Rgdkxgq7rqadwvdvyewm_mae 195 Report 1

8/16/2019 Rgdkxgq7rqadwvdvyewm_mae 195 Report 1

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University of California, Irvine

Henry Samueli School of Engineering

Department of Mechanical and Aerospace Engineering

Laminar Pipe & Channel Flow AnalysisUsing ANSYS Fluent

Author:

Jaspal Sidhu

ID:

18107158

Professor:

Said Elghobashi

Class:

MAE 195

April 10, 2015


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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Flow Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Mechanics of Laminar Flow Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1 Laminar Velocity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2 Laminar Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Laminar Flow in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Computational Domain & Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7 Computational Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7.1 Circular Pipe (Axisymmetric 2D Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107.2 Channel (Planar 2D Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Initialization of Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.1 Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.1.1 Simulation Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.1.2 Computational Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.2 Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.2.1 Simulation Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.2.2 Computational Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9.3 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

10 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610.1 Pipe Flow (N x = 100 and N y = 20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

10.1.1 Velocity Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1610.1.2 Centerline Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1710.1.3 Coefficient of Skin Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1810.1.4 Outlet Velocity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10.2 Channel Flow (N x = 100 and N y = 20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.2.1 Velocity Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.2.2 Centerline Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2010.2.3 Coefficient of Skin Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.2.4 Outlet Velocity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

11 Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211.1 Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2211.2 Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.1 Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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List of Figures

1 Streak-line Profile for Laminar Flow in a Pipe/Duct . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Effect of Reynolds Number On Time-averaged Axial Velocity Component . . . . . . . . . . . . . . 53 Development of Laminar Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Simulated Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Laminar Flow in a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Side View of Mesh (N x = 100 and N y = 10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Preliminary Mesh Configuration For Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Mesh Configuration Check For Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Isometric View of Mesh (N x = 100 and N y = 10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Preliminary Mesh Configuration For Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Mesh Configuration Check For Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112 Initialization for Second Order Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Residual Convergence for Pipe Flow for 100 x 10 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 1314 Scaled Residual Computation for Pipe Flow with 100 Iteration Limit . . . . . . . . . . . . . . . . . 1315 Residual Convergence for Channel Flow in 100 x 10 Mesh . . . . . . . . . . . . . . . . . . . . . . . 1416 Scaled Residual Computation for Channel Flow with 100 Iteration Limit . . . . . . . . . . . . . . . 1417 Grid Refinement using Circular Pipe Centerline Velocity . . . . . . . . . . . . . . . . . . . . . . . . 1518 Grid Refinement using Channel Centerline Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519 Velocity Vectors in Pipe Flow (Half ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1620 Velocity Vectors in Pipe Flow (Full) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1621 Centerline Velocity in Pipe Flow (Full Range) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1722 Centerline Velocity in Pipe Flow (Truncated Range) . . . . . . . . . . . . . . . . . . . . . . . . . . 1723 Coefficient of Skin Friction for Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824 Outlet Velocity Profile using ANSYS Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825 Velocity Vectors in Channel Flow (Half) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926 Velocity Vectors in Channel Flow (Full) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1927 Centerline Velocity in Channel Flow (Full Range) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2028 Centerline Velocity in Channel Flow (Truncated Range) . . . . . . . . . . . . . . . . . . . . . . . . 2029 Coefficient of Skin Friction for Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

30 Outlet Velocity Profile using ANSYS Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2131 Axial Velocity Comparison for Laminar Flow in Circular Pipe . . . . . . . . . . . . . . . . . . . . . 2232 Axial Velocity Comparison for Laminar Flow in a Channel . . . . . . . . . . . . . . . . . . . . . . . 2333 Contour Plots for Laminar Flow in a Circular Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . 2434 Contour Plots for Laminar Flow in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435 N x = 100 and N y = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536 N x = 100 and N y = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2537 Inlet Velocity Contour in a Circular Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2638 Outlet Velocity Contour in a Circular Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639 Inlet Velocity Contour in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2740 Outlet Velocity Contour in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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Nomenclature

γ Specific Heat Ratio

µ Dynamic Viscosity

ν Kinematic Viscosity

ρ Fluid Density

τ Shear Stress

τ w Wall Shear Stress

V Fluid Velocity

A Cross-sectional Area

C f Coefficient of Skin Friction

D Hydraulic Diameter

h Channel Height

le Entrance Length

P Ambient Pressure

Q Volumetric Flowrate

R Pipe Radius

r Incremental Pipe Radius

Re Reynolds Number

T Ambient Temperature

u Axial Velocity Component

v Vertical Velocity Component

V avg Average Axial Velocity (also U z)

