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Fluid Dynamics (Chemical Engineering) 61

One of the better versions of the modified mixing length being investigated, one can obtain various degrees of de-
model is tailed information about the turbulent motions. Most of
the more complex formulations require large computing
L = k t R(1 − ξ)(1 − E) (135)
facilities and may result in extreme numerical stability
∗
E = exp[−φ (1 − ξ)] (136) and convergence problems. All of the different methods
√ of computing the turbulent field structure cannot be dis-

∗
∗
φ = R − R ∗ c 2 2B (137) cussed here. Therefore, only one of these other methods,
the so-called κ–ε method, which is a two-equation type
∗
R = Re f (138)
of closure model, is outlined. Models such as this have to
The parameter R is the laminar–turbulent transition value
∗
c date been applied only to Newtonian flow problems.
of R and has the numerical value 183.3 for Newtonian The idea involved in the κ–ε model is to assume that
∗
fluids. For non-Newtonian fluids it would have to be com- the Reynolds’ stress tensor can be written as
puted from the various results presented above.
The parameter B is called a dampening parameter, as its ¯ 2
ρv v = 2µ t D − κδ, (142)

physical significance is associated with dampening turbu- 3
lent fluctuations in the vicinity of a wall. For Newtonian
pipeflowithasthenumericalvalue22.Fornon-Newtonian where κ is the turbulent kinetic energy,
fluids it has been found to be a function of various rheo-
logical parameters as follows: κ = v · v , (143)
1

2
Bingham Plastic
2
B BP = 22[1 + 0.00352He/(1 + 0.000504He) ] (139) and µ t is a turbulent or eddy viscosity function quite anal-
ogous to the eddy diffusivity discussed earlier. Just as in
Power Law
the zero-equation modeling situation, one cannot write
B PL = 22/n (140) down a general defining equation for µ t , but must resort
to modeling. In the present case the model used is
Herschel–Bulkley
B HB = B BP /n (141) µ t = c 1 ρκ 2 ε, (144)
No correlation has as yet been developed for the Casson
model. where c 1 is a (possibly) Reynolds number-dependent coef-
These simple models of turbulent pipe flow for var- ficient that must be determined empirically. The function ε
ious rheological models do not produce accurate details is the turbulent energy dissipation rate function. The func-
regarding the structure of the turbulentflow. They do, how- tions κ and ε are determined by the pair of simultaneous
ever, offer the practicing design engineer the opportunity differential equations
to predict the gross engineering characteristics of interest
with reasonable correctness. They are called zero-order Dκ µ t

equations because no differential equations for the turbu- = ∇ · c 2 ∇κ + τ : ∇v − ε (145)
Dt ρ
lence properties themselves are involved in their solutions.
Rather, one must specify some empirical model, such as Dε µ t ε 2

= ∇ · c 3 ∇ε + c 4 τ : ∇v − c 5 ε κ (146)
the mixing length, to close the equations. Dt ρ κ
In this model the coefficients c 1 to c 5 are commonly
D. Other Closure Models
given the numerical values c 1 = 0.09, c 2 = 1.0, c 3 = 0.769,
Actually, all methods of closure involve some type of mod- c 4 = 1.44, and c 5 = 1.92, although these values can be var-
eling with the introduction of adjustable parameters that ied at will by the user and are definitely problem spe-
must be fixed by comparison with data. The only ques- cific. They can also be made functions of any variables
tion is where in the hierarchy of equations the empiricism necessary.
should be introduced. Many different systems of modeling Here ends this article’s discussion of this model, but
have been developed. The zero-equation models have al- extensive detail is available in numerous books on the
ready been introduced. In addition there are one-equation subject. Some of these models present very accurate, de-
and two-equation models, stress-equation models, three- tailed descriptions of the turbulence in some cases, but
equation models, and large-eddy simulation models. De- may be very much in error in others. Considerable skill
pending on the complexity of the model and the problem and experience are required for their use.