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The linearly stretching wall jet
Journal Pre-proof The linearly stretching wall jet Amin Jafarimoghaddam
PII: DOI: Reference:
S0997-7546(19)30433-9 https://doi.org/10.1016/j.euromechflu.2019.12.001 EJMFLU 103574
To appear in:
European Journal of Mechanics / B Fluids
Received date : 25 July 2019 Revised date : 27 November 2019 Accepted date : 3 December 2019 Please cite this article as: A. Jafarimoghaddam, The linearly stretching wall jet, European Journal of Mechanics / B Fluids (2019), doi: https://doi.org/10.1016/j.euromechflu.2019.12.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Masson SAS.
Journal Pre-proof
The Linearly Stretching Wall Jet
2
Amin Jafarimoghaddam
3
Independent Researcher, Tehran, Iran
4
Previously at the Department of Aerospace Engineering, K. N. Toosi University of Technology,
5
Tehran, Iran
6
Email Address: [email protected]; Phone No.: +98 935 665 1957
pro of
1
Abstract
8
It is constructed a set-up comprising a jet from a slit at the leading edge, discharged over a
9
linearly stretching wall. The non-similar flow can be interpreted as a combination of two
10
distinct similarity regions; Akatnow-Glauert flow at the leading edge and Crane flow far away
11
from it. In this respect, it is employed appropriate coordinate expansions to explore
12
perturbatively the behavior of the flow near the similarity regions. A suitable composite
13
transformation amalgamated with an abridgement of the stream-wise coordinate, facilitated an
14
immaculate numerical simulation of the involved nonlinear partial differential equation over
15
the entire spatial domain ( 0 X , 0 ˆ ) followed by quasi-linearization technique
16
together with an implicit algorithm of a tridiagonal form. As a result, shear stress at the wall is
17
accurately predicted through a proposed formulation, valid all the way along the wall. It is also
18
exhibited that there exists a transition region with a critical coordinate in the stream-wise
19
direction in which the shear stress at the wall becomes zero. This universal coordinate (namely,
20
the turning point) is determined as, reasonably close to, X
21
coordinate measuring distance along the wall and ˆ is the dimensionless non-similarity
22
variable).
23
Keywords: The Stretching Wall Jet; Perturbation Analysis; Numerical Solution; Wall Jet;
24
Stretching Sheet; Non-Similar Flow
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. r c
13 ( X is a dimensionless 23
1‐ Introduction
Jo
25
re-
7
26
The wall jet flow appears in many industrial processes. Knowledge
of
the wall
jet
27
characteristics such as the rate of spread and the decay rate of the maximum velocity can
28
potentially extend the life of gas turbines, provide greater maneuverability of aircraft and
29
increase the efficiency of the automobile defrosters. It seems that a wall jet set-up has been
30
initially considered by Akatnow [1] (in Russian). Later, independently, Glauert [2] considered a 1
Journal Pre-proof comprehensive wall jet set-up including radial and plane wall jets in laminar or turbulent
32
regimes. He was able to solve the pure wall jet flow in similarity terms; a pure wall jet flow is
33
that with stationary and impermeable wall.
34
It was derived an integral condition, namely the Flux of Exterior Momentum Flux for the pure
35
wall jet flow. Later on, in many other subsequent papers, researchers tried to extend the work
36
by Glauert; e.g., it was analyzed that a suction rate together with a moving wall condition could
37
also result in the preservation of the Glauert invariant [3, 4].
38
Thermal characteristics of the Glauert flow were initially studied in [5]. Some approximate
39
solutions were developed for the energy equation associated with the pure wall jet with heat
40
dissipation [6]. In [7], for an adiabatic surface and also an isothermal set-up, some similarity
41
solutions for the energy equation were derived in the context of nanofluids. The
42
Magnetohydrodynamics (MHD) wall jets in similarity terms were analyzed in several resources
43
such as [8- 11]. The MHD wall jet flows in the context of nanofluids were initially explored by
44
the present author in [12]. In that paper, it was briefly shown that it is NOT possible to study the
45
MHD wall jets in similarity terms taking into account the Glauert constraint if the wall is fixed
46
and no transpiration velocity is allowed (only a specific space dependent magnetic field is a
47
volunteer for the similarity reduction in the MHD wall jet; however, even with this scheme, the
48
Glauert integral condition is NOT preserved if the wall is stationary and impermeable).
49
For the 1st time, Jafarimoghaddam carried out an in-depth analysis of the MHD wall jets and in
50
particular, it was shown techniques for rendering similar and non-similar solutions [13].
51
Specifically, it was shown that the MHD wall jets (with uniform and non-uniform magnetic
52
fields) with the wall being stationary and impermeable are inherently non-similar flows (the
53
specific space dependent magnetic field mentioned earlier, stands as a singular case in a regular
54
non-similar transformation). Other similarity solutions for the wall jet flows subject to various
55
conditions may be tracked through [14- 17]; e.g., in [17] the present author has presented
56
closed form solutions for the energy equation associated with the Glauert type wall jets with
57
heat dissipation and subject to various thermal conditions, those may admit a similarity
58
reduction and specifically, it was shown that that there is NO physically-valid similarity solution
59
for the energy equation associated with the pure wall flow with heat dissipation. More precisely,
60
it was shown that for the original Glauert case (the pure wall jet flow), a surface with a
61
prescribed wall temperature in the form of Tw (x ) mx
62
the prescribing parameters below
63
instabilities implying that the imposed singularity is not physically interpretable.
Jo
urn a
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re-
pro of
31
1 4
T is the Induced Heat Shield where
1 would result in the appearance of spectrums of thermal 4
2
Journal Pre-proof The non-similar wall jet flows have been also studied in some cases; e.g. the jet over a moving
65
wall with a uniform speed [18], the jet over a wall with suction/injection [19-22] and the MHD
66
wall jet (with a uniform transverse magnetic field ) over a fixed and impermeable wall [23, 24].
67
In the present work, it is studied the non-similar linearly stretching wall jet flow seemingly for
68
the 1st time. A schematic is sketched in Fig. 1.
re-
pro of
64
69
2‐ Governing Equations
71
For 2D, incompressible and steady state boundary layer flows, the governing equations are:
u v 0 x y
urn a
73
lP
Fig. 1 Schematic of the Problem
70
72
u u 2u v 2 x y y
(1)
u
75
In above, is kinematic viscosity. In addition, u and v are tangential and normal velocity
76
components respectively.
77
For a jet discharged from a slit, located at the leading edge, over a linearly stretching wall, the
78
boundary conditions are:
79
u ( x , 0) ax , v ( x , 0) 0, u ( x , y ) 0
80
In above, a is a positive factor.
81
Following the previous works by the present author e.g. [24] it is introduced:
Jo
74
(2)
(3)
3
Journal Pre-proof
F 12 F 1 F ) , y ( 3 ) 2 Y , x X U ref . U ref . U ref2 .
82
(
83
Here, F is the actual/dimensional Flux of Exterior Momentum Flux (measured at the
84
leading edge) and U ref . is an arbitrary reference velocity to be specified later. For the
85
pure wall jet flow, the reference velocity would be the average velocity at the jet slit.
86
Physically, it is notable that the above transformation already contains the discharge
87
velocity in the term F and hence, U ref . can be chosen in such a way to drop the
88
stretching factor. For this purpose it is also defined:
89
U ref . (
90
Using 1-5, one gets:
91
U V 0 X Y
92
U
93
Along with:
94
U ( X , 0) X , V (X , 0) 0, U (X ,Y ) 0
pro of
(5)
re-
U U 2U V X Y Y 2
lP
1
)3
urn a
95
aF
(4)
(6)
(7)
(8)
3‐ Perturbation Solution: Small 'X' Analysis
96
Near the leading edge, the flow is essentially that given by Glauert [2]. This suggests:
97
(X ,Y ) X 4 F (X , ), X
1
U X
99
U X Y
100
F
Jo
98
1 2
5 4
2F 2
3 4
Y
(9)
(10)
(11)
This delivers:
4
Journal Pre-proof
101
3F 1 2 F 1 F 2 F 2 F F 2 F F ( ) X ( ) 3 4 2 2 X X 2
102
F (X , 0) 0,
103
The perturbation solution should be constructed around the Glauert similarity origin
104
(leading edge). This implies:
105
F (X , ) F0 ( ) F1 ( ) 2 F2 ( ) 3F3 ( ) ... i Fi ( )
3 F F (X , 0) X 2 , (X , ) 0
(12)
pro of
(13)
n
(14)
i 0
3 2
X
107
Upon substitution and simplification one gets:
108
(15)
re-
106
n n 1 n 3 n 1 n 3 n [ i Fi ( )]2 [ i Fi ( )][ i i Fi ( )] [ i Fi ( ) i i Fi ( )][ i Fi ( )] 2 i 0 2 i 0 4 i 0 2 i 0 i 0 i 0 n
[ i Fi ( )]
lP
i 0
(16)
109
Fi (0) Fi ( ) 0, i 0,1, 2,... F1(0) 1, Fi (0) 0, i 0, 2,...