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Abstract

The ANSYS Fluent tutorial presented in this report is designed to introduce the computational

methods needed to solve a Laminar Pipe and Channel Flow problem. The goal of the simulation

tutorial was to guide users through the comparison of analytical and computational results of a

well-studied, low Reyonolds Number fluid flow. The simulation analysis presented below, brings

light to the importance of using Computational Fluid Dynamics to solve Fluid Mechanic problems

that do not have analytical solutions. The following report illustrates the well known velocity

profile solutions for Laminar flow through a Circular Pipe and a Planar Channel. The compu-

tational results for the axial velocity through a Circular Pipe and a Channel compare well with

the desired analytical solutions of U max = 2 m

sand U max = 1.5

m

s, respectively. This report details

the simulation Geometry, Mesh, and Flow Physics configuration required to obtain the accurate

results. Furthermore, to verify the accuracy of the computational results, mesh refinement was

carried out and the simulation was repeated to test for grid independence.

1 Introduction

Computational Fluid Dynamics utilizes numerical methods to solve the governing equations of mass, momentumand energy in order to predict flow properties in a vast variety of cases. Instead of attempting to analyticallysolve the governing equations, CFD replaces the conservation equations with a mesh of discretized grid points thatcover the entire flow domain. The discretized mesh is then solved by a computer to determine the flow properties

at individual grid points.

Why does CFD matter? Prior to the introduction of super-computers, determining the flow properties in aflow domain was done by solving the exact governing equations analytically, or through experimental techniquesthat measured the local properties. With a limited number of analytical solutions, many real fluid mechanicproblems were difficult to solve. Of the limited flow problems with analytical solutions, the Laminar flow througha pipe/duct (shown in Figure 1) is one that provides analytical results that compare well with computationalsolutions.

Figure 1: Streak-line Profile for Laminar Flow in a Pipe/Duct

In this report, a Laminar flow through a circular pipe and channel are studied. These two flow cases provideinsight into the accuracy of the computational solution by simulating a Newtonian fluid in a 2D geometry that isspacially axisymmetric for the circular pipe and planar for the channel. ANSYS Fluent will be used to carry outthe computational simulation by configuring the simulation Geometry, Mesh, and Flow Physics. After completingthe configuration of the flow simulation, the remainder of the report will cover the simulation results and result

verification.

The goal of this report is to solve this problem of Laminar Flow through a Circular Pipe and Channel usingANSYS Fluent. We expect the viscous boundary layer to grow along the pipe starting at the inlet. It will even-tually grow to fill the pipe completely. When this happens, the flow becomes fully-developed and there is novariation of the velocity profile in the axial direction, x. Knowing this, I will obtain the plots for the centerlinevelocity, wall skin-friction coefficient, and velocity profile at the outlet and validate the simulation results.

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2 Flow Description

2.1 Reynolds Number

The Reynolds Number is a non-dimensionalized form of velocity that can be defined as the ratio of inertial forcesto viscous forces or more formally:

Re = INERTIAL

V ISCOUS =

ρU D

µ (1)

In the equation above, ρ is the fluid density, U is the fluid velocity, D is the hydraulic diameter of the pipe andµ is the fluid viscosity. The Reynolds Number is the primary parameter for quantifying when and whether a flowwill be Laminar (dominant viscous effects), Turbulent (dominant inertial effects) or somewhere in between.

2.2 Laminar Flow

In Laminar flow, the fluid flows smoothly down the pipe due to low Reynolds Number (Re) because viscous effectsdominate the flow. The result of the highly viscous flow causes streak-lines to be straight lines (shown below inFigure 2) largely because momentum transfer between fluid particles is not as effective at low Re. For circularpipes, the flow is Laminar when Re


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When Newton’s Second Law is applied to a fluid element inside the circular pipe, we find that the shear stressexerted on the fluid element depends only on the relative distance as a fraction of the diameter and the fluidviscosity:

τ = 2τ wr

D (3)

Without fluid viscosity, the wall shear stress (τ w), and therefore the total shear stress (τ ), would be zero. As aresult, the presence of fluid viscosity dissipates kinetic energy which forces the pressure of the flow to drop alongthe axial direction where:

∆P = F A

= 2lτ 2

= 4lτ wD

(4)

