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BOUNDARY LAYER ANALYSIS TOOLKIT

An Independent Study ReportPresented to the

Faculty ofCalifornia State Polytechnic University, Pomona

In Partial FulfillmentOf the Requirements for the Degree

Master of ScienceIn

Mechanical Engineering

By

Richard Getze

December, 2003


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SIGNATURE PAGE

INDEPENDENT STUDY: Boundary Layer Analysis Toolkit

AUTHOR: Richard Getze

DATE SUBMITTED:

Dr. Kevin Anderson _____________________________________Ind. Study Committee ChairMechanical Engineering

Dr. Hassan M. Rejali _____________________________________Mechanical Engineering

Dr. Thuan K. Nguyen _____________________________________Chemical Engineering


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TABLE OF CONTENTS

Section Page

List of Tables ii

List of Figures iii

Abstract 1

Introduction 2

Definitions 2

The Boundary Layer Equations 4

Turbulence Models 8

Discretization 10

Software Operation 14

Software Validation 25

Bibliography 27

Appendix A: SIL Code 28

Appendix B: SCL Code 36

Appendix C: SIT Code 44

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LIST OF TABLES

Table Page

2.1 Definition of Variables 3

3.1 Boundary Layer Approximations 5

5.1 Discretized Forms of Important Differentials 10

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LIST OF FIGURES

Figure Page

6.1 SIL Input Form 16

6.2 SIL Form With Output Choices 20

6.3 SIL Output Example 21

6.4 SCL Form With Output Choices 22

6.5 SIT Form With Output Choices 24

7.1 Validation of SIT Module 25

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Boundary Layer Analysis ToolkitAn Independent Study ReportRichard Getze

ABSTRACT

The purpose of this project was to provide an intuitive and integrated software

toolkit for boundary layer calculations. The toolkit is constrained to flow over flat

and gently curving surfaces and is able to model incompressible laminar,

compressible laminar and incompressible turbulent boundary layers. Modelling

of compressible turbulent boundary layers may be included at a later time.

Microsoft Visual Basic was used to create the software toolkit. Code for the

compressible laminar and incompressible turbulent flow modules was translated

from FORTRAN code provided in Schetz [1] and a graphical user interface (GUI)

was added for intuitive use. Code for the incompressible laminar flow module

was written from fundamental equations, discretized by the author.

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BOUNDARY LAYER ANALYSIS TOOLKIT

1. Introduction

For most situations faced by practicing engineers, the boundary layer equations

are too complex to be solved analytically. To address that problem, many

numerical techniques have been devised for solving the boundary layer

equations. However, computer implementation of those techniques has been

largely relegated to complex and expensive software packages. Research has

indicated that an intuitive, integrated software toolkit for solving boundary layer

equations is not available as freeware. A toolkit of the type provided by this

project is warranted, since many practicing engineers require a desktop aid,

which can be used to perform quick simulations and offer fast turn-around of

results. This is an attractive alternative when contrasted with having to build an

overly complex and computationally top heavy CFD (Computational Fluid

Dynamic) model to perform a desktop calculation.

2. Definitions

A. Definition of the Boundary Layer

When a fluid flows over a solid surface, frictional drag between the fluid and the

solid creates a layer of slower-moving fluid near the solid. That layer is termed

the boundary layer. In the boundary layer, momentum is transferred between

molecules moving with different velocities. This momentum transfer is

characterized by the viscosity, a physical property of the fluid. The fluid viscosity

plays an important part in the differential equations describing the fluids behavior

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in the boundary layer. Together, those equations are termed the boundary layer

equations. Outside the boundary layer, molecules are all moving with a similar

velocity (the free-stream velocity), no significant momentum transfer occurs and,

thus, the fluid viscosity does not appear in the differential equations. For that

reason, the region outside the boundary layer is termed the inviscid region.

B. Definition of Variables

Throughout this report, and in the software, certain variables are used without

explanation. The definition of those variables is provided in table 2.1.

Table 2.1 Definition of Variables

x Distance along plate

y Distance across boundary layer

u Velocity component in x-direction

v Velocity component in y-direction

T Temperature

K Turbulent kinetic energy

Velocity boundary layer thickness

T Thermal boundary layer thickness

Ue Velocity at the edge of the boundary layer

Te Temperature at the edge of the thermal boundary layer

Density

Viscosity

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Kinematic viscosity

Second coefficient of viscosity

Shear stress

Cf Skin friction [dimensionless shear stress at the wall (y=0)]

St Stanton number (dimensionless heat transfer coefficient at the wall)

Re Reynolds number (ratio of inertial forces to viscous forces)

Gamma, the ratio of specific heats P

V

C

C

h Enthalpy

pe Pressure at the edge of the velocity boundary layer

k Thermal conductivity

q Heat flux

3. The Boundary Layer Equations

A. Fundamental Equations

The fundamental equations of viscous flow are consituted by three conservation

laws conservation of mass (continuity), conservation of momentum (Navier-

Stokes), and conservation of energy. Those equations are presented without

derivation below. For the purposes of this project, buoyancy terms have been

neglected. An excellent derivation of these equations is presented in White [2].

Continuity: 0D

UDt

+ =

i (2.1)

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Momentum:ji

ij

j j i

uuDUg p U

Dt x x x

= + + +

i (2.2)

Energy: ( ) ji iij ijj i

uuDh Dp k T p U Dt Dt x x x

= + + + + +

i i

j

u (2.3)

Definitions of operators such as the total derivativeD

Dt

, the Kronecker delta

, and the del operator ( can be found in any Calculus text. Also, tensor

notation is used in the above equations to present them in an efficient manner.

) )

( ij

B. Boundary Layer Approximations

The assumption of steady-state and the application of certain approximations

consistent with boundary layers yield the boundary layer equations. These

approximations are valid for Re >> 1 and are listed in table 3.1.

Table 3.1 Boundary Layer Approximations

(2.4)x

(2.5)v u

u

x y

u

(2.6)

v

x y

v

(2.7)

C. Boundary Layer Equations

The two-dimensional boundary layer equations for steady-state, compressible

flow are:

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Continuity: ( ) ( )u vx y

+

0= (2.8)

x momentum:1 1

e

pu uu vx y x

+ + y (2.9)

y momentum: 0p

y

(2.10)

Energy: edph h q

u v ux y dx y

+ +

u

y (2.11)

Equations (2.9) and (2.11) can be simplified using Bernoulis equation along the

streamline that forms the boundary between the viscid and inviscid regions.

