critical Reynolds number,as shown for ξ = 2.

The results obtained including the fluid accelera-tion force are similar to those shown before when allterms are used. Thus, the fluid acceleration is the dom-inant force in the system. The fluid acceleration forceis proportional to Dui

Dtwhereas the Added Mass to

(Dui

Dt− dupi

dt). The fact that the fluid acceleration force

is the most dominant of the two indicates that Dui

Dtand

dupi

dttend to be of the same order for light particles.

Meccanica

Fig. 4 Critical Reynolds number vs. relaxation time S for par-ticles of density ratio ξ=[0.1, 0.5, 2] and mass fraction f = 0.1.As reference, we display the results for clean flow and heavyparticles (Stokes drag only). In (a) Stokes drag and added mass,in (b) Stokes drag and fluid acceleration are used as interactionterms

3.2 Non-modal analysis

Results of the non-modal analysis are presented inFig. 5. The only cases presented are those denotedfluid → fluid and all → all. The first case investigatesthe transient growth of the fluid energy when the initialcondition consists of fluid perturbation only; the opti-mal growth of the fluid disturbance velocity is studied.The particles are in the system and can gain energy,but this is not apparent. In the case all → all, the to-tal energy of the two-phase system is investigated andparticles may have some initial disturbance velocity. InFig. 5(a), we display the optimal growth as a functionof spanwise wave-number β , the streamwise wave-number α is set to zero. The data represent the max-imum over the final optimization time. The computa-tions are performed for 2D waves as well; since thetransient growth of these streamwise waves is two or-

Fig. 5 (a) Optimal growth (Gmax ) as a function of spanwisewave-number β at S = 5 × 10−5, f = 0.1 and R = 2000.The cases fluid → fluid and all → all are presented forξ = [0.1,0.5,2]. (b) Optimal growth as a function of ξ us-ing β = 2 at S = 5 × 10−5, r/L = 0.01 and R = 2000 forΦ = [0.005,0.01,0.02]

ders smaller than for spanwise waves these results willbe presented later. The density ratios under consider-ation are those used also in the modal analysis, ξ =[0.1, 0.5, 2]. For all → all we see an increase of theoptimal growth by a factor (1+f −Φ)2. The increasecan be explained by using the modified Reynolds num-ber Rm. The optimal growth is proportional to R2,and with the modified Reynolds number defined previ-ously, we see that the optimal growth therefore shouldincrease by (1 + f − Φ)2. For fluid → fluid we findtransient growth smaller than for a clean flow. This iscaused by the fact that there is less fluid per unit vol-ume, (1 −Φ), part of the fluid volume is taken by par-ticles. Thus, a modified Reynolds number based on thefluid density alone is smaller and therefore the distur-bance growth is also smaller. If we include the den-sity variations in the system in a modified Reynoldsnumber, the results with all extra forces included are

Meccanica

similar to those for heavy particles, namely there isno significant effect of particles on the streak transientgrowth. In the second figure, Fig. 5(b), the particlesize and total volume is set, while the density ratio ischanged along the horizontal axis. Note that the massfraction is a function of density ratio and volume frac-tion, f = Φ/ξ . For small values of ξ , the mass frac-tion is large and thus the growth for the case all → allis larger. This effect is the same as in Fig. 5(a): theoptimal growth increases by (1 + f − Φ)2. The casefluid → fluid hardly changes: as the volume fraction isfixed and the growth scales as (1 − Φ), particles havenegligible effect on the amplification of the fluid ki-netic energy.

The effect of the added mass and fluid accelera-tion on the streak transient growth is further inves-tigated for several values of S using the spanwisewavenumber β = 2, where the optimal growth is atthe maximum. These results can be found in Fig. 6.First we note in 6(a) that for the cases all → all themaximum possible amplification starts to diverge forS > 1 × 10−2; this is due to the decoupling of particleand fluid behavior at high Stokes number, discussedfor heavy particles in [10]. This indicates that the twoadditional forcing terms present in the case of lighterparticles do not influence the growth at large values ofS and therefore the results are consistent with those forheavy particles.

