Differential equations for integral scales of semiempirical developed-turbulence тheory
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Цитувати: | Differential equations for integral scales of semiempirical developed-turbulence тheory / V.I. Karas’, V.P. Maljkhanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 200-204. — Бібліогр.: 10 назв. —англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859668839470465024 |
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| author | Karas, V.I. Maljkhanov, V.P. |
| author_facet | Karas, V.I. Maljkhanov, V.P. |
| citation_txt | Differential equations for integral scales of semiempirical developed-turbulence тheory / V.I. Karas’, V.P. Maljkhanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 200-204. — Бібліогр.: 10 назв. —англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| first_indexed | 2025-11-30T12:37:23Z |
| format | Article |
| fulltext |
200
UDK 532.517.4
DIFFERENTIAL EQUATIONS FOR INTEGRAL SCALES OF
SEMIEMPIRICAL DEVELOPED-TURBULENCE THEORY
V.I. Karas’, V.P. Maljkhanov*
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov 61108,
Ukraine,
* CRYOCOR, Moscow 105043, Russia
The calculation of turbulent shear flows and of
turbulent diffusion in these flows calls for the theory that
enables the anisotropic properties of the flows to be
adequately taken into account. The information provided
by classical semiempirical theories of Boussinesque,
Prandtle, Taylor, Carman is limited to mere averaged
velocities, and appears insufficient, as it gives no way of
calculating the spatial and temporal distributions of
turbulence intensity and turbulent diffusion coefficients.
In this regard, of interest are the semiempirical theories,
where in parallel with the Reynolds equations use is
made of the equations for turbulent characteristics, in
particular, the turbulent energy (b)-balance equation [1].
Similarly to classical semiempirical theories,
these theories also comprise the notion of the coefficient
of turbulent diffusion for different substances
(momentum, energy, etc.), however, the dependence of
this coefficient on statistical characteristics of turbulence
is quite different and is based on the following obvious
considerations. It is known that the kinematic coefficient
of molecular viscosity is related to the average
molecular velocity Um and the average free path length
of molecules lm by v=Umlm. It can be assumed that the
coefficients of turbulent viscosity would be described by
similar relations, where the corresponding
characteristics of disordered turbulent flows will play
the role of molecular flow characteristics Um and lm.
Thus the mean-square value of pulsation
velocity, i.e., √b, is obviously analogous to Um. Instead
of lm, one should use the turbulence scale l, i.e., the
length dimensionality describing the mean distance that
the turbulent formations can be displaced without
disturbing their individuality (i.e., the Prandtle “way of
mixing”, which is coincident in the order of magnitude
with the “correlation length” calculated from the spatial
correlation function). In this case, the anisotropic
turbulence can be characterized by different scales in
different directions.
At the same time, it is the semiempirical
theories that are now gaining acceptance [2]. In these
theories, the anisotropy of the flow can be considered in
terms of pulsation velocity components, differing in
direction, but at one (isotropic) scale. The very
construction of the semiempirical theory in this way
shows its limited usefulness, because in reality the
anisotropy in shear flows must exert an effect not only
on pulsation velocity values, but on the scales as well.
Strictly speaking, at each point of the flow,
along with the pulsation velocity components, a scale
ellipsoid must be determined, i.e., a symmetric second-
rank tensor lij (scale tensor) must be assigned, the
components of which have the length dimensionality.
Therefore, to hold the rigidity in the determination of
anisotropic properties of the flow, one needs not only
the equations for the pulsation velocity components but
also the equations for turbulence scales; however, this
will result in a bulky set of equations.
The set of equations for turbulence energy and
scales may appear more compact to consider the
anisotropic properties of flows. Then the tensor of
turbulent viscosity coefficients can be expressed as
ij ijv bl= (1)
This approach appears more natural, because the
turbulence energy is the parameter quite real for shear
flows.
With the tensor vij, we can express the
Reynolds stresses in terms of the turbulence energy,
scales and derivatives of averaged velocities, assuming
(as is generally the case) that the dependence of these
stresses on the derivatives ∂Ui/∂xj is linear, namely,
( )2 1
3 2i j ij i j j iu u b b l lα α α αδ= − Φ + Φ (2)
where ji
ij
j i
UU
x x
∂∂
Φ = +
∂ ∂
is the strain tensor, Ui is the
ith pulsation velocity component value, Uj is the jth
averaged velocity component value, δij is the Kronecker
symbol, <...> is the probability-theoretic averaging.
