CC BY
11Magazine of Civil Engineering. 2020. 94(2). Pp. 65-80
Magazine of Civil Engineering issn
2071-0305
journal homepage: http://engstroy.spbstu.ru/
DOI: 10.18720/MCE.94.6
Intake rate through openings in the side wall of the duct
D.V. Maklakova, V.N. Posohinb, R.G. Safiullinb, J.R. Kareeva*b
a Kazan (Volga region) Federal University, Kazan, Russia b Kazan State University of Architecture and Engineering, Kazan, Russia * E-mail: jkareeva2503@gmail. com
Keywords: intake ducts, openings in the side wall, intake rate, ideal fluid, conformal mappings, numerical calculation
Abstract. Subject. In technological and general ventilation ducts with a given uniformity of intake is often needed to be designed. The calculation of pressure losses in such ducts is complicated by the lack of reliable information about the characteristics of the flows at the inlet to the intake openings and slots. Intensity of air intake through slot openings located on one and two opposite walls of the duct in a series of sequentially placed slots, which determines the presence of a transit air stream passing by the hole is calculated. The slots are perpendicular to the generatrix panel and can be opposite or offset relative to each other. The presence of a stagnant zone formed when the flow is cut off from a sharp edge at the inlet is taken into account. Methods. The search for a solution is carried out in the framework of ideal fluid jets theory using the Kirchhoff scheme and Chaplygin method of singularities, as well as by the numerical method using Flow3d software package, where the system of equations of plane turbulent motion was ended with "standard" k-e model. Results. The flow rates of the air entering through the slots were found, depending on their width and value of the transit flow. Dependencies for the attached flow with and without flow separation are obtained. The shape of the free streamline separating the jet and vortex zones, the compression coefficients of the jet are determined. Current flow lines are constructed for different values of the geometric parameters of the duct and the hole. Conclusion. Analytical and numerical calculations showed that the kinematics of currents and values of the attached flow rate are very similar, but the size and shape of the stagnant zone are significantly different. A numerical solution gives more physics of stagnant zone formation. It was found that flow separation reduces the associated flow rate. It was also found that the intensity of absorption is minimal with the opposite order of cracks.
1. Introduction
The intake of air through slots or a series of local openings of various shapes (round, square, slotted) in the side wall when exposed to intense transit flow is a typical task for many technical devices. For example, it is used when calculating cooling of the turbine blades surface [1], during the operation of ejectors [2, 3]. In technological applications of ventilation, there is often a need to design ducts with a given uniformity (nonuniformity) of intake along the length. Failure to comply with the specified intake conditions can lead to ineffective local intake from sources of harmful emissions, a violation of regulations for the operation of technological equipment, etc. [3-10]. Uniform intake ducts (USD) are also used in public buildings to remove air from the upper zone by general ventilation.
Depending on the current task, uniform intake along the entire length can be achieved by selecting the optimal hole sizes and distances between them. With sufficient length, such air ducts are more technologically advanced and economical in comparison with air ducts having variable parameters. The main characteristic of the energy-efficient choice of geometry and location of the openings in the USD is the local resistance coefficient (LRC) of its main perturbing elements. The analytical determination of USD flow parameters, which are important for calculating the LRC (associated flow rate, intake rate, stream compression ratio, flow stream lines, shape and size of stagnant zones), is very difficult, because when entering through openings and slots, air loses energy in the opening itself, and when passing through the duct near such a hole. Therefore, the LRC values are determined, as a rule, experimentally. A large amount of LRC data is contained in the well-known Reference book on hydraulic resistance by I.E. Idelchik [11], which has already been published in four editions in 1960, 1975, 1992 and 2005. But this fundamental work is far from exhausting the whole variety of
Maklakov, D.V., Posohin, V.N., Safiullin, R.G., Kareeva, J.R. Intake rate through openings in the side wall of the duct.
Magazine of Civil Engineering. 2020. 94(2). Pp. 65-80. DOI: 10.18720/MCE.94.6
This work is licensed under a CC BY-NC 4.0
constructive situations encountered in practice. In these handbooks there is no data on separation zones, compression ratios of flow at the entrance and passage through the side openings of the air-blast.