3 Governing Equations

3.1 Continuity Equation

For 2D Steady-State Flow:

∇ · (ρṼ) =< ∂

∂x,

∂

∂y > · < ρu, ρv >= ρ

∂u

∂x + ρ

∂v

∂y = 0 (5)

For 2D Steady-State Incompressible Flow:

∇ · (Ṽ) =< ∂

∂x,

∂

∂y > · < u, v >=

∂u

∂x +

∂v

∂y = 0 (6)

3.2 Momentum Equation

Navier-Stokes Equation in Planar Cylindrical for Steady-State Pipe Flow:

ρ

ur

∂ur

∂r + uz

∂ur

∂z

= −

∂p

∂r + µ

1

r

∂

∂r

r

∂ur

∂r

+

∂ 2ur

∂z2 −

ur

r2

+ ρgr (7)

ρ

ur

∂uz

∂r + uz

∂uz

∂z

= −

∂p

∂z + µ

1

r

∂

∂r

r

∂uz

∂r

+

∂ 2uz

∂z2

+ ρgz (8)

Navier-Stokes Equation in 2D Cartesian for Steady-State Channel Flow:

ρ

ux

∂ux

∂x + uy

∂ux

∂y

= −

∂p

∂x + µ

∂ 2ux

∂x2 +

∂ 2ux

∂y2

+ ρgx (9)

ρ

ux

∂uy

∂x + uy

∂uy

∂y

= −

∂p

∂y + µ

∂ 2uy

∂x2 +

∂ 2uy

∂y2

+ ρgy (10)

Compact Form for Steady-State Incompressible Navier-Stokes Equation:

(u · ∇)u − ν ∇2u = −∇w + g. (11)

4 Analytical Solution

4.1 Laminar Velocity Profile

In order to find the velocity profile of a Laminar Flow, the fluid must be assumed to be a Newtonian Fluid. Bydefinition, a Newtonian Fluid is a fluid in which ”the viscous stresses arising from its flow, at every point, arelinearly proportional to the local strain rate—the rate of change of its deformation over time” [2, p. 31]. Moreformally, a Newtonian Fluid can be defined as:

τ = −µ∂u

∂y (12)

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4.2 Laminar Flow in a Pipe

Using Equation 10, and the definition shear stress exerted on the fluid, we can obtain a differential equation inthe form:

du

dr = −

∆P

2µlr −→ u(r) =

1

4µ(−

dP

dx) + C 1ln(r) + C 2

The wall shear stress (τ w) is proportional to ∆P , thus we can define the friction factor in terms of ∆P :

f = ∆P D

12ρDU 2L=

64

Re −→ c

f =

|τ w|

12ρU 2 =

16

Re

By applying the boundary conditions where ∂u∂r

= 0 at r = 0 because of symmetry about the centerline andu(r = R) = 0 at the the wall because of the no-slip condition, therefore:

u(r) = −R2

4µ(

dP

dx)[1 − (

r

R)2] (13)

In order to non-dimensionalize the axial velocity for the flow in a pipe, the average flow velocity must be definedas the following:

V avg = 2

R2

R0

u(r)rdr = −R2

8µ(

dP

dx) (14)

So by combining, Equations 11 and 12, we get that:

u(r) = 2V avg[1 − r2

R2] (15)

Which ultimately shows that:U max = u(0) = 2V avg (16)

For the pipe flow simulation, the average velocity was given as 1 ms

therefore, the maximum velocitywithin the flow is expected to be 2 m

s .

4.3 Laminar Flow in a Channel

By applying the definition of volumetric flow rate, and the aforementioned axial velocity equation, we can derivethe non-dimensionalized form of velocity for the channel flow. Know that:

Q =

h0

udy = 1

2µ(−

dP

dx)

h0

[h2 − y2]dy = 1

µ(−

dP

dx)[

h3

3 ] (17)

We also know that Q = u ∗ h therefore:

u = Q

h =

1

2µh(−

dP

dx)

h0

[h2 − y2]dy = 1

2µh(−

dP

dx)[h2y −

y3

3 ]h0 (18)

To solve for the non-dimensionalized velocity of a channel flow, average (y = 0 → h) and maximum (y=0) axiavelocity need to be determined therefore:

U avg = 1

2µh(−

dP

dx)[h2y −

y3

3 ]h0 =

1

µ(−

dP

dx)[

h2

3 ] (19)

U max = 1

2µ(−

dP

dx)[h2] (20)

Which ultimately shows that:

U max

U avg=

12µ

(−dP dx

)[h2]

1µ

(−dP dx

)[h2

3 ]

= 3

2 (21)

For the channel flow simulation, the average velocity was given as 1 ms

therefore, the maximumvelocity within the flow is expected to be 1.5 m

s .