As noted above, the energy amplification is largerwhen considering both fluid and particle energy (all →all) than for fluid alone (fluid → fluid). Figure 6(a)shows that at fixed mass fraction, the energy growthfor the fluid → fluid case is almost equal to that forclean fluid and independent of the particle density ra-tio. Only at very large values of S the optimal growthdeviates a few percent in the presence of particles. Fig-ure 6(b) confirms that at fixed volume fraction andparticle size, the maximum growth for all → all goesas (1 + f − Φ)2. This could be seen also by plottingG/(1 + f − Φ)2 instead of G. Note that in the figurethe largest values of f occur at large S.

To investigate the structures of the disturbances,we display the optimal initial condition and responsein Fig. 7 for the case fluid → fluid, S = 1 × 10−3.The initial condition for the fluid consists of vortices,which can be seen in (a). The particle disturbance ve-locities are all zero. This results in streaks at the fi-nal time, where the streamwise disturbance velocity u

is larger than the spanwise and wall-normal velocities

Fig. 6 Optimal growth (Gmax ) as a function of S, us-ing β = 2, R = 2000 and (a) f = 0.1, ξ = [0.1,0.5,2]and (b) Φ = [0.005,0.01,0.02], r/L = 0.01. The casesfluid → fluid and all → all are presented

present. The velocity field of particles and fluid at fi-nal time are almost identical to each other and to thecase of clean fluid (not shown here). Since differencesin velocity induce a loss in energy, particles and fluidhave similar velocity in optimal configurations; lightparticles do not induce any additional gain or loss indisturbance energy.

We therefore demonstrated how light particles havelittle effect on the non-modal stability of spanwisewaves, confirming the results previously obtained forheavy particles. We now investigate streamwise andoblique disturbances. Even though these disturbancesgrow less than purely spanwise disturbances, theyare important for transition in real configurations,e.g. when transition is initiated by localized distur-bances. The effect of particles on these is thereforeworth investigation. Figure 8 shows the results for per-turbations with several streamwise α and spanwisewavenumbers β . When both are nonzero, the wavesare oblique. The values of the maximum possible am-

Meccanica

Fig. 7 Initial condition (a) and optimal response (b) forS = 1 × 10−3. Case fluid → fluid with ξ = 2, β = 2, R = 2000and f = 0.1

plification for a clean fluid flow are reported in the ta-ble, whereas the curves in the figure display the rel-ative variation with respect to the case of clean fluid.We first consider small values of the dimensional re-laxation time SR: for two-dimensional waves (α = 1,β = 0), particles induce a decrease in the transientgrowth. In our previous paper on heavy particles [10],we have shown that this reductions scales as (1 + f ).Streaky modes with (α = 0, β = 2), conversely, dis-play no difference between laden and unladen flow inthe case of heavy particles, ξ = 0.001. When the den-sity ratio increases, the growth is reduced by a factor(1−Φ)2 for the case fluid → fluid in the figure, and asshown above in Fig. 5. Note however, that an increaseof the streaks transient growth by a factor (1+f −Φ)2

would be observed when studying the total energy inthe system. For oblique waves, the results lie in be-tween these two extremes. For slightly oblique waves(β = 2, α = 0.1), we observe the lowest relative lossof energy growth. The magnitude of this loss increasesfor larger values of α.

