Stability of Fluids withShear-Dependent Viscosity in the

Lid-Driven Cavity

Simon Haque

Thesis Submitted to The Royal Institute of Technologyfor the Master’s Degree

Supervisor:Luca Brandt

August 26, 2011

1

Abstract

The classical problem of the lid-driven cavity extended infinitely inthe spanwise direction is considered for non-Newtonian shear-thinningand shear-thickening fluids where the viscosity is modeled by the Carraeu-Yasuda model. Linear stability is used to determine the critical Reynoldsnumber at which the two-dimensional base-flow becomes unstable tothree-dimensional spanwise-periodic disturbances. We consider a squarecavity, characterized by steady unstable modes, and a shallow cavityof aspect ratio 0.25, where oscillating modes are the first to becomeunstable for Newtonian fluids. In both cases, the critical Reynoldsnumber first decreases with decreasing power-index n (from shear-thickening to shear-thinning fluids) and then increase again for highlypseudoplastic fluids. In the latter case, this is explained by the thinnerboundary layers at the cavity walls and less intense vorticity inside thedomain. Interestingly, oscillating modes are found at critical condi-tions for shear-thickening fluids in a square cavity while the shallowcavity supports a new instability of lower frequency for large enoughshear-thinning. Analysis of kinetic energy budgets and structural sen-sitivity are employed to investigate the physical mechanisms behindthe instability.

2

1 Introduction

1.1 The Lid-Driven Cavity

The incompressible and Newtonian flow inside a lid-driven cavity is one ofthe most studied fluid mechanics problem since the establishment of compu-tational fluid dynamics (CFD). It describes the flow inside a rectangular boxdue to the tangential translation of one side wall. Much attention has beendrawn to the cavity flow, both from an industrial and academic standpoint.For example, practical applications can be found in abundance in coatingand mixing devices [1]. The geometric simplicity of the problem makes itideal for numerical discretization and computations, while the discontinuousmathematical boundary condition for the velocity leads to singular proper-ties close to the corners of the moving lid at the two neighboring stationarywalls. These features has made the lid-driven cavity a popular method ofverification for Navier-Stokes solvers [2] and different numerical techniques.Furthermore, this flow has interesting behavior from the point of view ofinstabilities and bifurcations in closed system behavior.

Previous work have been conducted both numerically and experimen-tally. Burggraf [3] aimed to investigate the Prandtl-Batchelor theorem fora square cavity. He used a relaxation method to compute solutions forReynolds number (Re) between 0 and 400. The results suggests that forhigher Re an inviscid core vortex is formed, but secondary eddies developnear the bottom corners of the square for all Re.

Later, Pan and Acrivos [4] also used relaxation techniques to find creep-ing flow solutions close to bottom corners of the cavity for aspect ratios, Γ,between 0.25 and 5 where Γ is defined as the ratio between the depth of thecavity and the length of the moving lid. Moreover, through experiments,they studied the flow for Re ranging from 20 to 4000 and concluded that forfinite cavities and as Re→∞ a single inviscid core will form and the cornereddies reduces significantly in size. For infinitely deep cavities the size ofthe primary vortex hinders the core from becoming fully inviscid even as Re→∞ resulting in a balance between viscous and inertial forces in the cavity.The main assumption in [4] is two-dimensional flow.

Ghia et al. [5] and Schreiber and Keller [6] focused on numerical methodswhich they tested on the two-dimensional cavity with Γ= 1 for Re up to10000. Both studies indicate that at such high velocities the position of thecore vortex moves toward the center of the square and that the bottom left(BL) and right (BR) eddies grow in size. The BR is often referred to as thedown stream secondary eddy (DSE).

Extensive three-dimensional work was performed by Koseff and Streettogether with others [7]-[8]. They carried out experiments with spanwiseaspect ratio, Λ, between 1 and 3 where Λ is defined as the ratio betweenthe width and length of the moving lid. The results showed that the flow

3

becomes turbulent at an Re in the interval 6000-8000 and that fluid motionin the spanwise direction is considerable [7]. In subsequents investigationsthey also established local three-dimensionality as corner vortices at the endwalls are formed as well as longitudinal Taylor-Grtler-like (TGL) vorticesdue to instabilities close to the DSE [9], [10]. The experimental results werelater used to confirm numerical result obtained by the same research group[8]. The main conclusion that can be drawn from the work of Koseff et al.is that two-dimensional flow (for Λ up to 3 ) at high Re cannot occur, it isstrictly fictitious.

Kim and Moin used a time-splitting scheme together with the approximate-factorization technique to investigate the three-dimensional driven cavitywith periodic boundary condition in the transversal (spanwise) direction,i.e. without end wall effects [11]. They added small random perturbationsin the spanwise direction to the two-dimensional solutions and found thata pair of TGL vortices appeared at Re around 900. This result was impor-tant because it proved the flow becomes three-dimensional at high Re evenwithout the presence of end walls in the spanwise direction.

Chiang et al. set out to map the eddy structure as function of the Reup to 1300 [12]. They used a finite volume method on a cavity with squarecross section and Λ = 3. It is shown in the paper that the BL and DSLbecome developed at around Re = 50. At an Re close to 100, the cornereddies adjacent to the lid are formed. As the Re is increased it becomesevident that the two-dimensional nature of the flow undergoes a transitionto a fully three-dimensional character in the form of a bifurcation from thesteady state to an oscillating periodic state.

Additional information can be found in a review paper by Shankar andDeshpande [13].

1.2 Stability

After the work of Koseff, Street and their colleagues much attention hasbeen drawn to a new aspect of the lid-driven cavity, namely stability. Sinceit is generally accepted that the flow goes through a transition a from two-dimensional to a three-dimensional state, research have been conducted tofind at what Re this conversion occurs. As Albensoeder et al. describesin [2] the flow undergoes symmetry breaking instabilities before becomingturbulent. The task is to find at what Re the first instability takes placewhen increasing the Re from 0. This Re is often referred to as the criticalRe.

An innovative experiment was carried out by Aidun et al. in which asmall amount of through flow was injected into a cavity with Λ = 3 tocompensate for the liquid which was removed by the moving lid [14], [15].They concluded that this addition would not alternate the flow character-istics significantly. Two sets of experiments were set up. In the first one,

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the transition from steady to unsteady state was studied when increasingRe from 100 to 2000. From the results they concluded that the core vortexand DSE is stable up to a Re of around 825, but then small-amplitude time-periodic waves form at the DSE, extending towards the end walls. Theyestimated a critical Re between 825 and 925. In [15] they also found super-critical bifurcations from steady state to a pair of spiral waves at Re = 966with a dimensionless frequency of 0.1112. Upon increasing the Re, travel-ing waves developed for Re between 1000 and 1300 leading to much morecomplex dynamics. As the Re was increased further, non-uniformly sizedand irregular spaced mushroom-like spikes appear. In the second set of ex-periments the velocity was decelerated from high speeds (Re ∼ 2000) to low(Re < 500) and showed that the steady state are not unique.

