Theoretical and Computational
Fluid Dynamics
Theoret. Comput. Fluid Dynamics (2002) 15: 265–281
 Springer-Verlag 2002
Numerical-Asymptotic Expansion Matching for Computing
a Viscous Flow Around a Sharp Expansion Corner
Takumi Hawa
Institute for Mathematics and Its Applications, University of Minnesota,
Minneapolis, MN 55455, U.S.A.
hawa@ima.umn.edu
Zvi Rusak
Department of Mechanical, Aerospace and Nuclear Engineering,
Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A.
rusakz@rpi.edu
Communicated by A. I. Ruban
Received 14 August 2000 and accepted 25 October 2001
Abstract. A computational technique which is based on a numerical-asymptotic expansion matching for
computing the local singular behavior of a viscous flow around a sharp right-angle expansion corner is
presented. Moffatt’s (1964) asymptotic solution is extended and a matching with a time-marching finitedifference scheme of the Navier–Stokes equations is formulated. Local mesh refinement around the corner
is required to meet the validity of the asymptotic solution. Flows in an expanding channel with expansion
ratio D/d = 3 at various Reynolds numbers 1 ≤ Re ≤ 700 are simulated. The results are compared with
those from a standard finite-difference scheme that uses second-order forward/backward differences near
the corner. It is found that the results of the standard scheme converge toward those of the present technique as the level of local refinement near the corner is increased. The time-dependent parameters of the
first two terms of the asymptotic solution at the steady-state solution are also described for various cases
of Re and D/d. It is demonstrated that the present method enhances the accuracy of the simulations and
requires less refinements near the corners to achieve converged numerical results.
1. Introduction
Variety of viscous flow problems include sharp corners at the boundaries of the flow domain. Examples of
such cases are the flow around a forward- or backward-facing step, the flow over and inside a cavity, the flow
in an expanding channel, and the flow near the trailing edge of an airfoil. As a result of the sharp change in
the slope of the boundary surface at the corner, vorticity and stress singularities appear in the flow around
the corner (see Moffatt, 1964). These singularities may dominate the behavior of the flow near the corner
as well as in the entire domain. They are typically related to the appearance of flow separation and reversed
flow regions which can have significant engineering implications. The sharp corner singularities also create
serious numerical difficulties in accurately resolving the flow field around the corner. Unless special numeri265
266
T. Hawa and Z. Rusak
cal algorithms such as mesh refinement techniques are used, these difficulties may affect the accuracy of the
overall numerical simulation in the computational domain.
Demuren and Wilson (1994), among many others, studied the flow around a forward- or backward-facing
step. They investigated the sources of uncertainties in their numerical solution schemes of the Navier–Stokes
equations. They focused on the effects of truncation errors, discretization errors, outflow boundary conditions, incomplete iterative convergence, and the computational grid aspect ratio. Their study estimated the
optimum grid-cell aspect ratio for computational accuracy and efficiency. However, like in many other studies, they applied a Poiseuille flow inlet condition right at the channel expansion cross section without having
the upstream part of the channel in front of the expansion. In this way they avoided a solution of the full corner singularity since there is no physical corner in their computational domain. It is clear that such an inlet
condition does not accurately describe the physical situation. In the present work we compute a flow in an
expanding channel including the upstream part of the channel in front of the expansion section and look for
an accurate calculation of the viscous flow around the expansion corners.
Wang (1994) also studied a laminar flow in a channel with forward- and backward-facing steps. His numerical scheme used an eigenfunction expansion and point-match method. Complex flow domains can be
decomposed into contiguous simpler subdomains and along their common boundaries the solutions to each
subregion can be matched. The use of point-matching along the boundary between a subregion around the
corner and the rest of the flow domain made the numerical algorithm of solution simple and efficient.
Gatski and Grosch (1985) investigated a laminar flow over an embedded cavity with sharp corners. In
order to handle the corners singularities and to compute the pressure-induced forces on the vertical walls of
the cavity, they used in their numerical computations average values of the dependent variables over each
edge of the cell. Such an averaging technique typically smears the corner singularity. Therefore, many levels
of mesh refinements around the corners are needed to improve the accuracy of the numerical solution.
The numerical studies of Fearn et al. (1990), Shapira et al. (1990), Battaglia et al. (1997), Alleborn
et al. (1997), and Drikakis (1997) constructed numerical solutions of flows in channels with sharp expansions. They studied the flow behavior as the Reynolds number is increased and demonstrated the bifurcation
of asymmetric states at a certain critical Re. These studies also used mesh refinement techniques to improve the numerical treatment of the solution around both the upper and lower expansion corners of the
channel.