Ignoring gravity, that equation states2

2

Up Constant

+ = along a streamline of

the flow. Then, taking a derivative and inserting subscripts to indicate edge

conditions, that equation becomes

e

e

dp dU U

dx dx= e (2.12)

Thus, equation (2.9) becomes

x momentum:1e

e

dUu uu v U

x y dx y

+ +

(2.13)

Equations (2.8) through (2.11) are equally valid for laminar and turbulent flow. In

application, the difference arises from the way in which the shear stress, , and

the heat flux, q, are modeled. For laminar flow,u

y

=

and

Tq k

y

=

, resulting

in equations (2.14) through (2.21). For turbulent flow, the transport coefficients

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and k are modified to include additional terms so that ( )Tu

y

= +

and

( )TT

q k k y

= + . The boundary layer equations for incompressible turbulent flow

are given by equations (2.22) through (2.25). Compressible turbulent flow is not

included in the scope of this report.

Steady-state, compressible, laminar flow

Continuity: ( ) ( )u vx y

+

0= (2.14)

x momentum:1e

e

dUu u uu v U

x y dx y y

+ +

(2.15)

y momentum: 0p

y

(2.16)

Energy:

2

ep e

dUT T T uc u v uU k

x y dx y y y

+ + +

(2.17)

Steady-state, incompressible, laminar flow

Continuity: 0u v

x y

+ =

(2.18)

x momentum:2

2

ee

dUu uu v U

x y dx y

+ +

u

(2.19)

y momentum: 0p

y

(2.20)

Energy:

22

2

p p

T T k T uu v

x y c y c y

+ +

(2.21)

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Steady-state, incompressible, turbulent flow

Continuity: 0u v

x y

+ =

(2.22)

x momentum: ( )1e

e

dUu u uu v U

x y dx y y

+ + +

T

(2.23)

y momentum: 0p

y

(2.24)

Energy: ( ) ( )2

p T

T T T uc u v k k

x y y y y

+ + + +

T

(2.25)

4. Turbulence Models

Though it is listed above, the energy equation was not included in the scope of

this projects turbulence module. Three turbulence models were implemented in

the software the eddy viscosity model, the mixing length model and the

turbulent kinetic energy (TKE) model.

Turbulent boundary layers are typically decomposed into an inner layer, in which

viscous shear dominates, and an outer layer, in which turbulent (eddy) shear

dominates. The turbulence models use different parameters for the different

layers.

D. Eddy viscosity

The parameterT is defined as the eddy viscosity. For convenience and ease of

comparison, it can be used with the other models also.

(3.1)*,T outer eC U 0.016C=

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(3.2)*,T inner u y 0.41 =

where the friction velocity is * wu

= (3.3)

E. Mixing length

The mixing length parameter is l and is related to as follows.m T

2T mu

ly

(3.4)

where

*

, 1yuA

m innerl y e

(3.5)26A =

where and u*are as described in equations (3.2) and (3.3).

And (3.6), 0.09m outerl

F. Turbulent Kinetic Energy

This model introduces a new variable, the turbulent kinetic energy K, and a new

equation that is coupled with the continuity and momentum equations [(2.22) and

(2.23)]. The equation for K is

32

2T T

D

k

K K K u Ku v C

x y y y y

+ + l

(3.7)

where CD is not the drag coefficient, but an empirical constant, . Also,

and is related to l by

0.09DC

1k l m

1

4DC l l

= m (3.8)

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and 2T mu

ly

(3.9)

so that equations (3.5) and (3.6) can be used to estimate and .T l

5. Discretization

All flow types treated in the software were discretized using the finite difference

method. Implicit solutions were implemented in all cases to avoid instability. The

incompressible turbulent flow and compressible laminar flow modules were

translated from FORTRAN code in Schetz [1]. However, the incompressible

laminar equations were discretized by the author. The method of discretization is

presented below. Throughout the development, subscript i represents a nodes

position in the x-direction, and subscript j represents a nodes position in the y-

direction. Table 5.1 lists the discretized forms of important differentials.

Table 5.1 Discretized Forms of Important Differentials (Adapted from Chung [3])

Method Approximations Error Equation

Forward Difference, ,i j+1 i ju uu

y y

( )

2, ,

22

2i j+2 i j+1 i ju uu

y y

+

,u

( )O x (4.1)

(4.2)

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Backward Difference, ,i j i j -1u uu

y y

( )

2

, ,22

2i j+2 i j+1 i ju uuy y

+ ,u

( )O x (4.3)

(4.4)

Central Difference, ,

2

i j+1 i j -1u uu

y y

( )

2, ,

22

2i j+1 i j i j-1u u uu

y y

+

,

2( )O x (4.5)

(4.6)

G. Continuity Equation

This equation is modelled with a forward difference approximation inu

x

and a

backward difference approximation forv

y

. However, while the simplest

application of those approximations would result in

, , , ,0

i+1 j i j i+1 j i+1 j-1u u v v

x y

+

=

, numerical accuracy would suffer because the

first term is level j, while the second term is level j - . In order to place both

terms on the same level and thereby improve accuracy, the first term is

decomposed into the average of its value at level j and its value at level j-1.

Then, the discretized equation becomes

, , , , , ,10

2

i+1 j i j i+1 j-1 i j-1 i+1 j i+1 j-1u u u u v v

x x y

+ +

=

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and (1, 1, 1 , , , ,2

i j i j i+1 j i j i+1 j-1 i j -1

yv v u u u u

x+ +

= +

) (4.7)

H. Momentum Equation

Using forward difference in the x-derivatives and central difference in the y-

derivatives, momentum equation (2.19) becomes

( )

2 2

, , , , , , ,, ,

, , 2

2

2 2

i+1 j i j i+1 j+1 i+1 j-1 i+1 j+1 i+1 j i+1 j-1e i+1 e i

i j i j

u u u u u u uU Uu v

x y x y

+ + = +

where Ive used the fact that2

12

e ee dU dU

dx dx=U .

Then, gathering like terms,

( ) ( ) ( ), , ,

, , ,2 2

2 2 2

, , ,

2

2 2

2

i j i j i j

i+1 j+1 i+1 j i+1 j 1

i j e i+1 e i

v u vu u u

y x yy y

u U U

x x

+ + +

= +

2y

(4.8)

Equation (4.8) can be expressed in matrix format like below.

2 22 2

33 3 3

44 4 4

55 5 5

0 0

0 0

0 0 0

0 0

. .. . . . . . .

0 0

0 0

i+1,2 i+1,1

i+1,3

i+1,4

i+1,5

i+1,N -2 N -2N -2 N -2 N -2

i+1,N -1 N -1 N -1 i+1,N N -1 N -1

u u

u

u

u

u

u u

=

(4.9)

Where

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( )

( )

( )

,

2

,

2

,

2

2 2 2

, , ,

2

2

2

2

i j

j

i j

j

i j

j

i j e i+1 e i

j

v

y y

u

x y

v

y y

u U U

x x

=

= +

=

= +

(4.10)

Equation (4.9) can then be solved using the Thomas algorithm for tri-diagonal

matrices.

