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On the downstream boundary conditions for the vorticity-stream function formulation of two-dimensional incompressible flows
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 85 (1991) 207-217 NORTH-HOLLAND
ON THE DOWNSTREAM BOUNDARY CONDITIONS FOR THE
VORTICITY-STREAM FUNCTION FORMULATION OF TWO-DIMENSIONAL INCOMPRESSIBLE FLOWS* T.E. T E Z D U Y A R and J. LIOU Department of Aerospace Engineering and Mechanics and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, USA Received 28 September 1989
Downstream boundary conditions equivalent to the homogeneous form of the natural boundary conditions associated with the velocity-pressure formulation of the Navier-Stokes equations are derived for the vorticity-stream function formulation of two-dimensional incompressible flows. Of particular interest are the zero normal and shear stress conditions at a downstream boundary.
1. Introduction
Imposing proper downstream (or outflow) boundary conditions for incompressible flow problems has been and still is the subject of substantial investigation. A popular choice in the velocity-pressure formulation of the Navier-Stokes equations is to use the homogeneous form of the natural boundary conditions associated with this formulation (see for example [1,2]). The 'traction-free' (i.e., zero normal and shear stress) conditions can easily be imposed within this framework [1]. What to do in the vorticity-stream function formulation has been less clear; a number of investigators looked into this and related issues. For example, Bristeau et al. [3] introduced an 'absorbing boundary condition' applicable to vorticity-stream function and velocity-pressure formulations. Among the boundary conditions considered (in the context of Stokes' flow) by Pironneau [4] and Hughes and Franca [5] is one which is equivalent to specifying the vorticity and stream function. The downstream boundary conditions used for the computations performed in [6, 7] are the homogeneous form of the natural boundary conditions associated with the governing equations of the vorticity-stream function formulation; this meant zero normal derivatives for the vorticity and stream function. For advection dominated flows, setting the normal derivative of the vorticity at the downstream boundary equal to zero is quite reasonable. Zero normal derivative for the stream function, on the other hand, implies zero tangential velocity, and this is too restrictive. Still, comparisons (for flow past a cylinder at Reynolds number 100) based on the near-field solution, drag and lift coefficients, and the Strouhal number show that there is no significant difference between the solutions obtained by * This research was sponsored by NASA-Johnson Space Center under contract NAS9-17892 and by NSF under grant MSM-8796352. 0045-7825/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
208
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
using the vorticity-stream function and velocity-pressure formulations with homogeneous natural boundary conditions [7]. However, more visible differences in the computed flow field were observed in the vicinity of the downstream boundary. In this paper we derive, for the vorticity-stream function formulation, the downstream boundary conditions equivalent to the homogeneous natural boundary conditions associated with the velocity-pressure formulation. We consider two different forms of the velocitypressure formulation. The first form is obtained by incorporating the incompressibility constraint into the momentum equation but, without expanding the divergence of the stress tensor; the homogeneous natural boundary conditions corresponding to this form are the 'traction-fr,.e' conditions. The second form is obtained by replacing the divergence of the stress tensor with the terms involving the pressure gradient and the Laplacian of the velocity vector; the homogeneous natural boundary conditions for this form are slightly different, but only in the normal direction. These natural boundary conditions can be combined into a unified form by introducing a parameter which is equal to one for the first form and zero for the second; the equivalent boundary conditions in the vorticity-stream function formulation are derived for this unified form. 2. Formulation
Let O and (0, T) denote the spatial and temporal domains, with x and t representing the coordinates associated with O and (0, T). The boundary F of the domain O consists of an external boundary F, and possibly several internal boundaries. We assume that the external boundary F. has four subsets: F,., F. 2, F..~ and F,4. The upstream and downstream boundaries are represented by F.~ and F.~, while the upper and lower boundaries are represented by F,, and F.4 (see Fig. 1). Only the downstream boundary is assumed to be a straight line. The unit normal and tangential vectors at a boundary are denoted by n and 1". We first consider the following velocity-pressure formulation of the incompressible Navier-Stokes equations:
Main Flow
--~
F01
IV Fig. 1. The computational domain.
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-streant
on~x(0,
p(Ou/Ot+u.Vu)-V.o'=O
V.u=0
onOx(0,
T),
T),
209
(1) (2)
where p and u are the density and velocity, and or is the stress tensor given as o" = - p l + 2/xe(u),
(3)
,(u) = ½(vu + (Vu)').
