Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
Viscous flow in a channel with a sudden expansion
T. Hawa^ & Z. Rusak^
i //zjfzWz/br Maf/zemfm'cs <rW z'
University of Minnesota, USA
Abstract
A theoretical investigation of two-dimensional and viscous flows in a symmetric
channel with a sudden expansion with right angles is presented. The analysis explores the flow states around a critical Reynolds number Rec where asymmetric
states appear in addition to the basic symmetric states when #e > #6c- The size
of the asymmetric perturbation changes like \/#e - #6c- Linear stability studies
of the various equilibrium states show that the symmetric states have a stable mode
of perturbation when #e < #6^ and are unstable when Ae > Ee^. The asymmetric states have an asymptotically stable mode of disturbance. As a result, the
symmetric flow states always evolve into the asymmetric states when #e > #e^.
Space-and time-accurate numerical simulations using the unsteady Navier-Stokes
equations are used to demonstrate the evolution of perturbations in the flow. A
weakly nonlinear analysis of the flow dynamics also shows that the evolution of
the perturbation's amplitude can be described by the Landau equation. The analytical solution shows good agreement with the unsteady simulations. The present
work provides the possible mechanism by which two-dimensional flows in a suddenly expanding channel transition from symmetric to asymmetric states.
1 Introduction
The flow of a laminar and incompressible fluid in a symmetric channel with a sudden expansion shows an interesting nonlinear transition phenomenon from symmetric to asymmetric equilibrium states as the Reynolds number of theflow,/?e,
(or the incoming flow flux) is increased (see the experiments of Durst et al.[l].
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
466
Advances in Fluid Mechanics III
Cherdron et al.[2], Sobey and Drazin [3], Fearn et al.[4], and Durst et al.[5]). When
Re is relatively low, steady, two-dimensional, and symmetric states with two separation zones attached to the expansion corners are observed. However, for higher
values of Re, the flow becomes asymmetric with two recirculation zones of different lengths on either the upper or the lower wall of the channel.
The theoretical study of flow states in the channel focuses on numerical simulations and linear stability analyses. The numerical simulations of Fearn et al.[4],
Battaglia et al.[6]. Drikakis [7], and Alleborn et al.[8] demonstrated the existence
of a pitchfork symmetry-breaking bifurcation at a certain Rec with a fixed channel
expansion ratio, D/d. Pairs of stable and asymmetric solutions of opposite senses
with respect to the channel centerline were found in addition to the symmetric
states when #e > #e,,. Battaglia et al.[G] and Drikakis [7] also found that the
pitchfork symmetry-breaking bifurcation may occur at all expansion ratios greater
than 1 and that the critical Reynolds number, Ae^ decreases with increasing D/d.
The linear stability analyses of the symmetric states in the channel by Shapira
et al.[9] showed that, for a fixed D/d, the flow becomes unstable as Re is increased above #6^ or, for a fixed #e, the flow becomes unstable as the expansion
angle increases. Alleborn et al.[8] conducted numerical linear stability studies and
found similar results. They used the linearized unsteady Navier-Stokes equations,
which formulate a complicated eigenvalue problem, to solve the growth rate of the
eigenmode of perturbation to the symmetric state at any given Re.
The previous experimental and numerical investigations demonstrate that the
transition phenomenon described above may be related to the bifurcation of asymmetric steady states at #6c and to the loss of stability of the symmetric states when
Re > R&C- A theoretical and computational approach which clarifies this transition behavior may shed more light on the flow dynamics and the mechanism by
which asymmetric states are established as the flow Reynolds number is increased.
The objective of this paper is to present a review of our recent bifurcation analyses, linear stability studies, and numerical simulations of the evolution of twodimensional viscous flows in a symmetric channel with a sudden expansion (see
Hawa and Rusak [10] and Rusak and Hawa [11]). We carefully explore the balance
between the viscous and the convection effects on the dynamics of perturbations.
The results explain the relationship between the static bifurcation of solutions of
the steady Navier-Stokes equations and the change of stability related with the bifurcation point. They provide more insight into the relationship between the linear
stability results and the unsteady flow simulations. The results help to identify the
possible mechanism by which asymmetric states appear out of the flow dynamics.
