An Analysis of Hiemenz Flow - COMSOL Multiphysics · An Analysis of Hiemenz Flow E. Kaufman and E....
Transcript of An Analysis of Hiemenz Flow - COMSOL Multiphysics · An Analysis of Hiemenz Flow E. Kaufman and E....
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
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
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
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
Velocity Profiles
Wall
Flow Direction
Inviscid and Viscous Hiemenz Flow: Velocity
Symmetry Line
Inviscid and Viscous Hiemenz Flow: Pressure
Pressure Contours
Wall
Flow Direction
•Inviscid heat flux at the wall is ~ 2X Viscous
Inviscid and Viscous Hiemenz Flow: Temperature
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
COMSOL Mesh
FLUENT Mesh
COMSOL Velocity
FLUENT Velocity
COMSOL Pressure Field
FLUENT Pressure Field
COMSOL & FLUENT: Case Study - Results
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
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
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