ABSTRACT: Fully developed incompressible turbulent flow in a conical diffuser having a total divergence an- gle of 8 ~ and an area ratio of 4:1 has been ...
ACTA MECHANICA SINICA, Voi.8, No.2, May 1992 Science Press, Beijing, China Alterton Press, INC., New York, U.S.A.
ISSN 0567 - 7718
NUMERICAL PREDICTION OF TURBULENT FLOW CONICAL DIFFUSER USING k-~ MODEL
IN A
He Yongsen ( ~ j ' 7 ~ ) (Changsha Institute of Mining Research , Changsha 410012, China )
Toshio Kobayashi
Youhei Morinishi
(Institute of Industrial Science, University of Tokyo, Japan) ABSTRACT: Fully developed incompressible turbulent flow in a conical diffuser having a total divergence angle of 8 ~ and an area ratio of 4:1 has been simulated by a k-E turbulence model with high Reynolds number and adverse pressure gradient. The research has been done for pipe entry Reynolds numbers of 1.16 x 105 and 2.93x 105. The mean flow velocity and turbulence energy are predicted successfully and the advantage of Boundary Fit Coordinates approach is discussed. Furthermore, the k-n turbulence model is applied to a flow in a conical diffuser having a total divergence angle of 30 ~ with a perforated screen. A simplified mathematical model, where only the pressure drop is considered, has been used for describing the effect of the perforated screen. The optimum combination of the resistance coefficient and the location of the perforated screen is predicted for high diffuser efficiency or the uniform velocity distribution. KEY WORDS: numerical simulation, diffuser, turbulent flow, k-e turbulence model, HSMAC scheme BFC approach
I. INTRODUCTION Since Venturi and others tried to determine the geometry for the most efficient diffuser, a vast number of experimental and theoretical studies have been devoted to the diffuser flow phenomena {I, 21. Flow through diffusers, especially wide angle diffusers,does not usuallypossess the simple features of a fully developed flow, and it may be expected that any theoretical approach with reference to the existing experimental data for symmetric equilibrium flow will be inadequate when used to predict diffuser flows in moderate to strong adverse pressure gradients. In recent years, the advent of powerful digital computers has paved a way to study complicated turbulent flow fields. In accordance with the practice, the present study has proposed Generatrix Ladderlike Approximate-HSMAC method TM, Power Law-BFC method (Boundary Fit Curvilinear Coordinate Transformation )13, 51 and the Wall function-BFC method TM, which are applied to k-e turbulence model for high Reynolds number in diffuser flow. Furthermore, it has proposed No-slip Boundary conditions-BFC method I41 that is applied to k-e turbulence model for low-Reynolds-number flow in diffuser. Numerical prediction of turbulent flows in a conical diffuser has been made accordingly. The present paper discusses the numerical prediction and results of the first two methods applied to fully developed incompressible turbulent flow in a conical diffuser. The stress is placed on the first method as well as the numerical prediction of the k-e model applied to a turbulent flow in a wide angle conical diffuser with a perforated plate. A simplified mathematical model is proposed to describe the effect of the perforated plate and the optimal combination of the resistance coefficient and the location of the perforated plate is predicted numerically. II. NUMERICAL SIMULATION
1. Basic equations The computational flow field is the inside of an axisymmetric conical diffuser as shown in Fig. 1. The following assumptions are made for the computation: (1) the flow is incompressible Received 13 September 1990, revised 15 March 1991
118 ACTA MECHANICASINICA 1992 and axisymmetric, (2) no swirling flow exists, (3) effect of gravity is negli~ble and (4) turbulence is isotropic in the flow field. Velocities are non-dimensionalized by inflow velocity Uo, coordinates by the inlet duct diameter Do, turbulence energy by U02, turbulence dissipation ratio by U~/Do , pressure by p U02 (p: fluid density )and time t by Do / U 0 . Therefore, the k-e turbulence model with high Reynolds number and adverse pressure gradients flow is obtained and its basic equations in cylindrical coordinate system can be written as follows:
~ , U.
