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THEORETICAL SOLUTIONS FOR TURBULENCE GENERATED BY VIBRATING
GRIDS AND BY WALL FLOWS USING THE k- MODEL IN LINEAR FORM
Harry Edmar Schulz Johnnes Grson Janzen Karine Cristiane de
Oliveira Souza [email protected] Departamento de Hidrulica e
Saneamento, Escola de Engenharia de So Carlos, Universidade de So
Paulo Av. do Trabalhador Sancarlense 400, 13566-590, So Carlos S.P.
Brasil Abstract. Theoretical solutions for turbulence are presented
considering the basic equations of the k- model. The equations are
used with the terms of diffusion, generation and dissipation,
together with the "eddy viscosity" or "turbulent viscosity" defined
by the k- model. In the present formulation only stationary
(steady) situations are considered and solutions are obtained for
different flow conditions. The deduction procedures followed to
obtain linear governing equations for different flow conditions are
presented in detail, and the proposed solutions may be obtained
through traditional mathematical tables or adequate softwares.
Experimental results are compared with the presented theoretical
predictions. The obtained results permit to suggest, for example,
that "wall flows" and "grid flows" are well represented through the
proposals of the present study. Based on the results it is
concluded that: a) theoretical solutions are important for the
better understanding of turbulent flows (possibility to verify
basic assumptions, which is not always the case in numerical
simulations); b) the k- assumption represents an adequate ad-hoc
approximation for flows such as those considered in this study.
Keywords:Turbulent flows, Turbulence models, k- model, Grid
turbulence, Wall Turbulence
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1. INTRODUCTION AND OBJECTIVES
The k- model is perhaps the most largely model used to predict
mean characteristics of turbulent flows. Its use is mostly coupled
with numerical procedures, applied directly to the original
equations for k and . The original equations are non-linear, and
relatively less effort has been directed in the literature to find
linear forms which permit theoretical predictions of flow
characteristics. Theoretical solutions are important because they
may furnish details of the behavior of turbulence parameters in the
flow. Moreover, theoretical solutions may be used to check
mathematical procedures used in numerical codes, through the
comparison between numerical and theoretical predictions. In this
sense, the main objective of this study is to present theoretical
solutions for flows described by the k- model. The terms of
diffusion, generation and dissipation of turbulent kinetic energy
are considered, together with the "eddy viscosity" or "turbulent
viscosity". Through the use of mathematical tools, linear governing
differential equations are first obtained. Further, the program
Mathematica was used to obtain the solutions presented here for
different situations. For wall flows, a finite-difference scheme is
used to show the form of the solution of the governing linear
equation for k and . The theoretical predictions are compared with
experimental results of several sources, showing very good
agreement between data and theory. 2. k- MODEL FOR TURBULENCE WITH
DIFFUSION-DISSIPATION TERMS
The general equations for turbulent flows, considering the k-
approximation, are giving by:
The turbulent kinetic energy (k) equation:
=
+
j
iji
ik
t
iii x
uuuxk
xxku
tk '' (1)
The equation for the dissipation rate () of turbulent kinetic
energy:
kC
xu
xu
xu
kC
xxxu
t
j
i
i
j
j
it
i
t
iii
2
21
+
+
+
=
+
(2)
The equation for the turbulent viscosity (t):
2kCt = (3)
In Eq. (1), (2) and (3) iu represents the i component of the
mean velocity, i'u represents
the i component of the velocity fluctuation, C1, C2, C, C, and k
are constants of the model. For the cases studied here, stationary
conditions and no mean motion are considered, so that the left side
of both equations vanish. When considering only diffusion and
dissipation
-
phenomena, also the middle term of the right side in both
