The document summarizes a study of air flow through a two-dimensional channel with a sudden expansion. It describes the physical system of an air flow with different inlet velocity profiles through a channel that expands suddenly. It presents the mathematical model and governing equations used to model the laminar, incompressible flow. The numerical method and boundary conditions applied in the Fluent software are described. Key results are presented on streamlines, temperature contours, velocity and pressure profiles, and how heat transfer metrics vary with inlet velocity. It is concluded that vortices form after expansion and increase in size with velocity, while heat transfer depends on flow regimes and local conditions.

1. Study of Air Flow Through Two-
dimensional Channel with Sudden
ExpansionAli Jraisheh
164103117
Department of Mechanical
Engineering
Indian Institute of Technology

2. Road
Map
Introduction
Objectives of the work
Physical System
Mathematical Model
Boundary Conditions
Numerical Method Details
Governing Equations
Results and Discussion
Conclusion

3. Introduction
 When the fluid flows over a heated surface, it causes forced convection.
 Forced convection can be seen in many engineering application such as
heat exchangers.
 We can reach high values of heat transfer coefficient, but it requires a fluid
flow equipment such as pump, compressor or blower.
 We need to know the flow field, temperature and heat transfer in the
domain of interest.

4. Physical System
• Air flow in suddenly expanded two-
dimensional channel.
• The narrow part is kept adiabatic.
• The wide part is isothermally heated
to 330° K on the bottom wall and to
300° K on the top wall
• The inlet temperature of the air is
300° K
• The inlet velocity profile is governed
by the equation:
𝑈 = 𝑈0 1.0 −
𝑦
ℎ
2
𝑈0 = 0.01, 0.02, … , 0.1
Schematic of the
problem

5. Mathematical Treatment
Assumptions:
• The flow inside the channel is laminar and incompressible.
• Viscous dissipation is negligibly small.
• Radiation effects are assumed to be negligible.
• The buoyancy term is assumed to be negligible.
• The physical properties are assumed to be constant with respect to
temperature.

6. Governing Equations:
o Continuity Equation
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
= 0
o x-momentum Equation 𝜌 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= −
𝜕𝑝
𝜕𝑥
+ 𝜇(
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2)
o y-momentum Equation 𝜌 𝑢
𝜕𝑣
𝜕𝑥
+ 𝑣
𝜕𝑣
𝜕𝑦
= −
𝜕𝑝
𝜕𝑦
+ 𝜇(
𝜕2 𝑣
𝜕𝑥2 +
𝜕2 𝑣
𝜕𝑦2)
o Energy Equation 𝜌𝑐 𝑝 𝑢
𝜕𝑇
𝜕𝑥
+ 𝑣
𝜕𝑇
𝜕𝑦
= 𝑘(
𝜕2 𝑇
𝜕𝑥2 +
𝜕2 𝑇
𝜕𝑦2)

7. Boundary Conditions:
• The Narrow Channel:
- The bottom wall (𝑦 = 0): 𝑢 = 0, 𝑣 = 0,
𝜕𝑇
𝜕𝑦
= 0
- The top wall (𝑦 = ℎ): 𝑢 = 0, 𝑣 = 0,
𝜕𝑇
𝜕𝑦
= 0
• The Wide Channel:
- The bottom wall (𝑦 = −ℎ): 𝑢 = 0, 𝑣 = 0, 𝑇 = 𝑇 𝐻
- The top wall (𝑦 = 2ℎ): 𝑢 = 0, 𝑣 = 0, 𝑇 = 𝑇𝐶
• The inlet (𝑥 = 0): 𝑇 = 300°𝐾, 𝑈 = 𝑈0 1.0 −

8. Numerical Method Details:
 Finite Volume Method (FVM) is used
 SIMPLE algorithm is opted for pressure-velocity coupling
 Power law scheme is used for momentum and energy equations for
convection-diffusion terms
 Maximum residual limit for continuity, x-momentum and y-momentum is
taken as 10-5 and for energy as 10-7 .
 Under relaxation factor: for pressure 0.3, for momentum 0.4, for energy 0.8.

9. Inputs to Fluent software:
 FLUENT software version 14.5.
 Finite Volume Method (FVM).
 Pressure-velocity coupling- SIMPLE algorithm.
 Convection-diffusion terms- Power law.
 The physical properties are assumed to be constant with respect to
temperature.
 Double precision used throughout all the computations.

10. Results and Discussion
The flow and heat transfer characteristics were analysed for various inlet
velocity profiles using:
• Streamline and temperature contours,
• x-velocity and temperature profile along the vertical line at a distance 12h
from the inlet,
• Pressure variation along the middle horizontal line,
• Variation of local Nusselt number and skin friction coefficient along the
bottom heated wall,
• Variation of average Nusselt number on the bottom heated wall with
respect to Uo,
• Pressure drop across the channel with respect to Uo,
• Temperature variation at the point (0.12 , 0.01) with respect to Uo.
• Reattachment length behind the sudden expansion on bottom and top wall
with respect to U0;

11. Streamline contours
Streamline contours for various inlet velocity profiles

12. Temperature contours
Temperature contours for various inlet velocity profiles

13. Temperature and x-velocity profiles
Temperature profile along the
vertical line at a distance 12h from
x-velocity profile along the vertical
line at a distance 12h from the inlet

14. Pressure variation along the middle horizontal line
Pressure variation along the horizontal line at a distance 0.015 from the bottom
heated wall

15. Local Nusselt number and Skin friction coefficient
Variation of skin friction coefficient
along the bottom heated surface
Variation of Local Nusselt number
along the bottom heated surface

16. Average Nusselt number along the bottom wall
0
2
4
6
8
10
12
14
16
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Variation of Average Nusselt number versus Uo
Avg.
Nu
Uo

17. Pressure drop
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ΔP
Uo
Pressure drop across the channel versus Uo

18. Temperature at (0.12 , 0.01)
300
302
304
306
308
310
312
314
316
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Uo
T (°K)
Variation of temperature at a point (0.12 m, 0.01 m) versus Uo

19. Reattachment Length
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Top Wall Bottom Wall
Reattachment length behind the sudden expansion versus Uo
L (m)
Uo

20. Conclusion
• The sudden expansion causes vortices at both bottom and top edges and the
vortices grow as the inlet velocity increases.
• The heat transfer consists of various regimes starting from conduction dominated
regime to convection dominated regime.
• The local Nusselt number along the heated wall shows low values immediately
after the expansion and increases through the vortex until the maximum value at
reattachment location where it starts to decrease smoothly throughout the channel.
• The skin friction coefficient has high values before the center of vortex and after the
flow reattachment. However, the friction decreases after the vortex center.
• The pressure is decreasing through the channel due to pressure losses and it is
rapidly decreasing at the expansion location.