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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.
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Acta Mechanica manuscript No.
(will be inserted by the editor)
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/ro1. 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 +WU
∂z V2
r=1
ρ
∂P
∂r +ν2U
∂z2,(1b)
U∂V
∂r +WV
∂z +U V
r=ν2V
∂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 +Uu
∂r +V
r
∂u
∂θ +wU
∂z +uU
∂r +Wu
∂z np 2V v
rc
=
∂p
∂r +1
Re 2u
∂r2+1
r2
2u
∂θ2+2u
∂z2+1
r
∂u
∂r u
r22
r2
∂v
∂θ c+fu,(2b)
∂v
∂t +Uv
∂r +V
r
∂v
∂θ +wV
∂z +uV
∂r +Wv
∂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 +Uw
∂r +V
r
∂w
∂θ +uW
∂r +wW
∂z +Ww
∂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., WU, 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 +Uu
∂r +V
r
∂u
∂θ +wU=∂p
∂r +1
Re 2u
∂r2+1
r2
2u
∂θ2+2u
∂z2+fu,(3b)
∂v
∂t +Uv
∂r +V
r
∂v
∂θ +wV =1
r
∂p
∂θ +1
Re 2v
∂r2+1
r2
2v
∂θ2+2v
∂z2+fv,(3c)
∂w
∂t +Uw
∂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 Reoθospace 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+),
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+Uˆw=iαˆp+1
Re D2k2ˆu+ˆ
fu,(4b)
ˆv
∂t + i (αU +βV )ˆv+Vˆw=iβˆp+1
Re D2k2ˆv+ˆ
fv,(4c)
ˆw
∂t + i (αU +βV )ˆw=−Dˆp+1
Re D2k2ˆw+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
Mˆw
∂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
fuiαˆ
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#+"iαM1DiβM1D M1k2
iβiα0#
|{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
q
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(tLq0.(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(tLq0||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=(iωIL)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
||(iωIL)1Bˆ
f||2
E
||ˆ
f||E
=||(iωIL)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||FAF1Fˆq||2
||Fˆq||2=||FAF1||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)F1||2
2,(14a)
R(ω) = ||F(iωIL)1BF1||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=VΛV1. 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(1),...,exp(2N))V1,(15a)
(iωIL)1=Vdiag 1
iωλ1
,..., 1
iωλ2NV1.(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= 8and 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= 0represents 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= 90as 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 yzplane, 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= 90in the yzplane, 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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