图中所示为管道流动中的基流。 假设经扰动后的流速为
u
=
(
U
(
z
)
+
u
′
(
x
,
z
,
t
)
,
0
,
w
′
(
x
,
z
,
t
)
)
{\displaystyle \mathbf {u} =\left(U(z)+u'(x,z,t),0,w'(x,z,t)\right)}
其中
(
U
(
z
)
,
0
,
0
)
{\displaystyle (U(z),0,0)}
u
′
∝
exp
(
i
α
(
x
−
c
t
)
)
{\displaystyle \mathbf {u} '\propto \exp(i\alpha (x-ct))}
流函数
μ
i
α
ρ
(
d
2
d
z
2
−
α
2
)
2
φ
=
(
U
−
c
)
(
d
2
d
z
2
−
α
2
)
φ
−
U
″
φ
{\displaystyle {\frac {\mu }{i\alpha \rho }}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi }
其中
μ
{\displaystyle \mu }
动力黏度 ，
ρ
{\displaystyle \rho }
密度 ，
φ
{\displaystyle \varphi }
瑞利方程
无量纲形式的奥尔－索末菲方程为：
1
i
α
R
e
(
d
2
d
z
2
−
α
2
)
2
φ
=
(
U
−
c
)
(
d
2
d
z
2
−
α
2
)
φ
−
U
″
φ
{\displaystyle {1 \over i\alpha \,Re}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi }
其中
R
e
=
ρ
U
0
h
μ
{\displaystyle Re={\frac {\rho U_{0}h}{\mu }}}
雷诺数 （
U
0
{\displaystyle U_{0}}
h
{\displaystyle h}
z
=
z
1
{\displaystyle z=z_{1}}
z
=
z
2
{\displaystyle z=z_{2}}
α
φ
=
d
φ
d
z
=
0
{\displaystyle \alpha \varphi ={d\varphi \over dz}=0}
φ
{\displaystyle \varphi }
或
α
φ
=
d
φ
d
x
=
0
{\displaystyle \alpha \varphi ={d\varphi \over dx}=0}
φ
{\displaystyle \varphi }
方程的特征值为
c
{\displaystyle c}
φ
{\displaystyle \varphi }
c
{\displaystyle c}
参考文献
编辑
Orr, W. M'F. The stability or instability of the steady motions of a liquid. Part I. Proceedings of the Royal Irish Academy. A. 1907, 27 : 9–68. Orr, W. M'F. The stability or instability of the steady motions of a liquid. Part II. Proceedings of the Royal Irish Academy. A. 1907, 27 : 69–138. Sommerfeld, A. Ein Beitrag zur hydrodynamische Erklärung der turbulenten Flüssigkeitsbewegungen. Proceedings of the 4th International Congress of Mathematicians III . Rome. 1908: 116–124. 