Significant simplifications introduced in the USD analysis in the well-known works on this topic [13] do not make it possible to consider this problem as a solved one, and theoretical and experimental studies of transit ducts with constant parameters of channel and slot or channel and side openings are relevant. Therefore, now there still appear works devoted to the study of USD currents, for example. In these works, the authors use numerical experiment using computational fluid dynamics methods. Modern software systems, such as Fluent [13], Flow3d [14], which implement computational fluid dynamics methods, allow one to study such flows quite accurately and in detail. However, the accuracy and adequacy of results obtained by such methods should be monitored by comparison with already known and reliable methods.
The paper considers methods for calculating flows during intake through a series of slit-like openings, which can be located on one or two opposite walls of the duct. The purpose of the calculation is to determine the intensity of absorption through the holes for different geometric situations, as well as the flow parameters necessary to calculate the CLR of the flow in the turbine fouling (connected flow rate, absorption rate, stream compression ratio, flow stream lines, shape and size of stagnant zones). The analysis is carried out within the framework of the ideal fluid flow model and numerically taking into account viscosity C using Flow3d software package.
2. Methods 2.1. Hole on one side of the intake duct wall
A flow diagram is given in Figure 1a where l - long CA slot in the wall of the intake duct with height h (Figure 1a). Transit airflow from overlying openings moves with velocity v», accordingly, specific transit flow is q0 = v»h. It is necessary to determine the intensity of air intake Aqo through the gap (adjoined flow rate) and construct the currents flow lines.
2.1.1. Ideal fluid, unseparated flow
First, we consider the unseparated flow model (Figure 1, a), which does not take into account the fact of formation of a weakly-vortex (stagnant) zone when stream flows around a sharp edge A. We will find the solution using the conformal mapping method [15, 16].
As a parametric region, we take the upper half-plane t = % + in with the location of points indicated in Figure 1, a, b. The flow region in the plane of complex potential w = $ + iy is shown in Figure 1, c.
Figure 1. Flow areas for unseparated flow model: a is physical plane z = x + iy; b is parametric plane t = in; c is plane of complex potential w = iy
The matching between the flow areas in the z and t planes is established using the Christoffel-Schwartz formula [15]
dz t2 - b2
— = -,
dt t2 -1
where c and b are the mapping parameters.
After integration the following expression is obtained:
z = c
1 - b\ t -1 1 - b2
t +--ln---m
2 t +1 2
(1)
At the point D (t = -1), the function dz/dt has a first order pole. By determining the residue of function at this point, we obtain
b2 -1
h = cn-
2
and find the final form of the mapping formula
h
z = — n
2t , t +1 + ln
b2 -1 t -1
+ iH.
At points A and C we have
ZA = Z (b) = '
f 1 - b\ b -1 1 - b2 ^
b + -
-ln-
zc = z (-b ) = '
b +1
-m
J
2
7 1-b\ b+1 1-b
-b +--ln---ni
2 b -1
J
Consequently,
z a Zc l
f b2 -1, b +1
b + -
-ln-
b-1
and further
f A2
c = — 2
b +
b2 -1, b +1 ln-
A-1
b -1
(2)
(3)
(4)
From equations (2), (4) we obtain the formula for determining the parameter b > 1
l=1 + ln b+11.
h n I b2 -1 b -1)
(5)
Further we find the equation of the AC line in the parametric plane t = % + irf. It is a consequence of the expression
Im
2t , t +1 + ln
b2 -1 t -1
= 0.
Having completed the necessary transformations, we obtain
(6)
^ = +1 -n2, 0 <n<riu,
where rM > 0 is determined from the equation 2rMctg 2 M +1 -rM = 0.
Note that for large values of b, from formula (6) it follows that the line BE in the parametric plane is determined by equation (at b ^ œ) Ç = ±yjb2 -r2, that is, this line is almost a circle of radius b.
On the segment AC
- y i - x 4.1
y = — = 1, x = — = ±— h h n
2/ -1) + 4n2
- + ln-
b2 -1
(/-1)2
\ç\>b, o<n<nM.
