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4.8 TREATMENT OF TURBULENT FLOWS
˙ 11:7 87
u ∂p/∂x represents the pressure–work effect in steady flow, and Q md =

∂/∂y {( all k ρ D k ∂ω k /∂y)h k } represents the contribution of species diffusion
mass transfer having specific enthalpy h k .If h k equals mixture enthalpy h then
˙
Q md = 0.
When no chemical reaction is present, R k = 0. However, for a reacting boundary
layer, R k will be finite for each species because each may be generated via some
reactions and destroyed via some other reactions among the postulated chemical
reactions. Very often, for gaseous fuels and for highly volatile solid/liquid fuels,
an SCR can be assumed [73]. The SCR is specified as
1kg of fuel + R st kg of oxidant → (1 + R st )kg of product, (4.86)
where R st is the stoichiometric ratio for the fuel under consideration. Thus, there
are three species and one must specify R fu , R ox , and R pr . However, in an SCR,
R fu = R ox /R st =−R pr /(1 + R st ) so that no net mass is generated or destroyed as a
result of chemical reaction. This enables construction of a conserved scalar variable
= ω fu − ω ox /R st = ω fu + ω pr /(1 + R st ) when mass diffusivities of all species
are taken equal. Thus, one may now solve only for ω fu and with R = 0 instead
˙
of three variables. Further, Q cr =|R fu | H c where H c is the heat of combustion
of the fuel. The value of R fu is obtained from a reaction rate law

E
n
m
R fu = R fu,kin =− A exp − ω ω , (4.87)
ox
fu
R u T
where, preexponential constant A and constants E, m, and n are specified for the
fuel [82] and R u is the universal gas constant.
If turbulent reacting flow is considered then the effective R fu is given by a variant
[44] of the eddy-breakup model due to Spalding [74],
ω ox ω prod
R fu =−ρ m min A ω fu , A , A , R fu,kin , (4.88)
e R st (1 + R st )

where A = 4 and A = 2. The postulated arguments in favour of this expression
are beyond the scope of this book.

4.8 Treatment of Turbulent Flows
In turbulent flows,
in Table 4.1 will assume an effective value. Thus, following
Equation 4.63, we have
µ µ t

,eff = + , (4.89)
Pr Pr t,
where suffix t denotes the turbulent contribution. The task now is to represent µ t and
Pr t, via modelled expressions. This exercise, called turbulence modelling, implies
validity of the Boussinesq approximation for turbulent viscosity. Although there