Ramanan and Homsy performed a numerical linear stability analysis ofa square cavity with periodic boundary condition in the spanwise directionby applying a finite difference approach [16]. Similar to Kim and Moin,they first solved for the two-dimensional base flow and then perturbed itwith three-dimensional disturbances using a uniform mesh of 65×65 gridpoints. The result indicated a Grtler type instability close to the separatingstreamline between the core vortex and the DSE. The flow became unstableat Re ∼ 594 to a stationary mode with a transversal wave number κ ∼ 2.12

Adding compressibility effects to the problem, Ding and Kawahara useda finite element method to calculate the critical Re in a square cavity ofinfinite span [17]. They detected an oscillatory mode with non-dimensionalfrequency ω = 0.08 at Re = 920 for κ = 7.4.

During the last decade, Kuhlmann and coworkers have produced a sig-nificant amount of work on problems of the lid-driven cavity. For example,utilizing a cavity where two lids move in different direction, they were able toshow non-uniqueness in the two-dimensional steady flow [18]. In a followingstudy they employed a finite volume discretization to establish the criticalRe for a cavity with different Γ but with Λ → ∞ [2]. With a non-uniform141×141 grid and by linear stability analysis it was shown that the baseflow becomes unstable to four different three-dimensional modes at higherRe depending on the aspect ratio. For example, the square cavity suffers astationary instability at Re around 786 and a κ of 15.4, hence quite differentfrom results obtained by earlier research. These values were confirmed byexperiments in a cavity with dimensions Γ = 1 and Λ = 6.55 and are nowgenerally considered to be correct. Kuhlmann et al. asserted that this modehad been suppressed by end wall effects (due to short spanwise length) inprevious work.

To the best of the author’s knowledge, very little work concerning sta-bility of non-Newtonian fluids in the lid-driven cavity exists. Pakdel et al.experimentally analyzed two-dimensional cavity flow for elastic liquids [1].With spanwise aspect ratios between 0.25 and 4 they used an ideal elasticfluid and Deborah number (De) ranging from 0 to 35 to conduct their in-

5

vestigation. At low De the flow was two-dimensional and the core vortexmoved upstreams as Re was increased. For low De the flow became unstableand three-dimensional. When the lid motion was stopped a distinct recoileffect caused the direction of the core vortex to reverse.

The aim of this thesis project is to numerically establish bench mark re-sults for non-Newtonian shear-thinning and, to some extent, shear-thickeningfluids in the classical lid-driven cavity, assuming Λ→∞. A two-dimensionalbase flow solution is initially acquired and then perturbed with three-dimensionaldisturbances. Linear stability analysis then leads to an eigenvalue problemwhich is solved using the Arnoldi iteration method.

2 Problem Formulation

2.1 The Geometry

The geometry and generic flow behavior of the lid-driven cavity is depictedin Figure 1. The lid has a width Lx and is moving with a constant velocityV in the x-direction, hence the the Reynolds number Re is defined as:

Re =ρLxV

µ0,

where ρ and µ0 is the density and zero shear rate viscosity, respectively, ofthe fluid. Furthermore, the aspect ratio is given by Γ = Ly/Lx. If a thirdz-direction is added then the spanwise aspect ratio is Λ = Lz/Lx, where Lzis the length of the cavity in the corresponding direction.

Core Vortex

Corner Vortices

V

Ly

Lx

Figure 1: Geometry and notation of the lid-driven cavity.

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2.2 The Viscosity Model

As mentioned, an important aspect of this work which differentiates it frommany earlier investigations is that it concerns Non-Newtonian fluids. Toexamine the non-Newtonian effects, a simple model where viscosity is de-pendent on shear rat is adopted. Fluid elasticity is neglected. Hence, forthe the relation can mathematically be written as:

µ = µ(γ), (1)

where γ is the rate-of-shear tensor and will simply be referred to as γ.

Thus, many empirical relations can be used to fit experimental data. Someexamples, including the Cross and Bingham model can be found in [19].

For this project the so called Carreau-Yasuda model is employed fromwhich the connection between viscosity and rate of strain is given by:

µ(γ) = µ∞ + (µ0 − µ∞)[1 + (γλ)a]n−1a . (2)

Here, µ∞ is the infinite shear rate viscosity. In this work, µ∞ and µ0 areset to 0.001 and 1 respectively. Dividing equation (2) by µ0 yields the non-dimensional viscosity:

µ(γ) =µ∞µ0

+ (1− µ∞µ0

)[1 + (γλ)a]n−1a . (3)

The exponent a is fixed at 2 in this analysis since then the rheological behav-ior of many polymeric solutions can be fitted to the Carreau-Yasuda model[20]. For a more detailed description of the parameters featured in equation(3) see [21].

2.3 Governing Equations

The incompressible Non-Newtonian lid-driven cavity flow is considered. Theflow is induced by the tangential velocity of the top wall in the positive x-direction. All remaining walls are rigid. The cavity length in the transversez-direction is assumed to be infinite. Thus, the flow is governed by theNavier-Stokes and continuity equation which is expressed in non-dimensionalform as:

∂u

∂t+ u · ∇u = −∇p+

1

Re· ∇[µ(∇u +∇uT )] (4a)

∇ · u = 0. (4b)

The no-slip boundary conditions imposed on the problem are the follow-ing:

7

u =

{1 ex at y =

Ly

Lx= Γ (5a)

0 at x = 0, x = 1 and y = 0. (5b)

Note that for equation (5), the origin of the coordinate system has beenplaced in the bottom left corner of Figure 1. Also, the boundary conditionsare dimensionless in accordance with equation (4).

2.4 Linear Stability Analysis

Since the cavity length is presumed infinite in the z-direction, equation (4)has a steady two-dimensional solution [Ub, pb] = [Ub(x, y), pb(x, y)]. A per-turbation [U, p] = [U(x, y, z, t), p(x, y, z, t)] is added to this time indepen-dent state in order to perform the linear stability analysis. Thus, it ispossible to decompose the flow variables into:

u = Ub + U (6a)

p = pb + p. (6b)

Furthermore, a similar formation is introduced for the viscosity:

µ = µb + µ, (7)

where µb = µb(x, y) and µ = µ(x, y, z, t) are the base flow (steady state)and perturbation viscosity respectively. According to [22] the perturbationviscosity is equal to the first term of the Taylor series expansion of (3):

µ = ˙γij(U)∂µ

∂ ˙γij(Ub). (8)

Now, substituting equations (6) and (7) into (4), subtracting the baseflow variables and linearizing around [Ub, pb] yields a linear stability problemfor the perturbations which can be formulated compactly as:

∂U

∂t+ L(Ub, Re)U +∇p = 0 (9a)

∇ · U = 0. (9b)

Above, L(Ub, Re)U is:

L(Ub, Re)U = U · ∇Ub + Ub · ∇U

− 1

Re∇ · [µb(∇U + (∇U)T ) + µ(∇Ub + (∇Ub)

T )].