(7)
In the plane t, we have sources at points D (t = -1); B (t = a) and drain at E (t = 1). The complex potential of such a flow will be
w = illn(t +1)-q° +Aq° ln(t-1) = ^[ln(t +1)-(1 + c )ln(t-1)1 (8)
n 71 n L J
where Cq = Aq0/q0 is the dimensionless adjoined flow. The complex adjoined velocity is
dw dt _q0 (t - l)-(l + Cq )(i +1)
^ 1Vy dt dz n t2 - b2 '
By defining residues of function dw/dt at points D and B, we find the dimensionless adjoined flow
(9)
2
C =-. (10)
q b -1
By separating the real and imaginary parts of equalities (8), (9), we find the equations for the flow function
q°
y = — n
Z + 1 ^^ ^ Z-1
arcCos , _-(1 + C ) arcCos
/ „ t V qZ
\I(z+1)2+n 4(t-1)2+n
(11)
and for the flow velocity components:
q°
v = —
x
Z- 1 -(1 + Cq )(Z + 1)]-(Z2 -n - b2 ) + 2tfCq
nc (Z2 -n2 -b2) + 4Z2n2
q°
v = ■ y nc
2Zn[1 -Z+(1+cq )(1+Z)]-cqn(z2-n2 - b2)
(Z2 -n2 - b2) + 4Z2n2
(12)
(13)
where Z ^ °, n > 0.
We let us consider the problem of an ideal fluid flow, taking into account the separated flow (the diagram is shown in Figure 2 a). At point A, the flow breaks away from the wall, forming a stagnant (weakly vortex) zone. The velocity at the free AE boundary is vo. At point C, the leakage rate is finite. As before, it is necessary to determine the adjoined flow rate and construct the flow stream lines. We will find the general solution to the problem by the method of singular points of S.A. Chaplygin [15]. As a parametric region, we choose the upper right quadrant with the corresponding points shown in Figures 2, a, b.
We construct the function dw/dt, which is a the complex conjugate velocity of the imaginary flow in the parametric plane. We take into account that the fluid flows from the infinitely distant points B, D and is absorbed at the infinitely distant point E, that is, the streamlines start at points B, D and close at point E. There exist a separation point C, where the general streamline is perpendicular to the axis Z-
We define the features of the function dw/dt. At points A and C, the right angles are flowed around, so tA = 0, tc = c are zeros of the first order. Points E lie at infinity; therefore, they do not participate in the construction of the function dw/dt. At points B (tB = b) and D (tD = 1) we have the poles of the first order.
7/7 f dw
We analytically continue dw/dt on the entire complex plane. On the real axis Zthe condition Im— = 0
dt
is satisfied. According to the principle of symmetry, we analytically continue dw/dt through this axis to the
dw
lower right quadrant, with no additional features. On the imaginary axis rjthe condition Re— = 0 is satisfied
dt
that allows one to analytically continue dw/dt on the whole complex plane. Moreover, at the point C (tc = -c) we get zero, and at the points B (tB = -b) and D (tD = -1) we obtain the first order poles.
Magazine of Civil Engineering, 94(2), 2020
As a result of these continuations, the function dw/dt is built in the form
dw
t
( t2 - c 2 )
dt~ 0 ( t2 -1)( t2 - b2 )
= k f ( t ).
(14)
dw
On ADE Im— = 0 and therefore, is a real number.
dt
The picture of the flow lines of an imaginary flow in the parametric plane, constructed using the StreamPlot option of the Mathematica software [14], is shown in Figure 3.
Figure 2. Flow areas for separated flow model: a is physical plane z = x + iy; b is parametric plane t = %+ in; c is plane of complex potential w = < + iy.
Figure 3. Imaginary flow.
dw v vy 1
We construct a function-= — -i —. This function has a unique zero at the point t = b, which, after
vodz vo
analytic continuation through the imaginary axis, turns into a pole at the point t = -b, since
dw
imaginary axis. Given that with t ^ »-= 1, we obtain
dw
v0 dz
= 1 on the
v0dz
dw t - b
v0dz t + b
Next, we find the derivative of the mapping function z = z(t)
, dz dz dw < t(t2 - c2) t -b < \
q = v0 h — =---= -^7-^-p---= F (t).
dt dw dt v0 (t2 - 1)(t2 - b2) t + b v0 W
(15)
The expressions (14), (15), in principle, allow constructing streamlines in the parametric and physical planes.
We proceed to determine the flow rate of air sucked through the gap. The flow rate at point D is q0 = vœh, and at point E it is q = v0h1, so flow through the gap is Aq0 = q - q0.