111
Collecting like powers of , it is easy to obtain the governing recursive structure as:
112
1 1 F0 F0 F0 F0 2 0 4 2
(18)
113
1 1 7 F1 F0 F1 F0F1 F0F1 0 4 2 4
(19)
114
1 3 1 3 Fm F0 Fm (1 m )F0Fm ( m )F0Fm m ,0 4 2 4 2
(20)
115
m ,0
116
Here, it is solved the 1st order equation analytically; whilst, the rest will be tackled numerically.
Jo
urn a
110
1 m 1 1 m 1 3 m 1 3 m 1 F F F F kF F kFk Fmk , m 2 k m k 4 k m k 2 k m k 2 k 1 2 k 1 k 1 k 1
(17)
(21)
5
Journal Pre-proof 117
In order to solve the 1st order equation, by differentiating from the zeroth order equation one
118
gets:
119
5 1 F0iv F0F0 F0 F0 0 4 4
(22)
Now, it is easy to note that the 1st order equation is convertible to the above equation by:
121
F1 F0
122
From the boundary conditions:
123
124
This gives:
125
F1 (0) 0
126
3‐1 The Conserved Quantity and the Zeroth Order Solution
127
The momentum equation can be written in the following integral form (see e.g. [22]):
128
X
129
HavingU ( X , 0) X , V ( X , 0) 0 :
130
X
131
The above integral is the so-called Flux of Exterior Momentum Flux and the constant F , as
132
brought into consideration earlier, should be that, connected to X 0 (the zeroth order
133
equation, m 0 ). Besides, the specific dimensionless variables (4) give:
134
F [u (0, y ) u 2 (0, y ) dy ]dy F [U (0,Y ) U 2 (0,Y ) dY ]dY [U (0,Y ) U 2 (0,Y ) dY ]dY
(24)
1 2 2 2 0 [U YU dY ] dY 2U w V w 0 U dY
(26)
urn a
1 2 2 0 [U YU dY ]dY 2 X 0
Jo
0
136
(23)
(25)
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re-
1 F0 (0)
135
pro of
120
y
(27)
0
Y
0
Y
(26)
Therefore, the solutions for the zeroth order term F0 ( ) is that given in [3, 4] as:
6
1
Journal Pre-proof
137
2 2
1
log[
(F0 ( ) F0 ( ) 2 2 ) 1 2 2
( F0 ( ) )
1
2 3 F0 ( ) 2
1
] 2 3 tan [ ], 40 8 1 1 (2F0 ( ) 2 ) 1
(29)
6
138
F0(0)
139
3‐2 Numerical Solution for Higher Orders
140
The equations for m 2 are nonhomogeneous linear equations and were discretized
141
employing Runge-Kutta-Fehlberg (RKF45). In order to ease the solution procedure for the linear
142
equations, and principally to avoid shooting technique, it was followed a procedure described in
143
[24].
144
Eventually, for the main quantity of interest it was obtained:
145
2F 40 4 4 2 ( X , 0) (2+ ) X 3 (6+ ) X 2 72 19 5
146
A truncation up to the 5th order revealed a very good accuracy for this quantity up to
147
X
148
divergent, giving rise to a very slow coverage speed (a very small radius of convergence).
(30)
9/2
(32+
63 19 ) X 6 (196+ )X 320 942
15/2
...
(31)
7 ; however, we also checked that the solutions of the linear equations were 50
lP
149
3 2
re-
3
pro of
72
4‐ Perturbation Solution: Large 'X' Analysis Far away from the leading edge, the jet effects vanish and the flow is essentially governed by the
151
stretching wall. This implies the following transformation:
152
(X ,Y ) XF (X , ), Y
153
U X
154
U 2F X Y 2
155
Upon substitution:
156
3F 2 F F 2 F 2 F F 2 F F X ( ) ( ) 3 2 X X 2
(32)
(33)
Jo
F
urn a
150
(34)
(35)
7
Journal Pre-proof
F F (X , 0) 1, (X , ) 0
157
F ( X , 0) 0,
158
It is assumed:
159
F (X , ) F0 ( ) F1 ( ) 2 F2 ( ) 3F3 ( ) ... i Fi ( )
(36)
n
(37)
160
(X ) X
161
This gives:
n
n
n
n
i 0
i 0
i 0
pro of
i 0
n
n
i 0
i 0
(38)
[ i Fi ( )] [ i Fi ( )][ i Fi ( )] [ i Fi ( )]2 [[ i Fi ( )][ i i Fi ( )] i 0
162
n
n
i 0
i 0
[ i Fi ( )][ i i Fi ( )]]
re-
(39)
163
Collecting like powers of , it is easy to obtain the governing recursive structure as:
165
F0 F0 F0 F0 2 0
166
F0 (0) 0, F0(0) 1, F0( ) 0
(41)
167
F1 F0 F1 F1F0 2F0 F1 F0 F1 F0F1
(42)
168
F1 (0) 0 F1(0) F1( ) 0
(43)
169
Fm F0 Fm (2 m )F0Fm (1 m )F0Fm m ,0
(44)
170
m ,0 FkFm k FkFm k kFkFm k kFk Fmk , m 2
(45)
171
Fm (0) 0 Fm (0) Fm ( ) 0
(46)
172
In order to determine (X ) X and further to prevent eigen-solutions for the 1st order
173
equation, it is used:
m 1
174
m 1
m 1
m 1
k 1
k 1
k 1
(40)
Jo
k 1
urn a
lP
164
1 3 2 0 [U YU dY ]dY 6 X 1
(47)
8
Journal Pre-proof Having (X , 0) 0 , one is able to write:
176
U
2
0
177
1 dY X 3 1 6
(48)
This gives:
pro of
175
2 F0 F0 d [F1 F0 2F0F0F1]d [F0 F1 2F0F0F2 2F1F0F1 F2 F0 ]d 2
0
178
2
0
1 3 [... ]d ... X 6 0
0
3
180
The solution for the zeroth order reads:
181
F0 ( ) 1 e F0 F0 d
2
182
Therefore, it can be assumed:
183
X
184
This yields:
185
3
[ F F 1
0
2
2F0 F0F1]d 1
0
urn a
3
lP
0
1 6
re-
(49)
179
2
2
(50)
(51)
(52)
The 1st order equation with condition (52) was solved by RKF45; and the main quantity of
187
interest was determined as:
188
2F 49 (X , 0) 1 (1 )X 2 125
189
Eventually:
Jo
186
3
190
U 49 (X , 0) X (1 )X Y 125
191
5‐ Numerical Solution
192
...
2
...
(53)
(54)
In order to conduct a numerical analysis, a composite transformation is used as: 9
Journal Pre-proof
193
1
3
3 4
3 4
X 4 (1 X ) 4 f (X ,ˆ )
ˆ X
(55)
(1 X ) Y
The above composite transformation has the properties of the foregoing perturbative analysis
195
at the leading edge and far away from it; besides, it will be shown soon how the above
196
composite form allows one to eliminate singularity from the transformed non-similar equation
197
by introducing a suitable mapping function.
