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Acta Mechanica manuscript No.

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Stability and receptivity of boundary layers in a swirl ﬂow channel

B. Herrmann-Priesnitz ·W. R. Calder´on-Mu˜noz ·R. Soto

May 28, 2018

Abstract The analysis of the disturbances on a spiraling base ﬂow are relevant for the design, operation and

control of technological devices such as parallel-disc turbines and swirl ﬂow channel heat sinks. Spiraling inﬂow

inside an annular cavity closed at the top and bottom is analyzed in the framework of modal and nonmodal

stability theories. Local and parallel ﬂow approximations are applied, and the inhomogeneous direction is

discretized using the Chebyshev collocation method. The optimal growth of initial disturbances and the optimal

response to external harmonic forcing are characterized by the exponential and the resolvent of the dynamics

matrix. As opposed to plane Poiseuille ﬂow, transient growth is small, and consequently, it does not play a

role in the transition mechanism. Transition is attributed to crossﬂow instability that occurs because of the

change in the shape of the velocity proﬁle due to rotational eﬀects. Agreement is found between the critical

Reynolds number predicted in this work and the deviation of laminar behavior observed in the experiments

conducted by Ruiz and Carey [15]. For the harmonically driven problem, energy ampliﬁcation of O(100) is

observed for spiral crossﬂow waves. Transition to turbulence should be avoided to ensure the safe operation of

a parallel-disc turbine, whereas large forcing ampliﬁcation may be sought to promote mixing in a swirl ﬂow

channel heat sink. The analysis presented predicts and provides insight on the transition mechanisms. Due to

its easy implementation and low computational cost, it is particularly useful for the early stages of engineering

design.

1 Introduction

The spiraling inﬂow inside an annular cavity closed at the top and bottom has recently regained interest for

its relevance to applications in turbomachinery and high heat ﬂux dissipation systems. Scaling of parallel-disc

turbines for uses in the microscale has been investivated by Krishnan et al., Pfenniger et al., and Sengupta and

B. Herrmann-Priesnitz ( ) ·W. R. Calder´on-Mu˜noz

Department of Mechanical Engineering, FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile.

Tel.: +569-96791014

E-mail: bherrman@ing.uchile.cl

W. R. Calder´on-Mu˜noz

Energy Center, FCFM, Universidad de Chile, Av. Tupper 2007, Santiago, Chile.

E-mail: wicalder@ing.uchile.cl

R. Soto

Physics Department, FCFM, Universidad de Chile, Av. Blanco Encalada 2008, Santiago, Chile.

E-mail: rsoto@dﬁ.uchile.cl

2 B. Herrmann-Priesnitz et al.

Guha among others [7, 12,19]. Ruiz and Carey proposed a novel swirl ﬂow microchannel heat sink for high heat

ﬂux applications such as cooling electronics and concentrated solar photovoltaics [14]. In subsequent work, a

dye injection experiment revealed a considerable increase in dye diﬀusion when the ﬂow rate going through

the channel exceeded 190 ml/min. In the same work, the authors also observed that, for ﬂow rates over 190

ml/min, laminar theory signiﬁcantly underpredicted their experimental measurements for pressure drop and

total heat ﬂux, they attributed these results to hydrodynamic instabilities [15].

Steady state ﬂow was studied numerically using integral methods by Herrmann-Priesnitz et al. [4] and

diﬀerent boundary layer structures were observed depending on the governing parameters. Parabolic velocity

proﬁles are found for low Reynolds numbers, while large inﬂection of the radial velocity component occurs for

higher values. Whether these highly inﬂected proﬁles can occur in the real world device is a matter of stability.

Stability and receptivity analyses are also motivated by its relevance to design, operation and control of the

parallel-disc turbine and the swirl ﬂow channel heat sink.

Velocity proﬁles found in this type of channel are similar to those observed in other rotating boundary layer

ﬂows, such as von K´arm´an and B¨odewadt ﬂows. The ﬁrst experimental observation of stationary crossﬂow

vortices and the ﬁrst theoretical stability analysis for the rotating disc ﬂow were presented by Gregory et al.

