Mechanics
Nickolay Zosimovych, Manoj Kumar
Sharda University (India, UP)
PROBLEM
STATEMENT FOR STATISTICAL MODELLING OF
NON-STATIONARY
TURBULENT FLOW IN THE PIPES
Summary. In this paper studies the task
of the statistical model unsteady turbulent flow through a pipe using the local
quasistationary performance of turbulence.
Key words: flow, pipe, turbulence, experimental
data, statistical model, pressure, characteristics of a flow, unsteady
turbulent moving, pulsations, viscous fluid, a pressure gradient, component of velocity, energy equation.
Introduction. The flow in a long pipe on enough
big distance from its ends is one of examples of shift turbulent flow [1]. Unlike
a flow in an interface or as a stream without compressibility, the average
characteristic of movement, the distribution of speed on radius, depends only
from one spatial coordinate [2]. Models of turbulent flows in pipes are
semiempirical, with use of the constants usually received by practical
consideration [3, 4]. Mathematical models of unsteady flows become more and
more complicating. In them there are no experimental data about influence unsteady
flows on the constants entering into semiempirical dependences. Therefore it is
necessary to use the assumption about quasi steady flow [5].
In the
given work it is necessary to investigate the statistical model of unsteady
turbulent flow in a pipe with use of local quasi steady characteristics of
turbulence. The offered model
allows to investigate more detailed the features of unsteady turbulent flow,
and in particular, local kinematic characteristics of turbulence, energy
dissipation, instant values of pressure of a friction about walls and so on. Calculations
on statistical model give us the chance to compare with the data of experiments
both integrated, and local characteristics of a flow that is rather interesting
from the point of view of check of correctness of the most quasi stationary
model.
Problem
statement. Following the work [6], we
will consider an unsteady turbulent moving of an incompressible viscous fluid
in a round cylindrical pipe, in the assumption that the flow is not turbulent
moving (there is an axial symmetry) and is statistically homogeneous along a
pipe axis, that is average values of speed, and product of pulsations of speed
do not depend on from coordinate 
Using these assumptions
and conditions of tightness of a wall of a pipe, from the equation of
indissolubility [7] it is received that 
Taking into
account stated above assumptions, the system of the equations of a turbulence
moving assumes the following air [6, 8]:
|
|
(1) |
Having
excluded the second equation for a pressure gradient in a radial direction in
system (1) and integrating on radius, we will receive
(2)
where 
value of pressure
at the wall at 
From
dependence (2) follows that pressure changes on pipe radius. If we
differentiating the equation (2) with the account that according to the
accepted assumptions also
do not depend from
that 
is function only and
as well as size 
If to analyze the first equation in system (1) it will
be found out that according to assumptions all members, except the last, do not
depend from 
Hence, from
should not depend on last member, i.e. 
the function only time.
Parts
enter into the equation of turbulent energy (2) (in square brackets), pressure
of viscous forces friction describing work. It is known that work of forces of
viscous pressure in a turbulent stream is essential only in immediate proximity
from a wall. Thus the basic contribution to work of viscous pressure brings part
[9]. Accordingly, the
other parts in the brackets
can be ignored. After simplifications [7], the
system equations reduce in two equations - Reynolds
equation for a longitudinal component
of velocity and turbulent energy
equation. However, in these
equations, the number of unknowns greater than the number of equations. To
obtain a closed system equations by following the from the paper G.S. Glushko
[10] we introduce the semiempirical relations obtained by in the assumptions
[11]:
1.
Transfer of the momentum by turbulent pulsations by
diffusion of gradient type (Bussinesque hypothesis)
(3)
where 
coefficient of turbulent
viscosity.
2. Transferring the
total of turbulent energy also
carried out by the diffusion gradient
type
(4)
where 
the summary diffusion
coefficient.
3.
The process of dissipation of
turbulence energy described by the
relationship
(5)
where 
the scale of turbulence; 
universal
constant according to the [10], 
Equations (3) - (5) include
three empirical parameters and
for determining which necessary to involve experimental results.
Using the results obtained in [10], which show that coefficient of eddy viscosity
can be expressed as a function of
turbulent Reynolds number 
of writers
[6] proposed another formula, which approximates the dependence of a turbulent viscosity:
(6)
Equation (6) agrees well with a piecewise smooth
function, suggested in the article
[10].
When calculating the coefficients in
equation (6) were used the results of experiments [12]
for circular tubes. For the summary diffusion
coefficient the authors of [6]
used the linear empirical dependence
(7)
where 
constant coefficient. Quantity
inverse is an analog of the turbulent
Prandtl number. According to the
G.S. Glushko [10] 
For the path length displacement
for flow in a pipe (similar to the known formula Nikuradse) in [6] be
used polynomial for a scale 
(8)
Constants
are determined from the empirical dependence
for the resulting in G.S. Glushko [10]
and from the Laufer’s experimental data
[12]: 
Considering all the
above, equation (2) and (1) can be transformed to the following form:
(9)
(10)
where the
coefficients 
and the scale 
are defined by (6) - (9) and averaging
of signs omitted for simplicity. From the symmetry condition
on the tube axis and the adhesion conditions of the liquid on the wall the boundary conditions written to (BC):
under 
and 
under 
(11)
In general of BC (11)
must be added the initial conditions (IC):
under 
(12)
and set to change flow rate or
the pressure gradient over time.