Function dw/dt is analytical in the upper right quadrant, excluding points B, C, E. Points B, C on the parametric plane are marked by semicircles of infinitely small radius; we surround the remote point E with a quarter of a circle of infinitely large radius (Figure 2, b). When switching from DC to DE in a semicircle Cd Im w experiences a jump qo (Figure 2, c).
Using the residue theorem, we find
^0 = $ ~~dt =nires kf (t)] = ni^0 1 c 2 \ cd dt *=1L WJ 2(1 -b2)
whence it follows that the transit flow intensity is
1 2
1 - c
q0 = n0—(—(16) 2 (1 - b2 )
When moving from BC to BA in a semicircle Cb Im w experiences a jump Aq0, hence
dw r / \i b2 - c2
iAq0 = j — dt = ni res \j/>0) f ( t) J = ni ф0
r Hi - , L I r0J\ ) \- V0 / 2 V
cb dt t=b L WJ 2 (b2 -1) and the intensity of flow entering the gap is
, 2 2 b - c
Aqo =пфо—г~2—v (17)
2 (b2 -1)
when switching from DE to AE in a quarter circle Cr Im w experiencing a jump q and
dw n r- /S.-1 n iq = — dt = — ires I ^f(t) I = , cr dt 2 t=aL 0 WJ 0 2
which means the intensity of the total flow
q (18)
2
From formulas (16)^(18) we find the dimensionless associated flow
F 2
Aqr. q 1 - b
Cq = Ji° = 1 = - 1. (19)
q0 q0 1 - c
Now we determine the sizes h and M. When switching from DC to DE by semicircles cd Im z experiences a jump h. From here
2
dz t/>0 r \ $0 1 - c
ih = <p — dt = — ni res F (t) = — ni-
у КЛ1 --/t t / C.O i III --/И ,
c dt v0 t=1 v0 2 (1 - b2 )
, фо 1 - c2
h = — -, (20)
V0 2 (1 - b2 )'
when switching from DE to AE in a quarter circle cr Im z experiences a jump h1 and it means that
dz <bc, ni , v <bc, ni
h = j — dt = res F (t) =
1 cR dt v0 2 t=œ w v0 2
Ф0 n
h = --■ (21)
V0 2
Combining expressions (20), (21) we find the compression ratio of the flow
*=hi=iLb£ (22)
h 1 - c
It remains to determine the mapping parameters b, c and the coefficient 90. Since point B in the physical plane at infinity to the left and to the right of the gap is at the same level, then
- dz / n.
Im i — dt = 0 ^ resF (t ) = 0. (23)
t=b dt t =b W
On the other hand
res F ( t ) = — t=b V ' dt
-(t2 - c2)" t2 -1
t=b
We take the logarithmic derivative
d t ( t2 - c2 )
dt t -1
t=b
1 2t 2t
t t2 - c2 t2 -1
= 0
t=b
From the previous equalities it follows
1 2b 2b
b + b2 - c2 b2 -1
= 0
and
c = b*
'r?
1 + b
(24)
From the graph of function (24) (Figure 4) it can be seen that for 0 < b < 1, the following is always true: b < c < 1.
The function F ( t ) =
f(t2 -c2)
(t2 -1)(t2 - b2 )
entering the mapping formula (15), will be decomposed to a sum
of partial fractions
MNP K
1V1 1\
t )=—+—
t -1 t +1
( t - b )2
t - b
Figure 4. Plot of function c = c(b). In view of (23) the coefficient K = 0, for other coefficients we get
. dz , v 1 - c
M = § — dt = res F ( t )=--
t=1 dt t=i w 2 (1 - b )2
dz 1 - c2
B = § — dt = res F ( t )=--
t=-1 dt t2 (1 + b )2
f dz b (b 2 - c2 )
C = § — dt = -
t=b dt b -1
(25)
We find the primitive for function F(t)
G ( t ) = § F ( t )dt = M ln ( t -1) + N ln ( t +1)-
P t - b
(26)
Magazine of Civil Engineering, 94(2), 2020
For the parametric plane the slot of the width l corresponds to a segment AC
I = — Re [G ( 0 )-G ( C )].