[2]. Work on the modal and spatial stability continued with Malik, who computed the neutral curves for sta-

tionary disturbances using the parallel ﬂow approximation [10]. Lingwood followed by studying the absolute or

convective nature of the instabilities [8]. More recently, Serre et al. and Lopez et al. used DNS and found that

the B¨odewadt layer is unstable to axisymmetric circular radial waves and three-dimensional multi-armed spiral

waves [20,9]. In a follow up study, Do et al. showed that in the absence of any external forcing, the circular

waves are transitory, but low amplitude forcing can sustain them indeﬁnitely [1].

Over the past two decades, nonmodal stability theory has emerged to provide a more complete picture

of the linear perturbation dynamics for ﬂuid ﬂows using an initial-value problem formulation [18,16, 17]. The

modal approach characterizes a ﬂow as stable if all the eigenvalues of the dynamics operator are located on the

left half of the complex plane. Nevertheless, for eigenvalues with negative real part, the ﬂow may experience a

large ampliﬁcation due to linear transient growth before decaying asymptotically [21]. The formulation as an

initial-value problem allows the incorporation of an external harmonic forcing term that may represent free-

stream turbulence, wall roughness, acoustic perturbations or body forces among others. The response of the

system to these external disturbances, i.e. receptivity of the ﬂow, is determined by the particular solution to

the harmonically driven problem. Neither modal nor nonmodal stability and receptivity analyses for the swirl

ﬂow channel have been reported elsewhere.

In this study, the formulation of a linear initial-value problem for the perturbation dynamics in a swirl

ﬂow channel is detailed. The methodology to apply the local and parallel ﬂow approximations based on order

of magnitude arguments is presented. We calculate the optimal energy growth of initial disturbances, and we

examine the dependence of the maximum growth on the wavenumbers. We also calculate the optimal response

to external harmonic forcing and examine its dependence on the wavenumbers. Unlike in plane Poiseuille ﬂow,

rather than transient energy growth of disturbances, the transition to turbulence is attributed to the change

from parabolic to inﬂected base ﬂow velocity proﬁles due to rotational eﬀects when increasing the Reynolds

number.

Stability and receptivity of boundary layers in a swirl ﬂow channel 3

2 Governing equations

2.1 The base ﬂow

The swirl ﬂow channel consists in an annular cavity, which is open at the outer and inner radii, roand ri,

the top and bottom boundaries are solid walls with a separation of 2h, and it has a very small aspect ratio

h/ro≪1. Incompressible ﬂuid enters the channel at rowith an inlet angle θowith respect to the tangent,

spirals radially inward, and exits through ri. A schematic of the swirl ﬂow channel as well as the cylindrical

system of coordinates used are shown in Fig. 1a. Steady state ﬂow is axisymmetric and presents a boundary

layer nature, it is therefore governed by

1

r

∂(rU )

∂r +∂W

∂z = 0,(1a)

U∂U

∂r +W∂U

∂z −V2

r=−1

ρ

∂P

∂r +ν∂2U

∂z2,(1b)

U∂V

∂r +W∂V

∂z +U V

r=ν∂2V

∂z2,(1c)

0 = −1

ρ

∂P

∂z ,(1d)

U= 0 at z=±h, U=Uo(1,cot(θo),0)Tat r=ro, P = 0 at r=ri,(1e)

where U= (U, V , W )Tis the steady state velocity ﬁeld in cylindrical coordinates (ˆr,ˆ

θ, ˆz), Pis the steady

state pressure, and ρand νare the density and kinematic viscosity. Boundary conditions are shown in Eq.