I. Energy Equation

The energy equation is much like the momentum equation in form. Using

forward difference in the x-derivatives and central difference in the y-derivatives,

equation (2.21) becomes

( )

2

, , , , , , , , ,

, , 22

2 2

i+1 j i j i+1 j+1 i+1 j-1 i+1 j+1 i+1 j i+1 j-1 i j+1 i j-1

i j i j

p

T T T T T T T u uu vx y cy

+ + = + y

where Ive used the thermal diffusivity,p

k

c= . Gathering like terms,

( ) ( ) ( ), , ,

, , ,2 2

2

, , , ,

2

2 2

2

i j i j i j

i+1 j+1 i+1 j i+1 j 1

i j i j i j+1 i j-1

p

v u vT T T

y x yy y

u T u ux c y

+ + +

= +

2y

(4.11)

Equation (4.11) can be expressed in matrix format like below

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2 22 2

33 3 3

44 4 4

55 5 5

0 0

0 0

0 0 0

0 0

. .. . . . . . .

0 0

0 0

i+1,2 i+1,1

i+1,3

i+1,4

i+1,5

i+1,N -2 N -2N -2 N -2 N -2

i+1,N -1 N -1 N -1 i+1,N N -1 N -1

T T

T

T

T

T

T T

=

(4.12)

With

( )

( )

( )

,

2

,

2

,

2

2

, , , ,

2

2

2

2

i j

j

i j

j

i j

j

i j i j i j+1 i i-1

j

p

v

yy

u

x y

v

y y

u T u u

x c y

=

= +

=

= +

(4.13)

6. Software Operation

J. Overview

The Boundary Layer Analysis Toolkit (BLT) is a desktop tool allowing engineers

to solve the boundary layer equations for many different flow parameters

including velocity components, temperature, thermal and velocity boundary layer

thickness and skin friction.

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K. Constraints

Flow must be over a flat or gently curving plate. Flow must be in the laminar

regime or, if turbulent, flow must be incompressible. For turbulent flows, solving

for temperature is not included.

L. Steady-state, Incompressible, Laminar (SIL) Module

Module SIL can be accessed at any time by pressing Ctrl-L or by pulling down

the File menu, then choosing New Form, then Steady Incompressible

Laminar. Figure 6.1 shows the SIL input form.

The SIL input form contains check boxes and text boxes in which to specify theflow. There are 5 frames grouping these controls by type. They are:

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Figure 6.1 SIL Input Form

Computational Parameters Frame

Control 1. Include temperature in calculations? Because the energy equation

is decoupled from the other equations for incompressible flow, the user has the

option to ignore temperature considerations. Checking the No box will disable

all controls specific to the energy equation, such as thermal conductivity and

temperature boundary conditions.

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Control 2. Extent to carry calculation in y-direction If Auto is checked, the

program estimates the extent of the boundary layer using the Blasius solution

5.0

Rex

x = .

Controls 3 and 4. Number of nodes Use these controls to specify how many

points in the flow field are to be calculated. Generally speaking, larger numbers

mean greater accuracy but longer calculation time.

Physical Properties Frame

Controls 5 and 6. Limits of calculation in x-direction.

Fluid Properties Frame

Control 7. Viscosity Fluid viscosity, normally indicated by the symbol .

Control 8. Density Fluid density. If temperature is not considered, only the

kinematic viscosity is used in the calculations. Any combination of viscosity and

density with the same ratio

will produce the same solution.

Control 9. Thermal conductivity Only enabled if temperature calculations are

requested.

Control 10. Specific heat Only enabled if temperature calculations are

requested.

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Inlet Conditions Frame

Control 11. Source of inlet conditions If user chooses Auto, inlet conditions

for u (streamwise velocity) are developed in a cubic profile. Inlet conditions for v

are taken equal to v(x=xi,y=0) and for T, if required, inlet conditions are equal to

T(x=xi,y=0).

Controls 12, 13, 14. User-specified inlet conditions These controls will accept

either constant values or functions of y.

Boundary Conditions Frame

Control 15. Freestream velocity - This control will accept a constant or a

function of x.

Control 16. Wall velocity Velocity of injected fluid. This control will accept a

constant or a function of x.

Control 17. Wall temperature Temperature of the plate surface. This control

will accept a constant or a function of x.

Control 18. Freestream temperature Temperature of the freestream fluid. This

control will accept a constant or a function of x.

Control 19. First transition position If the boundary conditions change during

the calculation, this control can be used to specify the point at which the change

occurs. In the example of figure 6.1, the freestream velocity changes at x=1.

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Controls 20 through 23. New boundary conditions at the first transition In the

example of figure 6.1, the freestream velocity changes from 10 to a linearly

decreasing function of x at x=1.

Controls 24 through 28. Second transition position and new boundary conditions

If boundary conditions change at a second point after the first transition, the

point of transition and the new boundary conditions are specified in these

controls.

Control 29. Run button Pressing this button begins the calculation.

Output

After calculations have been completed, the SIL form expands to reveal output

choices. As shown in figure 6.2, there are two choices. One choice is an internal

two-dimensional line plot, shown in figure 6.3. The second choice is to export

data to an ASCII data file, designed for TecPlot, but readable by any program

that can import ASCII data (MS Excel and MatLab are examples). All variables

can be exported, but line plots cant be drawn for variables that are functions of

both x and y.

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20

Figure 6.2 SIL Form With Output Choices


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Boundary Layer Analysis ToolkitAn Independent Study Report

21

Richard Getze

Figure 6.2 SIL Output Example


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M. Steady-state, Compressible, Laminar (SCL) Module

Module SCL can be accessed at any time by pressing Ctrl-C or by pulling down

the File menu, then choosing New Form, then Steady Compressible

Laminar. Figure 6.4 shows the SCL input form with output choices. Many of the

input controls on the SCL form are the same as described above for the SIL

form. Only controls exhibiting differences will be described below.

Figure 6.4 SCL Form With Output Choices

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Physical Properties Frame

Mach Number User enters Mach number of freestream flow.

Fluid Properties Frame

Gas pull-down This control allows user to select type of gas. This information

is used in computing viscosity as a function of temperature.

Prandtl number, Gamma (Cp/Cv), Kinematic viscosity These controls are self-

explanatory

Boundary Conditions Frame

Freestream density The density of the freestream flow. This control will accept

either a constant or a function of x.

Temperature units Required in this module only for determining values of

certain constants, intrinsic to the program.

Output

Variables available for output now include density, the Stanton number and the

ratio of specific heats (gamma).

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N. Steady-state, Incompressible, Turbulent (SIT) Module

Module SIT can be accessed at any time by pressing Ctrl-T or by pulling down

the File menu, then choosing New Form, then Steady Incompressible

Turbulent. Figure 6.5 shows the SIT input form with output choices. Many of

the input controls on the SIT form are the same as described above for the SIL

form. Only controls exhibiting differences will be described below.