2 The problem formulation
A two-dimensional, viscous, and incompressiblefluidis studied in a long channel
of width d which suddenly expands symmetrically, with right angles, to a long
channel of width D where D > d (see Figure 1). Axial and transverse lengths
x and y are the rescaled lengths by the upstream channel width d. The y — 0
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
467
Advances in Fluid Mechanics I
cl
Fieure 1: A two-dimensional channel with a symmetric sudden expansion
axis is the centerline of the channel and the expansion section is at Z = 0. The
inlet section to the channel is located at -oo. The outlet section of the channel is
located at +00. The flow domain inside the expanding channel is defined as ,4. All
velocities are normalized by Ua^. which is the averaged velocity of the Poiseuille
flow far upstream of the channel expansion at the inlet section. The time t is scaled
by d/l''„,,,,. The Reynolds number characterizing the flow problem is defined as
#e = [%(,c((/;/ where // is the kinematic viscosity. The non-dimensional XavierStokes equations in the vorticitx -stream function formulation are given by
(3)
Here (](2\(/.f) denotes the vorticity, %'(z\z/.f) denotes the stream function, and
w(.r. z/. ^), r(.z\ (/, ^) are the axial and transverse velocity components, respectively.
Eqn (1) is the vorticity transport equation of motion and eqn (2) is the Poisson
equation relating between the stream function and the vorticity. These equations
are used to solve the dynamics of the entire flow field. The boundary conditions
along the lower and upper walls are the tan gen cy and no-slip conditions. A steady
Poiseuille flow velocity profile is considered far upstream at the inlet of the channel. The outlet flow at far downstream of the channel is assumed to have a fully
developed velocity profile for all time t.
The problem defined above describes the evolution of the flow in a channel for
every Re and D/d. It may have steady-state solutions which describe equilibrium
flow states, some of those may be symmetric about the ,i'-axis and others may be
asymmetric states. Also, some of the equilibrium states may be stable and others
may be unstable to two-dimensional perturbations. In the following sections we
use this problem to define the critical Reynolds number Re<.. to analyze the flow
states around Re,-* and to clarifv their stability characteristics.
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
468
Advances in Fluid Mechanics III
3 Bifurcation and stability analyses
3.1 The bifurcation of asymmetric flow states
The stream function and vorticity for steady asymmetric flow at Re close to Re<;
may be described by
where 0 < |e^| <K |e^| <K 1 and e^ » |A#e|/#er. Subscripts a and /3 denote
perturbations to the base symmetric state at Re? which is described by ?/>* and fl*.
Inserting eqns (4) and (5) into the steady version of the eqns (l)-(3), results in a
linearized homogeneous system of the equations of motion [10],
^
+
-^
- vn
- n,,,, + ^
/^e, = o,
The boundary conditions result in the homogeneous conditions:
UK - Va — 0a = 0
Q^ = 0
along all boundaries,
at the inlet and outlet sections.
(8)
(9)
Similar conditions apply to 0^ and fl/j.
The problem described by eqns (6)-(9) formulates an eigenvalue problem for
the solution of the critical Reynolds number, #e^ and the corresponding eigenfunction ^. Notice that this is a very complicated problem to solve since •</>* and
n* are also complicated functions of Re^ The numerical solution of this eigenvalue problem by Alleborn et el. [8] demonstrates the possible existence of a finite,
first, positive eigenvalue of the problem defined as the critical Reynolds number
Rec. It is also found that ifj^ describes an asymmetric flow perturbation to the base
symmetric state. Therefore, it can be concluded that the solution of eqns (6)-(9)
provides the critical point for the bifurcation of asymmetric states.
The second-order problem suggests that for the richest case
^e.
(10)
It is found that e/#/3 is given by e/#f; = (V^i + 6^1,1) A#e/#6c where ^i is
the symmetric perturbation to the basic symmetric state and ^i^ is the perturbation
due to the eigenmode ?/>« (see Hawa and Rusak [10]).
The constant ko in eqn (10) can be determined from the third-order problem as
(see Hawa and Rusak [10] for more details):
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
Advances in Fluid Mechanics III
469
The above asymptotic analysis demonstrates that the solution of the leadingorder term exists only when Re > R€C. Therefore, when Re > Rec* steady
asymmetric states, ^s = ^ -f- /cn\/A^e/^ec^ 4- • • % exist in addition to the
steady symmetric states, ?/;,, = V^I + (AJ?e/^er)^i H----. However, when #e <
Rec, only the symmetric states exist. These arguments clarify the nature of the
critical state at Re? as a pitchfork bifurcation point for steady asymmetric states.
