M Eng (2)

Improvement in drag
reduction in
thermally modulated
channel
August 15, 2016
Mechanical and Materials Engineering

Nomenclature
α – Heating wave number
Ω – Phase shift between upper and lower periodic heating
θuni – Uniform heating, scaled with Tuni
θP, U – Periodic heating, scaled with TP, U
θP, L – Periodic heating, scaled with TP, L
Ra – Rayleigh Number, associated with uniform heating
RaP, U – Upper wall periodic Rayleigh Number
RaP, L – Lower wall periodic Rayleigh Number
Rap – Periodic Rayleigh Number, wherever written indicates equal
periodic Rayleigh Number for both upper and lower walls
Nuav – Average Nusselt Number, indicates vertical heat transfer
Nuh, L – Horizontal Nusselt Number for lower plate
Nuh, U – Horizontal Nusselt Number for upper plate
Improvement in drag reduction
in thermally modulated channel

Problem Formulation
• A horizontal channel on which spatial-
periodic heating and uniform heating along
the upper and lower walls have been applied
to analyze the effect on drag reduction by
using numerical technique.

Problem Formulation
• Industrial know-how for heating the pipes
using electrical wires and controls
Courtesy: M/s Chromolax

Problem Formulation
• Above technique can be used in the
formulated problem by heating the channel in
intervals to achieve similar periodic pattern.
• Create alternate hot and cold spot along the
channel length
Courtesy: M/s Thermon

Background
• Earlier studies on shear drag reduction
techniques:
 By changing the wall topography (Mohammadi
& Floryan, 2013a, 2013b, 2014, 2015; Moradi &
Floryan, 2013)
 Using hydrophobic materials with irregular
wall topography (Rothstein, 2010; Ou et al, 2004;
Ou & Rothstein, 2005; Joseph et al, 2006; Truesdell
et al, 2006; Samaha et al, 2011; Zhou et al, 2011;
Quere, 2008; Reyssat et al, 2008)

Background
 Using spatial heating (Floryan, 2012; Hossain
et al, 2012; Hossain & Floryan, 2013a; Yamato et al,
2013; Hossain & Floryan, 2014; Floryan & Floryan,
2015; Hossain & Floryan, 2015a)

Methodology and
Analysis
 Governing Equations
  1
211
1
0
1
1
10 u
x
p
y
u
v
dy
du
vRe
x
u
uuRe 









  UP,
1
UP,LP,
1
LP,uni
1
1
1
1
211
1
1
10 θPrRaθPrRaθPrRaθPrv
y
p
y
v
v
x
v
uReu 










  1
211UP,
UP,
LP,
LP,
uni
1
1UP,
UP,
LP,
LP,10 θPr
y
θ
y
θ
Ra
y
θ
Ra
dy
dθ
Rav
x
θ
x
θ
Ra
x
θ
RauReu 



























 
0
y
v
x
u 11







Methodology and
Analysis
 Above equations are solved numerically
using MATLAB code (tested)
 Fast Fourier Transform (FFT) is used
 Sinusoidal heating patterns, when applied
with FFT, simplifies discretization and
numerical analysis

Results and Discussion
• Flow topologies
(a) (b) (c)
Figure 1: Flow topologies resulting from the same periodic heating patterns applied at both
walls with RaP,L = RaP,U =1000, Re = 5, α= 2.5, Pr = 0.71 with uniform heating Ra=0 and
phase shift of (a) Ω=0, (b) Ω=π/2 and (c) Ω=π
0.0
0.30.50.9
0.9
0.7
0 0.5 1 1.5 2
-1
0
1
y
x/
 = 0
>
>
>
<
>
>
0>
>
>
>
0.1
0.3
0.5
0.7
0.9
0.9
0.9
0 0.5 1 1.5 2
-1
0
1
y
x/
 = /2
>
>
>
< >
>
0
0
>
>
>
>
>
>
>
>
0.1
0.3
0.4
0.6
0.7
0.9
0.5
0 0.5 1 1.5 2
-1
0
1
y
x/
 = 
>
>
<
>
>
0
0
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>

