Geology Reference
In-Depth Information
p(x,r,T) + ½ U (I
x
2
+ I
r
2
+ 1/r
2
I
T
2
) = p
0
(7.3.6)
Since Equation 7.3.6 is nonlinear in I(x,r,T), a simple formula connecting ) and
p cannot be obtained. Special treatment is needed so that the radially-averaged
potential variable )(x,T) can be used. First, we expand I= I
0
+ I
1
where |I
0
| >>
|I
1
|. Here “0” represents the uniform oncoming flow and “1” the disturbance
flow induced by the presence of the siren. Neglecting higher order terms,
p(x,r,T) + ½ UI
0 x
2
+ UI
0 x
I
1x
+ . . . = p
0
(7.3.7)
As before, multiply throughout by 2Sr and integrate over (R
i
,R
o
) to obtain
³p(x,r,T) 2Sr dr + ½ UI
0 x
2
³2Sr dr + UI
0 x
³I
1x
2Sr dr
#
p
0
³2Sr dr
(7.3.8a)
noting that I
0 x
and p
0
are constants. Introducing
³p(x,r,T) 2Sr dr = A p
avg
(x,T)
(7.3.8b)
³I
1x
2Sr dr = A)
1x
(x,T)
(7.3.8c)
³2Sr dr = S(R
0
2
- R
i
2
)
(7.3.8d)
we find that Equation 7.3.8a becomes
Ap
avg
(x,T) + ½ UI
0 x
2
S(R
0
2
- R
i
2
) + UI
0 x
A)
1x
(x,T)
#
p
0
S(R
0
2
- R
i
2
)
(7.3.9)
Without loss of generality, we now set the reference area to A = S(R
0
2
- R
i
2
) so
that Equations 7.3.5 and 7.3.9 simplify as follows,
)
xx
+ 1/R
m
2
)
TT
#
2U
f
{R
i
V
i
- R
o
V
o
}/(R
0
2
- R
i
2
) (7.3.10)
p
avg
(x,T) + ½ UI
0 x
2
+ UI
0 x
)
1x
(x,T)
#
p
0
(7.3.11)
Equations 7.3.10 and 7.3.11 are the final governing partial differential equations
solved. We next discuss auxiliary conditions used to obtain specific solutions.
The normalization used for A was selected so that Equation 7.3.11 takes the
form of Bernoulli's equation linearized about the mean speed I
0x
. The quadratic
terms neglected in this approximation can be added back to Equation 7.3.11
without formally incurring error to this order. If this is done, the modified
formulation is advantageous since it is exact for planar cascade flow.
7.3.3 Upstream and annular boundary condition.
Auxiliary conditions are key in solving the two-dimensional elliptic
Poisson equation prescribed by Equation 7.3.10. Despite the reduced order of
the partial differential equation, geometric complexities can render fast solutions
difficult. However, motivated by the “thin airfoil” and “thin engine nacelle”
approaches offered in Section 2, in which exact flow tangency conditions are
approximated along mean lines, planes, and cylindrical surfaces, we adopt a
similar approach but for three-dimensional mud sirens. Before describing the
Search WWH ::
Custom Search