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An Analysis of Hiemenz Flow
E. Kaufman and E. Gutierrez-MiraveteDepartment of Engineering and Science
Rensselaer at Hartford
Presented at the COMSOL Conference 2008 Boston
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OBJECTIVE
To solve a simple flow field for which an exact solution is available using COMSOL & FLUENT
HIEMENZ FLOW• Planar • Laminar • Viscous • Incompressible• Close to a stagnation point x
y
x
y
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BACKGROUND
• Exact Solution Exists for Hiemenz Flow• Viscous Solution is Derived from the Inviscid Solution
Viscous
jxayfixafV ˆ)('ˆ)( +−=
( )[ ]yFxapp +=− 2221
0 ρ
jayiaxV ˆˆ +−= Inviscid
( )22221
0 yxapp +=− ρ
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HIEMENZ SOLUTION
• Substitute into 2D Incompressible Navier-Stokes Equations
•Similarity Solution Yields Hiemenz Equation
•Boundary Conditions
•Solve Using Shooting Method
01'''''' 2 =+−+ φφφφ
( ) ''''''22 axfxpgfffxa x µρρ +∂∂
−=− '''2 afypgffa y µρρ −∂∂
−=
0)0(')0( ==φφ
1)(' =∞φ
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Velocity Profiles
Wall
Flow Direction
Inviscid and Viscous Hiemenz Flow: Velocity
Symmetry Line
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Inviscid and Viscous Hiemenz Flow: Pressure
Pressure Contours
Wall
Flow Direction
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•Inviscid heat flux at the wall is ~ 2X Viscous
Inviscid and Viscous Hiemenz Flow: Temperature
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Exit Pressure
0
1
2
3
4
5
6
60 70 80 90
Pa
m
COMSOL & FLUENT: Case Study - Inputs
• Compare Analytical Solution to Computational Results
• Density = 1 kg/m3
• Viscosity = 1 kg/ms• Conductivity = 1 W/moC
Material Properties
Similarity Factors• η = y/1•ϕ = f/1
• Set up Dimensional Problem for Easy Comparison
Flow Field Scale Factor• a = 1 s-1
Computational Domain• 6 meters by 6 meters
Inlet Velocity:(u, v) = (x, 5.3521)Inlet Temperature: T=100Inlet
Exit
Wall: T=1000
Symmetry
(0,0)
(0,6)
(6,0)
(6,6)
Boundary Conditions• Consistent with analytical solution at boundaries• Assume stagnation pressure of 100 Pascals
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COMSOL Mesh
FLUENT Mesh
COMSOL Velocity
FLUENT Velocity
COMSOL Pressure Field
FLUENT Pressure Field
COMSOL & FLUENT: Case Study - Results
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Pressure Profile at Symmetry Plane (X=0)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
98 99 100 101 102
Pressure (Pa)
Y co
ordi
nate
(m)
COMSOL FLUENT
Hiemenz Inviscid
Pressure Profile at Symmetry Plane (X=0)
0
1
2
3
4
5
6
80 85 90 95 100 105
Pressure (Pa)
Y co
ordi
nate
(m)
COMSOL FLUENT Hiemenz Inviscid
Wall
Inlet
Flow Direction
COMSOL & FLUENT: Case Study – ResultsStatic Pressure Along Symmetry Plane
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Temperature Profile
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Normalized Temp (T-Twall)/(Tinf-Twall)
Y co
ordi
nate
(m)
COMSOL FLUENT Hiemenz Inviscid
COMSOL Temperature Distribution
FLUENT Temperature Distribution
COMSOL & FLUENT: Case Study – ResultsTemperature
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CONCLUSIONS
• Both COMSOL and FLUENT have been shown to reliably reproduce the flow field
• Velocities, Pressures and Temperatures match the analytical predictions closely
•Selection of appropriate boundary conditions and adequate domain size are critical to accurately predict flow field