198
With this, one reaches:
199
3f 1 4X 1 2 f 1 1 2X f 2 f 2 f f 2 f { } f { }( ) X ( ) ˆ 3 4 X 1 ˆ 2 2 X 1 ˆ ˆ ˆ X X ˆ 2
200
f (X , 0)
201
Moreover, the quantities of interest are:
202
U X
203
X
3
(1 X ) 2
f ˆ
9 U 2f (1 X ) 4 Y ˆ 2
re-
1 2
3 3 f f (X ,ˆ ) 0, (X , 0) X 2 (1 X ) 2 ˆ ˆ
lP
5 4
pro of
194
204
1 3 1 1 V {( X 4 (X 1) 4 X 4 4
205
It is introduced a new variable as:
206
ˆ
207
With this, Eq. 56 and its boundary conditions become:
3
(X 1) 4 )f (
urn a
Jo
X X 1
3 4
3 1 f 3 f (X 2 X ) 1ˆ ) (X 1) 4 X 4 } X 4 ˆ
208
2 2 2 3f 1 ˆ} f f 1 {1 3ˆ}( f )2 ˆ(1 ˆ) ( f f f f ) {1 3 ˆ 3 4 ˆ 2 2 ˆ ˆ ˆ ˆ ˆ ˆ 2
209
3 f ˆ f ˆ ˆ ˆ ˆ ( , ) 0, ( , 0) 2 f ( , 0) ˆ ˆ
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
10
Journal Pre-proof 210
The quantities of interest become:
211
ˆ 2 f U 1 ˆ ˆ
212
3 1 1 3 3 1 f ˆ 4 f } V {( ˆ 4 ˆ 4 )f ( (1 ˆ) 2 ˆ (2 ˆ ) ) 4 4 4 ˆ ˆ 1 ˆ ˆ
213
Obviously, 0 X gives 0 ˆ 1 . From Eq. 62, it now becomes vivid that ˆ 0 retains the
214
pure wall jet flow and ˆ 1 is linked to the linearly stretching sheet flow.
215
5‐1 Numerical Algorithm
216
Here, Eq. 62 is solved numerically employing quasi- linearization technique combined with an
217
implicit algorithm. It is initially assumed:
218
f ˆ F f F d ˆ
1
(64)
ˆ
0
re-
f w 0
pro of
1
(65)
(66)
The quasi-linearization consists:
220
FFˆ F n 1Fˆn F n Fˆn 1 F n Fˆn
(67)
221
F 2 2F n 1F n F n F n
(68)
222
Using Eq. 66-68 into Eq. 62 one obtains:
223
n 1 n n 1 F 2n F n 1 3n Fˆn 1 4n ˆ ˆ 1 Fˆ
224
1n (1 3ˆ) f
225
2n (1 3ˆ) F n ˆ(1 ˆ) Fˆn
(71)
226
3n ˆ(1 ˆ) F n
(72)
227
4n ˆ(1 ˆ) F n Fˆn (1 3ˆ) ( F n ) 2
urn a n
ˆ(1 ˆ) f ˆn
Jo
1 4
lP
219
1 2
(69)
(70)
(73)
11
Journal Pre-proof
228
The derivatives in ˆ and ˆ directions are discretized using forward and backward schemes
229
respectively and the tridiagonal system becomes:
230
a j F n 1 (i , j ) b j F n 1 (i , j 1) c j F n 1 (i , j 2) d j
231
a j 1 X 1n (ˆ ) X 2n (ˆ ) 2 X 3n
232
b j 2 X 1n (ˆ )
233
cj 1
234
d j X 4n (ˆ ) 2 X 3n
235
A schematic of the numerical domain is sketched in Fig. 2.
pro of
(ˆ ) 2 ˆ
(74)
(76) (77)
(78)
237
Fig. 2 Schematic of the Numerical Domain: 0 ˆ L , 0 ˆ W , 1 i n , 1 j m
Jo
236
urn a
lP
re-
(ˆ ) 2 n 1 F (i 1, j ) ˆ
(75)
238
The involved boundary conditions are:
239
F (i , m ) f (i ,1) 0, F (i ,1) {ˆ(i ,1)}2
240
The above tridiagonal system was coded in MATLAB employing tridiagonal matrix algorithm
241
(TDMA). It was obtained convergent solutions by predefining a suitable initial solution as well
3
(79)
12
Journal Pre-proof 242
as a compatible updating procedure. It was checked that the grid dependency of the numerical
243
solution reasonably vanishes if n 50, m 600 ; however, in order to approach highly accurate
244
solution, a web of 400 2000 was considered and the presented results are based on this mesh.
245
Moreover, the convergence criterion was defined as:
246
ERR MAX {ABS {(Fˆn 1 Fˆn )}} 10 6
pro of
247
(80)
6‐ Results and Discussion
248
In this section, it is initially compared the results by the perturbation analysis with that
249
quantified numerically for the main quantity of interest. Fig. 3 shows X
250
obtained perturbatively and numerically. More precisely, this figure compares Eq. 31 with Eq.
251
59, scaled to ˆ coordinate. It is clear that the radius of convergence is very small. It should be
252
pointed out that Eq. 31 is the perturbation result, truncated at the 5th order. Fig. 4 compares
253
U (X , 0) (from Eq. 59) with Eq. 54 (the same quantity, obtained perturbatively for large 'X'). Y
254
From this figure, it seems that the radius of convergence is larger, compared with that obtained
255
from small 'X' perturbation analysis. It should be considered that Eq. 54 represents a 1st order
256
perturbation analysis.
257
Fig. 5 shows some plots from
258
component from the leading edge to downstream. It is observed that the pure jet flow gradually
259
becomes the pure stretching sheet flow as it goes downstream. In this respect, the transition
260
region is the linkage between the 2 similarity flows.
261
Obviously, there exists a turning point within the transition region; this is to be dealt later. Fig. 6
262
shows the above quantity (
U (X , 0) in ˆ scale Y
lP
re-
5 4
urn a
f ˆ ( ,ˆ ) . This figure reveals the behavior of the main velocity ˆ
ˆ
f ˆ ( ,ˆ ) ) in contour form for the entire spatial region. Fig. 7 ˆ
1 2
f in ˆ scale, excluded, near the similarity regions for a better 1 ˆ ˆ
exhibits U
264
representation of the flow behavior.
265
Fig. 8 shows f (ˆ,ˆ ) for several ˆ stages. This quantity is linked to the vertical velocity
266
component (see Eq. 65). From this figure, it becomes clear that the quantity f (ˆ,ˆ ) shows
Jo
263
13
Journal Pre-proof a non-monotonic behavior as ˆ increases. Interestingly, it was checked that the inward velocity,
268
into the boundary layer, decreases, approaching a minimum level and then slightly increases to
269
become that, as featured by the Crane flow far away from the leading edge; however, it should
270
be pointed out that the velocity far away from the wall is inward everywhere (see Fig. 9 and 10).
271
From Fig. 10 it becomes clear that close to the leading edge the inward velocity is much stronger
272
compared with that, at the downstream. Mathematically, the flow characteristics are indefinite
273
at the leading edge. That is to say that the jet origin is singular in a mathematical point of view;
274
however, as the flow is convected downstream, the inward velocity decreases and then becomes
275
slightly stronger for the far field adjustment. In other words, the mass conservation mechanism
276
is to decrease the inward velocity into a minimum level and to enhance it afterwards as to
277
become compatible with the far field condition.
278
Fig. 11 betrays the quantity
279
Fig. 13 and 14 indicate the main quantity of interest
280
respectively. From Fig. 13 it was found that the turning point (the critical ˆ coordinate in which
281
the shear stress at the wall becomes zero) is reasonably close to ˆcr .
282
X cr .
283
Retrieving the dimensional coordinate, the above information is linked to:
x cr .
lP
re-
2f ˆ ( ,ˆ ) . Fig. 12 also shows the same quantity in contour form. ˆ 2
13 23
urn a
284
pro of
267
13 13 23 13 a F 23
2f ˆ ( , 0) as a function of ˆ and log 10 (X ) ˆ 2
13 . This gives: 36 (81)
(82)
1 2 It should be pointed out that F may be written as: F U jet Q jet ; U jet is the jet average 2
286
velocity at the leading edge (the discharge velocity) and Q jet is the volumetric flow rate
287
from the slit.
