Page 89 - Computational Fluid Dynamics for Engineers

74 2. Conservation Equations

dui dui 1 dp dan „ . ^ ^ ^ N
7J7 + U H^ = --7T- + JT l + fi P2.2.2
at J oxj g oxi OXJ
show that the kinetic energy equation, (P2.1.2) can also be written as
D / UiUi\ dp da ik , D O O Q ,
u
Q
-
' -
m K ^r) = *dx~ + ^ ~ ^ (P2 2 3)
/fmt: Take the scalar product of Uj and the momentum equation for Ui, Eq.
(P2.2.2), with its left-hand side in conservation form

— + —(u-u ) = --^- + ^ ^ + f-
dt dxjc l gdxi dxk
together with the scalar product of U{ and the momentum equation for Uj. Add
the resulting two expressions to get
d ( N i d ^ ^ dp dp da^
Q {UiUj) + W f c ) u Ui + u
m W = - ^ % - d^ ^ k

+ u i^-+u jf i + u if j (P2.2.4)

j
and then set i = .

2-3. Noting the definition of aij in the form given by Eq. (2.3.7) and following
the procedure of Problem 2.2, show that the mean kinetic energy equation of
the mean motion can be written as
+
1+sf
--
§i ^) - -£ -S- -'i'^ - (P231)
(
which is identical to Eq. (P2.2.3) if we assume that a^ represents both viscous
and turbulent stresses. The third term in Eq. (P2.3.1) can be written as

-^(»,K<) K<H (P2 32)
-
+
The first term in this equation represents the spatial transport of mean kinetic
energy by the turbulent fluctuations; it is sometimes called the "gain from en-
ergy flux" or "the divergence of the energy flux transmitted by the working of
the mean flow against the Reynolds stress." The second term represents the
"loss to turbulence" or the production of turbulent energy from the mean flow
energy.

2-4. Defining a differential length of a streamline by ds, which for a Cartesian
coordinate system is
ds = dxi + dyj + dzk (P2.4.1)