Fig. 8 Transient growth of oblique waves normalized with thegrowth rate of a clean flow as a function of SR for fluid → fluidand f = 0.1 for ξ = [0.001, 0.1, 0.5, 2]. The amplificationvalues for a single phase fluid are reported in the table

Meccanica

We now consider the flow behavior at intermedi-ate values of the Stokes number, cf. Fig. 8. A cleardecrease of the transient energy growth relative to thecase of single phase fluid is apparent for a range of par-ticle relaxation times. This occurs at lower SR for two-dimensional disturbances and at larger values of SRfor disturbances approaching spanwise streaky waves.This stabilization is observed for values of the stabil-ity Stokes number, defined by the ratio between theparticle relaxation time and the final time giving thelargest amplification, of order one. Therefore, the re-duction of the amplitude of oblique disturbances is ob-served at lower Stokes number for modes with β = 0for which the transient growth is faster, and at higherStokes number for modes with α = 0 for which thetransient growth is a slower process. This stabilizationis the counterpart of the increase of critical Reynoldsnumber shown by the modal analysis. Note however,as discussed before, that the values of the Stokes num-ber necessary to observe this stabilization can be re-lated to parameters outside the range of validity of ourmodel, especially for large S and therefore waves withβ ≈ 0. Finally, When further increasing the particle re-laxation time, we observe an increase of the transientgrowth (values larger than 1 in the figure). Again, wenote that this is however happening for particle sizetoo large for our model to be valid.

4 Basset history force

In this section we include the Basset history term inthe set of equations. First we show how this term isimplemented and in the second part we discuss resultsfor both heavy and light particles.

4.1 Numerical implementation

The Basset history term has not been considered pre-viously for linear stability computations. The historyterm is different from the terms discussed so far as itappears as a convolution integral, where the total his-tory of the fluid and particle flow has to be known. ForDirect Numerical Simulations this is expensive andtherefore only the recent history is usually considered.Sometimes a model to account for large times is used[40]. For stability analysis, the integral is also difficultto deal with and is therefore rewritten so that it can beimplemented in a linear system.

The Basset history term is expressed as a convolu-tion:

q̄i (t) = 1

Sb·∫ t

−∞dτ

d/dτ {upi− ui}

[t − τ ]1/2

= 1

Sb·∫ t

−∞F(t − τ)q(τ )dτ, (23)

with i = 1 − 3 and where we approximate∫ t

−∞ F(t −τ)dτ by an exponential; having an exponential filter

q̄i (t) =∫ t

−∞C exp

(− t − τ

�

)q(τ)dτ, (24)

we can use the differential form:

dq̄i(t)

dt= Cq − q̄

�. (25)

In this way, we are able to solve an eigenvalue prob-lem similar to that without Basset history term, onlyextended with the three extra equations for q̄i , for-mally similar to Eq. (25). The matrix representationof this system is given in the Appendix. Sb is a newdimensionless number, similar to the Stokes numberSR: Sb = SRLπ1/2

rR1/2 .Next, we need to approximate the square root de-

pendence of the Basset term with an exponential. Fig-ure 9 shows the difficulty with the approximation: ei-ther the exponential is more accurate for small times,or for larger times. Several combinations of C and� are tested to investigate whether any combinationchanges the critical Reynolds number or the transientenergy growth in a significant way.

Fig. 9 The approximation of a 1/√

(t) by an exponential in theform C exp(−t/�)

Meccanica

Fig. 10 Basset History with heavy particle approximation (onlyStokes drag as interaction term apart from the Basset Historyterm), f = 0.1 and ξ = 0.001. (a) Critical Reynolds number asa function of S. (b) Maximum transient growth as a function ofS with R = 2000

4.2 Results

The Basset history term has not been implemented instability of a particle laden flow before. Therefore weshow first the results without added mass and fluid ac-celeration: the heavy particle approximation, and theninclude these terms. In both cases, we investigate theinfluence of the Basset history on both modal and non-modal stability.

For heavy particles, only a small difference isfound when the history term is used, Fig. 10(a): Thecritical Reynolds number is decreased a few per-cent using C = 3,� = 1. In non-modal analysis,Fig. 10(b), hardly anything changes for all filter pa-rameters investigated. This could be expected due to

the definition of SR and Sb. Sb is Lπ1/2

rR1/2 times largerthan the Stokes number, thus 1/Sb is much smallerthan 1/SR when ξ = 0.001. If we consider SR = 1,R = 2000, the size fixed at r/L = 0.0015, Sb isabout 30 times larger than SR; thus the Basset his-

Fig. 11 Basset History with light particles with f = 0.1.(a) Critical Reynolds number as a function of S. (b) Maximumtransient growth as a function of SR with R = 2000

tory term is less important than Stokes drag in thisregime.