Figure 6.5 SIT Form With Output Choices

Computational Parameters Frame

Model Here the user selects a computational model. See section 4,

Turbulence Models, for a description of the different types.

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Boundary Conditions Frame and Inlet Conditions Frame

Controls allowing specification of v are not included because of a large number of

intrinsic computational constants whose value changes dramatically based on

the value of injection or suction velocity.

7. Software Validation

The steady, incompressible, turbulent model was validated against Spaldings log

law of the wall. The results of that validation are displayed in figure 7.1.

Figure 7.1 Validation of SIT Module

0

5

10

15

20

25

30

1 10 100 1000 10000

y+ = yu*/nu

u+

=

u/u*

SIT Module Spalding's Log Law of the Wall

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The log law of the wall is given by

( ) ( )2 3

1 2 6

B uu u

y u e e u

++ +

+ + +

= +

where and are defined in figure 7.1 and and B are empirical constants

with values 0.4 and 5.5, respectively.

y+ u+

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BIBLIOGRAPHY

1. Schetz, J.A., Boundary Layer Analysis, Prentice Hall, 1993

2. White, F.M. Viscous Fluid Flow, 2nd Ed., McGraw-Hill, 1991

3. Chung, T.J., Computational Fluid Dynamics, Cambridge University Press,

2002

4. Wilcox, David C., Turbulence Modeling for CFD, 2nd Ed., DCW Industries,

1998

5. Potter, Merle C. and Wiggert, David C., Mechanics of Fluids, 2nd Ed.,

Prentice Hall, 1997

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APPENDIX A

CODE FOR STEADY INCOMPRESSIBLE LAMINAR FLOW (SIL) MODULE

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ReDim x(m), y(n)

'*********************************************'Insert boundary conditions in matrices

'*********************************************'First, wall conditions and free-stream conditionsFor i = 1 To m

u(1, i) = 0v(1, i) = vxy0(i)u(n, i) = Ue(i)

Next i'Next, inlet conditionsIf auto Then 'If user asks for auto velocity profile use cubic

'If starting at x=0, then no profileIf xi > 0 Then

del = EST_DELTA(xi, Ue(1), nu)For j = 1 To n

yod = (j - 1) * dy / delIf yod > 1 Then

u(j, 1) = Ue(1)Else

u(j, 1) = Ue(1) * (1.5 * yod - 0.5 * yod ^ 3)End Ifv(j, 1) = 0

Next jElse

For j = 1 To nu(j, 1) = Ue(1)v(j, 1) = 0

Next jEnd If

ElseFor j = 1 To n

u(j, 1) = ux0y(j)v(j, 1) = vx0y(j)

Next jEnd If

'initialize x and y arraysx(1) = xiy(1) = 0

'*********************************************'Start moving downstream'*********************************************

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For i = 1 To m - 1'Update progress barfm_progress.progbar.value = i'*********************************************

'NSLx yields streamwise velocity (u) distribution.x(i + 1) = x(i) + dx'matrix elements for TDMAFor j = 2 To n - 2

'lower diagonala(j - 1) = -1 * v(j + 1, i) / (2 * dy) - nu / ((dy) ^ 2)'upper diagonalc(j - 1) = v(j, i) / (2 * dy) - nu / ((dy) ^ 2)'main diagonalb(j - 1) = u(j, i) / dx + 2 * nu / ((dy) ^ 2)'right side vector

d(j - 1) = (u(j, i) ^ 2) / dx + (Ue(i + 1) ^ 2 - Ue(i) ^ 2) / (2 * dx)Next j'add last element to main diagonalb(n - 2) = u(n - 1, i) / dx + 2 * nu / ((dy) ^ 2)'modify first element and add last element to'right side vectora1 = -1 * v(2, i) / (2 * dy) - nu / dy ^ 2d(1) = d(1) - a1 * u(1, i + 1)cn = v(n - 1, i) / (2 * dy) - nu / dy ^ 2d(n - 2) = (u(n - 1, i) ^ 2) / dx + (Ue(i + 1) ^ 2 - Ue(i) ^ 2) / (2 * dx) - cn * u(n, i

+ 1)

'Solve for u(i+1) using TDMAu_iter = TDMA(c, a, b, d)'*********************************************'Check for transition to turbulence REx 1000000# And Not turb Then

turb = Truemsg = "Warning! Transition to turbulence at x = " & x(i) _

& ". Results beyond that point may not be valid."check = MsgBox(msg, vbOKCancel)If check = 2 Then Exit Sub

End If'*********************************************'Record wall shear, skin friction and check for separationwallshear(i) = mu / (2 * dy) * (4 * u(2, i) - u(3, i))cf(i) = wallshear(i) / (0.5 * rho * Ue(i) ^ 2)If wallshear(i) < 0.0001 Then

MsgBox "Separation occurs at x=" & (i - 1) * dxUnload fm_progress

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Exit SubEnd If'*********************************************'Transfer values into u matrix and record boundary layer thickness

For z = 2 To n - 1u(z, i + 1) = u_iter(z - 1)If u_iter(z - 1) > 0.99 * Ue(i) And deltind(i + 1) = 0 Then

deltind(i + 1) = zdelta(i + 1) = z * dy

End IfNext z'*********************************************'COM yields transverse velocity (v) distribution.For j = 2 To n

v(j, i + 1) = v(j - 1, i + 1) - dy / (2 * dx) * _

(u(j, i + 1) - u(j, i) + u(j - 1, i + 1) - u(j - 1, i))y(j) = y(j - 1) + dy

Next jNext i'*********************************************'Store the data'*********************************************'store_uv_data x, y, u, v, delta, cfglobal_x = xglobal_y = yglobal_u = u

global_v = vglobal_delta = deltaglobal_Cf = cf'*********************************************'Recover memory'*********************************************Erase u, v, deltind, delta, wallshear, cfErase a, b, c, d, u_iterErase x, y

End Sub

Public Sub SIL_T(u() As Single, v() As Single, Tx0y() As Single, Txy0() AsSingle, Te() As Single, nu As Single, k As Single, rho As Single, cp As Single, _

xi As Single, dx As Single, dy As Single)'*********************************************'Solution for Steady, Incompressible, Laminar boundary layer flow.'Temperature solution only. Requires velocity solution as input.'Solution for velocity distribution is implemented separately.