8

As introduced, for example, by Albensoeder et al. the general solutionto equation (9) can be written as complex modes of the form:

U(x, y, z, t) = U(x, y)exp[σt+ iκz] + compl. conj. (10a)

p(x, y, z, t) = p(x, y)exp[σt+ iκz] + compl. conj., (10b)

where κ is the transverse wavenumber of the disturbance. A two-dimensionalinstability corresponds to κ = 0. Inserting the ansatz (10) into (9) finallyproduces a linearized general eigenvalue problem:

σU + L(Ub, Re)U +∇p = 0 (11a)

∇ · U = 0. (11b)

The complex eigenvalue σ contains information about the growth rate (RE{σ})and frequency (IM{σ}) of the instability. Subsequently, q = (U, p) is theeigenmode. The main task for this analysis is to determine for which Reand κ the growth rate becomes zero for a given Γ, n and λ. These Re and κwill be referred to as the critical Reynolds number (Rec) and wavenumber(κc).

2.5 Structural Sensitivity

Investigating the sensitivity of the unstable modes will give further insightabout the origin of the instability. The theoretical framework is based on thework of Giannetti and Pralits [23] who introduce and define the wavemakerof the instability as the region in space where a change in the structure ofthe problem causes the largest drift in the eigenvalues. Compared to theiranalysis, terms for the perturbation viscosity are added here. Introducing asmall momentum force in the stability equations yields the following eigen-value problem:

σ′U′+ L(Ub, Re)U

′+∇p′ = δH(U

′, p′) (12a)

∇ · U′ = 0. (12b)

As explained in [24], δH is a differential operator representing a force pro-portional to the local perturbation velocity:

δH(U′, p′) = δM(x, y) · U′ = δ(x− x0, y − y0)δM0 · U

′

Here, δM0 is a coupling matrix and δ(x− x0, y − y0) is the Kronecker deltafunction. The eigenvalue and eigenmode drifts are given as the expansion

σ′ = σ + δσ, U′

= U + δU and p′ = p + δp. Inserting these into equation

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(12) and excluding higher order terms yields an equation for the eigenvaluedrift:

σδU + L(Ub, Re)δU +∇δp = −δσU + δM · U (13a)

∇ · δU = 0 (13b)

The Lagrange identity is introduced as a function of the differentiable directfield q = (u, p) and its corresponding adjoint field g+ = (f+,m+). More onadjoint methods can be found in [25] and [26]. By taking the inner productof equation (11) and the adjoint field and using differentiation by parts oneacquire:

[(σU + L(Ub, Re)U +∇p) · f+ + (∇ · U) · m+]

+ [U · (−σf+ + L+(Ub, Re)f+

+∇m+) + p∇f+] = ∇ · J(q, g+), (14)

where J is a bilinear concomitant and L+ is the adjoint operator of thelinearized Navier-Stokes equation.

J(q, g+) = Ub(U · f+

) +1

Re[µb(∇f

++ (∇f+)T ) · U−µb(∇U+ (∇U)T ) · f+

− µ(∇Ub + (∇Ub)T ) · f+] + m+U + pf

+,

and

L+(Ub, Re)f+

= Ub · ∇f+ −∇Ub · f

+

+1

Re[µb(∆f

++ (∆f

+)T ) + (∇Ub + (∇Ub)

T ) · ∇f+B(Ub)].

where B(Ub) is an operator on the form:

B(Ub) =

2 ∂µ∂γ11

(Ub) · ∂∂x + 2 ∂µ∂γ12

(Ub) · ∂∂y

2 ∂µ∂γ21

(Ub) · ∂∂x + 2 ∂µ∂γ22

(Ub) · ∂∂y

Additionally, the adjoint mode g+(x, y) = (f

+, m+) satisfies the following

system of equations:

−σf+ + L+(Ub, Re) +∇m+ = 0 (15a)

∇ · f+ = 0 (15b)

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Considering equations (13) and (14) and integrating over the entire do-main D gives:

−δσ∫

Df+ · UdS +

∫Df+ · δM · UdS =

∮∂D

J(q, g+) · ndl. (16)

The boundary conditions are chosen in such a way that the right hand sideof equation (16) is zero. The sensitivity tensor S is introduced:

S(x0, y0) =f+

(x0, y0)U(x0, y0)∫D f

+ · UdS. (17)

Note that f+U represents a dyadic product between the adjoint and direct

mode. Combining equation (16) and (17) generates the final expression forthe eigenvalue drift:

δσ(x0, y0) =

∫D f

+ · δM · UdS∫D f

+ · UdS

=f+ · δM0 · U∫D f

+ · UdS= S : δM0 = SijδM0ij . (18)

The core of the instability can be found by studying different norms of thetensor S.

2.6 Kinetic Energy Analysis

By performing an energy analysis, additional information about the instabil-ity mechanism can be extracted. Multiplying equation (9a) with the complexconjugate of the perturbation velocity U

∗produces the perturbation kinetic

energy which in index notation can be written as:

U∗i∂Ui∂t

+ U∗i Uj∂Ubi∂xj

+ U∗i Ubj∂Ui∂xj

= −U∗i∂p

∂xi

+2

ReU∗i

∂

∂xj(µbeij) +

2

ReU∗i

∂

∂xj(µEij). (19)

Note here that Ubi and Ubj are the i:th and j:th component of the base flowvelocity Ub. Furthermore, eij and Eij are the perturbation and base flowshear rate tensors respectively:

11

eij =1

2(∂U∗i∂xj

+∂U∗j∂xi

) (20)

Eij =1

2(∂Ubi∂xj

+∂Ubj∂xi

) (21)

Subsequently, the kinetic energy budget is:

d(Ekin)

dt=

∂

∂xj[UbjUiU

∗i +

1

2(U∗j p+Uj p

∗)+1

Reµb(U

∗i eij+Uie

∗ij)+

1

ReEij(U

∗i µ+Uiµ

∗)]

− 1

2(U∗i Uj + UiU

∗j )∂Ubi∂xj

− 2

Reµb(eije

∗ij)−

1

Re(µe∗ijEij + µ∗eijEij) (22)

Here Ekin = 12(UiU

∗i ) is the kinetic energy and the superscript * denotes the

complex conjugate of the corresponding quantity. The first term on the righthand side of equation (22) can be interpreted as the kinetic energy transferthroughout the domain. However since there is no net flux in a closed flowsystem like the cavity, it vanishes. The second and third contribution arethe production and dissipation of the perturbation kinetic energy, whereasthe last expression is an additional term due the non-Newtonian propertiesof the flow. Note that the three dimensional effects only come into play inthe energy dissipation since Ub3 = W = 0 and ∂

∂z = 0 for the base flow Ub.