The corner flow can also be found at the trailing edge of plates and airfoils. The Blasius solution of the
boundary layer on the plate is not valid in the neighborhood of the trailing edge because of the breakdown
of the assumptions made in the theory, similar to flow singularities found around a sharp corner. McDonald
and Briley (1983) presented a numerical algorithm which is based on viscous–inviscid interactions, in which
the boundary layer and wake solutions are matched with the external inviscid flow by interactive means. The
singularity at the trailing edge and at the point of separation was avoided by the interaction with the external
flow.
Chen and Patel (1987) constructed numerical solutions of the Navier–Stokes equations for a laminar flow
around and beyond the trailing edge of a flat plate at high Re. Comparison of their solutions with those obtained by the triple-deck and interactive boundary layer theories showed agreement and demonstrated the
regions of validity of these theories.
Holstein and Paddon (1981) studied the corner singular behavior using a finite-difference scheme. They
applied the first mode of Moffatt’s (1964) corner solution to construct a vorticity related function in which
the singularity at the corner is removed with the aid of terms from the vorticity in the corner solution.
Central-difference approximation of this function was then applied to estimate the vorticity derivatives and
the fictitious vorticity at the corner. This technique allowed the use of a standard numerical scheme for the
entire flow computation.
Botella and Peyret (1998) have recently computed lid-driven cavity flows by using a Chebyshev collocation method. To achieve accurate calculations, they used the Moffatt (1964) solution in the vicinity of the 90◦
compression corners and matched it with the global numerical scheme for the flow. However, their method
was not extended to compute viscous flows in domains with expansion corners. The flow around an expansion corner is different in its physical behavior and is more complicated than that around a compression
corner.
Most of the previous numerical solutions of flow problems with sharp expansion corners at the boundaries concentrated primarily on refinement of the mesh step sizes around the corners to improve solution
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
267
accuracy. However, it is clear that many levels of local mesh refinements are needed to achieve this goal,
specifically when the Reynolds number is high. Other techniques used complicated matching algorithms or
assigned a finite value of the vorticity and stresses at the corner. Such methods may not provide sufficient
accuracy to the computations.
In this work we present a computational technique which is based on numerical-asymptotic expansion
matching for computing the local singular behavior of a viscous flow around a sharp right-angle expansion
corner. It should be noted that the flow singularity at the expansion corners is very different from that around
a compression corner. We use the two leading-order modes of the asymptotic solution of the flow around
the corner by following Moffatt’s (1964) asymptotic solution. The first-order mode describes a smooth turning of the flow around the corner without any flow separation and the second-order mode describes a flow
separation oriented along a 135◦ line from both walls creating the right corner. We also identify the range
of validity of the singular behavior around the corner as function of the Reynolds number. The extended
corner solution and the resulting scaling are used in a time-marching finite-difference scheme of solution
of the Navier–Stokes equations to compute the flow in a symmetric channel with a sudden expansion. The
inner analytical solution around the corners is matched with the outer numerical solution. Symmetric flow
in an expanding channel with expansion ratio of 3 at various Reynolds numbers 1 ≤ Re ≤ 700 are simulated. The results are compared with those from a standard finite-difference scheme that uses second-order
forward/backward differences near the corner.
The present technique provides accurate calculations of a viscous flow in a domain with right-angle expansion corners. Such a method was not given before. The present computations show that the results of
a standard scheme converge toward those of the present technique as the level of local refinement near the
corner is increased. This is an interesting behavior that supports the accuracy of the present method and also
demonstrates that when using the present technique, less refinements near the corners are needed to achieve
converged numerical results.
2. The Right-Angle Expansion Corner Solution
A two-dimensional and incompressible viscous flow problem of a Newtonian fluid in which a sharp right corner exists at the rigid boundary of the flow domain (such as a flow in an expanding channel with right-angles
or a flow around a backward-facing step) is considered. This problem is characterized by a global length
scale l and speed U. The distances are scaled with l, the speeds with U, time with l/U, and the Reynolds
number is Re = lU/ν (where ν is the kinematic viscosity). The local flow around the right corner is studied
using a local polar coordinate
system (r, θ), where the system’s origin is at the corner (the point r = 0 is at
the expansion corner), r = x 2 + y 2 , θ = arctan(y /x ), and the axes x and y are aligned with the walls
constructing the corner (the line θ = 0 is parallel to the x -axis facing downstream, see Figure 1).