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'Implicit scheme is implemented.'*********************************************

Dim m As Integer, n As Integer, i As Integer, j As IntegerDim T() As Single, alpha As Single, z As Integer

Dim a() As Single, b() As Single, c() As Single, d() As SingleDim a1 As Single, cn As SingleDim T_iter() As Single, T_del_flag As BooleanDim T_deltaind() As Integer, T_delta() As SingleDim x() As Single, y() As Single

'alpha is thermal diffusivityalpha = k / (rho * cp)'m is number of stations in streamwise (x) directionm = UBound(Txy0)'n is number of stations across boundary layer (y)

n = UBound(Tx0y)'Dimension arraysReDim T(n, m)ReDim a(n - 3), b(n - 2), c(n - 3), d(n - 2), T_iter(n - 2)ReDim T_deltaind(m), T_delta(m)ReDim x(m), y(n)

'Construct x and y arraysx(1) = xiFor i = 2 To m

x(i) = x(i - 1) + dx

Next iy(1) = 0For j = 2 To n

y(j) = y(j - 1) + dyNext j

'*********************************************'Insert boundary conditions in matrix'*********************************************'First, wall conditions and free-stream conditionsFor i = 1 To m

T(1, i) = Txy0(i)T(n, i) = Te(i)

Next i'Next, inlet conditionsFor j = 1 To n

T(j, 1) = Tx0y(j)Next j'*********************************************

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'Start moving downstream'*********************************************For i = 1 To m - 1

fm_progress.progbar.value = m + i

'*********************************************'COE yields temperature distribution.'matrix elementsFor j = 2 To n - 2

'lower diagonala(j - 1) = -1 * v(j + 1, i) / (2 * dy) - alpha / dy ^ 2'upper diagonalc(j - 1) = v(j, i) / (2 * dy) - alpha / dy ^ 2'main diagonalb(j - 1) = u(j, i) / dx + 2 * alpha / dy ^ 2'right side vector

d(j - 1) = u(j, i) * T(j, i) / dx + nu / cp * ((u(j + 1, i) - u(j - 1, i)) / (2 * dy)) ^ 2Next j'add last element to main diagonalb(n - 2) = u(n - 1, i) / dx + 2 * alpha / dy ^ 2'modify first element and add last element to'right side vectora1 = -1 * v(2, i) / (2 * dy) - alpha / dy ^ 2d(1) = d(1) - a1 * T(1, i + 1)cn = v(n - 1, i) / (2 * dy) - alpha / dy ^ 2d(n - 2) = u(n - 1, i) * T(n - 1, i) / dx + nu / cp * ((u(n, i) - u(n - 2, i)) / (2 * dy))

^ 2 - cn * T(n, i + 1)

'Solve for T(i+1) using TDMAT_iter = TDMA(c, a, b, d)'Transfer values into T matrix and mark position of boundary layerT_del_flag = FalseFor j = 2 To n - 1

T(j, i + 1) = T_iter(j - 1)If Abs(Te(i) - T_iter(j - 1)) >= 0.01 * Te(i) Then

T_del_flag = TrueElseIf T_del_flag And Abs(Te(i) - T_iter(j - 1)) < 0.01 * Te(i) And

T_deltaind(i + 1) = 0 ThenT_deltaind(i + 1) = j

T_delta(i + 1) = j * dyEnd If

Next jNext i'*********************************************'Store the data'*********************************************'store_T_data x, y, T, T_delta

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global_T = Tglobal_T_delta = T_delta'*********************************************'Recover memory

'*********************************************Erase TErase a, b, c, d, T_iterErase T_deltaind, T_deltaErase x, y

End Sub

Public Function EST_DELTA(x As Single, u As Single, nu As Single) As SingleDim REx As Single, d As Single

'Esimate boundary layer thickness using Blasius equationREx = u * x / nud = x * 5 / Sqr(REx)EST_DELTA = d

End Function

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APPENDIX B

CODE FOR STEADY COMPRESSIBLE LAMINAR FLOW (SCL) MODULE

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Option Base 1Option Explicit

Dim xm1 As Single, gamma() As Single

'Dim xmif As Single, Tinf As Single

Public Sub CLBL6(xi As Single, xf As Single, Tw() As Single, Te() As Single, _Ue() As Single, xmif As Single, ux0y() As Single, vx0y() As Single, vxy0() As

Single, _cnuinf As Single, nmax As Integer, mmax As Integer, dy As Single, Pr As

Single, _gamma1 As Single, rhoe() As Single, S As Single)

'2D compressible boundary layer computation from Schetz.'1st order implicit, lagged coefficient.'Laminar with Sutherland viscosity law

'Equations are dimensionless, fluid is assumed ideal gas

Dim u() As Single, v() As Single, delta() As SingleDim H() As Single, a() As Single, b() As Single, c() As SingleDim R() As Single, Cmu() As Single, rho() As SingleDim cf() As Single, St() As Single, temp() As SingleDim L As Single, Re As Single, dx As Single, ncon As SingleDim T1 As Single, nnx As Integer, deltaind() As IntegerDim deno As Single, mt As Integer, rhoua As Single, rhoub As Single, r1 As

SingleDim He() As Single, pe() As Single, T_delta() As Single

Dim i As Integer, j As Integer, Tref As Single, RR As Single, Haw() As SingleDim mest As Integer, mmm As Integer, T_deltaind() As IntegerDim u_temp() As Single, H_temp() As Single, temp_temp() As SingleDim Uinf As Single, Tinf As Single, rhoinf As SingleDim x() As Single, y() As Single, T_del_flag As Boolean

L = 1dx = (xf - xi) / (nmax - 1)Re = L * Ue(1) / cnuinf'S is Sutherland constant for viscosityT1 = S / Te(1) 'for Sutherland viscosity law

'Redim variablesReDim u(mmax, nmax), v(mmax, nmax), H(mmax, nmax), u_temp(mmax),

H_temp(mmax)ReDim a(mmax), b(mmax), c(mmax), R(mmax), Cmu(mmax), rho(mmax,

nmax)ReDim cf(nmax), St(nmax), gamma(mmax), temp(mmax, nmax),

temp_temp(mmax)

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ReDim He(nmax), Haw(nmax), pe(nmax), delta(nmax), deltaind(nmax)ReDim T_delta(nmax), T_deltaind(nmax), x(nmax), y(mmax)

'Gamma is a function of temperature and equals the free stream ratio of

'specific heatsgamma(1) = gamma1xm1 = (gamma(1) - 1) * xmif ^ 2Uinf = Ue(1)Tinf = Te(1)rhoinf = rhoe(1)For i = 1 To nmax

Ue(i) = Ue(i) / UinfTw(i) = Tw(i) / TinfTe(i) = Te(i) / TinfHe(i) = Te(i) / xm1

rhoe(i) = rhoe(i) / rhoinfNext ipe(1) = 1 / (gamma(1) * xmif ^ 2)

For j = 1 To mmaxu(j, 2) = ux0y(j)u(j, 1) = Ue(1)v(j, 2) = vx0y(j)v(j, 1) = vx0y(j)H(j, 2) = He(2)H(j, 1) = He(1)

Tref = xm1 * H(j, 1)temp(j, 2) = Treftemp(j, 1) = Trefgamma(j) = gamma(1)Cmu(j) = (1 + T1) * Tref ^ 1.5 / (Tref + T1)rho(j, 1) = 1 / (xm1 * H(j, 2))rho(j, 2) = rho(j, 1)