3 Numerical Method

The numerical computations have been performed using a variant of thesecond order finite difference code developed by Giannetti and Luchini anddescribed in [23]. To begin with the two-dimensional steady base flow iscalculated by discretizing the flow variables on a staggered grid. Equations(4)-(5) are then solved using Newton-Raphson iteration, where the linearequations produced are inverted by means of a sparse LU decomposition.Next, the base flow solution is inserted into the perturbation equations (9)and the linear stability analysis is executed. Eigenvalues and modes of boththe direct and adjoint field are computed by employing the Arnoldi shiftand invert method. For all calculations a shift of 2 + 0i has proved sufficientin order to find correct results (including oscillatory modes). Moreover, aeigenvalue tolerance of 10−8 has been chosen to guarantee converged re-sults. As mentioned above, the instability occurs when the real part of theeigenvalue is larger than zero. A non-zero imaginary part corresponds towave propagating in the z-direction, hence, the perturbation mode is non-stationary. Finally, the sensitivity, and thereby core of the instability, isinvestigated by multiplying the direct and adjoint fields.

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The eigenvalue strongly depends on Re, Γ, κ, the power-law index n andthe time constant λ. The critical Re was found by applying the followingstrategy. Firstly, Γ, n and λ are set. Secondly, a range in κ is selected,often starting from 0 to 20 (the wavenumber is assumed to be less then30 since modes with higher κ are expected to be strongly damped [2]) andan incremental step size Dκ usually initially selected to be 1. Thirdly, abisection method is utilized to find an unstable Re. Two values of Re arechosen, one which is sure to be unstable (Reunstable) and one of which issure to be stable (Restable). Then, the eigenvalues are checked for all κat Re(1) = (Reunstable + Restable)/2. If the real part of all eigenvalues arestable then Restable = Re(1) and the next iteration will start with Re(2) =(Re(1) + Reunstable)/2. However, if an unstable eigenvalue is found for aspecific κ(k) then Reunstable = Re(1) and the next run will start at Re(2) =(Restable+Re(1))/2 and κ will only be scanned from κ = κ(k)−Dκ. A similarclassification of Re(2) is made and so on until Reunstable − Restable ≤ ∆Re

where ∆Re is a specified tolerance. After this, the parameters are refinedwith the intention to narrow down the interval between stable and unstableRe and thus finding the critical value.

3.1 Mesh

The square cavity of adimensional size Lx = Ly = 1 will be mainly in-vestigated in this work. Unless otherwise stated, a mesh size (nx × ny) of250×250 has been chosen for this case. Furthermore, a parabolic mapping isselected with the mesh being stretched by a ratio of 4 towards all boundariesof the geometry in both the x and y direction. Grid points are more denselyclustered at the walls than in the center of the cavity as seen in 2(a). Thisturned out to be necessary to obtain correct results in these high-gradientregions.

Analysis of a shallow cavity with aspect ratio Γ = 0.25 has also beenperformed. The dimensions are therefore set to Lx = 1, Ly = 0.25 wherethe number of grid points used is 300 × 75. Again parabolic stretching isused. Figure 2(b) shows the grid used for these cases. Note that for the sakeof clarity only every fourth point is shown in the figure.

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0 0.5 10

0.5

1

(a)

0 0.5 10

0.125

0.25

(b)

Figure 2: Mesh used square and shallow cavity. (a) Γ = 1 and (b) Γ = 0.25

3.2 Code Validation

The linear stability analysis has been validated by reproducing the resultsfor Newtonian fluid in [2]. It can be seen from Table 1 that the presentresults based on the stretched 150×150 grid are already in good agreement,with an error about or below 1%.

Table 1: Comparison between the critical values presented by Albensoeder et al.(141×141 grid points)[2] and the present results for a square cavity withresolution (150× 150).

Γ ReC , [2] ReC Error % κC , [2] κC Error % ωC , [2] ωC Error %

0.25 1152.7 1165 1.07 20.63 20.6 -0.15 2.27 2.25 -0.821 786.3 789.1 0.36 15.43 15.2 -1.30 0 0 0

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Grid dependence for the non-Newtonian cases is checked by finding thecritical Re and κ when Γ = 1, n = 0.5 and λ = 10 for the several meshsizes, see Table 2. The values reported in the Table confirm that the choiceof grid points yields satisfactory results.

Table 2: Grid independence study. The tables reports the critical values at differ-ent resolution for shear-thinning fluids with n = 0.5 and λ = 10.

Grid Resolution ReC Error % κC Error %

150× 150 330.7 6.23 13.9 -6.08200× 200 317.5 1.99 14.5 -2.03250× 250 (Reference) 311.3 - 14.8 -300× 300 307.5 -1.22 14.8 0

4 Results

In this section we examine the stability of the square and shallow cavity. Re-sults are presented for values of the power-law index between 0.4 < n < 1.4and time constant λ =1, 10 and 100. As already mentioned, the remainingconstants in the constitutive equation (3) are set to a = 2, µ∞ = 0.001 andµ0 = 1. When n < 1, the fluid is shear thinning and the viscosity decreaseswith the strain rate. The opposite is true for shear-thickening fluids definedby values n > 1, while Newtonian fluids are retrieved when n = 1 and theviscosity becomes independent of the strain rate.

4.1 The Square Cavity, Γ = 1

The base-flow streamlines computed by Newton iterations has been initiallycompared with solutions obtained by direct numerical simulations (DNS)with the code Nek5000 [27]; the data reported in Figure 3 reveal good agree-ment. In the figure, we report as example the streamlines for shear-thinningfluid with n = 0.4, λ = 10 and Re = 456.

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Nekton code

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

CPL code

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

Figure 3: Comparison of the nonlinear base flow computed with direct numericalsimulations (Nekton code) and Newton iterations (CPL code) for n =0.4, λ = 10 and Re = 456. The contours indicates streamlines.

Once the numerical calcualtions of the base flow is further validated,we proceed to the stability analysis. The neutral curve (critical Re vs. n)is reported in Figure 4(a) for the square cavity and different values of theparameter λ. Shear-thickening effects induce a significant increase of ReC ,this effect being more pronounced for larger values of λ. The opposite applieswhen n < 1; in this case however the critical Re first decreases and thenincreases when increasing the shear-thinning properties (decreasing n).

The viscosity varies locally inside the fluid and one can therefore definea local Reynolds number

Reloc =ρLxV

µ.

This is displayed in Figure 5 at critical condition for 4 different values ofthe power index n. The plots reveal that the region of largest Reloc, lowestviscosity, are located at the centre of the cavity and at the lower cornersfor shear-thickening fluids and for values of n close to 1, whereas they arelocated close to the upper wall and in a ring all around the cavity whenthe fluid is characterized by significant shear-thinning, n < 0.6. It is alsointeresting to consider the average Reynolds number Reavg, defined as

Reavg =

∫Relocdxdy

A(23)

where Reloc is defined above and A is the total area of the cavity. Again,µ(γ) is given by equation (3). This average Reynolds number is shownin Figure 4(b) versus the index n. The large decrease of the critical Rewhen decreasing n is significantly reduced when considering Reavg instead.As discussed below, this suggests that the same instability mechanism is atwork for 1.4 > n > 0.6. Conversely, the increase of critical Reynolds numberat low n is now more evident. Interestingly, the trend is consistent for all

16

values of λ considered and the difference almost disappear when re-scalingthe neutral curves with the local viscosity.