(1e): no-slip at the channel walls, velocity Uo(1,cot(θo),0)Tat the inlet, and a reference pressure is set at the

outlet. The parameters governing the steady state ﬂow are the inlet angle θoand a ﬂow rate Reynolds number

deﬁned as Reo=Uoh2/(νro). Solution to Eqs. (1) is approximated using the method developed in Ref. [4],

which considers the formulation of a ﬂow model for the mass and momentum transfer coupled between viscous

boundary layers and an inviscid core region. The resulting equations are solved using an integral method and

a space-marching technique, and the main advantage of this method is the low computational cost that allows

fast exploration of the parameter space [4, 5]. Rotation of the ﬂuid induces a crossﬂow and entrainment of ﬂuid

towards the channel walls, therefore, three diﬀerent boundary layer structures may develop depending on the

ﬂow rate Reynolds number and the ﬂow inlet angle: merged, entraining, or non entraining. These structures

are classiﬁed based on the shape of the radial velocity proﬁle, as shown in Fig. 1c.

2.2 Linearized perturbation equations

Through this section we cover in detail the formulation of the stability and receptivity problems for the swirl

ﬂow channel. The approximations presented allow these problems to be solved using standard nonmodal tech-

niques taken from, e.g., Schmid and Henningson, 2001 [18].

Linearizing the incompressible Navier-Stokes equations in cylindrical coordinates about an axisymmetric

base ﬂow yields the following system of equations

∂u

∂r +u

rc+1

r

∂v

∂θ +∂w

∂z = 0,(2a)

4 B. Herrmann-Priesnitz et al.

∂u

∂t +U∂u

∂r +V

r

∂u

∂θ +w∂U

∂z +u∂U

∂r +W∂u

∂z np −2V v

rc

=

−∂p

∂r +1

Re ∂2u

∂r2+1

r2

∂2u

∂θ2+∂2u

∂z2+1

r

∂u

∂r −u

r2−2

r2

∂v

∂θ c+fu,(2b)

∂v

∂t +U∂v

∂r +V

r

∂v

∂θ +w∂V

∂z +u∂V

∂r +W∂v

∂z np

+uV +Uv

rc

=

−1

r

∂p

∂θ +1

Re ∂2v

∂r2+1

r2

∂2v

∂θ2+∂2v

∂z2+1

r

∂v

∂r −v

r2+2

r2

∂u

∂θ c+fv,(2c)

∂w

∂t +U∂w

∂r +V

r

∂w

∂θ +u∂W

∂r +w∂W

∂z +W∂w

∂z np

=

−∂p

∂z +1

Re ∂2w

∂r2+1

r2

∂2w

∂θ2+∂2w

∂z2+1

r

∂w

∂r c+fw,(2d)

where u= (u, v, w)Tis the perturbation velocity and pis the perturbation pressure. We have added an external

forcing term (fu, fv, fw)Tto the momentum equations, which will later be used for the receptivity analysis. The

terms inside the parentheses ( )cand ( )np correspond to curvature and non-parallel ﬂow eﬀects respectively.

Equations (2) have been nondimensionalized using the half-height of the channel, h, as the characteristic length

scale, and the velocity magnitude √U2+V2at the midplane of the channel as the characteristic velocity scale.

In this study, we are concerned with the local behavior of ﬂow structures of length scale h, around a certain

radial station r. If the radius is suﬃciently large compared to the length scales of interest i.e., ε=h/r ≪1, the

ﬂow can be regarded as locally Cartesian in the coordinates (ˆr, r ˆ

θ, ˆz), therefore the curvature terms ( )ccan be

neglected. Due to the boundary layer nature of the ﬂow, the normal velocity component is much smaller than

those parallel to the walls i.e., W≪U, V , and the velocity gradient in the radial direction is much smaller than

the gradient normal to wall i.e, ∂ U/∂r, ∂ V/∂ r ≪∂ U/∂z , ∂V /∂z. In fact, these are smaller by a factor of order

O(ε), and by neglecting them we get U≈(U(z), V (z),0)T. This results in the elimination of the non-parallel

ﬂow eﬀects denoted by ( )np, which is known in literature as the parallel ﬂow approximation. Both, the local

system of coordinates and a schematic of the parallel base ﬂow are shown in Fig. 1b.