(4)
with Here p and tt represent the pressure and viscosity, while I denotes the identity tensor. Another commonly used form of the momentum equation is written as p(Ou/Ot + u .Vu) + Vp - / . t V2u = 0 .
(5)
The homogeneous natural boundary conditions corresponding to (1) are the 'traction-free' (i.e., zero normal and shear stress) conditions, and it is reasonable to assume that these conditions can be imposed at the downstream boundary F.3. That is, n . or. n = - p + 2 g Ou,,/On - 0
on F03 x (0, T ) ,
n.o'.~'=lz(Ou,/On+Ou,,/O~')=O
onFo3 x ( 0 , T).
(6)
(7)
The homogeneous natural boundary conditions associated with (5), on the other hand, are - p + ~ Ou,,/On = 0 t~ Ou,/On = 0
on F03 x (0, T ) ,
(8)
(9)
on Fo3 x (0, T ) .
We can combine the natural boundary conditions for (1) and (5) into a unified form by introducing a parameter z which is equal to 1 for (1) an(a O for (5); this unified form can be written as - ( l / p ) p + (1 + z)vOu,.IOn = 0 OUr~On+ ZOu,,/O'r=O
on r03 x (0, T),
one)3×(0, T),
(1o) (11)
where v is the kinematic viscosity. Of course the second condition makes sense only for viscous flows, and therefore the boundary condition given by (11) is discarded for inviscid flows. We would like to derive, for the vorticity-stream function formulation, the downstream boundary conditions equivalent to (10) and (11). In two-dimensional space, the vorticity-stream function formulation of the incompressible Navier-Stokes equations consists of an advection-diffusion equation for the vorticity and a Poisson equation for the stream function: Oto/Ot + u .Vto - vV2to = 0
V2$ + to = 0
on 12 x (0, T ) ,
on O x (0, T ) ,
(12) (13)
210
T.E. Tezduyar, J. Liou, On the downstream boundary conditions .for the vorticity-stream
where the stream function and vorticity are defined as u = {aq,/ax2,-aqJ/ox,},
to = (~U210X
(14)
,
1 -- t~Ul/t~X2
(15)
•
The components of the velocity vector u in the directions denoted by the unit vectors n and 7 can be expressed as u,,=n.u=-aO/,9"r,
(16)
u , = ,,r. u = O ~ l O n .
(17)
To derive the boundary condition equivalent to (11), we start with the following expression for the vorticity at the downstream boundary: Ou,.I a,r - a u , I On
= to on Fo3 x (0, T) ;
(18)
by combining this with (11), we obtain (1 + z ) Ou,,/Or = to
on Fo3 x (0, r ) .
(19)
From (19) and (16) we reach the second-order differential equation given below: (1 + z)cg"-Olar 2 --- - w
on
~,.~ x (0, T).
(20)
In conjunction with this differential equation, we assume that the following boundary conditions are given:
0 = (qJ)l,, at 'r = (~'),,l, q" = ( O ) , v
(21)
at ¢ = (,r), v .
(22)
This is a reasonable assumption, because if for example Fo2 is a symmetry line then the stream function is invariant along F02, and therefore the value of 0 at r = (r)m is known. On the other hand, if F02 is a boundary where we specify both components of the velocity, then, by integrating the velocity field along Fo2, the value of 0 at r = (r)m can be determined. For the derivation of the boundary condition equivalent to (10), we first consider the following form of the momentum equation: ou/ot + v ( l l u l l " / 2 ) - u x
+ (1/p)Vp +
x
=o.
(23)
By taking the projection of this equation in the direction of the unit vector r, we obtain
Ou, 10(u~+u~) Ot + 2
Or
--U.tO+
1 0p
p Or
+ v
0oo
On
----0 o n F o 3 × ( 0 , T).
(24)
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
211
Next we differentiate (10) with respect to ~- and get
1 0 p = (1 + z)v O(Ou,,IOn)
p Or
a~"
on Fo3 x (0, T ) .
(25)
We use the continuity equation to replace Ou,,/On with - 0 u , / 0 % and obtain 1 bp = - ( 1
p ¢~T
+ z)v o2u" 19T2
on Fo3 x (0, T ) .