0.40.60.811.21.4

500

1000

1500

2000

2500

3000

3500

4000

λ = 1λ = 10λ = 100

(a)

Re

n0.40.60.811.21.4

500

1000

1500

2000

λ = 1

λ = 10

λ = 100

(b)

Reavg

n

Figure 4: (a) Critical Reynolds number versus the index n for different values ofλ. (b) Average Reynolds number Reavg at neutral conditions versus n.

n=1.4, Rec=2570.31

0 0.5 10

0.5

1

500

1000

1500

2000

n=0.9, Rec=577.344

0 0.5 10

0.5

1

500

1000

1500

2000

n=0.6, Rec=313.77

0 0.5 10

0.5

1

500

1000

1500

2000

n=0.4, Rec=456.055

0 0.5 10

0.5

1

500

1000

1500

2000

Figure 5: Distribution of the local Reynolds number along the neutral curve Γ = 1and λ = 10.

The effect of the shear-dependent viscosity on the base flow is visualizedby the streamwise component Ub displayed in Figure 6. The boundarylayer at the lid becomes thinner and thinner when decreasing n while themagnitude of the negative counterflow at the lower wall decreases. This isalso associated to weaker vorticity in the centre of the cavity.

17

n=1.2

0 0.5 10

0.5

1

0

0.5

1n=0.8

0 0.5 10

0.5

1

0

0.5

1

n=0.6

0 0.5 10

0.5

1

0

0.5

1n=0.4

0 0.5 10

0.5

1

0

0.5

1

Figure 6: Distribution of the x-component of the baseflow velocity Ub at fixedRe = 600 for Γ = 1 and λ = 10 for the indicated values of n.

The circular frequency and the spanwise wavenumber pertaining to thefirst instability are reported in Figure 7. Steady modes are the first tobecome unstable in the case of Newtonian fluid and this is still valid forshear-thinning fluids, see Figure 7(a) . However, non-stationary modes arefirst unstable for n > 1.2 for all λ investigated. These modes have a lowerspanwise wavenumber as displayed in Figure 7(b). Interestingly, a modeof even higher frequency and lower spanwise wave-number κ appears as themost dangerous when n = 1.4. The frequency and wave-number are more orless independent of λ and of the power index n when steady modes appearfirst.

18

0.40.60.811.21.40

0.2

0.4

0.6

0.8

λ = 1λ = 10λ = 100

(a)

ω

n0.40.60.811.2

6

8

10

12

14

16

λ = 1

λ = 10

λ = 100

(b)

κ

n

Figure 7: (a) Frequency ω and (b) critical spanwise wavenumber κ of the firstinstability mode for different values of λ plotted versus the power indexn.

Direct and adjoint modes indicate where in the flow field the perturbationamplitude is maximized and the location of highest receptivity, respectively.The modulus of the unstable steady modes for n = 1 (Newtonian) and n =0.4 are displayed in Figure 8. For the Newtonian fluid, velocity perturbationsare mainly found on the left side of the cavity; a finding common to modesin the range 1.1 > n > 0.4, where the critical Reynolds number Reavg isalmost constant. When further decreasing the power-index n we see thatthe perturbation is now located on a thinner region evident also on the lowerwall and partially on the right side. A second peak is formed in the bottomright corner of the cavity for the v-component of the velocity perturbation.

19

umode

, n=1

0 0.5 10

0.5

1

0

0.01

vmode

, n=1

0 0.5 10

0.5

1

0

0.01

wmode

, n=1

0 0.5 10

0.5

1

0

0.01

umode

, n=0.4

0 0.5 10

0.5

1

0

0.01

0.02

vmode

, n=0.4

0 0.5 10

0.5

1

5

10

15

x 10−3

wmode

, n=0.4

0 0.5 10

0.5

1

0

0.005

0.01

Figure 8: Magnitude of the x-, y- and z-velocity components (u, v and w) of thefirst instability mode for (a) n = 1 and (b) n = 0.4, λ = 10.

The adjoint modes concerning the first instability in shear-thinning fluidare shown in Figure 9. Considering the Newtonian fluid, we see that theregion of highest receptivity is located on the corner opposite to that wherethe disturbance is largest. The adjoint v+ and w+ is stronger on the rightwall of the cavity. In the case of strong shear-thinning, the flow is mostsensitive to forcing in the x-direction at the upper wall, while normal forcingis most efficient on the left side of the cavity.

20

umode+ , n=1

0 0.5 1

0.20.40.60.8

24681012x 10

−3

vmode+ , n=1

0 0.5 1

0.20.40.60.8

0.0050.010.0150.02

wmode+ , n=1

0 0.5 1

0.20.40.60.8

246810x 10

−3

umode+ , n=0.4

0 0.5 1

0.20.40.60.8

51015

x 10−3

vmode+ , n=0.4

0 0.5 1

0.20.40.60.8

24681012x 10

−3

wmode+ , n=0.4

0 0.5 1

0.20.40.60.8

246810x 10

−3

Figure 9: Magnitude of the x-, y- and z components (u+, v+ and w+) of theadjoint of the first instability mode for (a) n = 1 and (b) n = 0.4,λ = 10.

The distribution of the time-periodic modes observed for shear-thickeningfluids is reported in Figure 10. Here we show results for n = 1.2 and n = 1.4,the latter associated to higher frequency and longer spanwise scale, as dis-cussed above. For n = 1.2, the velocity fluctuations are mainly located onthe left side of the cavity, as for the unstable steady modes; the area oflarge disturbance being now wider. When further increasing n = 1.4, themode appears now as a large vortex in the centre of the cavity. This modeenhances and decreases the amplitude of the base flow vortex periodicallyin time and spanwise direction.

21

umode

, n=1.4

0 0.5 10

0.5

1

246810x 10

−3

vmode

, n=1.4

0 0.5 10

0.5

1

0

0.005

0.01

wmode

, n=1.4

0 0.5 10

0.5

1

0

5

x 10−3

umode

, n=1.2

0 0.5 10

0.5

1

0

5

10

15x 10

−3

vmode

, n=1.2

0 0.5 10

0.5

1

24681012x 10

−3

wmode

, n=1.2

0 0.5 10

0.5

1

0

5

10

x 10−3

Figure 10: Magnitude of the x-, y- and z-velocity components (u, v and w) of thefirst instability mode for (a) n = 1.4 and (b) n = 1.2, λ = 10.

The adjoint of the non-stationary modes are reported in Figure 11. Inboth cases, the receptivity to forcing in the y-direction is strongest andlocated on the right side of the cavity, as for shear-thinning fluids. Thereceptivity to momentum forcing in x is instead more diffuse, peaking atthe lower or upper wall when varying n. Forcing in the spanwise directionis more effective at the right wall.