When light particles are considered, the Basset his-tory term might have an effect because the density ra-tio changes to ξ ∼ 1. However, we show in Fig. 11(a)that the history term has a small effect on the criticalReynolds number. For both ξ = 2 and ξ = 0.5, the crit-ical Reynolds number varies: The Basset history forceslightly decreases the critical Reynolds number. Themodel used for the Basset history in this case assumes� = 1 and C = 3.

Figure 11(b) shows the transient growth for lightparticles both with and without the Basset historyterm. In the figure only one set of filter parametersis used (� = 1,C = 3), more data with different fil-ters have been tested, but no significant difference isfound.

These results imply that the Basset history affectsboth modal and non-modal stability for light particlesin a negligible way.

Meccanica

5 Discussion and conclusions

We presented results for linear modal and non-modalstability of channel flow laden with particles of vary-ing density.

Concerning modal stability, the presence of lightparticles changes the behavior of single-phase flowand flows seeded with heavy particles. Particles lighterthan the surrounding fluid decrease the criticalReynolds number, whereas particles heavier than thefluid increase the critical Reynolds number. The de-crease of the critical Reynolds number is due to thefluid acceleration term. When the fluid accelerationterm is not used, but added mass and Stokes dragare, particles do increase the critical Reynolds num-ber. The latter two interaction terms are proportional to

(Dui

Dt− dupi

dt) and (ui − upi

) respectively, whereas the

fluid acceleration is proportional to Dui

Dt. This shows

that particle and fluid velocity and their derivatives areof the same order and that therefore added mass andStokes drag have less influence in the case of light par-ticles. When particles are heavier, for a density ratio ofξ = 0.1, the results are similar to those obtained withStokes drag alone, see Fig. 4.

The Reynolds number used in the modal-analysis isthe modified Reynolds number as defined in Sect. 2.1.The increase in density due to the addition of particlesis incorporated in this modified Reynolds number.

Non-modal analysis shows a more limited effect ofparticles. As in a clean fluid, streamwise-independentmodes can undergo large non-modal growth; this is as-sociated to the amplification of streamwise velocitystreaks induced by counter-rotating streamwise vor-tices. The mechanisms and flow structures are equalto those observed in a single phase fluid. The interac-tion terms added do appear in the energy budget but donot affect the whole process; this can be explained bythe long time scale associated to the streak lift-up thatallows particles to follow the fluid structures. Table 2shows the optimal growth as function of mass fractionfor particle laden flows versus clean fluid for spanwisewaves at small values of S. The shift by (1 + f − Φ)2

can be explained by the increased density of the totalsystem: optimal growth is still proportional to R2 withR based on the global suspension density. The optimalcondition is that particles and fluid have equal veloc-ities in order to minimize energy losses. Independentof which interaction forces one takes into account, theoptimal growth is equal up to S = 10−2, see Fig. 5. At

Table 2 Dependence of the optimal growth with the mass frac-tion f for streamwise independent disturbances

Case Dependence

all → all (1 + f − Φ)2

fl → fl (1 − Φ)2

larger values of S, the equations for fluid and particlesdecouple and the particle energy can grow infinitely inthe linear model, because the only dissipative terms inthe particle momentum equation stems from the forcesexchanged with the fluid. Limitations in size howevershow that these results lie outside the validity of thepresent model.