This formula was proposed by Monin [3]. It
explicitly takes into account the symmetry of the
Reynolds stress tensor, yet it often neglects the
significant fact that generally (except for the isotropic
turbulence) normal stresses differ between themselves in
magnitude. In a number of cases, one can simply take as
a first approximation that lij = lδij is the isotropic tensor,
i.e., take the hypothesis expressed by the formula
2
3i j ij iju u b l bδ= − Φ (3)
as the base (the Reynolds stress tensor just the same
remaining anisotropic).
It should be noted that in the literature, because
of the lack of data on the scale tensor lij, it is formula (2)
that is used to calculate turbulent flows by the set of
equations including the turbulent energy equation. If the
use is made of the common hypothesis that the turbulent
diffusion coefficients of different substances are equal to
the turbulent viscosity coefficient to an accuracy of
numerical factors, then the introduction of the tensor lj
allows one to write in terms of principal unknowns (e.g.,
201
[4]) both the third moments of the velocity field
1
2 i b i
a
b
u u u bl
xβ β αα
∂
= −
∂
(4)
(where bα is a certain numerical coefficient) and the
density components of the turbulent impurity flow
i i
a
u bl
xϑ αϑ α
∂Θ
= −
∂
(5)
where ϑ is the concentration pulsation, Θ is the
concentration average.
The construction of closed semiempirical
theory of inhomogeneous turbulence around the set of
Reynolds equations, the continuity equation and the
turbulent energy-balance equation calls for the presence
of data on space-time variation of scales, that cannot be
obtained from the principal equations of the set. These
data can be obtained either by processing the
experimental data or by solving differential equations
for scales. Currently some studies have been advanced,
which propose the equations for scales. Let us consider
a homogeneous isotropic turbulence, for which all points
and directions are of equal importance, therefore, all
single-point characteristics are constant in space. In this
case, two integral scales are of importance: longitudinal
Λf and transverse Λg. They are expressed in terms of
longitudinal and transverse correlation coefficients f(x)
and g(x) by the following formulas
( )
0
g g x dx
∞
Λ = ∫ , ( )
0
f f x dx
∞
Λ = ∫ , (6)
However, in the consideration of inhomogeneous
nonisotropic turbulence these definitions become
meaningless because of nonequivalence of different
points and directions in the flow. Generalizing the
isotropic case, Rotta [5] has defined the “averaged
scale” of turbulence L by the following
relationship ( )
( )
( )
0
3
, , ,
8 ,
dk
L x t F k x t
b x t k
π ∞
= ∫! !
! , (7)
where k is the wave vector modulus, x
→
is the radius
vector of the point under consideration, ),,( txkF
→
is
the function of three-dimensional spectrum (sum of
diagonal elements of the spectral function tensor,
integrated over all possible directions of the wave vector
→
k ). The differential equation derived for the thus
introduced “average scale” inadequately describes its
behavior in the vicinity of a hard wall. In [6], Glushko
has eliminated this drawback by modifying formula (7)
relationship
( )
( )
( )2
2
2
1 2 3 2
2
, ,3
, , ,
16 ,
x
x
Q x t
L x x x t d
b x t
ξ
ξ
π ξ
∞ ∞
−∞ − −∞
= ∫ ∫ ∫
!!
! ,(8)
where Q is the half-sum of diagonal moments of the
correlation function tensor for the points
2
x ξ−
!
!
and
2
x ξ+
!
!
.
The physical meaning of the modification
consists in the explicit consideration of the condition of
attachment on the wall (mathematically, this means the
restriction of the region of variation ξ
→
). Expressions
(7) and (8) are equivalent at the centre of the flow. In
fact, as ),,( txQ
→→
ξ is essentially different from zero for
→
ξ <L, then at x2 >>L (this holds at the centre of the
flow) the upper and lower limits can be considered
infinite, and thus the restriction on possible
→
k values is
removed.
Note that formulae (7) and (8) take into account
only the inhomogeneity of the flow, leaving quite apart
its anisotropy. This situation is not wholly satisfactory,
because turbulence can be characterized by different
scales in different directions.
The flow anisotropy can be taken into account
by introducing the scale tensor. Here, of importance is
the issue which interpretation should be given to these
scales. If the coordinate frame is chosen so that only
diagonal components of the scale tensor are nonzero,
then each principal direction in space has the sole scale.