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5 Computational Domain & Fluid Properties

Figure 4: Simulated Geometry

We are considering a fluid flowing through a circular pipe of constant radius and through a planar channel ofconstant height. The pipe diameter and channel height is 0.2 m and the length is 10 m.

Geometry Dimensions:

• Horizontal Length=10m • Vertical Height (from centerline)=0.1 m

Fluid Properties:

• Newtonian Fluid

• Density: ρ = 1 kgm3

• Viscosity: µ = 2e−3 kgm·s

• Reynolds Number: Re = ρUDµ

= 100

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6 Boundary Conditions

The edges of the geometry will be given names in order to assign boundary conditions in ANSYS Fluent. The leftside of the pipe will be called ”Inlet” and the right side will be called ”Outlet”. The top side of the rectangle willbe called ”PipeWall” and the bottom side of the rectangle will be called ”CenterLine” as shown below in Figure5.

Figure 5: Laminar Flow in a Pipe

1. For fully developed flow at the pipe/channel exit:

∂u

∂x = 0 therefore from the Continuity Equation

∂v

∂y = 0

2. From symmetry of plane, we get that:

y = 0 v = 0 and ∂u

∂y = 0

3. From the no-slip condition at the wall, we get that:

u = 0 and v = 0

4. Since ∂v∂y

= 0 therefore v = 0 everywhere

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7 Computational Mesh

7.1 Circular Pipe (Axisymmetric 2D Space)

Figure 6: Side View of Mesh (N x = 100 and N y = 10)

Figure 7: Preliminary Mesh Configuration For Pipe

Figure 8: Mesh Configuration Check For Pipe

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7.2 Channel (Planar 2D Space)

Figure 9: Isometric View of Mesh (N x = 100 and N y = 10)

Figure 10: Preliminary Mesh Configuration For Channel

Figure 11: Mesh Configuration Check For Channel

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8 Initialization of Dependent Variables

For this simulation tutorial, a second order discretization was used to to approximate the computational solutionIn particular, the second order scheme was implemented by using the ”Solution Methods” option in ANSYS Fluentand selecting the ”Second Order Upwind” feature under the ”Momentum” section.

Figure 12: Initialization for Second Order Discretization

Carrying out these steps will allow the flow field to be initialized to the values at the inlet of the Pipe and Channel

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9 Numerical Details

Before running the flow simulation, we checked to make sure the residual of the governing equations convergedat approximately 1e−6. ANSYS Fluent reports a residual for each of the governing equations being solved. Thecomputational residual is a measure of how well the current solution satisfies the discrete form of each governingequation. Fluent will iterate the solution until the residual for each equation falls below 1e−6.

9.1 Pipe Flow

9.1.1 Simulation Convergence

Figure 13 (below) shows that the residuals for the Pipe flow converged after 64 iterations. As expected, ourflow simulation took a greater number of iterations for the residual to converge when compared to the simulationtutorial, largely because we used a finer mesh.

Figure 13: Residual Convergence for Pipe Flow for 100 x 10 Mesh

9.1.2 Computational Residuals

Using the XY-Plot feature in ANSYS Fluent, I was able to produce the graphs shown in Figure 14 of the pipeflow’s residual convergence as a function of the number of computational iterations for the governing equations.

(a) First Iteration Set (b) Second Iteration Set

Figure 14: Scaled Residual Computation for Pipe Flow with 100 Iteration Limit

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9.2 Channel Flow

9.2.1 Simulation Convergence

Similar to the previous section, Figure 15 (below) shows that the residuals for the Channel flow converged after77 iterations. As expected, our flow simulation took a greater number of iterations for the residual to convergewhen compared to the simulation tutorial, largely because we used a finer mesh.

Figure 15: Residual Convergence for Channel Flow in 100 x 10 Mesh

9.2.2 Computational Residuals

Using the XY-Plot feature in ANSYS Fluent, I was able to produce the graphs shown in Figure 16 of the channelflow’s residual convergence as a function of the number of computational iterations for the governing equations.