Next j

u(1, 2) = 0u(1, 1) = 0

H(1, 2) = Tw(2) / xm1H(1, 1) = H(1, 2)Tref = xm1 * H(1, 1)temp(1, 2) = Treftemp(1, 1) = TrefCmu(1) = (1 + T1) * Tref ^ 1.5 / (Tref + T1)rho(1, 1) = 1 / (xm1 * H(1, 2))rho(1, 2) = rho(1, 1)

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'Initial profilesmest = 2mmm = mest + 3

For j = 2 To mmaxu(j, 1) = ux0y(j)v(j, 1) = vx0y(j)H(j, 1) = He(1)rho(j, 1) = 1 / (xm1 * H(j, 1))

Next j

cf(1) = (4 * u(2, 1) - u(3, 1)) * Cmu(1) / (Ue(1) ^ 2 * dy * Re * rhoe(1))RR = Pr ^ 0.5For i = 1 To nmax

Haw(i) = He(i) * (1 + 0.195 * RR * xmif ^ 2)

Next iSt(1) = (3 * H(1, 1) - 4 * H(2, 1) + H(3, 1)) * Cmu(1) / (2 * Pr * Ue(1) * dy * Re *

rhoe(1) * (H(1, 2) - Haw(1)))

For i = 2 To nmaxfm_progress.progbar.value = innx = iu(mmax, i - 1) = Ue(i - 1)H(mmax, i - 1) = He(i - 1)rho(mmax, i - 1) = 1 / (xm1 * H(mmax, i - 1))pe(i) = 1 / (gamma(1) * xmif ^ 2)

v(1, i - 1) = vxy0(i - 1)

b(1) = 1c(1) = 0R(1) = 0a(mmax) = 0b(mmax) = 1R(mmax) = Ue(1)deno = Re * dy ^ 2

For j = 2 To mmax - 1

a(j) = -0.5 * rho(j, i - 1) * v(j, i - 1) / dy - Cmu(j - 1) / denob(j) = rho(j, i - 1) * u(j, i - 1) / dx + 0.5 * (Cmu(j - 1) + Cmu(j)) * 2 / denoc(j) = 0.5 * rho(j, i - 1) * v(j, i - 1) / dy - Cmu(j) / denoR(j) = rhoe(i) * 0.5 / dx * (Ue(i) ^ 2 - Ue(i - 1) ^ 2) + rho(j, i - 1) * u(j, i - 1) ^

2 / dxNext j

u_temp = trid(mmax, a, b, c, R, u_temp)

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For j = 1 To mmaxu(j, i) = u_temp(j)If u_temp(j) > 0.99 * Ue(i) And deltaind(i) = 0 Then

deltaind(i) = jdelta(i) = j * dy

End IfNext j

b(1) = 1c(1) = 0R(1) = H(1, i - 1)a(mmax) = 0b(mmax) = 1R(mmax) = He(i - 1)

deno = Pr * Re * dy ^ 2

For j = 2 To mmax - 1a(j) = -0.5 * rho(j, i - 1) * v(j, i - 1) / dy - Cmu(j - 1) / denob(j) = rho(j, i - 1) * u(j, i - 1) / dx + 0.5 * (Cmu(j - 1) + Cmu(j)) * 2 / denoc(j) = 0.5 * rho(j, i - 1) * v(j, i - 1) / dy - Cmu(j) / denoR(j) = -rhoe(i) * 0.5 / dx * (Ue(i) ^ 2 - Ue(i - 1) ^ 2) * u(j, i - 1) + rho(j, i - 1) *

u(j, i - 1) * H(j, i - 1) / dx + _Cmu(j) * (u(j + 1, i) - u(j - 1, i)) ^ 2 / (4 * Re * dy ^ 2)

Next j

H_temp = trid(mmax, a, b, c, R, H_temp)

For j = 1 To mmaxH(j, i) = H_temp(j)

Next j

mt = mest - 8If mt < 1 Then mt = 1For j = mt To mmax

If u(j, i) > 0.88 * Ue(i) Thenmest = j

mmm = mest + 3Exit For

End IfNext j

For j = 1 To mmaxtemp_temp(j) = temp(j, i - 1)

Next j

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temp_temp = interp(nnx, mmm, gamma, temp_temp, H_temp, xmif, Tinf)

T_del_flag = False

For j = 1 To mmaxtemp(j, i) = temp_temp(j)If Abs(Te(i) - temp_temp(j)) >= 0.01 * Te(i) Then

T_del_flag = TrueElseIf T_del_flag And Abs(Te(i) - temp_temp(j)) < 0.01 * Te(i) And

T_deltaind(i) = 0 ThenT_deltaind(i) = jT_delta(i) = j * dy

End IfNext j

For j = 2 To mmax - 1Cmu(j) = (1 + T1) * (0.5 * (temp(j, i) + temp(j - 1, i))) ^ 1.5 / _

(0.5 * (temp(j, i) + temp(j - 1, i)) + T1)rho(j, i) = pe(i) * gamma(j) / (0.5 * (H(j - 1, i) + H(j, i)) * (gamma(j) - 1))

Next j

For j = 2 To mmax - 1rhoua = rho(j, i - 1) * u(j, i - 1)rhoub = rho(j - 1, i - 1) * u(j - 1, i - 1)r1 = rho(j - 1, i) / rho(j, i)v(j, i) = r1 * v(j - 1, i) - (0.5 * dy / dx) * (u(j, i) + r1 * u(j - 1, i) - (rhoua +

rhoub) / rho(j, i))If u(j, i) = u(j - 1, i) And u(j, i - 1) = u(j - 1, i - 1) And v(j, i) = v(j - 1, i) Then

v(j, i) = 0End If

Next j

cf(i) = (4 * u(2, i) - u(3, i)) * Cmu(1) / (Ue(i) ^ 2 * dy * Re * rhoe(i))St(i) = (3 * H(1, i) - 4 * H(2, i) + H(3, i)) * Cmu(1) / (2 * Pr * Ue(i) * dy * Re *

rhoe(i) * (H(1, i) - Haw(i)))Next i

x(1) = xiFor i = 2 To nmax

x(i) = x(i - 1) + dxNext iy(1) = 0For j = 2 To mmax


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'Reconstitute dimensional variablesFor i = 1 To nmax

For j = 1 To mmax

u(j, i) = u(j, i) * Uinfv(j, i) = v(j, i) * Uinftemp(j, i) = temp(j, i) * Tinfrho(j, i) = rho(j, i) * rhoinf

Next jNext i

'Transfer procedure-level variables into global-level variablesglobal_x = xglobal_y = yglobal_u = u

global_v = vglobal_T = tempglobal_rho = rhoglobal_Cf = cfglobal_St = Stglobal_gamma = gammaglobal_delta = deltaglobal_T_delta = T_delta