The asymptotic analysis of the flow around a sharp right corner follows the paper by Moffatt (1964). The
Navier–Stokes equations in the vorticity-stream function formulation take the form:
∂Ω
∂Ω vθ ∂Ω
1 ∂ 2 Ω 1 ∂Ω 1 ∂ 2 Ω
+
(1)
+ vr
+
=
+ 2 2 ,
∂t
∂r
r ∂θ
Re ∂r 2
r ∂r
r ∂θ
2
∂ ψ 1 ∂ψ
1 ∂2ψ
+
,
(2)
+
Ω=−
∂r 2
r ∂r r 2 ∂θ 2
1 ∂ψ
∂ψ
vr =
,
vθ = −
,
(3)
r ∂θ
∂r
where vr (r, θ, t) and vθ (r, θ, t) denote the velocity components in the radial and circumferential directions,
respectively (see Figure 1).
The boundary conditions at the walls are the no-slip and no penetration conditions, i.e., at any time t and
distance r the conditions are
vr (r, θ = π, t) = 0,
vr (r, θ = −π/2, t) = 0,
vθ (r, θ = π, t) = 0 ,
vθ (r, θ = −π/2, t) = 0 .
(4)
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T. Hawa and Z. Rusak
Figure 1. Local polar coordinate system at the lower expansion corner.
The behavior of the corner solution must match that of the outer solution as r increases.
A local asymptotic solution around the right-angle corner in the region 0 < r 1 is considered. It is
expected that for any time t, the stream function variations near the corner are relatively small. Therefore,
according to Moffatt (1964), for time t the stream function ψ(r, θ, t) is given by the following asymptotic
expansion:
ψ(r, θ, t) = ψ0 + r α Fα (θ, t) + r β Fβ (θ, t) + · · · ,
(5)
where ψ0 is a constant reference value of the stream function at the wall. From (5), the velocity components
vr (r, θ, t) and vθ (r, θ, t) can be computed. Also, for finite speeds near the corner, the power α must be greater
than 1.
Substituting (5) into (1)–(3) and neglecting the higher-order time derivative terms and convective terms
in the vorticity transport equation, results in the Stokes flow equation in the leading order O(r α−4 ):
2
∂ 4 Fα 2
2 ∂ Fα
+
α
+
(α
−
2)
+ α2 (α − 2)2 Fα = 0 .
∂θ 4
∂θ 2
(6)
From the wall boundary conditions (4), the two leading-order terms of the asymptotic expansion of the
stream function around the corner are found,
(7)
ψ(r, θ, t) = ψ0 + Dα1 (t)r α1 f α1 + Dα2 (t)r α2 f α2 + O Dα2 1 (t)Re r 2α1 ,
where the α1 = 1.5444 · · · term describes a smooth turning of the flow around the sharp corner without
any flow separation and the α2 = 1.9085 · · · term describes a symmetric flow separation at the corner. The
method for computing these special powers for the expansion corner problem was originally given by Moffatt (1964) and was first used in numerical computations by Holstein and Paddon (1981). Also, for i = 1, 2
we define
f αi (θ) =
Aαi
Bα
Cα
sin(αi θ) + i cos(αi θ) + i sin((αi − 2)θ) + cos((αi − 2)θ) .
Dαi
Dαi
Dαi
(8)
Here the relations between the various parameters are given by
Aα1
Dα1
Bα1
Dα1
Cα1
Dα1
= 0.5430 · · · ,
= 0.2030 · · · ,
= −0.3738 · · · ,
Aα2
Dα2
Bα2
Dα2
Cα2
Dα2
= −0.2189 · · · ,
= 3.0420 · · · ,
(9)
= 13.8956 · · · ,
where Dα1 (t) and Dα2 (t) are parameters to be determined from matching the solution (7)–(9) with the outer
numerical solution of the specific problem studied.
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
269
The solution (7)–(9) indicates that the first two terms are governed by the viscous effects only. It also
shows that the asymptotic expansion near the corner is valid when the distance r from the corner satisfies
the condition
0≤r 1
.
Re1/α1
In addition, the local asymptotic solution of vorticity and pressure are described by
d2 f α1
d2 f α2
α1 −2
2
α2 −2
2
Ω(r, θ, t) = − Dα1 (t)r
α1 f α1 +
− Dα2 (t)r
α2 f α2 +
dθ 2
dθ 2
+ O Dα2 1 (t)Re r 2α1 −2 ,
1/Re α1 −2 2 d f α1 d3 f α1
p(r, θ, t) = p0 + Dα1 (t)
r
α1
+
(α1 − 2)
dθ
dθ 3
1/Re α2 −2 2 d f α2 d3 f α2
+ Dα2 (t)
α2
+ O Dα2 1 (t)r 2α1 −2 .
r
+
3
(α2 − 2)
dθ
dθ
(10)
(11)
(12)
Note that locally the pressure gradients balance the viscous forces. It can be seen that the velocity components vanish as r → 0. On the other hand, the components of acceleration of the fluid, the vorticity, the
pressure, and the viscous stresses vary like r −0.4555 and are singular at the corner. Yet integration of the local
forces exerted on the planes θ = π and θ = −π/2 shows that they are of order O(r 0.5444 ), tend to zero as r
approaches zero, and no singular forces act on the corner. Moreover, in practice the two planes constructing
the corner never make a perfect sharp contact and there is always a small but finite radius of curvature at the
corner and the maximum acceleration, vorticity, pressure, and viscous stresses are finite.