3.2 Linear stability analyses
To consider the stability of the symmetric states, i\,(x,y, Re: D/d). let
where 0 < (5 <K |A7Ze|/^ec, (/' is an unsteady stream function disturbance, and
H is an unsteady vorticity disturbance. The evolution of the disturbances in the
channel as described in section 2 is studied. Substituting expansions (12) into
the unsteady equations (l)-(3), neglecting second-order terms, and considering a
suitable mode analysis of the following form:
gives an eigenvalue problem for the solution of cr. 0, and <&. In general, the perturbation growth rate a — CTRS + i&is may be a complex number and 0 and <l are
complex functions. Using asymptotic analysis, we find (Hawa and Rusak [10]):
-•- ,
(14)
,
( 15)
#e
where k^ is the rate of change of stability for the symmetric states,
It is found that kg > 0 for all D/d. This result shows that the symmetric states lose
their stability as Re is varied around Re^ and become unstable when Re > Re,,.
The stability analysis of the asymmetric states ^ is conducted in a similar
way. It is found (Hawa and Rusak [10]) that the perturbation to the base asymmetric state is also described by eqns (13) with a — (jRas + icrias and that
o-Ra.s = /Cn,,A#e/#fc where /c%.s = -2&s < 0 for every D/d > 1. This result
shows that the asymmetric states have an asymptotically stable mode of perturbation but they lose their stability characteristics as Re is reduced toward Re<,.
The stability analyses suggest the mechanism by which the symmetric states
lose their stability. There are three major effects which are the convection of the
vorticity perturbation by the axial velocity of the base flow, the convection of the
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
470
Advances in Fluid Mechanics III
base flow vorticity by the axial flow perturbations, and the viscous dissipation
of the vorticity perturbation. The first effect tends to convect the perturbations
downstream and has a stabilizing influence. The second effect is related to the
shape of the asymmetric eigenmode ipa- This perturbation creates an upstream
axial speed in one side of the channel and a downstream axial speed in the other
side of the channel. Therefore, it has a destabilizing effect, specifically in the side
of the channel where an upstream speed is induced. The viscous dissipation has a
stabilizing effect. The combination of these effects determines the stability of the
base symmetric flow. The viscous dissipation is large and dominant when Re is
relatively small and. therefore, the flow is stable. As Ae is increased, the viscous
dissipation effect decreases and, therefore, the symmetric flow becomes less stable.
At Rer. there is a critical balance between the destabilizing upstream convection
effects by the asymmetric perturbation and the combined stabilizing effects of the
viscous dissipation and the downstream convection by the base flow. When Re is
further increased the upstream convection effects by the asymmetric perturbation
become dominant and, therefore, the symmetric flow becomes unstable.
The stability analyses also explain the mechanism by which asymmetric states
become stable when Be > Ee^. In this case, the asymmetric perturbation to the
symmetric stale tends to grow as described above. As the perturbation grows, the
convection of the vorticity perturbation by the axial velocity perturbations also
grows and has a stabilizing influence. This nonlinear effect stabilizes the growth
of the asymmetric perturbation and establishes the asymmetric steady state.
4 Numerical studies
This section demonstrates the asymptotic results and compares it with the results of
direct numerical simulations using problem formulated in section 2. The numerical
scheme is described in details in Hawa and Rusak [10]. The scheme is based
on a uniform mesh which is constructed in the flow domain A, with grid points
labeled by (ij). The time derivative term in eqn (1) is approximated by a firstorder forward-difference expression and the space derivatives in the convective
and viscous terms are approximated by second-order central-difference schemes.
The finite-difference equation of eqn (1) is used to compute the evolution of the
vorticity field. At any time level, n, we use a Poisson equation solver, that is
based on second-order central-differences in space, to solve eqn (2) and describe
the stream function field related with the vorticity field at that time level.
For every Re and D/d, we consider the base, steady and symmetric states at
those given parameters, ^ and £V At the initial time level, t — 0, we introduce
perturbations to the base state. Typically, the initial states are:
^(z, vy, 0; #e; D/d)
=
(l^,
,y; #e; D/d) + (X/%rW
where 5 is the size of the initial perturbation. In this section we study the flow in a
channel with D/d = 3 for which #e^ = 53.8 ([4], [0], [7], [8], [9]).
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
Advances in Fluid Mechanics 111
471
t=o
Figure 2: Time-history plots of the stream function ^(z^f) f
Re - 40 in a channel with D/d - 3 and with 6 - 2.0.
We first study a flow with #e = 40 < jRe^. In this case the base symmetric
solution is unconditionally stable to any two-dimensional disturbance. Starting
from the initial conditions (17) with () = 2.0 we can see (Figure 2) that even a
large asymmetric disturbance decays relatively quickly and the flow returns to a
symmetric state and stays there as expected.