(26)
By combining (26) with (24) we reach the following equation:
#u,
2 1 O(u2, + u,)
#t -I 2
Or
OZu,
Oto
- u.to - ( 1 + z)v ~ 0 r~ + v 7 n = 0
on Fo3 x (0, T) .
(27)
Rewriting this equation by using (16) results in the following differential equation: 0u,
( 1 + z)v -02u" -=--
bu,
Ot + u, 0-'-¢-
Or 2
10(cgO/Or) 2 _ ( 00) Oto 2 Or -~r t o - v On
on Fo3 × (0, T ) .
(28)
It should be noted that ~, along Fo3 was already determined by using (20)-(22). For the differential equation (28), we assume that the following boundary conditions are given: u, -- (u,),,,
at ,r - (¢),,,,
(29)
u, --- ( u , ) , v
at
(30)
= 0"),v •
This assumption too is a reasonable one, because whether F0, is a boundary with symmetry conditions or a boundary where we specify both components of the velocity, the value of u, is known along F02, including the point ~-= 0")~. In this paper we will provide only the variational formulations associated with the implementation of the downstream boundary conditions. The rest of the variational formulations employed in the finite element discretization of the problem can be found in [6, 8]. We introduce the following solution spaces defined on the interval [0")m, (r)w] for ~, and u, along F0a: S~ = { 0 h I 0 h e H 'a, 0h(('r),,,) = (0),,,, 0h(('r),v) = (0),v},
(31)
h S h141. = {u~ lu.h E H lh, u.((¢)m ) - (u.),,,, u.h((¢),v) --(u.),v} •
(32)
We need only one variational space, and that is V h = { w h l w h ~. H 'h, wh((r),,,)= O, wh(('r),v)= 0}.
(33)
The variational formulation corresponding to (20)-(22) and (28)-(30) can be stated as
212
T.E. Tezduyar. J. Liou, On the downstream boundary conditions for the vorticity-stream h
h
h
follows: given to h, find 0 h E S, and u~ E S,,,, such that (1 + z )
fly
0 Wh a d,jh d z =
xx
Oz
Oz
f/v
whto h d z
and Wh
,, at. U r
~x
=
Ot
Oz /
f,v 0w,/00, II
or
(34)
VW h ~ V h
n
d'r + (1 + z)u
dr-
f:v
flTM tl
19W h
OU h
0"r
0~"
w h --~r O~bh to h d ' r - v
II
d~" 1v W h 0to h u
~ d z
V Wh E V h
(35)
The discrete equation system associated with (34) and (35) can be solved either (in the context of a block-iteration scheme [6, 8]) as an independent block by assuming that to h is given, or (in the context of a fully coupled formulation [9]) simultaneously with the rest of the discrete equations. Of course, in the context of a block-iteration scheme the blocks corresponding to (34) and (35) can also be solved independent of each other. REMARK. We can also ascertain, for two-dimensional problems, the equivalent boundary conditions in the vorticity-stream function formulation for the four types of boundary conditions considered in [5]. (l) Specifying both components of the velocity vector is equivalent to specifying the normal and tangential derivatives of the stream function. This type of boundary conditions was covered in [6]. (2) Specifying the normal component of the velocity and the tangential component of the vorticity (that is the only vorticity component in two dimensions) is equivalent to specifying the tangential derivative of the stream function and the vorticity itself. This type of boundary conditions too was covered in [6]. (3) In cases when the pressure and the tangential velocity is specified, for (24) we can obtain a second-order differential equation for the stream function. This differential equation, together with the end-point conditions of the form (21)-(22), can be used to determine the stream function along the boundary. Since the tangential velocity is also specified, as boundary conditions, we effectively have both the stream function and its normal derivative. (4) When the pressure is specified together with the vorticity, we can use (24) to obtain a differential equation for the tangential velocity. This differential equation, together with the end-point conditions of the form (29)-(30), can be used to determine the normal derivative of the stream function along the boundary.
3. Numerical test: Flow past a circular cylinder
In this test problem the dimensions of the computational domain, normalized by the cylinder diameter, are 30.5 and 16.0 in the flow and cross-flow directions, respectively. The
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
213
mesh employed consists of 5,220 elements and 5,329 nodes; around the cylinder there are 58 elements in the radial and 80 elements in the circumferential directions (see Fig. 2). Our previous convergence studies based on successive mesh refinement [7] suggest that solutions obtained with this mesh will be sufficiently accurate.
|
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l I l i l l l
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.........i
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g ~ m i V l l ~ l n i m
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m H H l i 6 l l ~ i b ~ R
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Fig. 2. Flow past a circular cylinder: the finite element mesh (5220 elements and 5,329 nodes).
