1
黏性流體通過運動圓柱支流場分析研究(I)
Viscous flow across a translating circular cylinder (I)
NSC- 90-2611-E110-001
90/08/01-91/07/31
Bang-Fuh Chen
Abstract
A time-independent finite difference scheme is developed to simulate the viscous flow across a stream-wise oscillating circular cylinder. Large Reynolds number and high oscillating frequency of moving cylinder are used in the analysis. The finite difference approximation and algorithm were validated by the reported numerical simulation and flow visualization of phenomenon (Re=3000) and phenomenon (Re=9500). Detail vorticity and streamline patterns developments were described and discussed. The surface vorticity distribution and position of separation point versus phases of oscillating stages were discussed. The flow behaviors of various amplitudes of exciting velocity and frequency of moving cylinder are simulated and compared. The in-line forces on cylinder during cylinder oscillation were calculated. The application and limitation of Morison’s equation was discussed.
Introduction
The viscous flow across a circular cylinder has draw significant attention during the last two decades. As early as 15th century, pairs of wakes behind a circular cylinder were captured in the drawings, see Fig. 1, of Leonardo, Da Vinci (1452-1519). In the 20th century, the earliest study of flow around circular cylinder was reported by Blasius in 1908, who used boundary layer theory to give a second order time series solution. Since then, many researchers spent their efforts to find higher order solutions and discussed the flow patterns behind the separation points. Among those, the most important investigations could be credited to Collins & Dennis (1973a,b), they developed analytical theory and implemented their theory numerically, and also to Bar-Lev and Yang (1975) who used matched asymptotic expansions to solve for vorticity equation up to third order solutions.
In the present study, the flow field around a high-rise offshore cylinder under a cross flow and a periodic ground motion for Re = 3000 and exciting frequencies ranged from 0.1 to 2 were made. The detailed description of vorticity development and the
streamline patterns comparison among various exciting frequencies were given. Besides, a real earthquake analysis with one-tenth of magnitude of El Centro 1940 earthquake is studied. The following parameters are given: the cylinder diameter = 0.1m, the peak ground acceleration of El Centro earthquake is scaled down one tenth and the maximum Reynolds number is around 3500 (since the maximum velocity is 0.035m/sec now). While, the exciting period is unchanged, i.e. around 0.4 sec and the corresponding frequency = 2.5 (1/sec), and the KC is as small as 0.14 and symmetrical flow phenomenon is presented throughout the simulation. A time-independent finite difference scheme is introduced that the time-dependent moving cylinder boundary is transformed to an independent one by proper mapping functions. The implicit finite difference approximation and higher order upwind scheme are used in solving stream function and vorticity transport equation and corresponding diffusive terms. The numerical scheme was validated by many comparisons with existing numerical studies and flow visualizations.
Basic Equations
1
1
Department of Marine Environment and Engineering, National Sun Yat-sen University
d v0 r r xc v yc v b2 ar x y r r
2
Fig. 1 The definition of sketch of the problem( please re-plot these figures) In this study, we assume the cross flow and
cylinder are suddenly starts to move in an incompressible viscous fluid. The cross flow is moving with a constant speed vo and the cylinder is stream-wise oscillating. The motion can be described in terms of the two simultaneous equations satisfied by the stream function and the scalar vorticity. One is the stream function equation, written in polar coordinate
(
~
r
,
)
system, as]
~
1
)
~
~
(
~
~
1
[
)
(
~
2 2 2 2
r
r
r
r
r
(1)and the other is the so called vorticity transport equation written as
~
)
~
~
~
~
(
~
1
~
2
r
r
r
t
(2)where is the stream function; and
w
~
is the vorticity, and represents the kinematic viscosity. The definition sketch of the problem is conceptually shown in Fig. 1, where a is the diameter of the cylinder, ar is the distance between the outer boundaryand the origin of the polar coordinate system.
r
is the radial distance measured from the instant center of moving cylinder and is the angle betweenr
and horizontal direction and among those parameters, the following relationship can be given
cos
cos
~
r
r
x
(3)in which x is the corresponding horizontal
displacements of the cylinder in stream-wise direction.
