A direct one-step pressure actualization for incompressible ow
with pressure Neumann condition
Elba Bravo∗, Julio R. Claeyssen, Rodrigo B. Platte
URI-PROMEC, Universidade Federal do Rio Grande do Sul, P.O. Box 10673, 90.001-000 Porto Alegre - RS, Brazil Received 30 October 1997
Abstract
We develop a velocity–pressure algorithm, in primitive variables and nite dierences, for incompressible viscous ow with a Neumann pressure boundary condition. The pressure eld is initialized by least-squares and up-dated from the Poisson equation in one step without iteration. Simulations with the square cavity problem are made for several Reynolds numbers. We obtain the expected displacement of the central vortex and the appearance of secondary and tertiary eddies. Dierent geometry ratios and a 3D cavity simulation are also considered. c 1999 Elsevier Science B.V. All rights reserved.
Keywords:Pressure update; incompressible ow; square cavity
1. Introduction
In this work, we consider incompressible viscous ow in primitive variables by using nite dif-ferences and a Neumann pressure boundary condition, as discussed by Gresho and Sani [13]. This allows us to develop a direct one-step pressure actualization.
The discretization by dierence methods of the Navier–Stokes equations on a staggered grid, as made by Casulli [8], when formulated in matrix terms, allows to identify a singular evolutive matrix system. When we derive the Poisson equation for the pressure and perform its integration, we can observe that a clear inuence of the Neumann condition arises. From this we can extract a nonsingular system for determining the pressure values at the interior points. The initialization process of the pressure, by a least-squares procedure, somehow incorporates an optimal pressure as a starting point, instead of employing an arbitrary constant as it usually occurs with iterative methods. The values of the velocity at interior points can then be well determined by a forward Euler or Adams–Bashforth method. For the pressure we solve a nonsingular Poisson equation without iteration. The latter means that we incorporate the values of the pressure and velocity as soon as they are computed.
∗Corresponding author.
This velocity–pressure algorithm with central dierences has been tested with the cavity problem for a wide range of Reynolds numbers and geometric ratios that include square, deep and shallow cavities. For a square cavity the displacement of the central vortex to the geometrical center of the cavity was obtained by increasing the Reynolds number, as earlier established by Burggraf [5], Ghia et al. [12] and Schreiber and Keller [20], among others. Also, the apparition of secondary and tertiary vortices can be observed.
The proposed algorithm, described for 2D regions, can be appropriately modied for 3D regions. Simulations were also made for a 3D cavity.
2. The continuum equations for incompressible ow
We consider the Navier–Stokes equations
@u
@t +u·3u+3p=3
2u; t ¿0; (1)
3·u= 0; (2)
u(x;0) =u0(x); x∈=⊕ ; (3)
u=w(x; t)∈ =@ : (4)
For the pressure, we have the Poisson equation
32p=3·(32u−u·3u) =·3·(u·3u) (5)
together with the Neumann condition [13]
@p @n =3
2u n−
@u
n
@t +u·3un
∈ for t¿0; (6)
where un=u·n is the normal velocity component.
The determination of the solution of the Poisson equation with Neumann boundary conditions requires that the following compatibility relation holds:
Z Z
−3·(u·3u) d= I
pnd ; (7)
where pn=n·3p, and n is an exterior normal unit vector to .
3. Discretization of the Navier–Stokes Equations
The primitive equations for a 2D incompressible viscous ow are
@u @t +u
@u @x +v
@u @y =−
@p @x +
@2u
@x2 +
@2u
@y2
!
; (8)
@v @t +u
@v @x +v
@v @y =−
@p @y +
@2v
@x2 +
@2v
@y2
!
@u @x +
@v
@y = 0; (10)
where u(x; y; t) and v(x; y; t) denote the velocity components in x and y directions, p(x; y; t) the pressure and¿0 the kinematic viscosity coecient. Following Casulli [8], we use central dierences for approximating the spatial derivatives and the explicit Euler method for approximating the time derivative. Thus, with reference to a staggered grid, we have
uk+1
4. The pressure equation discretization
The Poisson equation for the pressure is given by
p=−3·(u·3u)−Dt; (17)
where the dilation term
D=ux+vy
The following Neumann boundary conditions for the pressure are obtained from the momentum equation on a solid boundary x= 0, 06y6B, 06x6A, y=B, x=A, 06y6B, 06x6A, y= 0:
−px =ut+u ux+v uy+ (vxy−uyy) in x= 0; A; (18) −py =vt+u vx+v vy− (vxx−uxy) in y= 0; B: (19)
The Poisson equation (17) and the boundary conditions (18) and (19) for the pressure are now approximated on a staggered grid with x = y = h. The spatial derivatives in (17)–(19) shall be now approximated by second-order central dierences for interior cells and cells adjacent to the boundary.
4.1. Interior cells
We consider the Poisson equation
pxx+pyy =−
@ @x
u@u @x
− @
@x
v@u @y
− @
@y
u@v @x
− @
@y
v@v @y
−Dt: (20)
As usual, the dilatation term Dt is approximated by
Dt≃
Dk+1−Dk
t ; (21)
where the superscript indexes k and k + 1 refer to the time levels t and t+ t. In order to satisfy the continuity equation (10), Dk+1 is made equal to zero.