The local asymptotic behavior of the flow near the corner for various values of the parameters Dα1 and
Dα2 is studied (see Figure 2). When Dα1 (t) = 1 and Dα2 (t) = 0, the second-order term vanishes and the firstorder term dominates the behavior of streamlines. The streamlines pattern in this case shows that the flow
turns smoothly around the right-angle sudden expansion corner without any separation (see Figure 2(a)). On
the other hand, when Dα1 (t) = 0 and Dα2 (t) = −1, the first-order term vanishes and the second-order term
dominates the behavior of the streamlines. The streamlines pattern in this case shows that the flow separates
from the sudden expansion corner with a separation line inclined along the line θ = 45◦ . The flow is symmetric about this line (see Figure 2(h)). Since the local solution for ψ is a linear combination of these two basic
solutions, it is found that the behavior of the streamlines near the expansion corner is, in the leading orders,
a combination of the mode of smooth turning flow around the corner and the mode of a symmetric separated
flow.
When Dα1 (t) = 1 and Dα2 (t) becomes negative, the flow tends to develop a separation region that approaches the corner as Dα2 becomes more negative (see Figure 2(b)–(d)). As Dα2 is further decreased, the
effect of the first-order term becomes smaller and the flow pattern is dominated by the second-order term.
This situation describes a nearly symmetric separated flow where the separation line starts very close to the
corner (see Figures 2(e)). A similar behavior happens when Dα1 is decreased to zero and Dα2 is fixed (see
Figures 2(e)–(h)). When Dα1 is further decreased to negative values, the separation line moves upstream
along the horizontal wall (see Figure 2(i)).
We comment here that when both Dα1 and Dα2 are positive the flow pattern is reversed, i.e., the flow runs
from right to left. Such cases are beyond the interest of the present study.
3. Numerical Technique
The asymptotic solution (7)–(9) is used now in numerical computations to establish the values of the flow
properties around the corner. A numerical-asymptotic expansion matching for computing the local singular
270
T. Hawa and Z. Rusak
Figure 2. Streamline patterns near an expansion corner according to the asymptotic solution for various combination of Dα1 (t) and
Dα2 (t).
behavior of a viscous flow around a sharp right-angle corner is constructed. In the present numerical computations, a uniform finite-difference mesh (∆x, ∆y) is used in the flow domain with grid points labeled by
(i, j). Also, equal time steps ∆t are considered where each time level is labeled by n.
To avoid computation of the value of the vorticity at the corner point, the asymptotic solution is applied
at the seven grid points surrounding the corners. These grid points are denoted by ◦ in Figure 3. The asymptotic solution is expected to match the numerical outer solution computed from a global scheme along the
intermediate grid points, denoted by ♦ in Figure 3. The two time-dependent parameters Dα1 (t) and Dα2 (t)
in the asymptotic solution (7) are computed from matching both solutions for ψ along the intermediate grid
points. It should be emphasized that this matching has to be done under the required condition (10). From
(7), the relationship between the location and ψ at each one of the 13 intermediate points and at each time
level n may be described in a matrix form by
(n)
ψ1 − ψ0
(n)  ψ − ψ 
0
Dα1 (t)
 2
X
=
 ,
.
Dα2 (t)


..

ψ13 − ψ0
(13)
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
271
Figure 3. Grid geometry near an expansion corner.
where the matrix X is given by

r1α1 f α1 (θ1 )
r1α2 f α2 (θ1 )

 r α1 f (θ ) r α2 f (θ ) 
α
2

2 α2 2 
X =  2 .1
.
..
..


.
α1
α2
r13 f α1 (θ13 ) r13 f α2 (θ13 )
(14)
Multiplying both sides of (13) from the left by the transpose matrix of X, X T , it is found that

(n)
ψ1 − ψ0
(n)
 ψ2 − ψ0 
Dα1 (t)


= XT 
Y
 ,
..
Dα2 (t)


.