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o
0
"~
ca
b
o o
0
-0.50
...~
--0T25
..._,
~
0.00
I
~'
~
i
0;25
0.5{}
o o
o
o
o o o
0
1.15
1.20,,
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1.30
~
1.35
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J
1.40
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~
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~
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~o',~
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T.E. Tezduyar, J. Liou, On the downstream boundary conditions jor the vorticity-stream
215
inviscid at the downstream boundary; therefore we discard the downstream boundary condition given by (11) and replace it with the zero normal derivative condition for the vorticity. Since at the downstream boundary the vorticity transport equation is nearly hyperbolic, this zero normal derivative condition is essentially equivalent (in the context of the variational formulation) to not specifying any condition for the vorticity. Figure 3 shows the time history of the drag and lift coefficients. The periodic how patterns corresponding to the crest and trough values of the lift coefficient are shown in Figs. 4 and 5. By examining the time history of the lift coefficient, we estimate the Strouhal number to be 0.171. (a)
VortLcLt~
(b)
VortLcLt~
VortLcLt~
( Stream ?unctLon
5treom functLon
Stream ?unctLon
S~otLonor~ s~reom ?unc~Lom
Stotuor,or~ streom ?unctLon
5totLomor~ streom functuon
Fig. 4. Flow past a circular cylinder at Reynolds number 100: periodic solution (corresponding to the crest value of the lift coefficient) for (a) the full computational domain; (b) regions near the cylinder and the downstream boundary.
216
T.E. Tezduyar, J. Liou, On the downstrean, boundary conditions for the vorticity-stream
(a)
VortLcLt~
(b)
VortLcLt~
VortLcCt~
,
i
Stream functLon
5treom functcon
Streom functLon
5~.ot~onor~ s~reom fumC~LOn
Sto~onor~ streom CumctLom
S~otLono~ stream functLon
i .
.
.
.
ii
.
g
Fig. 5. Flow past a circular cylinder t~t Reynolds mlmber I00: periodic solution (corresponding to the trough value of the lift coefficient) for (a) the full computational domain; (b) regions near the cylinder and the downstream boundary.
References [1] A.N. Brooks and T.J.R. Hughes, Streamline-upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259.
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
217
[2] P.M. Gresho and S.T. Chan, Semi-consistent mass matrix techniques for solving the incompressible NavierStokes equations, Lawrence Livermore National Laboratory Preprint, UCRL-99503, 1988. [3] M.O. Bristeau, R. Glowinski and J. Periaux, Numerical methods for the Navier-Stokes equations; Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Rep. 6 (1987) 73-187. [4] O. Pironneau, Conditions aux limites sur la pression pour les equations de Stokes et de Navier-Stokes, C.R. Acad. Sc. Paris 303, Serie I (9) 403-406. [5] T.J.R. Hughes and L.P. Franca, A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987) 85-96. [6] T.E. Tezduyar, J. Liou and R. Glowinski, Petrov-Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 8 (1988) 1269-1290. [7] T.E. Tezduyar, J. Liou and D.K. Ganjoo, Incompressible flow computations based on the vorticity-stream function and velocity-pressure formulations, Comput. & Structures 35 (1990) 445-472. [8] T.E. Tezduyar, Finite element formulation for the vorticity-stream function form of the incompressible Euler equations on multiply-connected domains, Comput. Methods Appl. Mech. Engrg. 73 (1989)331-339. [9] T.E. Tezduyar, J. Liou, D.K. Ganjoo and M. Behr, Solution techniques for the vorticity-stream function formulation of two-dimensional unsteady incompressible flows, Internat. J. Numer. Methods Fluids l 1 (1990) 515-539.
ON THE DOWNSTREAM BOUNDARY CONDITIONS FOR THE
VORTICITY-STREAM FUNCTION FORMULATION OF TWO-DIMENSIONAL INCOMPRESSIBLE FLOWS* T.E. T E Z D U Y A R and J. LIOU Department of Aerospace Engineering and Mechanics and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, USA Received 28 September 1989
Downstream boundary conditions equivalent to the homogeneous form of the natural boundary conditions associated with the velocity-pressure formulation of the Navier-Stokes equations are derived for the vorticity-stream function formulation of two-dimensional incompressible flows. Of particular interest are the zero normal and shear stress conditions at a downstream boundary.