V0
Using the expression (20), we obtain
I _ 2 (1-b)2
h n 1 - c2
Re
[G ( 0)- G ( c )]. (27)
So, to find b and c, which meet the criteria 0 < b < c < 1, one should solve a system of equations (24)-(27). By combining them, we get
(1 _ b)2 4b2c + (b - c)[(1 + b2 ) ln (l - c) + (l - b2 ) ln (l + c)] l
(l + b)2 n(c-b)(l-c2) ~h ' (28)
The equation (28) is solved using the Find_Root of the MathematiCa software, and the parameter c is defined using the formula (24).
Let vo =1, h =1. We introduce notations:
Ml = M0, N = N00, - P = P0o-Taking into account (20), (25), (28) we obtain
1 (b -1)2 4b2
_ A T__i_L_ D__
Ml —, N = V \ 2 , P =■ , ,2
n n( b +1) n( b +1)
As point A has coordinates (1/2,0), we have a conformal mapping of the parametric plane onto the physical plane in the form
z ( t ) = — ln ( t -1)+ ( b 1) 2 ln ( t +1)
4b2 1
n " ' n(b +1) n(b +1)2 2
-i (29)
We proceed to the construction of streamlines. This is easier to perform in the parametric plane, and then, using the conformal mapping (29), to transfer them to the physical plane. The imaginary flow in the parametric plane has velocities
V^ = v^(^,n)=^0Re [ f (# + in)], vn = vn (^,n) = -^0Im[ f (% + in)].
Since the imaginary flow is steady, the streamlines coincide with the particle trajectories. Therefore, the differential equations of streamlines will be
£'W= ViM') ,n(T)], n (t)= vn y(r) ,n(T)], (30)
where ris the time of particles motion in the parametric plane.
We construct N streamlines of the transit flow. Initial conditions must be added to the system of differential equations (30). On the semicircle Cd of a small radius s\, we arrange N points. The initial conditions are formulated as
ni „ . ni
E(0) = scos-+1, n(0) = ssin-, j = 1...N,
v ' 1 N +1 ' ,w 1 N +1
(31)
Using NDSolve program of the Mathematica software the Cauchy problem (30), (31) was solved on the interval t e [0, rmax ]. As a result, we obtain the coordinates of streamlines in the parametric plane
The values Tmax were determined from a numerical experiment.
The problem described above for a viscous fluid was solved numerically by Flow3d software package. The system of equations for plane turbulent motion of a viscous fluid was closed using the "standard" k-s model (k is the kinetic energy of turbulence, sis the dissipation rate k). For flow zones in the immediate vicinity of the duct walls, standard wall functions were used.
At the boundaries of the computational domain, the following boundary conditions were adopted (see Figure 1, a):
- On DD, the condition for uniform distribution and constancy of velocity v« in the positive direction of the x axis corresponding to a given specific intensity qo;
- On EE, the condition for a smooth continuation of the flow through the boundary (normal derivatives at the boundary for all quantities are equal to zero);
- At the permeable boundaries of the area of air leakage to the gap, the excessive static pressure AP = 0;
- On the boundaries BC, AB, DE, solid impermeable walls on which the condition of adhesion (nonslip) is fulfilled vT = 0.
The temperature at all boundaries is equal to air temperature of 293K.
The total number of grid cells in the channel is 80 thousand, along the length of the slotted hole it is at least 20, as it is recommended in [14]. The channel length to the slot hole was taken equal to 3h in order to exclude the influence of boundary conditions in the DD section on the flow near the slot hole. The length of the channel after the slot hole was taken to be at least 10h to smooth the flow deformation due to adjoining of the flow through the slot and separation of flow from the sharp edge, which affects the accuracy of determining the total flow rate q = qo + Aqo.
2.2. Bilateral arrangement of slotted openings in the duct
An element of the intake panel with a two-sided arrangement of slotted openings is shown in Figure 5 a. Transit flow from upstream openings has velocity v«. Here it is also necessary to determine the flow rates of air entering through the slots and to build the flow stream lines.
a)
/i
—A,
-A,
- A<
A,
b)
n
(t
/ w
/ \ ,
1 -1 a2 "3 «4 1 ] 5
Figure 5. Flow areas: a) physical plane z = x + iy; b) parametric plane t = in.
Ideal fluid, unseparated flow. We find the solution using the theory of potential flows of ideal fluid by the method of conformal mappings. We do not take into account the presence of vortex zones in the places of
boundaries fractures (points A4, A6). We assume that at points A0 and A2 the Kutta-Joukowski condition of the limited velocity [20] is satisfied.