Using the local base ﬂow (U(z), V (z),0)Tand the parallel ﬂow approximation yields the following system

of equations

∂u

∂r +1

r

∂v

∂θ +∂w

∂z = 0,(3a)

∂u

∂t +U∂u

∂r +V

r

∂u

∂θ +wU′=−∂p

∂r +1

Re ∂2u

∂r2+1

r2

∂2u

∂θ2+∂2u

∂z2+fu,(3b)

∂v

∂t +U∂v

∂r +V

r

∂v

∂θ +wV ′=−1

r

∂p

∂θ +1

Re ∂2v

∂r2+1

r2

∂2v

∂θ2+∂2v

∂z2+fv,(3c)

∂w

∂t +U∂w

∂r +V

r

∂w

∂θ =−∂p

∂z +1

Re ∂2w

∂r2+1

r2

∂2w

∂θ2+∂2w

∂z2+fw,(3d)

where ′denotes diﬀerentiation of the base ﬂow with respect to to z. The local temporal evolution problem for the

perturbations is completed with appropriate initial conditions and no-slip boundary conditions on the channel

walls i.e., u=v=w= 0 at z=±1. Equations (3) have coeﬃcients that do not depend on rand θ, this allows for

Stability and receptivity of boundary layers in a swirl ﬂow channel 5

(a)

h

h

(b)

0.25 1 4 16 64

0.1

1

10

100

1000

θ0

(degrees)

Reo

non

entraining

entraining

merged

(c)

Fig. 1 (a) Schematic of the swirl ﬂow channel and cylindrical system of coordinates. (b) Base ﬂow schematic and local system

of coordinates. (c) Classiﬁcation of the steady state boundary layer structures on the Reo–θospace in log scale, based on the

radial velocity proﬁles [5].

the perturbation variables to be expanded as Fourier modes in these directions u(r, θ, z, t) = ˆu(z , t) ei(αr+mθ),

where αis the radial wavenumber, and mis the integer azimuthal wavenumber. Identical expansions are carried

out for the perturbation pressure and external forcing terms. The governing equations are simpliﬁed to

iαˆu+ iβˆv+Dˆw= 0,(4a)

∂ˆu

∂t + i (αU +βV )ˆu+U′ˆw=−iαˆp+1

Re D2−k2ˆu+ˆ

fu,(4b)

∂ˆv

∂t + i (αU +βV )ˆv+V′ˆw=−iβˆp+1

Re D2−k2ˆv+ˆ

fv,(4c)

∂ˆw

∂t + i (αU +βV )ˆw=−Dˆp+1

Re D2−k2ˆw+ˆ

fw,(4d)

where β=m/r, the total wavenumber is k= (α2+β2)1

2, and Ddenotes diﬀerentiation with respect to zof

the perturbation variables. We want to rewrite the system in terms of the normal vorticity ˆηand the normal

velocity ˆwinstead of the primitive variables. To do this, we ﬁrst derive the transport equation for ˆηby taking the

zcomponent of the curl of the momentum Eqs. (4b)–(4d). Secondly, we obtain an expression for the pressure

by taking the divergence of the momentum Eqs. (4b)–(4d) and using the continuity Eq. (4a). Substituting the

resulting expression into Eq. (4d) we eliminate ˆpfrom the system and get

M∂ˆw

∂t + iM(αU +βV )ˆw+ i αU ′′ +β V ′′ ˆw+1

Re M2ˆw= iαDˆ

fu+ iβDˆ

fv+k2ˆ

fw,(5a)

∂ˆη

∂t + i (αU +βV )ˆη+1

Re Mˆη= i αV ′−βU ′ˆw+ iβˆ

fu−iαˆ

fv,(5b)

where M=k2− D2. The no-slip boundary conditions for the normal vorticity and normal velocity become

Dˆw(±1) = ˆw(±1) = ˆη(±1) = 0. Equations (5) are discretized using the Chebyshev collocation method and

the operator Dis replaced with the Chebyshev diﬀerentiation matrix D. We obtain a linear dynamical system

where the state variables are the normal velocity and normal vorticity evaluated at the collocation points. The

matrix representation of the system is

d

dt"ˆw

ˆη#="LOS 0

LCLSQ #

|{z }

L

"ˆw

ˆη#+"iαM−1DiβM−1D M−1k2

iβ−iα0#

|{z }

B

ˆ

fu

ˆ

fv

ˆ

fw

,(6)