When we look at streamwise-dependent distur-bances, for which modal analysis reveals an increaseof the critical Reynolds number, the non-modal re-sponse is smaller when compared to that in a cleanfluid flow. We recall that the transient growth is atleast one order of magnitude larger for spanwise-periodic streamwise-independent modes. However,oblique modes are important in the case of localizedinitial disturbances and at the non-linear stages of tran-sition and thus stabilization of oblique waves can havean impact on transition as shown for the case of heavyparticles. [11]

The extra interaction terms have no influence on thenon-modal stability and this appears less obvious thanfor heavy particles. [10] The energy analysis of heavyparticles shows that particle-fluid interaction alwaysinduces a loss in kinetic energy. To optimize energy,both the fluid and particles should have equal veloci-ties, which then results in an (1 + f − Φ)2 increase ofthe energy gain when the particles act as passive trac-ers and increase the fluid density. In non-modal growththe particles act basically as passive tracer up to largevalues of S because transient growth is a slow process.Therefore, the stability Stokes number is small indicat-ing that particles react to the flow faster than the timeneeded for the streaks to grow. These apply to bothlight and heavy particles.

Finally, we have used a differential form of the ap-proximated kernel and show that the Basset historyforce has a negligible effect for both heavy and lightparticles.

Considering also the first part of this investigation[10], we see that the presence of a solid phase has nosignificant effect on the non-modal growth responsible

Meccanica

for subcritical transition in channel flows, especiallyin the case of elongated structures. As a next step,both particle-particle interactions and finite-size par-ticles should be considered. For finite-size particles,we expect significant effects also for neutrally buoyantparticles, see for example Matas et al. [31] and Picanoet al. [41]. Also, a non-linear model should be used toinvestigate the effect of particles on secondary insta-bilities and final breakdown.

Appendix: Matrix operators for the eigenvalueproblem

The equations for the particle laden flow can be writtenin compact form as follows:

Md

dtq = Lq, (26)

with q = [v, η, up, vp, wp]′ without Basset Historyforce Eqs. ((8), (13) and (14)) and

q = [v, η, up, vp, wp, q1, q2, q3]′

with the Basset history force (Eqs. (8), (13) and (14)with the q̄ defined in Eq. (25)).

The matrices M and L without the history force aredefined in Eq. (27), with all the terms for velocity v

given in Eqs. ((28)–(35)).For the inclusion of the Basset history force, we

do not solve any integro-differential equations. Theintegral associated to the Basset history term hasbeen replaced by 3 new unknowns: [q1, q2, q3], seeEq. (36) where the matrices are given including theBasset history force. In here I is the identity ma-

trix and c = C ∗ Sb with Sb = SRLπ1/2

rR1/2 the dimen-sionless variable for the Basset history term. C and� are constants used to approximate the differen-tial form of the integral in the Basset history force.

⎛⎜⎜⎜⎜⎜⎝

Mvv 0 MvU MvV MvW

0 Mηη MηU 0 MηW

MUv MUη MUU 0 0

MVv 0 0 MVV 0

MWv MWη 0 0 MWW

⎞⎟⎟⎟⎟⎟⎠

︸ ︷︷ ︸M

d

dt

⎛⎜⎜⎜⎜⎜⎝

vη

up

vp

wp

⎞⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎝

Lvv 0 LvU LvV LvW

Lηv Lηη LηU LηV LηW

LUv LUη LUU LUV 0

LVv 0 0 LVV 0

LWv LWη 0 0 LWW

⎞⎟⎟⎟⎟⎟⎠

︸ ︷︷ ︸L

⎛⎜⎜⎜⎜⎜⎝

vη

up

vp

wp

⎞⎟⎟⎟⎟⎟⎠

(27)

Mvv =(

1 + 1

2f ξ

)∇2 (28)

MvU = 1

2f ξiαD (29)

MvV = 1

2f ξk2 (30)

MvW = 1

2f ξiβD (31)

Lvv = −(

1 + 1

2f ξ

)(iαU∇2 − iαU ′′) − f

SR∇2 + 1

R∇4 (32)

LvU = −iαf

SRD − 1

2f ξUiαiαD − 1

2f ξU ′iαiα (33)

LvV = − f

SRk2D − 1

2f ξUiαk2D − 1

2f ξU ′iαD − 1

2f ξU ′′iα (34)

LvW = −iβf

SRD − 1

2f ξUiαiβD − 1

2f ξU ′iαiβ (35)