II II II
L~
II II II II II ,
II
tl
'0
--o,
!to x L
i
Fig. 1 Model flow field in conical diffuser
8u
+
1
a
o--7 S-TT Ou +
(1)
(rv)=O
O (uu)+
1
a--7- 7 7
0
7
o-T
(r v u ) =
0
+ 1 _~_r[r( 1 7
2
--~--
-kT-e +v'
+T
)(~-'#
~+-~
a-~ )]
(2)
-
v a (uv)+ 0"-'7-+ -~x
1 r
0 (rv'?)= ar
+~#
+~,
7-~-
-
~ -R7 +v'
0 a--#-
+ 2 T
-~- + -~-
-~-+-~(3)
r
ak+
a
a--F -~-
(-~-k)+ 1
7W
r
a (r~-k) =
+,7- -~-[r(a ~ a-T -~-
7-~-
0 [( 1
-aT
Yt
-~7 + -O"I
) a~]
+ --m -&- +a-~
77
+--
(4)
Vot. 8, No. 2
He Yongsen et al.: Turbulent Flow in Conical Diffuser
+ 1
-7-
_~_rrr(_~_+ ~--~vt ) 77gle ] + C I TeC - C 2 T s2
v,= Co ~
119
(5)
(6)
where x is axial distance, r, radial distance from the centre, y , radial distance from the wall, p, pressure, u, v and w, velocity components, vt eddy viscosity, and all variables are nondimensional. Re is Reynolds number (= Uo Do/v ), t, time, v, kinetic viscosity, k, turbulence energy and e, turbulence dissipation ratio.
2. Computational Procedure and Boundary Conditions (1) Generatrix Ladder-like Approximate-HSMAC Method Staggered mesh system is shown in Fig. 2, ~r X where, u and v are defined on mesh edges, p, k, e and vt are defined at the mesh centre. Let bx = ~x 0.125; by = 0.05. E q s . (1) - - (6) are transformed into a series of difference equations and the solution is obtained. Based on the staggered mesh system, Adams-Bashforth and central differencings 8y ~,J are applied with regard to time and space, res+ k,.j E~,j pectively. It should be noted that the upwind vtl,j differencing is occasionally used for the convective Fig. 2 Grid mesh terms of momentum equations, k and e transport equations. Apply Eqs. (1) and ( 4 ) - - ( 6 ) to the grid centre and Eqs. (2) and (3) to the defined points of ~" and ~" respectively. The differencing of the momentum Eq. (2): all the terms are moved from left side to the right side except ~u /t3t and / ~ / a x and then all terms on the right side are rewritten in a difference form on the defined point , denoted by Li,j.
T
The differ_encing of the momentum Eq. (3): all terms are moved from left side to the right side except t~v/~t and ~p"/t~r and then all terms on the right side are rewritten in a difference form on the defined point, denoted by Mi 4 " The time differencing: Let time increment be bt , all the physical parameters after time nbt are denoted by superscript n. So u ,+1 can be expressed by Taylor series as follows:
~"+l=~-~+bt -N-
2
Ot
where, the right side terms
-~-
i,j = -~- (Phi Pi+l,j)+Li, j
(8)
therefore, time differencing can be transformed into space differencing. By substituting ( 8 ) into (7), and applying backward differencings to the time differential of the third term on the right side, we have
i,j -- ,
-~ bt "~- (Pi, j - P i + l , j ) + L n j
--~--
--~- ~ , j -Pi+l,j)+Li, j
(9)
and
~.n+l=~, 3 (~t I ~ 1 q ~ j - ? n j + l ) + g n j ] - Wb t [ "~1 q~'j . - 1 --~P-i1' j + l ) + g i '.J~ l ] i,j , j+ -~
(10)
The differencing of the k , e transport equation: all the terms in (4) and (5) are moved from left side to the right side except the time differential terms, then all the right side terms
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ACTA MECHANICA SINICA
1992
are rewritten in a difference form on the defined point of ki, j and ei,j, and expressed by Fi, j and G;,j . The time differencing: expanding /:+~, ~"+~ into Taylor series up to the third term, we have
k i,n+~ j =kin, y + ~3 6tFn _ T6t F7'31
(II)
n-1
sn+l_ n 3 ~tG[j t~t ~,: --~,j+ ~ - T G~,j
(12)
The upwind differencing of each convective terms is as follows:
dif k ax
~,j+u~+l,:
=4--'~-"
+lu-~,j
/-u~+l,j
-(~,-1, j +u,,x)2- lui-~,j+u,,xl(ffH, j-u,,j)] [ ~;~ I = ~ ki- [(-""J+"'+"; - )cE,. ,+;,,§ dif L T J . . ,~,,, .+z:+,,icE,,-~,,,+,)
(13)
)('U,,j.1.-I-u--hj)--IV~.,j.I-I-v~+I,]-II (Ui.j- l --U-,,j]
(14)
--(-~i,j_l"l'l~"-i+l,j:l dif L ax
'I G"J+u"J+I)CE"J+-Ei+"J)+I-~"J+u"J+II@F-E~+I'~) -
=4-~-
--(Ui-l,jq--Ui-l,j+l)(-~i-l,j'lt'-~i,j) dif
E#]
= " ~1y
- -Ui-1
j
+U,-l,j+,l(v~'-,,j-u--i, jt
(15)
[
(vi,j + E , j + l )2 + Iv~,j'4-V~ , j+l I ~ i , j - E , j+l ) 1
--(-~i,j-I -{-E,j) 2- E j-I"I-Ejl(-Vhj-l--Vi, j)] dif