equations vanish (which corresponds to production). Eq. (1) and (2)
simplify to:
=
iki xkkC
x
2
(4)
kC
xkC
x ii
2
2
2
=
(5)
Basic situations of turbulent flows may be studied as
one-dimensional cases. Considering
one-dimensional problems, the first integration of Eq. (4),
furnishes (Schulz and Chaudhry, 1998):
( )1332 wkCdxd k += (6)
w1 is an integration constant. Although Eq. (6) is still
nonlinear, and k appear isolated (only one variable at each side of
the equation), and suggest that
)(k = and (7) )(xkk =
Matsunaga et al.(1999) suggest the transformation (8), which
simplify the representation
of the original equations:
t
k
xdFd
= (8)
Equations (4), (5) and (8) appear as:
22
2
kFdkd = and kj
Fdd =2
2
(9)
where the following definitions apply:
kC
kk= and
k
Cj 2= (10)
The first integration of Eq. (4), after the suggestion of Schulz
and Chaudhry (1998, 1999)
and Schulz (2001), furnishes:
32 3
2 kwFdkd += (11)
w2 is an integration constant. Equation (6) is still valid, so
that Eq. (7) may be expressed as:
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)(k = and )(Fkk = (12) From Eq. (12) it follows immediately
that:
2
22
2
2
2
2
Fdkd
kdd
Fdkd
kdd
Fdd +
= (13)
Using Eq. (8), (9), (12) into Eq. (13) leads to:
032 2
2
23
2 =+
+ kjkd
dkkd
dkw (14)
This result is important, because the original problem, composed
by two nonlinear
coupled equations for ),( Fk= and )(Fkk = , is transformed into
only one linear equation for )(k = . To generalize the equation and
its results, a nondimensional form (Eq. 15) is
presented, considering the values of the turbulent kinetic
energy (k0) and its dissipation rate (0) at the origin.
0*****
***
32* 22
23 =+
+ jkkd
dkkd
dkw (15)
0
* = and
0
*kkk =
w* is a nondimensional constant. A solution of this linear
equation is presented using Hypergeometric functions (Hyp2F1 in Eq.
16), useful to represent turbulence generated in the region between
two oscillating grids.
( )
+
=*
667.0,667.0,734.0,567.012
**667.0,333.1,067.1,234.012
*667.0,667.0,734.0,567.012*
*667.0,333.1,067.1,234.0121
**667.0,667.0,734.0,567.012
**
3
4
3
3
wFHyp
wkFHyp
wFHypwk
wFHypw
wkFHyp
k
(16)
w3 and w4 are integration constants. Figures 1 and 2 show the
agreement between the proposed solution and experimental data
obtained by Janzen (2003) for turbulence generated between two
oscillating grids, in nondimensional form. Such analysis was also
performed, in dimensional form, by Souza (2004). The constants of
the k- model used here are C,=1.44, C2=1.92, C=0,09, =1.3 and
k=1.0. The boundary condition adopted is: for k* = 1, * = 1. The
remaining integration constants were evaluated through adjustment
with the experimental data. Table 1 presents some of the studied
experimental conditions.
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2 5 10 20 50k*
510
50100
5001000
*
Figure 1 * versus k* (S = 2.0 cm, f = 2.0 Hz of table 1). The
line is the prediction of Eq.
(16). Dots are measured values.
2 5 10 20 50k*
510
50100
5001000
*
Figure 2 * versus k* (S = 5.0 cm, f = 2.0 Hz of table 1). The
line is the prediction of Eq.
(16). Dots are measured values. Table 1: Some of the
experimental conditions studied for turbulence generated by a
pair
of oscillating grids
Frequency f (Hz)
Stroke S (cm) w3=w4 w*
2.0 2.0 4.5 1.0 3.0 2.0 6.0 1.0 2.0 5.0 2.0 1.0 3.0 5.0 2.5
1.0
Hypergeometric functions as presented by Eq. (16) are tabulated
in the literature and may
also be evaluated through adequate softwares. The obtained
solution also permits to calculate other variables (although
numerically). As an example, from definition (8) and Eq. (3), (11)
and (16) it is possible to obtain the evolution of k* with the
nondimensional distance x*, in the form of Eq. (17):
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325.0
32**)(
+
=
kwk
kCdkdx
k , where 2/3
0
0*kxx = (17)
*(k*) is given by Eq. (16). A graph of k* for the same condition
of fig. 1 is given by fig. 3. Experimental points were obtained by
Janzen (2003). Equation. (17) was solved using a fourth order
Runge-Kutta scheme, with boundary condition k* = 1 for x* = 0.