We will construct the flow in the parametric region, which in our case will be the upper half-plane with the correspondence of points shown in Figure 7a, b. The relationship between the flow regions in the z and t planes is established using the Christoffel - Schwartz formula
z = AS F (u )du ; F (t )= (t 2 - 1)(t - - ^ , (32)
-1 (t - a )(t - a3) (t - a5)
where u is the integration variable.
The mapping parameters al, a2, a3, a4, a5 and coefficient A must be defined. We expand the function
F(t) at the sum of simple rational fractions
1 P A ,
1 1 -r- +-— + 1,
a a
F (t ) =-+ —+ 2 .
t - a1 t - a5 (t - a3 ) t - a3
where coefficients a, a, P, P1 are real. The coefficient A is also real and positive, since for A6A7 we have dxld%> 0.
In order to obtain the same duct width at points A1 and A5, the following conditions must be met
dz dz
Im J — = ih and Im J — = -ih, c1 dt c5 dt
where C1 and C5 are semicircles of infinitely small radius, surrounding points A1 and A5 in the parametric plane t (see Figure 7, b).
Using the residue theorem, we obtain
Aadt h
Im J-= ih ^ Aani = ih ^ Aa = — ;
C t - a1 n
Aadt h
Im J -= -ih ^ Aa1ni = ih ^ Aa1 =--..
c5 t - a5 n
It follows that
h h
a = — =--; (33)
an a1n
a1 +a= 0. (34)
In order to have the straight lines A7A0, A6A7 at the same level in the physical plane, we additionally use the condition
- dz
Im J — = 0 ^ Im J F (t) dt = 0, (35)
Cr dt cR
or otherwise
Im
, adt , adt , P,dt , Pdt
J -+ + J — + J + J dt
cRt - a1 cRt - a5 cRt - a3 cr (t - a3 ) cr
= 0,
where Cr is the semicircle of an infinitely large radius surrounding the point A7 in the parametric plane (see Figure 5, b).
Pdt
According to the residue theorem J-- = 0, and besides, Im J dt = 0, which means
Cr (t - a3) Cr
ma
+ niax +nipx = 0. Taking into account expression (34), Pi = 0, and the function F(t) takes the form
a a P
F (t) =---+-+1.
t - aj t - a5 (t - a3 )
Having determined the residues of function F(t) at points Al, A5 and the limit of function value F(t) (t - a3 )2 at t ^ a3, we express the coefficients a, a1 , p through mapping parameters a1, a2, a3, a4, a5
(a,2 - 1)(a, - a9)(a, - a4) (a52 - 1)(a5 - a2)(a5 -a4)
a = —1-L^-2—1-—; a =—5-^^—5-—;
(aj - a3) (aj - a5) (a5 - a3) (a5 - aj)
p = (a32 - 1)(a3 - a2)(a3 - a4) (a3 - aj )(a3 - a5)
Using condition (3) from the first two expressions we find
(a2 - 1)(aj - a2 )(ax - a4) (a52 - 1)(a5 - a2 )(a5 - a4)
-2---2-= 0 • (36)
(aj - a3) (a5 - a3)
Now consider the condition Im S F(t)dt = 0 ^ res F(t) = 0. By determining the residue of the function
c t
F(t) at infinity, we get
- aj + a2 - 2a3 + a4 - a5 = 0. (37)
If the mapping parameters satisfy conditions (36), (37), then
P
f (t) = S F(t)dt = a ln(t - aj) - a ln(t - a5)---— +1.
t - a3
The well-known geometric characteristics of the flow region make it possible to write down three more equalities
I, = Re[ f (1) - f (-1)]• A; l2 = Re[f (a4) - f (a2)]• A; l3 = Re[f (a2) - f (-1)]• A.
Given that A = h/an, and introducing new notations jjh = lj, h = l^,l^h = I3 we get three equations that are missing to determine the five mapping parameters aj , j=1,2,...5
Re [ f (1) - f (-1) ]- \an = 0; (38)
Re [ f (a4) - f (a2) ]- ^ an = 0; (39)
Re [ f (a2) - f (-1) ]- T3 an = 0. (40)
Now it is necessary to solve a system of five nonlinear equations (35), (36), (38)-(40) with respect to five mapping parameters. The main difficulty in solving this system by the Newton method is the ordering of
parameters:
-1 < a, < a2 < a3 < a4 < a5 < 1
The point is that the method iterations regularly violate this ordering. To preserve it, we use the technique proposed in [18] (see also [19]). We introduce new variables
aj - a-i —
bi = ln-—, i = 1,5, a0 =-1, a6 = 1. (41)
ai+1 - ai
We turn the last equalities to variables ai, for this we solve the following system of linear equations with a tridiagonal matrix
b — at - ai-l - e i (ai +j - at) = 0, where i = j, 5.