-- "~
(16)
i, j (k i,
-u~-l,j(ki-l,j+ki,j) - Ui-l,y (ki-1 j - k i , j ) ] / ~ x [-dvk I
difL~
=--flI-~i ,j(ki, j+ki, j+,
)+-~iyl(kij-k ,
i,j+l
)
(ki, j-l-l-ki,j) - '-~i,,-,'(k,,,-,-k,,,)J/,y dif
~
,j(~i,j+~i+l,j)+l~i,j
---~-
1
)+Iv-~,:l (e~,j-e~,
)[/6x
(19)
) ~
~ ~
(18)
(si,j-~i+l,j)
--U""i-l,j(?'i-l,j+'Si, j ) - I u i - l , j l ( e i - , , j - e i , j
~e
(17)
t
(20)
jl 1
The solution for pressure is obtained with the HSMAC scheme (simultaneous iteration.for
u, v and p ). The convergence condition of iterative calculation is: Max
dif - ~
+-~
+-r
e N
(21)
Vol. 8, No. 2
He Yongsen et al.: Turbulent Flow in Conical Diffuser
121
where, dif.: short for difference, which means differencing the continuity equations, N: number of iterative calculation for pressure and velocity of flow, Max: maximum value in the solution region, e : truncation error. The boundary conditions: At the inlet section, uniform velocity profile and Laufer's experimental h 9 ~i +1, m + 2 profile are used for the longitudinal velocity, and ~i +I, m + 2 Laufer's experimental values of fully developed pipe i / / / / / / . J / / / / // flow are used for k and e ( k i n = 3 . 2 x 10 -3, ei~ = m + l Ui +1, m+l ~ 0 7.2 x 10 -4). Divergence free conditions for velocity, 9 ki, m+l 9 ki + 1 , m + l k and e are imposed at the outlet section. The i -1 +1 Ui , m• Ei +1, nz+l 9m ~ i , m+l ladderlike geometry is used to describe the wall surI~]--/il/ , Z / l l l l l i face generatrix approximately; and the wall boundai+~ ry conditions are processed as in Fig. 3. The normal components of the velocity to the wall (Ki, m) and the parallel component of the velocity in the mesh of the ladderlike section (u//. m+l ) i--I are assumed to be zero , and the one-seventh powFig. 3 Wall boundary conditions er law is assumed for the relations between the inside and outside one of the grid on the wall surface in the no-ladderlike section (~J-l, = , ui-1, =+1). As regards k and e , they are very large and with very steep variations near the wall surface, and when the radial grid number is few, and mean values are imposed on the inlet boundary condition for k and e , let & / a r =0, & / & = O , as shown in Fig. 3, i. e. ki, = + 1 =ki, =; k i + l , m + 2 = k i + l , m + l ; ~i, m + l : ~ i , m ; / ~ i + 1 , m + 2 = e i + l , m + l .The algebraic auxiliary formula for e derived from the mixing length theory is examined. The auxiliary formula for e is expressed as follows:
e=
ry
(22)
where r is the Karman constant ( r = 0 . 4 ) and y is the distance from the wall. (2) Power Law-BFC method BFC is used, in which the generatrix geometry of a diffuser is described by the coordinate transformation, a n d the second group of model constant (see Table 1 ) is chosen, where the one-seventh power law is employed.
3.Typical Exampi6s The turbulent flow in an axisymmetric conical diffuser of 8 o total expansion angle is computed, as shown in Fig. 1. The Reynolds numbers used for the computation are 2.93 x 105 and 1.16x 105. The turbulent flow is calculated by the above two methods with two inlet boundary conditions respectively. The computational conditions and the boundary conditions in cases A, B, C, D are shown in Table 1. Generatrix ladderlike approximate-HSMAC method and the Power Law-BFC method are applied to cases A, B and cases C, D respectively. Uniform velocity profile and experimental values of fully developed pipe flowi61 are applied to the inlet boundary conditions of u in cases A, C and cases B, D. III. C O M P U T A T I O N A L RESULTS AND DISCUSSIONS The calculated results are shown in Figs. 4 - 8, and the results are compared with each other and also with experimental data by Okwubi [7l and Singh TM. The extinction of the influence of the initial conditions and the establishment of the steady state solution can be judged ,by observing the longitudinal velocities at three fixed points, that
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is,
1992
ACTA MECHANICA SINICA
a(x/Do=O.063, r/Do=0.475),
b(3.562, 0.475) and c(7.062, 0.475), see Fig. 4.