To demonstrate the convergence to symmetric states when #e < ^e^ we describe the time-history plots of a norm of the difference of the temporal solution
from the symmetric state, defined as A|j; = moz,j(|^ - ^,j|)
(see figure 3).
It can be seen that for the two cases with #e = 40 and 6 = 0.1, 2.0 the dynamics
of the disturbance is similar and has the same rate of decay for both values of (5.
Also shown in figure 3 is the dynamics of disturbances for several Re. It shows
that the decay rate of a disturbance decreases as Re is increased toward the critical
value and the symmetric flow loses its stability characteristics.
We study now a Mow with 7?e = 65 > #e^ In this case, the symmetric state
is unstable to two-dimensional disturbances. Starting from the initial conditions
(17) with 5 — 0.1 we can see (Figure 4) that the relatively small initial asymmetric
disturbance grows in time and evolves in a nonlinear process into an asymmetric
state which reaches a steady state, as expected by the stability study.
Figure 5 shows time-history plots of the norm dA^,/df for values of Reynolds
number above the critical value Re<: - 53.8 with same initial disturbance of 8 —
0.1. The comparison between the lines shows that the growth rate of a disturbance
increases as Re is increased above the critical value.
The time-history plots in figures 3 and 5 are used to compute the decay or
growth rates of disturbances, a^ for the symmetric states and ana., for the asymmetric states, as function of Re for a fixed expansion ratio D/d = 3. It is found
that (jfi, grows monotonically with Re and CTR,, = 0 at the critical state where
Re — Re,, ~ 53.8, in complete agreement with Battaglia et al.[6], Drikakis
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
472
Advances in Fluid Mechanics III
A"
^-\ \u
^
,
,
Figure 3: Time-history plots of the norm A'J; for various flows with Re < Rec in
a channel with D/d — 3. Initial disturbance is indicated in each case.
t=8O
t=28O
Figure 4: Time-history plots of the stream function 0(.T,y,£) for a flow with
Re = 65 in a channel with D/d - 3 and with <5 = 0.1.
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
Advances in Fluid Mechanics III
473
dt
Figure 5: Time-history plots of the norm d/\",Jdt for various flows with Re > Re,,
in a channel with D/d — 3. Initial disturbance is indicated in each case.
[7], and Alleborn et al.[8]. Moreover. (?#,, < 0 when #e < ^6c and (?#,, > 0
when #e > ^e^. We also find that near ^e^ the rate of stability parameter
/Cg = (%Rg/(AAe/Bec) = 0.105 when D/d = 3. This value is in good agreement with the value k,, — 0.102 found according to the formula (16). Similarly,
we find &%,, = cr^,/(AEe/^ec) = -0.195 when D/d = 3. This value is in
agreement with the value k,^ — -0.204 predicted by the theory. Also, it shows
that kas ~ — 2/c.^ as predicted by the theory.
5 A weakly nonlinear approach
In order to study the dynamics of the flow near a critical Reynolds number. Re^,
we consider the following asymptotic expansions:
The perturbations to this base flow state are given by a combination of first- and
second-order perturbations where 0 < e^\ <C e,*| 4C 1 and are functions of
ec and of time f. The first- and second-order problems are same as those
problems described in section 3.1. However, the third-order problem gives a non-
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
474
Advances in Fluid Mechanics III
linear homogeneous first-order differential equation for the evolution of the perturbation's amplitude (see Rusak and Hawa [11] for more details):
Eqn (20) is Landau equation for e«. In eqn (20)
The numerical computations show that ^ > 0 and 6 < 0 for every channel expansion ratio D/d. Equation (20) describes the evolution of e^ for a given A#e/#6c,
staring from an initial perturbation, e^o- The solution of (20) shows that in the
leading order the flow perturbation evolution is described by:
__
(22)
Equation (22) demonstrates again that when A#e < 0 (or #e < 7?6r) the asymmetric perturbation decays in time and the flow returns to a steady and stable
symmetric state described by ?As = VC + (A^e/J?eJV;i + • • •. However, when
A#e > 0 (or #e > 7?e,;) the asymmetric perturbation grows in time, the symmetric state becomes unstable, and it evolves into an asymmetric steady and stable
state describedbyi&s = ^I+koyA^^^
The asymptotic solution (22) describes the essence of the nonlinear dynamics of
the transition process at Reynolds numbers near the critical value.