For the steady the periodic motion in a pipe IC (12)
is not needed and the instantaneous
values of flow 
or an average velocity in the cross section 
are related to the
pressure gradient. Indeed, using
definition of the
average velocity
[7]:
(13)
and integrate the
equation (9) over the radius of
taking into account the conditions (11)
and (13), we obtain:
(14)
where 
wall shear stress, 
Equation (14) is a
special case depending on [7]: 
without convective terms.
The continuity equation it follows that instantaneous
values of the average velocity and
flow rate for an incompressible liquid are identical in any cross section of pipe. Setting the law of variation 
or 
from the equation (14)
we can find the pressure which in this
framework will be the desired function.
Conversely, setting the law of
change of pressure gradient over
time from (14) will be find 
and 
The analysis of results. Given above system
equations was solved numerical in at a given medium
MathCAD harmonious law of change of rate 
To confirm the effectiveness
of both proposed model same,
and calculation method were conducted
the numerical calculations by the
establish to a
stationary flow the results are in satisfactory agreement with the
experimental the data of F. Sedat Tardu, Rogeiro Maestri and J. Laufer [1,12].
For nonstationary flow consumption
was specified the change law of according a
harmonious [7]:
(15)
where 
average discharge; 
the
relative amplitude of flow rate fluctuations; 
constants given quantity.
As well as in case of laminar pulsed flow a dimensionless criterion of characterizing relative influence of inertia and viscosity parameter is 
and for the
average of the turbulent flow -
parameters 
and 
where 
References
1. F. Sedat Tardu, Rogeiro Maestri.
Wall shear stress modulation in a turbulent flow subjected to imposed
unsteadiness with adverse pressure gradient. Fluid Dyn. Res. 42, 2010, 21 pp.
2. Khaleghi, A., Pasandideh-Fard, M., Malek-Jafarian, M., Ghung, Y.M.
Assessment of common turbulence models under conditions of temporal
acceleration in a pipe. Journal of Fluid Mechanics, Vol. 3, No. 1, pp. 25-33, 2010.
3. Zandi, I. and Govatos, G. Heterogeneous flow
of solids in pipelines. ASCE J. Hyd. Div., 93, 1967, 145-159.
4. Lazarus, J.H., Neilson, I.D. A generalized correlation for friction head
losses of settling mixtures in horizontal smooth pipelines. Hydrotransport 5. Proc.
5th Int. Conf. on Hydraulic Transport of Solids in Pipes.,1, 1978, B1-1-B1-32.
5. Amies, G., Greene, B. Aircraft hydraulic systems dynamic analysis. Volume
IV. Frequency response (HSFR). Wright-Patterson Air Force Base Technical Report
AFAPL-TR-76-43, IV, 1977.
6. Âàñèëüåâ Î.Ô., Êâîí Â.È.
Íåóñòàíîâèâøååñÿ òóðáóëåíòíîå òå÷åíèå â òðóáå. – ÏÌÒÔ, 1971, ¹ 6.
7. Ãëèêìàí Á.Ô. Ìàòåìàòè÷åñêèå
ìîäåëè ïíåâìîãèäðàâëè÷åñêèõ ñèñòåì. – Ì.: Íàóêà, 1986. – 368 ñ.
8. Áóêðååâ Â.È., Øàõèí Â.Ì.
Ñòàòèñòè÷åñêè íåñòàöèîíàðíîå òóðáóëåíòíîå òå÷åíèå â òðóáå. – Äåï. ðóêîïèñü,
ÂÈÍÈÒÈ, ¹ 866-81, Íîâîñèáèðñê, 1981.
9. Ìîíèí À.Ñ., ßãëîì À.Ì.
Ñòàòèñòè÷åñêàÿ ãèäðîìåõàíèêà. Ìåõàíèêà òóðáóëåíòíîñòè. ×. 1., Ì.: Íàóêà, 1965.
10. Ãëóøêî Ã.Ñ. Òóðáóëåíòíûé ïîãðàíè÷íûé ñëîé íà
ïëîñêîé ïëàñòèíå â íåñæèìàåìîé æèäêîñòè. – Èçâ. ÀÍ ÑÑÑÐ, Ìåõàíèêà, 1965, ¹ 4.
11. Âàñèëüåâ Î.Ô., Àòàâèí À.À., Âîåâîäèí À.Ô.
Ìåòîäû ðàñ÷åòà íåóñòàíîâèâøèõñÿ òå÷åíèé â ñèñòåìàõ îòêðûòûõ ðóñåë è êàíàëîâ /
×èñëåííûå ìåòîäû ìåõàíèêè ñïëîøíîé ñðåäû. – Íîâîñèáèðñê, 1975. Ò. 6, ¹ 4.
12. Laufer J. The
structure of turbulence in fully developed pipe flow. – Nat. Advic. Com.
Aeronaut., 1954, Rept. ¹ 1175.