Derivation of RANS equations Reynolds-averaged Navier–Stokes equations
^ tennekes, h.; lumley, j. l. (1992). first course in turbulence (14. print. ed.). cambridge, mass. [u.a.]: mit press. isbn 978-0-262-20019-6.
^ splitting each instantaneous quantity averaged , fluctuating components yields,
∂
(
u
i
¯
+
u
i
′
)
∂
x
i
=
0
{\displaystyle {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{i}}}=0}
∂
(
u
i
¯
+
u
i
′
)
∂
t
+
(
u
j
¯
+
u
j
′
)
∂
(
u
i
¯
+
u
i
′
)
∂
x
j
=
(
f
i
¯
+
f
i
′
)
−
1
ρ
∂
(
p
¯
+
p
′
)
∂
x
i
+
ν
∂
2
(
u
i
¯
+
u
i
′
)
∂
x
j
∂
x
j
.
{\displaystyle {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial t}}+\left({\bar {u_{j}}}+u_{j}^{\prime }\right){\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{j}}}=\left({\bar {f_{i}}}+f_{i}^{\prime }\right)-{\frac {1}{\rho }}{\frac {\partial \left({\bar {p}}+p^{\prime }\right)}{\partial x_{i}}}+\nu {\frac {\partial ^{2}\left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{j}\partial x_{j}}}.}
time-averaging these equations yields,
∂
(
u
i
¯
+
u
i
′
)
∂
x
i
¯
=
0
{\displaystyle {\overline {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{i}}}}=0}
∂
(
u
i
¯
+
u
i
′
)
∂
t
¯
+
(
u
j
¯
+
u
j
′
)
∂
(
u
i
¯
+
u
i
′
)
∂
x
j
¯
=
(
f
i
¯
+
f
i
′
)
¯
−
1
ρ
∂
(
p
¯
+
p
′
)
∂
x
i
¯
+
ν
∂
2
(
u
i
¯
+
u
i
′
)
∂
x
j
∂
x
j
¯
.
{\displaystyle {\overline {\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial t}}}+{\overline {\left({\bar {u_{j}}}+u_{j}^{\prime }\right){\frac {\partial \left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{j}}}}}={\overline {\left({\bar {f_{i}}}+f_{i}^{\prime }\right)}}-{\frac {1}{\rho }}{\overline {\frac {\partial \left({\bar {p}}+p^{\prime }\right)}{\partial x_{i}}}}+\nu {\overline {\frac {\partial ^{2}\left({\bar {u_{i}}}+u_{i}^{\prime }\right)}{\partial x_{j}\partial x_{j}}}}.}
note nonlinear terms (like
u
i
u
i
¯
{\displaystyle {\overline {u_{i}u_{i}}}}
) can simplified to,
u
i
u
i
¯
=
(
u
i
¯
+
u
i
′
)
(
u
i
¯
+
u
i
′
)
¯
=
u
i
¯
u
i
¯
+
u
i
¯
u
i
′
+
u
i
′
u
i
¯
+
u
i
′
u
i
′
¯
=
u
i
¯
u
i
¯
+
u
i
′
u
i
′
¯
{\displaystyle {\overline {u_{i}u_{i}}}={\overline {\left({\bar {u_{i}}}+u_{i}^{\prime }\right)\left({\bar {u_{i}}}+u_{i}^{\prime }\right)}}={\overline {{\bar {u_{i}}}{\bar {u_{i}}}+{\bar {u_{i}}}u_{i}^{\prime }+u_{i}^{\prime }{\bar {u_{i}}}+u_{i}^{\prime }u_{i}^{\prime }}}={\bar {u_{i}}}{\bar {u_{i}}}+{\overline {u_{i}^{\prime }u_{i}^{\prime }}}}
^ follows mass conservation equation gives,
∂
u
i
∂
x
i
=
∂
u
i
¯
∂
x
i
+
∂
u
i
′
∂
x
i
=
0
{\displaystyle {\frac {\partial u_{i}}{\partial x_{i}}}={\frac {\partial {\bar {u_{i}}}}{\partial x_{i}}}+{\frac {\partial u_{i}^{\prime }}{\partial x_{i}}}=0}
Comments
Post a Comment