22

umode+ , n=1.4

0 0.5 1

0.20.40.60.8

2468x 10

−3

vmode+ , n=1.4

0 0.5 1

0.20.40.60.8

0.0050.010.0150.02

wmode+ , n=1.4

0 0.5 1

0.20.40.60.8

2468

x 10−3

umode+ , n=1.2

0 0.5 1

0.20.40.60.8

2468x 10

−3

vmode+ , n=1.2

0 0.5 1

0.20.40.60.8

0.0050.010.0150.02

wmode+ , n=1.2

0 0.5 1

0.20.40.60.8

2468x 10

−3

Figure 11: Magnitude of the x-, y- and z components (u+, v+ and w+) of theadjoint of the first instability mode for (a) n = 1.4 and (b) n = 1.2,λ = 10.

Next, the structural sensitivity is presented. As introduced above, itsdistribution provides important information about the instability mecha-nism. Results for both shear-thinning and thickening fluids are presentedin Figure 12. As we increase shear-thinning, the region at the core of theinstability is getting thinner but still consists of a ring inside the cavity.In the case of shear-thickening fluid and unsteady modes, we see that thewave-maker is located on the lower-left corner.

23

n=1.4, Rec=2570.31, κ=5.9

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0

1

2

3

4

5

n=1.2, Rec=1492.2, κ=7.6

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0

1

2

3

4

5

n=0.7, Rec=353.516, κ=13.9

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0

1

2

3

4

5

n=0.4, Rec=456.6, κ=15.9

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0

1

2

3

4

5

Figure 12: Structural sensitivity for the first instability along the neutral curvefor λ = 10 and the indicated values of n.

We now present the analysis of the perturbation kinetic energy budget.As introduced in section 2.6 the kinetic energy budget, i.e. equation (22),is given by the sum of production, dissipation and additional productionterms related to the varying viscosity. The density of energy productionassociated to the base flow shear is reported in Figure 13, again for λ = 10.We choose here to display 4 representative cases: two unsteady modes inshear-thickening fluids, n = 0.7 indicative of Newtonian and weakly shear-thinning fluids and n = 0.4 representing the region of increasing criticalReynolds number. One can see from the figure that the Newtonian fluidactually has the largest energy production and that its spatial distributionis almost independent on the power index n. The spatial distribution ofthe additional energy production due to non-Newtonian effects is shown inFigure 14. The additional term is, per definition, positive for shear-thinningfluids and negative for shear-thickening fluids. It is interesting to note thatthis additional term is most relevant on the left and lower side of the cavity.

24

n=1.4, Rec=2570.3, κ=5.9

0 0.2 0.4 0.6 0.80

0.5

1

0

5

10

15

n=1.2, Rec=1492.2, κ=7.6

0 0.2 0.4 0.6 0.80

0.5

1

0

5

10

15

n=0.7, Rec=353.5, κ=13.9

0 0.2 0.4 0.6 0.80

0.5

1

0

5

10

15

n=0.4, Rec=456.6, κ=15.9

0 0.2 0.4 0.6 0.80

0.5

1

0

5

10

15

Figure 13: Density of production of perturbation kinetic energy for selected valuesof n and λ = 10 at neutral conditions.

n=1.4, Rec=2570.3, κ=5.9

0 0.2 0.4 0.6 0.80

0.5

1

−0.5

0

0.5

1

1.5

n=1.2, Rec=1492.2, κ=7.6

0 0.2 0.4 0.6 0.80

0.5

1

−0.5

0

0.5

1

1.5

n=0.7, Rec=353.5, κ=13.9

0 0.50

0.5

1

−0.5

0

0.5

1

1.5

n=0.4, Rec=456.6, κ=15.9

0 0.50

0.5

1

−0.5

0

0.5

1

1.5

Figure 14: Density of the additional energy production term related to varyingviscosity. It is reported for the indicated values of n and λ = 10 atneutral conditions.

The sum of all production and dissipation terms is displayed in Figure 15for the same cases as before and at neutral conditions. Production is dom-inating in the lower left corner for intermediate values of n. In the case ofstrong shear-thinning, the peak of total production is on the lower right cor-ner, whereas for unsteady modes (n = 1.4) the peak of positive production

25

is more diffuse towards the lower left corner. Negative production, dissipa-tion, appears usually in this layers close to the regions of highest positiveproduction and on the upper left corner. It is also interesting to note thatin this close configuration the wave-maker of the instability and the regionof largest production of perturbation kinetic energy almost overlap; this wasnot the case for the cylinder flow as shown in [24].

n=1.4, Rec=2570.3, κ=5.9

0 0.2 0.4 0.6 0.80

0.5

1

−5

0

5

10

15

n=1.2, Rec=1492.2, κ=7.6

0 0.2 0.4 0.6 0.80

0.5

1

−5

0

5

10

15

n=0.7, Rec=353.5, κ=13.9

0 0.2 0.4 0.6 0.80

0.5

1

−5

0

5

10

15

n=0.4, Rec=456.6, κ=15.9

0 0.2 0.4 0.6 0.80

0.5

1

−5

0

5

10

15

Figure 15: Density of the total energy production for the indicated values of nand λ = 10 at neutral conditions.

Finally, we integrate the densities of the different energy terms over thedomain. The results are shown in Figure 16 for λ = 10 at neutral condi-tion and for fixed Re = 600 and κ = 15. In 16(a) we see that productionassociated to the base flow shear and dissipation have maximum aroundn = 1 and decrease when adding shear-dependent viscosity. The decreasein dissipation magnitude can be associated to the vorticity of the instabilitymode, while the reduction in production to the localization of the perturba-tion when n < 1 and to the weaker base shear when n > 1. The additionalproduction term becomes relevant when n < 0.6. It is also instructive tostudy the energy budget at fixed Reynolds number. In this case, the sumof the different terms reveals whether the mode is stable or unstable sinceit can be related to the real part of the eigenvalue. For the case depictedin the figure, the flow is unstable for 0.9 > n > 0.4. In addition to theexpected increase of the additional production for lower values of n, we no-tice a decrease of the classic production with shear-thinning. This is due tothe spatial de-correlation between the region of largest disturbance (lowerand left wall, cf. figure 8) and the region of largest shear (upper wall, cf.figure 6).

26

0.40.60.811.21.4−2

−1

0

1

2

(a)

E

n0.20.40.60.81

−3

−2

−1

0

1

2

Production

Dissipation

Additional Production

Sum

(b)

E

n

Figure 16: Budget of production of perturbation kinetic energy for λ = 10. (a)Budget at neutral condition and (b) at fixed Re = 600 and κ = 15.