1. Introduction
Imposing proper downstream (or outflow) boundary conditions for incompressible flow problems has been and still is the subject of substantial investigation. A popular choice in the velocity-pressure formulation of the Navier-Stokes equations is to use the homogeneous form of the natural boundary conditions associated with this formulation (see for example [1,2]). The 'traction-free' (i.e., zero normal and shear stress) conditions can easily be imposed within this framework [1]. What to do in the vorticity-stream function formulation has been less clear; a number of investigators looked into this and related issues. For example, Bristeau et al. [3] introduced an 'absorbing boundary condition' applicable to vorticity-stream function and velocity-pressure formulations. Among the boundary conditions considered (in the context of Stokes' flow) by Pironneau [4] and Hughes and Franca [5] is one which is equivalent to specifying the vorticity and stream function. The downstream boundary conditions used for the computations performed in [6, 7] are the homogeneous form of the natural boundary conditions associated with the governing equations of the vorticity-stream function formulation; this meant zero normal derivatives for the vorticity and stream function. For advection dominated flows, setting the normal derivative of the vorticity at the downstream boundary equal to zero is quite reasonable. Zero normal derivative for the stream function, on the other hand, implies zero tangential velocity, and this is too restrictive. Still, comparisons (for flow past a cylinder at Reynolds number 100) based on the near-field solution, drag and lift coefficients, and the Strouhal number show that there is no significant difference between the solutions obtained by * This research was sponsored by NASA-Johnson Space Center under contract NAS9-17892 and by NSF under grant MSM-8796352. 0045-7825/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
208
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
using the vorticity-stream function and velocity-pressure formulations with homogeneous natural boundary conditions [7]. However, more visible differences in the computed flow field were observed in the vicinity of the downstream boundary. In this paper we derive, for the vorticity-stream function formulation, the downstream boundary conditions equivalent to the homogeneous natural boundary conditions associated with the velocity-pressure formulation. We consider two different forms of the velocitypressure formulation. The first form is obtained by incorporating the incompressibility constraint into the momentum equation but, without expanding the divergence of the stress tensor; the homogeneous natural boundary conditions corresponding to this form are the 'traction-fr,.e' conditions. The second form is obtained by replacing the divergence of the stress tensor with the terms involving the pressure gradient and the Laplacian of the velocity vector; the homogeneous natural boundary conditions for this form are slightly different, but only in the normal direction. These natural boundary conditions can be combined into a unified form by introducing a parameter which is equal to one for the first form and zero for the second; the equivalent boundary conditions in the vorticity-stream function formulation are derived for this unified form. 2. Formulation
Let O and (0, T) denote the spatial and temporal domains, with x and t representing the coordinates associated with O and (0, T). The boundary F of the domain O consists of an external boundary F, and possibly several internal boundaries. We assume that the external boundary F. has four subsets: F,., F. 2, F..~ and F,4. The upstream and downstream boundaries are represented by F.~ and F.~, while the upper and lower boundaries are represented by F,, and F.4 (see Fig. 1). Only the downstream boundary is assumed to be a straight line. The unit normal and tangential vectors at a boundary are denoted by n and 1". We first consider the following velocity-pressure formulation of the incompressible Navier-Stokes equations:
Main Flow
--~
F01
IV Fig. 1. The computational domain.
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-streant
on~x(0,
p(Ou/Ot+u.Vu)-V.o'=O
V.u=0
onOx(0,
T),
T),
209
(1) (2)
where p and u are the density and velocity, and or is the stress tensor given as o" = - p l + 2/xe(u),
(3)
,(u) = ½(vu + (Vu)').
(4)
with Here p and tt represent the pressure and viscosity, while I denotes the identity tensor. Another commonly used form of the momentum equation is written as p(Ou/Ot + u .Vu) + Vp - / . t V2u = 0 .
(5)
The homogeneous natural boundary conditions corresponding to (1) are the 'traction-free' (i.e., zero normal and shear stress) conditions, and it is reasonable to assume that these conditions can be imposed at the downstream boundary F.3. That is, n . or. n = - p + 2 g Ou,,/On - 0
on F03 x (0, T ) ,
n.o'.~'=lz(Ou,/On+Ou,,/O~')=O
onFo3 x ( 0 , T).