Solving this system of equations in the Mathematica package, we obtain
at = Gt(bj,b2,...,b5), i = 1,5. (42)
where functions Gi are built explicitly.
Now, taking into account equalities (42), we transform the system of equations (35), (36), (38)-(40), solving which we find the mapping parameters. Note that variables b, are unordered, but any set of them gives an ordered set of variables a . When applying the Newton method to the transformed system, the zero values of variables bi were simply chosen as the zeroth approximation, and the iteration process was always converged.
Thus, we have an opportunity to construct a function
P
f (t) = J F (t )dt = aln(t -aj) -aln(t ^5)--+ t
t - a3
and conformal mapping
z (t ) = A [f (t)-f (-1)] (43)
Now we calculate the adjoined flow rate through the slotted openings in the walls of duct. In the parametric plane, we have sources at points Ai (t = ai), A3 (t = a3), A7 (t = <x>) and stock at point A5 (t = a5). The complex
potential w = $ + iy of such a flow will be equal to
q (q + q1 + q2) q2
w = — ln(t - aj)--1-2— ln(t - a5) + — ln(t - a3), (44)
n n n
where p, yare potential and function of flow, qi and q2 are flow rates through the corresponding slotted openings, q = vJh is the transit flow rate. We find the derivative
dw _ q qj + q2 + q + q2
dt n(t - a1) n(t - a5) n(t - a3) At points Ao, A2 velocities are limited and therefore dw/dt, dw/dt = 0. We solve the system of equations
d^ (t=-1) = q - qj+q2+q + q2 = 0.
dt n(-1 - aj) n(-1 - a5) n(-1 - a3)
dw (t = a2) = __qj + q2 + q + q2 = 0
dt n(a2 - aj) n(a2 - a5) n(a2 - a3)
with respect to qi and q2. As a result, we obtain
= q(a3 - aj)(a5 - aj) = q(a3 + j)(a3 - a2)(aj - a5) W1 --, W2 --•
(1 + ai )(a2 - ai) (1 + ai )(a2 - ai )(a3 - a5)
Now we can calculate the adjoined flow rate, which we present in the dimensionless form
q + q2 (a3 - a1)(a5 - a1) + (a3 + 1)(a3 - a2)(a1 - a5)
Cq =-=--+--• (45)
q (1 + a1)(a2 -a1) (1 + a1)(a2 -a1)(a3 -a5)
The problem for a viscous fluid was also solved numerically in the Flow3d software package. The boundary conditions were formulated in the same way as in Problem 1. The calculation results are presented below.
3. Results and Discussion
3.1. Hole on one side of the intake duct wall
During the calculations, we also determined the dimensionless adjoined flow rate Cq and the jet contraction coefficient K. Figure 6 shows graphs of the corresponding dependences (19) and (22). Recall that the parameters b and c at given values l/h are determined by formulas (24), (28). The dashed line shows the graph Cq for the flow, where the flow separation at point A is not taken into account.
Comparison of graphs shows that separation of flow reduces the adjoined flow. We also see that the jet contraction coefficient tends to 0.5 with an increase in the length of the gap, that is, the flow in the channel becomes close to the flow in the Bord nozzle [17].
0.0 0.5 1.0 1.5 l/h
Figure 6. Function graphs: 1 - adjoined flow taking into account flow separation (solid line), without taking into account flow separation (dashed line), 2 - jet contraction coefficient, (Cq values obtained numerically in the Flow3d program are shown by circles).
Using the mapping formula (29), the streamlines' coordinates in the physical plane are determined. Similarly, the intake flow streamlines are constructed, but a semicircle of small radius Cb is constructed around point B (t = b). Figure 5a shows the streamlines constructed in the described way for various values of l/h.