Combining the results presented above, we can conclude that the in-stability mechanism is not significantly changed for weak shear-thinningand shear-thickening effects. Indeed, the Reavg is almost constant in thisrange, steady modes are the most unstable and the instability wave-makeris similar. When further increasing the shear-thinning properties, we see asurprising increase of the critical Reynolds number based on the zero-shear-rate viscosity. This can be explained by the fact that the region of largestshear become more and more localized close to the wall and the base-flowvorticity is weaker towards the centre of the cavity and on the left side: inthis case very large local Reynolds number (very low local viscosity) is nec-essary to overcome dissipation with the relevant contribution from the extraproduction terms associated to the shear-thinning effects. Interestingly, wenotice that unsteady modes are the first to become unstable when the powerindex n is above 1.2 with modes significantly longer in the spanwise direc-tion. Finally, we note that the instability characteristics are found to varywith the power index n, while no significant qualitative variations are foundwith respect to the time constant λ.

4.2 The Shallow Cavity, Γ = 0.25

We now consider the shallow cavity of aspect ratio Γ = 0.25, as in [2], wherethe first instability mode is time-periodic in Newtonian fluids. Computationshave been performed with λ = 10; as shown above this parameter does notseem to significantly affect the physics of the instability, the main variationscoming from the power index n.

The neutral curves are first displayed in Figure 17 both in terms ofzero-shear rate viscosity, Reynolds number Re, and average viscosity, aver-age Reynolds number Reavg. The critical Reynolds number decreases from

27

shear-thickening to shear-thinning fluids (decreasing n) whereas critical val-ues of the average Reynolds number first decrease and then increase with aminimum at n ≈ 0.7, similarly to what observed for the square cavity.

0.40.60.811.2

1000

2000

3000

4000

(a)

Re

n0.40.60.811.2

500

1000

1500

2000

(b)

Reavg

n

Figure 17: (a) Critical Reynolds number versus the index n and (b) AverageReynolds number Reavg at neutral conditions versus n for Γ = 0.25and λ = 10.

The effect of shear-thinning viscosity on the two-dimensional base flowis depicted in Figure 18 where we report the x-component of the baseflowvelocity Ub for 4 values of n. As n decreases we clearly see that the mainvortex moves to the right side of the cavity. In addition, and in analogy tothe case of the square cavity, the boundary layer at the moving lid becomesthinner at lower n.

n=1

0 0.5 10

0.10.2

−0.500.51

n=1.3

0 0.5 10

0.10.2

−0.500.51

n=0.8

0 0.5 10

0.10.2

−0.500.51

n=0.5

0 0.5 10

0.10.2

−0.500.51

Figure 18: Distribution of the x-component of the baseflow velocity Ub at fixedRe = 500 for Γ = 0.25, λ = 10 and the values of n indicated.

Unlike the square cavity, for the shallow cavity all critical modes areoscillatory. The frequency and spanwise wavenumber of the neutral modesare reported in Figure 19; here one can see that the value for κc and ωcdrops continuously with decreasing power-law index , from about κ = 21.2and ω = 2.34 respectively, until suddenly at n = 0.4 the first instability

28

appears to have a long-wavelength (κ ≈ 11) and low-frequency (ω ≈ 0.7).

0.40.60.811.2

1

1.5

2

2.5

(a)

ω

n0.40.60.811.2

12

14

16

18

20

22

(b)

κ

n

Figure 19: (a) Frequency ω and (b) critical spanwise wavenumber κ of the firstinstability mode plotted versus the power index n for Γ = 0.25 andλ = 10

The shape of the direct modes does is shown in Figure 20 for shear-thinning fluids. The region of largest fluctuations is seen to move from thetop-right corner to the centre of the cavity (left side of the big vortex dis-played in Figure 18) when shear-thinning increases and the instability modehas a significantly lower frequency. In all cases, velocity fluctuations aredetectable on the right-half of the cavity. The corresponding adjoint modes,depicted in Figure 21, reveal a certain overlap with the direct modes; inter-estingly, forcing in the y-direction is one order of magnitude most effectivethan forcing in the other two directions when n = 0.9, whereas acting inthe x-direction at the lower wall is the most effective way to excite the slowoscillatory mode at n = 0.4.

29

umode

, n=0.9

0 0.5 10

0.10.2

00.0050.01

vmode

, n=0.9

0 0.5 10

0.10.2

00.0050.01

wmode

, n=0.9

0 0.5 10

0.10.2

00.0050.01

umode

, n=0.4

0 0.5 10

0.10.2

00.0050.01

vmode

, n=0.4

0 0.5 10

0.10.2

00.0050.01

wmode

, n=0.4

0 0.5 10

0.10.2

00.0050.01

Figure 20: Magnitude of the x-, y- and z-velocity components (u, v and w) of thefirst instability mode for (a) n = 0.9 and (b) n = 0.4 for Γ = 0.25 andλ = 10

umode+ , n=0.9

0.2 0.4 0.6 0.80

0.10.2

246

x 10−3

vmode+ , n=0.9

0.2 0.4 0.6 0.80

0.10.2

0.0050.010.0150.02

wmode+ , n=0.9

0.2 0.4 0.6 0.80

0.10.2

51015

x 10−3

umode+ , n=0.4

0.2 0.4 0.6 0.80

0.10.2

0.010.020.03

vmode+ , n=0.4

0.2 0.4 0.6 0.80

0.10.2

246810

x 10−3

wmode+ , n=0.4

0.2 0.4 0.6 0.80

0.10.2

246810

x 10−3

Figure 21: Magnitude of the x-, y- and z components (u+, v+ and w+) of theadjoint of the first instability mode for (a) n = 0.9 and(b) n = 0.4 forΓ = 0.25and λ = 10

The direct and adjoint modes pertaining a shear-thickening fluid aredisplayed in Figures 22 and 23 in comparison with the Newtonian case. Itis seen here that the shape of the direct modes is not affected by the non-

30

Newtonian properties of the fluid, while it is interesting to note that for thecase with n > 1, the adjoint modes are significant also in the left-half ofthe cavity. This can be explained by the fact that the base-flow vortex nowextends throughout the cavity and it not only localized on the right side ofit, as seen in Figure 18.

umode

, n=1.3

0 0.5 10

0.10.2

00.0050.01

vmode

, n=1.3

0 0.5 10

0.10.2

00.0050.01

wmode

, n=1.3

0 0.5 10

0.10.2

00.0050.01

umode

, n=1

y

0 0.5 10

0.10.2

00.0050.01

vmode

, n=1

0 0.5 10

0.10.2

00.0050.01

wmode

, n=1

0 0.5 10

0.10.2

00.0050.01

Figure 22: Magnitude of the x-, y- and z-velocity components (u, v and w) of thefirst instability mode for (a) n = 1.3 and (b) n = 1 for Γ = 0.25 andλ = 10