Let (i; j) refer to an interior cell, that is, without common sides with the boundary. By using second-order central dierences for approximating the derivatives pxx and pyy, the Poisson equation
(20) is approximated by
pi+1; j+pi−1; j+pi; j+1+pi; j−1−4pi; j
=−1
2ui+1=2; j (ui+3=2; j−ui−1=2; j)
+1
2ui−1=2; j (ui+1=2; j−ui−3=2; j)−
1
2vi+1=2; j (ui+1=2; j+1−ui+1=2; j−1)
+1
2vi−1=2; j (ui−1=2; j+1−ui−1=2; j−1)− 1
2ui; j+1=2 (vi+1; j+1=2−vi−1; j+1=2)
+1
2ui; j−1=2 (vi+1; j−1=2−vi−1; j−1=2)−
1
2vi; j+1=2 (vi; j+3=2−vi; j−1=2)
+1
2vi; j−1=2 (vi; j+1=2−vi; j−3=2) +
h
t (ui+1=2; j−ui−1=2; j+vi; j+1=2−vi; j−1=2);
5. Cells adjacent to the boundary
The boundary condition (18) is computed at u3=2; j by using a central dierence approximation.
Thus
Similar expressions are obtained by using (18) at un−1=2; j and computing the boundary condition
(19) at vi;3=2 andvi; m−1=2, that is
The addition of terms on both sides of (22)–(26) can be interpreted as a discrete divergence theorem [1]. In our case, both add up to zero which tell us that the compatibility equation (7) is exactly satised on a staggered grid.
We should observe that the viscous terms in the momentum equations (8) and (9) do not appear in the source term for the Poisson equation (17). However, they are present within the Neumann boundary conditions (18), (19). In order to satisfy the compatibility condition (7), the integral of the viscous terms over the boundary must cancel. This is obtained by writing the viscous terms in a convenient way. More precisely, by using the continuity equation (10), we can write uxx+uyy = −vxy+uyy in (18) and vxx+vyy =vxx−uxy in (19). The additional term does not ocassionate any
6. The velocity–pressure algorithm
We now give an algorithm for integrating the Navier–Stokes equations. First, the pressure is initialized by applying least squares to the singular system that arises from the discretization of (17) with the Neumann conditions (18) and (19). Second, the momentum Eqs. (8) and (9) are solved for the velocity eld at each time step. Third, the pressure is up-dated from (17)–(19) by giving a special treatment for the interior points that correspond to interior cells and to the adjacent cells in such a way that the compatibility condition is veried. The pressure at interior points of interior cells are computed in a direct manner, that is, by incorporating the already known pressure values at cause points in (22).
6.1. Pressure initialization
From (22) at the time level k = 0 and discretizing the Neumann condition we obtain the matrix system
Ap0=b; (27)
where A is the singular matrix
A=
and I is the identity matrix of order n.
At time k = 0, the vector p0 contains all associated values of the pressure at interior points, that
is,
The vector b contains all values u0
i+1=2; j, v0i; j+1=2; from the right-hand side of (22)–(26), which are
given initial values, and this has the particular form,
where
Hence, b is a nonzero vector.
The singular system (27) is then solved by least squares.
6.2. Pressure equation
Once the pressure is initialized, the interior pressure values pi; j at time t+ t are computed with
the following criteria:
1. At the interior points corresponding to adjacent boundary cells we use (23) and (36).
2. At interior points of the interior cells, we employ (22) to compute the pressure values at each time level by incorporating previous values of the velocity and pressure elds. This modication lead us to
The algorithm for solving an incompressible viscous ow with prescribed Neumann condition for the pressure is as follows.
1. Introduction of the initial velocity components u0
i+1=2; j, v0i; j+1=2 at time t0 = 0, corresponding to
level k = 0, and the boundary conditions for the velocity eld.
2. Initialization of the pressure through least squares, that is, to solve a singular linear system of the type
Ap0 =b:
3. Computation of the velocity eld uk+1i+1=2; j and vk+1i; j+1=2 by using (11)–(13) and (14)–(16). 4. Direct computation of the pressure p at level time k+ 1 through (28).
6. To perform steps (3)–(5) for k = 1;2; : : : . 7. End the calculations.
7. Pressure correction of a multi-step velocity integration
By using Adams–Bashforth for the time derivatives, the momentum equation can be written in discretized form
where F only contains convective and diusive terms.
By applying the divergence operator to (29), we have that the incompressibility condition at time level k+ 1 is characterized by
32pk=3·F(uk) +3·uk
Following the steps of the pressure discretization for Euler integration, we obtain
pk+1i; j =1
By Taylor expansion about pk+1
i; j and replacing 32pk+1 in 3·uk+1 turns out that
This shows that we have an articial compressibility of order O((t2=h2)(@p=@t)).
8. Simulations
Numerical simulations were carried out for the cavity problem for a broad range of Reynolds numbers. Figs. 1 and 2 show the velocity for Re = 400, 1000, 5000 and 10 000 on a square grid with x = 0.01 and time steps t = 0.001, 0.002. This values meet the stability criteria t=h ¡1 and t6h2=4 as suggested by Roache, [18], Casulli [8], among others.
Fig. 1. Normalized velocity eld: square cavity (a) Re = 400; (b) Re = 1000; (c) Re = 5000 and (d) Re = 10 000.
9. Conclusions
Fig. 2. Tridimensional cavity at Re = 400 (a) perspective view; (b)y–z plane; (c)x–z plane, (d) x–yplane.
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