Apparently, in the determination of these tensor
components one can proceed from the following
qualitative considerations:
• the scale in the given direction must characterize
the same-direction correlation of both longitudinal
and transverse components of pulsation velocity;
• the scale must make allowance for the presence of
flow-limiting surfaces, on which the averaged
velocity and the pulsation velocity become zero;
• in the case of isotropic homogeneous flow, the scale
must be coincident with either transverse or
longitudinal integral scale.
In view of the mentioned requirements, let us
define the scale in the direction xi as a mean integral
scale of correlation between like pulsation-velocity
components in the given direction, i.e.,
( )
( )
( ) ( )
( )( )
3
1
3 1
, , ,
8 , 2ii kk
kx i
l x t R x t d
b x t
δ ξ
ξ ξ
δ ξ=Ω
= ∑∫∫∫
!
!
! !! !
! ,(9)
where b is the specific turbulent energy, Ω (x) is the
domain of ξ variation
*
2 2ij i j j iR u u u x u x
ξ ξ′ ′= = − +
! !
! ! ,
( )1 2 3, ,ξ ξ ξ ξ=
! is the distance between the points under
consideration, )()()()( 321
→
= ξδξδξδξδ is the Dirac delta
function having the following property
( ) 1d
ε
ε
δ ξ ξ
−
=∫ , where 0ε > .
We now show that this definition of lij in the
202
case of isotropic homogeneous turbulence is coincident
with the definition of the transverse integral scale Λg. It
is known that for the turbulence of this kind
( )
2
2
3ij i j ij
f g
R b gξ ξ ξ δ
ξ
−
= +
!
,
where f and g are, respectively, the longitudinal and
transverse correlation coefficients, ijδ is the Kronecker
symbol.
Taking the half-sum of diagonal elements of the
tensor Rij and making use of the known relationship [7]
between f and g, we obtain
( )
3
1
1
3
2 3kk
k
b f
R fξ ξ
ξ=
∂
= +
∂
∑
! !
! ,
and since
i i
f fξ
ξ ξξ
∂ ∂=
∂∂
!
! ,
then
( )
( )
2
1
3
8
1
3
8
ii
i i i
i i
i
f
f d
f
f d
l ξ δ ξ
ξ
ξ ξ δ ξ
ξ ξ
ξ
∞ ∞ ∞
−∞ −∞ −∞
∞
−∞
∂
= + =
∂
∂
= +
∂
∫ ∫ ∫
∫
! !
!
Taking into consideration that f is the even function of
ξi, we find
( )
0
0
1
3
4
3
1
4 2
ii i i
i
f
f
g
i i
f
l f d
f d
ξ ξ
ξ
ξ ξ
∞
∞
∂
= + =
∂
Λ −
Λ
= = = Λ
−
∫
∫
,
where Λf(g) is, respectively, the longitudinal (transverse)
integral scale. By definition (9), the functional
dependences of lij on the coordinates can be obtained in
three ways, namely:
- from the experimental data on correlation functions;
- with the use of theoretical dependences for correlation
functions;
- by equations formed for lij.
Let us illustrate the first way of setting up
functional dependences lij on the coordinates using the
experimental data of Conte-Bellaux [8] who measured,
among other important turbulence characteristics, the
spatial distributions of turbulent energy b in the plane
flow, the average squared components of pulsation
velocities 〈ui
2〉 , the correlation coefficients
( ) ( ) ( )
( ) ( )
1 2 3 1 2 3
2 2
1 2 3 1 2 3
, , , ,
, , 0, 0
, , , ,
i j
ij
i j
u x x x u x x x
r x
u x x x u x x x
γ
γ
γ
+
=
+
!
As is seen from expression (9), the calculation
of lij requires the correlation functions of the form
( ),
2 2
ij j iR x u x u x
ξ ξ
ξ == − +
! !
!! ! ! .
They can easily be related to the correlation coefficients
rij(x,ξ) measured in experiment, namely:
( ) ( ) ( ) ( ), , 0 , 0 ,
2 2 2
ee ee ee ee
R x R x R x r x
ξ ξ ξ
ξ ξ= − + −
! ! !
! !! ! ! ! .
Since the experiment was conducted in the plane flow,
all single-point characteristics, including lij , are merely
the functions of one coordinate x2 (the axis x2 is directed
along the averaged velocity variation), and for two-point
characteristics the dependence on the first argument is
related only to the function of x2.
The experimental data were used to plot
( ) ( )
3
1
1
,
2 kk i
k
R x b xξ
=
∑ ! !
versus ξi for different x2/D values (where D is the
half-width of the channel). Finding the area under the
curve (e.g., with a planimeter) that corresponds to the
direction i and to a certain x2/D value and multiplying by
a constant factor, we obtain the 2
ii
x
l
D
values.