31

umode+ , n=1.3

0.2 0.4 0.6 0.80

0.10.2

00.0050.01

vmode+ , n=1.3

0.2 0.4 0.6 0.80

0.10.2

00.0050.01

wmode+ , n=1.3

0.2 0.4 0.6 0.80

0.10.2

00.0050.01

umode+ , n=1

0.2 0.4 0.6 0.80

0.10.2

246

x 10−3

vmode+ , n=1

0.2 0.4 0.6 0.80

0.10.2

0.0050.010.0150.02

wmode+ , n=1

0.2 0.4 0.6 0.80

0.10.2

24681012

x 10−3

Figure 23: Magnitude of the x-, y- and z components (u+, v+ and w+) of theadjoint of the first instability mode for (a) n = 1.3 and(b) n = 1 forΓ = 0.25and λ = 10

The structural sensitivity, the core of the instability, is shown in Figure24 at neutral conditions and different values of n. The area of largest sensi-tivity is always located on the right-side of the cavity and decreases as thepower-law index is reduced. Further, the maximum moves from the regionclose to the upper wall for n > 0.6 to that close to the lower wall. Thiseffect is found to be associated to the increase of the critical Reavg, shownin Figure 17(b).

n=1.2, Rec=2776.25, κ=20.9

0.2 0.4 0.6 0.8

0.050.1

0.150.2

0

10

20

30

n=0.8, Rec=514.062, κ=20.3

0.2 0.4 0.6 0.8

0.050.1

0.150.2

0

10

20

30

n=0.6, Rec=253.438, κ=19.9

0.2 0.4 0.6 0.8

0.050.1

0.150.2

0

10

20

30

n=0.4, Rec=181.25, κ=11.4

0.2 0.4 0.6 0.8

0.050.1

0.150.2

0

10

20

30

Figure 24: Structural sensitivity for the first instability along the neutral curvefor Γ = 0.25 and λ = 10. The critical values are reported in the plots.

The spatial distribution of the sum of the different terms in the budget forthe perturbation kinetic energy are presented in Figure 25. It can be seen

32

that the production mainly stems from the binocular- and banana-shaperegions close to the upper and lower wall respectively, whereas dissipationoccurs in a thin stripe between the regions of largest positive production.As for the square cavity, we note that the wave-maker identified by thestructural sensitivity and the areas more energetically active overlap. Asdiscussed above, the core of the instability shifts from the upper to thelower wall as shear-thinning increases.

n=1.2, Rec=2776.3, κ=20.9

0 0.2 0.4 0.6 0.80

0.1

0.2

−20

0

20

n=1, Rec=1165, κ=20.6

0 0.2 0.4 0.6 0.80

0.1

0.2

−20

0

20

n=0.7, Rec=353.8, κ=19.9

0 0.2 0.4 0.6 0.80

0.1

0.2

−20

0

20

n=0.4, Rec=181.25, κ=11.4

0 0.2 0.4 0.6 0.80

0.1

0.2

−20

0

20

Figure 25: Density of the total energy production for the indicated values of nand λ = 10 at neutral conditions.

Finally, as for the square cavity, we examine the total budget of theperturbation kinetic energy, Figure 26. As for the square cavity, we notethat the additional production term related to shear-thinning becomes de-terminant for the instability only when n < 0.6. In this case the averagedReynolds number is increasing and the core of the instability has moved tothe lower half of the cavity. Interestingly, at neutral conditions the dissipa-tion magnitude first increases for decreasing n, indicating increasing vorticityof the instability mode, while it decreases for the long-wavenumber modefound for n = 0.4. Looking at the budget at fixed Re and κ, Figure 26(b),we see how the pocket of instability is mainly created by increasing the pro-duction induced by the work of the Reynolds stresses against the base-flowshear.

33

0.40.60.811.2−3

−2

−1

0

1

2

3

(a)

E

n0.50.60.70.80.91

−5

0

5

10

ProductionDissipationAdditional ProductionSum

(b)

E

n

Figure 26: Budget of production of perturbation kinetic energy for the shallowcavity Γ = 0.25 and λ = 10. (a) Budget at neutral condition and (b)at fixed Re = 500 and κ = 20.

5 Conclusions

Linear stability analysis of the lid-driven cavity containing non-Newtonianfluid has been performed for two different aspect ratios, namely Γ = 1(square cavity) and Γ = 0.25 (shallow cavity). The former characterizedby steady unstable modes at critical conditions, the latter by oscillatinginstabilities. The Carreau-Yasuda model has been chosen to model shear-thinning and shear-thickening fluids and the rheological parameters exam-ined in the range 0.4 < n < 1.4 and λ = 1, 10 and 100. To investigate theinstability mechanisms we consider both the classic equation for the evolu-tion of the perturbation kinetic energy and the structural sensitivity of theinstability, as introduced in [23].

In general, shear-thickening effects stabilize the flow, i.e. increase thecritical Re both for the square and shallow cavity. Conversely, shear-thinningcreates instabilities at lower Reynolds numbers. However we see that at thelowest values of n considered the critical Reynolds number increases again.In particular, we observe for the square cavity that there exists an intermedi-ate range of values of the power index n at which the instability mechanismis unaffected by non-Newtonian effects. This is demonstrated by examiningthe wave-maker of the instability as well as the spatial distribution of thekinetic energy production and dissipation. In addition, we show that thecritical average Reynolds number, based on the average value of the localviscosity inside the cavity, is almost constant in this regime and, interest-ingly, becomes almost independent of the time constant λ (a result thatactually applies to all values of n considered).

The increase of the critical Reynolds number for large shear-thinningcan be explained by considering the non-Newtonian effects on the base flow:

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formation of thinner boundary layers close to the walls and reduction ofthe intensity of base-flow shear inside the cavity. In these cases, the extra-production of kinetic energy due to shear-thinning becomes determinant forthe instability occurrence. For square cavities, we report a change fromunsteady to oscillating critical modes already at moderate values of thepower index n (n > 1.2) associated to a significant increase of the spanwisescale of the unstable disturbance.

The neutral stability curve for the shallow cavity also show an increaseof the critical Reynolds number at the lowest values of n considered, asmentioned above. In this case however, we find a new instability modecharacterized by lower frequency and longer spanwise scale to be the firstto become unstable when n < 0.5. The core of the instability has shiftedfrom regions close to the upper driving wall to areas close to the lower wall.The same physical mechanisms as for Newtonian fluids appears to drivethe instability for both shear-thinning and shear-thickening fluids otherwise.Finally, we note that unlike in open flows, as for the flow past a circularcylinder [24], the analysis of the energy budget and the structural sensitivity,based on the superposition of the unstable mode and its adjoint, indicatethe same critical region for the instability mechanism.

The linear analysis conducted here reveals the first flow bifurcation toa steady or oscillating three-dimensional flow. The present work is there-fore being continued by considering the appearance of secondary instabilitiesand the effect of the shear-dependent viscosity on the unsteady regime atReynolds number above the critical threshold. In addition, the flow sensi-tivity can be used to design passive control strategies to manipulate the flowinside the cavity.

Acknowledgments

The author would like to his express gratitude to L. Brandt for supervisionand inspiration throughout the work. Also, great appreciation is shown toF. Giannetti for providing and instructing the use of the CPL-code and toI. Lashgari for all of his valuable input.

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