///¡=1.0 !/h=0.5
Figure 7. Flow lines near the hole in one wall of the duct: a - analytical solution in NDSolve program of Mathematica software, b - numerical solution in the Flow3d program
3.2. Double-sided intake panel element
Figure 8 presents a graph of dependence of Cq on the device geometry. Minimum of Cq is achieved for opposed openings. As the gap between the slots increases, the adjoined flow tends to a constant value.
Magazine of Civil Engineering, 94(2), 2020
The intake rate is minimal with the opposite arrangement of openings, with an increase in the spacing it increases. Starting from a distance l3 = 2, it reaches its maximum value and then remains constant. That is,
it can be argued that at l3 > 2 the mutual influence of the openings disappears.
-3-2-1 0 1 2 13
Figure 8. The graph of dependence of the adjoined flow from the spacing between openings \ =1, l2= 0.5.
Knowing the complex potential (44) and the conformal mapping formula (43), we can further construct the flow streamlines in the physical domain. Figure 9a shows the flow map for different locations and sizes of slot openings.
Figure 9. Flow streamlines in openings on opposite walls of the duct: a - analytical solution in the NDSolve program of Mathematica package, b - numerical solution in the Flow3d program.
The intake rate is minimal with the opposite arrangement of holes, with an increase in the spacing it increases. Starting from a distance l3 = 2, it reaches its maximum value and then remains constant. I.e. it
can be argued that with l3 > 2 the mutual influence of the cracks disappears. Considering the complex
potential (44) and the conformal mapping formula (43), we can further construct the flow streamlines in the physical domain. In Figure 9a a flow pattern for different locations and sizes of slit openings is shown. Pictures of flows constructed by Flow3d software package are shown in Figure 9, b.
The results of calculating the attached flow are shown by dots in Figure 8. We can state the accordance between analytical and numerical calculations
The numerical method allows to obtain more physics of the flow and, in particular, determines the geometry of the stagnant zone formed when the flow is cut off from sharp edges. Within the framework of the used analytical model, the shape of the stagnant zone is not determined.
The problem was also solved numerically in the Flow3d software package. The boundary conditions were formulated in the same way as in Problem 1. The flow maps constructed using the Flow3d software package are shown in Figure 9, b. The results of calculating the adjoined flow are shown by dots in Figure 8. We can state a good agreement between analytical and numerical calculations.
The numerical method allows one to obtain a more physical picture of the flow and, in particular, it determines the geometry of the stagnant zone formed when the flow is cut off from sharp edges. Within the framework of the used analytical model, the shape of the stagnant zone is not determined.
4. Conclusions
In the work, the velocity field created by the action of the slotted hole on the duct wall, as well as at the intake panel with a two-sided arrangement of slotted holes, is analytically and numerically calculated. In the framework of the inseparable and detached models of potential ideal fluid flows, equations for the flow stream lines are obtained. The flow patterns are compared with the results of numerical calculations performed for a viscous fluid. The intensity of absorption is determined depending on the width of the slit and the flow rate.
Important results have been obtained that can be used in the design of air duct systems:
• the flow rates of air entering through the slots in the walls of the duct are determined, depending on their width, relative position and transit flow value;
• current flow lines are constructed;
• it was found that the intake intensity is minimal with the opposite arrangement of slots;
• The shape and size of the stagnant zones are determined numerically.
It should be noted that the qualitative patterns of the flows obtained by analytical and numerical methods are slightly different, since the flow model underlying the analytical calculation does not imply that the boundaries of the stagnant zone are closed on the duct wall. The values of the attached flow, determined analytically and numerically, practically coincide.
The results of the experimental determination of the flow characteristics and the intensity of absorption through the openings for the considered geometric situations are being prepared for publication. In particular, Figure 10 shows photographs of the boundaries of the separation zones at the middle suction inlet in a rectangular duct at l/h = 0.32. The experiments were run by Ph.D. Ziganshin A.M. in the laboratory of Kazan State University of Architecture and Engineerin (KSUAE).
Figure 10. Boundaries of the separation zones at the middle intake hole in a rectangular duct at l/h = 0.32 at different velocity of the transit flow.
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Contacts:
Dmitry Maklakov, [email protected] Vladimir Posohin, [email protected] Rinat Safiullin, [email protected] Julia Kareeva, [email protected]
© Maklakov. D.V., Posohin, V.N., Safiullin, R.G., Kareeva, J.R., 2020