ψ13 − ψ0
(15)
Y = XT X .
(16)
where the matrix Y is given by
Then, multiplying both sides of (15) from the left by the inverse matrix of Y, Y −1 , it is found that

(n)
ψ1 − ψ0
(n)
 ψ2 − ψ0 
Dα1 (t)


= Y −1 X T 
 .
..
Dα2 (t)


.
ψ13 − ψ0
(17)
Therefore, the numerical solution for ψ at the time level n is used to estimate the two time-dependent parameters Dα1 and Dα2 at time level n. With these parameters, the asymptotic solution (7) provides all the
flow properties in the region inside the intermediate points, denoted by ◦. The flow properties outside that
region and at the 13 intermediate points are computed by the global numerical scheme. This is a least-squares
method of solution for Dα1 (t) and Dα2 (t).
A multiple grid refinement near the expansion corner (see Figure 4) is introduced to increase further the accuracy of the numerical results due to the rapid spatial behavior of vorticity and to satisfy the
condition (10). The method for the grid refinement is based on the third-order tensor product polynomial
to satisfy the consistency of the numerical scheme between the global and local grids. This approach can
272
T. Hawa and Z. Rusak
Figure 4. Grid refinement geometry near an expansion corner.
be applied to the composite overlapping grids (also called overlaid grids), where multiple grids are allowed
to overlap and functions defined on the grids are matched by interpolation. One of the major advantages of
composite grids is the grid generation around a complicated geometry when finite-difference methods are
used to solve partial differential equations on such a region. The approach has been successfully applied to
a wide range of problems. In our present problem it is not necessary to use this method since the region
contains a simple right-angle corner geometry. In some sense our present local grid can be thought of as an
overlapping grid with exactly zero overlap. Henshaw (1985) showed that it is necessary to use the third-order
interpolation technique in solving second-order partial differential equations with second-order accuracy.
Therefore, we use the third-order tensor product polynomial to generate a finer mesh (see Chesshire and
Henshaw (1990), or Cole and Schwendeman (1990), for more details of this technique). According to the
required condition (10), when Re of the incoming flow is increased, the region where the asymptotic corner
solution is valid becomes smaller and more grid refinement levels are typically needed.
The present local numerical technique is applied to a direct numerical simulation of a viscous flow in
a symmetric long channel of width d, which suddenly expands symmetrically, with right-angles, to a long
channel of width D where D > d (see Figure 5). The flow field is described in a Cartesian coordinate system
(x̄, ȳ), where ȳ = 0 is the centerline of the channel, and x̄ = 0 is the expansion section of the channel. The
inlet section to the channel is located at x̄ = −x̄ 0 (where x̄ 0 1) and tends to −∞. The outlet section of the
channel is located at x̄ = x̄ 1 (where x̄ 1 1) and tends to +∞. Along the upstream section of the channel,
x̄ < 0, the lower wall is at ȳ = −d/2 and the upper wall is at ȳ = d/2. At x̄ = 0, the lower wall is given along
the segment −D/2 ≤ ȳ ≤ −d/2 and the upper wall along the segment d/2 ≤ ȳ ≤ D/2. Along the channel
section where x̄ > 0, the lower wall is at ȳ = −D/2 and the upper wall is at ȳ = D/2.
Figure 5. A two-dimensional channel with a symmetric sudden expansion.
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
273
0.2
0.15
0.1
0.05
0
w/ corner solution
y'
-0.05
1 level refinement
-0.1
w/o corner solution
-0.15
1 level refinement
3 level refinement
5 level refinement
-0.2
-0.25
-0.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 6. Comparison of streamline patterns according to the numerical scheme using the asymptotic corner solution and to a
standard scheme with different levels of refinement for Re = 1. Here x = x̄/d and y = ( ȳ + d/2)/d.
In this problem the characteristic length and speed are l = d and U = Uave , the averaged velocity of the
Poiseuille flow far upstream of the channel expansion at the inlet section, and Re = Uave d/ν. Let x = x̄/d,
y = ȳ/d, and t = t¯Uave /d. The nondimensional Navier–Stokes equations in the vorticity-stream function
formulation are given by
∂Ω
∂Ω
∂Ω
1 ∂2Ω ∂2Ω
+
,
(18)
+u
+v
=
∂t
∂x
∂y
Re ∂x 2
∂y 2
2
∂ ψ ∂2ψ
Ω=−
+ 2 ,
(19)
∂x 2
∂y
∂ψ
∂ψ
u=
,
v=−
.