Based on the definition (1) and the data
obtained, one can ascertain the known fact about the
excess of the longitudinal coefficient v11 over the
transverse coefficient v22 by factors of about 3 to 4.
For comparison, the data measured by Conte-
Bellaux are used for the correlation lengths defined as
It should be noted that the definition (10) provides a
consideration of correlation in one direction only, this
being insufficient for finding the diagonal elements of
scale tensor, as two of the scales must characterize
certain “averaged correlation lengths” in transverse
directions, and this is not provided by the definition.
Besides, it masks the anisotropy of the flow, because
even in the case of homogeneous isotropic turbulence
the correlation length L1 will be twice as large as the
lengths L2 and L3.
Using the introduced integral turbulence scale
(9) as the base, differential equations can be set up to
take into account the anisotropic properties of the flow.
In the derivation of equations for diagonal
components of integral scale tensor (DCIST), according
to (9), we use the equation for ),,( txQ
→→
ξ , that is found
by folding the equations for two-point moments Rij),
multiply it by 3/8 ( , )b x t
→
and integrate over )(
→
Ω x .In
the integration one should take into account that the
average velocity and the pulsation velocity are equal to
zero at the boundary of the region. It is assumed that all
two-point characteristics are dependent on the ratio
ξi/li
→ tx, . Then, similarly to [2,5], we write down each
203
characteristic as follows:
( ) ( ) ( ) 1 2 3
11 22 33
, , , 0, , ,x t x t f
l l l
ξ ξ ξ
ξ ΓΓ = Γ
!! ! ,
where f(r) satisfies the condition
( )
( ) ( )
( )( )
( )3
8 , x i
f d Const
b x t
δ ξ
ξ ς
δ ξ
Γ Γ
Ω
= =∫∫∫
!
!
!
! .
This means that the dependence on the
direction in space is determined by the dependence on
lii( tx ,
→
). The mentioned manipulations lead to the set of
equations for b lii( tx ,
→
).
( ) ( ) ( )
( ) ( ) ( ) ( )
, 0,
, 0,
ii ii i
k ki ki ii
k k
b bii
ki ki ii
k k
s s ii t
bl bl U
U R x t l
t x x
bl
v I x t l
x x
T l
ς
ς
ς ε
∂ ∂ ∂
+ + =
∂ ∂ ∂
∂∂
= + +
∂ ∂
+ −
!
! , (11)
( )
3
2
1
1 ik i
k
s ii i i
k ikkk
ii ii
T vbl
fl
l l
δ β
α
ξ ξ
δ=
− +
= −
+
∑ , i Sξ ∈ (12)
where v is the kinematic viscosity coefficient; ςkiς(b)
kiςs,
αk, βi are the constants, S is the surface of the region
Ω , ( ) ( ) 2
/ 2b
k k i i kI u u u pu
ρ
= + , p - p is the
pressure pulsation ,
2
i
t
k
u
v
x
ε
∂
=
∂
.
The turbulent energy balance equation has the form
( )
( ) ( )
, 0,
, 0,
i
k ki
k k
b
ki t
k k
b b U
U R x t
t x x
b
v I x t
x x
ε
∂ ∂ ∂
+ + =
∂ ∂ ∂
∂ ∂
= + −
∂ ∂
!
!
. (13)
To derive equations for DCIST, we need only
take the difference between (11) and (13) multiplied by
lii and transform it in accordance with the above-given
closure hypotheses. In the general case, to calculate the
turbulent shear flow, the derived equation should be
solved along with the equation for averaged velocities
and turbulent energy. Besides, it is essential to
determine the unknown constants entering into the
closure equation. Further, we consider experimental
studies of a steady-state plane-parallel turbulent flow
between two walls spaced at 2D. For convenience, we
transform the problem into a formally nonstationary task
with the boundary conditions
0il E
y y
∂ ∂
= =
∂ ∂
at y=1 (middle of the flow)
il =E=0 at y=0 (wall)
where 2x
y
D
= , ii
i
ll
D
= , 2
f
bE
U
= , fU
t t
D
′ = ,
Re fU D
ν
= is the analog of the Reynolds number, Uf is
the dynamic velocity.
Along with the boundary conditions, we assign
the initial conditions for li, E, approaching the
experimental values obtained from ref. [7] by means of
corresponding transformations. Using the physical
properties of the plane-parallel shear flow with respect
to the balance between the terms of turbulent energy
equation, and based on the initial values for li and E, we
obtain the constants klσ , 3σ , c , æ , ( )bα , ( )b
kς , δ ,
iβ , kα .
