Page 71 - Computational Fluid Dynamics for Engineers

56 2. Conservation Equations

Q + 2 +efy e {v u ] {v 2) Q v w,) (234)
wt = -% ^ " ~ t ' '' % ' - h' --

e^ = -f z+ MV «) + oh - e ^ ( w ) - Q^tvft) - Q§-J^) (2-3-5)
2

T~V/-T-I r\ r\ r\
2
gc p— =q w + kV T - QC P—TW - gc p—TQ - Qc p—TW (2.3.6)
It is common to drop the overbars on the basic time variables; this results
in a continuity equation identical to that given by Eq. (2.2.1), and the left-
hand sides of the momentum and energy equations, Eqs. (2.3.3) to (2.3.6),
become identical to the equations for laminar flow. The right-hand sides of the
momentum and energy equations also resemble the right-hand sides of Eqs.
(2.2.2) to (2.2.4) and (2.2.11) with the addition of the Reynolds normal, shear
stress, and heat flux terms: in our previous notation the Reynolds stresses in Eq.
(2.3.3) represent the turbulent contributions to G XX, a xy and a xz, respectively.
The mean viscous contributions are still given by Eq. (2.2.7) and are based on
the mean-velocity components. Equations (2.2.2) to (2.2.4) thus apply to both
laminar and turbulent flows, provided that the so-called "stress tensor", cr^,
including the viscous contributions, is written as

= - ^ ^ _ ^ j (2.3.7a)
, y + +
or
(Tij = 4 + 4 (2.3.7b)

where now o\- denotes the Reynolds stresses so that for three-dimensional
l
/2
f2
l
l
floWS a xx = -QU , (T xy = G yx = -QU'V , 1 G xz = (J zx = -Qu'w', G yv = - QV ,
= = —QV'W', l — —QW' , and G\- is the viscous stress tensor as given
2
<jy Z a zy a zz
by Eq. (2.2.7) for a Newtonian fluid.
We note that, as is the case for the momentum equations, additional terms
appear on the right-hand side of the energy equation, (2.3.6). These terms,
which are the thermal analogs of the Reynolds-stress gradients in Eqs. (2.3.3) to
, f
(2.3.5), are called the turbulent heat-flux gradients. For example. gc pT v is the
rate of flux of enthalpy in the ^-direction per unit area in the (x, z) plane, due to
turbulent fluctuations. These terms, together with the Reynolds-stress terms in
the momentum equations, introduce additional unknowns into the conservation
equations. To proceed further, additional equations for these unknown quanti-
ties, or assumptions regarding the relationship between the unknown quantities
to the time-mean flow variables, are needed. This is referred to as the "closure"
problem in turbulent flows: we shall discuss turbulence modeling in some detail
in Chapter 3.