(20)
∂y
∂x
Here Ω(x, y, t) denotes the vorticity, ψ(x, y, t) denotes the stream function, and u(x, y, t), v(x, y, t) are the
nondimensional axial and transverse velocity components (scaled with Uave ), respectively.
A steady Poiseuille flow axial velocity profile with no transverse velocity component is considered at
the inlet section (x̄ = −x̄ 0 ). The boundary conditions along the lower and upper walls are the tangency
and no-slip conditions. The outlet flow downstream of the channel (x̄ = x̄ 1 ) is assumed to have a fully developed (steady and columnar) velocity profile with no transverse velocity component. We use x̄ 0 = 2d and
274
T. Hawa and Z. Rusak
0.2
0.15
0.1
0.05
0
w/ corner solution
y'
-0.05
1 level refinement
-0.1
w/o corner solution
-0.15
1 level refinement
3 level refinement
5 level refinement
-0.2
-0.25
-0.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 7. Comparison of streamline patterns according to the numerical scheme using the asymptotic corner solution and to a standard scheme with different levels of refinement for Re = 10. Here x = x̄/d and y = ( ȳ + d/2)/d.
x̄ 1 = 200d in the numerical examples shown below. In this paper we also concentrate only on the symmetric solution of the problem by imposing a symmetry condition along the channel centerline. We do so since
we are interested only in demonstrating the numerical effectiveness of our method and not in investigating
flow physics. It should be noted here that the symmetric solution becomes unstable above a certain critical
Reynolds number Rec = 53.8 (for D/d = 3) and evolves into an asymmetric state that can also be computed
by our method of simulations (see more details in Rusak and Hawa (1999) and Hawa and Rusak (2001)).
In the present numerical computations, a uniform finite-difference mesh is constructed in the flow domain. The vorticity transport equation, in a Cartesian coordinate system, is approximated by a first-order
forward-difference expression for the time derivative and a second-order centered-difference expression for
the spatial derivatives in the convective and viscous terms. The scheme of marching in time is given by (see
also Hawa 1999)
n
Ωn+1
i, j − Ωi, j
∆t
+
u ni+1, j Ωni+1, j − u ni−1, j Ωni−1, j
+
vi,n j+1 Ωni, j+1 − vi,n j−1 Ωni, j−1
2∆x
2∆y
n
n
n
n
n
n
1 Ωi+1, j − 2Ωi, j + Ωi−1, j
1 Ωi, j+1 − 2Ωi, j + Ωi, j−1
=
+
.
Re
(∆x)2
Re
(∆y)2
(21)
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
275
0.2
0.15
0.1
0.05
0
w/ corner solution
y'
-0.05
2 level refinement
-0.1
w/o corner solution
-0.15
1 level refinement
3 level refinement
5 level refinement
-0.2
-0.25
-0.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 8. Comparison of streamline patterns according to the numerical scheme using the asymptotic corner solution and to a standard scheme with different levels of refinement for Re = 50. Here x = x̄/d and y = ( ȳ + d/2)/d.
This finite-difference scheme is used to compute the dynamics of the flow field in the channel. In this equation the velocity components are computed according to (20) using central finite-difference approximations
of the derivatives of ψ. An iterative Poisson equation solver, which is based on second-order central differences in space, is used at any time level n to compute the stream function field related with the vorticity field
at that time step. For each iteration level k the scheme of integration is
1
n,k
n,k+1
n,k
n,k+1
2 n,k
2
ψi,n,k+1
(∆x)
=
Ω
+
ψ
+
ψ
+
κ
+
ψ
ψ
.
j
i, j
i+1, j
i−1, j
i, j+1
i, j−1
2 1 + κ2
(22)
Here κ = ∆x/∆y.
The initial conditions for the computations are the Poiseuille flow profile for the axial velocity with no
transverse speed for the range −x 0 ≤ x ≤ x 1 and − 12 ≤ y ≤ 12 and zero velocity at any other point. As the
flow evolves, the fluid expands around the corners.
In the present numerical computations, a uniform finite-difference mesh with step sizes ∆x = ∆y = 0.05
is constructed in the flow domain. Mesh convergence studies in Hawa (1999), using a finer mesh with
∆x = ∆y = 0.025, showed that the above is a sufficiently dense mesh for converged results of the flow
dynamics.
276
T. Hawa and Z. Rusak
0.2
0.15
0.1
0.05
0
w/ corner solution
y'
-0.05
2 level refinement
-0.1
w/o corner solution
1 level refinement
3 level refinement
5 level refinement
-0.15
-0.2
-0.25
-0.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 9. Comparison of streamline patterns according to the numerical scheme using the asymptotic corner solution and to a standard scheme with different levels of refinement for Re = 100. Here x = x̄/d and y = ( ȳ + d/2)/d.
