An Analysis of Hiemenz Flow
E. Kaufman and E. Gutierrez-Miravete
Department of Engineering and Science
Rensselaer at Hartford
OBJECTIVE
To solve a simple flow field for which an exact solution is
available using COMSOL & FLUENT
HIEMENZ FLOW U8
• Planar
• Laminar
• Viscous
• Incompressible
• Close to a stagnation point
y
x
BACKGROUND
• Exact Solution Exists for Hiemenz Flow
• Viscous Solution is Derived from the Inviscid Solution
 Inviscid
V  axiˆ  ayˆj

p0  p  12 a 2 x 2  y 2
Viscous


V  af ( x)iˆ  ayf ' ( x) ˆj


p0  p  12 a 2 x 2  F  y 
HIEMENZ SOLUTION
• Substitute into 2D Incompressible Navier-Stokes Equations
p
a x f '  ff ' '  g x   axf ' ' '
x
2

2

•Similarity Solution Yields Hiemenz Equation
 ' ' ' ' ' ' 1  0
2
•Boundary Conditions
 (0)   ' (0)  0
 ' ( )  1
•Solve Using Shooting Method
p
a ff '  g y   af ' '
y
2
Inviscid and Viscous Hiemenz Flow: Velocity
Velocity Profiles
Flow
Direction
Symmetry Line
Wall
Inviscid and Viscous Hiemenz Flow: Pressure
Pressure Contours
Flow
Direction
Wall
Inviscid and Viscous Hiemenz Flow: Temperature
•Inviscid heat flux at the wall is ~ 2X Viscous
COMSOL & FLUENT: Case Study - Inputs
• Compare Analytical Solution to Computational Results
• Set up Dimensional Problem for Easy Comparison
Material Properties
• Density = 1
• Viscosity = 1 kg/ms
• Conductivity = 1 W/moC
kg/m3
Inlet
Inlet Velocity:(u, v) = (x, 5.3521)
Inlet Temperature: T=100
(0,6)
Exit Pressure
(6,6)
6
Flow Field Scale Factor
5
• a = 1 s-1
Symmetry
4
m
Similarity Factors
• η = y/1
•ϕ = f/1
Exit
3
2
1
Boundary Conditions
• Consistent with analytical solution at
boundaries
• Assume stagnation pressure of 100
Pascals
(0,0)
(6,0)
Wall: T=1000
Computational Domain
• 6 meters by 6 meters
0
60
70
80
Pa
90
COMSOL & FLUENT: Case Study - Results
COMSOL Mesh
FLUENT Mesh
COMSOL Velocity
FLUENT Velocity
COMSOL Pressure Field
FLUENT Pressure Field
COMSOL & FLUENT: Case Study – Results
Static Pressure Along Symmetry Plane
Pressure Profile at Sym m etry Plane (X=0)
Pressure Profile at Symmetry Plane (X=0)
COMSOL
FLUENT
Hiemenz
Inviscid
Inlet 6
COMSOL
FLUENT
Hiemenz
Inviscid
1
0.9
5
0.8
0.7
4
Y coordinate (m)
Y coordinate (m)
Flow
Direction
3
2
0.6
0.5
0.4
0.3
0.2
1
0.1
0
Wall 0
80
85
90
95
Pressure (Pa)
100
105
98
99
100
Pressure (Pa)
101
102
COMSOL & FLUENT: Case Study – Results
Temperature
Temperature Profile
COMSOL
FLUENT
Hiemenz
Inviscid
6
Y coordinate (m)
5
4
3
COMSOL Temperature Distribution
2
1
0
0
0.2
0.4
0.6
0.8
1
Normalized Temp (T-Twall)/(Tinf-Twall)
FLUENT Temperature Distribution
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