Meccanica

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Mvv 0 MvU MvV MvW 0 0 00 Mηη MηU 0 MηW 0 0 0

MUv MUη MUU 0 0 0 0 0MVv 0 0 MVV 0 0 0 0MWv MWη 0 0 MWW 0 0 0c iα

k2 D −ciβ

k2 D −cI 0 0 I 0 0cI 0 0 −cI 0 0 I 0

ciβ

k2 D c iα

k2 D 0 0 −cI 0 0 I

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸ ︷︷ ︸M

d

dt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vη

up

vp

wp

q1q2q3

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Lvv 0 LvU LvV LvW f ciαD f ck2I f ciβD

Lηv Lηη LηU LηV LηW −f ciβI 0 f ciαI

LUv LUη LUU LUV 0 −cI 0 0LVv 0 0 LVV 0 0 −cI 0LWv LWη 0 0 LWW 0 0 −cI

−U ′ + U α2

k2 D −Uαβ

k2 iαU 0 0 −1/�I 0 0−iαU 0 0 iαU 0 0 −1/�I 0

Uαβ

k2 D U α2

k2 0 0 iαU 0 0 −1/�I

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸ ︷︷ ︸L

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vη

up

vp

wp

q1q2q3

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(36)

Fig. 12 Spectrum for the most unstable eigenmode forR = 3000, S = 5e − 3, ξ = 0.1 and f = 0.1 with (red cross)and without (blue circle) Basset history term (Color figure on-line)

The eigenvalue spectra with and without Basset his-tory term are similar to each other, as can be seen inFig. 12 and discussed in the paper.

References

1. Toschi F, Bodenschatz E (2009) Lagrangian properties ofparticles in turbulence. Annu Rev Fluid Mech 41:375–404

2. Balachander S, Eaton JohnK (2010) Turbulent dispersedmultiphase flow. Annu Rev Fluid Mech 42:111–133

3. Sproull WT (1961) Viscosity of Dusty Gases. Nature190:976–978

4. Rossetti SJ, Pfeffer R (1972) Drag reduction in dilute flow-ing gas-solid suspensions. AIChE J 18(1):31–39

5. McCormick ME, Bhattacharyya R (1973) Drag reduc-tion of a submersible hull by electrolysis. Naval Eng. J.85(2):11–16

6. Jacob B, Olivieri A, Miozzi M, Campana EF, Piva R (2010)Drag reduction by microbubbles in a turbulent boundarylayer. Phys Fluids 22(11):115104

7. Ferrante A, Elghobashi S (2003) On the physical mecha-nisms of two-way coupling in particle-laden isotropic tur-bulence. Phys Fluids 15:315–329

8. Xu J, Maxey MR, Karniadakis GEM (2002) Numericalsimulation of turbulent drag reduction using micro-bubbles.J Fluid Mech 468:271–281

9. Zhao LH, Andersson HI, Gillissen JJJ (2010) Turbulencemodulation and drag reduction by spherical particles. PhysFluids 22:081702

10. Klinkenberg J, de Lange HC, Brandt L (2011) Modal andnon-modal stability of particle-laden channel flow. PhysFluids 23:064110

11. Klinkenberg J, Sardina G, de Lange HC, Brandt L (2013)Numerical study of laminar-turbulent transition in particle-laden channel flow. Phys Rev E 87:043011

12. Saffman PG (1962) On the stability of laminar flow of adusty gas. J Fluid Mech 13:120–128

13. Michael DH (1964) The stability of plane Poiseuille flowof a dusty gas. J Fluid Mech 18:19–32

14. Rudyak VYa, Isakov EB, Bord EG (1997) Hydrodynamicstability of the poiseuille flow of dispersed fluid. J AerosolSci 82:53–66

15. Asmolov ES, Manuilovich SV (1998) Stability of a dusty-gas laminar boundary layer on a flat plate. J Fluid Mech365:137–170