4. Results
In this section the use of the numerical scheme developed above is demonstrated and results of computations are compared with those from numerical simulations using the same finite-difference scheme described
above, but without the matching with the local corner solution. In this second (standard) scheme the derivatives at the grid points just around the corners were calculated using second-order backward differences (at
mesh points upstream of the corners) or second-order forward differences (at mesh points downstream of the
corners). In this way we avoid in the standard scheme the need to compute the values of Ω at the corners. The
standard scheme computations also used the same meshes, as well as the same mesh refinement technique,
around the corners as described in the previous section.
The results of the streamlines pattern around the corners according to the standard scheme for Re = 1 and
expansion ratio D/d = 3 were compared with the results according to the present scheme (see Figure 6). It
is found that, unlike the present numerical method where only one level of refinement is needed near the corners, the standard scheme needs several levels of refinement near the corners (at least five levels) to achieve
numerical convergence of the solution near the corners. Moreover, it is very interesting to see that the results of the standard scheme converge toward the results of the present scheme as the level of refinement in
the standard scheme is increased. Similar results are also found for higher Re; the results at Re = 10, 50,
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
277
0.2
0.15
0.1
0.05
0
w/ corner solution
y'
-0.05
3 level refinement
-0.1
w/o corner solution
1 level refinement
3 level refinement
5 level refinement
-0.15
-0.2
-0.25
-0.3
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 10. Comparison of streamline patterns according to the numerical scheme using the asymptotic corner solution and to
a standard scheme with different levels of refinement for Re = 700. Here x = x̄/d and y = ( ȳ + d/2)/d.
100, and 700 are shown in Figures 7–10, respectively. We should note here that the location of the flow separation along the vertical segment of the corner wall is not accurately specified due to the use of MATLAB
contour plots. Figures 6–10 show that the local flow just around the corners is dominated by the asymptotic
corner solution. Also, the flow separation line approaches the corner and the recirculation zone under this
line becomes larger as Re increases. In other words, the effect of the asymptotic solution (7), especially the
first-order term, on the flow behavior near the corner becomes smaller and this can be predicted from the requirement condition (10). Also note that downstream effects become important as Re is increased and those
effects interact with the corner solution to determine the position of the separation line and its inclination.
Figure 11 demonstrates the streamline plots around the expansion corner according to the asymptotic
solution (7) for D/d = 3 and Re = 1, 10, 50, 100, and 700. The steady-state values of the time-dependent
parameters Dα1 and Dα2 in (7) are obtained from the results of the present numerical-asymptotic matching
scheme and are used to produce the streamline plots in Figure 9. These values are Dα1 = 0.8357 · · · and
Dα2 = −0.2321 · · · for Re = 1, Dα1 = 0.6360 · · · and Dα2 = −0.2899 · · · for Re = 10, Dα1 = 0.4005 · · ·
and Dα2 = −0.3211 · · · for Re = 50, Dα1 = 0.3373 · · · and Dα2 = −0.3190 · · · for Re = 100, and Dα1 =
0.2354 · · · and Dα2 = −0.2997 · · · for Re = 700. The asymptotic corner solution shows that when Re = 1
the flow turns smoothly around the expansion corner and separates along the vertical wall at about
y = −0.07 down from the expansion corner (r = 0). The separation line gradually inclines away from the
wall (see Figure 11(a)). Similar results are found for higher Re and the results at Re = 10 and 50 are shown
in Figure 11(b),(c). It can be seen that as Re increases, the flow separation location approaches the cor-
278
T. Hawa and Z. Rusak
Figure 11. Streamline plots according to the asymptotic solution for D/d = 3 and (a) Re = 1, (b) Re = 10, (c) Re = 50, (d) Re = 100,
and (e) Re = 700. Here x = x̄/d and y = ( ȳ + d/2)/d.
ner and the second-order term of the asymptotic solution becomes dominant (compare the different parts of
Figure 11).
The steady-state values of the time-dependent parameters Dα1 and Dα2 for various Re in the range
0.1 ≤ Re ≤ 100 and at a fixed expansion ratio D/d = 3 are summarized in Figure 12. It can be seen that as Re
approaches zero (a Stokes flow), Dα1 tends toward a limit value of about 0.87 and Dα2 tends toward a limit
value of about −0.23. In this limit of a creeping flow, the first-order term in (7) is dominant and the flow
turns smoothly around the corner with no separation. On the other hand, it can also be seen that as Re is increased, the steady-state values of the parameter Dα1 decreases and the term describing the smooth turning
of the flow around the corner becomes weaker. Also, as Re is increased, the parameter Dα2 increases and the
term describing the flow separation from the corner becomes more dominant. This means that the location of
the flow separation near the sudden expansion corners approaches the sharp corner, as was observed in both
experimental and other numerical studies.