Experimental energy dissipation rates were obtained using
=CDk3/2/L, where CD is an empirical constant and L is the integral
turbulence length scale. The value of CD used here is 0.5.
100 200 300 400 500k*
0.2
0.4
0.6
0.8
1
z*
Figure 3 k* as function of the normalized distance (S = 2.0 cm,
f = 2.0 Hz, CD = 0.5). The
line is the prediction of Eq. (17). Dots are measured
values.
3. k- MODEL FOR TURBULENCE WITH PRODUCTION-DIFFUSION-
DISSIPATION TERMS
Considering flows parallel to walls and using the turbulent
viscosity defined with the mean velocity gradient, general Eq. (1)
and (2) are now simplified to:
Turbulent kinetic energy (k) equation:
02
=
+
i
jt
ik
t
i xu
xk
x (18)
Energy dissipation rate () equation:
02
2
2
1 =
+
kC
xu
kC
xx ij
ti
t
i
(19)
Usual definition of the turbulent viscosity:
i
jt xd
ud = (20)
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is the shear stress and is the density of the fluid. Traditional
studies of wall-flows consider a constant , related to the shear
velocity u* as /* =u . The production terms of Eq. (18) and (19)
may then be represented as:
2
42 *kC
uxdud
i
jt
=
(21)
Equations (20) and (21) assume the forms:
0* 242
=+
kC
uxkkC
x iki (22)
0*2
23
24
1
2
=+
kC
kCuC
xkC
x ii
(23)
Once more a set of two nonlinear coupled equations must be
solved. As for the diffusion-
dissipation problem, algebraic simplifications were elaborated
to transform this problem into only one linear equation relating k
and . The following nondimensional forms simplify the presentation
of the equations:
For the turbulent kinetic energy:
Cu
kk k2#
*= (24)
For the energy dissipation rate:
444444444444 3444444444444 21 2###
##3
# ,,* k
yddFandhyy
uh === (25)
First possibility, using a fluid depth h.
444444444444 3444444444444 21 2###
##3
# *,* k
yddFanduyyu
=== (26)
Second possibility, using only the variables already defined.
Any form conducts to the following nondimensional equations:
AkFdkd = 2#2#
#2
(27)
0##2##
12#
#2
=+ BkCk
ABCFd
d (28)
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Where 32
C
A k= and k
B = are known constants. Following the procedures of Schulz
and Chaudhry (1998, 1999) and Schulz (2001), the first
integration of Eq. (27) leads to:
#3###
#
232 Akkw
Fdkd += (29)
w# is a nondimensional integration constant. Following Eq. (12),
we have:
)( ### k = and (30) )( ### Fkk =
Equation(13) may be reproduced using the nondimensional k# and
#, which together with
Eq. (27), (28) and (29), lead to the linear governing Eq. (31),
for # as a function of k#: ( ) ( ) 02
32 #2#
21#
##3#
2#
#2##2#4# =++
+ kCACB
kddAkk
kddkwAkk (31)
As for the diffusion-dissipation case, this result is important,
because the original
problem, composed by two nonlinear coupled equations is
transformed into only one linear equation. Particular solutions may
be found for different combination of values of the constants
involved in this equation. A graph using C,=1.44, C2=1.92, C=0,09,
=1.3 and k=1.0 is quickly obtained using a numerical
finite-difference scheme, for example with:
2#
#1
##1
2#
#2 2kdk
d iii
+= + and ##
1#
#
#
kdkd ii
= (32)
Equations (31) and (32) lead to:
( ) ( ) 02
232
#2#21#
#1
##3#
2#
#1
##12#4###
=+
+
+
+
+
+
iiii
ii
iiiiii
BkCABCk
Akk
kAkkkw
(33) A value of k=1.0 was used. The optimal w# for channel flows
is evaluated to be 67850.
