Data processing: measuring – calibrating – or testing – Measurement system in a specific environment – Mechanical measurement system
Reexamination Certificate
2000-09-29
2002-07-23
Hilten, John S. (Department: 2863)
Data processing: measuring, calibrating, or testing
Measurement system in a specific environment
Mechanical measurement system
C703S002000, C703S008000, C703S009000, C137S804000, C073S147000, C073S195000
Reexamination Certificate
active
06424923
ABSTRACT:
BACKGROUND OF THE INVENTION
(1) Field of the Invention
This invention generally relates to the analysis of fluid flow past a three dimensional object and more particularly to a method and apparatus for calculating three-dimensional unsteady flows past such an object by direct solution of the vorticity equation on a Lagrangian mesh.
(2) Description of the Prior Art
Understanding the characteristics of fluid as it flows past an object, such as an airfoil, is important both from the standpoint of understanding and improving the designs of such objects and in understanding the nature of any turbulence introduced as a result of relative motion of a fluid an airfoil, either by moving of the airfoil through the fluid or by moving the fluid past the airfoil.
Eventually additional studies determined that vorticity was useful as a basis for understanding fluid flow. Vorticity is produced at a solid boundary because at the surface the fluid has no velocity (i.e., the fluid exhibits a no slip condition). Once generated at the surface, vorticity diffuses into the volume of the fluid where it is advected by local flow. Conventional vortex methods generally mime this process. In accordance with such methods, the strengths of the vortex elements or segments originating on the body surface are determined by requiring that the velocity induced by all the vortex elements on the surface be equal and opposite to the velocity at the surface. It is assumed that this vorticity is contained in an infinitely thin sheet at the surface. In these methods a resulting matrix equation is solved for the surface vorticity at all points on the body simultaneously. Vorticity transfer to the flow is then accomplished by placing the vortex elements above the surface.
It has been recognized that these vortex methods have several shortcomings. When computational methods use point vortices in their simulations, mathematical singularities can produce divergent solutions. This has been overcome by using a kernel function that contains a regularized singularity. However, this kernel function depends on certain ad hoc assumptions such as the value of the cutoff velocity and core radius. While the no-slip and no-flow boundary conditions provide information regarding the strength of the surface vorticity and subsequent strength of the vortex element, their use often neglects the effects of all other vortex sheets on the surface. Other implementations of such methods neglect the effects of coupling between the surface vortex sheets and surface sources. Finally, many methods assume, a priori, a separation point to analyze shedding of vorticity from the surface into the flow that generally requires experimental knowledge of the flow. More recent prior art has utilized computer modeling based upon the nature of vortex elements at the surface of an object, such as an airfoil. These models then track the motion of each element as it moves into the flow over time to calculate the velocity of each element. While this prior art produces acceptable results, the direct calculation of the velocity of each vortex element produces an N
2
increase in the required time for processing where N is the number of vortex elements for each time step. Such increases can become unacceptable when high resolution demands the calculation of a large number of vortex elements.
The evolution of fluid flow of uniform density is prescribed by the vortcity equation
∂
ω
∂
t
+
u
·
▽ω
=
ω
·
▽
u
+
v
▽
2
ω
(
1
)
where v is the kinematic viscosity and the velocity u and vorticity&ohgr; are related by
∇=
u
=&ohgr; (2)
Since the density is taken to be uniform, the equation for conservation of mass reduces to the condition on the velocity field
∇·
u
=0 (3)
At the no-slip surface of a body immersed in the fluid and moving with velocity U
body
, the velocity of the fluid satisfies the boundary condition
u =u
body
(4)
on the body surface. In this frame of reference, the velocity of the fluid at infinity is typically taken to be zero. The vorticity boundary condition at the body surface is established by requiring the satisfaction of (4), as described below. When the pressure distribution on the body surface is desired, it is found by solution of the matrix equation stemming from the integral form of the pressure Poisson equation (Uhlman, 1992)
β
B
+
∯
S
B
∂
∂
n
(
1
r
)
ⅆ
S
=
-
∯
S
(
n
r
·
∂
u
∂
t
+
v
r
·
n
×
ω
r
3
)
ⅆ
S
+
∫
∫
∫
V
r
·
u
×
ω
r
3
ⅆ
V
(
5
)
where B is defined as
B
=
p
-
p
∞
ρ
+
1
2
(
u
·
u
-
U
∞
·
U
∞
)
(
6
)
and
a
^
=
{
4
∂
′
→
V
2
∂
′
→
S
0
→
V
c
(
7
)
with V defined as the volume exterior to the body whose surface is S, and Vc is the complement of V. The matrix equation has as its unknown B on the surface; for pressure in the interior of the flow, the surface integral on the left-hand side of (5) is known from the matrix solution.
To determine the velocity field associated with the instantaneous vorticity field calculated by solution of (1) and thus advance the solution in time, we employ the vector identity for a sufficiently integrable and differentiable vector field a defined within a volume V bounded by a surface S with normal unit vector n:
β
a
=
-
∯
S
(
(
n
·
a
)
G
-
G
×
(
n
×
a
)
)
ⅆ
S
+
∫
∫
∫
V
(
(
▽
·
a
)
G
-
G
×
(
▽
×
a
)
)
ⅆ
V
(
8
)
with&bgr; as given above, and G is any vector Green'function of the form
G
=
r
r
3
+
H
(
r
)
(
9
)
Here r is the vector from integration element to field point, and H is regular. When a is the velocity u, after substituting (2) and (3) into (6) and taking H to be zero, we have the familiar Biot-Savart integral
u
(
x
)
=
1
4
π
∫
∫
∫
V
ù
×
(
x
-
x
′
)
&LeftBracketingBar;
x
-
x
′
&RightBracketingBar;
3
ⅆ
3
x
′
(
10
)
plus surface terms depending on the particular application.
This integral is to be computed given the solution&ohgr; (x
m
) on the calculational points x
m
, m=1, 2, . . . , N, where N is the number of points. An important feature of (6) is that the farfield boundary condition on velocity is explicit, so that computational points are needed only where&ohgr; is nonzero. An approach to the velocity calculation based on an inversion of the Poisson equation for velocity would require a volume of calculational points extending to where the farfield velocity can be accurately approximated by an analytic expression.
The vortex blob method is an effective way to carry out integration on scattered points, and is quite useful for many unbounded flow investigations; however it has some drawbacks for flow past surfaces. The principal limitation stems from the blob geometry. Flow near a surface is well known to be anisotropic in scale, strongly so at high Reynolds numbers. Resolution of the vorticity field in such flows demands a large number of blobs. Additionally, anisotropic blobs to have limited utility in computing flow past an object because it is difficult to maintain overlap to properly resolve the flow. In addition, regions near the object surface can arise where the anisotropy is complicated, such as in separated flow.
A recent prior art innovation some of these computational methods was taught in U.S. Pat. No. 5,544,524 filed by Grant, Huyer and Uhlman which is incorporated by reference herein. This prior art teaches the use of
Grant John R.
Huyer Stephen A.
Marshall Jeffrey S.
Uhlman James S.
Cherry Stephen J.
Hilten John S.
Kasischke James M.
Lall Prithvi C.
McGowan Michael J.
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