288
A highly accurate formulation was derived for prediction of the main quantity of
289
interest. The relation is expressed as:
Jo
285
14
Journal Pre-proof 3
3
290
40 4 7 51 28 40 4 7 51 28 2f ˆ ( , 0) ( 1) ˆ ˆ 2 ˆ3 ˆ 4 ; 0 ˆ 1 2 ˆ 72 81 44 59 72 81 44 59
291
Fig. 15 shows the error spectrum of the above formulation with that obtained
292
numerically. It is readily seen that almost in the entire region, the error barely exceeds
293
10 3 . Eventually, for the shear stress at the wall, valid for the entire region ( 0 X ),
294
it is developed:
295
U (X , 0) X Y
9 4
pro of
5 4
3
3
40 4 7 51 28 40 4 X 7 X 2 51 X 3 28 X 4 (1 X ) { ( 1) ( ) ( ) ( )} 72 81 44 59 72 X 1 81 X 1 44 X 1 59 X 1 (84)
298
Fig. 3 X
5 4
U (X , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) Y
Jo
297
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re-
296
(83)
15
Journal Pre-proof
re-
pro of
299
U (X , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) Y
urn a
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Fig. 4
Jo
300
16
Journal Pre-proof
re-
pro of
301
f ˆ ( ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ ˆ
urn a
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Fig. 5
Jo
302
17
Journal Pre-proof
re-
pro of
303
f ˆ ( ,ˆ ) as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) ˆ
urn a
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Fig. 6
Jo
304
18
Journal Pre-proof
re-
pro of
305 1 2
f as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) 1 ˆ ˆ
urn a
lP
Fig. 7 U
Jo
306
ˆ
19
Journal Pre-proof
re-
pro of
307
urn a
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Fig. 8 f (ˆ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ
Jo
308
20
Journal Pre-proof
re-
pro of
309
Fig. 9 Vertical velocity component, as denoted via Eq. 65, (X-Coordinate) as a function of ˆ (Y-
311
Coordinate) in different ˆ
Jo
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310
21
Journal Pre-proof
re-
pro of
312
Fig. 10 Vertical velocity component, as denoted via Eq. 65, as a function of ˆ (X-Coordinate) and
314
ˆ (Y-Coordinate)
Jo
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313
22
Journal Pre-proof
re-
pro of
315
2f ˆ ( ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ ˆ 2
urn a
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Fig. 11
Jo
316
23
Journal Pre-proof
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pro of
317 Fig. 12
2f ˆ ( ,ˆ ) as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) ˆ 2
lP
318
Jo
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319
24
Journal Pre-proof
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pro of
320
2f ˆ ( , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) ˆ 2
urn a
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Fig. 13
Jo
321
25
Journal Pre-proof
re-
pro of
322
2f (X , 0) (Y-Coordinate) as a function of log10 (X ) (X-Coordinate) ˆ 2
urn a
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Fig. 14
Jo
323
26
Journal Pre-proof
324 Fig. 15
326
7‐ Conclusion
Eq .75
2f ˆ ( , 0) ˆ 2
Num .
(Y-Coordinate) as a function of ˆ (X-Coordinate)
lP
2f ˆ ( , 0) ˆ 2
325
re-
pro of
In the present work, the linearly stretching wall jet was studied. Near the similarity regions, it
328
was obtained series solutions; however, the perturbation solutions close to the leading edge
329
were divergent with a small radius of convergence. The pure numerical solution was followed
330
by a composite transformation and an additional variable change. Employing quasi-linearization
331
technique combined with an implicit algorithm, convergent solutions were reached. It was
332
exhibited behavior of the flow through plotted graphs. A critical point was marked, that the
333
shear stress at the wall becomes zero. This universal location was quantified in a highly
334
accurate manner and further a formulation was proposed for accurate prediction of the shear
335
stress at the wall which is valid for the entire surface.
336
It is hopeful that this work inspires further analysis in this field including stability analysis,
337
thermal characteristics, nanofluids flows, non-linearly stretching wall jet flows and also with
338
considerations such as magnetohydrodynamics, porosity, slip boundary, suction/injection and
339
etc.
340
Author Contribution
Jo
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327
27
Journal Pre-proof Amin Jafarimoghaddam (an independent researcher) is the sole author of the present work.
342
Declaration of Conflict of Interest
343
No conflict of interest of any type exists.
344
References
345
[1] Akatnow, N. I., Polytechn. Inst. Maschgis., 5 (1953) 24
346 347 348 349 350 351 352 353 354
[2] M.B. Glauert, The wall jet, J. Fluid Mech. 1 (1956) 625–643.
pro of
341
[3] J.H. Merkin, D.J. Needham, A note on the wall-jet problem, J. Eng. Math. 20 (1986) 21–26. [4] D.J. Needham, J.H. Merkin, A note on the wall-jet problem, II, J. Eng. Math. 21 (1987) 17–22.
re-
[5] WILLIAM H. SCHWARZ and BRUCE CASWELL, Some heat transfer characteristics of the two-dimensional laminar incompressible wall jet, Chemical Engineering Science, 1961, Vol. 16, pp. 338-391.
355 356 357 358 359
[6] J. L. Bansal, S. S. Tak, Approximate Solutions of Heat and Momentum Transfer in Laminar Plane wall Jet, Appl. Sci. Res 34 (1978) 299-312.
360 361
[8] N. Sandeep, I.L. Animasaun, Heat transfer in wall jet flow of magnetic-nanofluids with variable magnetic field, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.12.019.
362 363 364
[9] S.Z. Ali Zaidi, S.T. Mohyud‐Din, Convective heat transfer and MHD effects on two dimensional wall jet flow of a nanofluid with passive control model, Aerosp. Sci. Technol. (2015), http://dx.doi.org/10.1016/j.ast.2015.12.008.
365 366 367 368
[10] Syed Zulfiqar Ali Zaidi, Syed Tauseef Mohyud‐din, Bandar Bin-Mohsen, (2017) "A comparative study of wall jet flow containing carbon nanotubes with convective heat transfer and MHD", Engineering Computations, Vol. 34 Issue: 3, pp.739-753, https://doi.org/10.1108/EC-03-2016-0087
369 370
[11] S.T. Mohyud‐Din & S.Z.A. Zaidi, Neural Comput & Applic (2016). doi:10.1007/s00521016-2366-9
371 372
[12] Amin Jafarimoghaddam, Two-phase modeling of magnetic nanofluids jets over a Stretching/shrinking wall, Thermal Science and Engineering Progress 8 (2018) 375–384
373 374
[13] Amin Jafarimoghaddam, The magnetohydrodynamic wall jets: Techniques for rendering
375
similar and perturbative non-similar solutions, European Journal of Mechanics / B Fluids 75
376
(2019) 44–57
Jo
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[7] M. Turkyilmazoglu, Flow of nanofluid plane wall jet and heat transfer, European Journal of Mechanics B/Fluids (2016), http://dx.doi.org/10.1016/j.euromechflu.2016.04.007
28
Journal Pre-proof [14] Amin Jafarimoghaddam, Closed form analytic solutions to heat and mass transfer characteristics of wall jet flow of nanofluids, Thermal Science and Engineering Progress 4 (2017) 175–184
380 381 382 383
[15] Amin Jafarimoghaddam, Ioan Pop, (2018) "Numerical modeling of Glauert type exponentially decaying wall jet flows of nanofluids using Tiwari and Das’ nanofluid model", International Journal of Numerical Methods for Heat & Fluid Flow, https://doi.org/10.1108/HFF-08-2018-0437
384 385 386
[16] Amin Jafarimoghaddam, Fatemeh Shafizadeh, Numerical modeling and spatial stability analysis of the wall jet flow of nanofluids with thermophoresis and brownian effects, Propulsion and Power Research 2019;8(3):210-220
387 388 389
[17] Amin Jafarimoghaddam, Wall jet flows of Glauert type: Heat transfer characteristics and the thermal instabilities in analytic closed forms, European Journal of Mechanics / B Fluids 71 (2018) 77–91
390
[18] T. Mahmood, A Laminar Wall jet on a Moving Wall, Aeta Mechanica 71, 51--60 (1988)
391 392
[19] Fukusako, S. Laminar Wall Jet with Blowing or Suction, Journal of Spacecraft and Rockets, 7, 91-92 (1970).
393 394
[20] Gorla, R.S.R., 1996. Nonsimilar solutions for heat transfer in wall jet flows. Chemical Engineering Communications 140, 139–156.
395 396
[21] Asterios Pantokratoras, The nonsimilar laminar wall jet with uniform blowing or suction:New results, Mechanics Research Communications 36 (2009) 747–753
397 398 399
[22] Amin Jafarimoghaddam, I. Pop and J.H. Merkin, On the Propagation of the Non-Similar Wall Jet Flows with Suction/Injection, Eur. Phys. J. Plus (2019) 134: 215, DOI 10.1140/epjp/i2019-12647-5
400 401
[23] J.L. BANSAL, M. L. GUPTA, On the Hydromagnetie Laminar Plane-Wall Jet, IL NUOVO CIMENTO, 1978, VOL. 50 B, N. 2
402 403
[24] Jafarimoghaddam, A. Arab J Sci Eng (2019). https://doi.org/10.1007/s13369-019-03859x
406 407 408 409
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405
Jo
404
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377 378 379
410 411 29
Journal Pre-proof 412 Conflict of Interest
414
The author declares NO conflict of interest of any kind within this submission.
