Thermally modulated heating is applied to a horizontal channel to analyze its effect on drag reduction using numerical techniques. Periodic heating patterns are applied to the upper and lower walls using a sinusoidal function. The governing equations are solved using MATLAB. Results show that drag reduction is highest when the phase shift between the upper and lower wall heating is zero. Increasing uniform background heating along the walls also increases drag reduction, up to 110% higher for some cases. However, the improvement in drag reduction due to uniform heating is only significant at lower Reynolds numbers and for higher values of uniform heating. Vertical heat transfer, measured by the average Nusselt number, also increases with background heating.

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

3. 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

4. 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.
Improvement in drag reduction
in thermally modulated channel

5. Problem Formulation
• Industrial know-how for heating the pipes
using electrical wires and controls
Improvement in drag reduction
in thermally modulated channel
Courtesy: M/s Chromolax

6. 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
Improvement in drag reduction
in thermally modulated channel
Courtesy: M/s Thermon

7. 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)
Improvement in drag reduction
in thermally modulated channel

8. 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)
Improvement in drag reduction
in thermally modulated channel

9. Methodology and
Analysis
 Governing Equations
Improvement in drag reduction
in thermally modulated channel
  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







10. 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
Improvement in drag reduction
in thermally modulated channel

11. Results and Discussion
• Flow topologies
Improvement in drag reduction
in thermally modulated channel
(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
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>

12. Results and Discussion
• Flow topologies
Improvement in drag reduction
in thermally modulated channel
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

13. Results and Discussion
• Drag reduction is maximum for zero phase
shift and increases with uniform heating
Improvement in drag reduction
in thermally modulated channel
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

14. Results and Discussion
• Drag reduction is increased by 110 % when
uniform heating is increased
Improvement in drag reduction
in thermally modulated channel
(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


15. Results and Discussion
• Decreasing growth rate in drag reduction as
heating intensities are increased further on
Improvement in drag reduction
in thermally modulated channel
(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


16. Results and Discussion
• Drag reduction effect due to uniform
heating is appreciable at lower Re only
Improvement in drag reduction
in thermally modulated channel
(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

17. Results and Discussion
Improvement in drag reduction
in thermally modulated channel
(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

18. Results and Discussion
• Effect of uniform heating is appreciable for
its higher values and lower Re only
Improvement in drag reduction
in thermally modulated channel
(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

19. Results and Discussion
Improvement in drag reduction
in thermally modulated channel
(a) (b) (c)
Figure 9: Variation of Pressure Gradient Correction (A/Re) as a function of the uniform
Rayleigh Number (Ra) and the periodic Rayleigh Number (RaP) for α=3 with (a) Re=1,
(b) Re=5 and (c) Re=10
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

20. Results and Discussion
• Vertical heat transfer Nuav
Improvement in drag reduction
in thermally modulated channel
(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


21. Results and Discussion
• Effect on vertical heat transfer is appreciable
at higher uniform heating and lower Re
Improvement in drag reduction
in thermally modulated channel
(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

22. Results and Discussion
• Effect on vertical heat transfer is appreciable
at higher uniform heating and lower Re
Improvement in drag reduction
in thermally modulated channel
(a) (b) (c)
Figure 12: Variation of average Nusselt number Nuav 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
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

23. Results and Discussion
• Horizontal heat transfer at lower plate Nuh,L
Improvement in drag reduction
in thermally modulated channel
(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


24. Results and Discussion
• Effect on horizontal heat transfer is appreciable
at higher uniform heating and lower Re
Improvement in drag reduction
in thermally modulated channel
(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

25. 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.
Improvement in drag reduction
in thermally modulated channel

26. 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?
Improvement in drag reduction
in thermally modulated channel

27. Conclusion
• Scenarios:
 Large pump to overcome larger drag, or
 Small pump with lesser drag (reduced via
application of heating)
Improvement in drag reduction
in thermally modulated channel