Page 51 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

34 SLENDER STRUCTURES AND AXIAL FLOW

K-E model,’ ut cx K2 /E. Equations may be written for K and E, namely
aK aK a vi
- pc + - pm- - ;pmq - 4 (2.102)
pat + puj- = 7;j- ax , ,
a
axj ax, ax, aK

a -
+ Ui.kUi.lUk,l + VmGZGi} + aX,[PmUjUi,/Ui,l - 2UmP,lUj,l], (2.103)

in which p is the fluctuating pressure and 9, the Reynolds stress tensor,

7ij = 2pte;, - $p~&;,, (2.104)
with e;j being the mean strain-rate tensor [cf. relations (2.65)]; ui,k = au;/axk, p,l =
ap / axl and so on. Since the correlations in (2.102) and (2.103) are effectively impossible
to measure, these very complex equations have been simplified by various approximations.
The ‘standard form’ of the K-E model is expressed in terms of the following equations
and relationships (Wilcox 1993):
Eddy viscosity
pt = pC,K2/e; (2.105a)
Turbulence kinetic energy
a
aK aK au. ax [(.-+E) 51;
P- + puj- = tij- - pE + - (2.105 b)
at ax ax,
Dissipation rate

Closure coeflcients

c,, = 1.44, Ct2 = 1.92, C, = 0.09, OK = 1.0, O, = 1.3; (2.105d)

Auxiliary relations
o = E/ (C,K) and 1 = C,k3I2/c, (2.105e)

w being the so-called specific dissipation rate and 1 the turbulence length scale. Thus,
the K and E equations contain five empirical constants which have been inferred from
standard measurements.
It has been found necessary to adjust the closure coefficients somewhat to agree with
different classes of measurements, but in the hands of a skilled practitioner this approach is
usually much better than integral methods. [In integral methods, equations for entrainment,
momentum, mechanical energy and so on are written integrated-up across the flow at any

~~
+This is usually written as the k-e model, but an upper case K is used here to avoid confusion with the
wavenumber k.