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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

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