(a) First Iteration Set (b) Second Iteration Set

Figure 16: Scaled Residual Computation for Channel Flow with 100 Iteration Limit

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9.3 Mesh Refinement

Shown below in Figures 17 and 18 are the Mesh refinement comparisons for the circular pipe and channel flow.Mesh refinement was carried out in an increasing order (N x and N y): 100 x 5 (red), 100 x 10 (green), 100 x 20(blue) and 100 x 40 (white). For both simulations, the coarse mesh (red) deviates significantly from the analyticasolutions of the flow. Comparing the 100 x 20 and 100 x 40 meshes, the solution convergence takes much longerfor the finer mesh with minimal improvements in centerline velocity accuracy. Therefore for the remainder of thereport, I will be using the 100 x 20 mesh configuration for the pipe and channel flow simulations.

Figure 17: Grid Refinement using Circular Pipe Centerline Velocity

Figure 18: Grid Refinement using Channel Centerline Velocity

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10 Results

10.1 Pipe Flow (N x = 100 and N y = 20)

10.1.1 Velocity Vectors

Half and Full Velocity vectors plots are shown to prove that the flow is indeed symmetric about the pipe centerline.

Figure 19: Velocity Vectors in Pipe Flow (Half)

Figure 20: Velocity Vectors in Pipe Flow (Full)

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10.1.2 Centerline Velocity

Figure 21: Centerline Velocity in Pipe Flow (Full Range)

Figure 22: Centerline Velocity in Pipe Flow (Truncated Range)

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10.1.3 Coefficient of Skin Friction

Figure 23: Coefficient of Skin Friction for Pipe Flow

10.1.4 Outlet Velocity Profile

Figure 24: Outlet Velocity Profile using ANSYS Fluent

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10.2 Channel Flow (N x = 100 and N y = 20)

10.2.1 Velocity Vectors

Half and Full Velocity vectors plots are shown to prove that the flow is indeed symmetric about the channelcenterline.

Figure 25: Velocity Vectors in Channel Flow (Half)

Figure 26: Velocity Vectors in Channel Flow (Full)

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10.2.2 Centerline Velocity

Figure 27: Centerline Velocity in Channel Flow (Full Range)

Figure 28: Centerline Velocity in Channel Flow (Truncated Range)

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10.2.3 Coefficient of Skin Friction

Figure 29: Coefficient of Skin Friction for Channel Flow

10.2.4 Outlet Velocity Profile

Figure 30: Outlet Velocity Profile using ANSYS Fluent

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11 Validations

11.1 Pipe Flow

To verify the accuracy of the computational solution for the radial velocity in a laminar flow, I wrote a MATLABscript to compute the analytical solution for radial velocity. Shown below is the output of my MATLAB codewhich contains the radial increment, r, and the associated radial velocity component, u(r):

>> u = 2.0000 1.9950 1.9800 1.9550 1.9200 1.8750 1.8200 1.7550 1.6800 1.5950 1.5000 1.3950 1.2800 1.1550 1.0200

0.8750 0.7200 0.5550 0.3800 0.1950 0.0000

>> r = 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500 0.0550 0.0600 0.0650 0.0700

0.0750 0.0800 0.0850 0.0900 0.0950 0.1000

The MATLAB output above was saved as a .xy file and uploaded into ANSYS Fluent and plotted against thecomputational solution. Shown below are the results of this comparison.

(a) Full Graph of Axial Velocity Comparison (b) Truncated Graph of Axial Velocity Comparison

Figure 31: Axial Velocity Comparison for Laminar Flow in Circular Pipe

Figure 31a shows the entire radial velocity range comparison as a function of radial pipe position and proves thecomputational accuracy of the simulation. Figure 29b depicts the truncated range of radial velocity where thecomputational solutions begin to deviate from the analytical solution.

In Figure 31b, the green dots correspond to the analytical solution, the red dots correspond to the coarse mesh(10 x 100), and the white dots correspond to the refined mesh (20 x 100). The truncated plot shows that thecomputational solutions improve in accuracy with grid refinement with minor deviations from the analytical solu-tion. For the 20 X 100 mesh, the computational solution for velocity yields a value of 1.98 m

s which is 1% less than

the analytical value of 2 ms , therefore any further mesh refinement (as shown in Section 7.3) would be wasted effort

Lastly, when comparing the computational solution for the skin friction coefficient (cf = 0.158) in Figure 21to the analytical solution we get that:

(cf )analytical = 16

Re =

16

100 = 0.160 ∴

(cf )computational(cf )analytical

= 0.9875

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11.2 Channel Flow

To verify the accuracy of the computational solution for the axial velocity in a laminar flow, I wrote a MATLABscript to compute the analytical solution for channel flow velocity. Shown below is the output of my MATLABcode which contains the axial increment, h, and the associated lateral velocity component, u(h):

>> u = 1.5000 1.4963 1.4850 1.4663 1.4400 1.4063 1.3650 1.3162 1.2600 1.1963 1.1250

1.0463 0.9600 0.8662 0.7650 0.6562 0.5400 0.4163 0.2850 0.1463 0.0000

>> h = 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0.0450 0.0500

0.0550 0.0600 0.0650 0.0700 0.0750 0.0800 0.0850 0.0900 0.0950 0.1000

The MATLAB output above was saved as a .xy file and uploaded into ANSYS Fluent and plotted against thecomputational solution. Shown below are the results of this comparison.

(a) Full Graph of Axial Velocity Comparison (b) Truncated Graph of Axial Velocity Comparison

Figure 32: Axial Velocity Comparison for Laminar Flow in a Channel

Figure 32a shows the full axial velocity range comparison as a function of channel position and proves the computa-tional accuracy of the simulation. Figure 32b depicts the truncated range of axial velocity where the computationalsolutions begin to deviate from the analytical solution.

In Figure 32b, the green dots correspond to the analytical solution, the red dots correspond to the coarse mesh(10 x 100), and the white dots correspond to the refined mesh (20 x 100). The truncated plot shows thatthe computational solutions improve in accuracy with grid refinement with minor deviations from the analyticasolution. For the 20 X 100 mesh, the computational solution for velocity yields a value of 1.49255 m

s which is 0.5%

less than the analytical value of 1.5 ms , therefore any further mesh refinement (as shown in Section 7.3) would be

wasted effort.

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12 Conclusions

In summary, we expected the viscous boundary layer to grow along the distance of the pipe and channel startingfrom the inlet. In doing so, we also expected the boundary layer to fill the pipe and channel completely until theflow became fully-developed and there was no longer any variation of the velocity profile in the axial direction.Additionally, my computational results were expected to satisfy the analytical solutions for the Laminar Flow ina Pipe and Channel.

The graphs shown below in Figures 33 and 34 illustrate the expected boundary layer growth (through the in-crease in axial velocity) and span-wise drop in static pressure. Figure 33a shows that the computational model forthe Laminar Flow in a Pipe achieves the analytical axial velocity value of 2 m

s . Likewise, Figure 34a illustrates the

growth of the boundary layer where maximum axial velocity is approximately 1.5 ms . Furthermore, as mentioned

in class, both simulations for circular pipe and channel exhibit the static pressure drop along the axial directionof the pipe (shown in Figures 33b and 34b), which is the flow’s driving force.

(a) Inlet Axial Velocity Contour Plot (b) Static Pressure Contour Plot

Figure 33: Contour Plots for Laminar Flow in a Circular Pipe

(a) Inlet Axial Velocity Contour Plot (b) Static Pressure Contour Plot

Figure 34: Contour Plots for Laminar Flow in a Channel

Detailed in Section 7.3 and Section 11 are verifications showing that ample grid refinement helps increase the ac-curacy of the flow simulation by making sure the governing equations are properly discretized over the geometry.Additionally, further verifications such as comparing the skin friction coefficient between analytical and compu-tational solutions, provides conclusive evidence for the accuracy of the computational simulations of a LaminarFlow through a Circular Pipe and Channel.

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A Appendices

A.1 Additional Figures

Figure 35: N x = 100 and N y = 10

Figure 36: N x = 100 and N y = 20

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Figure 37: Inlet Velocity Contour in a Circular Pipe

Figure 38: Outlet Velocity Contour in a Circular Pipe

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Figure 39: Inlet Velocity Contour in a Channel

Figure 40: Outlet Velocity Contour in a Channel

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References

[1] Fox, Robert W., Robert W. Fox, Philip J. Pritchard, and Alan T. McDonald. Fox and McDonald’s Introductionto Fluid Mechanics . Hoboken, NJ: John Wiley & Sons, 2011. Print.

[2] Munson, Bruce Roy, T. H. Okiishi, Wade W. Huebsch, and Alric P. Rothmayer. Fundamentals of Fluid Me-chanics . Hoboken, NJ: John Wiley & Sons, 2013. Print.

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