415
Best Regards,
416
Amin Jafarimoghaddam, the author
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413
417
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418
30
PII: DOI: Reference:
S0997-7546(19)30433-9 https://doi.org/10.1016/j.euromechflu.2019.12.001 EJMFLU 103574
To appear in:
European Journal of Mechanics / B Fluids
Received date : 25 July 2019 Revised date : 27 November 2019 Accepted date : 3 December 2019 Please cite this article as: A. Jafarimoghaddam, The linearly stretching wall jet, European Journal of Mechanics / B Fluids (2019), doi: https://doi.org/10.1016/j.euromechflu.2019.12.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Masson SAS.
Journal Pre-proof
The Linearly Stretching Wall Jet
2
Amin Jafarimoghaddam
3
Independent Researcher, Tehran, Iran
4
Previously at the Department of Aerospace Engineering, K. N. Toosi University of Technology,
5
Tehran, Iran
6
Email Address: [email protected]; Phone No.: +98 935 665 1957
pro of
1
Abstract
8
It is constructed a set-up comprising a jet from a slit at the leading edge, discharged over a
9
linearly stretching wall. The non-similar flow can be interpreted as a combination of two
10
distinct similarity regions; Akatnow-Glauert flow at the leading edge and Crane flow far away
11
from it. In this respect, it is employed appropriate coordinate expansions to explore
12
perturbatively the behavior of the flow near the similarity regions. A suitable composite
13
transformation amalgamated with an abridgement of the stream-wise coordinate, facilitated an
14
immaculate numerical simulation of the involved nonlinear partial differential equation over
15
the entire spatial domain ( 0 X , 0 ˆ ) followed by quasi-linearization technique
16
together with an implicit algorithm of a tridiagonal form. As a result, shear stress at the wall is
17
accurately predicted through a proposed formulation, valid all the way along the wall. It is also
18
exhibited that there exists a transition region with a critical coordinate in the stream-wise
19
direction in which the shear stress at the wall becomes zero. This universal coordinate (namely,
20
the turning point) is determined as, reasonably close to, X
21
coordinate measuring distance along the wall and ˆ is the dimensionless non-similarity
22
variable).
23
Keywords: The Stretching Wall Jet; Perturbation Analysis; Numerical Solution; Wall Jet;
24
Stretching Sheet; Non-Similar Flow
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. r c
13 ( X is a dimensionless 23
1‐ Introduction
Jo
25
re-
7
26
The wall jet flow appears in many industrial processes. Knowledge
of
the wall
jet
27
characteristics such as the rate of spread and the decay rate of the maximum velocity can
28
potentially extend the life of gas turbines, provide greater maneuverability of aircraft and
29
increase the efficiency of the automobile defrosters. It seems that a wall jet set-up has been
30
initially considered by Akatnow [1] (in Russian). Later, independently, Glauert [2] considered a 1
Journal Pre-proof comprehensive wall jet set-up including radial and plane wall jets in laminar or turbulent
32
regimes. He was able to solve the pure wall jet flow in similarity terms; a pure wall jet flow is
33
that with stationary and impermeable wall.
34
It was derived an integral condition, namely the Flux of Exterior Momentum Flux for the pure
35
wall jet flow. Later on, in many other subsequent papers, researchers tried to extend the work
36
by Glauert; e.g., it was analyzed that a suction rate together with a moving wall condition could
37
also result in the preservation of the Glauert invariant [3, 4].
38
Thermal characteristics of the Glauert flow were initially studied in [5]. Some approximate
39
solutions were developed for the energy equation associated with the pure wall jet with heat
40
dissipation [6]. In [7], for an adiabatic surface and also an isothermal set-up, some similarity
41
solutions for the energy equation were derived in the context of nanofluids. The
42
Magnetohydrodynamics (MHD) wall jets in similarity terms were analyzed in several resources
43
such as [8- 11]. The MHD wall jet flows in the context of nanofluids were initially explored by
44
the present author in [12]. In that paper, it was briefly shown that it is NOT possible to study the
45
MHD wall jets in similarity terms taking into account the Glauert constraint if the wall is fixed
46
and no transpiration velocity is allowed (only a specific space dependent magnetic field is a
47
volunteer for the similarity reduction in the MHD wall jet; however, even with this scheme, the
48
Glauert integral condition is NOT preserved if the wall is stationary and impermeable).
49
For the 1st time, Jafarimoghaddam carried out an in-depth analysis of the MHD wall jets and in
50
particular, it was shown techniques for rendering similar and non-similar solutions [13].
51
Specifically, it was shown that the MHD wall jets (with uniform and non-uniform magnetic
52
fields) with the wall being stationary and impermeable are inherently non-similar flows (the
53
specific space dependent magnetic field mentioned earlier, stands as a singular case in a regular
54
non-similar transformation). Other similarity solutions for the wall jet flows subject to various
55
conditions may be tracked through [14- 17]; e.g., in [17] the present author has presented
56
closed form solutions for the energy equation associated with the Glauert type wall jets with
57
heat dissipation and subject to various thermal conditions, those may admit a similarity
58
reduction and specifically, it was shown that that there is NO physically-valid similarity solution
59
for the energy equation associated with the pure wall flow with heat dissipation. More precisely,
60
it was shown that for the original Glauert case (the pure wall jet flow), a surface with a
61
prescribed wall temperature in the form of Tw (x ) mx
62
the prescribing parameters below
63
instabilities implying that the imposed singularity is not physically interpretable.
Jo
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pro of
31
1 4
T is the Induced Heat Shield where
1 would result in the appearance of spectrums of thermal 4
2
Journal Pre-proof The non-similar wall jet flows have been also studied in some cases; e.g. the jet over a moving
65
wall with a uniform speed [18], the jet over a wall with suction/injection [19-22] and the MHD
66
wall jet (with a uniform transverse magnetic field ) over a fixed and impermeable wall [23, 24].
67
In the present work, it is studied the non-similar linearly stretching wall jet flow seemingly for
68
the 1st time. A schematic is sketched in Fig. 1.
re-
pro of
64
69
2‐ Governing Equations
71
For 2D, incompressible and steady state boundary layer flows, the governing equations are:
u v 0 x y
urn a
73
lP
Fig. 1 Schematic of the Problem
70
72
u u 2u v 2 x y y
(1)
u
75
In above, is kinematic viscosity. In addition, u and v are tangential and normal velocity
76
components respectively.
77
For a jet discharged from a slit, located at the leading edge, over a linearly stretching wall, the
78
boundary conditions are:
79
u ( x , 0) ax , v ( x , 0) 0, u ( x , y ) 0
80
In above, a is a positive factor.
81
Following the previous works by the present author e.g. [24] it is introduced:
Jo
74
(2)
(3)
3
Journal Pre-proof
F 12 F 1 F ) , y ( 3 ) 2 Y , x X U ref . U ref . U ref2 .
82
(
83
Here, F is the actual/dimensional Flux of Exterior Momentum Flux (measured at the
84
leading edge) and U ref . is an arbitrary reference velocity to be specified later. For the
85
pure wall jet flow, the reference velocity would be the average velocity at the jet slit.
86
Physically, it is notable that the above transformation already contains the discharge
87
velocity in the term F and hence, U ref . can be chosen in such a way to drop the
88
stretching factor. For this purpose it is also defined:
89
U ref . (
90
Using 1-5, one gets:
91
U V 0 X Y
92
U
93
Along with:
94
U ( X , 0) X , V (X , 0) 0, U (X ,Y ) 0
pro of
(5)
re-
U U 2U V X Y Y 2
lP
1
)3
urn a
95
aF
(4)
(6)
(7)
(8)
3‐ Perturbation Solution: Small 'X' Analysis
96
Near the leading edge, the flow is essentially that given by Glauert [2]. This suggests:
97
(X ,Y ) X 4 F (X , ), X
1
U X
99
U X Y
100
F
Jo
98
1 2
5 4
2F 2
3 4
Y
(9)
(10)
(11)
This delivers:
4
Journal Pre-proof
101
3F 1 2 F 1 F 2 F 2 F F 2 F F ( ) X ( ) 3 4 2 2 X X 2
102
F (X , 0) 0,
103
The perturbation solution should be constructed around the Glauert similarity origin
104
(leading edge). This implies:
105
F (X , ) F0 ( ) F1 ( ) 2 F2 ( ) 3F3 ( ) ... i Fi ( )
3 F F (X , 0) X 2 , (X , ) 0
(12)
pro of
(13)
n
(14)
i 0
3 2
X
107
Upon substitution and simplification one gets:
108
(15)
re-
106
n n 1 n 3 n 1 n 3 n [ i Fi ( )]2 [ i Fi ( )][ i i Fi ( )] [ i Fi ( ) i i Fi ( )][ i Fi ( )] 2 i 0 2 i 0 4 i 0 2 i 0 i 0 i 0 n
[ i Fi ( )]
lP
i 0
(16)
109
Fi (0) Fi ( ) 0, i 0,1, 2,... F1(0) 1, Fi (0) 0, i 0, 2,...