The steady-state values of the parameters Dα1 and Dα2 for various expansion ratios in the range
1 ≤ D/d ≤ 4 and at a fixed Re = 10 are summarized in Figure 13. It can be seen that when D/d ≥ 3 both
Dα1 and Dα2 are nearly constant with D/d. This means that when D/d > 3 the effect of the expansion ratio
on the flow behavior around the corner is not important, especially in determining the position of the separation line and its inclination. As D/d increases, Dα1 tends toward a limit value of about 0.63 and Dα2 tends
toward a limit value of about −0.29. These values may indicate the limit values of these parameters in the
limit as D/d tends to ∞ and Re = 10, i.e., the case of flow expansion into an infinite domain. In this case
the flow around the corner may be similar to that described by the converged solution in Figure 7.
On the other hand, as the channel expansion ratio is decreased below 2, there is an effect on the corner
solution. When D/d = 1.5, the value of the Dα1 decreases whereas that of Dα2 remains nearly constant. The
279
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
0.9
0.8
0.7
Dα1 ,
0.6
-D α2
0.5
0.4
0.3
0.2
0
10
20
30
40
50
Re
60
70
80
90
100
Figure 12. The change of (◦) Dα1 and (∗) Dα2 with Re for D/d = 3.
0.65
0.6
0.55
0.5
Dα1
0.45
-D α2
0.4
0.35
0.3
0.25
1.5
2
2.5
3
3.5
4
D/d
Figure 13. The change of (◦) Dα1 and (+) Dα2 with D/d for Re = 10.
effect of the channel walls as a result of a smaller channel expansion becomes strong enough to change the
flow behavior around the expansion corner. To demonstrate this behavior, streamline plots according to the
present numerical simulation for Re = 10 and D/d = 1.5, 3, and 4 are compared in Figure 14.
It can be seen that when 1.5 ≤ D/d ≤ 4, the general flow behavior around the corner is not significantly
affected by the change of the expansion ratio. However, the flow separation line runs away from the corner
due to the wall effect of an expanding channel section as D/d is decreased from 4 to 1.5.
280
T. Hawa and Z. Rusak
0.25
0.2
0.15
0.1
y'
0.05
0
-0.05
-0.1
D/d=1.5
D/d=3
D/d=4
-0.15
-0.2
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
x'
Figure 14. Comparison of streamline plots according to simulation with the corner solution for Re = 10 and D/d = 1.5, 3, and 4.
Here x = x̄/d and y = ( ȳ + d/2)/d.
5. Conclusions
A new computational technique based on numerical-asymptotic expansion matching for describing the local
singular behavior of a viscous flow around a sharp right-angle expansion corner was constructed. The first
two terms of the asymptotic solution are used and a matching with a time-marching finite-difference scheme
of the Navier–Stokes equations is formulated. Local mesh refinements around the corner are required to
meet the validity condition of the asymptotic solution. The present numerical technique is applied to the
symmetric flow in a suddenly expanding channel with an expansion ratio D/d = 3 at various Reynolds numbers 1 ≤ Re ≤ 700. The results are compared with those from a standard finite-difference scheme that uses
second-order forward/backward differences near the corner. It is found that the standard scheme results converge toward those of the present technique as the level of local refinement near the corner is increased.
These computations demonstrate that the use of the local corner solution within the numerical simulation of
the flow in the channel enhances the accuracy of the simulations and requires less computational effort (less
refinements near the corners) to achieve the same converged numerical results.
The steady-state values of the time-dependent parameters Dα1 (t) and Dα2 (t) in the asymptotic corner solution (7) are described for various cases of Re and D/d. It is found that when Re decreases to zero, with
a fixed D/d, these parameters reach certain limit values and the flow is dominated by the first-order mode
in (7), describing a smooth turning around the corner. On the other hand, as Re is increased, with a fixed
Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner
281
D/d, the parameter Dα1 decreases and the term describing a smooth flow turning around the corner becomes
weaker. Also, in this case, the parameter Dα2 increases in its absolute value and the term describing flow separation becomes dominant. Therefore, as Re is increased, the location of the flow separation near the sudden
expansion corner approaches the sharp corner, as was observed in both experimental and other numerical
studies. It is also found that at expansion ratios above 3 the parameters Dα1 and Dα2 are not affected by the
channel walls. However, when 1 < D/d < 3, the walls affect the flow behavior around the expansion corners.
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