6 B. Herrmann-Priesnitz et al.

where LOS,LSQ , and LCare the familiar Orr-Sommerfeld, Squire, and coupling operators for a base ﬂow that

has velocity components along both, the spanwise and streamwise directions. The rate of change of the state

vector ˆq= ( ˆw, ˆη)Tis related to its current state by operator L, and to the input forcing ˆ

f= ( ˆ

fu,ˆ

fv,ˆ

fw)Tby

operator B. The system can be written in compact notation as follows

dˆq

dt=Lˆq+Bˆ

f.(7)

Equation (7) governs the dynamics of the perturbation variables, and we are interested in two particular

cases: the temporal evolution of initial disturbances and the long-time response to external harmonic forcing.

2.2.1 Response to initial conditions

For a prescribed initial condition ˆq(0) = ˆq0, the solution to Eq. (7) without any external forcing is given by

ˆq= exp(tL)ˆq0.(8)

In order to study the stability of the system, we calculate the maximum energy ampliﬁcation over a speciﬁed

time interval and optimized over all initial conditions

G(t) = max

ˆ

q06=0 ||ˆq(t)||2

E

||ˆq0||2

E

= max

ˆ

q06=0 ||exp(tL)ˆq0||2

E

||ˆq0||2

E

=||exp(tL)||2

E,(9)

where || · ||Eis a norm that measures the kinetic energy of the perturbations [16]. The ﬂow is asymptotically

unstable when G(t)→ ∞ as t→ ∞, which will occur when at least one eigenvalue of Lhas a positive real part.

On the counterpart, the ﬂow is called asymptotically stable when G(t)→0 as t→ ∞.

2.2.2 Response to external harmonic forcing

For an external harmonic forcing ˆ

f(t) = ˜

fexp(−iωt), the particular solution to Eq. (7) is given by

ˆq=(iωI−L)−1Bˆ

f.(10)

This solution provides insight on the receptivity process, and for an asymptotically stable ﬂow it represents

the long-time response of the system [6]. In order to study the receptivity of the system, we calculate the

maximum energy ampliﬁcation of the output optimized over all shapes of input forcing

R(ω) = max

ˆ

f6=0

||ˆq||E

||ˆ

f||E

= max

ˆ

f6=0

||(iωI−L)−1Bˆ

f||2

E

||ˆ

f||E

=||(iωI−L)−1B||E.(11)

Therefore, the optimal response R(ω) is the resolvent norm [18].

2.3 Energy norm

A physically relevant quantity to measure growth is the kinetic energy of the perturbations. For a vector written

in the normal velocity and normal vorticity formulation, the kinetic energy is calculated as follows [3]

||ˆq||2

E=1

2k2Z1

−1|D ˆw|2+k2|ˆw|2+|ˆη|2dz= ˆqHQˆq,(12)

where Qis the energy weight matrix that contains the appropriate weighting of the variables ( ˆw, ˆη)Tat the

collocation points, as well as the integration weights between the channel walls. A Cholesky decomposition of

Stability and receptivity of boundary layers in a swirl ﬂow channel 7

Q=FHFallows us to relate this norm to an equivalent standard (Euclidean) L2-norm ||ˆq||E=||Fˆq||2. The

energy norm of a matrix Ais easily derived using the deﬁnition of a vector-induced norm

||A||E= max

ˆ

q||Aˆq|E

||ˆq||E

= max

ˆ

q||FAF−1Fˆq||2

||Fˆq||2=||FAF−1||2.(13)

Going back to Eqs. (9) and (11), we can rewrite the optimal growth rate G(t) and the optimal response

R(ω) as L2-norms

G(t) = ||Fexp(tL)F−1||2

2,(14a)

R(ω) = ||F(iωI−L)−1BF−1||2.(14b)

Before we can compute G(t) and R(ω) from the above expressions, we have to perform a spectral decom-

position of the operator L=VΛV−1. Here Vis the matrix whose columns are the eigenvectors of L, and