• Flow topologies
0.0
0.1
0.3
0.5
0.7
0.9
0 0.5 1 1.5 2
-1
0
1
y
x/
 = 0
>
>
>
<
>
>
>
>
>
>
0.1
0.5
0.3
1.0
0.9
0.9
0.9
0.1
0.3
0.5
0.7
0.9
0.9
0 0.5 1 1.5 2
-1
0
1
y
x/
 = /2
>
>
><
>
>
0
0
>
>
>
>
>
>
>
>
0.1
0.3
0.4
0.5
0.6
0.7
0.9
0 0.5 1 1.5 2
-1
0
1
y
x/
 = 
>
>
>
<
>
>
0
0
>
>
>
>
>
>
>
>
>
>
>
>
>
>
(a) (b) (c)
Figure 2: Flow topologies resulting from the same periodic heating patterns applied at
both walls with RaP,L = RaP,U =1000, Re = 5, α= 2.5, Pr = 0.71 with uniform heating Ra=200
and phase shift of (a) Ω=0, (b) Ω=π/2 and (c) Ω=π

• Drag reduction is maximum for zero phase
shift and increases with uniform heating
Figure 3: Variations of the pressure gradient A as a function of the phase shift Ω for
RaP,L = RaP,U = 1000, α = 2.5, Re = 1, Pr = 0.71.
0 50 100 150 200 250 300 350
0
0.2
0.4
0.6
0.8
1
1.2
1.4
 (in degrees)
Pressuregradient
Ra=0
Ra=50
Ra=100
Ra=150
Ra=200

• Drag reduction is increased by 110 % when
uniform heating is increased
(a) (b) (c)
Figure 4 : Variation of Pressure Gradient Correction (A/Re) as a function of the heating
wave number (α) and the flow Reynolds No. (Re) for RaP,L = RaP,U =500 with (a) Ra =0,
(b) Ra =100 and (c) Ra =200
0.05
0.2
0.001
0.01
0.1
0.3
0.35
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

0.1
0.35
0.001
0.01
0.2
0.5
0.05
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

0.1
0.5
0.001
0.01
0.2
0.7
0.3
0.05
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re


• Decreasing growth rate in drag reduction as
heating intensities are increased further on
(a) (b) (c)
Figure 5: Variation of Pressure Gradient Correction (A/Re) as a function of the heating
wave number (α) and the flow Reynolds No. (Re) for RaP,L = RaP,U =1000 with (a) Ra =0,
(b) Ra =100 and (c) Ra =200
0.83
0.001
0.5
0.2
0.1
0.01
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re
 1
0.001
0.05
0.7
0.2
0.01
0.5
1.21 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

0.001
0.05
1
0.7
0.35
0.2
0.1
0.01
1.5
1.3
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re


• Drag reduction effect due to uniform
heating is appreciable at lower Re only
(a) (b)
Figure 6: Variation of Pressure Gradient Correction (A/Re) as a function of the
Reynolds number (Re) and the Uniform Rayleigh Number (Ra) for α=2 with Rayleigh
Number (a) RaP =500 and (b) RaP =1000
0.0001
0.001
0.004
0.02
0.1
0.2
0.35
0.5
0.4
0.3
10
0
10
1
10
210
0
101
10
2
Re
Ra
0.001
0.004
0.02
0.1
0.4
0.6
0.65
0.7
0.2
10
0
10
1
10
210
0
101
10
2
Re
Ra

(a) (b)
Figure 7: Variation of Pressure Gradient Correction (A/Re) as a function of the
Reynolds number (Re) and the Uniform Rayleigh Number (Ra) for α=3 with Rayleigh
Number (a) RaP =500 and (b) RaP =1000
0.0001
0.001
0.004
0.1
0.2
0.3
0.02
0.4
10
0
10
1
10
210
0
101
10
2
Re
Ra
0.001
0.004
0.02
0.3
0.8
1
0.1
1.2
0.6
0.9
10
0
10
1
10
210
0
101
10
2
Re
Ra