Table 1 Computational conditions
Grid
staggered
Scheme
B
C
(MAC type)
regular
A
CASE
(Time)
l) (Grid Generation )
Crank- Nicolson ( for 2nd diff. )
Adams- Bashforth
Adams- Bashforth (for other ) 3rd upwind (for cony. ) Central (for other)
Doner Cell (for conv.) Scheme (Space)
Central
Solution for pressure
Central (for other)
Simultaneous iteration
for u, v & p
SOR method for poisson eq. o f p (MAC scheme)
(HSMAC scheme) Inflow B.C
vel.
uniform
I
k, Outflow
[
exp. data
I
k i n = 3 . 2 x 10-3 , 8in=7.2xl0 - 4
B. C
divergence free
vel. Wall B.C
uniform
exp. data
1/7 th power law
k
free slip
8
auxiliary formula , Eq . (22)
P
free slip
Model constant
CD=0.09, C1 = 1.59
CD= 0.09, C1 = 1.44
trl= 1.0 ,
Ol = 1.0,
C2= 2.0
a2= 1.3 Time increment
1/1000
1/100
Number of Grids
C2 = 1.92
#2 = 1.3 1 /200
75 x 20
150 x 50
Co,v0r, condition
Max
dif
CXl
N
, increase with K. The increase of Xp reduces the mean velocity through the perforated plate and this means that A~-t )decreases as Xp increases. On the other hand when Xp exceeds a limit, flow velocities in the main flow region become considerably different from those in the neighborhood of the wall boundary, and this velocity difference causes an increase of A0] / [ 1-(Do/Dx )41 (25) Fig. 11 shows the contour lines of diffuser efficiency. In the range between K = 1.2 and K= 1.4 and between Xp=1.3 and Xp= 1.4, r/ takes approximate maximum values of about 0.51 . The experiment by Kline TM shows that r/ for the conical diffuser of 15 o expansion angle is
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ACTA MECHANICA SINICA
1992
approximately 0 . 4 , and thus there appears to be a range o f location and resistance which would give a high diffuser performance. Figure 12 s h o w s the contour lines o f g. F r o m the figure it is expected" that the combination of K= 2.7 and Xp= 1.6 is almost optimal for producing a g o o d uniform flow at diffuser exit section, in the region o f this numerical simulation. F i g s . l l and 12 reveal that optimal combination for diffuser efficiency is not similar to that for flow uniformity. V. C O N C L U S I O N S A fully developed incompressible turbulent flow in a conical diffuser o f 4 ~ expansion angle is predicted numerically by using a k-e turbulent model. M e a n flow velocity and turbulence energy inside a conical diffuser are predicted successfully, and B F C approach is advantageous for predicting such a diffuser flow. The k-e turbulence m o d e l is applied to a flow in a conical diffuser of 15 ~ angle with a perforated plate. A simplified mathematical model is used to describe the effect of the perforated plate. Typical charts are presented to select the location o f the perforated plate with a suitable resistance coefficient, from a viewpoint o f diffuser efficiency or flow uniformity. Calculations were carried .out on the F A C O M VP100 system in the Institute of Industrial Science, University of Tokyo. A c k n o w l e d g e m e n t : W e express our thanks to M r Liu Xiaolin and Mrs Liu Shaoying for their help in preparing this paper. REFERENCES Kline SJ, et al. Optimum design of straight-walled diffusers. Trans ASME, 1959, Ser, D, 81(3): 321 SchlichtingH. Boundary layer theory. 7th ed. McGraw-Hill, 1978. 626 He Yong Sen, Morinishi Youhei, kobayashi Toshio Numerical prediction of turbulent flow in a conical diffuser using k-8 model. SEISAN-KENKYU, 1988, 40(1 ): 39 (in Japanese) 4 KobayashiToshio, He Yong Sen, Morinishi Youhci. Numerical prediction of turbulent flow in a conical diffuser using k-e model--2nd Rcp WF-BFC method and LRN model-BFC method-- SEISAN-KENKYU,1989, 41 (1):28 (in Japanese ) 5 Kobayashi T, Morinishi Y. Numerical prediction of turbulent swirling flow in a rectangular channel ( 1st pep. ). SEISAN-KENKYU, 1986, 38(12): 60 (in Japanese) 6 Laufer J. The structure of turbulence in fully developed pipe flow, NACA Pep 1174, 1953. 1 7 Okwuobi PAC, Azad RS. Turbulence in a conical diffuser with fully developed flow at entry. J Fluid Mech, 1973, 57, part 3 : 603 8 Singh D , Azad RS. Turbulent kinetic energy balance in a conical diffuser in : Proc. of Turbulence, 1981.21 9 Kobayashi T, et al. Numerical prediction of effects of perforated plates in conical diffusers (lst Rep .). Bull of JSME, 1986, 29(247): 91 10 Kobayashi T et al. Numerical prediction of effect of perforated plates in conical diffusers (2nd pep). SEISAN-KENKYU, 1985, 37 (2): 72 (in Japanese) 1 2 3