(6)
(7)
The homogeneous natural boundary conditions associated with (5), on the other hand, are - p + ~ Ou,,/On = 0 t~ Ou,/On = 0
on F03 x (0, T ) ,
(8)
(9)
on Fo3 x (0, T ) .
We can combine the natural boundary conditions for (1) and (5) into a unified form by introducing a parameter z which is equal to 1 for (1) an(a O for (5); this unified form can be written as - ( l / p ) p + (1 + z)vOu,.IOn = 0 OUr~On+ ZOu,,/O'r=O
on r03 x (0, T),
one)3×(0, T),
(1o) (11)
where v is the kinematic viscosity. Of course the second condition makes sense only for viscous flows, and therefore the boundary condition given by (11) is discarded for inviscid flows. We would like to derive, for the vorticity-stream function formulation, the downstream boundary conditions equivalent to (10) and (11). In two-dimensional space, the vorticity-stream function formulation of the incompressible Navier-Stokes equations consists of an advection-diffusion equation for the vorticity and a Poisson equation for the stream function: Oto/Ot + u .Vto - vV2to = 0
V2$ + to = 0
on 12 x (0, T ) ,
on O x (0, T ) ,
(12) (13)
210
T.E. Tezduyar, J. Liou, On the downstream boundary conditions .for the vorticity-stream
where the stream function and vorticity are defined as u = {aq,/ax2,-aqJ/ox,},
to = (~U210X
(14)
,
1 -- t~Ul/t~X2
(15)
•
The components of the velocity vector u in the directions denoted by the unit vectors n and 7 can be expressed as u,,=n.u=-aO/,9"r,
(16)
u , = ,,r. u = O ~ l O n .
(17)
To derive the boundary condition equivalent to (11), we start with the following expression for the vorticity at the downstream boundary: Ou,.I a,r - a u , I On
= to on Fo3 x (0, T) ;
(18)
by combining this with (11), we obtain (1 + z ) Ou,,/Or = to
on Fo3 x (0, r ) .
(19)
From (19) and (16) we reach the second-order differential equation given below: (1 + z)cg"-Olar 2 --- - w
on
~,.~ x (0, T).
(20)
In conjunction with this differential equation, we assume that the following boundary conditions are given:
0 = (qJ)l,, at 'r = (~'),,l, q" = ( O ) , v
(21)
at ¢ = (,r), v .
(22)
This is a reasonable assumption, because if for example Fo2 is a symmetry line then the stream function is invariant along F02, and therefore the value of 0 at r = (r)m is known. On the other hand, if F02 is a boundary where we specify both components of the velocity, then, by integrating the velocity field along Fo2, the value of 0 at r = (r)m can be determined. For the derivation of the boundary condition equivalent to (10), we first consider the following form of the momentum equation: ou/ot + v ( l l u l l " / 2 ) - u x
+ (1/p)Vp +
x
=o.
(23)
By taking the projection of this equation in the direction of the unit vector r, we obtain
Ou, 10(u~+u~) Ot + 2
Or
--U.tO+
1 0p
p Or
+ v
0oo
On
----0 o n F o 3 × ( 0 , T).
(24)
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
211
Next we differentiate (10) with respect to ~- and get
1 0 p = (1 + z)v O(Ou,,IOn)
p Or
a~"
on Fo3 x (0, T ) .
(25)
We use the continuity equation to replace Ou,,/On with - 0 u , / 0 % and obtain 1 bp = - ( 1
p ¢~T
+ z)v o2u" 19T2
on Fo3 x (0, T ) .
(26)
By combining (26) with (24) we reach the following equation:
#u,
2 1 O(u2, + u,)
#t -I 2
Or
OZu,
Oto
- u.to - ( 1 + z)v ~ 0 r~ + v 7 n = 0
on Fo3 x (0, T) .
(27)
Rewriting this equation by using (16) results in the following differential equation: 0u,
( 1 + z)v -02u" -=--
bu,
Ot + u, 0-'-¢-
Or 2
10(cgO/Or) 2 _ ( 00) Oto 2 Or -~r t o - v On
on Fo3 × (0, T ) .