The corresponding flow velocity in
r
and directions are defined asv
r andv
and can be calculated through the definitions of velocity and stream function; ' , ' 1 ' r v r vr (5)Also shown in the definition sketch, b2 is the distance between outer boundary and the instant center of the moving circular cylinder and can be shown and expressed as 2 1 2 2 2
(
,
t
)
(
t
)
cos
{
a
[
(
t
)
sin
]
}
b
x
r
x (13) Since the oscillating cylinder surface is varying withtime, the first of the following equations is used to remove the dependence of b2 on time.
a t b a r r ) , ( 2 *The time-dependent moving cylinder face is transformed to a fixed value and the whole computational domain is mapped onto a rectangular region.
The Crank-Nicolson method is used to solve the time dependent terms at each time step and the Gauss-Seidel method is used to solve the systems of linear equations. The convergence conditions shown as follow are used to assure acceptable accuracy.
6 1
10
m m10 1
10
m
m
where the subscripts represent the iteration numbers.
Results and discussions
In this study we use R=1/250 with k1 = 3, = 1/200 and T = 0.001 for stability criteria and numerical accuracy.
Phenomenon and phenomenon
. The phenomenon is simulated in the present study for Re = 3000, while phenomenon is simulated for Re = 9500. For Re= 3000, Fig. 2 compares the present numerical results and the results reported by C&C and PL&B, and good agreements are noted. The phenomenon- can be clearly identified when T = 3. To generate phenomenon , the Reynolds number is increased to 9500, Fig. 3 lists the comparison of the present numerical results and those of C&C and PL&B, the agreement is also very good while the phenomenon is not clear in C&C’s analysis.
Flow over a back-forth oscillating
circular cylinder
The cross flow and cylinder are set to suddenly move at the same time. The velocity of the cross flow is a constant and is denoted as v0, the velocity of oscillating cylinder is given as u(t) = -v0 cos (t), where is frequency of oscillating cylinder and is a parameter to define the velocity amplitude of the moving cylinder.
Vorticity development and streamline
patterns development
For =1, when the cylinder moves westward, it is moving against to cross flow and experiences a higher Reynolds number, and the equivalent Re is denote as eRe=6000 at T=0+. Actually, there are four stages in one cycle of harmonic cylinder motion. In
2
Fig. 2 Comparison of streamline patterns for Re=3000, at T=5;(i)Phuoc Loc & Bouard (1985) flow visualization, (ii)Phuoc Loc & Bouard (1985) numerical result, and (iii) present analysis.
the first stage: T=2n to 2n+0.5 (n=0,1,2,..); the second stage: T = 2n+0.5 to 2n+1; the third stage: T = 2n+1 to 2n+1.5; and the fourth stage: the cylinder changes moving direction again from its most eastward position to origin during T = 2n+1.5 to T=2n+2. For ==1, Fig. 4 depicts the patterns of the streamlines and vorticity contour at different times. The first column of the figure is the streamlines and the second column is the corresponding vorticity contour.
The vorticity must be strong enough to induce a circulating wake, and positive vorticity generates clockwise eddy and negative vorticity produces clockwise and counter-clockwise eddy. Both positive and negative vorticities may distort the streamline patterns. A sketch of the streamlines for this flow field is also shown in Fig. 4. The impulsively moving circular cylinder results in the overall streamline pattern looks like flow across a larger cylinder whose size is 1.414 times larger in radius. After T = 4, the positive and negative vorticities are continuously generated alternately, and periodically phenomenon is presented as T increases.
Acknowledgement
This study is supported by a project of NSC under grant number NSC90-2611-E-110-01.
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Streamline at T = 2 0.00 0.50 1.00 1.50 2.00
Fig. 3 Comparison of streamline patterns for Re=9500, at T=2;(i)Phuoc Loc & Bouard (1985) flow visualization, (ii)Phuoc Loc & Bouard (1985) numerical result, and (iii) present analysis.
-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 Streamline at T = 5 0.00 0.50 1.00 1.50 2.00
1 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 0 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 1 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 0.25 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 0.75 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 0.5 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Streamline at T = 4 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
Fig. 4 The pstterns of the streamlines and vorticity contour at Re = 3000. -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 0 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 1 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 0.25 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 0.5 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 0.75 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Vorticity at T = 4 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00