This value is obtained considering experimental data for k and
extracted from figs. 6.4 and 6.5 of Nezu and Nakagava (1993),
measured for channel flows. Figure 4 shows the comparison between
the theoretical model and the experimental results. It must be
emphasized that the squares (experimental data) correspond to
lectures made on graphics, and may include deviations related to
this somewhat crude way to compile information. Anyway, it is
interesting to note that a strong change in the behavior of the
experimental data (for k# about 35) is well reproduced by the
theoretical linear model.
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020406080100120
0 10 20 30 40 50 60
k#
#
Figure 4- Solution of Eq. (31) (solid line) and experimental
data (squares)
collected in the literature for energy dissipation rate as a
function of the turbulent kinetic energy in channel flows.
4. CONCLUSIONS
In this paper, linear forms for the k- model are presented and
applied to flows generated between oscillating grids and flows
parallel to rigid walls. This linearity permits to find particular
solutions, with different boundary conditions, using adequate
combination of values of integration constants (and also k- model
constants, not always "universal") involved in the obtained
equations. The comparisons between theoretical solutions and
experimental data (for grid-turbulence and wall-turbulence) show
very good agreement and point to the validation of the use of the
ad hoc k- model for flows such as those studied here. It is
important to stress that the original k- problem is composed by two
non-linear coupled differential equations and that the present
solutions consider mathematical procedures which lead to only one
linear differential equation relating k and . The results show that
the theoretical procedures are well conducted and that the
propositions given by equations (7) and (30) are valid for the
flows considered here. In this sense, it is concluded that: a)
theoretical solutions are important for the better understanding of
turbulent flows (possibility to verify basic assumptions, which is
not always the case in numerical simulations); b) the k- assumption
represents an adequate ad-hoc approximation for the flows
considered in this study.
Acknowledgements
To FAPESP, CNPq and CAPES, Brazilian foundations for research
support. REFERENCES Janzen, J.G., 2003. Details of turbulence
properties in water agitated through a pair of
oscillating grids, Master degree thesis, School of Engineering
at So Carlos, University of So Paulo, Brazil (text in
portuguese).
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Janzen, J.G. & Schulz, H.E., 2003. Using PIV to determine
turbulence characteristics in tanks
with two oscillating grids, Proceedings of the XVth Brazilian
Simposium on Water Resource,, ABRH, Published in CD-ROM, Curitiba
PR, Brazil (text in portuguese), 23 to 27 november.
Matsunaga, N.; Sugihara, Y.; Komatsu, T. & Masuda, A., 1999.
Quantitative properties of
oscillating-grid turbulence in a homogeneous fluid, Fluid
Dynamics Research, v. 25, p.147-165.
Nezu, I. & Nakagava, H., 1993. Turbulence in Open-Channel
Flows, AA Balkema, IAHR,
Rotterdam. Schulz, H.E. & Chaudhry, F.H., 1998. A
Theoretical Solution for Turbulence Generated by
Oscillating Grids, Proceedings of the First Spring School of
Transition and Turbulence, ABCM, Brazilian Association of
Mechanical Sciences, COPPE, Rio de Janeiro, Brazil, 21 to 25/09,
pp. 181-194.
Schulz, H.E. & Chaudhry, F.H., 1999. Theoretical Solutions
for Turbulence Generated by
Two Oscillating Grids. In: XVth Brazilian Congress of Mechanical
Engineering, ABCM. Publicado em CD-ROM, ISBN-85-85769-03-3.
(Published in CD-ROM), guas de Lindia, SP, Brazil.
Schulz, H.E., 2001. Alternatives in Turbulence (in Portuguese).
Ed. EESC-EDUSP, ISBN 85-
85205-37-7, So Carlos, S.P., Brazil. Souza, K.C.O., 2004.
Analysis of Theoretical Solutions for Grid-Turbulence Flows.
Master
degree thesis, School of Engineering at So Carlos, University of
So Paulo, Brazil (text in portuguese).
Harry Edmar SchulzJohnnes Grson JanzenKarine Cristiane de
Oliveira Souza