(28)
It should be noted that ~, along Fo3 was already determined by using (20)-(22). For the differential equation (28), we assume that the following boundary conditions are given: u, -- (u,),,,
at ,r - (¢),,,,
(29)
u, --- ( u , ) , v
at
(30)
= 0"),v •
This assumption too is a reasonable one, because whether F0, is a boundary with symmetry conditions or a boundary where we specify both components of the velocity, the value of u, is known along F02, including the point ~-= 0")~. In this paper we will provide only the variational formulations associated with the implementation of the downstream boundary conditions. The rest of the variational formulations employed in the finite element discretization of the problem can be found in [6, 8]. We introduce the following solution spaces defined on the interval [0")m, (r)w] for ~, and u, along F0a: S~ = { 0 h I 0 h e H 'a, 0h(('r),,,) = (0),,,, 0h(('r),v) = (0),v},
(31)
h S h141. = {u~ lu.h E H lh, u.((¢)m ) - (u.),,,, u.h((¢),v) --(u.),v} •
(32)
We need only one variational space, and that is V h = { w h l w h ~. H 'h, wh((r),,,)= O, wh(('r),v)= 0}.
(33)
The variational formulation corresponding to (20)-(22) and (28)-(30) can be stated as
212
T.E. Tezduyar. J. Liou, On the downstream boundary conditions for the vorticity-stream h
h
h
follows: given to h, find 0 h E S, and u~ E S,,,, such that (1 + z )
fly
0 Wh a d,jh d z =
xx
Oz
Oz
f/v
whto h d z
and Wh
,, at. U r
~x
=
Ot
Oz /
f,v 0w,/00, II
or
(34)
VW h ~ V h
n
d'r + (1 + z)u
dr-
f:v
flTM tl
19W h
OU h
0"r
0~"
w h --~r O~bh to h d ' r - v
II
d~" 1v W h 0to h u
~ d z
V Wh E V h
(35)
The discrete equation system associated with (34) and (35) can be solved either (in the context of a block-iteration scheme [6, 8]) as an independent block by assuming that to h is given, or (in the context of a fully coupled formulation [9]) simultaneously with the rest of the discrete equations. Of course, in the context of a block-iteration scheme the blocks corresponding to (34) and (35) can also be solved independent of each other. REMARK. We can also ascertain, for two-dimensional problems, the equivalent boundary conditions in the vorticity-stream function formulation for the four types of boundary conditions considered in [5]. (l) Specifying both components of the velocity vector is equivalent to specifying the normal and tangential derivatives of the stream function. This type of boundary conditions was covered in [6]. (2) Specifying the normal component of the velocity and the tangential component of the vorticity (that is the only vorticity component in two dimensions) is equivalent to specifying the tangential derivative of the stream function and the vorticity itself. This type of boundary conditions too was covered in [6]. (3) In cases when the pressure and the tangential velocity is specified, for (24) we can obtain a second-order differential equation for the stream function. This differential equation, together with the end-point conditions of the form (21)-(22), can be used to determine the stream function along the boundary. Since the tangential velocity is also specified, as boundary conditions, we effectively have both the stream function and its normal derivative. (4) When the pressure is specified together with the vorticity, we can use (24) to obtain a differential equation for the tangential velocity. This differential equation, together with the end-point conditions of the form (29)-(30), can be used to determine the normal derivative of the stream function along the boundary.
3. Numerical test: Flow past a circular cylinder
In this test problem the dimensions of the computational domain, normalized by the cylinder diameter, are 30.5 and 16.0 in the flow and cross-flow directions, respectively. The
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
213
mesh employed consists of 5,220 elements and 5,329 nodes; around the cylinder there are 58 elements in the radial and 80 elements in the circumferential directions (see Fig. 2). Our previous convergence studies based on successive mesh refinement [7] suggest that solutions obtained with this mesh will be sufficiently accurate.
|
|
|
|
|
R
a
n
m n n
m
|
|
l
f
l
l
--,--------------------, i Z l l l
'
l
|
l I l i l l l
""'"'" ~:'"'""~ ~ ~
~i~i /:!! ;:;,., :ii~!!!i O !ill,i: ~]~]]~~i! ~:~i~ ,:
.........i
IBIUIIIIIIIIIIIIIIIIIUIIilIll
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~~n~~
I
.;;.==~:===~
'--'''""'"'''""'---" "-----'"'"'--------
g ~ m i V l l ~ l n i m
i
m H H l i 6 l l ~ i b ~ R
anuunuunnnnnnnunnnnuuunnuunun Imlllunimnilllnlmlniulmnil
IIUIIIIIIIIIIIIIUIIIIIIIIIll Inlllllilnlllllllllnlllilll
J
Fig. 2. Flow past a circular cylinder: the finite element mesh (5220 elements and 5,329 nodes).
