Data processing: structural design – modeling – simulation – and em – Simulating nonelectrical device or system – Fluid
Reexamination Certificate
1999-02-10
2003-02-04
Jones, Hugh M. (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Simulating nonelectrical device or system
Fluid
C703S005000, C210S512200, C345S419000
Reexamination Certificate
active
06516292
ABSTRACT:
FIELD AND BACKGROUND OF THE INVENTION
The present invention relates to the numerical simulation of fluid flow and, more particularly, to a numerical method of simulating barotropic fluid flow past a body by minimizing a potential functional.
Barotropic fluid flow is described by the Euler equations:
∂
v
→
∂
t
+
v
→
·
∇
→
v
→
+
∇
→
(
h
+
Φ
)
=
0
(
1
)
and of the continuity equation
∂
ρ
∂
t
+
∇
→
·
(
ρ
v
→
)
=
0
(
2
)
where {right arrow over (&ngr;)}(x
k
,t) is the velocity vector field, &PHgr;(x
k
,t) is a potential relating to an external force, &rgr;(x
k
,t) is the scalar density field, and h, the specific enthalpy, is a function of &rgr; through the equation of state of the fluid. {right arrow over (&ngr;)}, &PHgr; and &rgr; are functions of position x
k
=(x,y,z) in space and of time t. In the special case of a stationary flow with small viscosity, the time derivatives vanish:
{right arrow over (&ngr;)}·{right arrow over (∇)}{right arrow over (&ngr;)}+{right arrow over (∇)}(
h
+&PHgr;)=0; {right arrow over (∇)}·(&rgr;{right arrow over (&ngr;)})=0 (3)
Only in trivial cases can these equations be solved analytically. In cases of practical interest, for example, in aerodynamics and hydrodynamics, these equations must be solved numerically. The most common way to solve these equations is by integrating them. The possible numerical instabilities associated with numerical integration are well known and need not be detailed here. A numerical solution based on a variational principle would be inherently numerically stable. Ideally, such a numerical solution would involve finding the minima of a potential functional with respect to the associated variables. In the case of fluid flow, the variables are four scalar fields: &rgr; and the three Cartesian components of {right arrow over (&ngr;)}. Heretofore, no variational formulation of barotropic fluid flow has represented the solutions of equations (1) and (2) or of equations (3) as the values of {right arrow over (&ngr;)} and &rgr;, or of any other set of only four scalar fields, that minimize a potential functional. The smallest number of scalar fields obtained heretofore was seven, by R. L. Seliger and G. B. Witham (
Proc. Roy. Soc. London
, Vol. A305, p. 1, 1968). Because equations (1)-(3) depend on only four scalar fields, it should be possible to obtain a variational formulation of barotropic fluid flow in terms of only four scalar fields.
There is thus a widely recognized need for, and it would be highly advantageous to have, a method of finding numerical solutions of the equations of fluid flow by minimizing a potential functional of four scalar fields.
SUMMARY OF THE INVENTION
According to the present invention there is provided a numerical method of simulating fluid flow past a body having a boundary, including the steps of: (a) formulating the fluid flow in terms of a potential functional of at most four fundamental scalar fields and at most two scalar variables; and (b) extremizing the potential functional with respect to the at most four fundamental scalar fields and with respect to the at most two scalar variables.
According to the present invention there is provided a system for numerical simulation of fluid flow, including: (a) a software module including a plurality of instructions for computing, from discrete representations of at most four fundamental scalar fields, a potential functional representative of the fluid flow, and for varying the discrete representations to extremize the potential functional; (b) a processor for executing the instructions; and (c) a memory for storing the discrete representations.
According to the present invention there is provided a numerical method of simulating fluid flow, including the steps of: (a) formulating the fluid flow in terms of a potential functional of at most four fundamental scalar fields and at most two scalar variables; and (b) extremizing the potential functional with respect to the at most four fundamental scalar fields and with respect to the at most two scalar variables.
According to the present invention there is provided a numerical method of simulating fluid flow past a body having a boundary, including the steps of: (a) formulating the fluid flow in terms of an action which is the sum of: (i) an action integral, from an initial time t
0
to a final time t
1
, of a Lagrangian functional L of at most four fundamental scalar fields, (ii) a spatial integral of a function of the at most four fundamental scalar fields at the final time t
1
, and (iii) a negative of a spatial integral of the function of the at most four fundamental scalar fields at the initial time t
0
; and (b) extremizing the action with respect to the at most four fundamental scalar fields.
According to the present invention there is provided a numerical method of simulating fluid flow including the steps of: (a) formulating the fluid flow in terms of an action which is the sum of: (i) an action integral, from an initial time t
0
to a final time t
1
, of a Lagrangian functional L of at most four fundamental scalar fields, (ii) a spatial integral of a function of the at most four fundamental scalar fields at the final time t
1
, and (iii) a negative of a spatial integral of the function of the at most four fundamental scalar fields at the initial time t
0
; and (b) extremizing the action with respect to the at most four fundamental scalar fields.
According to the present invention there is provided a system for numerical simulation of fluid flow, including: (a) a software module including a plurality of instructions for computing, from discrete representations of at most four fundamental scalar fields, an action representative of the fluid flow, and for varying the discrete representations to extremize the action; (b) a processor for executing the instructions; and (c) a memory for storing the discrete representations.
It is shown in the Theory section below that stationary barotropic fluid flow can be formulated as the extremization of a potential functional of at most four scalar fields: the Clebsch scalar fields &agr;, &bgr; and &ngr; and the density field &rgr;; and also of at most two scalar variables &bgr;
1
and &ngr;
1
. Specifically, the functional is minimized with respect to &agr;, &bgr; and &ngr; and maximized with respect to &bgr;
1
and &ngr;
1
. The scalar fields are functions of the position vector x
k
. The velocity vector field {right arrow over (&ngr;)} is related to the Clebsch scalar fields through {right arrow over (&ngr;)}=&agr;{right arrow over (∇)}&bgr;+{right arrow over (∇)}&ngr;.
The potential functionals are as follows:
If the component of {right arrow over (&ngr;)} normal to the boundary of the region of fluid flow is zero, or if the density &rgr; on the boundary is zero, the potential functional is
∫
V
[
1
2
(
α
∇
→
β
+
∇
→
v
)
2
+
ϵ
(
ρ
)
+
Φ
]
ρ
ⅆ
3
x
-
β
1
(
∫
V
α
ρ
0
ⅆ
3
x
-
α
_
M
0
)
-
v
1
(
∫
V
ρ
ⅆ
3
x
-
M
0
)
(
4
)
(equations 56 and 62 of the Theory section below) for compressible flows and
∫
V
[
1
2
(
α
∇
→
β
+
∇
→
v
)
2
+
Φ
]
ρ
0
ⅆ
3
x
-
β
1
(
∫
V
α
ρ
0
ⅆ
3
x
-
α
_
M
0
)
(
5
)
(equations 65 and 66 of the Theory section below) for incompressible flows. &egr;(&rgr;) is the specific internal energy of the fluid, in the case of a comp
Friedman Mark M.
Jones Hugh M.
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