Λ= diag (λ1,...,λ2N)is a diagonal matrix containing its eigenvalues, where 2Nis the length of ˆq. This way,

the exponential and the resolvent of Lcan be easily calculated

exp(tL) = Vdiag (exp(tλ1),...,exp(tλ2N))V−1,(15a)

(iωI−L)−1=Vdiag 1

iω−λ1

,..., 1

iω−λ2NV−1.(15b)

Using Eqs. (15) we can compute the matrices in Eqs. (14), and their Euclidean norm which is given by their

largest singular value. Additionally, the principal right singular vector and principal left singular vector of each

of these matrices correspond to the maximum ampliﬁcation input and output disturbances, respectively. That

is, the optimal initial condition and the disturbance at time tfor the exponential, and the optimal forcing

and the response at a frequency ωfor the resolvent. Computing the eigenvalues and eigenfunctions requires

O((2N)3) arithmetic operations. To reduce the amount of computational work, we restrict our attention to the

Kleast stable modes instead of all 2N, thus requiring only O(K3) operations [13, 18]. In this study, N= 80

and K= 50 are found to be enough for convergence of the computed results.

3 Results and discussion

3.1 Base ﬂow

Our base ﬂow considers the steady state in a swirl ﬂow channel with an inlet angle θo= 8◦and an aspect

ratio h/ro= 0.02, evaluated at a local radius r. Equations (1) allow us to calculate the local boundary layer

thickness δ, the local ﬂow angle θcat z= 0 (midplane), and the local Reynolds number Re, for diﬀerent values

of r, as shown in Fig. 2 [5].

Boundary layers in the swirl ﬂow channel are merged (δ= 1) for low Reo, they separate over a certain value

Reo, and increments over that value result in thinning of the boundary layers, as shown in Fig. 2a. Figure 2b

shows how the direction of the local ﬂow at the midplane of the channel changes as Reoincreases. Starting

from radial ﬂow (θc= 90◦), the angle increases until we get tangential ﬂow (θc= 0◦) when the boundary layers

separate. Figure 2cshows the values of the local Reynolds number Re increasing as a function of the ﬂow rate

Reynolds number Reo. When the boundary layers are merged, the local ﬂow resembles a plane Poiseuille ﬂow

in the direction of θcand with a Reynolds number Re which is much lower than the critical value 5772 required

for the ﬂow to be asymptotically unstable [11]. As shown in Fig. 2, the overall behavior of the base ﬂow does

not depend strongly on the local radius, therefore we arbitrarily select r= 0.6roas a representative local radius

for the ﬂuid dynamics in a swirl ﬂow channel. Although our conclusions in this study are not aﬀected by this

particular value, we consider that a global stability analysis is a logical next step for future work.

8 B. Herrmann-Priesnitz et al.

(a)(b)(c)

Fig. 2 Base ﬂow variables as a function of the ﬂow rate Reynolds number Reoat diﬀerent local radii r/ro= 0.4,0.5,and

0.6.(a) Boundary layer thickness δ. (b) Flow angle at the midplane of the channel θc, measured starting from the tangential

direction. (c) Local Reynolds number Re, based on the half-height of the channel and the streamwise velocity component at

z= 0 (midplane).

3.2 Stability and receptivity

Optimal growth G(t) and optimal response R(ω) are calculated for ﬂow rate Reynolds numbers Reobetween

0.1–2. In order to study the inﬂuence of the wavevector, instead of using its radial and azimuthal components

αand β, we use its magnitude k=pα2+β2and the waveangle deﬁned as θk= arctan(α/β)−θc. In this

notation, an angle θk= 0◦represents a wave propagating in the direction of ﬂow in the midplane of the channel

(aligned with θc), which we refer to as the streamwise direction. Therefore, we refer to θk= 90◦as the crossﬂow

direction, and it represents a wave direction normal to the ﬂow at z= 0. This formulation allows for an easier

physical interpretation of the results, because the streamwise and crossﬂow directions of the base ﬂow change

with Reo, as shown in Fig. 2b.