111
Collecting like powers of , it is easy to obtain the governing recursive structure as:
112
1 1 F0 F0 F0 F0 2 0 4 2
(18)
113
1 1 7 F1 F0 F1 F0F1 F0F1 0 4 2 4
(19)
114
1 3 1 3 Fm F0 Fm (1 m )F0Fm ( m )F0Fm m ,0 4 2 4 2
(20)
115
m ,0
116
Here, it is solved the 1st order equation analytically; whilst, the rest will be tackled numerically.
Jo
urn a
110
1 m 1 1 m 1 3 m 1 3 m 1 F F F F kF F kFk Fmk , m 2 k m k 4 k m k 2 k m k 2 k 1 2 k 1 k 1 k 1
(17)
(21)
5
Journal Pre-proof 117
In order to solve the 1st order equation, by differentiating from the zeroth order equation one
118
gets:
119
5 1 F0iv F0F0 F0 F0 0 4 4
(22)
Now, it is easy to note that the 1st order equation is convertible to the above equation by:
121
F1 F0
122
From the boundary conditions:
123
124
This gives:
125
F1 (0) 0
126
3‐1 The Conserved Quantity and the Zeroth Order Solution
127
The momentum equation can be written in the following integral form (see e.g. [22]):
128
X
129
HavingU ( X , 0) X , V ( X , 0) 0 :
130
X
131
The above integral is the so-called Flux of Exterior Momentum Flux and the constant F , as
132
brought into consideration earlier, should be that, connected to X 0 (the zeroth order
133
equation, m 0 ). Besides, the specific dimensionless variables (4) give:
134
F [u (0, y ) u 2 (0, y ) dy ]dy F [U (0,Y ) U 2 (0,Y ) dY ]dY [U (0,Y ) U 2 (0,Y ) dY ]dY
(24)
1 2 2 2 0 [U YU dY ] dY 2U w V w 0 U dY
(26)
urn a
1 2 2 0 [U YU dY ]dY 2 X 0
Jo
0
136
(23)
(25)
lP
re-
1 F0 (0)
135
pro of
120
y
(27)
0
Y
0
Y
(26)
Therefore, the solutions for the zeroth order term F0 ( ) is that given in [3, 4] as:
6
1
Journal Pre-proof
137
2 2
1
log[
(F0 ( ) F0 ( ) 2 2 ) 1 2 2
( F0 ( ) )
1
2 3 F0 ( ) 2
1
] 2 3 tan [ ], 40 8 1 1 (2F0 ( ) 2 ) 1
(29)
6
138
F0(0)
139
3‐2 Numerical Solution for Higher Orders
140
The equations for m 2 are nonhomogeneous linear equations and were discretized
141
employing Runge-Kutta-Fehlberg (RKF45). In order to ease the solution procedure for the linear
142
equations, and principally to avoid shooting technique, it was followed a procedure described in
143
[24].
144
Eventually, for the main quantity of interest it was obtained:
145
2F 40 4 4 2 ( X , 0) (2+ ) X 3 (6+ ) X 2 72 19 5
146
A truncation up to the 5th order revealed a very good accuracy for this quantity up to
147
X
148
divergent, giving rise to a very slow coverage speed (a very small radius of convergence).
(30)
9/2
(32+
63 19 ) X 6 (196+ )X 320 942
15/2
...
(31)
7 ; however, we also checked that the solutions of the linear equations were 50
lP
149
3 2
re-
3
pro of
72
4‐ Perturbation Solution: Large 'X' Analysis Far away from the leading edge, the jet effects vanish and the flow is essentially governed by the
151
stretching wall. This implies the following transformation:
152
(X ,Y ) XF (X , ), Y
153
U X
154
U 2F X Y 2
155
Upon substitution:
156
3F 2 F F 2 F 2 F F 2 F F X ( ) ( ) 3 2 X X 2
(32)
(33)
Jo
F
urn a
150
(34)
(35)
7
Journal Pre-proof
F F (X , 0) 1, (X , ) 0
157
F ( X , 0) 0,
158
It is assumed:
159
F (X , ) F0 ( ) F1 ( ) 2 F2 ( ) 3F3 ( ) ... i Fi ( )
(36)
n
(37)
160
(X ) X
161
This gives:
n
n
n
n
i 0
i 0
i 0
pro of
i 0
n
n
i 0
i 0
(38)
[ i Fi ( )] [ i Fi ( )][ i Fi ( )] [ i Fi ( )]2 [[ i Fi ( )][ i i Fi ( )] i 0
162
n
n
i 0
i 0
[ i Fi ( )][ i i Fi ( )]]
re-
(39)
163
Collecting like powers of , it is easy to obtain the governing recursive structure as:
165
F0 F0 F0 F0 2 0
166
F0 (0) 0, F0(0) 1, F0( ) 0
(41)
167
F1 F0 F1 F1F0 2F0 F1 F0 F1 F0F1
(42)
168
F1 (0) 0 F1(0) F1( ) 0
(43)
169
Fm F0 Fm (2 m )F0Fm (1 m )F0Fm m ,0
(44)
170
m ,0 FkFm k FkFm k kFkFm k kFk Fmk , m 2
(45)
171
Fm (0) 0 Fm (0) Fm ( ) 0
(46)
172
In order to determine (X ) X and further to prevent eigen-solutions for the 1st order
173
equation, it is used:
m 1
174
m 1
m 1
m 1
k 1
k 1
k 1
(40)
Jo
k 1
urn a
lP
164
1 3 2 0 [U YU dY ]dY 6 X 1
(47)
8
Journal Pre-proof Having (X , 0) 0 , one is able to write:
176
U
2
0
177
1 dY X 3 1 6
(48)
This gives:
pro of
175
2 F0 F0 d [F1 F0 2F0F0F1]d [F0 F1 2F0F0F2 2F1F0F1 F2 F0 ]d 2
0
178
2
0
1 3 [... ]d ... X 6 0
0
3
180
The solution for the zeroth order reads:
181
F0 ( ) 1 e F0 F0 d
2
182
Therefore, it can be assumed:
183
X
184
This yields:
185
3
[ F F 1
0
2
2F0 F0F1]d 1
0
urn a
3
lP
0
1 6
re-
(49)
179
2
2
(50)
(51)
(52)
The 1st order equation with condition (52) was solved by RKF45; and the main quantity of
187
interest was determined as:
188
2F 49 (X , 0) 1 (1 )X 2 125
189
Eventually:
Jo
186
3
190
U 49 (X , 0) X (1 )X Y 125
191
5‐ Numerical Solution
192
...
2
...
(53)
(54)
In order to conduct a numerical analysis, a composite transformation is used as: 9
Journal Pre-proof
193
1
3
3 4
3 4
X 4 (1 X ) 4 f (X ,ˆ )
ˆ X
(55)
(1 X ) Y
The above composite transformation has the properties of the foregoing perturbative analysis
195
at the leading edge and far away from it; besides, it will be shown soon how the above
196
composite form allows one to eliminate singularity from the transformed non-similar equation
197
by introducing a suitable mapping function.