The set of equations was solved by the finite-
difference method using the implicit scheme. At each
time step, four equations for li and E are solved. The
equations are solved by the method of matrix run [7-8]
with the use of the Samarsky difference scheme for
solving parabolic-type quasilinear equations [9].
The DCIST and E values, found at each time
step, are used to calculate the related coefficients of
difference equations; the calculations are repeated in the
same sequence until the steady-state profiles for li and E
are obtained.
In calculations, the following constant values
were used: 0.15δ = ; 3.93c = ; æ=0.06 ;
( ) ( )
3
1
1
,
2 kk i
k
R x b xξ
=
∑ ! ! ; 12 0.21σ = ; 3 0.2σ = ; 1 0.1β = ;
3 0.42β = ; 2 0.16α = ;
( ) ( )2
20.6 0.000098 120b zα = − − ,
2
50, 0.1
500 , 0.1 0.25
120, 0.25
y
z y y
y
≤
= < <
≥
3
2
2
2
2
y l
f
l y
=
.
With these values, the functional dependences
for the DCIST were found. They were compared to the
dependences plotted using the experimental data from
ref. [7]. Comparison was also made between the bution
of the turbulent viscosity coefficient
( )11 222T b l l
δ
ν = + , the experimental distribution
plotted in [10] by the data of Lauder and Nunner for the
flow in the tube, and also the function for the velocity
deficit 1 0
f
U U
U
− based on the Conte-Bellaux data [10]
(where U0 is the velocity in the middle of the flow). The
comparison of calculated results with the measured data
shows that the proposed model provides a fair
description of the turbulent shear flow.
204
References
1. А.S.Моnin, А.М Yaglom. Statisticheskaya
gidromekhanika. - М.: Nauka, 1965. Part. 1.
2. J.C.Rotta // Z.Phys. 1951. vol. 129, p. 547; 1951.
vol. 131, p. 51.
3. Тurbulentnye techeniya / Sbornik nauchnykh trudov.
- М.: Nauka, 1970.
4. А.Т.Оnufriev // PMTF. 1970, т. 2, c. 62.
5. J.C.Rotta // Z. Angew. Math. Mech. 1970, vol. 50, p.
204.
6. G.Cont-Belaux. Тurbulentnoye techenie v kanale s
parallel`nymi stenkami. - М.: Мir, 1968.
7. О.S.Маzhorova, Yu.P.Popov // Preprint IPM im.
М.V.Кеldysha АN SSSR №134, 1974.
8. О.S.Маzhorova, B.N.Chetverushkin // Preprint IPM
im. М.V.Кеldysha АN SSSR №29,1973.
9. A.А.Samarsky. Vvedenie v teoriyu raznostnykh
skhem. - М.: Nauka, 1971.
10. I.О.Khintse. Тurbulentnost`. - М.: Physmatgiz, 1963.
|
| id | nasplib_isofts_kiev_ua-123456789-81669 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T12:37:23Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Karas, V.I. Maljkhanov, V.P. 2015-05-19T08:37:43Z 2015-05-19T08:37:43Z 2000 Differential equations for integral scales of semiempirical developed-turbulence тheory / V.I. Karas’, V.P. Maljkhanov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 200-204. — Бібліогр.: 10 назв. —англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81669 532.517.4 en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Differential equations for integral scales of semiempirical developed-turbulence тheory Article published earlier |
| spellingShingle | Differential equations for integral scales of semiempirical developed-turbulence тheory Karas, V.I. Maljkhanov, V.P. Нелинейные процессы |
| title | Differential equations for integral scales of semiempirical developed-turbulence тheory |
| title_full | Differential equations for integral scales of semiempirical developed-turbulence тheory |
| title_fullStr | Differential equations for integral scales of semiempirical developed-turbulence тheory |
| title_full_unstemmed | Differential equations for integral scales of semiempirical developed-turbulence тheory |
| title_short | Differential equations for integral scales of semiempirical developed-turbulence тheory |
| title_sort | differential equations for integral scales of semiempirical developed-turbulence тheory |
| topic | Нелинейные процессы |
| topic_facet | Нелинейные процессы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81669 |
| work_keys_str_mv | AT karasvi differentialequationsforintegralscalesofsemiempiricaldevelopedturbulencetheory AT maljkhanovvp differentialequationsforintegralscalesofsemiempiricaldevelopedturbulencetheory |