Meccanica

16. Ellingsen T, Palm E (1975) Stability of linear flow. PhysFluids 18:487–488

17. Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA(1993) Hydrodynamic stability without eigenvalues. Sci-ence 261:578–584

18. Reddy SC, Henningson DS (1993) Energy growth in vis-cous channel flows. J Fluid Mech 252:209–238

19. Schmid PJ, Henningson DS (2001) Stability and transitionin shear flows. Springer, Berlin

20. Maxey MR, Riley JJ (1983) Equation of motion for a smallrigid sphere in a nonuniform flow. Phys Fluids 26:883–889

21. Tchen CM (1947) Mean value and correlation problemsconnected with the motion of small particles suspended ina turbulent fluid. PhD thesis, Delft

22. Corrsin S, Lumley J (1956) On the equation of motion fora particle in turbulent fluid. Appl Sci Res A 6:114–116

23. Saffman PG (1992) Vortex dynamics. Cambridge UnivPress, Cambridge

24. Dandy DS, Dwyer HA (1990) A sphere in shear flow atfinite Reynolds number: effect of shear on particle lift, drag,and heat transfer. J Fluid Mech 216:381–400

25. Mei R (1992) An approximate expression for the shear liftforce on a spherical particle at finite Reynolds number. IntJ Multiph Flow 18:145–147

26. McLaughlin JB (1991) Inertial migration of a small spherein linear shear flows. J Fluid Mech 224:262–274

27. Boronin SA, Osiptsov AN (2008) Stability of a Disperse-Mixture Flow in a Boundary Layer. Fluid Dyn 43:66–76

28. Vreman AW (2007) Macroscopic theory of multicompo-nent flows: Irreversibility and well-posed equations. Phys-ica D 225:94–111

29. Boronin SA (2008) Investigation of the stability of a plane-channel suspension flow with account for finite particle vol-ume fraction. Fluid Dyn 43:873–884

30. Calzavarini E, Volk R, Bourgoin M, Lévêque E, Pinton J-F,Toschi F (2009) Acceleration statistics of finite-sized par-

ticles in turbulent flow: the role of Faxén forces. J FluidMech 630:179–189

31. Matas JP, Morris JF, Guazzelli É (2003) Transitionto turubulence in particulate pipe flow. Phys Rev Lett90:014501

32. Matas JP, Morris JF, Guazzelli É (2003) Influence of par-ticles on the transition to turbulence in pipe flow. PhilosTrans R Soc Lond A 361:911–919

33. Boffetta G, Celani A, De Lillo F, Musacchio S (2007) TheEulerian description of dilute collisionless suspension. Eu-rophys Lett 78:14001

34. Meiburg E, Wallner E, Pagella A, Riaz A, Härtel C, NeckerF (2000) Vorticity dynamics of dilute two-way-coupledparticle-laden mixing layers. J Fluid Mech 421:185–227

35. Elghobashi S (1994) On predicting particle-laden turbulentflows. Appl Sci Res 52:309–329

36. Ferry J, Balachander S (2001) A fast Eulerian method fordisperse two-phase flow. Int J Multiph Flow 27:1199–1226

37. van der Hoef Van MA, Annaland S, Deen NG, KuipersJAM (2008) Numerical simulation of dense gas-solid flu-idized beds: a multiscale modeling strategy. Annu RevFluid Mech 40(1):47–70

38. Guazzelli É, Morris JF (2011) A physical introduction tosuspension dynamics. Cambridge University Press, Cam-bridge

39. Reddy SC, Schmid PJ, Baggett JS, Henningson DS (1998)On the stability of streamwise streaks and transition thresh-olds in plane channel flows. J Fluid Mech 365:269–303

40. van Hinsberg MAT, ten Thije Boonkkamp JHM, ClercxHJH (2011) An efficient, second order method for the ap-proximation of the basset history force. J Comp Physiol230(4):1465–1478

41. Picano F, Breugem W-P, Mitra D, Brandt L (2013) Shearthickening in non-brownian suspensions: an excluded vol-ume effect. Phys Rev Lett 111:098302