• Effect of uniform heating is appreciable for
its higher values and lower Re only
(a) (b) (c)
Figure 8: Variation of Pressure Gradient Correction (A/Re) as a function of the uniform
Rayleigh Number (Ra) and the periodic Rayleigh Number (RaP) for α=2 with (a) Re=1,
(b) Re=5 and (c) Re=10
0.02
0.1
0.004
0.001
0.3
0.6
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
0.004
0.02
0.001
0.1
0.3
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
0.0005
0.003
0.0001
0.02
0.1
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra

(a) (b) (c)
Figure 9: Variation of Pressure Gradient Correction (A/Re) as a function of the uniform
0.02
0.1
0.003
0.3
0.8
0.0005
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
0.003
0.02
0.0005
0.1
0.3
0.0001
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
0.0005
0.003
0.0001
0.02
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra

• Vertical heat transfer Nuav
(a) (b) (c)
Figure 10: Variation of average Nusselt number Nuav as a function of the heating
wave number (α) and the flow Reynolds No. (Re) for RaP,L = RaP,U =500 with (a) Ra=0,
(b) Ra=100 and (c) Ra=200
8
50
2
25
100
135
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

8
50
2
25
150
175
100
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

8
100
2
25
225
240
190
50
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re


• Effect on vertical heat transfer is appreciable
at higher uniform heating and lower Re
(a) (b)
Figure 11: Variation of average Nusselt number Nuav as a function of the Reynolds
number (Re) and the Uniform Rayleigh Number (Ra) for α=2 with periodic Rayleigh
Number (a) RaP =500 and (b) RaP=1000
1
3
8
25
70
150
125
100
10
0
10
1
10
210
0
101
10
2
Re
Ra
3
8
25
70
250
360
320300
150
10
0
10
1
10
210
0
101
10
2
Re
Ra

• Effect on vertical heat transfer is appreciable
(a) (b) (c)
Figure 12: Variation of average Nusselt number Nuav as a function of the uniform
8
25
1
0.2
80
160
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
1
3
0.2
25
100
8
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra
0.2
1
0.05
8
25
3
10
0
10
1
10
210
0
101
102
10
3
RaP
Ra

• Horizontal heat transfer at lower plate Nuh,L
(a) (b) (c)
Figure 13: Variation of horizontal Nusselt number in lower plate, Nuh,L, as a function of
the heating wave number (α) and the flow Reynolds No. (Re) for RaP,=500 with (a) Ra =0,
(b) Ra=100 and (c) Ra=200
600
400
800
500
300
200
50
700
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

600
400
800
500
300
200
50
700
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re

600
400
800
500
300
200
50
700
1 2 3 4 5
10
-2
10-1
100
10
1
10
2
Re


• Effect on horizontal heat transfer is appreciable
(a) (b)
Figure 14: Variation of horizontal Nusselt number in lower plate, Nuh,L, as a function of
the Reynolds number (Re) and the Uniform Rayleigh Number (Ra) for α=2 with
periodic Rayleigh Number (a) RaP =500 and (b) RaP=1000
308
310
320
350
450
425
400
10
0
10
1
10
210
0
101
10
2
Re
Ra
618
625
650
725
1000
950
900
10
0
10
1
10
210
0
101
10
2
Re
Ra

Limitations
• Drag reduction due to heating is observed for
low Reynolds number flow; few applications
such as micro-channels
• Sinusoidal heating pattern is hard to
achieve in practice.
• Errors associated with numerical analysis
are minimized, however cannot be
eliminated completely.

Conclusion
• Drag reduction is increased by 110 % when
uniform heating is added over the periodic
heating
• Most effective wave number for drag
reduction is α~3
• Vertical heat transfer also increases with the
addition of uniform heating
• Net energy saving?

Conclusion
• Scenarios:
 Large pump to overcome larger drag, or
 Small pump with lesser drag (reduced via
application of heating)

M Eng (2)

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