Figure 3 shows the diﬀerent behaviors of G(t) and R(ω) that can be observed for diﬀerent k,θk, and Reo.

Flow may be asymptotically unstable and therefore have unbounded energy growth, it may be stable but

present some transient growth, or it can present monotonic energy decay, as shown in Fig. 3a. The ﬂow is found

to be highly receptive to stationary crossﬂow waves, as shown in Fig. 3b, and mildly receptive to streamwise

travelling waves with two diﬀerent frequencies, as shown by the two peaks in Fig. 3c.

For asymptotically stable ﬂows we calculate the maximum values of G(t) and R(ω) and denote them Gmax

and Rmax respectively. The dependence of Gmax and Rmax on the total wavenumber kand the waveangle θk

for Reo= 0.8 is shown in Fig. 4. Maximum growth is observed for k= 2.12 and the maximum response is

obtained for k= 1.63, both along the crossﬂow direction θk= 90◦, where the base ﬂow velocity proﬁles are

most inﬂectional.

Figure 5 shows vector ﬁelds of relevant disturbances in the y–zplane, where yis the coordinate along the

wave direction. The disturbance that achieves the largest transient ampliﬁcation, i.e. the optimal initial condi-

tion, corresponds to streamwise vortices, as shown in Fig. 5afor Reo= 0.8. Similarly to plane Poiseuille ﬂow,

Fig. 5bshows that these vortices change little as time evolves, however, high energy streamwise streaks will

form due to the lift-up eﬀect. The most responsive forcing and the most receptive disturbance, i.e. the optimal

forcing and the optimal response, are shown in Figs. 5cand 5dfor Reo= 0.8, and the leading eigenmode for

unstable ﬂow at Reo= 1.07 is shown in Fig. 5e.

Stability and receptivity of boundary layers in a swirl ﬂow channel 9

(a)(b)(c)

Fig. 3 (a) Optimal growth G(t) in the crossﬂow direction. Curves labelled unstable for Reo= 1.07 and k= 0.75, stable for

Reo= 0.62 and k= 2.25, and no energy growth for Reo= 0.30 and k= 2.25. (b,c) Optimal response R(ω) for Reo= 0.8 and

k= 1.25. (b) Crossﬂow direction θk= 90◦. (c) Streamwise direction θk= 0◦.

(a)(b)

Fig. 4 Contours of (a) maximum transient ampliﬁcation Gmax and (b) maximum frequency response Rmax as a function of

the total wavenumber kand the waveangle θkfor Reo= 0.8.

The Reodependence of Gmax in the crossﬂow direction, and the neutral curve for asymptotic stability

are shown in Fig. 6. Two important values for the ﬂow rate Reynolds numbers are calculated: ReE

o= 0.36,

below which there is no energy growth of disturbances, and Rec

o= 0.91 that delimits the onset of asymptotic

instability where there is unbounded energy growth. Between these values, perturbations experience transient

growth followed by asymptotic decay, where the kinetic energy is ampliﬁed up to O(10), as shown in Fig. 6.

For plane Poiseuille ﬂow, transient growth plays an important role in the transition process, whereas for the

swirl ﬂow channel we ﬁnd very small transient growth. This suggests that the mechanism for transition is the

change in the shape of the velocity proﬁle due to rotational eﬀects (crossﬂow instability). It is worth noting that

Ruiz and Carey in Ref. [15] observed that their experimental results deviated from laminar behavior for ﬂow

rates over 190 ml/min, which corresponds to Reo= 0.86 according to our deﬁnition of the ﬂow rate Reynolds

number. Therefore, we ﬁnd reasonable agreement between experimental observations and predictions from our

simpliﬁed analysis. The critical value for the Reynolds number, Rec

ois also useful to determine bounds for the

validity of assumptions and models of the ﬂow inside this type of channel.

10 B. Herrmann-Priesnitz et al.

(a)

(b)

(c)

(d)

(e)

Fig. 5 Disturbance vector ﬁelds for θk= 90◦in the y–zplane, where yis the coordinate along the wave direction. (a,b)

Optimal initial condition and its structure after maximum transient ampliﬁcation for Reo= 0.8 and k= 2.12. (c,d) Optimal

forcing and response for Reo= 0.8 and k= 1.63. (e) Most unstable eigenmode for Reo= 1.07 and k= 2.52.