°
o
0
"~
ca
b
o o
0
-0.50
...~
--0T25
..._,
~
0.00
I
~'
~
i
0;25
0.5{}
o o
o
o
o o o
0
1.15
1.20,,
1.25
1.30
~
1.35
f
J
1.40
].45
--
~
¢=
~r
~
° (~,.,,.,.
~o',~
r~"O
"~ '~ o ~-"o
~0r~
~.
~0°
°
T.E. Tezduyar, J. Liou, On the downstream boundary conditions jor the vorticity-stream
215
inviscid at the downstream boundary; therefore we discard the downstream boundary condition given by (11) and replace it with the zero normal derivative condition for the vorticity. Since at the downstream boundary the vorticity transport equation is nearly hyperbolic, this zero normal derivative condition is essentially equivalent (in the context of the variational formulation) to not specifying any condition for the vorticity. Figure 3 shows the time history of the drag and lift coefficients. The periodic how patterns corresponding to the crest and trough values of the lift coefficient are shown in Figs. 4 and 5. By examining the time history of the lift coefficient, we estimate the Strouhal number to be 0.171. (a)
VortLcLt~
(b)
VortLcLt~
VortLcLt~
( Stream ?unctLon
5treom functLon
Stream ?unctLon
S~otLonor~ s~reom ?unc~Lom
Stotuor,or~ streom ?unctLon
5totLomor~ streom functuon
Fig. 4. Flow past a circular cylinder at Reynolds number 100: periodic solution (corresponding to the crest value of the lift coefficient) for (a) the full computational domain; (b) regions near the cylinder and the downstream boundary.
216
T.E. Tezduyar, J. Liou, On the downstrean, boundary conditions for the vorticity-stream
(a)
VortLcLt~
(b)
VortLcLt~
VortLcCt~
,
i
Stream functLon
5treom functcon
Streom functLon
5~.ot~onor~ s~reom fumC~LOn
Sto~onor~ streom CumctLom
S~otLono~ stream functLon
i .
.
.
.
ii
.
g
Fig. 5. Flow past a circular cylinder t~t Reynolds mlmber I00: periodic solution (corresponding to the trough value of the lift coefficient) for (a) the full computational domain; (b) regions near the cylinder and the downstream boundary.
References [1] A.N. Brooks and T.J.R. Hughes, Streamline-upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259.
T.E. Tezduyar, J. Liou, On the downstream boundary conditions for the vorticity-stream
217
[2] P.M. Gresho and S.T. Chan, Semi-consistent mass matrix techniques for solving the incompressible NavierStokes equations, Lawrence Livermore National Laboratory Preprint, UCRL-99503, 1988. [3] M.O. Bristeau, R. Glowinski and J. Periaux, Numerical methods for the Navier-Stokes equations; Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Rep. 6 (1987) 73-187. [4] O. Pironneau, Conditions aux limites sur la pression pour les equations de Stokes et de Navier-Stokes, C.R. Acad. Sc. Paris 303, Serie I (9) 403-406. [5] T.J.R. Hughes and L.P. Franca, A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg. 65 (1987) 85-96. [6] T.E. Tezduyar, J. Liou and R. Glowinski, Petrov-Galerkin methods on multiply connected domains for the vorticity-stream function formulation of the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 8 (1988) 1269-1290. [7] T.E. Tezduyar, J. Liou and D.K. Ganjoo, Incompressible flow computations based on the vorticity-stream function and velocity-pressure formulations, Comput. & Structures 35 (1990) 445-472. [8] T.E. Tezduyar, Finite element formulation for the vorticity-stream function form of the incompressible Euler equations on multiply-connected domains, Comput. Methods Appl. Mech. Engrg. 73 (1989)331-339. [9] T.E. Tezduyar, J. Liou, D.K. Ganjoo and M. Behr, Solution techniques for the vorticity-stream function formulation of two-dimensional unsteady incompressible flows, Internat. J. Numer. Methods Fluids l 1 (1990) 515-539.