'Free up memoryErase u(), v(), H(), u_temp(), H_temp(), a(), b(), c(), R(), Cmu()

Erase rho(), cf(), St(), gamma(), temp(), temp_temp(), x(), y()Erase He(), Haw(), pe(), delta(), deltaind(), T_delta(), T_deltaind()

End Sub

Private Function interp(nnx As Integer, mm As Integer, gamma() As Single,temp() As Single, _

H() As Single, xmif As Single, Tinf As Single) As Single()

Dim xt(), ygam()Dim i As Integer, k As Integer, j As Integer, m As Integer

Dim xint As Single, yout As Single, term As Single

xt = Array(273, 373, 474, 573, 673, 773, 873, 973)ygam = Array(1.401, 1.397, 1.39, 1.378, 1.368, 1.357, 1.346, 1.338)

For m = 2 To mm + 10j = 0temp(m) = xm1 * H(m) * Tinf

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80:j = j + 1xint = temp(m)If j > 160 Then GoTo 70

yout = 0For k = 1 To 8

term = ygam(k)For i = 1 To 8

If k i Thenterm = term * (xint - xt(i)) / (xt(k) - xt(i))

End IfNext iyout = yout + term

Next k

temp(m) = (1.4 / yout) * (yout - 1) * xmif ^ 2 * H(m) * TinfIf Abs(temp(m) - xint) < 0.005 Then GoTo 70GoTo 80

70:gamma(m) = youttemp(m) = temp(m) / Tinf

Next minterp = temp

End Function

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APPENDIX C

CODE FOR STEADY INCOMPRESSIBLE TURBULENT FLOW (SIT) MODULE

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Option Base 1Option ExplicitDim Re As SingleDim tmu() As Single, xx() As Single

Dim tke0() As Single, ncon As IntegerDim rkap As Single, ypa As Single'Dim u0() As Single, v0() As Single

Public Sub ITBL2(xi As Single, xf As Single, cnu As Single, Uinf As Single, _mmax As Integer, nmax As Integer, Ue_full() As Single, model As Integer, _auto As Boolean, ux0y() As Single)

'****************************************'Incompressible, Turbulent Boundary Layers'From Schetz, "Boundary Layer Analysis"'****************************************

Dim u() As Single, v() As Single, cf() As SingleDim pi As Double, red As Single, L As Single, u_vec() As SingleDim del As Single, dy As Single, dx As Single, i As Integer, j As IntegerDim usue As Single, mest As Integer, fm1 As IntegerDim mmm As Integer, nmaxp As Integer, mmaxp As IntegerDim nnx As Integer, yod As Single, Ue() As SingleDim a() As Single, b() As Single, c() As Single, R() As SingleDim deno As Single, mt As Integer, delta() As SingleDim testnum As Integer, y() As Single

pi = 4 * Atn(1)

L = 1rkap = 0.41 'coefficients for modelling law of the wallypa = 9.7 'coefficients for modelling law of the walldel = xi * 0.375 / (Uinf * xi / cnu) ^ 0.2 'boundary layer thickness, Schetz eq. 7-43dy = del / (400 / 550 * mmax - 1) '(using (400/550)*mmax points across boundary layer.)dx = (xf - xi) / (nmax - 1)Re = Uinf * L / cnu

'*****************************************'Redim some arrays

'*****************************************ReDim tmu(mmax), u(mmax, nmax), v(mmax, nmax), y(mmax)ReDim a(mmax), b(mmax), c(mmax), R(mmax), delta(nmax)ReDim tke0(mmax), cf(nmax), xx(nmax), Ue(nmax), u_vec(mmax)

'Put this one down here because it must be dimensioned before usingcf(1) = 0.0456 * (Uinf * del / cnu) ^ (-0.25) 'Blasius law, Schetz eq. 7-13

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For i = 1 To nmaxxx(i) = xi + (i - 1) * dxUe(i) = Ue_full(i) / Uinf

Next i

'set up inlet conditionsFor j = 1 To mmax

u(j, 1) = Ue(1)v(j, 1) = 0tke0(j) = 0

Next j'set up no-slip conditionFor i = 1 To nmax

u(1, i) = 0Next i

'if user specifies initial velocity profile, use it; however'by default, establish initial velocity profiles using Coles wake lawusue = Sqr(cf(1) / 2) '=u*/Ue, u*=sqrt(wallshear/rho)red = Re * del * usue * Ue(1) 'Red=Ue*delta/nu?mest = CInt(400 / 550 * mmax) 'm index of initial boundary layer positionfm1 = mest - 1mmm = mest + 2If auto Then

For j = 2 To mestyod = (j - 1) / fm1 'y over delta

u(j, 1) = usue * Ue(1) * (1 / rkap * Log(Uinf * yod * del * usue * Ue(1) _/ cnu) + 4.9 + 0.51 / rkap * 2 * Sin(pi / 2 * yod) ^ 2)

v(j, 1) = 0Next j

ElseFor j = 2 To mmax

u(j, 1) = ux0y(j)Next j

End Ifnmaxp = nmaxmmaxp = mest + 100

ncon = 1

'Start stepping downstreamFor i = 2 To nmax

fm_progress.progbar.value = innx = iu(mmax, i) = Ue(i)v(mmax, i) = 0

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Select Case modelCase 1

mixing nnx, mmax, mest, dy, Ue, u, v, cf

Case 2eddy nnx, mmax, mest, dy, Ue, u, v, cf

Case 3testnum = tkes(nnx, mmm, mest, dy, dx, Ue, u, v, cf)If testnum = -1 Then Exit Sub

End Selectncon = 2

'prepare for tdma solutionb(1) = 1c(1) = 0

R(1) = 0a(mmax) = 0b(mmax) = 1c(mmax) = 0R(mmax) = Ue(i)deno = Re * dy ^ 2

For j = 2 To mmax - 1a(j) = -0.5 * v(j, i - 1) / dy - (1 + tmu(j - 1)) / denob(j) = u(j, i - 1) / dx + (2 + tmu(j - 1) + tmu(j)) / denoc(j) = 0.5 * v(j, i - 1) / dy - (1 + tmu(j)) / deno

R(j) = 0.5 / dx * (Ue(i) ^ 2 - Ue(i - 1) ^ 2) + (u(j, i - 1) ^ 2) / dxNext j

'run TDMAu_vec = trid(mmax, a, b, c, R, u_vec)

'transfer u_vec into uFor j = 1 To mmax

u(j, i) = u_vec(j)Next j

'calculate vFor j = 2 To mmax - 1

v(j, i) = v(j - 1, i) - (0.5 * dy / dx) * (u(j, i) - u(j, i - 1) + u(j - 1, i) - u(j - 1, i - 1))Next j

mt = mest - 10If mt < 1 Then mt = 1For j = mt To mmax

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'find boundary layerIf u(j, i) > 0.99 * Ue(i) Then

mest = jmmm = mest + 3

mmaxp = mest + 100Exit For

End IfNext jdelta(i) = (mest - 1) * dy

'get skin frictioncf(i) = (4 * u(2, i) - u(3, i)) / (Ue(i) ^ 2 * dy * Re)