198
With this, one reaches:
199
3f 1 4X 1 2 f 1 1 2X f 2 f 2 f f 2 f { } f { }( ) X ( ) ˆ 3 4 X 1 ˆ 2 2 X 1 ˆ ˆ ˆ X X ˆ 2
200
f (X , 0)
201
Moreover, the quantities of interest are:
202
U X
203
X
3
(1 X ) 2
f ˆ
9 U 2f (1 X ) 4 Y ˆ 2
re-
1 2
3 3 f f (X ,ˆ ) 0, (X , 0) X 2 (1 X ) 2 ˆ ˆ
lP
5 4
pro of
194
204
1 3 1 1 V {( X 4 (X 1) 4 X 4 4
205
It is introduced a new variable as:
206
ˆ
207
With this, Eq. 56 and its boundary conditions become:
3
(X 1) 4 )f (
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X X 1
3 4
3 1 f 3 f (X 2 X ) 1ˆ ) (X 1) 4 X 4 } X 4 ˆ
208
2 2 2 3f 1 ˆ} f f 1 {1 3ˆ}( f )2 ˆ(1 ˆ) ( f f f f ) {1 3 ˆ 3 4 ˆ 2 2 ˆ ˆ ˆ ˆ ˆ ˆ 2
209
3 f ˆ f ˆ ˆ ˆ ˆ ( , ) 0, ( , 0) 2 f ( , 0) ˆ ˆ
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
10
Journal Pre-proof 210
The quantities of interest become:
211
ˆ 2 f U 1 ˆ ˆ
212
3 1 1 3 3 1 f ˆ 4 f } V {( ˆ 4 ˆ 4 )f ( (1 ˆ) 2 ˆ (2 ˆ ) ) 4 4 4 ˆ ˆ 1 ˆ ˆ
213
Obviously, 0 X gives 0 ˆ 1 . From Eq. 62, it now becomes vivid that ˆ 0 retains the
214
pure wall jet flow and ˆ 1 is linked to the linearly stretching sheet flow.
215
5‐1 Numerical Algorithm
216
Here, Eq. 62 is solved numerically employing quasi- linearization technique combined with an
217
implicit algorithm. It is initially assumed:
218
f ˆ F f F d ˆ
1
(64)
ˆ
0
re-
f w 0
pro of
1
(65)
(66)
The quasi-linearization consists:
220
FFˆ F n 1Fˆn F n Fˆn 1 F n Fˆn
(67)
221
F 2 2F n 1F n F n F n
(68)
222
Using Eq. 66-68 into Eq. 62 one obtains:
223
n 1 n n 1 F 2n F n 1 3n Fˆn 1 4n ˆ ˆ 1 Fˆ
224
1n (1 3ˆ) f
225
2n (1 3ˆ) F n ˆ(1 ˆ) Fˆn
(71)
226
3n ˆ(1 ˆ) F n
(72)
227
4n ˆ(1 ˆ) F n Fˆn (1 3ˆ) ( F n ) 2
urn a n
ˆ(1 ˆ) f ˆn
Jo
1 4
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219
1 2
(69)
(70)
(73)
11
Journal Pre-proof
228
The derivatives in ˆ and ˆ directions are discretized using forward and backward schemes
229
respectively and the tridiagonal system becomes:
230
a j F n 1 (i , j ) b j F n 1 (i , j 1) c j F n 1 (i , j 2) d j
231
a j 1 X 1n (ˆ ) X 2n (ˆ ) 2 X 3n
232
b j 2 X 1n (ˆ )
233
cj 1
234
d j X 4n (ˆ ) 2 X 3n
235
A schematic of the numerical domain is sketched in Fig. 2.
pro of
(ˆ ) 2 ˆ
(74)
(76) (77)
(78)
237
Fig. 2 Schematic of the Numerical Domain: 0 ˆ L , 0 ˆ W , 1 i n , 1 j m
Jo
236
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(ˆ ) 2 n 1 F (i 1, j ) ˆ
(75)
238
The involved boundary conditions are:
239
F (i , m ) f (i ,1) 0, F (i ,1) {ˆ(i ,1)}2
240
The above tridiagonal system was coded in MATLAB employing tridiagonal matrix algorithm
241
(TDMA). It was obtained convergent solutions by predefining a suitable initial solution as well
3
(79)
12
Journal Pre-proof 242
as a compatible updating procedure. It was checked that the grid dependency of the numerical
243
solution reasonably vanishes if n 50, m 600 ; however, in order to approach highly accurate
244
solution, a web of 400 2000 was considered and the presented results are based on this mesh.
245
Moreover, the convergence criterion was defined as:
246
ERR MAX {ABS {(Fˆn 1 Fˆn )}} 10 6
pro of
247
(80)
6‐ Results and Discussion
248
In this section, it is initially compared the results by the perturbation analysis with that
249
quantified numerically for the main quantity of interest. Fig. 3 shows X
250
obtained perturbatively and numerically. More precisely, this figure compares Eq. 31 with Eq.
251
59, scaled to ˆ coordinate. It is clear that the radius of convergence is very small. It should be
252
pointed out that Eq. 31 is the perturbation result, truncated at the 5th order. Fig. 4 compares
253
U (X , 0) (from Eq. 59) with Eq. 54 (the same quantity, obtained perturbatively for large 'X'). Y
254
From this figure, it seems that the radius of convergence is larger, compared with that obtained
255
from small 'X' perturbation analysis. It should be considered that Eq. 54 represents a 1st order
256
perturbation analysis.
257
Fig. 5 shows some plots from
258
component from the leading edge to downstream. It is observed that the pure jet flow gradually
259
becomes the pure stretching sheet flow as it goes downstream. In this respect, the transition
260
region is the linkage between the 2 similarity flows.
261
Obviously, there exists a turning point within the transition region; this is to be dealt later. Fig. 6
262
shows the above quantity (
U (X , 0) in ˆ scale Y
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5 4
urn a
f ˆ ( ,ˆ ) . This figure reveals the behavior of the main velocity ˆ
ˆ
f ˆ ( ,ˆ ) ) in contour form for the entire spatial region. Fig. 7 ˆ
1 2
f in ˆ scale, excluded, near the similarity regions for a better 1 ˆ ˆ
exhibits U
264
representation of the flow behavior.
265
Fig. 8 shows f (ˆ,ˆ ) for several ˆ stages. This quantity is linked to the vertical velocity
266
component (see Eq. 65). From this figure, it becomes clear that the quantity f (ˆ,ˆ ) shows
Jo
263
13
Journal Pre-proof a non-monotonic behavior as ˆ increases. Interestingly, it was checked that the inward velocity,
268
into the boundary layer, decreases, approaching a minimum level and then slightly increases to
269
become that, as featured by the Crane flow far away from the leading edge; however, it should
270
be pointed out that the velocity far away from the wall is inward everywhere (see Fig. 9 and 10).
271
From Fig. 10 it becomes clear that close to the leading edge the inward velocity is much stronger
272
compared with that, at the downstream. Mathematically, the flow characteristics are indefinite
273
at the leading edge. That is to say that the jet origin is singular in a mathematical point of view;
274
however, as the flow is convected downstream, the inward velocity decreases and then becomes
275
slightly stronger for the far field adjustment. In other words, the mass conservation mechanism
276
is to decrease the inward velocity into a minimum level and to enhance it afterwards as to
277
become compatible with the far field condition.
278
Fig. 11 betrays the quantity
279
Fig. 13 and 14 indicate the main quantity of interest
280
respectively. From Fig. 13 it was found that the turning point (the critical ˆ coordinate in which
281
the shear stress at the wall becomes zero) is reasonably close to ˆcr .
282
X cr .
283
Retrieving the dimensional coordinate, the above information is linked to:
x cr .
lP
re-
2f ˆ ( ,ˆ ) . Fig. 12 also shows the same quantity in contour form. ˆ 2
13 23
urn a
284
pro of
267
13 13 23 13 a F 23
2f ˆ ( , 0) as a function of ˆ and log 10 (X ) ˆ 2
13 . This gives: 36 (81)
(82)
1 2 It should be pointed out that F may be written as: F U jet Q jet ; U jet is the jet average 2
286
velocity at the leading edge (the discharge velocity) and Q jet is the volumetric flow rate
287
from the slit.
