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1990,Howard,粘弹性圆柱绕流的数值模拟
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粘弹性圆柱绕流的数值模拟是非常经典的文章,利用数值模拟方法研究流场。
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Journal of Non-Newtonian Fluid Mechanics, 37 (1990) 347-377
Elsevier Science Publishers B.V., Amsterdam
347
NUMERICAL SIMULATION OF VISCOELASTIC FLOW PAST
A CYLINDER
HOWARD H. HU and DANIEL D. JOSEPH
Department of Aerospace Engineering and Mechanics, University of Minnesota,
I10 Union St. SE, Minneapolis, MN 55455 (U.S.A.)
(Received May 14, 1990)
Abstract
The flow of an upper-convected Maxwell fluid past a circular cylinder is
simulated numerically using the algorithm SIMPLER, which is based on a
finite volume discretization on a staggered grid of the governing equations
and an iterative solution to the nonlinearly coupled equations. The effect of
the viscoelasticity of the fluid on the flow is examined. The drag force on the
cylinder and the heat transfer from the cylinder to the surrounding fluid are
calculated and compared with those obtained in experiments. One feature
seen in the experiments is the existence of a critical velocity across which the
variation of the drag and heat transfer changes from a characteristic
Newtonian variation to a flat response. This feature is interpreted as a
change of type when the critical speed UC of flow becomes greater than the
material wave speed c = /a. The numerical computation is completely
consistent with this interpretation. Using an Einstein-type formula for the
relaxation time h = A+ with A independent of (p we find that the experi-
ments follow a scaling law U@#V2 =
constant, where C#J is the mass fraction of
polymers in water in the regime of extreme dilution, the drag reduction
range, consistent with the interpretation that UC = c.
Keywords: change of type; dilute polymer solution; relaxation; viscoelastic flow past cylinder; vorticity
shock; wave propagation
1. Introduction
It is known that shear waves can propagate in a viscoelastic fluid with
instantaneous elasticity. When the flow velocity is larger than the shear wave
velocity, the governing vorticity equation undergoes a change of type. It has
0377-0257/90/$03.50 0 1990 - Elsevier Science Publishers B.V.
348
been suggested by Joseph and coworkers [l-4] that some interesting visco-
elastic phenomena can be associated with this change of type. Some of the
examples are sink flow [5], delayed die swell [3] and flow around bodies [4].
For viscoelastic flow around a circular cylinder, James and Acosta [6]
reported a critical phenomenon in which the heat transfer from the cylinder
to the surrounding fluid becomes essentially independent of the Reynolds
number when the Reynolds number is beyond a critical value. James and
Gupta [7] observed the same kind of behavior for the drag coefficient. The
velocity measurements of Koniuta et al. [8] revealed a stagnant or nearly
stagnant region around the cylinder when the velocity was greater than a
critical velocity. Ultman and Denn [9] presented an analysis which suggested
that the equation for the forward component of velocity changes type when
the flow velocity exceeds shear wave speed, and they applied this idea to the
experiments of James and Acosta [6]. Actually, the forward velocity does not
change type but the vorticity does; see Joseph [4].
Crochet and Delvaux [lO,ll] analyzed numerically the flow of an upper-
convected Maxwell fluid past a circular cylinder. They used a finite element
method which was developed for calculating highly viscoelastic flows [12].
The algorithm is characterized by the bilinear subelements for the stresses
and the use of streamline-upwinding for discretizing the constitutive equa-
tions. They examined the consequences of the transition from sub- to
supercritical flow regime upon momentum and heat transfer properties.
Their simulation confirms the experimentally observed change from
Newtonian-like variations of the transport properties to constant transport
properties for velocities exceeding the shear wave speed.
In the present paper, we simulated the flow of an upper convected
Maxwell fluid around a cylinder using the algorithm SIMPLER (semi-im-
plicit method for pressure-linked equations revised) which was introduced
by Patankar [13] and successfully used in solving problems of heat transfer
and fluid flow of Newtonian fluids. We find that with minor modifications
in solving the stress equations SIMPLER works well for the viscoelastic
flows in this problem and converges for relatively high Weissenberg number.
The computed drag force on the cylinder and the heat transfer from the
cylinder to the surrounding fluid are compared with those obtained in
experiments of James and Acosta [6]. The numerical results display many
details of the flow.
2. Basic equations
Consider a two-dimensional uniform flow for a Maxwell fluid past a
circular cylinder of diameter d in the plane. We limit ourselves to flows of
low and medium Reynolds numbers which are steady, symmetric and
349
U
-
-
Fig. 1. Flow geometry and boundaries.
without instability. The geometry of the problem is shown in Fig. 1. In the
figure, l?,, is the inflow and I,, is the outflow boundary, I,, and I?,, are the
symmetry boundaries and I, is the solid boundary on the cylinder surface.
The outer boundary I?,,
and I,, is assumed to be far away from the
cylinder. In our computation the radius of this outer boundary rm is chosen
to be about 50 times the radius of the cylinder.
We need to solve
V.u=O
(I)
and
p(u*v>u= -Vp+f+V ‘7,
(2)
where p is the density of the fluid, u the velocity, p the pressure, f the body
force which is assumed to be zero in this problem, and T the extra stress
tensor which is given by the constitutive equation
X;+r=271D
(3)
for an upper-convected Maxwell fluid. In the constitutive equation A is the
relaxation time, n the shear viscosity, D the rate of deformation tensor and
7” denotes the upper-convected derivative of T defined by
7”=g+(u.v)+T-
(VU)T - T(VU)T,
where (vu),~ = au,/axj. In the present problem we also wish to evaluate the
effect of viscoelasticity upon the heat transfer from the cylinder to the
surrounding fluid. We assume that the viscous heating is negligible and that
temperature differences in the flow are small and such that the fluid
properties (p, X and n) do not change. Then the temperature field is
decoupled from the velocity field; the energy equation is simply
c,p(u*V)T= K AT,
(5)
351
We use the polar coordinate system shown in Fig. 1, writing the velocity
as
u = u,e, + ueeO
and the stress as
‘I%= 'Err
ee + rEOeeee6 + rErB(eret? + eOer>.
i- r
Equations (l), (6), (7)
and (8) in component form are
+
rEU
- rwe
r
)I
7 03)
05)
-rEoo+ zw[(% - ~)(rErB+rNrB)
a24
-2
ar
(rEee+rNee
(16)
+rN,,)+(%-T)
X(rErr + 'Nrr
)I i
- w ur+@ + $#), (17)
(18)
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