'check for near separationIf cf(i) < 0.0001 Then

MsgBox ("Near separation")nmaxp = iExit For

End IfNext i

y(1) = 0For j = 2 To mmax


global_x = xxglobal_y = yFor i = 1 To nmax

For j = 1 To mmaxu(j, i) = u(j, i) * Uinfv(j, i) = v(j, i) * Uinf

Next jNext iglobal_u = uglobal_v = vglobal_delta = delta

global_Cf = cf

Erase tmu(), u(), v(), y(), a(), b(), c(), R(), delta()Erase tke0(), cf(), xx(), Ue(), u_vec()

End Sub

Private Sub mixing(nnx As Integer, mmax As Integer, mest As Integer, dy As Single, _

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Ue() As Single, u() As Single, v() As Single, cf() As Single)Dim xlmo As Single, xlmm As Single, j As IntegerDim y As Single, yp As Single

'for outer region, mixing length is Lm=0.09*deltaxlmo = 0.09 * (mest - 1) * dytmu(1) = 0For j = 2 To mmax - 1

y = (j - 1) * dyyp = y * Ue(nnx) * Re * Sqr(0.5 * cf(nnx - 1))xlmm = rkap * (1 - exp(-yp / 26#)) * yIf xlmm > xlmo Then xlmm = xlmo

' tmu(j) = Re * xlmm ^ 2 * Abs(u0(j + 1) - u0(j - 1)) / (2 * dy)tmu(j) = Re * xlmm ^ 2 * Abs(u(j + 1, nnx - 1) - u(j - 1, nnx - 1)) / (2 * dy)

Next j

tmu(mmax) = tmu(mmax - 1)End Sub

Private Sub eddy(nnx As Integer, mmax As Integer, mest As Integer, dy As Single, _Ue() As Single, u() As Single, v() As Single, cf() As Single)

Dim delst As Single, rmut As Single, yp As Single, y As SingleDim j As Integer, arg As Single, tanh As Singledelst = 0For j = 2 To mest

' delst = delst + dy * (1 - 0.5 * (u0(j - 1) + u0(j)) / Ue(nnx))

delst = delst + dy * (1 - 0.5 * (u(j - 1, nnx - 1) + u(j, nnx - 1)) / Ue(nnx))Next jrmut = 0.018 * Re * Ue(nnx) * delstFor j = 1 To mmax

y = (j - 1) * dyyp = y * Ue(nnx) * Re * Sqr(0.5 * cf(nnx - 1))arg = yp / ypatanh = (exp(arg) - exp(-1 * arg)) / (exp(arg) + exp(-1 * arg))tmu(j) = rkap * (yp - ypa * tanh)If tmu(j) > rmut Then tmu(j) = rmut

Next j

End Sub

Private Function tkes(nnx As Integer, mmm As Integer, mest As Integer, dy As Single, dx As Single, _Ue() As Single, u() As Single, v() As Single, cf() As Single) As Integer

Dim uss As Single, fm1 As Integer, j As Integer, yod As SingleDim cdf As Single, xlmo As Single, xlmm As Single, y As Single

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Dim yp As Single, tlm() As Single, stkm As Single, stkm1 As Single, deno As SingleDim a() As Single, b() As Single, c() As Single, R() As SingleDim tke() As Single

Const sigk = 1#Const cd = 0.09

ReDim tke(mmm), tlm(mmm), a(mmm), b(mmm), c(mmm), R(mmm)

tke(mmm) = 0#If ncon < 2 Then

uss = 0.5 * Ue(nnx) ^ 2 * cf(nnx - 1)'Assume initial profile of K from fig. 7-20 in Schetzfm1 = mest - 1For j = 1 To mest

yod = (j - 1) / fm1If yod < 0.001 Then

tke0(j) = 5500 * yod * ussElseIf yod < 0.12 Then

tke0(j) = 0.5 * uss * (7# - 3# / 0.099 * (yod - 0.11))Else

tke0(j) = 0.5 * uss * (-7 / 0.9 * (yod - 1))End If

Next jEnd If

cdf = cd ^ 0.25xlmo = 0.09 * (mest - 1) * dy

For j = 1 To mmmy = (j - 1) * dyyp = y * Ue(nnx) * Re * Sqr(0.5 * cf(nnx - 1))xlmm = rkap * (1# - exp(-yp / 26)) * yIf xlmm > xlmo Then xlmm = xlmotlm(j) = xlmm * cdf

Next j

stkm = 0#deno = sigk * dy ^ 2b(1) = 1#c(1) = 0R(1) = 0a(mmm) = 0b(mmm) = 1R(mmm) = tke(mmm)

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For j = 2 To mmm - 1stkm1 = stkmIf tke0(j) >= 0 Then

stkm = Sqr(tke0(j))Else

MsgBox "Attempted square root of negative. Quitting."tkes = -1Exit Function

End If

a(j) = -0.5 * v(j, nnx - 1) / dy - stkm1 * tlm(j - 1) / denob(j) = u(j, nnx - 1) / dx + cd * stkm / tlm(j) + (stkm1 * tlm(j - 1) + stkm * tlm(j)) / denoc(j) = -stkm * tlm(j) / deno + 0.5 * v(j, nnx - 1) / dyR(j) = stkm * tlm(j) * (u(j + 1, nnx - 1) - u(j - 1, nnx - 1)) ^ 2 / (4 * dy ^ 2) + u(j, nnx - 1) * tke0(j) /

dxNext j

tke = trid(mmm, a, b, c, R, tke)

tmu(1) = 0For j = 2 To mmm

tmu(j) = Re * tlm(j) * Sqr(tke0(j))Next j

For j = 1 To mmm

tke0(j) = tke(j)Next j

tke0(1) = 0tke0(mmm) = 0

End Function

Public Function trid(mm As Integer, a() As Single, b() As Single, c() As Single, R() As Single, S() AsSingle) As Single()

'TDMA

Dim j As Integer, gam() As Single, rp() As Single, deno As Single

ReDim gam(UBound(S)), rp(UBound(S))gam(1) = c(1) / b(1)rp(1) = R(1) / b(1)For j = 2 To mm

deno = b(j) - a(j) * gam(j - 1)gam(j) = c(j) / deno

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rp(j) = (R(j) - a(j) * rp(j - 1)) / denoNext j

S(mm) = rp(mm)

For j = 1 To mm - 1S(mm - j) = rp(mm - j) - gam(mm - j) * S(mm - j + 1)

Next j

trid = S

End Function

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