Stability and receptivity of boundary layers in a swirl ﬂow channel 11

Fig. 6 Contours of Gmax (k, 90◦, Reo). Three regions can be distinguished: Asymptotic instability (dark gray), asymptotic

decay with transient growth (light gray), and asymptotic decay without energy growth (white). Dashed contour levels, from left

to right, Gmax = 2,5,10,15.

The maximum response of the system Rmax increases with the ﬂow rate Reynolds number, and when

approaching Rec

othe forcing energy ampliﬁcation is of O(100). The forcing frequency and the waveangle that

give Rmax are ω= 0 and θk= 90◦, and do not change with Reo. Nevertheless, the direction of the base

ﬂow at the miplane of the channel is aﬀected. Consequently, when increasing Reo, crossﬂow waves manifest as

spirals of decreasing number of arms. The onset of asymptotic instability coincides with the separation of the

boundary layers and the appearance of velocity proﬁles that are most inﬂectional in the radial direction. The

fastest growing mode corresponds to axisymmetric circular waves that travel radially inward, and its structure

is shown in Fig. 5e. The characteristics of the forcing or disturbances that are largely ampliﬁed by the ﬂow are

important for the design and operation of technological devices. A large response should be avoided to ensure

the safe operation of a parallel-disc turbine, but it may be sought to promote mixing in a swirl ﬂow channel

heat sink.

4 Conclusions

This study provides numerical solutions for the energy growth of initial disturbances and the response to ex-

ternal harmonic forcing in a swirl ﬂow channel. Methodology is emphasized with a clear presentation of the

simpliﬁcations that are used. Linearized perturbation equations are obtained using the local and parallel ﬂow

approximations based on order of magnitude arguments. The result is equivalent to the Orr-Sommerfeld and

Squire system for a three-dimensional boundary layer.

The eﬀect of the ﬂow rate Reynolds number on the base ﬂow is presented. When Reoincreases, the ﬂow

direction on the channel midplane changes gradually from radial to azimuthal and eventually boundary layers

separate. Results for the stability analysis show that there is no energy growth if the Reynolds number is less

than ReE

o= 0.36, and the ﬂow becomes asymptotically unstable over Rec

o= 0.91 which coincides with the

separation of the boundary layers. As opposed to plane Poiseuille ﬂow, there is very little transient growth

between ReE

oand Rec

o, with an energy ampliﬁcation only up to O(10). This suggests that the mechanism for

transition to turbulence is the crossﬂow instability that occurs due to the change in the shape of the base ﬂow

12 B. Herrmann-Priesnitz et al.

velocity proﬁles over Rec

o. Our results are able to predict transition, as they present reasonable agreement with

experimental observations made by other authors in Ref. [15].

The particular solution to the harmonically driven problem, i.e. the receptivity analysis, shows a large re-

sponse of the ﬂuid system to external disturbances for subcritical Reynolds numbers, with energy ampliﬁcation

up to O(100). As Reoincreases, the ﬂow becomes increasingly receptive to stationary spiral crossﬂow waves

of decreasing number of arms. For asymptotically unstable ﬂow, the fastest growing mode corresponds to ax-

isymmetric circular waves that travel radially inward, which is the direction of the most inﬂectional velocity

proﬁle. Future work considers including the heat equation and studying the componentwise frequency response

to determine heat transfer enhancement strategies in a swirl ﬂow channel heat sink.

Stability and receptivity of the ﬂow are not only of fundamental interest, but are also relevant to modeling,

design, operation, and control of parallel-disc turbines and swirl ﬂow channel heat sinks. The methodology

presented is particularly useful for early design stages due to its ease of implementation and low computational

cost, which allows an exploration of geometric parameters and operating conditions.

Acknowledgments

B. H-P. thanks CONICYT- Chile for his Ph.D. scholarship CONICYT-PCHA/Doctorado Nacional/2015-

21150139.

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