288
A highly accurate formulation was derived for prediction of the main quantity of
289
interest. The relation is expressed as:
Jo
285
14
Journal Pre-proof 3
3
290
40 4 7 51 28 40 4 7 51 28 2f ˆ ( , 0) ( 1) ˆ ˆ 2 ˆ3 ˆ 4 ; 0 ˆ 1 2 ˆ 72 81 44 59 72 81 44 59
291
Fig. 15 shows the error spectrum of the above formulation with that obtained
292
numerically. It is readily seen that almost in the entire region, the error barely exceeds
293
10 3 . Eventually, for the shear stress at the wall, valid for the entire region ( 0 X ),
294
it is developed:
295
U (X , 0) X Y
9 4
pro of
5 4
3
3
40 4 7 51 28 40 4 X 7 X 2 51 X 3 28 X 4 (1 X ) { ( 1) ( ) ( ) ( )} 72 81 44 59 72 X 1 81 X 1 44 X 1 59 X 1 (84)
298
Fig. 3 X
5 4
U (X , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) Y
Jo
297
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re-
296
(83)
15
Journal Pre-proof
re-
pro of
299
U (X , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) Y
urn a
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Fig. 4
Jo
300
16
Journal Pre-proof
re-
pro of
301
f ˆ ( ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ ˆ
urn a
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Fig. 5
Jo
302
17
Journal Pre-proof
re-
pro of
303
f ˆ ( ,ˆ ) as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) ˆ
urn a
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Fig. 6
Jo
304
18
Journal Pre-proof
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pro of
305 1 2
f as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) 1 ˆ ˆ
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Fig. 7 U
Jo
306
ˆ
19
Journal Pre-proof
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pro of
307
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Fig. 8 f (ˆ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ
Jo
308
20
Journal Pre-proof
re-
pro of
309
Fig. 9 Vertical velocity component, as denoted via Eq. 65, (X-Coordinate) as a function of ˆ (Y-
311
Coordinate) in different ˆ
Jo
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310
21
Journal Pre-proof
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pro of
312
Fig. 10 Vertical velocity component, as denoted via Eq. 65, as a function of ˆ (X-Coordinate) and
314
ˆ (Y-Coordinate)
Jo
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313
22
Journal Pre-proof
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pro of
315
2f ˆ ( ,ˆ ) (X-Coordinate) as a function of ˆ (Y-Coordinate) in different ˆ ˆ 2
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Fig. 11
Jo
316
23
Journal Pre-proof
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pro of
317 Fig. 12
2f ˆ ( ,ˆ ) as a function of ˆ (X-Coordinate) and ˆ (Y-Coordinate) ˆ 2
lP
318
Jo
urn a
319
24
Journal Pre-proof
re-
pro of
320
2f ˆ ( , 0) (Y-Coordinate) as a function of ˆ (X-Coordinate) ˆ 2
urn a
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Fig. 13
Jo
321
25
Journal Pre-proof
re-
pro of
322
2f (X , 0) (Y-Coordinate) as a function of log10 (X ) (X-Coordinate) ˆ 2
urn a
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Fig. 14
Jo
323
26
Journal Pre-proof
324 Fig. 15
326
7‐ Conclusion
Eq .75
2f ˆ ( , 0) ˆ 2
Num .
(Y-Coordinate) as a function of ˆ (X-Coordinate)
lP
2f ˆ ( , 0) ˆ 2
325
re-
pro of
In the present work, the linearly stretching wall jet was studied. Near the similarity regions, it
328
was obtained series solutions; however, the perturbation solutions close to the leading edge
329
were divergent with a small radius of convergence. The pure numerical solution was followed
330
by a composite transformation and an additional variable change. Employing quasi-linearization
331
technique combined with an implicit algorithm, convergent solutions were reached. It was
332
exhibited behavior of the flow through plotted graphs. A critical point was marked, that the
333
shear stress at the wall becomes zero. This universal location was quantified in a highly
334
accurate manner and further a formulation was proposed for accurate prediction of the shear
335
stress at the wall which is valid for the entire surface.
336
It is hopeful that this work inspires further analysis in this field including stability analysis,
337
thermal characteristics, nanofluids flows, non-linearly stretching wall jet flows and also with
338
considerations such as magnetohydrodynamics, porosity, slip boundary, suction/injection and
339
etc.
340
Author Contribution
Jo
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327
27
Journal Pre-proof Amin Jafarimoghaddam (an independent researcher) is the sole author of the present work.
342
Declaration of Conflict of Interest
343
No conflict of interest of any type exists.
344
References
345
[1] Akatnow, N. I., Polytechn. Inst. Maschgis., 5 (1953) 24
346 347 348 349 350 351 352 353 354
[2] M.B. Glauert, The wall jet, J. Fluid Mech. 1 (1956) 625–643.
pro of
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[3] J.H. Merkin, D.J. Needham, A note on the wall-jet problem, J. Eng. Math. 20 (1986) 21–26. [4] D.J. Needham, J.H. Merkin, A note on the wall-jet problem, II, J. Eng. Math. 21 (1987) 17–22.
re-
[5] WILLIAM H. SCHWARZ and BRUCE CASWELL, Some heat transfer characteristics of the two-dimensional laminar incompressible wall jet, Chemical Engineering Science, 1961, Vol. 16, pp. 338-391.
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[6] J. L. Bansal, S. S. Tak, Approximate Solutions of Heat and Momentum Transfer in Laminar Plane wall Jet, Appl. Sci. Res 34 (1978) 299-312.
360 361
[8] N. Sandeep, I.L. Animasaun, Heat transfer in wall jet flow of magnetic-nanofluids with variable magnetic field, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2016.12.019.
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[9] S.Z. Ali Zaidi, S.T. Mohyud‐Din, Convective heat transfer and MHD effects on two dimensional wall jet flow of a nanofluid with passive control model, Aerosp. Sci. Technol. (2015), http://dx.doi.org/10.1016/j.ast.2015.12.008.
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[10] Syed Zulfiqar Ali Zaidi, Syed Tauseef Mohyud‐din, Bandar Bin-Mohsen, (2017) "A comparative study of wall jet flow containing carbon nanotubes with convective heat transfer and MHD", Engineering Computations, Vol. 34 Issue: 3, pp.739-753, https://doi.org/10.1108/EC-03-2016-0087
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[11] S.T. Mohyud‐Din & S.Z.A. Zaidi, Neural Comput & Applic (2016). doi:10.1007/s00521016-2366-9
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[12] Amin Jafarimoghaddam, Two-phase modeling of magnetic nanofluids jets over a Stretching/shrinking wall, Thermal Science and Engineering Progress 8 (2018) 375–384
373 374
[13] Amin Jafarimoghaddam, The magnetohydrodynamic wall jets: Techniques for rendering
375
similar and perturbative non-similar solutions, European Journal of Mechanics / B Fluids 75
376
(2019) 44–57
Jo
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[7] M. Turkyilmazoglu, Flow of nanofluid plane wall jet and heat transfer, European Journal of Mechanics B/Fluids (2016), http://dx.doi.org/10.1016/j.euromechflu.2016.04.007
28
Journal Pre-proof [14] Amin Jafarimoghaddam, Closed form analytic solutions to heat and mass transfer characteristics of wall jet flow of nanofluids, Thermal Science and Engineering Progress 4 (2017) 175–184
380 381 382 383
[15] Amin Jafarimoghaddam, Ioan Pop, (2018) "Numerical modeling of Glauert type exponentially decaying wall jet flows of nanofluids using Tiwari and Das’ nanofluid model", International Journal of Numerical Methods for Heat & Fluid Flow, https://doi.org/10.1108/HFF-08-2018-0437
384 385 386
[16] Amin Jafarimoghaddam, Fatemeh Shafizadeh, Numerical modeling and spatial stability analysis of the wall jet flow of nanofluids with thermophoresis and brownian effects, Propulsion and Power Research 2019;8(3):210-220
387 388 389
[17] Amin Jafarimoghaddam, Wall jet flows of Glauert type: Heat transfer characteristics and the thermal instabilities in analytic closed forms, European Journal of Mechanics / B Fluids 71 (2018) 77–91
390
[18] T. Mahmood, A Laminar Wall jet on a Moving Wall, Aeta Mechanica 71, 51--60 (1988)
391 392
[19] Fukusako, S. Laminar Wall Jet with Blowing or Suction, Journal of Spacecraft and Rockets, 7, 91-92 (1970).
393 394
[20] Gorla, R.S.R., 1996. Nonsimilar solutions for heat transfer in wall jet flows. Chemical Engineering Communications 140, 139–156.
395 396
[21] Asterios Pantokratoras, The nonsimilar laminar wall jet with uniform blowing or suction:New results, Mechanics Research Communications 36 (2009) 747–753
397 398 399
[22] Amin Jafarimoghaddam, I. Pop and J.H. Merkin, On the Propagation of the Non-Similar Wall Jet Flows with Suction/Injection, Eur. Phys. J. Plus (2019) 134: 215, DOI 10.1140/epjp/i2019-12647-5
400 401
[23] J.L. BANSAL, M. L. GUPTA, On the Hydromagnetie Laminar Plane-Wall Jet, IL NUOVO CIMENTO, 1978, VOL. 50 B, N. 2
402 403
[24] Jafarimoghaddam, A. Arab J Sci Eng (2019). https://doi.org/10.1007/s13369-019-03859x
406 407 408 409
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405
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404
pro of
377 378 379
410 411 29
Journal Pre-proof 412 Conflict of Interest
414
The author declares NO conflict of interest of any kind within this submission.
415
Best Regards,
416
Amin Jafarimoghaddam, the author
pro of
413
417
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418
30