In order to demonstrate the results of the weakly nonlinear analysis, we also
conducted direct numerical simulations of the problem formulated in section 2.
The time-history plots of A^ of the solution show (see [11]) that the result (22)
agrees with the numerical simulations when |A/?e /Rec <K 16 Conclusions
The dynamics of a viscous flow in a symmetric channel with a sudden expansion
and the transition from symmetric to asymmetric states can be analyzed by an
asymptotic approach. The analysis is based on a study of the unsteady NavierStokes equations around the critical Reynolds number, Re^ where a static bifurcation occurs. The bifurcation of asymmetric states at Re,- has a pitchfork nature. It is also found that the amplitude of the asymmetric perturbation grows as
The linear stability analysis shows that when #e < #6^ the symmetric states have a stable mode of perturbation and that when Re > Re<, the
symmetric states become unstable. In addition, it is shown that the rate of change
of stability for the asymmetric states is /c,,.s = -2k, < 0 for every D/d > 1 when
Re > Rec. The stability analyses suggest the mechanism by which the symmetric
states lose their stability. The stabilizing effect of viscous dissipation decreases as
Advances in Fluid Mechanics III, C.A. Brebbia & M. Rahman (Editors)
© 2000 WIT Press, www.witpress.com, ISBN 1-85312-813-9
Advances in Fluid Mechanics III
475
Re is increased, and therefore, when Re > Rec. a symmetric state loses its stability
and evolves into a steady asymmetric state. The evolution of asymmetric perturbations in the expanding channel is demonstrated by direct numerical simulations
of unsteady Navier-Stokes equations and the results support the analyses.
The evolution of the perturbation's amplitude can also be studied by a weakly
nonlinear analysis. It is shown that the perturbation's amplitude as function of
Re near Re^ can be described by an ordinary, nonlinear, first-order differential
equation (similar to Landau equation). The Mow evolution, as described by the
analysis, shows nice agreement with time-history plots from simulations using
the unsteady Navier-Stokes equations, specifically when Re is around Rec- The
analysis provides insight into the previous experimental and numerical results and
sheds more light on the transition of a viscous flow in an expanding channel.
References
[I] Durst, F.. Melling, A. & White law, J.H. Low Reynolds number flow over a
plane symmetrical sudden expansion* Journal of Fluid Mechanics, 64. pp.
I l l , 1974.
[2] Cherdron, W.. Durst. F. & White law, J.H. Asymmetric flows and instabilities
in symmetric ducts with a sudden expansion. Journal of Fluid Mechanics, 84.
pp. 13-31, 1978.
[3] Sobey, I.J. & Drazin, P.G. Bifurcations of two-dimensional channel flows.
yo/u7z^/o/Hf/zWM^c/7o/7/c6, 171, pp. 263-287, 1986.
[4] Fearn, R.M., Mullin, T., & Cliffe. K.A. Nonlinear flow phenomena in a
symmetric sudden expansion, Journal of Fluid Mechanics, 211, pp. 595, 1990.
5] Durst. F., Pereira, J.C.F. & Tropea, C. The plane symmetric sudden expansion
flow at low Reynolds numbers. Journal of Fluid Mechanics, 248, pp. 567-581,
1993.
[G] Battaglia, F., Tavener, S.J.. Kulkarni, A.K. & Merkle, C.L. Bifurcation of low
Reynolds number flows in symmetric channels, AIAA JournaL 35(1),
pp. 99-105. 1997.
[7] Drikakis, D. Bifurcation phenomena in incompressible sudden expansion flows,
P/%\\Y/^q/T/a/W6\ 9, pp. 76-87, 1997.
[8] Alleborn, N., Nandakumar, K., Raszillier, H. & Durst, F. Further contributions
on the two-dimensional flow in a sudden expansion. Journal of Fluid
MfcW,/cs\ 330, pp. 169-188. 1997.
[9] Shapira, M.. Degani, D. & Weihs, D. Stability and existence of multiple
solutions for viscous flow in suddenly enlarged channels. Computers and
F/z,/Yk, 18(3). pp. 239-258, 1990.
[10] Hawa. T. & Rusak, Z. The dynamics of a laminar flow in a symmetric channel
with a sudden expansion. Journal of Fluid Mechanics (in Review), 1999.
fill Rusak, Z. & Hawa, T. A weakly nonlinear analysis of the dynamics of a
viscous flow in a symmetric channel with a sudden expansion. Physics of
11(12), pp. 3629-3636, 1999.